The thermal conductivity of magnesium silicon nitride ... · The Thermal Conductivity of Magnesium...

The thermal conductivity of magnesium silicon nitride,MgSiN2, ceramics and related materialsCitation for published version (APA):Bruls, R. J. (2000). The thermal conductivity of magnesium silicon nitride, MgSiN2, ceramics and relatedmaterials. Eindhoven: Technische Universiteit Eindhoven. https://doi.org/10.6100/IR535906

DOI:10.6100/IR535906

Document status and date:Published: 01/01/2000

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.

Download date: 25. Jun. 2020

https://doi.org/10.6100/IR535906

https://doi.org/10.6100/IR535906

https://research.tue.nl/en/publications/the-thermal-conductivity-of-magnesium-silicon-nitride-mgsin2-ceramics-and-related-materials(7fcee631-2cb5-462b-92a8-7b06b447dd9a).html

The Thermal Conductivity of Magnesium Silicon Nitride,

MgSiN2, Ceramics and Related Materials

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr. M. Rem, voor eencommissie aangewezen door het College voor

Promoties in het openbaar te verdedigenop donderdag 5 oktober 2000 om 16.00 uur

door

Richard Joseph Bruls

geboren te Sittard

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr. R. Metselaar

en

prof.dr. K. Itatani

Copromotor:

dr. H.T. Hintzen

Druk: Universiteitsdrukkerij, Technische Universiteit Eindhoven

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Bruls, Richard J.

The Thermal Conductivity of Magnesium Silicon Nitride, MgSiN2, Ceramics and

Related Materials / by Richard J. Bruls. - Eindhoven: Technische Universiteit

Eindhoven, 2000. - Proefschrift. -

ISBN 90-386-3011-5

NUGI 813

Trefwoorden: keramische materialen; nitriden / warmtegeleidbaarheid / phononen

Subject headings: ceramic materials; nitrides / thermal conductivity / phonons

Kaft: temperatuur-tijd afhankelijkheid van een thermische diffusiviteitsmeting met

een atomic force microscoop thermisch beeld als achtergrond.

Aan mijn oudersen grootouders

Aan Marianne

5

Table of contents

Chapter 1. Introduction 11

1. General introduction 11

2. Substrate materials 132.1. Requirements 132.2. Relation between heat conduction and material

characteristics 142.3. AlN as a promising substrate material 162.4. The new ceramic material MgSiN2 18

3. Objective and outline 20

References 22

Chapter 2. Preparation and characterisation of MgSiN2 powders 29

1. Introduction 30

2. Experimental section 302.1. Starting materials 302.2. Preparation 312.3. Characterisation 32

3. Results and discussion 333.1. Starting powder characteristics 33

3.1.1. Mg3N2 333.1.2. Si3N4 35

3.2. Phase formation of MgSiN2 363.3. Oxygen content of the MgSiN2 powders 413.4. X-ray diffraction data of MgSiN2 443.5. Powder characteristics 493.6. Oxidation behaviour of MgSiN2 powders 52

4. Conclusions 53

References 55

Table of contents

6

Chapter 3. Preparation, characterisation and properties of MgSiN2

ceramics 59

1. Introduction 59

2. Experimental 612.1. Preparation 612.2. Characterisation 642.3. Properties 66

3. Results and discussion 673.1. Characterisation 67

3.1.1. Phase formation and lattice parameters ofMgSiN2 67

3.1.2. Density 713.1.3. Chemical composition 723.1.4. Microstructure 743.1.5. TEM/EDS 77

3.2. Properties 803.2.1. Oxidation resistance 803.2.2. Hardness 813.2.3. Young's modulus 823.2.4. Thermal expansion 823.2.5. Thermal diffusivity/conductivity 83

4. Theoretical considerations 864.1. Secondary phases 864.2. Grain size 874.3. Defects 874.4. Maximum influence of secondary phases, grain size

and defects 88

5. Conclusions 89

References 89

Chapter 4. Anisotropic thermal expansion of MgSiN2 97

1. Introduction 97

2. Experimental procedure 99

3. Results and discussion 101

Table of contents

7

3.1. Neutron diffraction data refinement 1013.2. Thermal expansion 107

4. Conclusions 112

References 112

Chapter 5. The heat capacity of MgSiN2 117

1. Introduction 117

2. Experimental 1192.1. Adiabatic calorimeter measurements 1192.2. Differential scanning calorimeter measurement 120

3. Results and discussion 121

3.1. Cpo of MgSiN2 121

3.2. Debye temperature of MgSiN2 124

3.3. Thermodynamic functions STo, (HT

o - H0

o) and

(GTo - H0

o) of MgSiN2 127

3.4. H0o of MgSiN2 130

4. Conclusions 133

References 133

Chapter 6. The Young's modulus of MgSiN2, AlN and Si3N4 137

1. Introduction 137

2. Experimental section 138

3. Results and discussion 1403.1. Evaluation of the measurements 1403.2. Interpretation of the fitting parameters 144

3.2.1. E0 1443.2.2. B and T0 145

4. Conclusions 146

References 147

Chapter 7. The Grüneisen parameters of MgSiN2, AlN and ββββ-Si3N4 153

1. Introduction 153

2. Evaluation of the input parameters 156

Table of contents

8

2.1. Lattice linear thermal expansion coefficient α lat 1562.2. Molar volume Vm 1592.3. Adiabatic compressibility βS 1592.4. Heat capacity at constant pressure Cp 160

3. Evaluation of the Grüneisen parameter γ 162

4. Discussion 1664.1. The temperature dependence of the Grüneisen

parameter 1664.2. The absolute value of the Grüneisen parameter at the

Debye temperature 168

5. Conclusions 170

References 170

Chapter 8. Theoretical thermal conductivity of MgSiN2, AlN and

ββββ-Si3N4 using Slack's equation 177

1. Introduction 177

2. The Slack equation 179

3. Influence of input parameters 181

4. The modification of the Slack equation 185

5. Applicability, reliability and limitations of Slack modified 193

6. Conclusions 195

References 196

Chapter 9. A new method for estimation of the intrinsic thermal

conductivity 203

1. Introduction 203

2. The temperature dependence of the thermal diffusivity andconductivity 204

3. Experimental 208

4. Results for MgSiN2, AlN and β-Si3N4 2094.1. The temperature dependence of the thermal diffusivity

a 2094.2. Inverse thermal diffusivity a -1 versus temperature T

plots 212

Table of contents

9

5. Discussion 2205.1. Interpretation of the fitting parameters 2205.2. Thermal conductivity estimates for MgSiN2, AlN and

β-Si3N4 2215.3. Comparison with other estimates 2255.4. Limitations, accuracy and reliability 227

6. Conclusions 228

References 229

Chapter 10. Conclusions 237

List of symbols 241

Lower-case symbols 241

Upper-case symbols 242

Greek symbols 243

Summary 245

Samenvatting 247

Nawoord 251

Curriculum Vitae 254

List of publications 255

11

Chapter 1.

Introduction

1. General introduction

"So the Lord God banished him from the Garden of Eden to work the ground" [1].

Since then people try to improve their existence by making life more comfortable.

They used their intellect, knowledge and inventiveness to increase the standard of

living. It started with stone tools, the ability of making fire and the production of

food by farming and is now (after making a large step in history in only few

seconds of writing [2]) continuing in the age of the computer information and

automation.

More and more processes are computer controlled and/or guided. Due to the

increasing number and complexity of tasks in e.g. the industry, and in order to

Pentium® IIProcessor

Pentium® ProProcessor

Pentium®Processor

80486SX80486DX

80386SX80386DX80286

80888086

80084004

8080

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1970 1975 1980 1985 1990 1995 2000 2005t [year]

Num

ber o

f tra

nsis

tors

Fig. 1-1: Number of transistors per chip versus time (t ) (Data supplied

by Intel Corp.).

Chapter 1.

12

reduce the human intervention the complexity, speed and calculating power of

these machines is still increasing. E.g. the last decades the number of transistors per

chip and the processing power have increased tremendously (see Figs. 1-1 and

1-2).

Related to this development there is a tendency to increase the processing

power per unit volume by miniaturisation. E.g. the computing power of the house

size first computer in 1945 (ENIAC) containing 17468 vacuum tubes is nowadays

easily surpassed by a microprocessor with a size much smaller than a match box

containing 10000000 transistors [3, 4]. This resulted in the use of more and more

chip controlled electronic devices during the last decades. One of the best examples

is the introduction of the personal computer (PC) with a high computing power.

But also a (mobile) telephone, audio equipment, bank/credit card and microwave

oven contain one or more chips.

So microelectronics is playing an essential role in nowadays life. For several

applications in microelectronics the (bare) chip is directly attached to a substrate

[5] which is for unencapsulated chip design one of the most important parts.

Besides for substrates, also for an enormous amount of other applications the

thermal properties of a material are of crucial importance. For one type of

Pentium® ProProcessor

Pentium®Processor

8008

4004

80386DX

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1970 1975 1980 1985 1990 1995 2000 2005t [year]

Inst

ruct

ions

per

sec

ond

Fig. 1-2: Processing power in instructions per second versus time (t )

(Data supplied by Intel Corp.).

Introduction

13

application the heat flow should be minimised like insulation material for a

furnace, heat shield of the space shuttle, etc. whereas for another application like

heat exchangers, lamp envelope materials, substrates, etc. the heat flow should be

high.

2. Substrate materials

2.1. Requirements

A substrate in electronic integrated circuits has two main functions. The first is

obvious viz. carry the chip attached to it giving it mechanical stability. Second, the

substrate is used as a heat sink in order to avoid over-heating and eventually

damage of the electronic circuit attached to it. For that purpose, in general a

substrate should fulfil the following requirements [6 - 10]:

- high mechanical strength

- high thermal conductivity

- high electrical resistance

- low dielectric constant

- low dielectric loss

- thermal expansion coefficient similar to silicon

- good thermal shock resistance

- good metallisation properties

- chemical and electrical stability

- non-toxic

- smooth surface

- easy to produce, economic viability, cheap and convenient

processing

Recently, the requirements for substrates are becoming more strict [5, 10, 11]

due to the tendency of miniaturising the electronic circuit attached to the substrate,

resulting in a higher heat dissipation per surface area. This implies that the thermal

Chapter 1.

14

conductivity of the substrate should be typically above 100 W m-1 K-1 at room

temperature resulting in a sufficient heat transport and a good thermal shock

resistance. As a consequence the traditional substrate material Al2O3 (38 W m-1 K-1

at room temperature [10]) does no longer fulfil the requirements.

So the selection of a substrate material that complies with all the needs is of

crucial importance for the design and development of better and new

microelectronics. Because substrates should have a high electrical resistivity

combined with a high thermal conductivity only a limited number of materials come

into account.

2.2. Relation between heat conduction and material characteristics

Since substrate materials have to be electrical insulators with a high thermal

conductivity, only materials showing a high phonon conduction (viz. heat is

transported by lattice vibrations, the so-called phonons) are suitable candidates.

Materials with good phonon conductivity should fulfil the following requirements

[6, 7, 12, 13]:

- simple structure

- low atomic mass

- strong covalent bonding

- low anharmonicity

- high purity

Especially materials with the simple adamantine type (diamond related) crystal

structure show a high phonon thermal conductivity [12] in combination with a high

electrical resistivity, which makes this class of materials potentially interesting as

substrate materials. In Table 1-1 some values for the thermal conductivity (κ )

[W m-1 K-1] of some commonly used adamantine type materials are presented.

Considering the above mentioned requirements for obtaining a high lattice

thermal conductivity, it is not supprising that carbon (C) with its simple diamond

structure, low atomic mass and strong covalent bonding has the highest thermal

Introduction

15

conductivity. For the other group IV elements Si and Ge with the diamond structure

the thermal conductivity decreases due to the increase in atomic mass and the decrease

of the strength of the covalent bonding as compared to diamond. Also the binary

compounds (IV - IV, III - V, II - VI) having an adamantine crystal structure show a

relatively high thermal conductivity although lower than diamond due to the

increasing complexity of the crystal structure, decrease of covalent character of the

bonds and in most cases also increase of average mass.

Table 1-1: The thermal conductivity (κ ) of some unary and binary adamantine type

materials at room temperature (300 K).

unary material κ[W m-1 K-1]

binary Material κ[W m-1 K-1]

IV C (diamond)SiGe

2000 [13] 160 [14] 60 [14]

IV - IVIII - V

II - VI

SiCBN (cubic)BPAlNAlPGaNGaPBeO

490 [13, 15]1300 [13] 350 [13] 285 [16] 130 [6] 130 [17] 100 [18] 370 [19]

Each of the materials mentioned in Table 1-1 has its specific disadvantages

[6, 7, 20]. C and cubic BN are difficult to produce (high pressures and

temperatures) and therefore expensive. SiC, Si and Ge are semiconductors. GaN,

GaP and AlP have a large mismatch in thermal expansion coefficient with Si and

BP is thermally unstable. BeO has the disadvantage of its toxicity especially of its

dust. The compound that is most intensively studied during the last two decades,

viz. AlN ceramics, was considered expensive due to the necessity of using pure

raw materials, the relatively high processing temperature (around 1700 °C) and

poor metallisation properties as compared to the traditional substrate material

Al2O3. However, nowadays AlN ceramics can be processed at normal pressure, and

Chapter 1.

16

recently production costs are minimised by optimisation of processing routes and

metallisation process.

2.3. AlN as a promising substrate material

Not only is AlN ceramics used as a substrate material, it has also several

other interesting applications due to the potential superior thermal and mechanical

properties and chemical stability as compared to more traditional materials like

Al2O3 and stainless steel. For example crucibles, tube envelopes, heater plates for

chemical vapour deposition (CVD) applications and nozzles for extrusion

(Fig. 1-3).

The most important problem that had to be solved for obtaining AlN

ceramics with a high thermal conductivity were the oxygen impurities in the

starting powder and the resulting ceramics. Oxygen dissolved in the AlN lattice

results in the formation of Al vacancies [8, 13] that are very effective in scattering

Fig. 1-3: Several examples of commercially available AlN products

showing a heater plate, nozzles and several substrates. To

obtain an impression of the size a 3.5 inch diskette is

included.

Introduction

17

the phonons [21, 22] resulting in a low thermal conductivity. Moreover, oxygen

impurities present as secondary phases normally hamper the heat transport between

the AlN grains due to the formation of a thermally insulating layer [23 - 25].

Furthermore, the grain size itself can be of importance because phonon - grain

boundary scattering also reduces the thermal conductivity [26 - 28]. So for

obtaining a high thermal conductivity the intrinsic properties (lattice defects) as

well as the extrinsic properties (microstructure) have to be controlled [29]

(Fig. 1-4).

It is suggested that AlN has an intrinsic thermal conductivity of

320 W m-1 K-1 at 300 K [16, 31] (for comparison Cu 400 W m-1 K-1 at room

temperature [30]). Around 1975 it was possible to synthesise on lab-scale single

Unfavorable microstructure Favorable microstructure

Fig. 1-4: A schematic drawing of a microstructure showing left a

situation, with defects in the lattice and a grain boundary

phase, which is not beneficial for obtaining a high thermal

conductivity, and right a more favourable situation in which

the defect concentration within the grains is reduced and the

secondary phases are located at the triple points, resulting in

less phonon-defect scattering and a good thermal contact

between the grains.

Chapter 1.

18

crystals with a high thermal conductivity (> 200 W m-1 K-1, see Fig. 1-5) [31, 32].

However, it took several decades to improve and optimise the thermal conductivity

of AlN polycrystalline ceramics [6, 7, 29, 31, 33 - 45] until the above mentioned

problems considering the oxygen impurities were solved and samples approaching

the theoretical value could be synthesised (see Fig. 1-5). Around 1990 it was

possible to synthesise commercial polycrystalline ceramic samples (Fig. 1-3) with

an excellent thermal conductivity (> 200 W m-1 K-1) and the research subsequently

concentrated on reducing the processing costs by lowering the processing

temperature and using other (cheaper) additives [46 - 49], and improvement of the

metallisation properties [50]. Nowadays, commercial AlN substrates are available

with a thermal conductivity of about 200 W m-1 K-1.

2.4. The new ceramic material MgSiN2

Already in 1973 Slack [13] noted that, besides unary and binary, several ternary

compounds having an adamantine crystal structure might have a high thermal

0

50

100

150

200

250

300

350

1950 1960 1970 1980 1990 2000t [year]

κ [W

m-1

K-1

]

AlN theoreticalAlN single crystalAlN polycrystalline

Fig. 1-5: Development of the room temperature thermal conductivity

(κ ) of polycrystalline and single crystalline AlN as

compared to the theoretically predicted value versus the

time (t ) (Data from Ref. 6, 7, 29, 31 and 33 - 45.). The lines

are drawn as a guide to the eye.

Introduction

19

conductivity (> 100 W m-1 K-1). Although this value is lower than that of the

corresponding binary compounds (having the same average atomic mass) due to

the increasing complexity of the crystal structure, nevertheless these compounds

might be interesting. So, instead of the full optimisation of already known materials

for substrate application there is also a drive for finding ternary and even

quaternary adamantine compounds which can be suitable new substrate materials.

Recently, several materials derived from AlN (see Fig. 1-6, 2 Al substituted

by 1 Mg + 1 Si → MgSiN2 or by 1 Ca + 1 Si → CaSiN2, and 2 N substituted by

1 O + 1 C → Al2OC) and other adamantine type materials (AlCON (Al28O21C6N6))

[20, 51 - 55] were suggested as being new potentially interesting substrate

materials. In view of the requirements for good phonon conduction it can be

concluded that MgSiN2 is the most promising material as CaSiN2 has a higher

average mass, Al2OC is only stable as a solid solution [55 - 57] and AlCON has a

rather complex crystal structure (large crystallographic unit cell) [54, 55]. MgSiN2

is a covalent electrical insulator with a rather simple structure, comparable with

that of AlN [51]. Before starting this Ph. D. work the thermal conductivity value

measured for MgSiN2 was 17 W m-1 K-1 at room temperature [51], whereas a

BeO

C

Al2OC

MgSiN2

Si

BN

SiC

III-VBP

Ge

AlPGaN GaP

AlN

II-VI

IV-IVIV GaAs

II-IV-V2

Si Ge

B

N

O

Mg

P OAs

Be

C

N

C

Al

Si

Ga

GeIV

II

III

VI

V

IV VIVIII2 -VI-IV

Fig. 1-6: Diamond and diamond-structure related unary, binary and

ternary materials.

Chapter 1.

20

maximum value was predicted of approximately 120 W m-1 K-1 [53] (in

comparison AlN (single crystal) 320 W m-1 K-1, Al2O3 (single crystal (sapphire))

60 W m-1 K-1). So at that time considerable improvement of the thermal

conductivity of MgSiN2 was expected, if the processing could be optimised and

better starting materials were used.

3. Objective and outline

The objective of this work was to optimise the thermal conductivity of MgSiN2

ceramics. In order to obtain a high thermal conductivity the impurity content and

especially the oxygen content in the MgSiN2 lattice was considered to be of crucial

importance analogous to the situation with AlN. Therefore this work first

concentrated on the optimisation of the synthesis of pure MgSiN2 powder and

ceramics by suitable processing. Although, originally a high thermal conductivity

was expected for optimised MgSiN2, this value could by far not be confirmed

experimentally notwithstanding improvement of the processing resulting in pure

MgSiN2 powders and ceramics. Therefore, the available theoretical method to

predict the maximum achievable thermal conductivity by phonon conduction

(Slack's theory) was reconsidered. This resulted in an improved theory of Slack and

moreover the development of a new prediction method based on extrapolation of

temperature dependent thermal diffusivity measurements allowing discrimination

between phonon-phonon and phonon-defect scattering processes. This procedure is

generally applicable for ceramic materials showing heat conduction by phonons.

So, this procedure can be used for identifying the potential thermal conductivity of

(new) non-optimised materials and for guiding the process optimisation resulting in

a decrease of the time and effort needed to optimise the thermal properties. The

improved prediction methods were also applied to the commercially interesting

materials AlN and β-Si3N4 in order to check the general validity. So, an

experimental as well as a theoretical approach is described in this thesis. The

results of this work are presented in the different parts of this thesis as follows:

Introduction

21

This part (Chapter 1) provides a short overview why MgSiN2 was considered

a potential interesting material. Furthermore it becomes clear that good estimates

for the intrinsic thermal conductivity are very important to choose the most

promising materials. Furthermore, it is important to minimise the effort put in

material optimisation.

Usually the main problem in achieving a high thermal conductivity in

phonon conductors is phonon scattering due to defects. For nitrides these defects

are mainly caused by oxygen impurities in the nitride starting powders. Therefore it

is considered important to synthesise pure MgSiN2 powder which is discussed in

Chapter 2.

Chapter 3 deals with the processing of MgSiN2 ceramics by hot uni-axial

pressing and the resulting properties. By suitable processing it should be possible

to identify and eliminate the mechanism that is limiting the thermal conductivity of

MgSiN2. This was done by changing the processing conditions viz. temperature,

time and/or using an additive during processing.

The experimental determination and modelling of the thermal expansion,

heat capacity and Young's modulus of MgSiN2 are discussed in Chapters 4, 5 and 6

respectively. These properties and, in particular, their temperature dependence are

needed for calculating the Grüneisen parameter and Debye temperature that are

required for the theoretical estimation of the maximum achievable thermal

conductivity with the Slack equation. Both the specific heat as well as the Young's

modulus is used to evaluate the Debye temperature.

The evaluation of the Grüneisen parameter of MgSiN2 is discussed in

Chapter 7. The Grüneisen parameter is related to the complexity of the crystal

structure and the characteristics of the bonding between the atoms. The Grüneisen

parameter of MgSiN2 is compared with that of AlN, which has a similar

wurtzite-like crystal structure, and β-Si3N4, with a phenakite (Be2SiO4) structure.

Besides the Grüneisen parameter and the Debye temperature, also the number of

atoms per primitive unit cell is an important parameter for estimating the maximum

achievable thermal conductivity. The comparison with β-Si3N4 (having a relatively

high thermal conductivity) is made because it has about the same number of atoms

Chapter 1.

22

per primitive unit cell as MgSiN2 (with a relatively low thermal conductivity)

whereas it has much more atoms per as per primitive unit cell as AlN (having a

high thermal conductivity). Furthermore, recently β-Si3N4 substrates are

commercially applied triggering a detailed comparison with MgSiN2 and AlN.

In Chapter 8 the theory of Slack is discussed. This theory describes a

relatively simple method to predict the maximum achievable thermal conductivity

of non-metallic materials from the crystal structure, Debye temperature, Grüneisen

parameter and number of atoms per primitive unit cell. The assumptions made in

this theory are briefly discussed and some improvements are presented, resulting in

a modified Slack theory. The applicability of this adapted theory is discussed by

calculating the intrinsic thermal conductivity values at room temperature for

MgSiN2, AlN and β-Si3N4 and comparing them with experimental data for

validation.

Another new method for predicting the intrinsic thermal conductivity is

presented in Chapter 9. This method is based on extrapolation of thermal

diffusivity measurements as a function of the temperature. The general

applicability, validity and limitations of this method are discussed using MgSiN2,

AlN and β-Si3N4 as model compounds. Therefore the results were compared with

those obtained in Chapter 8 and experimental values.

The final conclusions of this thesis are summarised in Chapter 10.

References

1. The Holy Bible, The fall of Man, Genesis 3:23.

2. After N.A. Armstrong (the first man on the moon), "one small step for a man -

one giant leap for mankind", at 10:56 p.m. Eastern Daylight Time July 20,

1969.

3. D.J.W. Sjobbema, Geschiedenis van de elektronica; Van voltacel naar digitale

televisie, first edition (Kluwer BedrijfsInformatie b.v., Deventer, The

Netherlands, 1998).

Introduction

23

4. E. Braun and S. MacDonald, Revolution in miniature, second edition

(Cambridge University Press, Cambridge, UK, 1982).

5. Microelectronics Packaging Handbook; Part II: Semiconductor Packaging,

edited by R.R. Tummala, E.J. Rymaszewski and A.G. Klopfenstein, second

edition (Chapman and Hall, London, 1997).

6. W. Werdecker and F. Aldinger, Aluminum Nitride - An Alternative Ceramic

Substrate for High Power Applications in Microcircuits, IEEE Trans.

Compon., Hybrids, Manuf. Technol. CHMT-7 (1984) 399.

7. Y. Kurokawa, K. Utsumi, H. Takamizawa, T. Kamata and S. Noguchi, AlN

Substrates with High Thermal Conductivity, IEEE Trans. Compon., Hybrids,

Manuf. Technol. CHMT-8 (1985) 247.

8. A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and Kinetic Effects

of Oxygen Removal on the Thermal Conductivity of Aluminum Nitride, J. Am.

Ceram. Soc. 72 (1989) 2031.

9. G.W. Prohaska and G.R. Miller, Aluminum Nitride: A Review of the

Knowledge Base for Physical Property Development, Mat. Res. Soc. Symp.

Proc. 167, Advanced Electronic Packaging Materials, Boston, Massachusetts,

USA, November 27 - 29 1989, edited by A.T. Barfknecht, J.P. Patridge, C.J.

Chen and C.-Y. Li, (Materials Research Society, Pittsburg, 1990) 215.

10. A. Roosen, Modern Substrate Concepts for the Microelectronic Industry,

Electroceramics IV 2, Aachen, Germany, September 5 - 7 1994, edited by

R. Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann, (Augustinus

Buchhandlung, 1994) 1089.

11. H. Treichel, E. Eckstein and W. Kern, New Dielectric Materials and Insulators

for Microelectronic Applications, Ceramics International 22 (1996) 435.

12. D.P. Spitzer, Lattice Thermal Conductivity of Semi-Conductors: a Chemical

Bond Approach, J. Phys. Chem. Solids 31 (1970) 19.

13. G.A. Slack, Nonmetallic Crystals with High Thermal Conductivity, J. Phys.

Chem. Solids 34 (1973) 321.

Chapter 1.

24

14. C.J. Glassbrenner and G.A. Slack, Thermal Conductivity of Silicon and

Germanium from 3 °K to the Melting Point, Phys. Rev. 134 (1964) A1058.

15. G.A. Slack, Thermal Conductivity of Pure and Impure Silicon, Silicon

Carbide, and Diamond, J. Appl. Phys. 35 (1964) 3460.

16. G.A. Slack, R.A. Tanzilli, R.O. Pohl and J.W. Vandersande, The Intrinsic

Thermal Conductivity of AlN, J. Phys. Chem. Solids 48 (1987) 641.

17. E.K. Sichel, and J.I. Pankove, Thermal Conductivity of GaN, 25 - 360 K,

J. Phys. Chem. Solids 38 (1977) 330.

18. H. Wagini, Die Wärmeleitfähigkeit von GaP und AlSb, Z. Naturforschg. 21 a

(1966) 2096.

19. G.A. Slack and S.B. Austerman, Thermal Conductivity of BeO Single Crystals,

J. Appl. Phys. 42 (1971) 4713.

20. W.A. Groen, P.F. van Hal, M.J. Kraan, N. Sweegers and G. de With,

Preparation and Properties of ALCON (Al28C6O21N6), J. Mater. Sci. 30 (1995)

4775.

21. V. Ambegaokar, Thermal Resistance due to Isotopes at High Temperature, Phys.

Rev. 114 (1959) 488.

22. M.G. Holland, Phonon Scattering in Semiconductors From Thermal

Conductivity Studies, Phys. Rev. 134 (1964) A471.

23. A. Hafidi, M. Billy and J.P. Lecompte, Influence of Microstructural Parameters

on Thermal Diffusivity of Aluminium Nitride-Based Ceramics, J. Mater. Sci. 27

(1992) 3405.

24. C.F. Chen, M.E. Perisse, A.F. Ramirez, N.P. Padture and H.M. Chan, Effect of

Grain Boundary Phase on the Thermal Conductivity of Aluminium Nitride

Ceramics, J. Mater. Sci. 29 (1994) 1595.

25. P.S. de Baranda, A.K. Knudsen, and E. Ruh, Effect of CaO on the Thermal

Conductivity of Aluminum Nitride, J. Am. Ceram. Soc. 76 (1993) 1751.

26. A.K. Collins, M.A. Pickering and R.L. Taylor, Grain size dependence of the

thermal conductivity of polycrystalline chemical vapor deposited β-SiC at low

temperatures, J. Appl. Phys. 68 (1990) 6510.

Introduction

25

27. K. Watari, K. Ishizaki and T. Fujikawa, Thermal Conduction Mechanism of

Aluminium Nitride Ceramics, J. Mater. Sci. 27 (1992) 2627.

28. K. Watari, K. Hirao, M. Toriyama and K. Ishizaki, Effect of Grain Size on the

Thermal Conductivity of Si3N4, J. Am. Ceram. Soc. 82 (1999) 777.

29. Y. Kurokawa, K. Utsumi and H. Takamizawa, Development and

Microstructural Characterization of High-Thermal-Conductivity Aluminum

Nitride Ceramics, J. Am. Ceram. Soc. 71 (1988) 588.

30. CRC Materials Science and Engineering Handbook, second edition, edited by

J.F. Shackelford, W. Alexander and J.S. Park (CRC Press, Boca Raton,

Florida, USA., 1994).

31. M.P. Borom, G.A. Slack and J.W. Szymaszek, Thermal Conductivity of

Commercial Aluminum Nitride, Bull. Am. Ceram. Soc. 51 (1972) 852.

32. G.A. Slack and T.F. McNelly, Growth of High Purity AlN Crystals, J. Cryst.

Growth 42 (1977) 560.

33. K.M. Taylor and C. Lenie, Some Properties of Aluminum Nitride, J.

Electrochem. Soc. 107 (1960) 308.

34. N. Kuramoto and H. Taniguchi, Transparent AlN Ceramics, J. Mater. Sci.

Lett. 3 (1984) 471

35. I.C.Huseby and R.F. Bobik, U.S. Pat. No. 4547471 (High Thermal

Conductivity Aluminum Nitride Ceramic Body), Oct. 15, 1985, Nos. 4578232

- 4578234 (Pressureless Sintering Process to Produce High Thermal

Conductivity Ceramic Body of Aluminum Nitride), 4578384 and 4578365

(High Thermal Conductivity Ceramic Body of Aluminum Nitride), Mar. 25,

1986, see A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and

Kinetic Effects of Oxygen Removal on the Thermal Conductivity of Aluminum

Nitride, J. Am. Cer. Soc. 72 (1989) 2031.

36. A. Horiguchi, F. Ueno, M. Kasori, K. Shinozaki and A. Tsuge, 25th Ceramic

Basic Seminar Proceedings Abstract (1987) 155 (see also M. Okamoto,

H. Arakawa, M. Oohasi and S. Ogihara, Effect of Microstructure on Thermal

Chapter 1.

26

Conductivity of AlN Ceramics, J. Ceram. Soc. Jpn. Inter. Ed. 97 (1998)

1486).

37. R.R. Tummala, Ceramics in Microelectronic Packaging, Am. Ceram. Soc.

Bull. 67 (1988) 752.

38. N. Kuramoto, H. Taniguchi and I. Aso, Development of Translucent

Aluminum Nitride, Am. Ceram. Soc. Bull. 68 (1989) 883.

39. M. Okamoto, H. Arakawa, M. Oohashi and S. Ogihara, Effect of

Microstructure on Thermal Conductivity of AlN Ceramics, J. Ceram. Soc.

Jpn. Inter. Ed. 97 (1989) 1486.

40. F. Ueno and A. Horiguchi, Grain Boundary Phase Elimination and

Microstructure of Aluminium Nitride, Proceedings of the 1st European

Ceramic Society Conference (EcerS'89) 1, Processing of Ceramics,

Maastricht, The Netherlands, 18 - 23 June 1989, edited by G. de With,

R.A. Terpstra and R. Metselaar (Elsevier Applied Science, 1989) 383.

41. M. Hirano and N. Yamauchi, Development of As-Fired Aluminium Nitride

Substrates with Smooth Surface and High Thermal Conductivity, J. Mater.

Sci. 28 (1993) 5737.

42. K. Watari, K. Ishazaki and F. Tsuchiya, Phonon Scattering and Thermal

Conduction Mechanisms of Sintered Aluminium Nitride Ceramics, J. Mater.

Sci. 28 (1993) 3709.

43. J. Jarrige, P.J. Lecompte, J. Mullot and G. Müller, Effect of Oxygen on the

Thermal Conductivity of Aluminium Nitride Ceramics, J. Eur. Ceram. Soc. 17

(1997) 1891.

44. T.B. Jackson, A.V. Virkar, K.L. More, R.B. Dinwiddle, Jr., R.A. Cutler,

High-Thermal-Conductivity Aluminum Nitride Ceramics: The Effect of

Thermodynamic, Kinetic, and Microstructural Factors, J. Am. Ceram. Soc. 80

(1997) 1421.

45. A. Witek, M. Bockowski, A. Presz, M. Wróblewski, S. Krukowski,

W. Wlosinski and K. Jablonski, Synthesis of Oxygen-free Aluminium Nitride


Introduction

27

46. J. Jarrige, K. Bouzouita, C. Doradoux and M. Billy, A New Method for

Fabrication of Dense Aluminium Nitride Bodies at Temperatures as Low as

1600 °C, J. Eur. Ceram. Soc. 12 (1993) 279.

47. K. Watari, M.C. Valecillos, M.E. Brito, M. Toriyama and S. Kanzaki,

Densification and Thermal Conductivity of AlN Doped with Y2O3, CaO and

Li2O, J. Am. Ceram. Soc. 79 (1996) 3103.

48. G.M. Gross, H.J. Seifert and F. Aldinger, Thermodynamic Assessment and

Experimental Check of Fluoride Sintering Aids for AlN, J. Eur. Ceram. Soc.

18 (1998) 871.

49. K. Watari, M.E. Brito, T. Nagaoka, M. Toriyama and S. Kanzaki, Additives

for Low-Temperature Sintering of AlN Ceramics with High Thermal

Conductivity and High Strength, Key Engineering Materials 159-160, Novel

Synthesis and Processing of Ceramics, (Trans Tech Publications, Switzerland,

1999) 205.

50. A. Adlaßnig, J.C. Schuster, R. Reicher and W. Smetana, Development of

Glass Frit Free Metallization Systems for AlN, J. Mater. Sci. 33 (1998) 4887.

51. W.A. Groen, M.J. Kraan and G. de With, Preparation, Microstructure and

Properties of MgSiN2 Ceramics, J. Eur. Ceram. Soc. 12 (1993) 413.

52. W.A. Groen, M.J. Kraan and G. de With, New Ternary Nitride Ceramics:

CaSiN2, J. Mater. Sci. 29 (1994) 3161.

53. W.A. Groen, M.J. Kraan, G. de With, and M.G.A. Viegers, New covalent

ceramics: MgSiN2, Mat. Res. Soc. Symp. 237, Covalent Ceramics II: Non-oxides,

Boston, Massachusetts, U.S.A., November 29 - December 2 1993, edited by

Barron, A.R., Fischman, G.S., Furry, M.A. and Hepp, A.F. (Materials Research

Society, 1994) 239.

54. W.A. Groen, M.J. Kraan, P.F. van Hal and A.E.M. De Veirman, A New

Diamond - Related Compound in the System Al2O3-Al4C3-AlN, J. Sol. State

Chem. 120 (1995) 211.

55. W.A. Groen, P.F. van Hal, M.J. Kraan and G. de With, New High Thermal

Conductivity Ceramics, Fourth Euro Ceramics 3, Basic Science -

Chapter 1.

28

Optimisation of Properties and Performance by Improved Design and

Microstructural Control, Faenza, Italy, October 1995, edited by S. Meriani and

V. Sergo (Gruppo Editoriale Faenza Editrice S.p.A., Faenza, Italy, 1995) 343.

56. H. Yokokawa, M. Dokiya, M. Fujishige, T. Kameyama, S. Ujiie and K.

Fukuda, X-Ray Powder Diffraction Data for Two Hexagonal Aluminum

Monoxycarbide Phases, J. Am. Ceram. Soc. 65 (1982) C-40.

57. J.M. Lihrmann, T. Zambetakis and M. Daire, High-Temperature Behavior of

the Aluminum Oxycarbide Al2OC in the System Al2O3-Al4C3 and with

Additions of Aluminum Nitride, J. Am. Ceram. Soc. 72 (1998) 1704.

29

Chapter 2.

Preparation and characterisation of MgSiN2 powders

Abstract

The powder preparation of MgSiN2 was studied using several starting mixtures

(Mg3N2/Si3N4, Mg/Si3N4 and Mg/Si) in the temperature range 800 - 1500 °C in N2

or N2/H2 atmospheres. The phase formation was followed with thermo gravimetric

analysis and differential thermal analysis (TGA/DTA) and powder X-ray

diffraction (XRD). At 1250 °C Mg/Si mixtures did not yield single phase MgSiN2

whereas for Mg/Si3N4 and Mg3N2/Si3N4 mixtures nearly single-phase powders

were obtained. The Mg/Si3N4 mixtures yielded MgSiN2 at the lowest processing

temperature but the Mg3N2/Si3N4 mixtures yielded the most pure MgSiN2 powder

with respect to secondary phases. The main secondary phase detectable with XRD

was MgO when starting from Mg3N2/Si3N4 or MgO and metallic Si when starting

from Mg/Si3N4 mixtures. When the processing starting from Mg3N2/Si3N4 mixtures

was optimised MgSiN2 powders containing only 0.1 wt. % oxygen could be

prepared. Using XRD the solubility of oxygen in the MgSiN2 lattice was estimated

to be at maximum 0.5 wt. %. The MgSiN2 powder was oxidation resistant in air till

830 °C. The morphology and particle size were studied with the scanning electron

microscope (SEM) and the sedimentation method. Two different kinds of

morphology were observed, determined by the morphology of the Si3N4 starting

material.

Chapter 2.

30

1. Introduction

As a consequence of the ever increasing miniaturisation of integrated circuits

combined with a high energy dissipation, in recent years there is a strong need for

substrate materials with improved thermal conductivity [1]. Because the electrical

conductivity must be low, only non-metallic materials showing phonon conduction

are suitable.

The traditional material Al2O3 does not longer fulfil the recent requirements.

Several binary alternatives deduced from diamond, which has a high thermal

conductivity and electrical resistivity, were considered [2], each material having its

own disadvantages: SiC is electrically conducting, BeO is toxic, and the compound

which is most intensively studied during the last years, viz. AlN, is considered to

be expensive. Also the ternary compounds deduced from AlN were proposed e.g.

MgSiN2 (by replacing two Al3+ ions by a combination of Mg2+ and Si4+) and Al2OC

(by replacing two N3- ions by a combination of O2- and C4-) [3]. Recently, for

MgSiN2 ceramics a fairly high thermal conductivity was reported [4, 5].

For optimum thermal conductivity it is expected that the oxygen content of

the MgSiN2 ceramics should be low, similar to that in AlN [6]. Therefore, for

achieving MgSiN2 ceramics with a high thermal conductivity the oxygen

concentration of the starting material preferably should be low.

In this chapter the preparation, phase formation and characterisation of

MgSiN2 powders with a low oxygen content is reported. Preliminary results have

already been published [7]. The present situation concerning the preparation of

ceramic samples and thermal conductivity is described elsewhere [8, 9].

2. Experimental section

2.1. Starting materials

MgSiN2 powders were prepared from Mg3N2/Si3N4, Mg/Si3N4 or Mg/Si powder

mixtures. The influence of the composition and impurity content (quality) of

various starting materials on the characteristics of the resulting MgSiN2 powders


31

was investigated for Mg3N2 (Table 2-1) and Si3N4 (Table 2-2 and Table 2-3). Mg

powder of Merck (5815) and Si powder of Riedel de Haen AG were used.

Table 2-1: Characteristics of the Mg3N2 starting materials used

(data from the supplier).

Manufacturer Code [N][wt. %]

[N] + [Mg][wt. %]

AlfaCeractheoretical

932825M1014

—

27.426.027.8

99.5 99.5 100

Table 2-2: Characteristics of the Si3N4 starting materials used (data from this work; a: measured with

Kjeldahl method, b: measured with LECO O/N gas analyzer, and c: data given by

supplier).

Manufacturer Code [N]spec [N]ameas [O]spec [O]b

meas

[wt. %] [wt. %]SKW TrostbergCeracHCSTKema NordSylvaniaTosohUbetheoretical

Silzot HQS1177LC12N — —TS10SNE10

> 38.5 (38.74c)> 38.0> 38.5 — — —> 38.0

38.7 ± 0.3b

38.4 ± 0.4 39.2 ± 0.1 38.4 ± 0.3 29.5 ± 0.3 39.3 ± 0.5 37.7 ± 0.6 39.9

< 1.0 (0.34c) — 1.4 - 1.7 — — —< 2.0

0.7 ± 0.1 0.7 ± 0.1 1.4 ± 0.1 2.4 ± 0.1 4.1 ± 0.2 1.6 ± 0.1 1.2 ± 0.1 0

2.2. Preparation

The starting materials were mixed using a porcelain mortar and pestle in

stoichiometric amounts in a glove-box to prevent oxidation and hydrolysis of the

starting materials, especially Mg3N2. Subsequently, the mixed powders were put in

a closed stainless steel (AISI 304) tube. When further purification of the resulting

powders became necessary, molybdenum (Plansee, regular grade) tubes were used.

The tubes had a small gas inlet/outlet to prevent pressure built up. The starting

Chapter 2.

32

mixtures were normally fired at 1250 °C during 16 hours in a horizontal tube

furnace in a flowing N2 (99.95 % pure) or 85 vol. % N2 (99.95 % pure) / 15 vol. %

H2 (99.95 % pure) atmosphere. The firing temperature of 1250 °C was taken from

two earlier studies on the preparation of MgSiN2 [4, 10]. Also other firing

temperatures in the range of 900 °C - 1500 °C were used.

2.3. Characterisation

The starting powders and the powders resulting after firing the starting materials

were characterised with powder X-ray diffraction (XRD, Philips PW 1050/25,

Cu-Kα). The Mg3N2 starting materials were characterised in the range of 10 - 100 °

2θ (scan rate 1 °/min.) and the Si3N4 starting materials in the range of 10 - 80 ° 2θ

(scan rate 2 °/min.). The phase formation of the fired materials was investigated

with powder X-ray diffraction. They were investigated in the range of 10 - 100 ° 2θ

using standard continuous scans (1 °/min. or 2 °/min.) as well as step scans (0.1

°/min.).

The lattice parameters of MgSiN2 were calculated with the computer

program Refcel [11] using the fact that MgSiN2 has an orthorhombic cell (space

group Pna21 [12]). At least ten reflections (200, 002, 121, 201, 122, 202, 040, 320,

123, 203, 042, 241, 322, 401, 242 and 243) including a zero point correction were

used for calculating the lattice parameters.

The nitrogen content ([N] [wt. %]) of the Si3N4 starting materials was

determined by the Kjeldahl method or a O/N gas analyser (Leco TC 436). For the

former method the sample (0.1 g powder) was decomposed in molten LiOH. The

released ammonia was binded in a saturated boric acid solution. The amount of

ammonia was determined by titration with 0.1 M hydrochloric acid using

bromophenolblue as indicator. For the latter method the nitrogen present in the

sample was thermally converted at high temperatures to N2 which was measured

with a catharometer.

The oxygen content ([O] [wt. %]) for the Si3N4 starting materials and the

MgSiN2 powders was measured using a O/N gas analyser (Leco TC 436). The


33

powder sample was mixed with carbon whereafter the oxygen present in the

sample was carbothermally converted at high temperatures in an inert atmosphere

into CO, which after further oxidation to CO2 was measured with infra-red (IR)

absorption spectroscopy.

Thermo gravimetric analysis and differential thermal analysis (TGA/DTA)

was performed with a Netzsch STA 409 thermobalance to investigate the phase

formation and oxidation of MgSiN2. The phase formation was studied in flowing

N2 atmosphere using Al2O3 sample holders applying a constant heating rate of

10 °C/min. The oxidation study was performed in flowing air using Al2O3 sample

holders applying a constant heating rate of 5 °C/min. Also tube furnace oxidation

experiments in air were performed in combination with XRD and mass

measurements to determine the (intermediate) reaction products and to study the

oxidation kinetics.

Scanning electron microscopy (SEM, JEOL 840A) was used to study the

particle size and morphology of some of the prepared powders, and energy

dispersive spectrometry (EDS) to determine the chemical composition of the

powders, especially the presence of contamination(s).

The particle size distribution was measured with a Sedigraph 5100

Micromeritics using a 60 wt. % ethylene glycol / 40 wt. % water mixture. Before

measuring the particle size distribution the dispersed powder mixture was

ultrasonic treated for 20 min. to break up powder agglomerates.

3. Results and discussion

3.1. Starting powder characteristics

3.1.1. Mg3N2

At maximum 0.5 wt. % of impurities are present in the Mg3N2 starting materials

(Table 2-1). The major impurity is oxygen which is present as MgO as observed

with XRD. The significant difference between the nitrogen concentration given by

Chapter 2.

34

the supplier for Mg3N2 from Cerac (26.0 wt. %) and the expected value

(27.8 wt. %) gives evidence for the presence (besides MgO) of free Mg metal.

During the reaction of Mg metal with Si metal in a flowing nitrogen atmosphere an

Mg3N2 ceramic disk (∅ ≈ 20 mm × 0.4 mm) was formed (see 3.2. Phase formation

of MgSiN2). This disk was also investigated with XRD using the same conditions

as for the investigated Mg3N2 powders. Almost no MgO could be detected in this

sample with XRD.

Because Mg3N2 has a cubic lattice it is possible to calculate the true lattice

parameter by plotting the lattice parameter a calculated for each reflection versus

the function f(θ ), which is given by:

θθ

θθθ )(cos)sin()(cos)(

22

+=f (1)

and extrapolating to f(θ ) = 0 (see Ref. 13). In Fig. 2-1 the lattice parameter, a, for

each reflection of the Mg3N2 starting powders and the ceramic Mg3N2 disk is

plotted versus f(θ). For comparison data of the JCPDS card 35-778 for Mg3N2

powder (Cerac) are also included. From this figure it can be deduced that, although

marked differences occur for the lattice parameters calculated from the individual

9.92

9.94

9.96

9.98

10.00

10.02

10.04

10.06

0 2 4 6 8 10 12

f (θ ) [-]

a [Å

]

CeracAlfaJCPDS 35-778Ceramic

Fig. 2-1: Lattice parameter a of Mg3N2 powder of Cerac (+), Alfa

(◊), JCPDS 35-778 (), and Mg3N2 ceramic disk (∆) as a

function of f (θ ).


35

reflections, the extrapolated lattice parameter for all samples is the same viz.

9.963 ± 0.002 Å, which is comparable with the lattice parameter mentioned in

JCPDS card 35-778 (9.9657 Å). Because the lattice parameter was the same for all

investigated Mg3N2 samples and the impurity content of the powder samples was at

maximum 0.5 wt. % it can be concluded that the Mg3N2 lattice is saturated with

oxygen and that the solubility of oxygen in the Mg3N2 lattice is very low.

3.1.2. Si3N4

In Table 2-2 and Table 2-3 the measured powder characteristics of the used Si3N4

powders are presented. The nitrogen content measured for all investigated Si3N4

powders is in good agreement with the specification of the suppliers (see

Table 2-2). The oxygen concentration in the Si3N4 starting materials ranges from

0.7 to 4.1 wt. %. For the SKW Trostberg Si3N4 powder the measured oxygen

content (0.7 wt. %) is well within the specifications (< 1.0 wt. %) but considerably

higher than the content given by the supplier (0.34 wt. %). It can be seen that for

materials with a nitrogen concentration close to the theoretical value (> 39 wt. %),

the oxygen concentration is low. A considerable deviation of the nitrogen

concentration from the theoretical value combined with a low oxygen content was

measured for the Cerac, Ube and SKW Trostberg Si3N4 powders. This indicates

that some free silicon or silicon containing compound like SiC may be present.

According to the supplier, for the SKW Trostberg Si3N4 the free Si metal content is

smaller than 0.5 wt. % and some SiC (0.4 wt. %) is present.

The crystallographic modification of the Si3N4 powders, viz. amorphous, α

(JCPDS card 41-360), β (JCPDS card 33-1160) or tetragonal phase (JCPDS card

40-1129), was determined with XRD (Table 2-3). Only the Sylvania powder

appeared to be amorphous. Most powders mainly consist of the α-modification,

except for Si3N4 of Cerac which contained predominantly β. For three powders also

the presence of the tetragonal modification could be demonstrated. For the

crystalline powders the α /(α +β ) ratio was calculated (Table 2-3) using the

methods described in Refs. 14 - 18. The calculated α /(α +β ) ratio agrees quite well

Chapter 2.

36

with the specification of the suppliers. Only the measured α /(α +β ) ratio of SKW

Trostberg Si3N4 deviates about 15 % from the specified ratio.

Table 2-3: Characteristics of the used Si3N4 starting materials (data from this work; **: also some

Si3N4 with the tetragonal modification present).

Manufacturer Codespec

+ βαα

meas

+ βαα

(d50)spec (d50)meas (d90)meas

[-] [µm]

SKW TrostbergCeracHCSTKema NordSylvaniaTosohUbe

Silzot HQS1177LC12N——TS10SNE10

> 0.80 ± 0.1 0.94 —amorphous — > 0.95

0.66 0.08 0.89**

0.91**

amorphous 0.93 1.00**

1.7< 2.0 0.6 — — — 0.6

2.21.20.62.3—1.10.7

4.92.63.09.0—7.51.4

For all investigated Si3N4 powders the median particle size, d50, was less

than 2.5 µm (Table 2-3) and some are submicrometer size (Ube and HCST). The

Si3N4 powders of HCST, Kema Nord and Tosoh have a broad particle size

distribution (3d50 < d90) which indicates that the primary particles are most

probably agglomerated, even after ultrasonic treatment.

3.2. Phase formation of MgSiN2

The TGA/DTA experiments show that when starting with an Mg3N2/Si3N4 mixture

the temperature should surpass about 1100 - 1150 °C to get fast formation of

MgSiN2, in agreement with literature data [10]. In the DTA signal two endothermic

peaks are present. Which peak or whether both peaks can be ascribed to the

formation of MgSiN2 is not clear because both peaks are less than 50 °C separated

from each other. No attempts were made to discriminate between them because the


37

used standard synthesis temperature of 1250 °C is sufficiently high to obtain a fast

reaction and a fully reacted product.

For the Mg/Si3N4 starting mixture, the reaction mechanism is much more

complicated than for the previous case. Several exothermic DTA peaks are present

(Fig. 2-2), the strongest at 612 °C, and some smaller ones at 897 °C, 920 °C, and

(not visible in Fig. 2-2) 1061 °C. At about 612 °C nitridation of Mg takes place

accompanied by a mass gain of about 9.5 wt. %. The total mass gain at 1000 °C is

about 12.5 wt. % which is comparable with the expected mass gain of 13.1 wt. %

for the nitridation of the Mg present in the Mg/Si3N4 starting mixture. XRD

showed that a Mg/Si3N4 mixture fired at 700 °C in an N2 atmosphere resulted in a

mixture of Mg3N2 and Si3N4 whereas a mixture fired at 900 °C resulted in MgSiN2

giving further evidence that the DTA peaks at 897 °C and 920 °C are related with

the formation of MgSiN2.

Also the nitridation of metallic Mg powder was studied with TGA/DTA

(Fig. 2-3). At 648 °C an endothermic peak is observed which can be ascribed to the

melting of Mg metal. Two exothermic nitridation peaks were observed at 660 °C

-100

0

100

200

300

400

500

600

700

800

0 200 400 600 800 1000 1200

T [°C]

∆T [

µV]

-3

0

3

6

9

12

15

18

21∆m

/m0 ×

100

[%]

DTATGA

Fig. 2-2: TGA/DTA plot of an Mg/Si3N4 mixture in a nitrogen atmosphere

showing the temperature difference (∆T ) and relative mass

difference (∆m/m0) as function of the temperature (T ).

Chapter 2.

38

and 690 °C. The last one is related to the rapid nitridation of Mg. The observed

results are in good agreement with earlier published data [19] on the nitridation of

Mg. When these results are compared with those obtained for the Mg/Si3N4

mixtures, a lowering by about 50 °C of the nitridation temperature of Mg and no

melting peak of Mg are observed when using the Mg/Si3N4 mixtures. A possible

explanation might be a different reactivity of Mg in the presence of Si3N4.

When comparing the phase formation of MgSiN2 starting with Mg3N2/Si3N4

and Mg/Si3N4 mixtures it can be concluded that when starting with an Mg/Si3N4

mixture nearly single-phase MgSiN2 can already be obtained at a temperature of

about 900 °C, which is much lower than the minimal temperature of about 1150 °C

necessary for an Mg3N2/Si3N4 mixture. This difference in phase formation

temperature might be related to the fact that during nitridation of Mg an Mg3N2

phase is formed different from the room temperature modification [20] with a

higher reactivity. Also gas phase reactions may play an important role in the

observed difference in temperature. When Mg(g) condenses on the Si3N4 particles

the reactivity of the starting mixture might be increased due to the small particle

size of the condensed Mg resulting in a lower reaction temperature.

-100

0

100

200

300

400

500

600 650 700 750 800 850 900 950T [ºC]

∆T [ µ

V]

-10

0

10

20

30

40

50

∆m/m

0 × 1

00 [%

]

DTATGA

Fig. 2-3: TGA/DTA plot of Mg powder in a nitrogen atmosphere showing the

temperature difference (∆T ) and relative mass difference (∆m/m0)

as function of the temperature (T ).


39

In order to study the observed differences between the Mg3N2/Si3N4 and

Mg/Si3N4 starting mixtures in more detail the phase formation of MgSiN2 for

several Mg3N2/Si3N4 fired at 1250 °C and for Mg/Si3N4 starting mixtures fired at

900 - 1250 °C using different starting materials was studied with XRD. For

completeness also the phase formation for a Mg/Si starting mixture at 1250 °C was

studied.

Nearly single-phase grey-brown coloured MgSiN2 materials were obtained

when starting with Mg3N2/Si3N4 mixtures fired at 1250 °C or Mg/Si3N4 mixtures

fired at 900 - 1250 °C in an N2 atmosphere. For all Si3N4 starting materials

(amorphous, α- or β -modification, irrespective of the presence of tetragonal phase

or free Si), MgSiN2 is readily formed. In all cases some MgO (periclase, JCPDS

card 4-829) could be detected with XRD as a secondary phase. Sometimes white

powder was observed at the outside of the reaction tube. This powder was also

MgO, as observed with XRD, indicating that the oxygen in the starting materials or

the gas atmosphere reacts with Mg or Mg3N2 to MgO. The MgO contamination is

caused by oxygen impurities in the starting material and oxygen pickup during the

processing (mixing) and the synthesis (oxygen impurities in the N2 atmosphere /

reaction with oxides from the stainless steel tubes). The relative MgO content

(I/I0)MgO in the MgSiN2 powders was determined by dividing the intensity of the

strongest reflection of MgO (hkl = 200) by the intensity of the strongest reflection

of MgSiN2 (hkl = 121) multiplied by 100 %. As expected the observed MgO

content decreases for the purer Si3N4 starting materials. Almost no MgO could be

detected ((I/I0)MgO = 3) for the Mg3N2/Si3N4 and Mg/Si3N4 mixtures using oxygen

poor Si3N4 starting powders of SKW Trostberg and Cerac, respectively. In general

for the same Si3N4 starting material the least amount of MgO was observed when

using Mg instead of Mg3N2 indicating that the purity of the resulting MgSiN2 might

be improved by using a Mg/Si3N4 instead of a Mg3N2/Si3N4 mixtures. Another

advantage of using Mg/Si3N4 mixture is that, due to the lower firing temperature

necessary, a less non-stoichiometric product, caused by possible evaporation of

magnesium [4], will be formed. Moreover, a lower firing temperature yields less

contamination of the prepared materials with metals from the stainless steel (or

Chapter 2.

40

molybdenum) tubes, and a smaller particle size which will improve the sinterability

of the resulting powders. So using Mg/Si3N4 instead of Mg3N2/Si3N4 starting

mixtures might be beneficial for preparing a pure MgSiN2 powder because the

reaction temperature can be lowered. However, in the powders synthesised from

Mg/Si3N4 mixtures always some free Si metal (JCPDS 27-1402) was detected with

XRD. So, Mg can not only react with the N2 atmosphere to form Mg3N2 but also

with the Si3N4 powder to form Mg3N2 and metallic Si [12]. Because the nitridation

of metallic Si is kinetically hampered even at the standard processing temperature

of 1250 °C [21] removing of this secondary phase is a problem. So, the advantages

of the lower reaction temperature when using Mg/Si3N4 mixtures are cancelled by

the reaction of Mg with Si3N4 forming metallic Si which cannot be removed at low

reaction temperatures.

Starting with an Mg/Si mixture in a stainless steel reaction tube fired in a

flowing N2 atmosphere at 1250 °C MgSiN2 was formed. However, in this case no

single-phase MgSiN2 was obtained. The black coloured reaction product consisted

mainly of MgSiN2 and several not identified secondary phases. In the coldest part

of the reaction tube a light brown-orange coloured ceramic disk was formed. This

disk (∅ ≈ 20 mm × 0.4 mm) was investigated with XRD. It was concluded that Mg

condensed in the coldest part of the reaction tube as Mg3N2 ceramic. Considering

those difficulties, no further attempts were made to obtain single-phase MgSiN2

powder using a Mg/Si starting mixture.

In summary the phase formation study at 1250 °C using Mg3N2/Si3N4,

Mg/Si3N4 and Mg/Si starting mixtures showed that only the first two starting

mixtures resulted in nearly single phase MgSiN2. Although the TGA/DTA and

furnace experiments indicate that MgSiN2 can be synthesised at 900 °C using

Mg/Si3N4 starting mixtures, the use of a Mg3N2/Si3N4 starting mixture at 1250 °C is

preferred because the resulting MgSiN2 powder contains less Si impurities. In

general when a molybdenum tube was used instead of a stainless steel tube the

resulting MgSiN2 powder had a much more homogeneous and lighter colour

indicating that the powder contained less metallic contaminations (Fe, Cr and Ni as

detected with SEM/EDS). Based on the MgO found at the outside of the reaction


41

tube it can be assumed that MgO(g) can evaporate from the starting mixture. Using

these results it was tried to synthesise an oxygen poor MgSiN2 powder.

For this the Mg3N2/Si3N4 starting mixture with the lowest oxygen content

was used (Mg3N2 (Alfa)/Si3N4 (SKW Trostberg)). If the oxygen of the MgO at the

outside of the reaction tube originates from the starting mixture then it is possible

to purify the resulting MgSiN2 powder by adding an excess amount of Mg or

Mg3N2 to the starting mixture. The starting powder, with a small excess of Mg3N2

(± 1 wt. %) intentionally added, was fired in a 50 ml/h N2 (99.995 % pure)/5 ml/h

H2 (99.9999 % pure) atmosphere for 3 h at 1250 °C and subsequently 1 h at

1500 °C in a molybdenum tube using a heating and cooling rate of 3 °C/min. The

excess of Mg3N2 is used for maintaining the stoichiometry in the resulting MgSiN2

powder. The heat treatment at 1500 °C was performed to ensure that the starting

materials had fully reacted, to nitridate possible metallic Mg and Si impurities in

the starting powders, to evaporate the MgO present in the reaction mixture and to

remove the Mg3N2 excess present in the reaction product by decomposition into

Mg(g) and N2 [22]. So the stoichiometry of the reaction product is maintained

because MgSiN2 is stable at 1500 °C [22] and the added excess of Mg3N2 which

did not react to MgO is also removed from the reaction mixture.

Using this procedure single phase white MgSiN2 powder was formed. With

XRD using a scan rate of 0.033 °/min. only a small trace of MgO ((I/I0)MgO = 0.4)

could be detected. This indicates that the excess Mg3N2 does not increase the MgO

content in the resulting MgSiN2 powder under the given reaction conditions.

Because also no Mg3N2 could be detected this indicates that during the reaction

Mg3N2 or Mg3N2 and MgO evaporates from the reaction mixture.

3.3. Oxygen content of the MgSiN2 powders

The oxygen content of the MgSiN2 powders synthesised at 1250 °C is presented in

Table 2-4. The influence of the reaction temperature, in the range of

1000 - 1250 °C, on the oxygen content of the resulting MgSiN2 powders was

negligible. As expected, the overall oxygen content becomes lower when using

Chapter 2.

42

purer Si3N4 starting materials. Also, using Mg3N2 from Alfa (with the highest

nitrogen content, Table 2-1) instead of Mg3N2 from Cerac decreases the oxygen

content of the synthesised MgSiN2 powder. Use of Mg instead of Mg3N2 as a

starting material results in an even somewhat lower oxygen concentration in the

MgSiN2 powder. However, if the oxygen content in the used Si3N4 starting material

is low the difference in the oxygen content of MgSiN2 starting from Mg3N2/Si3N4

or Mg/Si3N4 mixtures appears to be negligible. So the oxygen content of the Si3N4

starting material is the dominating factor. For the standard synthesis temperature of

1250 °C the lowest oxygen content of about 0.9 - 1.0 wt. % is obtained for the

purest Si3N4 starting materials (Cerac S1068 and SKW Trostberg Silzot HQ). It is

significantly below the value of about 4 wt. % obtained in a previous study [4].

Table 2-4: Overall oxygen content of the MgSiN2 powders prepared from different

starting materials at 1250 °C.

Starting Materials Si3N4 Mg3N2 Mg-metal

[O]

[wt. %]

Cerac Alfa Merck

Si3N4

SKW TrostbergCeracHCSTKema NordSylvaniaTosohUbe

0.3 - 0.70.71.21.41.62.44.1

—1.4——2.03.05.3

0.91.01.61.61.72.26.1

—1.0—1.31.52.83.9

Fig. 2-4 shows the relative MgO content of the MgSiN2 powders synthesised

at 1250 °C as a function of the overall oxygen content. When the oxygen content is

high (> 2 wt. %), no strong correlation between the relative MgO content and the

oxygen content is observed because the oxygen can be present in several secondary

phases. Whereas, in case the oxygen content is low (≤ 2 wt. %), a correlation is

observed. Assuming that MgO is the only oxygen containing component at overall

oxygen concentrations ≤ 2 wt. %, a crude estimation of the maximum solubility of


43

oxygen in the MgSiN2 lattice was made by extrapolation to a relative MgO content

equalling 0, yielding a maximum oxygen concentration of about 0.5 ± 0.2 wt. %.

Above this solubility limit oxygen MgO is formed as a secondary phase, whereas

below this limit oxygen is assumed to incorporate in the MgSiN2 lattice. The

maximum solubility of 0.5 wt. % oxygen in the MgSiN2 lattice corresponds to

0.5 1021 O/cm3 at maximum, as compared with about 6 1021 O/cm3 reported for

AlN [3]. So the solubility of oxygen in the MgSiN2 lattice is much lower than in

AlN.

The MgSiN2 powder synthesised using the purest starting materials by firing

first at 1250 °C and subsequently at 1500 °C contained only 0.1 ± 0.1 wt. % O as

determined with the O/N gas analyser. This value is considerably lower than that

measured for the MgSiN2 powder synthesised at 1250 °C using the same starting

materials (0.9 wt. % O). This value is even lower than the value expected from the

oxygen content of the used starting mixture (∼ 0.6 wt. % O) indicating that during

the synthesis the oxygen content in the reaction mixture decreases. The unexpected

0

5

10

15

20

25

30

0 1 2 3 4 5overall oxygen content [wt. %]

(I/I 0)

MgO

[%]

Fig. 2-4: The relative MgO content ((I/I0)MgO) of several MgSiN2

powders synthesised at 1250 °C from Mg3N2/Si3N4 (+)

and Mg/Si3N4 (⊕) mixtures as a function of the overall

oxygen content (as determined with the O/N gas analyser

(Leco TC 436)).

Chapter 2.

44

low oxygen content might be caused, as discussed before, by a (partial) reaction of

the weighed-out Mg3N2 with the oxygen present in the starting mixture to MgO

which evaporates from the reaction mixture. An additional effect might be the

carbothermal nitridation reaction occurring at the higher firing temperature

between the trace SiC, present in the SKW Trostberg Si3N4 starting powder, and

the oxygen containing compounds present in the starting mixture. Also the use of a

purer gas atmosphere might be beneficial for the obtained oxygen content.

The low oxygen content in combination with the fact that still some MgO

was detectable with XRD ((I/I0)MgO = 0.4 %) indicates that the maximum solubility

of oxygen in the MgSiN2 lattice is most probably well below the estimated 0.5 wt.

% based on Fig. 2-4. So the estimation of the maximum solubility of oxygen in the

MgSiN2 lattice might be conservative.

The MgSiN2 powder sample with the low oxygen content of 0.1 wt. %

contained 34.2 ± 1.7 wt. % N which is considerably higher than the value obtained

in a previous study (30.7 wt. % [4]), and only somewhat lower than theoretical

value (34.8 wt. %). This is in agreement with the fact that due to the presence of

some residual oxygen and possibly other contaminations, the nitrogen content

should be somewhat lower than the theoretical value.

3.4. X-ray diffraction data of MgSiN2

In order to calculate reliable lattice parameters for MgSiN2 powders the

reflections of MgSiN2 should be correctly indexed. From the present XRD study of

MgSiN2 powders and another study of MgSiN2 ceramics [9] it is known that the

indexing of the MgSiN2 reflections given in JCPDS card 25-530 is not completely

correct. The data of ceramic samples were used to revise the published XRD data

of MgSiN2 powders because the ceramic samples gave a better signal-noise ratio

than the powder samples. The revised data (Table 2-5) were obtained from MgSiN2

ceramic samples (Ref. 9) with an average grain size of 0.25 - 1.5 µm in which no

preferential orientation was detectable with XRD using a cylindrical camera. The

data were used to identify the powder samples. The d-values, obsd , presented are


45

Table 2-5: List of d-values and relative intensities of pure MgSiN2 evaluated from ceramic samples.

hkl value obsd dobs 0/ II obs d-value according I/I0 according to

[Å] [Å] [%]to JCPDS 25-530

[Å]JCPDS 25-530

[%]110011111120200002210121201211112220130031221122202212310040013231132140320141321123203240213042241400/033322401150051004420151242332114313/421233402124251204

4.093.9493.1602.7582.63492.49222.44052.41332.32942.19192.12782.04341.99691.98031.89071.84911.81061.74371.69531.61841.60931.58301.55841.54711.54391.47751.47481.42321.40541.37901.37341.35731.32911.3175/1.31641.31251.27371.25731.25311.24611.22021.21911.20661.19541.19191.1866/1.18521.17761.16471.13561.13171.1265

4.083.9453.1582.7562.63322.49072.43842.41132.32832.19092.12582.04261.99521.97921.89091.84831.80981.74291.69471.61851.60891.58261.5578

1.5436

1.47451.42281.40511.37861.37401.35691.32881.31731.31221.27351.25691.25261.24581.22011.21861.20631.19501.19181.18681.17731.16431.13531.13091.1262

910

1 88 45 80 3

100 23 1 1

< 1 < 1

3 < 1 28 12 2 2

25 1 1 1

36

13411

71

1511

122

5112

< 1< 1

5< 1

1< 1

11211

4.13.963.142.762.6422.496

2.4152.336

1.983

1.8501.811

1.621

1.549

1.482

1.4251.4091.381

1.3591.328

1.3141.275

1.248

1.208

1.133

1.129

1214

88555

100

9530

3

3018

20

45

3

401810

1612

308

5

7

3

2

calculated using the average observed lattice parameters (orthorhombic lattice

a = 5.2698 ± 0.0013 Å, b = 6.4736 ± 0.0014 Å and c = 4.9843 ± 0.0010 Å)

Chapter 2.

46

determined for several ceramic samples. The relative intensities, obs0/ II , are the

average measured relative intensities for those ceramic samples. For determining

the lattice parameters the computer program Refcel with zero point correction was

used taking into account at least ten reflections. For comparison an observed

d-value list (dobs) of a ceramic sample is included in Table 2-5 and also the data of

JCPDS card 25-530 of MgSiN2, which refers to the results of David [23], are

presented.

As can be seen from Table 2-5 the d-value list of MgSiN2 was revised by

adding some low intensity peaks which are not mentioned in JCPDS card no.

25-530. We especially mention the 210 (d = 2.4405 Å), 212 (d = 1.7437 Å) and

310 (d = 1.6953 Å) reflections because they have a relative strong intensity

(I/I0 ≈ 2 - 3) as compared to the other low intensity peaks (I/I0 ≤ 1) which were

added. Another difference is that in JCPDS card no. 25-530 the d-values 1.549 Å

and 1.482 Å are indexed with hkl = 140 and hkl = 141 whereas in the present study

these reflections were indexed as 320 and 321, respectively. Furthermore, some

differences in observed intensity I/I0 are noticed, especially for the 111 reflection

for which I/I0 = 8 according to JCPDS 25-530 whereas the observed value is much

lower (I/I0 = 1).

However, David et al. [12] calculated a theoretical intensity of I/I0 = 0.7,

which is in excellent agreement with the intensity of I/I0 = 1 observed in the

present study. This mismatch in calculated and measured intensity by David et al.

is tentatively ascribed by the present author to the presence of free Si (JCPDS card

27-1402) in their MgSiN2 powder, which increased the intensity measured for the

111 reflection. The 111 reflections of Si and MgSiN2 have similar d-values of

3.136 Å and 3.160 Å, respectively.

The indexation of the 140 and 141 reflections was changed because if a

synthetic pattern (d-value list of all possible reflections) was generated using

calculated lattice parameters, the d-value of the 320 and 321 reflection matched

much better the experimentally found d-values than the calculated d-value of the

140 and 141 reflection. As an example the d-values observed for a ceramic MgSiN2

sample, dobs, can be compared with the calculated d-values using the average lattice


47

parameters, obsd , determined from the ceramic samples (Table 2-5). Using the

atomic positions for MgSiN2 taken from Ref. 24 and a computer program for

calculating X-ray diffraction intensities (Powder Cell [25]) we also concluded that

the 140 and 141 reflection should indeed be replaced by the 320 and 321 reflection,

respectively. The intensity calculated for the 140 and 141 reflection is << 1 %

whereas the calculated intensity of the 320 and 321 reflection were in good

agreement with the measured intensity. Furthermore David et al. [12] calculated a

much higher intensity for the 320 than for the 140 reflection (49.7 versus 0.1

respectively) whereas the calculated intensity of the 141 matched better than the

one calculated for the 321 reflection (3.4 versus 0.4 respectively). Wild et al. [26]

also used the 320 reflection instead of the 140 reflection.

In Fig. 2-5 the lattice parameters and in Fig. 2-6 the cell volume measured

for the MgSiN2 powders processed at 1250 °C are presented as a function of the

measured overall oxygen content. The error bar indicated in Fig. 2-5 and Fig. 2-6

equals 3 times the standard deviation of the Refcel calculation. From the figures we

can conclude that the powder samples processed at 1250 °C have the same lattice

parameters (aaverage = 5.275 ± 0.007 Å, baverage = 6.472 ± 0.009 Å and caverage = 4.987

± 0.011 Å) and cell volume (Vaverage = 170.25 ± 0.70 Å3). Within the limits of

accuracy the results are in agreement with the lattice parameters used in Table 2-5

for the ceramic samples. Because the lattice parameters for all samples synthesised

at 1250 °C are the same irrespective of the overall oxygen content ranging from

0.9 - 6.1 wt. % it is concluded that the maximum solubility of oxygen in the

MgSiN2 lattice is surpassed. This is in accordance with the estimated maximum

oxygen solubility of 0.5 wt. % in the MgSiN2 lattice; therefore no influence of the

overall oxygen concentration on the lattice parameters is expected above 0.5 wt. %

oxygen.

For the MgSiN2 powder with an oxygen content of about 0.1 wt. %, first

fired at 1250 °C and subsequently 1500 °C, the lattice parameters are

a = 5.276 ± 0.006 Å, b = 6.477 ± 0.006 Å and c = 4.990 ± 0.005 Å. This is

comparable with the calculated average lattice parameters observed for the ceramic

Chapter 2.

48

and powder samples indicating that the solubility of oxygen is most probably even

less than 0.1 wt. % oxygen.

4.90

5.00

5.10

5.20

5.30

5.40

0 1 2 3 4 5 6 7

overall oxygen content [wt. %]

a, c

[Å]

6.20

6.30

6.40

6.50

6.60

6.70

b [Å

]

c

b

a

Fig. 2-5: The calculated lattice parameters a, b and c determined for several

MgSiN2 powders versus the measured overall oxygen content.

166

168

170

172

174

176

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

overall oxygen content [wt. %]

V [Å

3 ]

Fig. 2-6: The calculated cell volume (V ) determined for several

MgSiN2 powders versus the measured overall oxygen

content.


49

3.5. Powder characteristics

The morphology and particle size as observed with SEM is similar for MgSiN2

powders prepared from Mg3N2/Si3N4 and Mg/Si3N4 mixtures when the same Si3N4

starting material is used (Fig. 2-7 and Fig. 2-8). This was confirmed by sedigraph

measurements of MgSiN2 powders prepared from Mg3N2/Si3N4 and Mg/Si3N4

mixtures. The MgSiN2 powder prepared from Mg has a broader particle size

distribution and a somewhat larger median particle size. In Fig. 2-9 the mass

cumulative particle size distribution of the Si3N4 powder (SKW Trostberg), and the

MgSiN2 powders synthesised thereof with Mg3N2 (Alfa) and Mg (Merck) at

1250 °C are presented. As can be seen in Fig. 2-9 the starting Si3N4 powder has a

narrow particle size distribution and a median particle size of 2.2 µm. The MgSiN2

powder synthesised from the Mg3N2/Si3N4 starting mixture has also a narrow

particle size distribution but the powder is coarser. The median particle size equals

3.2 µm. The MgSiN2 powder prepared from the Mg/Si3N4 starting mixture has a

broad particle size distribution but the median particle size, viz. 3.8 µm, is only

slightly larger than the powder synthesised from the Mg3N2/Si3N4 starting mixture.

Fig. 2-7: SEM picture of an MgSiN2 powder synthesised at 1250 °C from a

Si3N4 (SKW Trostberg)/ Mg3N2 (Alfa) starting mixture.

Chapter 2.

50

The powder consisted partially of hard agglomerates that could not be removed by

ultrasonic treatment and are probably related to the formation of free Si metal

during the synthesis.


Si3N4 (SKW Trostberg)/Mg (Merck) starting mixture.

0

10

20

30

4050

60

70

80

90

100

0 1 2 3 4 5 6 7 8 9 10particle size [µm]

cum

ulat

ive

mas

s [w

t. %

]

Fig. 2-9: The particle size distribution as determined by Sedigraph

measurements of Si3N4 (SKW Trostberg) starting material

(+) and MgSiN2 powder prepared thereof with Mg3N2

(Alfa) (∆) or Mg (Merck) ().


51

For the MgSiN2 powders two different morphologies were observed

(Fig. 2-10 and Fig. 2-11). The first one showed equi-axed grains with primary

particle sizes smaller than 3 µm and agglomerates smaller than 10 µm. The second

one consisted of large porous sponge like agglomerates (~ 100 µm). The observed

type of morphology was independent of the α /β ratio of the starting Si3N4 powder

used and whether a Mg/Si3N4 or Mg3N2/Si3N4 starting mixture was used. If the

Si3N4 starting powders of Tosoh or Ube were used then the sponge like MgSiN2

particles were observed. If the Si3N4 starting powders of HCST and SKW

Trostberg were used then small equi-axed MgSiN2 particles were observed. For

comparison the morphology of the Si3N4 starting material of HCST and Tosoh was

investigated with the SEM. The Si3N4 powder of HCST consisted of small grains

whereas the Tosoh powder consisted of large agglomerates of small grains. This

indicates that the morphology of the Si3N4 starting material determines the

morphology of the resulting MgSiN2 powder.


Si3N4 (HCST)/Mg (Merck) starting mixture.

Chapter 2.

52

3.6. Oxidation behaviour of MgSiN2 powders

TGA/DTA measurements and furnace experiments in combination with XRD were

used to study the oxidation behaviour of MgSiN2. TGA/DTA experiments show

that MgSiN2 powders are oxidation resistant in air up to 830 °C. At higher

temperatures 4 reaction peaks are observed (Fig. 2-12); 3 exothermic peaks at

904 °C, 1082 °C and 1362 °C, and 1 endothermic peak at 1459 °C. The total

weight gain for the first 2 DTA peaks is about 18 wt. %. This mass gain can be

represented by the following overall reaction:

8 MgSiN2 + 9 O2 → 4 Mg2SiO4 + 2 Si2N2O + 6 N2 (+ 18.6 wt. %)

The total weight gain after the third DTA peak is about 25 wt. %. This can be

represented by the following overall reaction:

2 MgSiN2 + 3 O2 → Mg2SiO4 + SiO2 + 2 N2 (+ 24.8 wt. %)

So after the third DTA peak the MgSiN2 powder is totally oxidised. This peak is

related to the fast oxidation of Si2N2O to SiO2. The reaction temperature of about

1362 °C is in favourable agreement with the temperature mentioned in the

Fig. 2-11: SEM picture of a MgSiN2 powder synthesised at 1250 °C from a

Si3N4 (Tosoh) / Mg3N2 (Alfa) starting mixture.


53

literature [27] for the fast oxidation of Si2N2O powder at about 1330 °C. So the

oxidation of MgSiN2 is a two step process as shown by the TGA/DTA

measurements. The fourth DTA peak at about 1459 °C is an endothermic one and

is related to the phase transformation of SiO2 from tridimite to cristobalite

(1477 °C as deduced from Fig. 5 of Ref. 28).

The isothermal oxidation behaviour of MgSiN2 powder was studied in air,

just above the oxidation temperature, at 850 °C. From the isothermal oxidation

study it was clear that MgSiN2 can be totally oxidised at 850 °C indicating that the

intermediate reaction products are not stable. No parabolic oxidation behaviour

[27] was observed probably due to a superposition of the two above mentioned

oxidation reactions.

4. Conclusions

The phase formation study of MgSiN2 showed that nearly single phase MgSiN2

powders can be obtained from Mg3N2/Si3N4 or Mg/Si3N4 mixtures. However, the

reaction paths are different as shown with TGA/DTA. Oxygen poor MgSiN2

-10

0

10

20

30

40

50

60

70

0 300 600 900 1200 1500T [°C]

∆T [ µ

V]

-7

0

7

14

21

28

∆m/m

0 ×

100

[%]

DTATGA

Fig. 2-12: TGA/DTA plot of the oxidation behaviour of MgSiN2 powder in

air showing the temperature difference (∆T ) and relative mass


Chapter 2.

54

powders can be prepared, not only by the conventional synthesis route starting with

Si3N4 and Mg3N2, but also by starting with Si3N4 and Mg in a flowing N2/(H2)

atmosphere. This alternative synthesis route has the benefit of a lower reaction

temperature and the disadvantage of a more critical processing due to the formation

of free Si metal during the synthesis.

When using the standard synthesis temperature of 1250 °C, the overall

oxygen content obtained for the MgSiN2 powders varied between 0.9 - 6.1 wt. %

O, mainly determined by the oxygen content of the Si3N4 starting material. The

lattice parameters of these powders do not depend on the overall oxygen

concentration indicating that the maximum solubility of oxygen in the lattice is

surpassed in accordance with the observed presence of some residual MgO. Its

concentration in these powders suggest that the maximum solubility of oxygen in

the MgSiN2 lattice does not exceed 0.5 ± 0.2 wt. %. By using improved processing

conditions it is possible to synthesise powders with an oxygen content of only 0.1

wt. % O. However, even in these powders containing only 0.1 wt. % O some MgO

could be detected with XRD indicating that the maximum solubility of oxygen in

the MgSiN2 lattice is probably even much lower than 0.5 wt. %.

The study of the MgSiN2 powders with SEM and the sedimentation method

showed that the morphology of the MgSiN2 powders is most probably determined

by the morphology of the used Si3N4 starting material. If the starting Si3N4 powder

was agglomerated, large sponge like MgSiN2 particles were observed.

Oxidation experiments showed that MgSiN2 powder is oxidation resistant in

air up to 830 °C as determined by TGA/DTA. At two different temperatures

(1082 °C and 1362 °C) fast oxidation takes place indicating that the oxidation of

MgSiN2 is at least a two step process. An isothermal oxidation experiment at

850 °C showed that MgSiN2 could be fully oxidised indicating that the

intermediate oxidation products are not stable.

Finally it can be concluded that it is possible to synthesise MgSiN2 powders

with a very low oxygen content which are very suitable for further processing to

ceramics with optimum thermal properties.


55

References


Electroceramics IV 2, Aachen, Germany, September 5 - 7 1994, edited by R.

Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann, (Augustinus


2. C.-F. Chen, M.E. Perisse, A.F. Ramirez, N.P. Padture, H.M. Chan, Effect of

Grain Boundary Phase on the Thermal Conductivity of Aluminium Nitride



Chem. Solids 34 (1973) 321.

4. W.A. Groen, M.J. Kraan, and G. de With, Preparation, Microstructure and


5. H.T. Hintzen, W.A. Groen, P. Swaanen, M.J. Kraan and R. Metselaar,

Hot-pressing of MgSiN2 Ceramics, J. Mater. Sci. Lett. 13 (1994) 1314.



7. H.T. Hintzen, R.J. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder

Preparation and Densification of MgSiN2, Ceram. Trans. 51, Int. Conf.

Ceramic Processing Science Technology, Friedrichshafen (Germany),

September 1994, edited by H. Hausner, G.L. Messing and S. Hirano (The

American Ceramic Society, 1995) 585.

8. H.T. Hintzen, R.J. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2

Ceramics, Fourth Euro Ceramics 2, Basic Science - Developments in

Processing of Advanced Ceramics - Part II, Faenza (Italy), October 1995,

edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, Italy,

1995) 289.

9. Chapter 3; R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar,

Preparation, Characterisation and Properties of MgSiN2 ceramics, to be

published.

Chapter 2.

56

10. J. David et J. Lang, Sur un nitrure double de magnésium et de silicium, C. R.

Acad. Sc. Paris 261 (1965) 1005.

11. Refcel, Calculation of cell constants and calculation of all possible lines in a

powder diagram by H.M. Rietveld, October 1972.

12. J. David, Y. Laurent et J. Lang, Structure de MgSiN2 et MgGeN2, Bull. Soc.

Fr. Minéral. Cristallogr. 93 (1970) 153.

13. B.D. Cullity, Elements of X-Ray Diffraction, second edition, (Addison-

Wesley Publishing Company, Inc., 1978).

14. C.P. Gazzara and D.R. Messier, Determination of Phase Content of Si3N4 by

X-ray Diffraction Analysis, Am. Ceram. Soc. Bull. 56 (1977) 777.

15. G. Petzow and R. Sersale, Characterization of Si3N4 Powders, Pure and Appl.

Chem. 59 (1987) 1673.

16. N. Matter, A. Riedel and A. Wassermann, Quantitative Phase Analysis of

Si3N4 Ceramics using the Powder Diffraction Standard Data Base, Mat. Sci.

Forum 133-136, EPDIC 2 (Trans Tech Publications, Switzerland, 1993) 39.

17. W. Pfeiffer and M. Schulze, A Method for the Determination of Weight

Factors for Quantitative Phase Analysis using Dual Phase Starting Powders

with Application to α/β -Silicon Nitride, Mat. Sci. Forum 133-136, EPDIC 2

(Trans Tech Publications, Switzerland, 1993) 39.

18. D.Y. Li, B.H. O'Conner, Q.T. Chen and M.G. Zadnik, Quantitative Powder

X-ray Diffractometry Phase Analysis of Silicon Nitride Materials by a

Multiline, Mean-Normalized-Intensity Method, J. Am. Ceram. Soc. 77

(1994) 2195.

19. T. Murata, K. Itatani, F.S. Howell, A. Kishioka and M. Kinoshita, Preparation

of Magnesium Nitride Powder by Low-Pressure Chemical Vapor Deposition,

J. Am. Ceram. Soc. 76 (1993) 2909.

20. I.S. Gladkaya, G.N. Kremkova and N.A. Bendeliani, Phase diagram of

magnesium nitride at high pressures and temperatures, J. Mater. Sci. Lett. 12

(1993) 1547.


57

21. M. Barsoum, P. Kangutkar and M.J. Koczak, Nitridation Kinetics and

Thermodynamics of Silicon Powder Compacts, J. Am. Ceram. Soc. 74

(1991) 1248.

22. R. Muller, Konstitutionsuntersuchungen und thermodynamischen

Berechnungen im system Mg, Si/N, O, Ph. D. Thesis, University of Stuttgart,

Stuttgart, Germany (1981) p. 107.

23. J. David, Étude sur Mg3N2 et quelques-unes de ses combinaisons, Rev. Chim.

Miner. 9 (1972) 717.

24. M. Wintenberger, F. Tcheou, J. David and J. Lang, Verfeinerung der Struktur

des Nitrids MgSiN2 - eine Neutronenbeugungsuntersuchung, Z. Naturforsch.

35b (1980) 604.

25. Powder Cell, Programm zur Manipulation von Kristallstrukturen und

Berechnung der Röntgenpulverdiffraktogramme. Werner Kraus, Dr. Gert

Nolze, Bundesanstalt für Materialforschung und -prüfung 12205 Berlin Unter

den Eichen 87 (1995).

26. S. Wild, P. Grieveson and K.H. Jack, The Crystal Chemistry of New Metal-

Silicon-Nitrogen Ceramic Phases, Spec. Ceram. 5 (1972) 289.

27. J. Persson, P.O. Käll and M. Nygren, Interpretation of the Parabolic and

Nonparabolic Oxidation Behaviour of Silicon Oxynitride, J. Am. Ceram. Soc.

5 (1992) 3377.

28. M. Hillert, S. Jonssen and B. Sundman, Thermodynamic Calculation of the

Si-N-O System, Z. Metallkd. 83 (1992) 648.

Chapter 2.

58

59

Chapter 3.

Preparation, characterisation and properties of

MgSiN2 ceramics

Abstract

MgSiN2 ceramics with and without sintering additives were prepared by hot

uni-axial pressing. For the sintered samples the lattice parameters, density, oxygen

and nitrogen content, microstructure, oxidation resistance, hardness, elastic

constants, linear thermal expansion coefficient and thermal diffusivity were

determined. By suitable processing fully dense MgSiN2 ceramics with an oxygen

content < 1.0 wt. % could be obtained. The size of the MgSiN2 grains increased

with increasing hot-pressing temperature and time. Transmission electron

microscopy (TEM) showed that no grain boundary phases were present on the

MgSiN2 grains and that secondary phases are present as separate grains in the

MgSiN2 matrix. Atomic force microscopy (AFM) thermal imaging revealed a

thermal barrier at the grain boundaries. However, the influence of the grain size /

microstructure on the thermal diffusivity was limited. Furthermore, the influence of

the oxygen content and defect chemistry on the thermal diffusivity was limited.

From these data it was concluded that the thermal conductivity of the MgSiN2

ceramic samples, which did not exceed 25 W m-1 K-1 at 300 K, is determined by

intrinsic phonon-phonon scattering.

1. Introduction

Non-electrically conducting materials with a high thermal conductivity are

potentially interesting as a heat sink material in integrated circuits. Due to the

Chapter 3.

60

miniaturisation of these integrated circuits there is a strong need for replacing the

traditional heat sink material Al2O3 (alumina) by new and better ones with a higher

thermal conductivity than Al2O3 (viz. 17 - 38 W m-1 K-1 [1]). Several binary

adamantine type compounds like SiC, BeO and AlN have been investigated for this

purpose.

Some years ago, also ternary and quaternary adamantine type oxy-carbo-

nitride, (oxy-)nitride, and (oxy-)carbide compounds have been suggested as

potential substrate materials [2 - 4]. Especially MgSiN2, which can be deduced

from AlN by replacing two Al3+ ions by a combination of Mg2+ and Si4+, was

considered to be very promising [2, 5]. Previous studies showed that the

mechanical and electrical properties of non-optimised MgSiN2 ceramics are

comparable with those of AlN and Al2O3 [2, 6]. A first estimate using the theory of

Slack [7] of the maximum achievable thermal conductivity at 300 K resulted in a

value of 75 W m-1 K-1 [8]. In a more recent study [9] we suggest that the maximum

achievable thermal conductivity of MgSiN2 does not exceed 50 W m-1 K-1 and

more probably is limited to a value of about 30 W m-1 K-1, which is comparable

with the highest reported experimental value of 25 W m-1 K-1 [10]. However, in the

literature no systematic research on the influence of the processing conditions on

the thermal conductivity of MgSiN2 ceramics has been reported which might give a

better insight in the thermal conductivity limiting mechanism in this material. If

this mechanism can be identified, possibly a better, more reliable estimate of the

maximum achievable thermal conductivity can be made. Moreover the thermal

conductivity of MgSiN2 can be optimised more effectively.

The thermal resistivity of phonon conductors is in general determined by the

sum of the occurring resistivities, that means the inverse phonon mean free paths of

all individual independent phonon scattering processes [11]. Therefore, for phonon

conductors like MgSiN2 phonon-phonon scattering, phonon-defect scattering and

phonon-grain boundary scattering are considered to influence the thermal

conductivity [12, 13]. Furthermore, the heat transport between the grains can be

hampered by (oxygen containing) secondary phases. So phonon scattering

processes in the MgSiN2 lattice itself (phonon-phonon, phonon-defect), or due to

Preparation, characterisation and properties of MgSiN2 ceramics

61

the microstructure of the ceramics ((phonon scattering at the grain boundaries

depending on the grain size), (secondary phases at the grain boundaries)) influence

the heat transport in MgSiN2 ceramics.

In this chapter the characterisation and properties of MgSiN2 ceramics

processed in different ways is described. The problem of obtaining not fully dense

MgSiN2 ceramics encountered during pressureless-sintering can be solved by using

the hot uni-axial pressing technique [14]. This technique was reported to give fully

dense ceramic samples necessary for obtaining a high thermal conductivity. Special

attention will be paid to the influence of the processing parameters and sintering

additives on the (thermal) properties in order to identify the mechanism that

determines the thermal conductivity of MgSiN2 ceramics. The synthesis of MgSiN2

powders with a low oxygen content, which is considered to be necessary for

obtaining a high thermal conductivity [2, 14], has been reported previously

[15 - 17]. Preliminary results concerning the ceramics have already been published

elsewhere [10, 15].

2. Experimental

2.1. Preparation

Dense MgSiN2 ceramic samples were prepared using the (reaction) hot uni-axial

pressing ((R)HUP) technique. The hot-press (HP 20, Thermal Technology Ind.)

was equipped with an Astro furnace (model 100-4560-FP) fitted with graphite

heating elements. The interior of the hot-press including the die and the ram is

made from graphite. During hot-pressing the temperature, force and displacement

were monitored.

Although HUP is a suitable method for obtaining fully dense MgSiN2

ceramics without sintering additives [14], during the present work additives were

still sometimes used. The additive was not only used to promote further

densification and/or grain growth but especially in an attempt to purify the MgSiN2

bulk by formation of a secondary phase. The intention is to minimise phonon-

Chapter 3.

62

defect scattering by (oxygen) impurities in the bulk causing vacancies, which is

reported to have a detrimental effect on the thermal conductivity of AlN [18, 19].

Y2O3 was used as an additive because it is known to be a suitable additive for AlN

and because it forms low melting compounds with MgO and SiO2 [20 - 22]. Also

CaO and 2Al2O3.Si3N4 were used because they form in combination with MgO and

SiO2 the low melting compounds CaMgSiO4 (1485 °C [23]) and Mg2SiAlO4N

(1600 °C [24]) or MgSi4Al2O6N4 (1450 - 1650 °C [24]), respectively. Also Si3N4

and Mg3N2 were used as an additive in order to influence the defect chemistry of

the MgSiN2 bulk [2]. Finally, the addition of Mg3N2 might be beneficial for

obtaining MgSiN2 ceramics with a low oxygen content as has been reported

previously for the synthesis of MgSiN2 powders [17] and ceramics [25] with a low

oxygen content.

The ceramic pellets were prepared with HUP starting from as prepared

MgSiN2 powder or RHUP starting from Mg3N2 (Alfa 932825)/Si3N4 (SKW

Trostberg Silzot HQ or Cerac S1177) powder mixtures with or without additives

(see Table 3-1). The MgSiN2 powders were prepared starting from either Mg

(Merck 5815)/Si3N4 (HCST LC12N) or Mg3N2 (Alfa 932825 or Cerac

M1014)/Si3N4 (Tosoh TS10, Cerac S1177 or Ube SNE10) mixtures. The

characteristics of the starting materials used and the experimental details for the

preparation of the MgSiN2 powders are described elsewhere [17]. To prevent

oxidation and hydrolysis of Mg3N2, the Mg3N2/Si3N4 starting mixture was handled

and put into the hot-press die in a glove-box purged with nitrogen. All parts of the

hot-press that were in contact with the starting powder during hot-pressing were

coated with boron nitride or protected with boron nitride coated graphite foil. The

following processing parameters (see Table 3-1) were varied: hot-pressing

temperature (1450 - 1750 °C), hot-pressing pressure (20 - 75 MPa) and time

applying the maximum pressure (0.5 - 5 h). After filling of the die (∅ = 34 mm)

with about 10 g of starting powder, the powder was pre-pressed at 10 MPa.

Subsequently the hot-press was closed and purged with nitrogen for at least 0.5 h

before heating. During heating, with a rate of 10 - 20 °C/min., a pressure of

3 - 4 MPa was applied. The maximum pressure was applied at the top temperature.


63

After cooling down, the resulting ceramic pellet (± ∅ 33 mm × 3 mm) was cleaned

and ground using a 250 µm abrasive diamond wheel and when necessary cut and

polished.

Table 3-1: Hot-pressing conditions used for processing the several MgSiN2 ceramic samples.

Code RB02 RB07 RB09 RB10StartingMaterial

MgSiN2(Merck & HCST)

MgSiN2(Merck & HCST)

MgSiN2(Alfa & Tosoh)

MgSiN2(Cerac & Cerac)

ReactionConditions

1550 °C2 h, 75 MPa

1550 °C2 h, 75 MPa

1550 °C2 h, 75 MPa

1550 °C2 h, 75 MPa


MgSiN2(Alfa & Ube)

Mg3N2 (Alfa)Si3N4 (Cerac)



ReactionConditions

1550 °C,2 h, 75 MPa

1550 °C,2 h, 75 MPa

1600 °C,2 h, 75 MPa

1550 °C,5 h, 75 MPa


MgSiN2(Mg & HCST)




Additive --- Si3N4 2.9 wt.% Mg3N2 2.1 wt.% Mg3N2 4.2 wt.%ReactionConditions

1550 °C2 h, 75 MPa

1600 °C,2 h, 75 MPa

1600 °C,2 h, 75 MPa

1600 °C,2 h, 75 MPa




Mg3N2 (Alfa)Si3N4 (SKW)


ReactionConditions

1650 °C,2 h, 75 MPa

1700 °C2 h, 75 MPa

1600 °C2 h, 75 MPa

1750 °C2 h, 75 MPa





MgSiN2(Alfa & SKW)

Additive Y2O3 6 wt.% 2Al2O3.Si3N4 5.8 wt.% CaO 2.3 wt.% Mg3N2 5.0 wt.%ReactionConditions

1600 °C2 h, 75 MPa

1600 °C1 h 4 MPa2 h 20 MPa½ h 25 MPa

1600 °C1 h 4 MPa2 h 20 MPa½ h 25 MPa

1600 °C1 h 4 MPa2 h 20 MPa½ h 25 MPa






Additive Mg3N2 5.0 wt.% Al2O3 4.3 wt.% --- ---ReactionConditions

1600 °C1 h 4 MPa2 h 20 MPa½ h 40 MPa

1600 °C1 h 4 MPa2 h 20 MPa½ h 25 MPa

1500 °C2 h, 75 MPa

1450 °C2 h, 75 MPa

Chapter 3.

64


The ceramic samples were investigated with X-ray diffraction (XRD, Philips

PW 1050/25) using Cu-Kα radiation. Standard continuous scans (1 °/min.) as well

as step scans (0.1 °/min.) were recorded in the range of 10 - 140 ° 2θ for all

samples.

The lattice parameters and unit cell volume of MgSiN2 were calculated with

the computer program Refcel [26] using at least ten reflections of the orthorhombic

cell (space group Pna21 [27]) including a zero point correction. The experimental

accuracy of the lattice parameters and unit cell volume was estimated to equal 3

times the standard deviation of the calculated lattice parameters and unit cell

volume.

The density of the samples was determined by the Archimedes method in

water and by using the lattice parameters as determined by XRD. The first method

results in the overall density (ρexp [kg m-3]) whereas the second procedure gives the

crystallographic density (ρcryst [kg m-3]) which does not take into account the

porosity and secondary phases present in the sample. The accuracy of ρexp was

estimated to equal the average standard deviation of 5 measurements and the

accuracy of ρcryst was obtained from the accuracy of the unit cell volume obtained

from the Refcel calculation.

The oxygen, nitrogen and carbon content were measured using a O/N gas

analyser (Leco TC 436). A small ceramic sample was powdered and mixed with

carbon, after which the oxygen present in the sample is carbothermally converted

at high temperatures in an inert atmosphere into CO, which after further oxidation

to CO2 is measured with IR-absorption spectroscopy. By further decomposition of

the sample at higher temperature the released N2 was measured with catharometry.

The carbon content of sample RB35 (see Table 3-1) was determined by heating the

sample at 1500 °C in pure O2. The carbon present in the sample is converted into

CO2, which is measured by IR spectroscopy. For this sample also the magnesium,

silicon and boron content were determined with Inductive Coupled Plasma Optical


65

Emission Spectroscopy (ICP/OES) by decomposing the sample in a Na2CO3 melt

after which the sample was dissolved in water.

Scanning electron microscopy (SEM, JEOL 840A) was used for

microstructural analysis of the ceramic samples. The SEM samples were prepared

by grinding a sample with abrasive diamond wheels (200, 63, 30 and 10 µm) and

subsequently polishing it on nylon cloth with diamond paste (1 and 0.25 µm). The

polished samples were thermally etched at 1300 °C for 18 minutes in vacuum using

a heating rate of 300 °C h-1 and a cooling rate of 600 °C h-1. Subsequently, the

samples were attached to a sample holder using conductive paste and finally

sputtered with gold to obtain an electrically conducting surface layer.

Transmission Electron Microscopy (TEM, JEOL 2000 FX) equipped with

Noran Energy Dispersive Spectroscopy (EDS) suitable for light element analyses

down to boron was used to study the microstructure in more detail, and moreover

to determine qualitatively the chemical composition. The TEM samples were

prepared by grinding a ceramic sample (8 × 6 × 2 mm) to a maximum thickness of

1 mm. Subsequently, a disk ∅ 3 mm was cut from the sample with an ultrasonic

disk cutter. Then, both sides of the disk were ground and subsequently polished on

a nylon cloth with diamond paste until ~ 100 µm thickness. With a dimple grinder

the thickness of the sample was further reduced to 5 - 10 µm. Finally, the thickness

of the sample was reduced using ion milling until a small hole appeared in the

centre of the sample. Before investigating the sample, a thin carbon layer was

sputtered on the samples to assure sufficient electrical conductivity.

Atomic Force Microscopy (AFM, Topometrix, Santa Clara, CA, USA) was

used to study simultaneously the topography and qualitatively the thermal

conductivity by thermal contrast imaging. In general this method provides

information about where the thermal barriers in a material are located (grains itself

or grain boundary). The sample RB31 used for measuring the thermal diffusivity

was investigated.

Chapter 3.

66

2.3. Properties

To check the oxidation behaviour of the densified MgSiN2 ceramics, DTA/TGA

(differential thermal analysis/ thermo gravimetric analysis) measurements were

performed (Netzsch STA 409). Samples with a high and low oxygen concentration

were investigated.

The Knoop and Vickers hardness (HK [GPa] and HV [GPa]) were measured

on several polished, fully dense ceramic samples. For each sample 5 - 10

measurements were performed. Small loads (< 500 g) did not result in a clear

indent whereas larger loads (≥ 1000 g) caused cracking and chipping of material.

The used load of 500 g is a compromise: the indentation is small but well shaped.

The average standard deviation was about 2 GPa.

The elastic constants (Young's modulus E [GPa] and Poisson's ratio ν [-]) at

293 K for sample RB43 were measured on a small ceramic disk (∅ 15.85 mm ×

1.00 mm) using the impulse excitation method [28] (GrindoSonic, Lemmens

Elektronica BV, Belgium). The fundamental natural flexural and torsional

frequency of the sample was measured. From this, the sample dimensions and

mass, the Young's modulus and Poisson's ratio were evaluated using the computer

program E-mod (Lemmens Elektronica BV, Belgium) based on the work of

Glandus [29]. The experimental accuracy for the Young's modulus and Poisson's

ratio was estimated to equal 5 GPa and 0.01, respectively.

The linear thermal expansion coefficient α [K-1] was measured with a dual

rod dilatometer (Linseis L 75) in nitrogen from 300 to 1573 K, and in air from

300 to 1173 K on a MgSiN2 ceramic bar (2 mm × 2 mm × 10.00 mm). Al2O3

(sapphire) was used as a reference material. Three heating and cooling cycles were

performed for each sample, one for setting of and two for measuring of the sample.

The used heating rate for setting was 10 ºC/min. and 2 ºC/min. for the actual

measuring of the sample. The experimental accuracy was estimated to be

± 0.4 10-6 K-1.

The thermal diffusivity a [m2 s-1] was measured on small carbon coated

ceramic samples (∅ 11 mm × 1 mm) with a uniform thickness and a low roughness


67

using photo and/or laser flash equipment (Compotherm Messtechnik GmbH). The

carbon coating was used to increase the absorptivity of the front surface, and the

emissivity of the back surface. Some samples were coated with a thin layer of gold

before the sample was coated with carbon. The thin gold layer prevents direct

transmission of the laser beam and aids the energy transfer to the sample. The

convective heat losses were minimised by measuring the samples in vacuum. The

experimental accuracy of the measurement was estimated to be within 5%.

The thermal conductivity κ [W m-1 K-1] of the samples was calculated from

the density ρexp [kg m-3], specific heat cV [J kg-1 K-1] and thermal diffusivity

(a [m2 s-1]) using:

κ = ρ cV a (1)

For each sample the density and thermal diffusivity data were taken from this work

and the specific heat of 767.38 J kg-1 K-1 at 300 K as given in a previous study

concerning the thermodynamic properties of MgSiN2 [30].



3.1.1. Phase formation and lattice parameters of MgSiN2

The XRD results indicate that nearly single phase MgSiN2 ceramics were obtained

(vide infra Table 3-3). No preferential orientation could be detected with

cylindrical camera measurements indicating that isotropic materials were obtained.

The XRD data were compared with data previously obtained for MgSiN2 powders

[17] and literature data [27, 31, 32]. In addition to the earlier presented results [17]

the reflections between 100 and 140 ° 2θ (d-value range 1.0063 - 0.8204 Å) were

established (Table 3-2). The tabulated d-values and intensities are evaluated from

the average observed lattice parameters (viz. a = 5.2697 ± 0.0014 Å,

b = 6.4734 ± 0.0011 Å, c = 4.9843 ± 0.0010 Å) and intensities of samples RB02,

RB07, RB09-RB14, RB25, RB30-RB36, RB40, RB41, RB43 and RB45,

Chapter 3.

68

respectively. If a reflection cannot be ascribed to a single set of hkl values both

possible sets are mentioned. An underlined set of hkl values indicates that this hkl

set is thought to be the correctly indexed one based on experimentally observed

d-value and/or theoretical intensity calculations.

Table 3-2: The hkl reflections and corresponding observed d-values and relative intensities (I/I0) for

MgSiN2 ceramics.

hkl d-value[Å]

I/I0

[%]hkl d-value

[Å]I/I0

[%]hkl d-value

[Å]I/I0

[%]110011111120200

4.093.9493.1602.758

2.6348

9 10 1 88 45

241400/033

322401150

1.32911.3174/1.3164

1.31241.27371.2573

11 1 22 5 1

205531/522

244334/360414/171

0.93230.9304/0.9298

0.92450.9194/0.91930.8965/0.8960

2 2 1 3<1

002210121201211

2.49212.44042.41322.32942.1918

80 3100 23 1

051004420151242

1.25311.24611.22021.21911.2066

1 2<1<1 5

171/225135/163532/154

353600

0.8960/0.89590.8919/0.89180.8852/0.8850

0.88290.8783

<1 4<1<1 1

112220130031221

2.12772.04341.99691.98021.8907

1 <1 <1 3 <1

332114

313/421233402

1.19531.1919

1.1866/1.18521.17761.1647

<1 1<1 1 1

610/443362523

235/263172

0.8703/0.87030.86250.8581

0.8559/0.85580.8555

4 4 1<1<1

122202212310040

1.84911.81051.74361.69521.6183

28 12 2 2 25

124251204161403

1.13551.13161.12641.03401.0323

2 1 1 6 4

254434/460

006602

461/533

0.84980.8348/0.8347

0.83070.8283

0.8232/0.8227

<1<1 1 1<1

013231132320321

1.60921.58301.55831.54391.4747

1 1 1 36 1

440/053153/520

441234/260

044

1.0217/1.02121.0026/1.0021

1.00090.9986/0.9984

0.9873

1 2 5<1 2

612 0.8216 <1

123203240213042

1.42311.40531.37901.37341.3573

34 11 7 1 15

521162324442125

0.98250.97310.96960.94530.9375

2 1 3 1 5


69

The lattice parameters and unit cell volume are the same within the

experimental accuracy for all samples, except for RB39 that was sintered with CaO

as an additive (see Table 3-3). For this sample somewhat smaller lattice parameters

were observed. This might be caused by the presence of dissolved Ca and O in the

MgSiN2 lattice. Since Ca has a significantly larger ionic radius than Mg (~ 1.0

versus 0.57 Å [33], respectively) this observation cannot be explained by only

replacing Mg2+ by Ca2+ in the MgSiN2 lattice. However, if Ca and O both dissolve

in the MgSiN2 lattice, the increase of the lattice parameters, due to the

incorporation of Ca on a Mg site, might be overcompensated by the substitution of

N3- by the smaller O2- ion (1.46 versus 1.38 Å [33], respectively) in combination

with the formation of cation vacancies, resulting as an overall effect in smaller

lattice parameters for MgSiN2. In the CaO doped sample no Ca containing

secondary phase could be detected supporting the assumption that CaO has

dissolved into the MgSiN2 lattice as indicated by the lattice parameter results.

In Table 3-3 also the observed secondary phases as detected with XRD are

presented. Between brackets the relative intensity is presented of the strongest

reflection of the detected secondary phase. In some of the samples without

additives MgO (Periclase JCPDS 4-829), Mg2SiO4 (Forsterite JCPDS 34-189),

α-Si3N4 (JCPDS 41-360) and β -Si3N4 (JCPDS 33-1160) could be detected as a

secondary phase. Only in RB43 a non-identified secondary phase was observed

indicated with a ‘Y’. Mg2SiO4 was only detected in the ceramic samples with a

high oxygen content (> 1.5 wt. % O) whereas MgO could be detected in oxygen

poor samples. The presence of Si3N4 can be explained by the evaporation of

magnesium [14, 16, 25] during hot-pressing, unreacted starting materials or the use

of a non-stoichiometric Mg3N2 deficient starting mixture. It is noted that both

α-Si3N4 and β -Si3N4 were found as a secondary phase in samples hot-pressed at

1550 °C whereas in samples hot-pressed at higher temperatures only the

thermodynamically more stable β -Si3N4 [34, 35] was found. In addition to the

earlier mentioned secondary phases the Al2O3 doped samples (RB38 and RB42)

contained AlN (JCPDS 25-1133) and a non-identified secondary phase indicated

with an 'X' in Table 3-3. The presence of AlN indicates that reaction between

Chapter 3.

70

Table 3-3: The lattice parameters, unit cell volume and secondary phases observed for the several

MgSiN2 ceramic samples (the experimental accuracy of the lattice parameters is indicated

between brackets and was estimated to equal 3 times the standard deviation of the

calculated lattice parameters and unit cell volume; * : 5.2672(18) equals 5.2672 ± 0.0018;

** : a question mark indicates that the presence of that phase is questionable).

Code a

[Å]

b

[Å]

c

[Å]

V

[Å3]

detected secondary phase(s)

RB02RB07RB09RB10

5.2672(18)*

5.2693(27)5.2692(15)5.2695(21)

6.4726(24)6.4732(33)6.4727(21)6.4721(27)

4.9829(18)4.9836(33)4.9829(15)4.9842(24)

169.88(12)169.98(24)169.95(12)169.98(18)

Mg2SiO4 (2), MgO (1)

Mg2SiO4 (1 ?**), MgO (2), α-Si3N4 (1)

Mg2SiO4 (2), α-Si3N4(2)

α -Si3N4(2), β -Si3N4 (1)RB11RB12RB13RB14

5.2677(12)5.2699(15)5.2691(15)5.2701(12)

6.4745(18)6.4711(21)6.4737(21)6.4743(15)

4.9841(15)4.9833(15)4.9848(15)4.9859(12)

169.99(9)169.94(12)170.03(12)170.12(9)

MgO (2)

α -Si3N4 (2), β -Si3N4 (2)—

α -Si3N4 (1), β -Si3N4 (2)RB25RB30RB31RB32

5.2676(21)5.2689(9)5.2701(12)5.2725(9)

6.4732(27)6.4738(9)6.4733(15)6.4740(12)

4.9826(18)4.9838(9)4.9844(12)4.9858(9)

169.90(18)169.99(6)170.04(9)170.19(6)

—

β -Si3N4 (3)MgO (1)MgO (3)

RB33RB34RB35RB36

5.2684(12)5.2699(9)5.2713(12)5.2692(15)

6.4734(15)6.4745(9)6.4746(21)6.4750(27)

4.9833(12)4.9840(9)4.9855(9)4.9836(15)

169.95(6)170.05(6)170.15(12)170.03(15)

β -Si3N4 (1 ?)—MgO (< 1 ?)—

RB37RB38RB39RB40

5.2705(27)5.2725(21)5.2684(21)5.2704(15)

6.4771(48)6.4715(21)6.4677(39)6.4744(27)

4.9852(27)4.9851(12)4.9803(18)4.9845(15)

170.18(24)170.09(15)169.70(21)170.08(15)

Y8Si4N4O14 (11)

β -Si3N4 (8), AlN (4), X(5)

β -Si3N4 (3)

β -Si3N4 (1), MgO (6)RB41RB42RB43RB45

5.2701(9)5.2745(30)5.2710(9)5.2717(21)

6.4737(18)6.4702(48)6.4724(12)6.4711(24)

4.9860(9)4.9849(30)4.9845(9)4.9843(18)

170.11(9)170.12(27)170.05(6)170.03(15)

MgO (0 ?)

MgO (2), β -Si3N4 (1), AlN (4), X(6)Y (5)

α -Si3N4 (1)

Al2O3 and N2 atmosphere and/or Mg3N2 has occurred. The presence of AlN is in

accordance with the fact that in RB38 also a substantial amount of (not reacted)

β -Si3N4 was detected and not as more likely expected a β -sialon. The Y2O3 doped

sample contained Y4Si2N2O7 (Y8Si4N4O14, JCPDS 32-1451) and not Y2Si3O3N4

(JCPDS 45-249) as previously reported [25] for pressureless sintered MgSiN2 with


71

Y2O3 addition. The amount of secondary phase present in the samples sintered with

an excess of Mg3N2 was not higher than for the undoped samples. This indicates

that during hot-pressing the excess Mg3N2 and/or the formed MgO evaporates, as

expected from a previous study [17] and the fact that the lattice parameters are

independent of the weighed-in Mg/Si ratio.

3.1.2. Density

The samples hot-pressed at low pressures (< 75 MPa) are not dense (relative

density ρexp/ρcryst < 99.5 % (see Table 3-4)), except RB39 which was sintered using

CaO as a sintering aid (Table 3-1). The high relative density of RB39 can be

Table 3-4: The overall density ρexp, crystallographic density ρcryst and relative density ρexp/ρcryst of the

MgSiN2 ceramic samples (between brackets the experimental accuracy is indicated).

Code ρexp

[g cm-3]

ρcryst

[g cm-3]

ρexp/ρcryst

[%]

Code ρexp

[g cm-3]

ρcryst

[g cm-3]

ρexp/ρcryst

[%]

RB02RB07RB09RB10

3.154(3)3.141(2)3.143(3)3.148(3)

3.144(2)3.143(4)3.143(2)3.143(3)

100.3(2) 99.9(2)100.0(2)100.2(2)

RB34RB35RB36RB37

3.144(1)3.144(1)3.142(1)3.168(1)

3.141(1)3.139(2)3.142(3)3.139(4)

100.1(1)100.2(1)100.0(2)100.9(2)

RB11RB12RB13RB14

3.147(1)3.145(2)3.145(1)3.149(1)

3.143(2)3.143(2)3.142(2)3.140(2)

100.1(1)100.1(1)100.3(1)100.3(1)

RB38RB39RB40RB41

3.060(3)3.131(2)3.074(3)3.074(1)

3.141(3)3.148(4)3.141(3)3.140(2)

97.4(2) 99.5(2) 97.9(2) 97.9(1)

RB25RB30RB31RB32RB33

—3.144(1)3.146(1)3.145(2)3.143(1)

3.142(3)3.143(1)3.141(2)3.139(1)3.143(2)

—100.0(1)100.2(1)100.2(1)100.0(1)

RB42RB43RB45

3.022(5)3.139(2)3.127(2)

3.140(5)3.141(1)3.142(3)

96.2(3) 99.9(1) 99.5(2)

explained by the fact that CaO most probably reacts with MgSiN2, as suggested by

the XRD results, which enhances the sintering process at lower hot-pressing

pressures. All samples hot-pressed at 75 MPa have a relative density

Chapter 3.

72

ρexp/ρcryst ≥ 99.5 % (see Table 3-4). As expected from the presence of a Y2O3

containing secondary phase (Y8Si4N4O14), with a higher density than MgSiN2,

RB37 has an overall density higher than the crystallographic density. Dense

samples (≥ 99.5 %) can be obtained at temperatures substantially below 1543 °C at

which liquid phase formation in the MgO-SiO2 system is expected to occur [36]

viz. RB43 (1500 °C) and RB45 (1450 °C). No systematic dependence of the

sintering behaviour on the oxygen content was observed. So, it can be concluded

that the applied pressure is more important than the temperature for obtaining fully

dense samples.

3.1.3. Chemical composition

For the ceramic samples it was quite difficult to obtain reliable oxygen and

nitrogen content data. Investigation showed that the measured oxygen and

especially nitrogen content varied with the method used to powder the ceramic

sample indicating the necessity of very careful sample preparation.

Table 3-5: Measured oxygen and nitrogen content for several MgSiN2 powders and ceramic

samples (* Oxygen and nitrogen content obtained after careful sample preparation

(samples powdered without introduction of oxygen). Note that no reliable nitrogen

content data were obtained; ** Nitrogen content for fully decomposed samples.).

Code wt. % oxygenpowder

wt. % oxygenceramic

Code wt. % oxygenceramic

wt. % nitrogenceramic

RB02RB07RB09RB10RB11RB25

1.62.63.42.72.71.4

3.81.63.12.01.8—

RB12RB13RB14RB30RB31RB32RB33RB34RB35

1.31.01.7 (0.8)*

1.21.11.01.11.02.5 (0.0 - 0.3)*

——— (25 - 31)*

——— 34.1**

— 35.2**

—31 (26 - 28)*


73

The oxygen content generally decreases (except for RB02) for MgSiN2

ceramics as compared to the corresponding starting powder (Table 3-5). This

decrease is probably due the graphite environment in which the samples are

sintered resulting in a carbothermal nitridation reaction of the oxygen containing

compounds present in the MgSiN2 starting powder. In the resulting ceramics still

some oxygen is present due to the fact that the densification process, which reduces

the contact surface of the MgSiN2 compact with the gas phase, is too fast for the

carbothermal nitridation reaction to complete. After densification of the sample, the

possibility of oxygen removal is hampered by solid state diffusion processes. The

oxygen content of the reaction hot-pressed ceramics is lower than for the hot-

pressed ceramics indicating that reaction hot-pressing results in purer samples. It is

noted that the measured oxygen content for the ceramics might be too high because

the method used to powder the ceramic sample can introduce oxygen impurities

into the sample.

The measured nitrogen content was in general much lower (25 - 31 wt. %)

than expected for MgSiN2 (theoretical value 34.8 wt. %) due to incomplete

decomposition of the sample. When careful sample preparation resulted in a full

decomposition of the sample (RB32 and RB33), reliable nitrogen content data were

obtained having values close to the theoretical value (see Table 3-5).

A complete chemical analysis of sample RB35 was made. The measured

magnesium and silicon content for this sample of 30.2 ± 0.9 wt. % Mg and

34.7 ± 1.0 wt. % Si matches very well with the theoretical amounts expected for

MgSiN2 (viz. 30.23 wt. % and 34.93 wt. % respectively). About 0.5 wt. % C was

detected which originates from SiC impurity present in the used Si3N4 starting

material and possibly from the graphite interior of the hot-press. Moreover, small

traces (< 0.01 wt. %) of boron were detected that originates from the boron nitride

coated graphite foils and dies used for hot-pressing of the sample. X-Ray

Fluorescence (XRF) revealed the presence of small traces of Fe and W. Other

impurities could not be detected.

Chapter 3.

74

These above results indicate that by suitable processing very pure MgSiN2

ceramics can be obtained with a purity comparable to that as previously reported

for MgSiN2 powder viz. < 0.1 wt. % O and 34.2 wt. % N [17].

3.1.4. Microstructure

The polished SEM samples showed grain boundaries only after thermal etching.

No residual porosity could be observed in the investigated samples in agreement

with the density measurements. The microstructural investigation with the SEM

and TEM showed that as expected the grain size increased with the hot-pressing

temperature from about 0.25 µm (Fig. 3-1, RB12, 1550 °C) to about 1.5 µm

(Figs. 3-2 and , RB34, 1700 °C). The grains of most SEM samples appeared as if

they were build up out of smaller grains but the TEM analyses showed that this

observation is a result of the used thermal etching procedure. The grain size also

increased with longer hot-pressing time (RB12: 1550 °C, 2 h about 0.25 µm and

RB14: 1550 °C, 5 h about 1.8 µm).

Fig. 3-1: SEM photograph of thermally etched (1300 °C, 18 min.)

surface of sample RB12 hot-pressed for 2 h at 75 MPa and

1550 °C, showing an average grain size of about 0.25 µm.


75

The microstructure was not influenced by the Mg/Si ratio in the starting

mixture (RB30 and RB32) and the oxygen content of the samples. Clean grain

boundaries were observed with TEM irrespective of the weighed-in Mg/Si ratio

and the oxygen content of the samples. In Fig. 3-4 a typical picture of a grain

boundary and triple point is shown. Only occasionally secondary phases could be

detected at a triple point. Although the TEM study did not reveal any grain

boundary phases, the AFM thermal contrast image study of RB31 revealed the

Fig. 3-2: TEM photograph of sample RB34 showing a grain size of

about 1.5 µm.

Chapter 3.

76

presence of a thermal resistance at the grain boundaries (see Fig. 3-5) resulting in

clear image of the microstructure of the sample whereas the conventional

topography image of the sample provided no information at all.

Fig. 3-3: SEM photograph of thermally etched surface of sample

RB34 hot-pressed for 2 h at 75 MPa and 1700 °C, showing

an average grain size of about 1.5 µm.

Fig. 3-4: TEM photograph of typical observed grain boundaries triple

points for MgSiN2 ceramics.


77

3.1.5. TEM/EDS

A qualitative EDS analysis of the phase at the triple points as observed with TEM

indicated only sometimes the presence of Mg, Si and O (suggesting MgSiO3 or

Mg2SiO4) or Mg and O with some trace Si (suggesting MgO). Although EDS

analysis with TEM is a qualitative method we concluded from the relative

intensities of the Mg and Si signal of a MgSiN2 grain (Fig. 3-6) and several Mg-Si-

O containing grains, that MgSiO3 grains (Fig 3-7) and Mg2SiO4 grains (Fig. 3-8)

are present as a secondary phase in the oxygen rich samples and Mg2SiO4 grains

and MgO grains are present as a secondary phase in the oxygen poor samples. The

Fig. 3-5: AFM thermal contrast image for sample RB31 showing a thermal barrier at the MgSiN2

grain boundaries.

Chapter 3.

78

presence of MgO and Mg2SiO4 as a secondary phase was confirmed by the XRD

measurements. This suggests that MgSiO3 is present as a glassy secondary phase

although crystalline MgSiO3 (proenstatite) is reported [16] as an oxidation product

of MgSiN2. Although Si3N4 was observed with XRD the presence of this secondary

phase could not be confirmed with EDS.

0

400

800

1200

1600

0.0 0.5 1.0 1.5 2.0

energy [keV]

Cou

nts

[-]

N

Si

Mg

C

O

Fig. 3-6: TEM/EDS analyses of a MgSiN2 grain (The "C" signal is

caused by the carbon coating to enhance the electrical

conductivity of the sample).

0

400

800

1200

1600

0.0 0.5 1.0 1.5 2.0energy [keV]

Cou

nts

[-]

O

Si

Mg

C

Fig 3-7: TEM/EDS analyses of a Mg-Si-O grain probably MgSiO3 as

suggested by the relative intensity of the Mg and Si peak as

compared to Fig. 3-6.


79

In order to study the secondary phases in more detail a MgSiN2 sample from

a previous study [14] with a high oxygen content (about 5 wt. % O) was

0

400

800

1200

1600

0.0 0.5 1.0 1.5 2.0

energy [keV]

Cou

nts

[-]

C

O

MgSi

N

Fig. 3-8: TEM/EDS analyses of a Mg-Si-O grain probably Mg2SiO4

as suggested by the relative intensity of the Mg and Si peak

as compared to Fig. 3-6 (The "C" signal is caused by the

carbon coating to enhance the electrical conductivity of the

sample).

Mg Si

N O

Fig. 3-9: TEM-EDS mapping of an MgSiN2 samples (area about 1.8 × 1.8

µm) for Mg, Si, N and O showing the location of magnesium (Mg),

silicon (Si), nitrogen (N) and oxygen (O) in the sample (in the upper

right corner the video image of the sample is given).

Chapter 3.

80

investigated. Also in this case neither grain boundary phases were observed nor did

most triple points contain any secondary phase. Several mappings of a

representative part of the sample (about 30 grains) for Mg, Si, N and O showed that

most grains consisted of Mg, Si and N (MgSiN2), whereas a few grains consisted of

Mg and O (MgO) mostly in combination with Si (MgSiO3 or Mg2SiO4). In Fig. 3-9

a typical result is presented. So the secondary phases like MgSiO3 or Mg2SiO4, and

MgO are present as separate grains in the MgSiN2 matrix. This indicates that the

wetting behavior of the MgSiN2 grains by secondary phases in the Mg-Si-N-O

system is very poor.

3.2. Properties

3.2.1. Oxidation resistance

Irrespective of the oxygen content, the MgSiN2 ceramics prepared by hot uni-axial

pressing are oxidation resistant in air up to about 1200 ºC (see Fig. 3-10,

Table 3-6), which is about 250 ºC higher than the value observed for MgSiN2

-10

-5

0

5

10

15

20

400 600 800 1000 1200 1400 1600T [ºC]

∆T [ µ

V]

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

∆m/m

0 × 1

00 [%

]

DTATGA

Fig. 3-10: TGA/DTA plot of the oxidation behaviour of MgSiN2 ceramics in

air showing the temperature difference (∆T ) and relative mass



81

powders [17]. The difference between the oxidation resistance of MgSiN2 powder

and ceramic can be ascribed to the much lower contact surface area of the ceramics

and the formation of protective layer during oxidation. For comparison the

oxidation resistance of AlN, γ -aluminium oxynitride (Alon) and Si3N4 ceramics are

given in Table 3-6.

Table 3-6: Some properties of MgSiN2 ceramics as compared to Al2O3, Alon, AlN and β -Si3N4

ceramics.

MgSiN2 MgSiN2 [2] Al2O3 Alon AlN β -Si3N4

HV [GPa] 14 - 20 14 - 16 19.5 [37] 20 [38] 12 [39] 16 - 22 [40]HK [GPa] 14 - 19 — 15.8 [41] — 12 [42] 10.9 [41]E [GPa] 284 235 393 [41] 322 [38] 315 [43] 304 [44]ν [-] 0.250 0.232 0.240 [41] 0.253 [38] 0.245 [43] 0.267 [44]Toxidation [°C] 1200 > 920 — 1200 [45] 900 [42, 46] 1400 [47]α(293 K) [K-1] 3.8 10-6 — 5.4 10-6 [48] 5.8 10-6 [49] 2.7 10-6 [50] 1.4 10-6 [51]

α(873 K) [K-1] 6.8 10-6 — 8.7 10-6 [48] 7.8 10-6 [49] 5.9 10-6 [50] 3.6 10-6 [51]

α [K-1] 5.8 10-6 5.8 10-6 7.8 10-6 [48] 7.4 10-6 [49] 4.8 10-6 [50] 2.5 10-6 [51]

3.2.2. Hardness

For the dense samples the measured Knoop and Vickers hardness varied from

13.9 to 19.9 GPa, in agreement with an earlier reported value for the Vickers

hardness of about 15 GPa [2] (see Table 3-6). In general, the reaction hot-pressed

samples starting from Mg3N2/Si3N4 mixtures have a lower hardness (∼ 15 GPa)

than those prepared starting from MgSiN2 powder (∼ 19 GPa). This difference in

hardness between the several MgSiN2 samples cannot be explained. The obtained

values are fairly high and the best values are comparable with the hardness

obtained for Al2O3, Alon and β -Si3N4 (see Table 3-6). The results indicate that

MgSiN2 ceramics with a high hardness can be quite easily obtained.

Chapter 3.

82

3.2.3. Young's modulus

The flexural and torsional frequency of RB43 equalled 62.4 kHz and 39.0 kHz,

respectively resulting in E = 284 GPa and ν = 0.250. The observed Young's

modulus is somewhat lower as compared to Alon, AlN and β -Si3N4 but

considerably higher than a previously observed value of 235 GPa [2] (see

Table 3-6). This relative large difference may be partially ascribed to the low

density (ρexp/ρcrys = 98.9 %) and purity (3.7 wt. % oxygen) of the sample described

in [2].

3.2.4. Thermal expansion

Within the experimental accuracy, the linear thermal expansion coefficient for the

sample measured in air and nitrogen are the same. For heating and cooling of the

sample no hysteresis was observed. The linear thermal expansion coefficient

increases with temperature (see Fig. 3-11) and becomes almost constant at about

1000 K. Subsequently the thermal expansion starts to increase again with

3.5E-06

5.0E-06

6.5E-06

8.0E-06

9.5E-06

300 500 700 900 1100 1300 1500 1700T [K]

α [K

-1]

Fig. 3-11: The isotropic linear thermal expansion coefficient (α) of

MgSiN2 ceramics as a function of the absolute

temperature (T ) (300 - 1573 K) as determined with

dilatometry in a nitrogen atmosphere.


83

increasing temperature. This increase at high temperatures is ascribed to the

thermal generation of defects in the MgSiN2 crystal structure causing macroscopic

length changes, which are measured with dilatometry. The value of 3.8 10-6 K-1 at

293 K, 6.7 10-6 K-1 at 827 K and the average value of 5.8 10-6 K-1 between 293 and

873 K (see Table 3-6) agrees reasonably well with the previously reported values

of respectively 4.4 10-6 K-1 [52], 6.5 10-6 K-1 [8] and 5.8 10-6 K-1 [2]. For

comparison the thermal expansion coefficients of Al2O3, AlN, Alon and Si3N4 are

presented (Table 3-6) indicating that Al2O3 and Alon have a higher, and AlN and

Si3N4 a lower thermal expansion coefficient than MgSiN2.

3.2.5. Thermal diffusivity/conductivity

In general the observed thermal conductivity of the fully dense samples

(ρexp/ρcryst ≥ 99.5 %) equals 17 - 21 W m-1 K-1 whereas the thermal conductivity of

the other samples is substantially smaller and varies between 12 - 16 W m-1 K-1

(Table 3-7).

Table 3-7: Measured thermal diffusivity (a) and resulting thermal conductivity (κ ) for several

MgSiN2 ceramic samples.

Code a κ Code a κ Code a κ[cm2 s-1] [W m-1 K-1] [cm2 s-1] [W m-1 K-1] [cm2

s-1] [W m-1 K-1]

RB02 0.076 18.4 RB30 0.076 18.3 RB38 0.058 13.6RB07 0.071 17.1 RB31 0.073 17.6 RB39 0.057 13.7RB09 0.082 19.8 RB32 0.076 17.8 RB40 0.069 16.3RB10 0.067 16.2 RB33 0.074 17.8 RB41 0.071 16.7RB11 0.061 14.7 RB34 0.082 19.2 RB42 0.052 12.1RB12 0.066 15.9 RB35 0.079 19.1 RB43 0.062 14.9RB13 0.080 19.3 RB36 0.086 20.7 RB45 0.054 13.0RB14 0.077 18.6 RB37 0.086 20.9

Despite the high relative density, also the CaO doped sample has a low thermal

conductivity, which is consistent with the assumption of incorporation of CaO into

Chapter 3.

84

the MgSiN2 lattice resulting in defects. The Y2O3 doped sample RB37 has a

thermal conductivity (20 W m-1 K-1) comparable with that of the best samples

indicating that the Y2O3 addition does not hamper the thermal conductivity.

The less dense Mg3N2 doped samples hot-pressed at a lower pressure (RB40

and RB41) have a low thermal conductivity (~ 16.5 W m-1 K-1) due to the lower

relative density. The 2Al2O3.Si3N4 and Al2O3 doped samples RB38 and RB42 have

the lowest thermal conductivity (12 - 14 W m-1 K-1), which is caused by the low

relative density of the samples and the presence of secondary phases.

For the fully dense samples, the influence of the overall oxygen content

(RB02, RB07, RB09-RB14 and RB30-RB36) is very limited (Fig. 3-12) indicating

that secondary phases are not hampering the heat transport between the MgSiN2

grains in agreement with the TEM investigation showing clean grain boundaries.

Also the influence of the processing temperature (RB12, RB13, RB33 and RB34,

and RB45, RB43, RB35 and RB36) on the thermal conductivity is small for

T ≥ 1600 ºC (Fig. 3-13). This indicates that the MgSiN2 grain size for the samples

processed at T ≥ 1600 ºC does not limit the heat transport. These results suggest

that the thermal conductivity of MgSiN2 is determined by phonon scattering

0.00

0.02

0.04

0.06

0.08

0.10

0 1 2 3 4overall oxygen content [wt. %]

a [c

m2 s

-1]

Fig. 3-12: The thermal diffusivity (a) versus the overall oxygen

content for fully dense MgSiN2 ceramics (line is drawn as

a guide to the eye).


85

processes within the MgSiN2 grains itself. From the fact that manipulation of the

defect chemistry (by changing the O concentration (Fig. 3-12) and the 'weighed-in'

Mg/Si ratio (RB30, RB13, RB31 and RB32) (Fig. 3-14)) only has a limited

influence it can be deduced that phonon-defect scattering is also not limiting the

thermal conductivity. Therefore the thermal conductivity is probably determined by

0.00

0.02

0.04

0.06

0.08

0.10

1400 1500 1600 1700 1800T [°C]

a [c

m2 s

-1]

Mg3N2 (alfa) & Si3N4 (cerac)Mg3N2 (alfa) & Si3N4 (SKW)

Fig. 3-13: The thermal diffusivity (a) versus the applied hot-pressing

temperature (T ) for fully dense MgSiN2 ceramics (line is

drawn as a guide to the eye).

0.00

0.02

0.04

0.06

0.08

0.10

0.90 0.95 1.00 1.05 1.10 1.15weighed-in Mg/Si ratio [-]

a [c

m2 s

-1]

Fig. 3-14: The thermal diffusivity (a) versus the weighted-in Mg/Si

ratio for fully dense MgSiN2 ceramics processed at

1600 °C (line is drawn as a guide to the eye).

Chapter 3.

86

intrinsic phonon-phonon scattering indicating that the thermal conductivity of

MgSiN2 cannot be increased substantially.

4. Theoretical considerations

The experimental results indicate that intrinsic scattering is determining the thermal

conductivity of MgSiN2 limiting its value to about 25 W m-1 K-1 at 300 K. This

conclusion can be supported by theoretical calculations considering the influence

of secondary phases, grain size, and defects in the grains on the thermal

conductivity.

4.1. Secondary phases

For simplicity the effect of a secondary phase at the grain boundary on the

experimental thermal conductivity κexp is approximated by a two-phase serial

system, which results in the most detrimental effect on the thermal conductivity,

using [53]:

κκκ 2

2

1

1

exp=1 VV + (2)

where V1 (= 1 - V2) and V2 [-] are the volume fraction and κ1 and κ2 [W m-1 K-1] the

intrinsic thermal conductivity of the MgSiN2 phase and the grain boundary phase,

respectively. We can rewrite the above formula as:

expexp22

2221 = κ

κκκκκ

V-V- (3)

V2 and κ2 was estimated to equal about 2 vol. % (≡ 1 wt. % O) and 2 W m-1 K-1,

respectively. For κexp = 25 W m-1 K-1 [10] we obtain κ1 = 33 W m-1 K-1. From this

it can be concluded that, even when secondary phases are present at the grain

boundaries, the influence on the thermal conductivity is limited due to the small

amount of secondary phase present.


87

4.2. Grain size

For grain sizes in the range of the phonon mean free path, phonon-grain boundary

scattering hampers effectively the heat transport. For the phonon mean free path, l

[m], applies [12, 54]:

ρκ

Vvcl exp3

= (4)

where κexp [W m-1 K-1] is the experimental thermal conductivity, v [m s-1] the

phonon group velocity, cV [J kg-1 K-1] the specific heat at constant volume and ρ

[kg m-3] the density. The specific heat at constant volume can be approximated by

the specific heat at constant pressure cp [55] and the group velocity was estimated

to equal the sound velocity vs. Taking the sound velocity and the specific heat from

our previous work (vs = 6.65 103 m s-1 [56] and cp = 767 J kg-1 K-1 [30]), the

resulting phonon mean free path at 300 K equalled about 4 - 5 nm. This is

substantially below the observed grain size of the MgSiN2 ceramic samples

suggesting that phonon-grain boundary scattering does not hamper the heat

transport.

4.3. Defects

Assuming that the defect chemistry of MgSiN2 analogous to that of AlN [19, 57]

incorporation of oxygen into the MgSiN2 lattice results in the formation of

vacancies on the metal sites according to:

2 MgSiO3 º 2 MgMg + 2 SiSi + 6•

NO + ''VMg + ''''SiV

These vacancies are very effective in scattering the phonons and for the thermal

conductivity then applies [19, 58, 59]:

2

2

2 2/3

ppexp

-121*)π(6+1 = 1

∑

i

iii

i mmm

ck

hθ

δκκ

(5)

where κexp is the measured thermal conductivity, κpp the intrinsic thermal

conductivity due to phonon-phonon scattering, h Planck’s constant (6.626 10-34

Chapter 3.

88

J s), δ 3 [m3] the average volume occupied by one atom, k the Boltzmann’s constant

(1.381 10-23 J K-1), θ [K] the Debye temperature, ci [-] the site fraction of the

isotope or foreign atom with mass mi at the ith lattice site, mi [g] the mass of

impurity on the ith lattice site and im [g] the average mass of the atoms at the ith

lattice site. The average mass at the nitrogen ( NM ), magnesium ( MgM ) and silicon

( SiM ) sites in the MgSiN2 lattice are taken constant and equal the atomic masses

so, the scattering term, ∆WI, for MgSiN2 is given by:

+

π∆

2

Mg

Mg

21

61

2

Si

Si

21

61

2

N

NO

NN

N2

2 2/3

I

-+

-

*N+O

O121* )(6 =

MgSi

MMM

MMM

MM - M

kh

W

VV

θδ

(6)

Introducing the input parameters for ∆WI and the Debye temperature of MgSiN2

(θ ≈ 830 K [8, 30]) we can write for the thermal conductivity of MgSiN2 with

oxygen dissolved into the lattice:

]O%.wt[ 0.0202 1 = 1ppexp

+κκ

(7)

Taking for κexp 25 W m-1 K-1 and for [wt. % O] the maximum solubility of oxygen

in the MgSiN2 lattice of 0.6 wt. % [17], an intrinsic thermal conductivity of

36 W m-1 K-1 results which indicates that a substantial improvement of the thermal

conductivity by minimisation of phonon-defect scattering can be excluded.

4.4. Maximum influence of secondary phases, grain size and defects

From the above theoretical considerations it can be concluded that the maximum

achievable thermal conductivity of MgSiN2 at 300 K will not exceed 35 W m-1 K-1.

These considerations confirm the experimental conclusion that the thermal

conductivity can not be significantly improved. However, it should be noted that

the two-phase serial and defect scattering formula are sensitive for the values used

for V2 and [wt. % O], respectively.


89

5. Conclusions

Dense MgSiN2 ceramic samples were prepared using the hot uni-axial pressing

technique. The influence of additives on the sintering behaviour was very limited.

The applied hot-pressing pressure appeared to be more important than the used

sintering temperature indicating that liquid phase sintering was not the most

important densification process. Oxygen poor samples (< 1.0 wt.%) could be

prepared by using pure starting materials and/or using Mg3N2 as an additive. The

grain size of the MgSiN2 grains increased with increasing hot-pressing temperature

and time. The ceramic samples have a good oxidation resistance, a fairly high

hardness and a low thermal expansion coefficient as compared to other ceramics.

The thermal conductivity of the samples is not determined by the grain size

or the presence of (inter-granular) secondary phases. From this it was concluded

that phonon scattering processes within the MgSiN2 grains determine the thermal

conductivity. It was concluded that the limiting factor is intrinsic phonon-phonon

scattering resulting in a maximum achievable thermal conductivity of MgSiN2

ceramics at 300 K of 20 - 25 W m-1 K-1.

From some simple theoretical considerations it was shown that the

maximum achievable value does not exceed about 35 W m-1 K-1. This value is

lower than the first reported theoretical estimate of 75 W m-1 K-1 with an

approximate accuracy of about 30 %.

References


Electroceramics IV 2, Aachen (Germany), September 1994, edited by

R. Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann (Augustinus




Chapter 3.

90

3. W.A. Groen, M.J. Kraan, and G. de With, New Ternary Nitride Ceramics:

CaSiN2, J. Mater. Sci. 29 (1994) 3166.

4. W.A. Groen, P.F. van Hal, M.J. Kraan, N. Sweegers and G. de With,

Preparation and Properties of ALCON (Al28C6O21N6) ceramics, J. Mater. Sci.

30 (1995) 4775.


Chem. Solids 34 (1973) 321.

6. W.A. Groen, M.J. Kraan, G. de With and M.P.A. Viegers, New Covalent

Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non-

Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron, G.S.

Fischman, M.A. Fury and A.F. Hepp (Materials Research Society, Pittsburgh,

1994) 239.

7. G.A. Slack, The Thermal Conductivity of Nonmetallic Crystals, Solid State

Physics 34, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic

Press, New York, 1979) 1.

8. G. de With and W.A. Groen, Thermal Conductivity Estimates for New

(Oxy)-Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science -

Optimisation of Properties and Performances by Improved Design and


V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405.

9. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modelling of the Thermal

Diffusivity / Conductivity of MgSiN2 Ceramics, ITCC 24 and ITES 12, 24th

International Thermal Conductivity Conference and 12th International Thermal

Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 1997, edited

by P.S. Gaal and D.E. Apostolescu (Technomics Publishing Co., Inc.,

Lancaster, Pennsylvania, 1999) 3.



Processing of Advanced Ceramics - Part II, Faenza, Italy, October 1995,


91

edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza,

1995) 289.

11. P.G. Klemens, The Thermal Conductivity of Dielectric Solids at Low

Temperatures, Proc. Roy. Soc. London A208 (1951) 108.

12. F.R. Charvat and W.D. Kingery, Thermal Conductivity: XIII, Effect of

Microstructure on Conductivity of Single-Phase Ceramics, J. Am. Ceram. Soc.

40 (1957) 306.

13. K. Watari, K. Ishizaki and F. Tsuchiya, Phonon Scattering and Thermal

Conduction Mechanisms of Sintered Aluminum Nitride Ceramics, J. Mater.

Sci. 28 (1993) 3709.

14. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan, Hot-

pressing of MgSiN2 ceramics, J. Mater. Sci. Lett. 13 (1994) 1314.

15. H.T. Hintzen, R.J. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder

Preparation and Desification of MgSiN2, Ceram. Trans. 51, Int. Conf. Cer.

Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by

H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society,

1995) 585.

16. H. Uchida, K. Itatani, M. Aizawa, F.S. Howell and A. Kishioka, Synthesis of

Magnesium Silicon Nitride by the Nitridation of Powders in the Magnesium-

Silicon System, J. Ceram. Soc. Japan 105 (1997) 934.

17. Chapter 2; R.J. Bruls, H.T. Hintzen and R. Metselaar, Preparation and

Characterisation of MgSiN2 Powders, J. Mater. Sci. 34 (1999) 4519.

18. M.P. Boron, G.A. Slack, and J.W. Szymazek, Thermal Conductivity of




20. Y. Suwa, S. Naka and T. Noda, Preparation and Properties of Yttrium

Magnesium Silicate with Apatite Structure, Mat. Res. Bull. 3 (1968) 139.

21. P.C. Martinengo, A. Giachello, P. Popper, A. Buri, F. Branda, Devitrification

Phenomena of a Pressureless Sintered Silicon Nitride, Proc. of International

Chapter 3.

92

Symposium on Factors in Densification and Sintering of Oxide and Non-oxide

Ceramics, Hakone, Japan, October 3 - 5, 1978, edited by S. Somiya and S.

Saito (Gakujutsu Bunken Fukyu-Kai, Tokyo, 1979) 516.

22. S. Kuang, M.J. Hoffmann, H.L. Lukas and G. Petzow, Experimental Study

and Thermodynamic Calculations of the MgO-Y2O3-SiO2 System, Key

Engineering Materials 89 - 91, Proceedings of the International Conference

on Silicon Nitride-Based Ceramics, Silicon Nitride 93, Stuttgart, Germany,

October 1993, edited by M.J. Hoffmann, P.F. Becker and G. Petzow (Trans

Tech Publications, Switzerland, 1994) 399.

23. E.M. Levin, C.R. Robbins and F.H. McMurdie, Phase Diagrams for

Ceramists, (comp. at the National Bureau of Standards and Technology -7th.

comp.- Columbus, American Ceramic Society, 1964) p. 210.

24. Z.K. Huang, S.D. Nunn, I. Peterson and T.Y. Tien, Formation of N-phase and

Phase Relationships in MgO-Si2N2O-Al2O3 System, J. Am. Ceram. Soc. 77

(1994) 3251.

25. I.J. Davies, H. Uchida, M. Aizawa and K. Itatani, Physical and Mechanical

Properties of Sintered Magnesium Silicon Nitride Compacts with Yttrium

Oxide Addition, Inorganic Materials 6 (1999) 40.

26. Refcel, Calculation of cell constants and calculation of all possible lines in a

powder diagram by H.M. Rietveld, October 1972.

27. J. David, Y. Laurent and J. Lang, Structure de MgSiN2 et MgGeN2, Bull. Soc.

fr. Minéral. Cristallogr. 93 (1970) 153.

28. K. Heritage, C. Frisby and A. Wolfenden, Impulse excitation technique for

dynamic flexural measurements at moderate temperatures, Rev. Sci. Instr. 59

(1988) 973.

29. J.C. Glandus, Rupture fragile et résistance aux chocs thermiques de

céramiques a usage mécaniques, Ph.D. Thesis, University of Limoges, France,

1981.


93

30. Chapter 5; R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,

Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102

(1998) 7871.

31. J. David, Étude sur Mg3N2 et quelques-unes de ses combinaisons, Rev. Chim.

Miner. 9 (1972) 717.

32. M. Wintenberger, F. Tcheou, J. David und J. Lang, Verfeinerung der Struktur


35b (1980) 604.

33. R.D. Shannon, Revised Effective Ionic Radii and Systematic Studies of

Interatomic Distances in Halides and Chalcogenides, Acta. Cryst. A32 (1976)

751.

34. D.R. Messier, F.L. Riley and R.J. Brook, The α/β Silicon Nitride Phase

Transformation, J. Mater. Sci. 13 (1978) 1199.

35. O.N. Carlson, The N-Si (Nitrogen-Silicon) System, Bull. Alloy Phase

Diagrams 11 (1990) 569.

36. S.F. Kuang, Z.-K. Huang, W.-Y.Sun, T.-S. Yen, Phase Relationships in the

System MgO-Si3N4-AlN, J. Mater. Sci. Lett. 9 (1990) 69.

37. R. Morrell, Handbook of Properties of Technical & Engineering Ceramics

(Part 2, Data Reviews) Her Majesty's Stationary Office, London, UK, 1987,

pp. 37 - 57.

38. H.X. Willems, P.F. van Hal, G. de With and R. Metselaar, Mechanical

properties of γ -aluminium oxynitride, J. Mater. Sci. 28 (1993) 6185.

39. Y. Kurokawa, K. Utsumi, H. Takamizawa, T. Kamata and S. Noguchi, AlN

Substrates with High Thermal Conductivity, IEEE Trans. Compon., Hybrids,

Manuf. Technol., CHMT-8 (1985) 247 - 252.

40. G. Ziegler, J. Heinrich and G. Wötting, Review Relationships between

Processing, Microstructure and Properties of Dense and Reaction-Bonded

Silicon Nitride, J. Mater. Sci. 22 (1987) 3041.

41. J. Piekarczyk, J. Lis and J. Bialoskórski, Elastic Properties, Hardness and

Indentation Fracture Toughness of β -Sialons, Key Engineering Materials

Chapter 3.

94

89 - 91, Proceedings of the International Conference on Silicon Nitride-Based

Ceramics, Silicon Nitride 93, Stuttgart, Germany, October 1993, edited by

M.J. Hoffmann, P.F. Becher and G. Petzow, (Trans Tech Publications,

Switserland, 1994) 541.


Substrate for High Power Applications in Microcircuits, 34th Electronic

Components Conference, May 1984, New Orleans, Louisiana, USA (Institute

of Electrical and Electronics Engineers, Inc., 1984) 402.

43. P. Boch, J.C. Glandus, J. Jarrige, J.P. Lecompte and J. Mexmain, Sintering,

Oxidation and Mechanical Properties of Hot Pressed Aluminium Nitride,

Ceram. Int. 8 (1982) 34.

44. I. Tomeno, High Temperature Elastic Moduli of Si3N4 Ceramics, Jpn. J. Appl.

Phys. 20 (1981) 1751.

45. P. Lefort, G. Ado and M. Billy, Comportement a l'oxydation de l'oxynitrure

d'aluminium transparant, J. Physique 47 (1986) C1, 521.

46. E.W. Osborne and M.G. Norton, Oxidation of aluminium nitride, J. Mater.

Sci. 33 (1998) 3859.

47. J. Persson, P.-O. Käll and M. Nygren, Parabolic - Non-Parabolic Oxidation

Kinetics of Si3N4, J. Eur. Ceram. Soc. 12 (1993) 177.

48. Y.S. Touloukian, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, Thermophysical

Properties of Matter 13, Thermal Expansion of Nonmetallic Solids,

(IFI/Plenum, New York, USA, 1977).

49. H.X. Willems, Preparation and Properties of Translucent γ-Aluminium

Oxynitride, Ph. D. Thesis, Eindhoven University of Technology, The

Netherlands (1992).

50. K. Wang and R.R. Reeber, Thermal Expansion of GaN and AlN, Mat. Res.

Soc. Symp. Proc. 482, Nitride Semiconductors, Boston, Massachusetts, USA,

December 1 - 5 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S.

Nakamura and S. Strite, (Materials Reseasrch Society, Warrendale,

Pennsylvania, 1998) 863.


95

51. C.M.B. Henderson and D. Taylor, Thermal Expansion of the Nitrides and

Oxynitride of Silicon in Relation to their Structures, Trans. J. Br. Ceram. Soc.

74 (1975) 49.

52. Chapter 4; R.J. Bruls, H.T. Hintzen, R. Metselaar and C.-K. Loong,

Anisotropic thermal expansion of MgSiN2 from 10 to 300 K as measured by

neutron diffraction, J. Phys. Chem. Solids 61 (2000) 1285.

53. D.W. Richerson, Modern Ceramic Engineering; Properties, Processing, and Use

in Design, second edition (Marcel Dekker, Inc., 1992) pp. 135 - 146.

54. P. Debije, in: Vorträge über die kinetische Theorie der Materie und

Elektrizität, edited by B.G. Teubner, Leipzig and Berlin (1914) p. 16.

55. See for example: R.A. Swalin, Thermodynamics of Solids, second edition,

John Wiley & Sons, New York, (1972) pp. 31 - 32 and 81 - 84.

56. Chapter 6; R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The

Temperature Dependence of the Young's Modulus of MgSiN2, AlN and Si3N4,

accepted for publication in J. Eur. Ceram. Soc.

57. A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and Kinetic Effects

of Oxygen Removal on the Thermal Conductivity of Aluminum Nitride, J. Am.

Ceram. Soc. 72 (1989) 2031.

58. V. Ambegoakar, Thermal Resistance due to Isotopes at High Temperature,

Phys. Rev. 114 (1959) 488.

59. G.A. Slack, Thermal Conductivity of MgO, Al2O3, MgAl2O3, and Fe3O4

Crystals from 3 ° to 300 °K, Phys. Rev. 126 (1962) 427.

Chapter 3.

96

97

Chapter 4.

Anisotropic thermal expansion of MgSiN2

Abstract

The lattice parameters of orthorhombic MgSiN2 as a function of the temperature

have been determined from time-of-flight neutron powder diffraction. The results

indicate that MgSiN2, just like several other adamantine-type crystals, exhibits a

relatively small thermal expansion coefficient at low temperatures. This is ascribed

to a strongly bonded three-dimensional, relatively open, crystal structure which is

characteristic for highly covalent bonded materials. The anisotropic linear thermal

expansion behaviour could be qualitatively related to the characteristics of the

crystal structure. The least dense packed crystallographic direction showed the

smallest anisotropic linear expansion coefficient.

1. Introduction

MgSiN2 is a relatively new ceramic nitride material that belongs to a class of

compounds with potentially interesting thermal, mechanical and luminescence

properties [1 - 6]. Therefore the crystal structure, the physical and the chemical

properties of MgSiN2 are of technological and scientific importance. The thermal

expansion is one of these properties, both for technological reasons (thermal

expansion mismatch between substrate and coating) and scientific reasons

(evaluation of the Grüneisen parameter, which is an important ingredient for

modelling the thermal conductivity [7]). This makes it necessary to study the

thermal expansion of MgSiN2 over a wide range of temperatures.

The thermal expansion of a material can be determined with dilatometry, or

diffraction measurements using X-rays or neutrons as a function of temperature.

Chapter 4.

98

Dilatometry provides only information about the change in the dimensions or

volume of the specimen with respect to the temperature, whereas X-ray and

neutron diffraction provides direct information about the lattice parameters and

atomic positions in the crystalline unit cell. When single crystals are not available,

as in the case for MgSiN2, the anisotropic behaviour cannot be studied with

dilatometry. However, the Rietveld refinement of powder diffraction data [8] is a

powerful and efficient method for crystal structure determination. An important

advantage of neutrons over X-rays forms the comparable yet different neutron

coherent scattering amplitudes among the elements in the periodic table (e.g.,

0.5375, 0.4149 and 0.936⋅10-12 cm for Mg, Si and N, respectively) thereby

providing the contrast and sensitivity needed for resolving the positions of different

atoms. Furthermore, neutron time-of-flight (TOF) technique permits the

measurement of the entire powder pattern at a fixed detector angle with a constant

∆d/d resolution (where d is atomic d-spacing) and minimal systematic errors. For

these reasons the method of temperature dependent neutron TOF powder

diffraction was employed for the measurement of the thermal expansion

coefficients of MgSiN2.

Structural refinement based on previous X-ray diffraction measurements

showed a wurtzite-like structure of MgSiN2 (space group Pna21, no. 33) [9, 10]

with Z = 4. Within this orthorhombic structure the Mg and Si metal atoms and the

two crystallographically different N atoms occupy the general position 4a (x, y, z;

x , y , z+½; x+½, y +½, z; x +½, y+½, z+½). Both metal atoms (Mg and Si) are

tetrahedrally co-ordinated by N (2× N(1) and 2× N(2)) and vice versa both N atoms

are tetrahedrally co-ordinated by the metal atoms (2× Mg and 2× Si). A neutron

diffraction study of MgSiN2 using fixed incident neutron wavelength was reported

in the literature [11]. However, in that study only a room-temperature measurement

was performed and given the limited number of reflections only the lattice

parameters and atomic positions were refined. The atomic positions for MgSiN2

from these previous measurements [9, 10, 11] are listed in Table 4-1.

Recently, the preparation technique of MgSiN2 was improved significantly

resulting in MgSiN2 powder [12, 13] and ceramic [14, 15] with a very low impurity


99

content, which makes the refinement of the neutron diffraction data easier because

no secondary phases have to be taken into account. Furthermore, the neutron

diffraction and Rietveld refinement techniques have been improved substantially

since the last neutron diffraction study of MgSiN2 [11].

Table 4-1: Atomic positions x, y, z for MgSiN2 as given in the literature

[9 - 11] (*: positions obtained after translation (0, ¼, ½) and

symmetry operation x +½, y+½, z+½).

XRD [9] XRD [10]*

x y z x y z

MgSiN(1)N(2)

0.0830.0700.0650.083

0.6000.1300.1250.650

0.0000.0000.3850.400

0.090.070.060.11

0.630.120.140.56

0.000.000.360.45

Constant wavelength neutron diffraction [11]

x y z

MgSiN(1)N(2)

0.076(2)0.072(2)0.0490(15)0.110(1)

0.625(5)0.131(5)0.095(2)0.652(4)

-0.005 0.0 0.356(3) 0.414(2)

In this chapter the Rietveld refinements of temperature dependent TOF

neutron diffraction data of a nearly single phase MgSiN2 powder is reported. The

calculated lattice parameters as a function of the temperature were used to calculate

the thermal expansion coefficients along the three crystallographic axes.

2. Experimental procedure

For the neutron diffraction measurements a nearly single phase MgSiN2 powder

(0.1 wt.% O and 34.2 wt.% N (theoretical value 34.8 wt.%)) was used. The

MgSiN2 powder was prepared starting from a Si3N4 (SKW Trostberg)/Mg3N2 (Alfa)

powder mixture, which was fired at a maximum temperature of 1500 °C in an

Chapter 4.

100

N2/H2 atmosphere. A more detailed description of the powder preparation method

is given elsewhere [13]. The powder was first characterised with X-ray diffraction

(Philips PW 1050/25, Cu-Kα, 20 - 50 ° 2θ-scan with step size 0.01 ° and scan rate

0.01 °/ 18 s) which identified the presence of some very small traces of MgO

(Periclase, JCPDS card 4-829, I/I0 < 0.5 %) as a secondary phase.

The neutron diffraction experiments were performed on the Special

Environment Powder Diffractometer (SEPD) of the Intense Pulsed Neutron Source

(IPNS) at the Argonne National Laboratories (ANL, U.S.A.). The MgSiN2 powder

sample (3.87 g) was put into a cylindrical vanadium sample holder under a helium

atmosphere that was used as a heat-exchange gas for thermal conduction. The

sample was cooled by a closed-cycle helium refrigerator and maintained at a

selected temperature within ∼ 0.5 K. Neutron TOF data were collected at 10, 20,

30, 40, 50, 75, 100, 150, 200, 250 and 300 K for about 1 h at each temperature in

the d-value range of 0.268 - 4.02 Å (2 - 30 msec). Only the data measured in back-

scattering configuration (of a mean detector angle 148 °) were refined because this

detector bank provides the best spatial resolution, ∆d/d ≅ 0.34%. The neutron

diffraction data were analysed in the d-spacing range of 0.5 - 3.88 Å using the

General Structure Analysis System (GSAS) [16] computer code which is based on

the Rietveld method. The experimental data below 0.5 Å were not included

because of the high background and the data above 3.88 Å contained one reflection

at about 3.96 Å which was not taken into account because of the lack of data points

at high d-spacing for modelling the background. The initial input parameters for

the refinement were taken from the literature [11]. The data were refined for 28

variables which include an overall scale factor, a background function and a

neutron pulse-shape profile (the convolution of two “back-to-back” exponentials

with a Gaussian), the lattice parameters, the atomic positions, an isotropic

temperature factor, and a sample absorption coefficient for MgSiN2. Furthermore,

the quality of the refinement was checked for disorder on the Mg and Si sites,

strain broadening and preferential orientation.

The anisotropic linear thermal expansion coefficients along the three

crystallographic axes, αa, αb and αc, from 10 to 300 K were evaluated by


101

determination of the lattice parameters as a function of the absolute temperature

using that αa = (da(T )/dT )/a(T ) [17], etc. The isotropic linear thermal expansion

coefficient, α, was evaluated using α = ⅓ (αa + αb + αc) = ⅓ (dV(T )/dT )/V(T ). The

lattice parameters and unit cell volume as a function of the absolute temperature

were evaluated by a polynomial fit of the obtained data. The lattice parameters

a(T ), b(T ) and c(T ), and unit cell volume, V(T ) were described with a third degree

polynomial: X(T ) = A + CT 2 + DT 3, where X(T ) denotes either a(T ), b(T ), c(T ) or

V(T ), using the data between 10 and 300 K. A linear term (BT ) was not included

for describing the lattice parameters and cell volume as a function of the absolute

temperature since the thermal expansion coefficient at 0 K equals 0 K-1 [17, 18].

Statistical F-testing [19] with a 95 % confidence interval of the variance ratios

showed that introduction of a linear or higher order terms to the polynomial fit did

not improve the fit of the data points. The thermal expansion was evaluated by

differentiation with respect to the temperature.


3.1. Neutron diffraction data refinement

Refinement of the neutron data confirmed the space group Pna21 for MgSiN2 and

the atomic positions of Mg and Si in the MgSiN2 lattice. Moreover, the results

showed that introduction of (anisotropic) strain broadening, preferential

orientation, or introduction of anisotropic temperature factors did not significantly

improve the refinement statistics.

Fig. 4-1 shows a typical observed and calculated powder pattern of MgSiN2,

and in Table 4-2 the lattice parameters and final refinement statistics at 10, 50, 250

and 300 K are presented. The observed statistics for all measurements are good.

The ‘goodness of fit’ in terms of χ 2, was about 1.6 - 1.7 for all temperatures except

for the measurement at 300 K for which the value was 1.98. The residual weighted

R-factor, wRp, equalled about 0.075 for all measurements except for the

measurement at 300 K for which the value was 0.062. Although the statistics of the

Chapter 4.

102

300 K measurement deviate somewhat from the other measurements, it can be

concluded that the observed statistics are, as expected, about the same for all

measurements.

Table 4-2: Lattice parameters a, b, c and V and refinement statistics Rp, wRp and χ 2 for MgSiN2 at

10, 50, 250 and 300 K.

Temperature 10 K 50 K 250 K 300 K

Lattice parametersa [Å]b [Å]c [Å]V [Å3]Data pointsReflectionsRp

wRp

χ 2

5.27078(5) 6.46916(7) 4.98401(5) 169.9425(28) 5000 833 0.0462 0.0752 1.618

5.27068(5) 6.46932(7) 4.98419(5) 169.9495(28) 5000 833 0.0463 0.0763 1.667

5.27171(5) 6.47175(7) 4.98542(5) 170.0883(28) 5000 833 0.0456 0.0745 1.639

5.27249(4) 6.47334(6) 4.98622(4) 170.1827(24) 5000 834 0.0312 0.0621 1.977

MgSiN2 on SEPD at 300 K Hist 1Bank 1, 2-Theta 144.8, L-S cycle 37 Obsd. and Diff. Profiles

D-spacing, A

Nor

m. c

ount

/mus

ec.

.5 1.0 1.5 2.0 2.5 3.0

.0

2

.0

4

.0

Fig. 4-1: A typical TOF neutron diffraction profile for MgSiN2 at 300 K (+),

fit (solid line) with tick marks indicated at the Bragg positions and

the difference plot (at the bottom on the same scale).


103

The lattice parameters and cell volume as a function of the absolute

temperature are presented in Fig. 4-2 and Fig. 4-3, respectively. The indicated error

bars equal 3 times the standard deviation obtained from the structural refinement.

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.0070

0 50 100 150 200 250 300 350T [K]

a, b

, c [Å

]

(a - 5.2700) Å(b - 6.4670) Å(c - 4.9825) Å

Fig. 4-2: The lattice parameters (a, b and c) of MgSiN2 as a function

of the absolute temperature (T ) as determined by the

Rietveld structure refinement using neutron diffraction data.

169.90

169.95

170.00

170.05

170.10

170.15

170.20

0 50 100 150 200 250 300 350

T [K]

V [Å

³]

Fig. 4-3: The unit cell volume (V ) of MgSiN2 as a function of

absolute temperature (T ) as determined by the Rietveld

structure refinement using neutron diffraction data.

Chapter 4.

104

Table 4-3: Atomic positions x, y, z and isotropic temperature factor Uiso for MgSiN2 at

10 and 300 K.

MgSiN2 10 K

MgSiN(1)N(2)

x0.08448(34)0.0693(5)0.04855(17)0.10859(18)

y0.62255(30)0.1249(4)0.09562(15)0.65527(14)

z-0.0134(5) 0 0.3472(4) 0.4102(4)

100 Uiso

0.298(33)0.14(4)0.196(20)0.234(20)

MgSiN2 300 K

MgSiN(1)N(2)

x0.08475(31)0.0687(4)0.04863(15)0.10873(16)

y0.62263(27)0.12535(34)0.09557(13)0.65519(13)

z-0.0135(4) 0 0.34822(34) 0.41130(37)

100 Uiso

0.628(32)0.302(35)0.358(19)0.384(19)

In Table 4-3 the atomic positions and isotropic temperature factors are

presented for Mg, Si, N(1) and N(2) at 10 and 300 K. To the extent of the accuracy

of the refinement, the atomic positions of Mg, Si, N(1) and N(2) do not vary with

temperature. The atomic positions at 300 K are in reasonable agreement with the

values reported previously in the literature (Table 4-1).

In the MgSiN2 lattice the tetrahedral co-ordination of Mg, Si, N(1) and N(2)

is distorted. Furthermore each cation (anion) is surrounded by 12 cations (anions)

among which 4 occupy the same crystallographic site and 8 do not. For example,

the nearest cation neighbours for Mg are 4 Mg and 8 Si atoms, for N(1) the nearest

anion neighbours are 4 N(1) and 8 N(2), etc. The important bond lengths and bond

angles were calculated at all temperatures and the results for 10 and 300 K are

presented in Table 4-4 and Table 4-5, respectively.

As expected, the average Mg–N and Si–N bond length (N = N(1), N(2))

increases with increasing temperature. The relative change for the average Si–N

bond length is smaller than that for the Mg–N bond length. The relative variation

between the 4 Si–N and 4 Mg–N bond lengths is smaller than that between the

4 N(1)–Me (Me = Mg, Si) and 4 N(2)–Me bonds. The tetrahedral co-ordination of

the N atoms is heavily distorted due to the presence of two types of N–Me bonds.


105

Table 4-4: Si–N and Mg–N bond lengths, N–Si–N, N–Mg–N, Me–N(1)–Me and Me–N(2)–Me

(Me = Mg, Si) angle, and N–N and Me–Me distance for MgSiN2 at 10 K.

bond length [Å] bond length [Å]

Mg–N(1)iv

Mg–N(1)ii

Mg–N(2)I

Mg–N(2)iv

2.0626 (1×)2.0733 (1×)2.1002 (1×)2.1256 (1×)

Si–N(1)v

Si–N(1)ii

Si–N(2)vi

Si–N(2)vii

1.7323 (1×)1.7442 (1×)1.7612 (1×)1.7667 (1×)

N–Me–N bond angle[°]

N–N dis. [Å]

Me–N–Me bond angle[°]

Me–Me dis. [Å]

N(1)ii–Mg–N(1)iv

N(1)ii–Mg–N(2)i

N(1)iv–Mg–N(2)iv


N(2)i–Mg–N(2)iv

N(1)iv–Mg–N(2)i

106.173105.396108.726121.865107.114106.673

3.3068 (2×)3.3402 (1×)3.3598 (1×)3.3832 (1×)3.3995 (2×)3.6478 (1×)

Mgx–N(1)–Siviii

Mgix–N(1)–Siv

Mgx–N(1)–Siv

Siv–N(1)–Siviii

Mgix–N(1)–Mgx

Mgix–N(1)–Siviii

104.935104.962105.439123.234 97.469117.103

3.0161 (1×)3.0345 (1×)3.0351 (1×)3.0586 (2×)3.1088 (2×)3.2515 (1×)

N(1)v–Si–N(1)ii

N(1)v–Si–N(2)vi

N(1)ii–Si–N(2)vi

N(1)v–Si–N(2)vii

N(1)ii–Si–N(2)vii

N(2)vi–Si–N(2)vii

108.919108.885108.953107.844111.244110.942

2.8289 (2×)2.8332 (1×)2.8479 (1×)2.8563 (1×)2.8834 (1×)2.9065 (2×)

Mgix–N(2)–Mgv

Siix–N(2)–Sixi

Mgv–N(2)–Siix

Mgv–N(2)–Sixi

Mgix–N(2)–Sixi

Mgix–N(2)–Siix

93.778122.480107.517107.337108.938112.766

3.0850 (2×)3.0927 (2×)3.1422 (1×)3.1429 (1×)3.1528 (1×)3.2211 (1×)

av. bond length [Å] average angle [°] av. distance [Å]Mg–NSi–NN(1)–MeN(2)–Me

2.0901.7511.9031.938

N–Mg–NN–Si–NMe–N(1)–MeMe–N(2)–Me

109.33109.46108.86108.80

N(1)–NN(2)–NMg–MeSi–Me

3.1293.1473.1153.108

Equivalent positions:(i) x, y, z-1(ii) -x, -y, z+½-1(iii) x+½, -y+½, z-1(iv) -x+½, y+½, z+½-1

(v) x+½, y+½, z(vi) -x+½, y+½-1, z+½-1(vii) -x, -y+1, z+½-1(viii) -x, -y, z+½

(ix) -x, -y+1, z+½(x) -x+½, y+½-1, z+½(xi) -x+½, y+½, z+½

The average Mg-N bond length at 300 K (2.0916 Å) is somewhat larger than the

previously reported value of 2.086 Å [11] but still substantially shorter than the

Mg–N distance (2.142 Å) in Mg3N2 [20]. Likewise, the average Si–N bond length

at 300 K (1.7520 Å) is somewhat smaller than the previously reported value

(1.760 Å) [11] but still substantially larger than the Si–N bond length observed in

β -Si3N4 (1.732 Å [21]). The average N–Mg–N angle (109.32 °) is slightly smaller

Chapter 4.

106

Table 4-5: Si–N and Mg–N bond lengths, N–Si–N, N–Mg–N, Me–N(1)–Me and Me–N(2)–Me

(Me = Mg, Si) angle, and N–N and Me–Me distance for MgSiN2 at 300 K.

bond length [Å] bond length [Å]Mg–N(1)iv

Mg–N(1)ii

Mg–N(2)I

Mg–N(2)iv

2.0598 (1×)2.0731 (1×)2.1013 (1×)2.1324 (1×)

Si–N(1)v

Si–N(1)ii

Si–N(2)vi

Si–N(2)vii

1.7322 (1×)1.7502 (1×)1.7576 (1×)1.7679 (1×)

N–Me–N bond angle[°]

N–N dis. [Å]

Me–N–Me bond Angle[°]

Me–Me dis. [Å]


N(1)ii–Mg–N(2)i

N(1)iv–Mg–N(2)iv


N(2)i–Mg–N(2)iv

N(1)iv–Mg–N(2)i

106.371105.279108.849121.943106.896106.575

3.3087 (2×)3.3428 (1×)3.3609 (1×)3.3845 (1×)3.4016 (2×)3.6500 (1×)

Mgx–N(1)–Siviii

Mgix–N(1)–Siv

Mgx–N(1)–Siv

Siv–N(1)–Siviii

Mgix–N(1)–Mgx

Mgix–N(1)–Siviii

105.017104.688105.425123.091 97.596117.388

3.0153 (1×)3.0332 (1×)3.0370 (1×)3.0617 (2×)3.1095 (2×)3.2561 (1×)

N(1)v–Si–N(1)ii

N(1)v–Si–N(2)vi

N(1)ii–Si–N(2)vi

N(1)v–Si–N(2)vii

N(1)ii–Si–N(2)vii

N(2)vi–Si–N(2)vii

108.718108.566108.933107.848111.549111.148

2.8301 (2×)2.8351 (1×)2.8484 (1×)2.8564 (1×)2.8855 (1×)2.9080 (2×)

Mgix–N(2)–Mgv

Siix–N(2)–Sixi

Mgv–N(2)–Siix

Mgv–N(2)–Sixi

Mgix–N(2)–Sixi

Mgix–N(2)–Siix

93.662122.508107.356107.142109.200112.860

3.0879 (2×)3.0910 (2×)3.1420 (1×)3.1456 (1×)3.1598 (1×)3.2209 (1×)

av. Bond length [Å] Average angle [°] Av.Distance

[Å]

Mg–NSi–NN(1)–MeN(2)–Me

2.0921.7521.9041.940

N–Mg–NN–Si–NMe–N(1)–MeMe–N(2)–Me

109.32109.46108.87108.79

N(1)–NN(2)–NMg–MeSi–Me

3.1203.1493.1173.110

Equivalent positions:(i) x, y, z-1(ii) -x, -y, z+½-1(iii) x+½, -y+½, z-1(iv) -x+½, y+½, z+½-1

(v) x+½, y+½, z(vi) -x+½, y+½-1, z+½-1(vii) -x, -y+1, z+½-1(viii) -x, -y, z+½

(ix) -x, -y+1, z+½(x) -x+½, y+½-1, z+½(xi) -x+½, y+½, z+½

than the average observed N–Si–N angle (109.46 °). Although these average bond

angles are close to that for an ideal tetrahedron (109.47 °), the individual bond

angles, especially for the N–Mg–N angles deviate considerably from the ideal

value. From these results it can be concluded that the Mg–N4 tetrahedra are more

distorted than the Si–N4 tetrahedra. Similarly, individual Me–N–Me angles deviate

substantially from the mean angles (108.86 ° and 108.80 °) as well as from the


107

ideal value of 109.47 °. The N–N distances vary from 2.83 to 3.65 Å whereas the

Me–Me (Me = Mg, Si) distances vary only from 3.02 to 3.26 Å. This also indicates

that the anion sublattice is distorted whereas the cation sublattice is still quite

regular. The N–(Si–)N and N–(Mg–)N distances in MgSiN2 (2.83 - 2.91 Å and

3.31 - 3.65 Å, respectively) are comparable with the N–(Me–)N distances observed

in Si3N4 (2.77 - 2.90 Å [21]) and in Mg3N2 (3.31 - 3.75 Å [20]). The same applies

for the Me-Me distances (3.06 - 3.09 Å for Si – Si, 3.09 - 3.11 Å for Mg – Mg and

3.02 - 3.26 Å for Si – Mg in MgSiN2) that vary between 3.00 and 3.05 Å [21] in

Si3N4 and 2.72 and 3.29 Å [20] in Mg3N2.

In a recent study of the crystal structures of Mg3N2 and Zn3N2 [20] the

Brese-O’Keeffe nitride bond valence parameters [22] for Mg–N were discussed. In

this study it is noted that it would be desirable to have a database of well-refined

nitride structures so that bond valence parameters could be directly determined. For

MgSiN2 the bond valence parameters for Mg–N and Si–N were calculated by using

the obtained bond lengths at 300 K. By varying the bond valence parameters for

Mg–N and Si–N the difference between the calculated and expected bond valence

sums of Mg, Si, N(1) and N(2) was minimised using a least-squares method. For

the Mg–N and Si–N bonds a bond valence parameter of 1.833 and 1.752

respectively was obtained. Both values are in reasonable agreement with the bond

valence parameters proposed by Brese and O’Keeffe for Mg–N (1.85) and Si–N

(1.77) [22].

3.2. Thermal expansion

Since the experimental diffraction data were collected under the identical

experimental configuration (other than the temperature change) without

interruption, and the refinements were performed in a consistent manner, it is

expected that the systematic errors, experimental or computational, to have very

small, if any, effects on the calculation of the thermal expansion coefficients. This

implies that the error in the observed dependence of the lattice parameters and cell

Chapter 4.

108

volume as a function of the absolute temperature (see Table 4-6) is very small and

equals about the accuracy of the structure refinement.

Table 4-6: The lattice parameters a, b and c and unit cell volume V of

MgSiN2 as a function of the absolute temperature T between

0 and 300 K.

T

[K]

a-axis

[Å]

b-axis

[Å]

c-axis

[Å]

V

[Å3]

10 15 20 25 30 35 40 45 50 75100150200250300

5.27078(5)5.27080(5)5.27064(5)5.27075(5)5.27072(5)5.27071(5)5.27079(5)5.27073(5)5.27068(5)5.27082(5)5.27083(5)5.27083(5)5.27118(5)5.27171(5)5.27249(4)

6.46916(7)6.46917(7)6.46914(7)6.46919(7)6.46924(7)6.46924(7)6.46917(7)6.46926(7)6.46932(7)6.46927(7)6.46953(7)6.46992(7)6.47065(7)6.47175(7)6.47334(6)

4.98401(5)4.98406(5)4.98415(5)4.98405(5)4.98408(5)4.98409(5)4.98404(5)4.98404(5)4.98419(5)4.98410(5)4.98424(5)4.98441(5)4.98473(5)4.98542(5)4.98622(4)

169.9425(28)169.9448(28)169.9422(28)169.9435(29)169.9449(29)169.9448(28)169.9439(28)169.9443(28)169.9495(28)169.9498(28)169.9617(28)169.9776(28)170.0189(28)170.0883(28)170.1827(24)

The coefficients of the third degree polynomial to describe the lattice

parameters and unit cell volume for all data points are presented in Table 4-7.

These coefficients were used to calculate the thermal expansion coefficients from

10 to 300 K for MgSiN2. The lattice parameters and cell volume at 293 K evaluated

from the polynomial fit are 5.27237 Å, 6.47308 Å, 4.98609 Å and 170.167 Å3 for

a(293 K), b(293 K), c(293 K) and V(293 K), respectively. These are in good

agreement with those obtained previously by XRD measurements at 293 K

a(293 K) = 5.2698 Å, b(293 K) = 6.4734 Å, c(293 K) = 4.9843 Å and

V(293 K) = 170.04 Å3 [13]. The isotropic and anisotropic linear thermal expansion


109

coefficients are presented in Table 4-8. The accuracy of the presented thermal

expansion data was estimated to be about 0.3 10-6 K-1.

Table 4-7: The coefficients used in the polynomial A + CT 2 + DT 3 for describing the lattice

parameters a, b and c and unit cell volume V of MgSiN2 as a function of the

absolute temperature T between 10 and 300 K.

coefficient a-axis b-axis c-axis V

ACD

5.27075 -6.16924 10-09

8.53434 10-11

6.46919 1.87691 10-08

9.07302 10-11

4.98407 5.83402 10-09

6.04839 10-11

169.9440 4.91604 10-07

7.20613 10-09

From Fig. 4-3 and Table 4-8 it can be concluded that the unit cell volume

increases monotonically as a function of the absolute temperature. The data points

at 50 and 100 K seem to be somewhat scattered but are well within the

experimental uncertainty. Similar to the case of Si3N4 [23] the observed thermal

expansion for MgSiN2 is quite small. The small thermal expansion coefficient in

these materials was attributed to an almost symmetric potential well for atomic

bonding [17].

It can be seen from Fig. 4-2 that the b and c-axis monotonically increase

with increasing temperature, whereas the a-axis in the temperature range of 10 to

150 K remains about constant. A low or negative thermal expansion coefficient is

often found for tetrahedrally bonded solids (adamantine type materials) below a

reduced temperature T/θ = 0.2 [24, 25] (where θ is the Debye temperature). E.g.,

Si, Ge, GaAs, GaSb, InAs, InSb and AlN exhibit a negative thermal expansion

coefficient at low temperatures [24, 26, 27]. The negative or small thermal

expansion coefficient in these tetrahedrally bonded solids may be related to

structural features such as [26, 28, 29]: i) a strongly covalently bonded three-

dimensional polyhedron network which hinders changes in bond length; and ii) a

relatively open structure which can absorb thermal energies via transverse modes

with atomic displacements perpendicular to the bond directions. In general, the

observed negative thermal expansion coefficient of these materials usually does not

Chapter 4.

110

exceed the value of 0.3 - 0.4 10-6 K-1 which is comparable to the accuracy of the

presented thermal expansion data for MgSiN2. Thus MgSiN2 exhibits a very low,

or possibly, negative thermal expansion coefficient for the a-axis in the

temperature range of 10 to 150 K.

Table 4-8: The anisotropic thermal expansion coefficients αa, αb and αc, and

linear isotropic thermal expansion coefficient α (= ⅓(αa + αb +

αc)) of MgSiN2 as a function of the absolute temperature T

between 0 and 300 K (*: The fit for α = ⅓(dV/dT )/V yields

identical results.).

T

[K]αa

[K-1]

αb

[K-1]

αc

[K-1]

⅓(αa + αb + αc)*

[K-1]

0 10 50100150200250300

0- 0.02 10-6

0.00 10-6

0.25 10-6

0.74 10-6

1.5 10-6

2.5 10-6

3.7 10-6

00.06 10-6

0.40 10-6

1.0 10-6

1.8 10-6

2.8 10-6

4.1 10-6

5.5 10-6

00.03 10-6

0.20 10-6

0.60 10-6

1.2 10-6

1.9 10-6

2.9 10-6

4.0 10-6

00.02 10-6

0.20 10-6

0.62 10-6

1.2 10-6

2.1 10-6

3.1 10-6

4.4 10-6

The difference in the thermal expansion behaviour along the a and b-axis

can be understood in terms of the crystal structure of MgSiN2. Fig. 4-4 shows the

crystal structure projected on the a-b plane. It can be seen that the MgSiN2

structure can be deduced from a hexagonal structure (with lattice parameters a' and

c') in which a ≈ √3 a', b ≈ 2 a'and c = c' ≈ √8/3a' [10, 30]. Along the

b-axis zigzag Mg–N–Si–N chains with bond angles of ∼ 120 ° are observed

whereas along the a-axis Mg–N····Mg–N·· and Si–N····Si–N·· chains are observed.

This gives rise to a low packing density along the a-axis. Since stretching force

constants are in general higher than bending forces [29, 31], elongation along the

b-axis will take place under a thermal load, resulting in some bending of the bonds

along the b-axis which counteracts the thermal expansion along the a-axis.


111

In the literature only the average thermal expansion coefficient of MgSiN2

between 293 and 873 K (5.8 10-6 K-1 [2]) and the thermal expansion coefficient at

T = θ = 827 K (6.5 10-6 K-1 [32]) were reported. A check for self-consistency of the

linear thermal expansion coefficient of MgSiN2 at 300 K can be made using the

Grüneisen relation γ = 3α Vm/βT CV. Here γ [-] is the Grüneisen parameter, α [K-1]

the linear thermal expansion coefficient, Vm [m3 mol-1] the molar volume, βT [Pa-1]

the isothermal compressibility and CV [J mol-1 K-1] the heat capacity at constant

volume. If the temperature is sufficiently high, the Grüneisen parameter is nearly

constant [25] and the thermal expansion can be evaluated from Vm, βT and CV.

Using the (almost) temperature independent values for Vm of 2.57 10-5 m3 mol-1, βT

of 6.84 10-12 Pa-1 and the average CV of 83.61 J mol-1 K-1 in the temperature range

of 293 and 873 K [33], the average Grüneisen parameter between 293 and 873 K

a

b

c

N(1)

N(2)

Si

Mg

Fig. 4-4: The crystal structure of MgSiN2 showing the a-b plane in

which along the a-axis Mg–N····Mg–N·· and Si–N····Si–N··

chains are observed and along the b-axis Mg–N–Si–N–

chains.

Chapter 4.

112

was calculated to be 0.78, in complete agreement with the γ of 0.78 at 827 K [32].

Therefore, using a constant Grüneisen parameter of 0.78 and CV at 300 K of

61.713 J mol-1 K-1 [33], the linear thermal expansion coefficient at 300 K was

estimated as 4.3 10-6 K-1. This calculated value agrees very well with the observed

value of 4.4 10-6 K-1.

4. Conclusions

Neutron TOF powder-diffraction data obtained between 10 and 300 K were used to

determine the volumetric thermal expansion coefficient and the linear thermal

expansion coefficient along the three different crystallographic directions of

orthorhombic MgSiN2. Rietveld analyses confirmed the crystal structure of

MgSiN2, space group Pna21. The results indicate that in-situ neutron diffraction is

very effective for accurate characterisation of the thermal expansion behaviour. A

relatively small thermal expansion coefficient is observed for MgSiN2 due to the

strongly bonded three-dimensional tetrahedra network and the open crystal

structure as a consequence of the high degree of covalency. The variation of the

Si–N bond lengths with temperature is smaller than that for the Mg–N bond

lengths. The anisotropic thermal expansion behaviour can be explained

qualitatively in terms of the characteristics of the crystal structure of MgSiN2. The

neutron diffraction data reveal a very low, or possibly, negative linear thermal

expansion coefficient along the a-axis at low temperatures which is the least dense

packed crystallographic direction.

References


Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II:

Non-Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron,

G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society,

Pittsburgh, 1994) 239.


113



3. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan, Hot-


4. G.K. Gaido, G.P. Dubrovskii and A.M. Zykov, Photoluminescence of

Europium-Activated MgSiN2, Inorganic Materials 10 (1974) 485 (translated

from Izv. Akad. Nauk. SSSR, Neorg. Mater. 10 (1974) 564).

5. G.P. Dubrovskii, A.M. Zykov and B.V. Chernovets, Luminescence of Rare

Earth Activated MgSiN2, Inorganic Materials 17 (1981) 1059 (translated from

Izv. Akad. Nauk. SSSR, Neorg. Mater. 17 (1981) 1421).

6. S. Lim, S.S. Lee and G.K. Chang, Photoluminescence Characterization of

MgxZn1-xSiN2:Tb for Thin Film Electroluminescent Devices Application,

Wissenschaft und Technik, Inorganic and organic electroluminescence,

Verlag, Berlin (1996), p. 363.




8. R.A. Young (ed), The Rietveld Method, Oxford University Press, (1996).



10. S. Wild, P. Grieveson and K.H. Jack, The Crystal Chemistry of New Metal-

Silicon-Nitrogen Ceramic Phases, Spec. Ceram. 5 (1972) 289.

11. M. Wintenberger, F. Tcheou, J. David und J. Lang, Verfeinerung der Struktur


35b (1980) 604.

12. H. Uchida, K. Itatani, M. Aizawa, F.S. Howell and A. Kishioka, Synthesis of

Magnesium Silicon Nitride by the Nitridation of Powders in the Magnesium-

Silicon System, J. Ceram. Soc. Japan 105 (1997) 934.


Characterisation of MgSiN2 Powders, J. Mater. Sci. 34 (1999) 4519.

Chapter 4.

114

14. H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder

Preparation and Densification of MgSiN2, Ceram. Trans. 51, Int. Conf. Cer.

Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by H.

Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995)

585.


Ceramics, Fourth Euro Ceramics 2, Basic Science – Developments in

Processing of Advanced Ceramics – Part II, Faenza, Italy, October 1995,

edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995)

289.

16. GSAS, General Structure Analysis System written by A.C. Larson and R.B. von

Dreele, LANSCE, MS-H805, Los Alamos National Laboratory, Los Alamos,

NM 87545, USA (1989).


Properties of Matter 13, Thermal Expansion of Nonmetallic Solids: Theory of

Thermal Expansion of Solids (IFI/Plenum, New York, USA, 1977),

pp. 3a - 16a.

18. R.A. Swalin, Thermodynamics of Solids, second edition (John Wiley & Sons,

Inc., New York, USA, 1972).

19. W.C. Hamilton, Statistics in Physical Science: Estimation, Hypothesis Testing

and Least Squares (The Ronald Press Company, New York, USA, 1964),

pp. 168 - 173.

20. D.E. Partin, D.J. Williams and M. O’Keeffe, The Crystal Structures of Mg3N2

and Zn3N2, J. Solid State Chem. 132 (1997) 56.

21. R. Grün, The Crystal Structure of β -Si3N4; Structural and Stability

Considerations Between α- and β -Si3N4, Acta Cryst. B35 (1979) 800.

22. N.E. Brese and M. O’Keeffe, Bond-Valence Parameters for Solids, Acta

Cryst. B47 (1991) 192.



(IFI/Plenum, New York, USA, 1977), pp. 1140 - 1141.


115

24. P.W. Sparks and C.A. Swenson, Thermal Expansion from 2 to 40 ºK of Ge,

Si, and Four III-V Compounds, Phys. Rev. 163 (1967) 779.

25. W.B. Daniels, The amomalous thermal expansion of germanium, silicon and

compounds crystallizing in the zinc blende structure, Int. Conf. on the Physics

of Semiconductors, Exeter, UK, July 1962, edited by A.C. Stickland (Institute

of Physics, London, 1962) 482.

26. G.A. Slack and S.F. Bartram, Thermal expansion of some diamondlike

crystals, J. Appl. Phys. 46 (1975) 89.



December 1 - 5, 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer,

S. Nakamura and S. Strite (Materials Research Society, Warrendale,


28. R. Roy, D.K. Agrawal and H.A. McKinstry, Very Low Thermal Expansion

Coefficient Materials, Annu. Rev. Mater. Sci. 19 (1989) 59.


Oxynitride of Silicon in Relation to their Structure, Trans. J. Br. Ceram. Soc.

74 (1975) 49.

30. F.F. Grekov, G.P. Dubrovskii and A.M. Zykov, Structure and Chemical

Bonding in Ternary Nitrides, Inorganic Materials 15 (1979) 1546 (translated

from Izv. Akad. Nauk. SSSR, Neorg. Mater. 15 (1979) 1965).

31. H.D. Megaw, Crystal Structures and Thermal Expansion, Mat. Res. Bull. 6

(1971) 1007.


(Oxy)-Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science –



V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) p. 405.

33. Chapter 5, R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg,


(1998) 7871.

Chapter 4.

116

117

Chapter 5.

The heat capacity of MgSiN2

Abstract

The heat capacity at standard pressure (Cpo) of MgSiN2 was determined by

adiabatic calorimetry in the range of 8 - 400 K and differential scanning

calorimetry in the range of 300 - 800 K. The measured Cpo data for T < 24 K

can be described using the Debye T 3 approximation: Cpo = AT 3 with

A = 1.3632 10-5 J mol-1 K-4. For temperatures between 350 and 650 K the Cpo can

be described with the Debye equation using a constant Debye temperature of

996 K. For temperatures between 24 and 350 K the Debye temperature is a

function of temperature and has a minimum value of 740 K at about 55 K. The Cpo

data for T ≥ 300 K were compared with those of AlN. As expected, the Cpo data of

MgSiN2 were about a factor 2 larger than those of AlN. The entropy STo, enthalpy

(HTo - H0

o) and the energy function (GT

o - H0

o) in the range of 0 - 800 K were

calculated using standard thermodynamic formulas. By extrapolating the Cpo data

to high temperatures at which GTo is known, H0

o was estimated to equal

- 534 kJ mol-1.

1. Introduction

MgSiN2 is a ternary adamantine type compound with tetrahedral coordination of

Mg and Si. It can be deduced from the well-known AlN by systematically

replacing two Al ions with one Mg and one Si ion. The properties of MgSiN2

ceramics have recently been reported [1]. Because the thermal and mechanical

Chapter 5.

118

properties of MgSiN2 ceramics look promising, an investigation of the preparation,

characterisation and properties of MgSiN2 was started. This chapter focuses in

more detail on one thermal property, viz. the heat capacity.

For understanding the thermal properties of a material, it is necessary to have

accurate and reliable specific heat data as a function of the sample temperature.

E.g., the heat capacity is needed for calculating the thermal conductivity from

thermal diffusivity data [2]. The heat capacity can also be used to estimate the

Debye temperature, which is an important parameter for theoretical modelling of

the thermal conductivity [3]. Furthermore, heat capacity data can be used to

evaluate the thermodynamic functions STo, (HT

o - H0

o) and (GT

o - H0

o).

The (GTo - H0

o) function can be used to evaluate the Gibbs energy, GT

o, if H0

o

is known, or vice versa, if (GTo - H0

o) and GT

o are known H0

o can be evaluated. The

GTo is necessary for predicting the thermodynamic stability of a compound in a

system. Therefore, it is a very strong tool when optimising the synthesis or

processing of a material.

As far as the (present) author knows, specific heat data and thermodynamic

functions for MgSiN2 have never been extensively reported. Only the room

temperature value of the specific heat has been published (738 J kg-1 K-1 [4]).

Concerning the thermodynamic functions of MgSiN2, only estimates of GTo at high

temperatures (T > 1600 K) have been reported [5, 6]. Specific heat data of MgSiN2

below 300 K have never been reported, and high temperature data (T ≥ 300 K) are

not available in the open literature [7].

In this chapter standard specific heat data at constant pressure (Cpo) of

MgSiN2 in the temperature range of 0 - 800 K are reported. These data were used

to calculate the Debye temperature and the thermodynamic functions STo,

(HTo - H0

o) and (GT

o - H0

o) of MgSiN2 in the same temperature range. Also the

value of H0o was estimated by extrapolation of the (GT

o - H0

o) function into the

temperature range where GTo data are known. In order to do this the Cp

o curve was

extrapolated up to 2000 K using standard thermodynamic functions.


119

2. Experimental

2.1. Adiabatic calorimeter measurements

The specific heat at constant pressure (Cp) for MgSiN2 in the range of 8 - 400 K

was measured with an adiabatic calorimeter. The experimental set-up of the

adiabatic calorimeter (CAL V) is described elsewhere [8]. For the measurement,

isostatically pressed MgSiN2 powder pellets were used. The synthesis of the

MgSiN2 powders is described elsewhere [9]. Isostatically pressed powder pellets

(diameter 6.5 mm, thickness 1 - 2 mm) with a total mass of about 12.12 g were put

into a sample holder (copper vessel) with a mass of about 20 g and an internal

volume of about 11 cm3. Before sealing the copper vessel, it was evacuated and

filled with 1000 Pa of He gas as a heat exchanger. The sample plus sample holder

was heated to the highest measuring temperature, before starting the Cp

measurement, to enhance possible energy relaxation. Subsequently, the sample plus

sample holder was cooled to the lowest measuring temperature. Then, stepwise a

known quantity of energy was added to the sample plus sample holder and the

temperature increase was measured. The temperature was measured with a Pt

resistance thermometer (100 Ω at 298.15 K, Oxford instruments) with an accuracy

of ± 0.003 K between 5 - 30 K and ± 0.005 K above 30 K. A temperature increase

of about 2 - 3 K was measured after the energy was added to the system. From the

temperature increase and the amount of energy added the specific heat of the

sample plus the sample holder was calculated. Another independent specific heat

measurement of the empty sample holder was used to correct for the specific heat

of it, and subsequently to determine the specific heat of the MgSiN2 sample.

When the powder pellets were first heated to the highest measuring

temperature energy relaxation was observed. This energy was introduced into the

system during isostatic pressing of the powder into pellets. After this energy

relaxation the measured specific heat of the sample was not influenced by the

thermal history of the measurement.

Chapter 5.

120

The reliability of the measured adiabatic calorimeter data was estimated

from the impurity content of the MgSiN2 pellets and the absolute accuracy of the

calorimeter. The major phase impurity in the MgSiN2 sample was MgO as a

secondary phase. The impurity content of the MgSiN2 sample was estimated to be

less than 1.0 wt. % based on the measured oxygen content and the measured X-ray

diffraction pattern of the sample. The internal precision of the adiabatic calorimeter

was about 0.02 %, and the absolute accuracy was estimated to be 0.1 %. From this

and the purity of the sample, the reliability of the measured adiabatic calorimeter

data was estimated to deviate no more than 1 % from the true values of pure

MgSiN2.

2.2. Differential scanning calorimeter measurement

For the DSC measurement MgSiN2 ceramic disks (diameter 4.90 mm, thickness

1.6 - 1.8 mm) cut from a sintered MgSiN2 tablet were used. The tablet was

prepared by hot uni-axial pressing. The hot-pressing method for obtaining the

MgSiN2 ceramics has been reported earlier [10, 11]. Nearly single phase, glassy

phase free, fully dense MgSiN2 ceramics with an isotropic microstructure were

obtained [11]. The Cp of the ceramic samples in the range of 300 - 800 K was

measured with a differential scanning calorimeter (DSC) Setaram 111. The sample

was first heated to the highest measuring temperature before starting the Cp

measurement to check for possible energy relaxation. Subsequently, the sample

was cooled to the lowest measuring temperature. Then, the measurements were

performed under a nitrogen atmosphere using a heating rate of 10 K/min. During

heating, the differences in energy input (q, Q, Q') to keep the temperature constant

were measured between: the reference sample holder and the sample holder (q); the

reference sample holder and the sample holder plus sample with mass m (Q); and

the reference sample holder and the sample holder plus a reference material with

mass m' (Q'). The differences between the differences in energy input (Q - q) and

(Q' - q) are directly related to the specific heat capacity in J g-1 K-1 of the sample

(cp) and the reference material (cp') according to:


121

mm

q-Qq-Qcc pp

')'()('= (1)

Sapphire rods (Calorimeter Conference Sample 720), supplied by the

National Bureau of Standards (NIST), were used as a reference material. The mass

of the MgSiN2 sample was 0.38659 g, and the mass of the sapphire reference

material was 0.19497 g. At every degree the Cp was determined.

No energy relaxation was observed in the ceramic sample for the DSC

measurement when it was heated to the highest measuring temperature. The

measured specific heat of the sample was not influenced by the thermal history of

the measurement.

The reliability of the DSC data was estimated from the impurity content of

the ceramic sample and the absolute accuracy of the DSC. The major phase

impurity in the MgSiN2 ceramic sample was MgO as a secondary phase. The

impurity content of the MgSiN2 sample was estimated to be less than 2.0 wt. %

based on the measured oxygen content and the measured X-ray diffraction pattern

of the sample. The absolute accuracy of the DSC was estimated to be 2 %. From

this and the purity of the ceramic sample, the reliability of the measured DSC data

was estimated to deviate no more than 3 % from the true values of pure MgSiN2.


3.1. Cpo of MgSiN2

For the results of both, adiabatic calorimeter and DSC, measurements it was

assumed that [∂Cp /∂p]T = -T [∂2V/∂T 2]p ≈ 0. This implies that the measured Cp is

pressure independent. So we may assume that the measured Cp equals the heat

capacity at standard pressure Cpo.

In Fig. 5-1 the measured Cpo values for MgSiN2 in the range of 8 - 400 K and

300 - 800 K are presented for the adiabatic calorimeter and DSC measurement,

respectively.

Chapter 5.

122

The Cpo data of the adiabatic calorimeter and the DSC in the overlapping

temperature range of 300 - 400 K are in excellent agreement with each other

considering the experimental accuracy of the equipment (largest deviation

≈ 0.5 %). The Cpo value measured at 293 K of 60.5 ± 0.6 J mol-1 K-1

(= 752 ± 8 J kg-1 K-1) is comparable with the earlier published value of 738 J kg-1

K-1 [4]. The S-shaped Cpo curve in Fig. 5-1 shows, as expected, a gradually increase

with the absolute temperature till 760 K. At temperatures above 760 K an

unexpected (small) decrease, as function of the temperature, is observed. The Cpo

for electronic insulators, like MgSiN2, is expected to increase only slowly at high

temperatures. So, it can be concluded that the DSC data are less reliable at high

temperatures. The systematic measurement of too low Cpo values can most

probably be ascribed to the difference in mass, shape and thermal properties like

thermal conductivity of the sapphire reference and the MgSiN2 sample.

The Cpo values between 0 and 8 K were obtained by extrapolation using the

Cpo data between 16 and 24 K and the Debye theory of the specific heat. This

theory states that if the temperature is sufficiently low the CV = AT 3 [12]. At low

0

20

40

60

80

100

0 100 200 300 400 500 600 700 800T [K]

Cp [

J m

ol-1

K-1

]

AdiabaticDSC

Fig. 5-1: The heat capacity at constant pressure (Cp) of MgSiN2 as a

function of the absolute temperature (T ) in the range of

0 - 800 K.


123

temperatures Cp ≈ CV. So, if the Cpo exhibits this T3 behavior then a Cp

o/T versus

T 2 plot results in a straight line through the origin with slope A. In Fig. 5-2 a Cpo/T

versus T 2 plot in the range of 0 - 50 K is shown. From the figure it can be

concluded that Cpo/T is proportional to T 2 if T 2 ≤ 600 K2 (T < 24 K). The Cp

o

values measured below 16 K (T 2 < 250 K2) are less accurate due to the small

contribution (< 1 %) of the MgSiN2 sample to the measured specific heat of the

sample plus sample holder. Therefore the Cpo/T values between 16 and 24 K were

used to evaluate the heat capacity data for MgSiN2 between 0 and 20 K, using that

Cpo/T = AT 2 with A = 1.3632 10-5 J mol-1 K-4 for temperatures between 16 and

24 K. This resulted in Cpo(T ) = 1.3632 10-5 T 3 for T ≤ 20 K. For T > 20 K the

Cpo(T ) function at every degree was constructed by polynomial fitting the Cp

o data

over several small temperature ranges.

The Cpo data of MgSiN2 at T ≥ 300 K were compared with tabulated Cp

o

values of AlN [13]. Because AlN and MgSiN2 have both a wurtzite-like crystal

structure, the same average atomic mass, about the same average volume per atom,

and about the same sound velocity [14], the Debye temperatures are about the same

and so are the specific heats per mole atoms. So it is expected that

Cp /T = 1.36315 10-5 T 2

0

0.01

0.02

0.03

0.04

0.05

0 500 1000 1500 2000 2500

T 2 [K2]

Cp/T

[J m

ol-1

K-2

]

Fig. 5-2: Cp /T versus T 2 for MgSiN2 at T ≤ 50 K

Chapter 5.

124

Cpo(MgSiN2) ≈ 2 Cp

o(AlN). In Table 5-1 the Cp

o values of MgSiN2 and AlN are

presented from 300 to 800 K for every 100 K. It can be seen that, as expected,

Cpo(MgSiN2)/Cp

o(AlN) ≈ 2.0. For T ≥ 600 K a systematic decrease of the

Cpo(MgSiN2)/Cp

o(AlN) ratio is observed. This is probably caused by systematically

measuring too low Cpo values at T ≥ 600 K.

Table 5-1: Cpo values of MgSiN2 and AlN from 300 to 800 K for every 100 K.

T Cpo(AlN) Cp

o(MgSiN2) Cp

o(MgSiN2)/Cp

o(AlN)

[K] [J mol-1 K-1] [J mol-1 K-1] [-]

300400500600700800

30.29136.40240.44843.68345.71947.083

61.7174.7882.4387.0990.1790.14

2.042.052.041.991.961.91

3.2. Debye temperature of MgSiN2

For calculating the Debye temperature we assumed that Cp = CV over the whole

temperature range of 0 to 800 K. In order to check this assumption we estimated

the maximum difference between the Cp and the CV for T ≤ 800 K using [15]:

TVp

TVC-Cβ

α m29= (2)

in which α [K-1] is the linear thermal expansion coefficient, Vm [m3 mol-1] the

molar volume, T [K] the absolute temperature and βT [Pa-1] the isothermal

compressibility. Vm and βT are almost constant as function of temperature, and from

the Grüneisen relation it is known that α is about proportional with CV. So

the (relative) difference between Cp and CV increases with temperature

(Cp - CV ~ CV2 T). The maximum difference between Cp and CV occurs at the


125

highest measuring temperature, viz. 800 K. For MgSiN2 this results in, taking the

values of [14] α(T = 800 K) ≈ α(T = 827 K) = 6.5 10-6 K-1, Vm(T = 800 K) ≈

Vm(T = 293 K) = 2.57 10-5 m3 mol-1 (= M/ρ of Ref. 14) and βT(T = 800 K)

≈ βT(T = 293 K) ≈ βS(T = 293 K) = 6.84 10-12 Pa-1 (calculated with equation (12) of

Ref. 14), a maximum difference between Cp - CV of about 1.2 J mol-1 K-1. So the

relative difference between Cp and CV at 800 K equals about 1.3 %. This is well

within the experimental accuracy of the measured DSC data, so it may be assumed

that Cp = CV.

In Fig. 5-3 the Debye temperature is presented as a function of the absolute

temperature. The Debye temperature was determined using tabulated values of the

specific heat per atom as function of T /θ [16]. The shape of the curve is similar to

that determined for other adamantine type compounds [17]: a decrease of the

Debye temperature with increasing temperature to a minimum value and then an

increase with temperature to a constant value. At T > 650 K an unexpected steep

increase of the Debye temperature is observed. This increase can be totally

ascribed to the less reliable measurement of the Cpo data at high temperatures as

will be discussed later in this section.

700

800

900

1000

1100

1200

0 100 200 300 400 500 600 700 800T [K]

θ [K

]

AdiabaticDSC

Fig. 5-3: The Debye temperature (θ ) as a function of the absolute

temperature (T ) of MgSiN2 in the range of 0 - 800 K.

Chapter 5.

126

The Debye temperature at 0 K (θ0) in Fig. 5-3 was evaluated from the Debye

T 3 expression for the specific heat per mol atoms at low temperatures [12]:

T T R CV334

A5

12 )( =π=θ

(3)

in which R is the gas constant 8.314 J mol-1 K-1. Using the value for A of

3.408 10-6 J mol-1 K-4 (= 1.3632 10-5/4; there are 4 mol atoms per mol MgSiN2) the

value for θ0 was calculated and equals 829 K. As expected, this value is very close

to the Debye temperature obtained from elastic data, θ E, of 827 K [14]. At about

55 K the Debye temperature has a minimum value of 740 K. If we express these

temperatures in terms of reduced values T /θ0 and θ /θ0 we obtain for the location

of the minimum T /θ0 = 0.07 with θ /θ0 = 0.9. The location of the minimum

is comparable with that of other adamantine like compounds [17, 18]

(T /θ0 ~ 0.05 - 0.07) whereas the minimum value is larger than for most other

adamantine type compounds like Si and Ge [17, 18] (θ /θ0 ~ 0.7). This difference is

most probably caused by the relative low mean mass of MgSiN2 [18]. As Fig. 5-3

shows, the Debye temperature at 350 K ≤ T ≤ 650 K has a constant value of about

996 ± 4 K. At T > 650 K an increase of the Debye temperature is observed. This

increase cannot be explained by the assumption made that Cp = CV because it

results in a decrease of the Debye temperature as a function of temperature. So, this

increase of the Debye temperature at T > 650 K is totally caused by systematically

measuring of too low Cpo values indicating that the DSC measurement is less

reliable at higher temperatures.

Using the constant Debye temperature of 996 K the Cpo of MgSiN2 at

T ≥ 350 K was calculated and compared with the experimentally measured values

(Table 5-2). From Table 5-2 it can be seen that between 350 - 700 K the measured

and calculated values are in good agreement with each other, and between

700 - 800 K the measured Cpo of MgSiN2 becomes less reliable. The maximum

deviation of the measured from the expected Cpo value is about 2.5 %. If the

assumption that Cp = CV is also considered, a maximum deviation of about 4 % is

expected between the true Cpo and the measured Cp

o.


127

Table 5-2: Comparison between the measured (Cpo

m) and calculated (Cpo

c) heat capacity using the

Debye equation with θ = 996 K.

T

[K]

Cpo m

[J mol-1 K-1]

Cpo c

[J mol-1 K-1]

Cpo

m - Cpo

c

[J mol-1 K-1]

(Cpo

m - Cpo

c)/Cpo

m * 100%

[%]

350400450500550600650700750800

69.0674.7879.1982.4485.0187.0988.7289.7690.1790.14

68.6974.5879.0482.4785.1487.2488.9690.3691.5292.47

0.37 0.20 0.15- 0.03- 0.13- 0.15- 0.24- 0.40- 1.35- 2.33

0.5 0.3 0.2- 0.0- 0.2- 0.2- 0.3- 0.4- 1.5- 2.6

3.3. Thermodynamic functions STo, (HT

o - H0

o) and (GT

o - H0

o) of MgSiN2

For T ≤ 20 K, the thermodynamic functions (STo - S0

o) and (HT

o - H0

o) as a function

of the absolute temperature, follow from the heat capacity function as

(STo - S0

o) = 3

1 AT 3 and (HTo - H0

o) = 1

4 AT 4. A non-zero S0o due to the (partially)

random occupation of the cation sites in MgSiN2 by Mg and Si is not expected

because Mg and Si are complete ordered in the MgSiN2 lattice [19]. So, the

absolute entropy, STo, can be calculated by taking S0

o = 0 J mol-1 K-1. The ST

o and

(HTo - H0

o) function at T > 20 K were calculated by numerical integration of the

Cpo(T )/T and Cp

o(T ) function. The Cp

o(T )/T function was constructed using the

polynomial fit of the Cpo(T ) curve in the corresponding temperature range. The

Gibbs energy, (GTo - H0

o), was calculated using (GT

o - H0

o) = (HT

o - H0

o) - T ST

o.

The Cpo, ST

o, (HT

o - H0

o) and (GT

o - H0

o) of MgSiN2 are presented in

Table 5-3 for every 10 K in the range of 0 - 800 K. It is noted that for T ≥ 700 K

Chapter 5.

128

the Cpo and the thereof calculated thermodynamic functions ST

o, (HT

o - H0

o) and

(GTo - H0

o) are less reliable.

Table 5-3: The Cpo, ST

o, (HT

o - H0

o) and (GT

o - H0

o) of MgSiN2 for every

10 K between 0 and 800 K.

T[K]

Cpo

[J mol-1 K-1]ST

o

[J mol-1 K-1](HT

o - H0

o)

[J mol-1](GT

o - H0

o)

[J mol-1] 0 10 20 30 40 50

0 0.014 0.109 0.403 1.133 2.367

0 0.0045 0.0364 0.120 0.325 0.701

0 0.034 0.545 2.716 9.997 27.069

0 -0.011 -0.183 -0.890 -2.990 -7.957

60 70 80 90 100

4.088 6.206 8.593 11.154 13.912

1.275 2.062 3.046 4.206 5.521

58.87 110.20 184.13 282.81 408.91

-17.66 -34.18 -59.56 -95.68 -144.20

110 120 130 140 150

16.650 19.484 22.337 25.183 28.001

6.975 8.545 10.217 11.977 13.811

560.66 741.30 950.41 1188.03 1453.98

-206.62 -284.13 -377.86 -488.77 -617.65

160 170 180 190 200

30.774 33.488 36.135 38.708 41.201

15.707 17.654 19.644 21.667 23.716

1747.9 2069.3 2417.4 2791.7 3191.3

-765.2 -932.0 -1118.4 -1324.9 -1551.8

210 220 230 240 250

43.612 45.938 48.181 50.340 52.416

25.785 27.868 29.959 32.056 34.153

3615.5 4063.3 4533.9 5026.6 5540.5

-1799.3 -2067.6 -2356.7 -2666.8 -2997.8

260 270 280 290 300

54.410 56.325 58.161 59.921 61.713

36.248 38.338 40.420 42.491 44.551

6074.7 6628.4 7200.9 7791.4 8399.1

-3349.8 -3722.8 -4116.6 -4531.1 -4966.4

310 320 330 340 350

63.32 64.86 66.32 67.73 69.06

46.60 48.63 50.65 52.65 54.63

9024 9665 10321 10993 11676

-5422 -5898 -6395 -6911 -7448

360 370 380 390 400

70.33 71.53 72.67 73.75 74.78

56.60 58.55 60.47 62.37 64.25

12373 13082 13803 14535 15278

-8004 -8580 -9174 -9789-10422


129

Table 5-3: (Continued) The Cpo, ST

o, (HT

o - H0

o) and (GT

o - H0

o) of

MgSiN2 for every 10 K between 0 and 800 K.

T[K]

Cpo

[J mol-1 K-1]ST

o

[J mol-1 K-1](HT

o - H0

o)

[J mol-1](GT

o - H0

o)

[J mol-1] 410 420 430 440 450

75.86 76.77 77.62 78.42 79.19

66.15 67.98 69.80 71.59 73.37

16044 16807 17579 18359 19147

-11076 -11747 -12436 -13143 -13868

460 470 480 490 500

79.91 80.59 81.24 81.85 82.43

75.11 76.84 78.54 80.23 81.88

19942 20745 21554 22370 23191

-14610 -15370 -16147 -16941 -17751

510 520 530 540 550

83.00 83.53 84.04 84.54 85.01

83.52 85.14 86.73 88.31 89.87

24019 24851 25689 26532 27380

-18578 -19421 -20281 -21156 -22047

560 570 580 590 600

85.46 85.89 86.31 86.71 87.09

91.40 92.92 94.42 95.90 97.36

28232 29089 29950 30815 31684

-22953 -23875 -24812 -25763 -26730

610 620 630 640 650

87.46 87.80 88.13 88.43 88.72

98.80 100.22 101.63 103.02 104.39

32557 33433 34313 35196 36082

-27710 -28706 -29715 -30738 -31775

660 670 680 690 700

88.98 89.21 89.42 89.61 89.76

105.75 107.09 108.41 109.72 111.01

36970 37861 38754 39649 40546

-32826 -33890 -34968 -36058 -37162

710 720 730 740 750

89.89 90.00 90.08 90.13 90.17

112.29 113.54 114.79 116.01 117.22

41445 42344 43244 44145 45047

-38279 -39408 -40549 -41703 -42870

760 770 780 790 800

90.18 90.18 90.17 90.15 90.14

118.42 119.60 120.76 121.91 123.04

45949 46851 47752 48654 49555

-44048 -45238 -46440 -47653 -48778

Chapter 5.

130

3.4. H0o of MgSiN2

In order to make fully use of the thermodynamic data, H0o should be known.

Because estimates of GTo values at high temperatures (T > 1600 K) are known

[5, 6] H0o, can be evaluated from (GT

o - H0

o) data in the same temperature range by

matching the (GTo - H0

o) function with the known estimates of the GT

o function.

In order to obtain (GTo - H0

o) data at high temperatures, the Cp

o curve at high

temperatures was calculated (up to 2000 K) using the Debye equation for the CV

with θ = 996 K and the expression for the difference between Cp - CV, assuming

that α and Vm/βT are constant for T ≥ 800 K. In order to obtain a smooth Cpo curve

between 500 and 800 K, and to minimize the error in the calculated thermodynamic

functions introduced by measuring to low Cpo values at T > 700 K, the

experimental data between 650 and 800 K were not used in the polynomial fit to

describe the Cpo data below 800 K. From the Cp

o curve the thermodynamic

functions STo and (HT

o - H0

o) for temperatures between 500 and 2000 K were

recalculated by numerical integration of the Cpo(T )/T and Cp

o(T ) function using the

polynomial fit of the Cpo(T ) curve for construction of the Cp

o(T )/T function. The

(GTo - H0

o) function was calculated in the same way as described before.

In the literature two estimates for the GTo of MgSiN2 at high temperatures are

reported [5, 6]. The first estimate results in a minimum and maximum value for

GTo, and is based on the following two reactions [5]:

4 Si2N2O + 2 MgSiN2 → 3 Si3N4 + Mg2SiO4

Si3N4 + 4 MgO → Mg2SiO4 + 2 MgSiN2

According to Müller [5] both reactions proceed to the right for temperatures

between 1673 and 2073 K. This results in the following conditions for the GTo of

MgSiN2 in the corresponding temperature range:

GTo MgSiN

2 > 1½ GTo Si

3N

4 + ½ GTo Mg

2SiO

4 - 2 GTo Si

2N

2O

GTo MgSiN

2 < ½ GTo Si

3N

4 + 2 GTo MgO - ½ GT

o Mg2SiO

4


131

Table 5-4: (GTo - H0

o) and the estimates of H0

o as function of the temperature.

T

[K]

(GTo - H0

o)

[kJ mol-1]

GTo max - (GT

o - H0

o)

[kJ mol-1]

GTo

min - (GTo - H0

o)

[kJ mol-1]

GTo *- (GT

o - H0

o)

[kJ mol-1]

800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

-48.840 -61.730 -75.680 -90.595-106.397-123.017-140.398-158.488-177.243-196.625-216.601-237.136-258.202

-523.000-521.280-519.397-517.407-515.346-513.246-511.127-509.008-506.900-504.816-502.760-500.744-498.773

-538.228-537.162-536.061-534.938-533.800-532.655-531.508-530.361-529.222-528.089-526.964-525.854-524.762

-539.791-539.168-538.509-537.829-537.134-536.431-535.726-535.022-534.326-533.635-532.953-532.285-531.636

average H0o

— -511 ± 8 -532 ± 4 -536 ± 3

From this a minimum and maximum value for the GTo function, GT

o min and GT

o max,

of MgSiN2 were obtained (Table 5-4). The second estimate is based on the ∆GTo

R

of the following reaction [6]:

Si3N4 + 4 MgO → Mg2SiO4 + 2 MgSiN2

with:

∆GToR = 3953 - 8.35 T [J mol-1] = GT

o MgSiN2 + ½GT

o Mg2SiO

4 - 2GTo MgO - ½GT

o Si3N

4

Kaufman et al. [6] used this expression, and the Gibbs energy of Mg2SiO4, MgO

and Si3N4 for thermodynamic calculations at temperatures above 1900 K resulting

in the GTo function of MgSiN2 based on Ref. 6 which will be referred to as GT

o *. It

was assumed that both estimates are valid for temperatures between 800 and

2000 K. For the calculation of GTo

max, GTo

min and GTo *, tabulated GT

o values of

Si3N4 (GTo Si

3N

4), Si2N2O (GTo Si

2N

2O), MgO (GT

o MgO) and Mg2SiO4 (GTo Mg

2SiO

4) were

Chapter 5.

132

used. GTo Si

3N

4 and GTo Si

2N

2O were taken from Ref. [20] and GT

o MgO and GTo Mg

2SiO

4

were taken from Ref. [21].

The estimates for GTo (GT

o max, GT

o min and GT

o *) and (GTo - H0

o) of MgSiN2

between 800 and 2000 K are graphically presented in Fig. 5-4. Although it is clear

from the figure that the GTo * function is not within the range of the GT

o max and

GTo

min function, but just below the GTo

min function, both estimates are in favourable

agreement with each other. As expected the GTo

max, GTo

min and GTo * function have

a similar shape as the (GTo - H0

o) function, indicating that the extrapolation of the

(GTo - H0

o) function is reliable and has been done correctly. In Table 5-4 the

(GTo - H0

o) function and the estimates of H0

o, based on the GT

o max, GT

o min and GT

o *

function, are presented as a function of the absolute temperature. Indeed, the H0o

values are nearly constant, as expected. The H0o values obtained from the GT

o *

function are the most constant and have the smallest standard deviation, whereas

the H0o values obtained from the GT

o max function fluctuate the most and have the

largest standard deviation. If it is assumed that the GTo

min and GTo * function are the

-8.0E+05

-7.5E+05

-7.0E+05

-6.5E+05

-6.0E+05

-5.5E+05

-5.0E+05

800 1000 1200 1400 1600 1800 2000T [K]

GTo [J

mol

-1]

-3.5E+05

-3.0E+05

-2.5E+05

-2.0E+05

-1.5E+05

-1.0E+05

-5.0E+04

(GTo -H

0o ) [J

mol

-1]

←

× × × × →

∆

Fig. 5-4: The Gibbs energies GTo max (), GT

o min () and GT

o * (∆), and the

energy function (GTo - H0

o) (×) as a function of the absolute

temperature T for MgSiN2 from 800 to 2000 K.


133

best estimates for the real GTo function, then a value for H0

o of -534 ± 3 kJ mol-1 is

obtained.

4. Conclusions

The heat capacity Cpo, the Debye temperature θ , and the thermodynamic functions

STo, (HT

o - H0

o) and (GT

o - H0

o) of MgSiN2 were determined for temperatures

between 0 and 800 K.

The experimental Cpo data for T < 24 K can be described by the Debye

T 3 approximation. The measured Cpo data for T ≥ 300 K were compared with those

of AlN. As expected the Cpo data of MgSiN2 were about a factor 2 larger than for

AlN.

The Debye temperature at 0 K equals 829 K and is comparable with the

Debye temperature obtained from elastic constants (827 K). The Debye temperature

below 350 K is a function of the absolute temperature and has a minimum value of

740 K at about 55 K. A constant Debye temperature of 996 K can be used to

describe the experimental Cpo data for T ≥ 350 K using the Debye equation.

By extrapolation of the Cpo data to high temperatures (> 1600 K), H0

o was

estimated to equal -534 kJ mol-1.

References



2. P. Debye, Zustandsgleichung und Quantenhypothese mit einem Anhang über

Wärmeleitung, in: Vorträge über die Kinetische Theorie der Materie und der

Electrizität (Teubner, Berlin, 1914), pp. 19 - 64.

Chapter 5.

134





Ceramics: MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II:

Non-Oxides, Boston, Ohio, USA, November 1993, edited by A.R. Barron,

G.S. Fischman, M.A. Fury and A.F. Hepp (Materials Research Society,

Pittsburgh, 1994) 239.

5. R. Müller, Kostitutionsuntersuchungen und thermodynamischen

Berechnungen im system Mg, Si/N, O, Ph. D. Dissertation University of

Stuttgart, 1981; pp. 32 - 34.

6. L. Kaufman, F. Hayes, and D. Birnie, Calculation of quasibinary and

quasiternary oxynitride systems, High Temp. High Pres. 14 (1982) 619.

7. W.A. Groen, Personal communication (Cpo values of MgSiN2 had been

determined between 300 - 850 K at Philips Research Laboratories but these data

have not been published).

8. J.C. van Miltenburg, G.J.K. van den Berg and M.J. van Bommel, Construction

of an adiabatic calorimeter. Measurements on the molar heat capacity of

synthetic sapphire and n-heptane. J. Chem. Thermodyn. 19 (1987) 1129.


characterisation of MgSiN2 powders, J. Mater. Sci. 34 (1999) 4519.

10. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen, M.J. Kraan,

Hot-pressing of MgSiN2 ceramics, J. Mat. Sci. Lett. 13 (1994) 1314.


Preparation, Characterisation and Properties of MgSiN2 Ceramics, to be

published.

12. See for example C. Kittel, Introduction to Solid State Physics, fifth edition

(John Wiley & Sons, Inc., New York, 1976), pp. 136 - 140, or R.A. Swalin,

Thermodynamics of Solids, second edition (John Wiley & Sons, Inc., New

York, 1972), pp. 57 - 62.


135

13. I. Barin, Thermochemical Data for Pure Substances, Part I, second edition

(VCH Verlagsgesellschaft mbH, Weinheim, FRG, 1993), p. 42.





V. Sergo (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) p. 405.

15. See for example Swalin, R.A. Thermodynamics of Solids, second edition

(John Wiley & Sons, Inc., New York, 1972).

16. W.M. Rogers and R.L. Powell, Tables of Transport Integrals, Natl. Bur. Stand.

Circ. 595 (1958) 1.

17. J.C. Phillips, Vibration Spectra and Specific Heats of Diamond-Type Lattices,

Phys. Rev. 113 (1959) 147.

18. T.C. Cetas, C.R. Tilford and C.A. Swenson, Specific Heats of Cu, GaAs,

InAs, and InSb from 1 to 30 °K, Phys. Rev. 174 (1968) 835.

19. R.K. Harris, M.J. Leach and D.P. Thompson, Nitrogen-15 and Oxygen-17

NMR Spectroscopy of Silicates and Nitrogen Ceramics, Chem. Mater. 4

(1992) 260.

20. M. Hillert, S. Jonsson and B. Sundman, Thermodynamic Calculation of the

Si-N-O System, Z. Metalkd. 83 (1992) 648.

21. I. Barin, Thermochemical Data for Pure Substances, second edition (VCH

Verlagsgesellschaft mbH, Weinheim, FRG, 1993).

Chapter 5.

136

137

Chapter 6.

The Young's modulus of MgSiN2, AlN and Si3N4

Abstract

The temperature dependence of the Young's modulus of MgSiN2 and AlN was

measured between 293 and 973 K using the impulse excitation method and

compared with literature data reported for Si3N4. The data could be fitted with

E = E0 - B·T exp (-T0/T ). The values of the fitting parameters E0 and T0 are related

to the Debye temperature and the parameter B to the harmonic character of the

bond.

1. Introduction

The relatively new ternary adamantine type compound MgSiN2, which can be

deduced from the well known AlN by replacing two Al3+ ions by one Mg2+ and one

Si4+ ion, might be interesting for specific applications because of its favourable

chemical, mechanical and thermal properties [1 - 8].

We pointed out [6] that in order to understand the (thermal) properties of

MgSiN2, AlN and other (new) potentially interesting materials more insight is

needed in the parameters that determine the intrinsic thermal conductivity. Two

important parameters that determine the intrinsic thermal conductivity are the

Debye temperature θ [K] and Grüneisen parameter γ [-] [6, 9 - 12]. The Debye

temperature can be evaluated from elastic constants E (Young's modulus) [Pa] and

ν (Poisson's ratio) [-] [13]. For evaluation of the Grüneisen parameter elastic

constants as a function of the temperature are needed. So far, only room

temperature values for the elastic constants have been published (MgSiN2:

Chapter 6.

138

E = 235 GPa and ν = 0.232 [2], and AlN: E = 308 - 315 GPa and ν = 0.179 - 0.245

[14 - 16]).

In this chapter the Young's modulus of MgSiN2 and AlN as a function of the

temperature between 293 - 973 K is reported. The temperature dependence of the

Young's modulus was described with the empirical expression E = E0 - B·T

exp (-T0/T ) which was previously shown to be valid by Wachtman for several

oxides [17]. Also temperature dependent Young's modulus data for the related

nitride compound Si3N4 [18] were fitted using this expression. For MgSiN2, AlN

and Si3N4 the fitting parameter E0 was used for calculating the Debye temperature

θ0. The values obtained for B and T0 from fitting of the experimental data are

discussed in view of the analytical expressions of Anderson [19] for B and T0.

2. Experimental section

The preparation of the MgSiN2 ceramic disks (∅ 33 mm × 3 mm) with hot-

pressing (1550 - 1650 °C, 75 MPa, N2 atmosphere, 2 h) is described elsewhere [5,

8, 20]. Three fully dense (ρ = 3.14 - 3.15 g cm-3) samples (RB10, RB31 and RB33)

processed in somewhat different ways, were selected to measure the temperature

dependence of the Young's modulus. X-ray diffraction (XRD) revealed that they

contain some (< 2 wt. %) α- and β-Si3N4 (RB10), MgO (RB31), and β-Si3N4

(RB33) as a secondary phase [8]. Clean grain boundaries were observed between

the MgSiN2 grains (~ 0.3 - 1.0 µm) with transmission electron microscopy (TEM).

The AlN ceramics were obtained from Xycarb ceramics (Helmond, The

Netherlands). The fully dense (ρ = 3.29 g cm-3) AlN ceramic disk (∅ 250 × 20

mm) was prepared by hot-pressing (1830 °C, 35 MPa, N2 45 min.) AlN powder

(ART, grade A100) containing about 4 wt. % Y2O3 as an additive. The resulting

ceramics contain some YAP (YAlO3, JCPDS 33-41) and YAG (Y3Al5O12, JCPDS

33-40) as detected with XRD, which are commonly found secondary phases for

AlN sintered with Y2O3 addition. The grain size of the AlN ceramics was about 4


139

µm as observed on a fractured surface with a field emission scanning electron

microscopy (FESEM).

For a fully dense MgSiN2 disk (∅ 15 mm × 2.89 mm) the room temperature

longitudinal vl [m s-1] and transverse sound velocity vt [m s-1] were measured at 10

and 20 MHz, respectively, using the pulse-echo method. From the sound velocities

and the density ρ [kg m-3] the room temperature Young's modulus E [Pa] and

Poisson's ratio ν [-] were calculated using the formulas for isotropic materials [21]:

= 222

2

t

l2

t

l -vv-

vvν (1)

( ))1(

)21)(1(12 2l

2t ν

ννρνρ-

-v v E +=+= (2)

The Young's modulus was measured from 293 to 973 K on three hot-pressed, fully

dense, MgSiN2 ceramic materials processed under somewhat different conditions

(for details see Ref. 8) and one hot-pressed, fully dense, AlN ceramic material

using the impulse excitation method [22] (GrindoSonic, Lemmens Elektronica BV,

Belgium). For MgSiN2 and AlN two different sample sizes of the same material

were measured (rectangular bars l × b × h ~ 18 mm × 8 mm × 2 mm and

~ 18 mm × 5 mm × 2 mm). For comparison also some room temperature

measurements on larger AlN bars (rectangular bars ~ 50 mm × 8 mm × 3 mm and

~ 50 mm × 6 mm × 3 mm) were performed. Each measurement was performed

twice in order to obtain an impression of the accuracy of the data points and of the

resulting fitting parameters.

The fundamental natural flexural frequency of the samples was measured

every 5 K during heating and cooling. From this frequency, the sample dimensions

and mass, the Young's modulus E [Pa] was evaluated using [23, 24]:

Abh

lmfE 3

32

9465.0= (3)

in which m [kg] is the sample mass, f [s-1] the flexural frequency, l [m] the sample

length, b [m] the sample width, h [m] the sample height and A a dimensionless

shape factor dependent on sample length, sample width and Poisson's ratio. As the

Chapter 6.

140

dependence of A on the Poisson's ratio is very limited, A can be approximated by:

A = 1 + 6.585(h/l)2 [24]. The sample dimensions were corrected for thermal

expansion in order to calculate the Young's modulus. The resolution of the flexural

frequency measurement was 10 Hz. For a typical resonance frequency of about 30

kHz this results in an experimental error in the Young's modulus introduced by the

frequency measurement of ∆E/E ≈ 2(∆f /f ) ≈ 0.07 %.

For comparison temperature dependent literature data for Si3N4

(β-modification according to the processing temperature of 1750 °C mentioned

[18]) were taken for sample H-1 with 0.5 wt. % MgO addition. The Young's

modulus data for this sample with the least amount of secondary phase were

corrected for porosity (E = Emeas/(2ρmeas/ρ the - 1) [18]) with ρmeas (3.104 g cm-3) and

Emeas (varying between 302 GPa and 291 GPa for temperatures between ~ 300 K

and ~ 1200 K, respectively) the experimental density and Young's modulus,

respectively, and ρ the (3.19 g cm-3) the theoretical density of β-Si3N4.

The data obtained as a function of the absolute temperature were described

using the empirical formula of Wachtman [17]:

)/(exp 00 T-TTB-EE ⋅= (4)

in which E0 [Pa] is the Young's modulus at 0 K, B [Pa K-1] and T0 [K] are fitting

parameters.


3.1. Evaluation of the measurements

For MgSiN2 a longitudinal sound velocity of 10.17 103 m s-1 and transverse sound

velocity of 5.90 103 m s-1 were measured. This resulted in a room temperature

value for the Poisson's ratio ν of 0.246 and the Young's modulus E of 273 GPa.

The Poisson's ratio is comparable with ν = 0.232 given in the literature measured

with the same pulse-echo technique [2]. The value of the Young's modulus

reported before is considerably lower E = 235 GPa [2], which may be (partially)


141

ascribed to the lower density (98.9 %) and purity (3.7 wt. % oxygen) of the sample

described in the literature [2].

For the impulse excitation experiments no hysteresis in the resonance

frequency was observed during heating and cooling. The reproducibility of the

measurements using the same sample was excellent (± 0.3 GPa). The slight

difference between the observed Young's moduli for the same material having

different dimensions is caused by experimental errors in the sample dimension

measurement, ∆l, ∆b and ∆h ≈ 0.02 mm leading to ∆E ≈ (3(∆l /l )2 + (∆b/b)2 +

3(∆h/h)2 + 2(∆f /f )2)1/2 E ≈ 0.016 E = 4.5 GPa). Considering the experimental

accuracy the Young's modulus at 293 K was the same for the various samples and

265

270

275

280

285

290

295

300

305

310

315

320

325

0 200 400 600 800 1000 1200T [K]

E [G

Pa]

Si3N4

AlN

MgSiN2

Fig. 6-1: A typical result obtained for the Young's modulus (E ) as a

function of the absolute temperature (T ) between 293 and

973 K for a MgSiN2 and AlN ceramic sample with fit

E = 284.8 - 0.0228⋅T exp (-424/T ) and E = 310.2 - 0.0247⋅T

exp (-533/T ), respectively. For comparison literature data

for Si3N4 [18] between 300 and 1200 K with fit

E = 320.4 - 0.0151⋅T exp (-445/T ) are included.

Chapter 6.

142

equal about 279 ± 4 GPa and 312 ± 4 GPa for MgSiN2 and AlN, respectively. The

room temperature value for MgSiN2 is in good agreement with the value measured

using the pulse-echo method (273 GPa) and the values for AlN (312 GPa) and

Si3N4 (319 GPa) are in excellent agreement with previously reported values (AlN:

308 - 315 GPa [14 - 16] and Si3N4: 290 - 335 GPa [25, 26]).

With increasing temperature the Young's modulus of MgSiN2, AlN and

Si3N4 slightly decreases (Fig. 6-1). As compared with the room temperature value

the Young's modulus at 973 K for the MgSiN2 samples has decreased with

12.6 ± 0.2 GPa (∆E/E293 = 0.045), for the AlN samples with 12.9 ± 0.2 GPa

(∆E/E293 = 0.041) and for the Si3N4 sample with 8.3 GPa (∆E/E293 = 0.026). So, the

temperature dependences of MgSiN2 and AlN are similar whereas Si3N4 shows a

smaller temperature dependence. As expected [17], the temperature dependence of

the experimental data is very well described by E = E0 - B·T exp (-T0/T ) (Fig. 6-1).

As the Young's modulus shows no anomalies, this indicates that for all three

materials the influence of microstructure and secondary phases on the temperature

dependence of the Young's modulus can be assumed to be negligible.

In Table 6-1 the values of the fitting parameters E0, B and T0 are presented.

The average E0 value for AlN of 314 GPa was calculated from the average

observed E293 and the average values of B and T0. Within the experimental

accuracy the values of E0, B and T0 of the AlN samples are the same as all samples

originate from one large homogeneous ceramic bar. Also for the several MgSiN2

samples processed under somewhat different conditions (see Ref. 8) the values of

E0, B and T0 are within the experimental error the same (see Table 6-1).

A relatively large variation in T0 is observed for the measurements

performed on the same sample having the same size. For T / T0 we can write for

E = E0 - B·T exp (-T0/T ) ≈ E0 - B·T (1-T0/T ) = (E0 + B·T0) - B·T resulting in a linear

relation between E and T (as observed in Fig. 6-1) showing that the slope B can be

easily evaluated whereas the constants E0 and B·T0 are correlated. As E0 >> B·T0

the fitting parameter T0 is relatively sensitive to small errors as compared to

E0 and B.


143

Table 6-1: Fitting parameters E0, B and T0 for describing the Young's modulus as a function of the

absolute temperature, and room temperature value of the Young's modulus for MgSiN2,

AlN and Si3N4. Between brackets the 95 % confidence interval of the fitting parameters

are presented. The experimental error was estimated to equal the standard deviation of the

average values.

Material / Sample E0

[GPa]B

[GPa K-1]T0

[K]E293

[GPa]MgSiN2

RB10 (17.29 × 8.05 × 2.15 mm)

(17.30 × 5.69 × 2.15 mm)

277.93 (7)278.24(16)286.64(10)286.76 (9)

0.02237(18)0.02190(38)0.02134(16)0.02136(15)

450(12)403(25)347(14)332(11)

276.5276.6284.7284.7

RB31 (17.76 × 8.06 × 2.14 mm)

(17.76 × 5.85 × 2.15 mm)

284.78 (6)285.01(10)276.45 (7)276.58 (8)

0.02281(14)0.02241(20)0.02251(17)0.02203(19)

424 (9)397(15)422(11)390(13)

283.2283.3274.9274.9

RB33 (17.54 × 8.06 × 2.15 mm)

(17.55 × 5.88 × 2.16 mm)

280.87(14)280.98(10)277.87(12)277.94 (5)

0.02133(24)0.02156(24)0.02161(25)0.02227(11)

349(19)366(16)386(18)403 (8)

279.0279.1276.2276.3

Average value MgSiN2 281 ± 4 0.0220 ± 0.0005 389 ± 34 279 ± 4AlN

(17.82 × 8.12 × 2.12 mm)

(17.80 × 5.89 × 2.12 mm)

(50.23 × 8.11 × 2.99 mm)

(50.06 × 5.88 × 2.99 mm)

310.11(12)310.14(14)310.20 (6)310.64(12)

————

0.02419(35)0.02451(39)0.02468(24)0.02404(37)

————

487(21)488(23)533(12)473(21)

————

308.8308.8309.0309.2318.8318.8312.9312.9

Average value AlN 314 ± 4 0.0244 ± 0.0003 495 ± 26 312 ± 4Si3N4

H-1 (∅ 30 mm × 12 mm) [18] 320.41(13) 0.01508(24) 445(28) 319.4

Chapter 6.

144

3.2. Interpretation of the fitting parameters

3.2.1. E0

The average E0 value (Young's modulus at 0 K) for MgSiN2, AlN and Si3N4 was

used to evaluate the Debye temperature at 0 K, θ0 [K]. The Debye temperature can

be calculated from the average sound velocity (vs [m s-1]) obtained from the

longitudinal vl [m s-1] and transverse sound velocity vt [m s-1] using the elastic

constants E and ν, and the density ρ [kg m-3] [12, 21, 27]:

)21)(1()1(

l ννν

ρ --E v

+= (5)

)1(21

t νρ += E v (6)

31

3t

3l

s ])21[31(

-

vv v += (7)

Subsequently the average sound velocity can be used to calculate the Debye

temperature using [12, 13, 27 - 29]:

31

As )

348(

2 MNs

khv ρθ

π= (8)

in which h is Planck's constant (6.626 10-34 J s), k the Boltzmann's constant

(1.381 10-23 J K-1), s [-] the number of atoms per formula unit, NA Avogadro's

number (6.023 1023 mol-1) and M [kg mol-1] the mole mass.

Using the Young's modulus and density at 0 K (E0 and ρ0, respectively) and

the room temperature value of the Poisson's ratio ν, the Debye temperature θ0 was

calculated (see Table 6-2). The resulting Debye temperatures of all three

compounds are in the same range (900 - 950 K) with θMgSiN2 = 900 K

. θAlN = 940 K . θSi3N4 = 955 K. The values agree reasonably well with previously

reported values for MgSiN2, AlN and Si3N4 determined in different ways (vide

infra Table 6-3).


145

Table 6-2: The sound velocities and Debye temperatures at 0 K for MgSiN2, AlN and Si3N4 (*:

assuming that ρ 0 = ρ 293 lacking the availability of low temperature data).

Compound E0 ρ0 ν vl vt vs s θ0

[GPa] [kg m-3] [-] [m s-1] [m s-1] [m s-1] [-] [K]

MgSiN2 281 3.142 103 [30] 0.246 1.033 104 5.99 103 6.65⋅103 4 900AlN 314 3.258 103 [31] 0.245 [14] 1.071 104 6.22 103 6.90 103 2 940Si3N4 320 3.202 103 [32]* 0.267 [18] 1.115 104 6.28 103 7.00 103 7 955

3.2.2. B and T0

Anderson [19] quantified the suggestion of Wachtman that the fitting parameters B

and T0 are related to the Grüneisen parameter and Debye temperature, respectively.

Using the equation of Anderson [19] and assuming that the Poisson's ratio is

temperature independent (dν/dT = 0) we can calculate B and T0 using:

0

3)21(3VR - B δγν= (9)

The parameter T0 is very approximately given as:

≈0T θ0/2 (10)

in which ν [-] is the Poisson's ratio, R (8.314 J mol-1 K-1) the gas constant, γ [-] the

Grüneisen constant [33], δ [-] the Anderson-Grüneisen constant [34] and V0

[m3 mol-1] the specific volume per atom at absolute zero. Using the expressions for

γ and δ [19] the equation for B can be written as:

TE

CRs Bp ∂

∂= 3 (11)

in which s [-] is the number of atoms per formula unit and Cp [J mol-1 K-1] the heat

capacity at constant pressure. It is directly clear that for calculating the value of the

fitting parameter B to describe the temperature dependence of the Young's

modulus these data themselves are needed. So, this makes an independent

evaluation of the fitting parameter B from the present data impossible.

Chapter 6.

146

For comparison with other compounds only few experimental data are

available. The experimentally observed value for the fitting parameter B of

0.0220 GPa K-1 for MgSiN2, 0.0244 GPa K-1 for AlN and 0.0151 GPa K-1 for Si3N4

are somewhat lower than the values reported for the three oxides investigated by

Wachtman (0.048 GPa K-1 for Al2O3 [17], 0.027 GPa K-1 for ThO2 [17] and

0.037 GPa K-1 for MgO [19]). The somewhat smaller value for B indicates that the

nitrides show a more harmonic bond character as compared to the oxides, as

expected from the more covalent nature of nitrides, and considering that the

Young's modulus of a fully harmonic bond is temperature independent.

Table 6-3: The Debye temperature θ 0 and θ T0 of MgSiN2, AlN and Si3N4 obtained from

the fitting parameters E0 and T0, respectively as compared to previously

reported values obtained from specific heat measurements (θ C), lattice

dynamic calculations (θ LD) and elastic constants (θ E).

Compound θ0 θ T0 θ C θ E θ LD

[K] [K] [K] [K] [K]

MgSiN2 900 778 829 [35] 827 [12]AlN 940 990 950 [1], 1010 [36] 800 [37]Si3N4 955 890 754 [38], 900 [39] 900 -1005 [40]

For MgSiN2, AlN and Si3N4 the experimentally obtained average T0 value

(see Table 6-1) was used to estimate the Debye temperature, resulting in 778, 990

and 890 K for MgSiN2, AlN and Si3N4, respectively. These θ T0 values are in rough

agreement with the θ0 value (obtained from E0) and the other reported Debye

temperatures for MgSiN2, AlN and Si3N4, respectively (see Table 6-3), indicating

the approximate nature of the Anderson equation θ0 ≈ 2T0 [19].

4. Conclusions

The temperature dependence of the Young's modulus of MgSiN2, AlN and Si3N4

can be described very well with E = E0 - B·T exp (-T0/T ). The Debye temperatures


147

estimated from E0 and T0 are in rough agreement with each other, and previously

reported values obtained in different ways. The values of the fitting parameter B

determined for our nitrides are lower than those previously reported for oxides.

This is ascribed to the more harmonic nature of bonds in nitrides as compared to

oxides resulting in a relatively temperature independent Young's modulus.

References


Chem. Solids 34 (1973) 321.







1994) 239.




Hausner, G.L. Messing and S. Hirano (The American Ceramic Society, 1995)

585 .





1995) 289.


Diffusivity/Conductivity of MgSiN2 Ceramics, Thermal Conductivity 24 and

Thermal Expansion 12, Pittsburgh, Pennsylvania, USA, October 1997, edited

Chapter 6.

148


Lancaster, Pennsylvania 1999) 3.






published.

9. G. Leibfried and E. Schlömann, Wärmeleitung in elektrisch isolierenden

Kristallen, Nach. Akad. Wiss. Göttingen, Math. Phys. Klasse 4 (1954) 71.

10. C.L. Julian, Theory of Heat Conduction in Rare-Gas Crystals, Phys. Rev. 137

(1965) A128.




12. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)-

Nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of

Properties and Performances by Improved Design and Microstructural

Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo

(Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405.

13. P. Debye, Zur Theorie der spezifischen Wärmen, Ann. Physik 39 (1912) 789.

14. P. Boch, J.C. Glandus, J. Jarrige, J.P. Lecompte and J. Mexmain, Sintering,


Ceram. Int. 8 (1982) 34.

15. D. Gerlich, S.L. Dole and G.A. Slack, Elastic Properties of Aluminum Nitride,


16. A.A. Khan, J.C. Labbe, Aluminium Nitride - Molybdenum Ceramic Matrix

Composites. Influence of Molybdenum Addition on Electrical, Mechanical

and Thermal Properties, J. Eur. Ceram. Soc. 17 (1997) 1885.


149

17. J.B. Wachtman, Jr., W.E. Tefft, D.G. Lam, Jr., and C.S. Apstein, Exponential

Temperature Dependence of Young's Modulus for Several Oxides, Phys. Rev.

122 (1961) 1754.


Phys. 20 (1981) 1751.

19. O.L. Anderson, Derivation of Wachtman's Equation for the Temperature

Dependence of Elastic Moduli of Oxide Compounds, Phys. Rev. 144

(1966) 553.



21. see for example E. Schreiber, O.L. Anderson and N. Soga, Elastic Constants

and Their Measurement (McGraw-Hill, Inc., 1973).



(1988) 973.

23. S. Spinner, T.W. Reichard and W.E. Tefft, A Comparison of Experimental

and Theoretical Relations Between Young's Modulus and the Flexural and

Longitudinal Resonance Frequencies of Uniform Bars, Journal of Research,

National Bureau of Standards 64A (1960) 147.

24. S. Spinner and W.E. Tefft, A Method for Determining Mechanical Resonance

Frequencies and for Calculating Elastic Moduli from these Frequencies,

Proceedings ASTM 61 (1961) 1221.

25. G. Woetting, G. Leimer, H.A. Lindner and E. Gugel, Silicon Nitride Materials

and Components for Industrial Application, Industrial Ceramics 15

(1995) 191.

26. D.-S. Park, H.-D. Kim, S.-Y. Lee and S. Kim, Gas Pressure Sintering of

Si3N4-Based Particulate Composites and their Wear Behavior, Key Eng.

Mater. 89 - 91 (1994) 439.

27. R.H. Fowler and E.A. Guggenheim, Statistical Thermodynamics (Cambridge

University Press, 1939), p. 126.

Chapter 6.

150

28. O.L. Anderson, A Simplified Method for Calculating the Debye Temperature

from Elastic Constants, J. Phys. Chem. Solids 24 (1963) 909.

29. D. Singh and Y.P. Varshni, Debye temperatures for hexagonal crystals, Phys.

Rev. B 24 (1981) 4340.





Soc. Symp. Proc. 482, Nitride Semiconductors, Boston (Massachusetts, USA),

December 1-5, 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S.

Nakamura and S. Strite (Materials Research Society, Warrendale,




74 (1975) 49.

33. E. Grüneisen, Zustand des festen Körpers, Handbuch der Physik 10, edited by

H. Geiger and K. Scheel (Springer, Berlin, Germany, 1926) 1.

34. Y.A. Chang, On the Temperature Dependence of the Bulk Modulus and the

Anderson-Grüneisen Parameter δ of Oxide Compounds, J. Phys. Chem. Solids

28 (1967) 697.


Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998)

7871.

36. V.I. Koshchenko, Ya. Kh. Grinberg and A.F. Demidenko, Thermodynamic

Properties of AlN (5 - 2700 ºK), GaP (5 - 1500 ºK), and BP (5 - 800 ºK),

Inorg. Mater. 20 (1984) 1550 (Translated from Izv. Akad. Nauk SSSR, Neorg.

Mater. 20 (1984) 1787).

37. J.C. Nipko and C.-K. Loong, Phonon Excitations and Related Thermal

Properties of Aluminum Nitride, Phys. Rev. B 57 (1998) 10550.

38. I.Ya. Guzman, A.F. Demidenko, V.I. Koshchenko, M.S. Fraifel'd and Yu. V.

Egner, Specific Heats and Thermodynamic Functions of Si3N4 and Si2ON2,


151

Inorg. Mater. 12 (1976) 1546 (Translated from Izv. Akad. Nauk SSSR, Neorg.

Mater. 12 (1976) 1879).

39. Personal communication J.C. van Miltenburg (unpublished results).

40. K. Watari, Y. Seki and K. Ishizaki, Temperature Dependence of Thermal

Coefficients for HIPped Sintered Silicon Nitride, J. Ceram. Soc. Jpn. Inter.

Ed. 97 (1989) 170.

Chapter 6.

152

153

Chapter 7.

The Grüneisen parameters of MgSiN2, AlN and

ββββ-Si3N4

Abstract

The temperature dependence of the Grüneisen parameter of MgSiN2 (80 - 1600 K),

AlN (90 - 1600 K), and β-Si3N4 (300 - 1300 K) was evaluated from thermal

expansion, elastic constants and heat capacity data of these materials. For all

compounds the Grüneisen parameter increases as a function of the reduced

temperature approaching a constant value at high temperatures (T /θ ≥ 0.8). The

high temperature limit of the Grüneisen parameter of the wurtzite type materials

MgSiN2 and AlN is about the same (0.98 and 0.95, respectively) whereas these are

much higher than that of the phenakite β-Si3N4 (0.63). This behaviour can be

understood quantitatively from the relation between the Grüneisen parameter and

the bond parameter W as established by Slack.

1. Introduction

The ternary adamantine type compound MgSiN2, which can be deduced from the

well known AlN structure by replacing two Al3+ ions systematically by one Mg2+

and one Si4+ ion, is considered a potentially interesting material because of its

mechanical and thermal properties [1 - 9]. AlN is known for its high intrinsic

thermal conductivity (320 W m-1 K-1 [1, 10] at 300 K) and therefore intensively

studied. Recently a very high thermal conductivity was reported for β-Si3N4

ceramics (> 100 W m-1 K-1 at 300 K) [11 - 16]. In order to understand the thermal

conductivity of these nitrides it is important to obtain more insight in the (thermal)

Chapter 7.

154

properties of these materials. Furthermore, a better understanding of the thermal

properties might result in more reliable models to predict the intrinsic thermal

conductivity of new potentially interesting materials.

An important parameter for the theoretical prediction of the intrinsic thermal

conductivity κ is the Grüneisen parameter at the Debye temperature γθ [-] [17 - 20]

as:

322

3

n

M

θγ

θδκ ∝ (1)

in which M [kg mol-1] is the mean atomic mass, δ 3 [m3] the average volume of

one atom in the primitive unit cell, θ [K] the Debye temperature, γθ [-] the

Grüneisen parameter at T = θ and n [-] the number of atoms per primitive unit

cell. Using the procedure as described by Slack [17] the Grüneisen parameter can

be evaluated from thermodynamic properties resulting in the so-called

thermodynamic Grüneisen parameter [21]:

==

VTpS CV

CV

βα

βα

γ mlatmlat 33(2)

in which α lat [K-1] is the lattice linear thermal expansion coefficient, Vm [m3 mol-1]

the molar volume, βS [Pa-1] the adiabatic compressibility, Cp [J mol-1 K-1] the heat

capacity at constant pressure, βT [Pa-1] the isothermal compressibility and CV

[J mol-1 K-1] the heat capacity at constant volume. The Grüneisen parameter, which

is related to the anharmonicity of the crystal structure, within the quasi-harmonic

approximation [22] is temperature independent. Therefore, the Grüneisen

parameter is often calculated at a temperature (in most cases room temperature) for

which the thermal expansion coefficient, molar volume, compressibility and heat

capacity are known. The resulting Grüneisen parameter is then assumed to equal

the Grüneisen parameter at the Debye temperature γθ. However, the few

experimental data that are available show that the Grüneisen parameter is a

function of temperature [23 - 26] and only becomes constant at high temperatures

[27]. Therefore it is essential to evaluate γ as a function of the temperature in order

The Grüneisen parameters of MgSiN2, AlN and ββββ -Si3N4

155

to investigate the temperature range for which the Grüneisen parameter can be

considered as a constant.

As far as the present author knows only one value for the Grüneisen

parameter of MgSiN2 was reported (viz. γθ = 0.78 with θ = 827 K [18]). For AlN

three values for γ are reported of which one deviates substantially from the others

viz. 0.71 (at T = θ = 941 K) [18], 0.77 (at T = θ = 950 K) [10] and 0.45 (room

temperature) [28]. The Grüneisen parameter of β-Si3N4 as a function of the

absolute temperature (300 - 1300 K) was already reported by Slack [26] and at T =

θ = 1140 K, γθ = 0.72. However, lacking the availability of reliable elastic

constants (Young's modulus E [GPa] and Poisson's ratio ν [-]) as a function of the

temperature,

an estimate of the temperature dependence of the compressibility

(-∂lnβS-1/∂T = 1.0 10-4 K-1) was used. For MgSiN2 all data required for evaluation

of the Grüneisen parameter as a function of the temperature have been previously

reported (thermal expansion [9, 29] providing also the values for the molar volume

Vm, elastic constants [30] resulting in values for the adiabatic compressibility βS,

and heat capacity [31]). For AlN thermal expansion and heat capacity data are

known from literature (αlat [32 - 35] and Cp [36 - 42]) and recently elastic constants

as a function of the temperature were reported [30]. In the meantime, for β-Si3N4

the temperature dependence of the elastic constants has been determined [43] and

modelled [30], and in the literature data for the thermal expansion coefficient

[44, 45] and specific heat are reported [40 - 48]. So, for MgSiN2, AlN and β-Si3N4

all required input parameters as a function of the temperature are available for

evaluation of the temperature dependence of the Grüneisen parameter.

In this chapter the experimental determination of the thermodynamic

Grüneisen parameter of MgSiN2, AlN and β-Si3N4 is reported. The choice of the

used input parameters is briefly discussed and using my best judgement some

irregularities were smoothed in order to obtain the most reliable data.

Chapter 7.

156

2. Evaluation of the input parameters

2.1. Lattice linear thermal expansion coefficient αααα lat

For MgSiN2 the lattice linear thermal expansion coefficient α lat [K-1] at T < 300 K

was taken from a previous neutron diffraction study [29] concerning the

temperature dependence of the lattice parameters. From 300 to 1573 K the linear

thermal expansion coefficient α lin of fully dense MgSiN2 ceramics was measured

with a dual rod dilatometer in nitrogen using Al2O3 (sapphire) as a reference

material. More experimental details are given elsewhere [9]. The length of the

ceramic rod as a function of the absolute temperature l(T ) [m] was fitted by a

polynomial l(T ) = A0 + A1·T + A2·T 2 + A3·T 3 + A4·T 4 + A5·T 5 (see Table 7-1).

Statistical F-testing with a 95% confidence interval of the variance ratios showed

that introduction of higher order terms to the polynomial did not improve the fit.

The linear thermal expansion coefficient, α lin [K-1], for T ≥ 300 K was calculated

from:

α lin(T ) = (dl(T )/dT )/l(T ) (3)

Table 7-1: The coefficients and statistics used for describing the dilatometer

MgSiN2 sample length l [mm] as a function of the absolute

temperature T [K] between 300 and 1573 K. Between brackets

the 95 % confidence intervals are given.

Coefficient Statistics

A0

A1

A2

A3

A4

A5

9.997729(1) -1.3375(66) 10-5

8.1942(16) 10-8

3.5418(19) 10-11

1.510(10) 10-15

3.9463(22) 10-18

Data pointsR2

χ2

ToleranceConfidence

999 0.99999938 7.15 10-14

0.0005 0.95

First the thermal expansion coefficient increases with temperature (Fig. 7-1)

to become almost constant at about 1000 K, and subsequently it increases again

with temperature. In order to obtain a smooth thermal expansion curve around


157

room temperature, the values between 250 K (measured by neutron diffraction) and

300 K (measured by dilatometry) were obtained by interpolation.

The further increase of the thermal expansion coefficient above 1000 K was

ascribed to the thermal generation of vacancies in the MgSiN2 crystal structure [9]

causing macroscopic length changes for constant unit cell size, which are measured

with dilatometry. Until now this point was only briefly discussed [9]. For

evaluation of the Grüneisen parameter discrimination between the lattice unit cell

and vacancy thermal expansion is required.

In order to do this the macroscopic linear thermal expansion coefficient α lin

for T ≥ 750 K was separated in a lattice contribution α lat and a vacancy

contribution αvac using the following expression:

vaclatlin ααα += (4)

with T )/(F Dlat θαα ∞= and

=

kTQ-

kTCQ vv exp3 2vacα

-2.0E-06

0.0E+00

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

0 500 1000 1500 2000T [K]

α [K

-1]

MgSiN2

AlNβ -Si3N4

Fig. 7-1: The linear thermal expansion coefficient (α ) of MgSiN2,

AlN and β -Si3N4 ceramics as a function of the absolute

temperature (T ). The dots indicate the calculated values and

the solid lines are drawn as a guide to the eye. The dashed

line represents the measured data for MgSiN2.

Chapter 7.

158

in which α∞ [K-1] is the high temperature limit of the lattice linear thermal

expansion coefficient, F(θD/T ) [-] the normalised Debye function (so, for T >> θD

applies F(θD/T ) = 1), θD [K] the Debye temperature, C [-] a constant, Qv [eV] the

energy required for the formation of a vacancy and k is the Boltzmann's constant

(8.62 10-5 eV K-1). An excellent fit of the data is obtained (Table 7-2) with

reasonable values for α∞ (7.502 10-6 K-1), θD (1248 K), C (19.2) and Qv (1.365 eV).

For comparison C and Qv have typical values of 101 - 103 (estimated from [49]) and

1 eV [50], respectively. This indicates that the increase of the macroscopic thermal

expansion coefficient above 1000 K can indeed be ascribed to thermal generation

of vacancies in the MgSiN2 lattice. It is noted that at 750 K the contribution of

vacancies to the thermal expansion coefficient is negligibly small.

Table 7-2: The values for α∞, θ D, C and Qv and statistics used for

describing the linear thermal expansion coefficient α lin [K-1]

for MgSiN2 as a function of the absolute temperature T [K].

Between brackets the 95 % confidence intervals are given.

Parameter Statistics

α∞ [K-1]θD [K]C [-]Qv [eV]

7.502(3) 10-6

1248(3) 19.2(4) 1.365(3)

Data pointsR2

χ 2

ToleranceConfidence

182 0.9999 2.8833 10-17

0.0005 0.95

For AlN [32 - 35] and β-Si3N4 [44, 45] several thermal expansion data sets

are reported. After evaluation, for AlN the thermal expansion data between

0 - 2000 K reported by Wang [35] based on lattice parameters were used. For

β-Si3N4 the thermal expansion coefficient between 293 - 1300 K was calculated

from the dilatometer data of Huseby [45]. Also for AlN and β-Si3N4 the lattice

thermal expansion coefficient increases as a function of the temperature

approaching a constant value at high temperatures (Fig. 7-1).


159

2.2. Molar volume Vm

The molar volume Vm of MgSiN2, AlN and β-Si3N4 as a function of the

temperature was evaluated from literature data of the molar volume at temperature

T (Vm T) and the lattice linear thermal expansion (α l a t ) using:

∫=*

*T

TT

T T VV

d3ln latm

m α (5)

For MgSiN2 Vm0 = 2.559 10-5 m3 mol-1 [29], for AlN Vm0 = 1.258 10-5 m3 mol-1 [35]

and for β-Si3N4 Vm293 = 4.381 10-5 m3 mol-1 [44]. Since for β-Si3N4 thermal

expansion data are reported for room temperature and above the molar volume for

β-Si3N4 could be calculated only in the same temperature region.

2.3. Adiabatic compressibility ββββS

For evaluation of the adiabatic compressibility βS elastic constants are needed. The

temperature dependence of the Young's modulus of MgSiN2 and AlN between

293 - 973 K [30] has been measured with the impulse excitation method [51] and

for β-Si3N4 literature data (293 - 1223 K) [43] were used. These Young's modulus

data obtained for MgSiN2 and AlN, and the literature data for β-Si3N4, can be very

well described [30] using the empirical formula [52]:

)/(exp 00 T-TTB-EE ⋅= (6)

in which E0 [GPa] is the Young's modulus at 0 K, and B [GPa K-1] and T0 [K] are

fitting parameters. For all temperatures this expression was used to describe the

Young's modulus of MgSiN2 (E0 = 281 GPa, B = 0.0220 GPa K-1 and T0 = 389 K),

AlN (E0 = 314 GPa, B = 0.0244 GPa K-1 and T0 = 495 K) and β-Si3N4

(E0 = 320 GPa, B = 0.0151 GPa K-1 and T0 = 445 K). The Poisson's ratio for these

materials was assumed to be constant (for MgSiN2: ν = 0.246 [30], AlN: ν = 0.245

[53] and β-Si3N4: ν = 0.267 [43] at room temperature).

Chapter 7.

160

For all temperatures (80 - 1600 K: MgSiN2, 90 - 1600 K: AlN and

300 - 1300 K β-Si3N4) the adiabatic compressibility βS [Pa-1] was calculated from

the Young's modulus and the Poisson's ratio using:

E- S

)21(3 νβ = (7)

The resulting adiabatic compressibility βS of MgSiN2, AlN and β-Si3N4 shows a

slight increase as a function of the absolute temperature (Fig. 7-2).

2.4. Heat capacity at constant pressure Cp

Below 400 K the heat capacity at constant pressure Cp has been previously

determined for MgSiN2 (8 - 400 K) [31], AlN and Si3N4 (20 - 400 K) [42]) with

adiabatic calorimetry (CAL V [54]). The Cp of MgSiN2 was measured on pure

(oxygen content < 0.1 wt. %), iso-statically pressed powder compacts (~ 35 % of

theoretical density), that of AlN on crushed hot-pressed pellets (fully dense)

prepared from Dow Chemical Aluminum Nitride Powder - XUS 35560, and that of

Si3N4 (Ube SN-E10) on crushed slip cast compacts sintered at high temperature in

4.0E-12

4.5E-12

5.0E-12

5.5E-12

6.0E-12

6.5E-12

0 500 1000 1500 2000T [K]

β S [P

a-1]

MgSiN2

AlNβ-Si3N4

Fig. 7-2: The adiabatic compressibility (βS) of MgSiN2, AlN and

β -Si3N4 as a function of the absolute temperature (T ). The

dots indicate the calculated values and the lines are drawn

as a guide to the eye.


161

N2 pressure furnace (~ 50 % of theoretical density). The sample was put into a

sample holder (copper vessel) and sealed using He gas as a heat exchanger. After

cooling of the sample and sample holder to the lowest measuring temperature,

stepwise a known quantity of energy was added and the temperature increase was

measured.

For low temperatures (T ≤ 350 K) these experimental Cp data for MgSiN2,

AlN and β-Si3N4 were used. For T ≥ 350 K the Cp for MgSiN2, AlN and β-Si3N4

was calculated with the Debye expression for the heat capacity at constant volume

CV [55, 56] and the expression for Cp - CV :

TVp

TV C-C

βα 2

latm9= (8)

The isothermal compressibility βT that was needed for obtaining Cp - CV was

calculated from the adiabatic compressibility βS with:

pST C

TV

2latm9 α

ββ += (9)

The Debye temperatures (θ∞) needed to describe the CV were evaluated using the

same procedure as previously described for MgSiN2 [31] resulting in 996 K, 989 K

and 1200 K for MgSiN2, AlN and β-Si3N4, respectively.

The Cp data calculated for MgSiN2 are in good agreement with

experimentally observed data (300 - 800 K), measured with differential scanning

calorimetry [31]. For AlN and Si3N4 the heat capacity is reported in a broad

temperature range (AlN: 5 - 2700 K [36]; 291 - 577 K [37]; 300 - 773 K [38];

0 - 800 K [39]; and β-Si3N4: 55 - 310 K [46]; 100 - 1250 K [47]; 0 - 680 K [48])

and presented in standard thermodynamic handbooks (AlN: 298 - 2000 K [40];

0 - 3000 K [41]; and Si3N4: 298 - 2200 K [40]; 298 - 3000 K [41]). No large

discrepancy was observed between the several heat capacity data of AlN and the

here presented experimental and calculated data. The data of β-Si3N4 deviate

substantially from the data presented in the standard thermodynamic handbooks

[40, 41] but agree very well with the data presented by Guzman [46], Watari [47]

Chapter 7.

162

and Rocabois [48]. The resulting Cp curves of MgSiN2, AlN and β-Si3N4 show the

expected S shaped increase with the absolute temperature (Fig. 7-3).

3. Evaluation of the Grüneisen parameter γγγγ

The Grüneisen parameters of MgSiN2 between 80 and 1600 K, AlN between

90 and 1600 K, and β-Si3N4 between 300 and 1300 K were calculated from α lat,

Vm, βS and Cp. The Grüneisen parameter of all three materials roughly has the same

temperature dependence, showing an increase of the Grüneisen parameter with

increasing temperature approaching a constant value at high temperatures

(Fig. 7-4). For all three materials the reliability of the absolute value of the

Grüneisen parameter was estimated to be within 10 % for T ≥ 300 K and 15 - 20 %

for T < 300 K considering the accuracy of the used input parameters. However, the

internal consistency of the resulting Grüneisen parameter is considered to be within

5 % providing a good indication of the true temperature dependence.

For MgSiN2 the Grüneisen parameter shows a minimum at about 125 K,

subsequently it significantly increases till 300 K and finally it becomes constant for

0

30

60

90

120

150

180

0 500 1000 1500 2000T [K]

Cp [

J m

ol-1

K-1

]

β-Si3N4

MgSiN2

AlN

Fig. 7-3: The heat capacity at constant pressure (Cp) of MgSiN2,

AlN and β -Si3N4 as a function of the absolute

temperature (T ). The dots indicate the calculated

values and the lines are drawn as a guide to the eye.


163

T ≥ 600 K having a value of 0.96 ± 0.02 (Table 7-3 and Fig. 7-4). The high

temperature limit is significantly higher than the previously reported value of 0.78

[18]. This difference is mainly caused by the too high input value used for the

isothermal compressibility that originates from a too low value measured for the

Young's modulus (see also [30]).

Table 7-3: The lattice linear thermal expansion coefficient α lat, molar volume Vm,

the heat capacity at constant pressure Cp, adiabatic compressibility βS,

and the Grüneisen parameter γ for MgSiN2 at several temperatures.

T[K]

α lat

[K-1]Vm

[m3 mol-1]Cp

[J mol-1 K-1]βS

[Pa-1]γ

[-] 80 100 110 120 130 140 150 200 250

0.426 10-6

0.617 10-6

0.725 10-6

0.842 10-6

0.967 10-6

1.10 10-6

1.24 10-6

2.08 10-6

3.13 10-6

2.5590 10-5

2.5591 10-5

2.5592 10-5

2.5592 10-5

2.5593 10-5

2.5594 10-5

2.5595 10-5

2.5601 10-5

2.5611 10-5

8.5913.9116.6519.4822.3425.1828.0041.2052.42

5.421 10-12

5.422 10-12

5.422 10-12

5.423 10-12

5.424 10-12

5.425 10-12

5.426 10-12

5.433 10-12

5.444 10-12

0.7010.6280.6170.6120.6130.6190.6280.7140.843

300 350 400 450 500 550 600 650 700 800 9001000120014001600

3.82 10-6

4.30 10-6

4.73 10-6

5.11 10-6

5.44 10-6

5.74 10-6

5.99 10-6

6.20 10-6

6.38 10-6

6.65 10-6

6.83 10-6

6.95 10-6

7.11 10-6

7.22 10-6

7.27 10-6

2.5625 10-5

2.5641 10-5

2.5658 10-5

2.5677 10-5

2.5697 10-5

2.5719 10-5

2.5742 10-5

2.5765 10-5

2.5789 10-5

2.5840 10-5

2.5892 10-5

2.5946 10-5

2.6057 10-5

2.6174 10-5

2.6307 10-5

61.7169.0675.0479.5783.0785.8588.0889.9291.4593.8595.6396.9998.88

100.05100.72

5.456 10-12

5.470 10-12

5.486 10-12

5.503 10-12

5.520 10-12

5.539 10-12

5.558 10-12

5.577 10-12

5.597 10-12

5.638 10-12

5.681 10-12

5.725 10-12

5.816 10-12

5.936 10-12

6.012 10-12

0.8730.8760.8840.8990.9150.9310.9450.9560.9650.9750.9770.9750.9660.9580.947

Chapter 7.

164

For AlN the Grüneisen parameter is negative at 100 K and shows a steep increase

till 200 K, subsequently it slowly increases till 700 K and finally for higher

temperatures T ≥ 700 K it is about constant equalling 0.95 ± 0.02 (Table 7-4 and

Fig. 7-4). The Grüneisen parameters determined in this work for AlN (0.95 at

T = 940 - 950 K and 0.70 at T = 300 K) deviates significantly from the value of

0.71 at 941 K previously reported by de With et al. [18], and the values reported by

Slack et al. of 0.77 at 950 K [10] and 0.45 at room temperature [28]. The value of

γ = 0.71 [18] was based on a mistake taking α lat = 4.8 10-6 K-1 instead of

α lat = 5.9 10-6 K-1 at T = θ as reported in his reference for the thermal expansion

coefficient [33].

Correction results in γ = 0.87 which is in much closer agreement with the value of

0.95 determined in this work. The low values of 0.77 [10] and 0.45 [28] can be

ascribed to the too low Poisson's ratio (ν = 0.179 [28]) reported by Slack resulting

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

300 600 900 1200 1500 1800

T [K]

γ [-]

MgSiN2

AlNβ -Si3N4

Fig. 7-4: The Grüneisen parameter (γ ) of MgSiN2, AlN and β -Si3N4

as a function of the absolute temperature (T ). The dots

indicate the calculated values and the lines are drawn as a

guide to the eye.


165

in a too high value for βS, and consequently, in a too small Grüneisen parameter.

Taking a more realistic value for the Poisson's ratio (ν = 0.245 [53, 57]) results in

γ = 0.99 at 950 K and γ = 0.58 at room temperature, which is in far better

agreement with our value of 0.95 at 950 K and 0.70 at 300 K.

Table 7-4: The lattice linear thermal expansion coefficient α lat, molar volume Vm, the

heat capacity at constant pressure Cp, adiabatic compressibility βS, and the

Grüneisen parameter γ for AlN at several temperatures.

T[K]

α lat

[K-1]Vm

[m3 mol-1]Cp

[J mol-1 K-1]βS

[Pa-1]γ

[-] 90 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 9001000110012001300140015001600

-0.140 10-6

-0.009 10-6

0.84 10-6

1.62 10-6

2.24 10-6

2.77 10-6

3.25 10-6

3.68 10-6

4.07 10-6

4.42 10-6

4.72 10-6

4.98 10-6

5.21 10-6

5.41 10-6

5.58 10-6

5.73 10-6

5.98 10-6

6.17 10-6

6.32 10-6

6.43 10-6

6.53 10-6

6.60 10-6

6.67 10-6

6.72 10-6

1.2580 10-5

1.2580 10-5

1.2580 10-5

1.2583 10-5

1.2586 10-5

1.2591 10-5

1.2597 10-5

1.2603 10-5

1.2611 10-5

1.2619 10-5

1.2627 10-5

1.2637 10-5

1.2646 10-5

1.2656 10-5

1.2667 10-5

1.2677 10-5

1.2700 10-5

1.2723 10-5

1.2747 10-5

1.2772 10-5

1.2796 10-5

1.2821 10-5

1.2846 10-5

1.2872 10-5

4.52 5.7712.5719.5025.5830.6434.6637.6839.8341.5642.9244.0144.9245.6746.3046.8547.7248.4048.9349.3449.6749.9350.1250.26

4.873 10-12

4.873 10-12

4.875 10-12

4.879 10-12

4.886 10-12

4.895 10-12

4.905 10-12

4.917 10-12

4.930 10-12

4.944 10-12

4.959 10-12

4.974 10-12

4.990 10-12

5.007 10-12

5.024 10-12

5.041 10-12

5.077 10-12

5.115 10-12

5.153 10-12

5.193 10-12

5.234 10-12

5.276 10-12

5.318 10-12

5.362 10-12

-0.241-0.012 0.520 0.643 0.677 0.698 0.723 0.753 0.785 0.814 0.840 0.863 0.882 0.899 0.912 0.923 0.940 0.951 0.958 0.962 0.964 0.964 0.964 0.963

Chapter 7.

166

Table 7-5: The lattice linear thermal expansion coefficient α lat, molar volume Vm,

the heat capacity at constant pressure Cp, adiabatic compressibility βS,

and the Grüneisen parameter γ for β -Si3N4 at several temperatures.

T[K]

α lat

[K-1]Vm

[m3 mol-1]Cp

[J mol-1 K-1]βS

[Pa-1]γ

[-] 300 350 400 450 500 550 600 650 700 750 800 9001000110012001300

1.19 10-6

1.59 10-6

1.94 10-6

2.23 10-6

2.47 10-6

2.66 10-6

2.83 10-6

2.96 10-6

3.07 10-6

3.17 10-6

3.25 10-6

3.37 10-6

3.47 10-6

3.54 10-6

3.59 10-6

3.64 10-6

4.3807 10-5

4.3816 10-5

4.3828 10-5

4.3842 10-5

4.3857 10-5

4.3874 10-5

4.3892 10-5

4.3911 10-5

4.3931 10-5

4.3952 10-5

4.3973 10-5

4.4016 10-5

4.4061 10-5

4.4108 10-5

4.4155 10-5

4.4203 10-5

90.68104.93115.91125.77133.48139.59144.52148.54151.88154.67157.04160.82163.67165.88167.62169.01

4.377 10-12

4.383 10-12

4.390 10-12

4.398 10-12

4.406 10-12

4.414 10-12

4.423 10-12

4.432 10-12

4.441 10-12

4.450 10-12

4.460 10-12

4.479 10-12

4.499 10-12

4.519 10-12

4.540 10-12

4.561 10-12

0.3930.4540.5000.5290.5520.5690.5820.5920.6000.6070.6120.6180.6220.6240.6250.626

For β-Si3N4 the Grüneisen parameter increases up to 700 K whereas above

this temperature it is about constant and equals 0.63 ± 0.02 (Table 7-5 and

Fig. 7-4). The observed temperature dependence of the Grüneisen parameter is in

good agreement with the previously reported temperature dependence [26]. Also

the observed Grüneisen parameter at T = θ∞ = 1200 K of 0.63 is in reasonable

agreement with the previously reported value of 0.72 [26].

4. Discussion

4.1. The temperature dependence of the Grüneisen parameter

For comparison, the reduced Grüneisen parameter (γ /γθ) as a function of the

reduced temperature (T /θ ) is presented in Fig. 7-5 for MgSiN2, AlN and β-Si3N4.


167

More or less a similar dependence of γ /γθ on the reduced temperature is found

indicating its universal behaviour, as it has also been observed for Ar [58], ZnTe

and RbI [24], Ge and Si [59], and several phenakites (Be2SiO4, Zn2SiO4 and

α-LiAlSiO4) [26].

The Grüneisen parameters of MgSiN2, AlN and β-Si3N4 are almost constant

for T /θ ≥ 0.8, at lower reduced temperatures (0.3 ≤ T /θ ≤ 0.8) it slowly decreases

followed by a faster decrease below about T /θ ≈ 0.3, and finally at still lower

temperatures (T /θ ≤ 0.15) different temperature dependencies are observed

(Fig. 7-5). For MgSiN2 the Grüneisen parameter shows a (positive) minimum value

of about 0.61 at T /θ ≈ 0.13. Also the Grüneisen parameter for AlN shows a

(negative) minimum at lower temperatures (0.03 < T /θ < 0.08) as the thermal

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.5 1.0 1.5 2.0

T/θ [-]

γ /γ θ

[-]

MgSiN2

AlNβ-Si3N4

Fig. 7-5: The reduced Grüneisen parameter (γ /γθ) of MgSiN2, AlN and β -

Si3N4 as a function of the reduced temperature (T /θ ), with γθ equal

to 0.98, 0.95 and 0.63 and θ equal to 996, 989 and 1200 K for

MgSiN2, AlN and Si3N4, respectively. The dots indicate the

calculated values and the lines are drawn as a guide to the eye.

Chapter 7.

168

expansion coefficient near 0 K has again a positive value. For several other

adamantine type compounds (Si, Ge, GaAs, GaSb, InAs and InSb) this (negative)

minimum in the Grüneisen parameter was also observed at about T /θ ≈ 0.04 - 0.07

[23]. It is important to note that the usual assumption that γθ = γ300 is not valid for

MgSiN2, AlN and β-Si3N4 (Fig. 7-5) as these materials have a high Debye

temperature (~ 1000 - 1200 K [30]).

Table 7-6: The average atomic mass M , density ρ, average volume per atom δ 3,

coordination number of the anions η, the average volume per anion bond W

and the Grüneisen parameter at T = θ γθ for several materials.

Material M[kg]

ρ

[kg m3]

δ 3

[Å3]

η

[-]

W

[Å3]

γθ

[-]

β-Si3N4

AlNMgSiN2

α-Al2O3

MgOThO2

0.0200410.0204950.020103

0.020390.020160.08801

3202 3256 3138

3986 [52] 3581 [52] 9991 [52]

10.636 [44]10.453 [35]10.391 [29]

8.494 9.34514.626

344

464

6.0565.2265.313

3.5393.1155.485

0.630.950.98

1.34 [52]1.52 [52]1.78 [52]

4.2. The absolute value of the Grüneisen parameter at the Debye

temperature

The Grüneisen parameters at the Debye temperature γθ for MgSiN2 (0.98) and AlN

(0.95) are about the same whereas for Si3N4 (0.63) it is considerably smaller

(Fig. 7-4). Furthermore, for MgSiN2 and AlN the absolute value of the Grüneisen

parameter as a function of the reduced temperature (T /θ ) is more or less similar for

T /θ ≥ 0.2 (Fig. 7-5). This is not surprising if we consider the similarity in crystal

structure, bond character, mean atomic volume and average mole mass per atom.


169

The differences between γθ for MgSiN2, AlN and Si3N4 can be quantitatively

explained in view of the empirical linear relation between the Grüneisen parameter

at T = θ (γθ) and the average volume per anion bond (W [Å3]) as found by Slack

[26]:

γθ = Γ∞[1 - W/W0] (10)

with Γ∞ [-] and W0 [m3] constants and the parameter W defined as:

W = stδ 3/saη (11)

in which st [-] is the number of atoms per formula unit, δ 3 [Å3] is the average

volume per atom, sa [-] the number of anions per molecule and η [-] the number of

bonds per anion. Using the data of Slack [26], some literature data for Al2O3, MgO

and ThO2 [52], and our data for MgSiN2, AlN and Si3N4 an updated plot of γθ

versus W was constructed (Table 7-6, Fig. 7-6). Again a linear relation is observed

between γθ and W (disregarding the point of ThO2) with Γ∞ = 2.11 and

W0 = 9.45 Å3 (dγθ /dW = -0.22 Å-3). The value of W0 equals the reported value by

Slack (W0 = 9.45 Å3 [26]). However, his reported values of Γ∞ = 2.91 and

ZnO

α-SiO2

β -SiAlONα-LiAlSiO4 Zn2SiO4

Be2SiO4

BeO

Zn2GeO4

β -SiO2

CdAl2O4

MgO

ThO2

α-Al2O3

MgSiN2AlN

β -Si3N4

0.0

0.5

1.0

1.5

2.0

2 4 6 8 10W [Å3]

γ θ [-

]

Slack [26]Wachtman [52]This work

Fig. 7-6: The Grüneisen parameter at T = θ, γθ versus the volume per anion

bond W for several materials.

Chapter 7.

170

dγθ /dW = -0.31 Å-3 appear to be erroneous as his figure of γθ versus W results in a

value for Γ∞ = 2.41 ± 0.05 and dγθ /dW = -0.26 ± 0.01 Å-3 which is close to the

values here presented. So, the general validity of the relation observed by Slack is

supported by the inclusion of the data for the oxides Al2O3 and MgO, and the

nitride materials MgSiN2, AlN and Si3N4. The point for ThO2 does most probably

not fit the relation due to the large ionic radius of Th resulting in an inverse

structure where the cation is larger than the anion providing a too high value for δ 3.

5. Conclusions

The reduced Grüneisen parameters (γ /γθ) of MgSiN2, AlN and β-Si3N4 show a

similar behaviour as a function of the reduced temperature (T /θ ). The Grüneisen

parameter increases as a function of the temperature approaching a constant value

at a reduced temperature of T /θ ≥ 0.8 indicating that the usual assumption that

γθ = γ300 is not valid for these materials as θ ≈ 1000 - 1200 K.

The absolute value of the Grüneisen parameter of AlN at the Debye

temperature (0.96) equals that of the structurally related MgSiN2 (0.98), whereas it

is much higher than that of Si3N4 (0.63). This can be quantitatively understood

from the relationship between the Grüneisen parameter and the average volume per

anion bond, established for other (most oxide) compounds.

References


Chem. Solids 34 (1973) 321.




Ceramics MgSiN2, Mat. Res. Soc. Symp. 327, Covalent Ceramics II: Non-



171


1994) 239.

4. H.T. Hintzen, P. Swaanen, R. Metselaar, W.A. Groen and M.J. Kraan, Hot-




Proc. Sci. Techn., Friedrichshafen, Germany, September 1994, edited by

H. Hausner, G.L. Messing and S. Hirano (The American Ceramic Society,

1995) 585.




edited by C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995)

289.




8. I.J. Davies, T. Shimazaki, M. Aizawa, H. Suemasu, A. Nozue and K. Itatani,

Physical Properties of Hot-Pressed Magnesium Silicon Nitride Compacts with

Yttrium Oxide Addition, Inorganic Materials 6 (1999) 276.


Preparation, Characterisation and Properties of MgSiN2 ceramics, to be

published.



11. K. Hirao, K. Watari, M.E. Brito, M. Toriyama and S. Kanzaki, High Thermal

Conductivity of Silicon Nitride with Anisotropic Microstructure, J. Am.

Ceram. Soc. 79 (1996) 2485.

12. N. Hirosaki, Y. Okamoto, M. Ando, F. Munakata and Y. Akimune, Thermal

Conductivity of Gas-Pressure-Sintered Silicon Nitride, J. Am. Ceram. Soc. 79

(1996) 2878.

Chapter 7.

172

13. N. Hirosaki, M. Ando, Y. Okamoto, F. Munakata, Y. Akimunde, K. Hirao, K.

Watari, M.E. Brito, M. Toriyama and S. Kanzaki, Effect of Alignment of

Large Grains on the Thermal Conductivity of Self-Reinforced β-Silicon

Nitride, J. Ceram. Soc. Jpn. Int. Ed. 104 (1996) 1187.

14. K. Watari, M.E. Brito, M. Toriyama, K. Ishizaki, S. Cao and K. Mori,

Thermal conductivity of Y2O3-doped Si3N4 ceramics at 4 to 1000 K, J. Mater.

Sci. Lett. 18 (1999) 865.

15. K. Watari, K. Hirao, M.E. Brito, M. Toriyama and S. Kanzaki, Hot Isostatic

Pressing to Increase Thermal Conductivity of Si3N4 Ceramics, J. Mater. Res.

14 (1999) 1538.

16. B. Li, L. Pottier, J.P. Roger, D. Fournier, K. Watari and K. Hirao, Measuring

the Anisotropic Thermal Diffusivity of Silicon Nitride Grains by

Thermoreflectance Microscopy, J. Eur. Ceram. Soc. 19 (1999) 1631.










Kristallen, Nachr. Akad. Wiss. Göttingen, Math. Physik. Kl. IIa (1954) 71.


Diffusivity/Conductivity of MgSiN2 Ceramics, ITCC 24 and ITES 12, 24th


Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 1997, edited


Lancaster, Pennsylvania, 1999) 3.

21. E. Grüneisen, Zustand des festen Körpers, Handbuch der Physik 10, edited by

H. Geiger and K. Scheel (Springer, Berlin, Germany, 1926), 1.


173

22. J.C. Slater, Introduction to Chemical Physics (McGraw-Hill, New York, 1939)

Chap. XIII.

23. P.W. Sparks and C.A. Swenson, Thermal Expansions from 2 to 40 ºK of Ge,

Si, and Four III-V Compounds, Phys. Rev. 163 (1967) 779.

24. J.F. Vetelino, K.V. Namjoshi and S.S. Mitra, Mode-Grüneisen Parameters and

Thermal-Expansion Coefficient of NaCl, CsCl, and Zinc-Blende-Type

Crystals, J. Appl. Phys. 41 (1970) 5141.

25. J.F. Vetelino, S.S. Mitra and K.V. Namjoshi, Lattice Dynamics of ZnTe:

Phonon Dispersion, Multiphonon Infrared Spectrum, Mode Grüneisen

Parameters, and Thermal Expansion, Phys. Rev. B 2 (1970) 967.

26. G.A. Slack and I.C. Huseby, Thermal Grüneisen parameters of CdAl2O4,

β -Si3N4, and other phenacite-type compounds, J. Appl. Phys. 53 (1982) 6817.

27. W.B. Daniels, The anomalous thermal expansion of germanium, silicon and

compounds crystallizing in the zinc blende structure, International Conference

on the Physics of Semiconductors, Exeter, UK, July 1962, edited by A.C.

Stickland (Institute of Physics, London, 1962) 482.

28. D. Gerlich, S.L. Dole and G.A. Slack, Elastic Properties of Aluminum Nitride,




neutron diffraction, J. Phys. Chem. Solids. 61 (2000) 1285.

30. Chapter 6; R. J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The

temperature dependence of the Young's modulus of MgSiN2, AlN and Si3N4,

accepted for publication in J. Eur Ceram. Soc.


Heat Capacity of MgSiN2 between 8 and 800 K, J. Phys. Chem. B 102 (1998)

7871.

32. G.A. Slack and S.F. Bartram, Thermal expansion of some diamondlike

crystals, J. Appl. Phys. 46 (1975) 89.

Chapter 7.

174



IFI/Plenum, New York, USA, 1977.



Components, Hybrids, Manuf. Technol. CHMT-7 (1984) 399.



December 1 - 5 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer, S.

Nakamura and S. Strite, (Materials Research Society, Warrendale,


36. V.I. Koshchenko, Ya.Kh. Grinberg and A.F. Demidenko, Thermodynamic

Properties of AlN (5 - 2700 ºK), GaP (5 - 1500 ºK), and BP (5 - 800 ºK),

Inorg. Mater. 20 (1984) 1550 (Izv. Akad. Nauk SSSR, Neorg. Mater. 20 (1984)

1787).

37. T.B. Jackson, K.Y. Donaldson and D.P.H. Hasselman, Temperature

Dependence of the Thermal Diffusivity/Conductivity of Aluminum Nitride, J.

Am. Ceram. Soc. 73 (1990) 2511.

38. P. S. de Baranda, A.K. Knudsen and E. Ruh, Effect of CaO on the Thermal


39. S.N. Ivanov, P.A. Popov, G.V. Egorov, A.A. Sidorov, B.I. Kornev, L.M.

Zhukova and V.P. Ryabov, Thermophysical properties of aluminum nitride

ceramics, Phys. Solid State 39 (1997) 81.

40. I. Barin, Thermochemical Data of Pure Substances, second edition, VCH,

Weinheim, FRG (1993).

41. M.W. Chase Jr., C.A. Davies, J.R. Downey Jr., D.J. Frurip, R.A. McDonald

and A.N. Syverud, JANAF Thermochemical Tables, third edition, American

Chemical Society and American Institute of Physics, Washington, USA

(1985).

42. R.J. Bruls and J.C. van Miltenburg, unpublished results.


175


Phys. 20 (1981) 1751.


Oxynitride of Silicon in Relation to their Structures, Trans. J. Br. Ceram. Soc.

74 (1975) 49.

45. I.C. Huseby, G.A. Slack and R.H. Arendt, Thermal Expansion of CdAl2O4,

β -Si3N4, and Other Phenacite-Type Compounds, Bull. Am. Ceram. Soc. 60

(1981) 919.

46. I. Ya. Guzman, A.F. Demidenko, V.I. Koshchenko, M.S. Fraifel'd and Yu.V.

Egner, Specific Heats and Thermodynamic Functions of Si3N4 and Si2ON2,

Inorg. Mater. 12 (1976) 1546. (Izv. Akad. Nauk SSSR, Neorg. Mater. 12

(1976) 1879).



Ed. 97 (1989) 170.

48. P. Rocabois, C. Chatillon and C. Bernard, Thermodynamics of the Si-O-N

System: High-Temperature Study of the Vaporization Behaviour of Silicon

Nitride by Mass Spectrometry, J. Am. Ceram. Soc. 79 (1996) 1351.

49. N.F. Mott and R.W. Gurney, Electronic processes in ionic crystals, edited by

R.H. Fowler and P. Kapitza, Chapter II, Lattice defects in thermal equilibrium

(Oxford, UK, Clarendon Press, 1940), pp. 27 - 33.

50. J.H. Crawford, Jr. and L.M. Slifkin, Point Defects in Solids 1, General and

Ionic Crystals (Plenum Press, New-York, USA, 1972), Chapter 1, Statistical

Thermodynamics of Point Defects in Crystals by A.D. Franklin.



(1988) 973.

52. J.B. Wachtman, Jr., W.E. Tefft, D.G. Lam, Jr., and C.S. Apstein, Exponential

Temperature Dependence of Young's Modulus for Several Oxides, Phys. Rev.

122 (1961) 1754.

Chapter 7.

176

53. P. Boch, J.C. Glandus, J.C. Jarrige, J.P. Lecompte and J. Mexmain, Sintering,


Ceram. Int. 8 (1982) 34.

54. J.C. van Miltenburg, G.J.K. van den Berg and M.J. van Bommel, Construction

of an adiabatic calorimeter. Measurements on the molar heat capacity of

synthetic sapphire and n-heptane. J. Chem. Thermodyn. 19 (1987) 1129.

55. P. Debije, Zur Theorie der spezifischen Wärmen, Ann. Physik 39 (1912) 789.

56. See for numerical calculation W.M. Rogers and R.L. Powell, Tables of

Transport Integrals, Natl. Bur. Stand. Circ. 595 (1958) 1.

57. A.A. Kahn and J.C. Labbe, Aluminium Nitride - Molybdenum CeramicMatrix

Composites. Influence of Molybdenum Addition on Electrical, Mechanical

and Thermal Properties, J. Eur. Ceram. Soc. 17 (1997) 1885.

58. O.G. Peterson, D.N. Batchelder and R.O. Simmons, Measurements of X-ray

Lattice Constant, Thermal Expansivity, and Isothermal Compressibility of

Argon Crystals, Phys. Rev. 150 (1966) 703.

59. R.R. Reeber, Thermal Expansion of Some Group IV Elements and ZnS, Phys.

Stat. Sol. a 32 (1975) 321.

177

Chapter 8.

Theoretical thermal conductivity of MgSiN2, AlN and

ββββ-Si3N4 using Slack's equation

Abstract

The maximum achievable thermal conductivity of MgSiN2, AlN and β-Si3N4

ceramics is estimated based on the theory of Slack. Using this procedure the

estimate obtained at the Debye temperature θ for MgSiN2 and β-Si3N4 appears to

be too high, whereas the value for AlN is in good agreement with the highest

experimentally observed value. Using better input parameters (especially the

Debye temperature) resulted in better estimates. In order to increase the validity of

Slack’s equation below the Debye temperature the temperature dependence of this

equation was modified, resulting in a fair agreement between the predicted and

experimentally observed values at 300 K for MgSiN2 and β-Si3N4. However, for

AlN the discrepancy between the predicted and calculated value is considerable.

Nevertheless, the modified Slack equation in combination with reliable input

parameters seems to result in an accurate (somewhat conservative) estimate of the

maximum achievable value making it suitable for materials selection.

1. Introduction

The nitride materials MgSiN2 [1, 2], AlN [3 - 5] and β-Si3N4 [6, 7] are (potentially)

interesting as high performance materials, able to resist severe thermal loads, e.g.

for substrates in microelectronics, engine parts, gas turbines, etc. One of the most

important physical properties for these ceramics is the thermal conductivity. In

view of guiding the optimisation of the processing of these materials, an estimate

Chapter 8.

178

of the maximum achievable thermal conductivity (especially at room temperature)

is highly desirable. The maximum achievable thermal conductivity κ [W m-1 K-1]

for these non-metallic materials is often discussed [6, 8 - 10] in view of Slack's

equation [11]:

Tn

MB θ

γ

θδκ3

22

2

= for T ≥ θ (1)

in which B [W mol kg-1 m-2 K-3] is a constant 3.04 107, M [kg mol-1] the mean

atomic mass, δ 3 [m3] the average volume of one atom in the primitive unit cell,

θ [K] the Debye temperature, γ [-] the Grüneisen parameter, n [-] the number of

atoms per primitive unit cell and T [K] the absolute temperature.

For MgSiN2 (with θ ≈ 900 K [12]) the above mentioned equation was used

to calculate the maximum achievable thermal conductivity at 300 K (i.e. below the

Debye temperature) resulting in 75 W m-1 K-1 [10]. However, the highest

experimental value at 300 K, observed in several studies concerning the influence

of the processing conditions on the thermal conductivity [2, 13 - 17], does not

exceed about 25 W m-1 K-1.

For AlN (with θ ≈ 940 K [12]) theoretical estimates of 157 W m-1 K-1 at

621 K [10] and 97 W m-1 K-1 at 516 K [8] have been reported, based on Slack's

equation. Beside the relative large difference between the two estimates in view of

the temperatures for which these estimates are reported, both values considerably

deviate from the values measured on pure AlN single crystals (about 91 W m-1 K-1

and 125 W m-1 K-1, respectively [8]).

For β-Si3N4 a first crude estimate of the intrinsic value at 300 K was made

based on Slack's equation at T = θ (≈ 955 K [12]), and subsequently extrapolating

this value to 300 K using the measured temperature dependence of the thermal

conductivity of SiC. This results in a value of 200 - 320 W m-1 K-1 [18]. This value

is considerably higher than the highest experimental value at 300 K of

122 W m-1 K-1 for isotropic β-Si3N4 reported by Hirosaki [9] or the averaged value

of 106 W m-1 K-1 measured on a single grain [19] (thermal conductivity along the

c-axis 180 W m-1 K-1 and along the a-axis 69 W m-1 K-1).

Theoretical thermal conductivity of MgSiN2, AlN and ββββ -Si3N4 using Slack's equation

179

For MgSiN2, AlN and β-Si3N4 a reliable prediction of the thermal

conductivity at the Debye temperature using Slack's formula is not reported

although some estimates at lower temperatures using this formula can be found in

the literature [10, 18]. These estimates at lower temperatures deviate all

substantially from the best experimentally observed values, being in most cases

higher than the experimentally observed value. This puts the question forward

whether reliable estimates at the Debye temperature and 300 K can be obtained

using Slack's equation as a starting point. For calculation of the thermal

conductivity at 300 K, i.e. below the Debye temperature the equation is used

outside its range of validity. This triggers a systematic evaluation of the

temperature dependence below the Debye temperature.

In this chapter the estimation of the maximum achievable thermal

conductivity will be described of MgSiN2, AlN and β-Si3N4 at the Debye

temperature θ and 300 K based on a modified formula of Slack, combined with the

use of more accurate input parameters. First the history of Slack's formula will be

briefly discussed, subsequently the theoretical values at the Debye temperature will

be calculated and the relevance of the accuracy of input parameters discussed.

Finally, an improved temperature dependence of the thermal conductivity as

calculated from Slack's formula is presented based on the temperature dependence

of the heat capacity and thermal diffusivity. The values calculated at 300 K

are compared with those obtained using the classical Slack theory and experimental

values. Some preliminary results concerning MgSiN2 have already been

reported [15, 20].

2. The Slack equation

The subject of the absolute magnitude of the thermal conductivity by lattice

vibrations (for T ≥ θ ) was first treated by Liebfried et al. [21] for a face centred

cubic (FCC) lattice having 1 atom per primitive unit cell (rare-gas crystals). Their

result was adjusted by Julian [22] by a factor 2 correcting a mistake in counting.

Slack [11] generalised Julian's equation making it suitable for complex lattices

Chapter 8.

180

(n > 1) for T ≥ θ assuming that the heat transport takes mainly place by acoustic

vibrations, resulting in the expression for the thermal conductivity as given in

equation (1).

In this equation B is taken as 3.04 107 W mol kg-1 m-2 K-3 [11], the mean

atomic mass M can be calculated from the stoichiometry of MgSiN2, AlN and

β-Si3N4 and the average volume of the atoms δ 3 and the number of atoms per

primitive cell n can be obtained from crystallographic data (see Table 8-1). Slack

[11] proposed to use for θ the Debye temperature evaluated from elastic constants

or heat capacity data near 0 K (θ0) and for γ the value of the thermodynamic

Grüneisen parameter at T = θ = θ0 (γθ). For MgSiN2, AlN and β-Si3N4 the Debye

temperature θ0 and Grüneisen parameter γ as a function of the absolute

temperature have been recently reported (θ0 [12] and γ [23]) (see Table 8-1). This

makes it possible to calculate the theoretical thermal conductivity (κThe) at T = θ

(see Table 8-1).

Table 8-1: The mean atomic mass M , the average volume occupied by one atom δ 3, the Debye

temperature θ , the Grüneisen parameter γ , the number of atoms per primitive unit cell n

and theoretical and experimental thermal conductivity at T = θ, κThe(θ ) and κExp(θ ) for

MgSiN2, AlN and β-Si3N4.

Material M δ 3 θ (= θ0) γ (= γθ) n κThe(θ ) κExp(θ )

[kg mol-1] [m3] [K] [-] [-] [W m-1 K-1] [W m-1 K-1]

MgSiN2

AlNβ-Si3N4

0.02010.02050.0200

1.064 10-29 [24]1.045 10-29 [25]1.039 10-29 [26]

900940955

0.980.950.62

16 [27] 4 [28]14 [29]

185354

10 [20, 30] 51 [8] 38 [9]

For MgSiN2 own data for the thermal diffusivity [20] and heat capacity [30]

were used to obtain the experimental thermal conductivity (κExp) at T = θ0

(Table 8-1). For AlN [8] and β-Si3N4 [9] literature data of the thermal conductivity

as a function of the temperature were used to obtain the value at T = θ0

(Table 8-1). For AlN the thermal conductivity was measured on almost pure AlN


181

single crystals (0.4 - 1800 K) [8] and for β-Si3N4 the highest experimental values

for isotropic material (300 - 1700 K) reported by Hirosaki [9] were used. As for

high temperatures the thermal resistance is largely determined by intrinsic thermal

phonon-phonon scattering processes, it can be assumed that the experimentally

obtained values (at T = θ ) equal about the intrinsic values. Except for AlN, the

predictions at T = θ are too optimistic (Table 8-1), triggering a detailed evaluation

of the input parameters.

3. Influence of input parameters

In order to understand the choice of the input parameters used for the Slack

equation, the 'exact' nature of B, θ and γ in the Slack equation should be discussed.

Actually the 'constant' B depends on the Grüneisen parameter γ, so B = B(γ )

[W mol kg-1 m-2 K-3]. The dependence of B on γ is given by [11, 22]:

B(γ) = )228.0514.01(2

849.0*10720.521

7

---

γγ +(2)

For γ = 2, which is approximately the case for many solids, the originally

used B value of 3.04 107 [W mol kg-1 m-2 K-3] results. As the Grüneisen parameter γ

0.0E+00

1.0E+07

2.0E+07

3.0E+07

4.0E+07

0.0 0.5 1.0 1.5 2.0 2.5 3.0

γ [-]

B( γ

)

B(γ )B(γ = 2)

Fig. 8-1: B(γ ) and B(γ = 2) plotted versus the Grüneisen parameter (γ ).

Chapter 8.

182

for solid substances can vary between 0.5 and 3, the true B value can vary with

about 10% (Fig. 8-1). Only for γ < 0.5, the B value drops significantly.

As it is assumed that only acoustic phonons contribute to the heat transport,

the Debye temperature θ and Grüneisen parameter γ should be based on the

acoustic vibration modes only. For the Debye temperature and Grüneisen

parameter this results in θ∞A [K] (the high temperature limit of the Debye

temperature based on the acoustic vibration modes only, as discussed by Domb et

al. [31]) and (γ A)2 [-] the value of γ 2 based on the acoustic branches [11],

respectively. Furthermore (γ A)2 is obtained from the average of γ 2 for all individual

modes. Because B(γ ) is a function of the Grüneisen parameter γ and κ ∼ θ 3/γ 2 it is

important to obtain reliable estimates for the Debye temperature θ and the

Grüneisen parameter γ.

The high temperature limit of the Debye temperature based on the acoustic

vibration modes only can be estimated from heat capacity data as will be discussed

below. In Fig. 8-2 the Debye temperature versus temperature plot is presented as

obtained from heat capacity measurements for MgSiN2 [30], and AlN and β-Si3N4

[23], respectively. For all three materials a similar temperature dependence is

700

800

900

1000

1100

1200

1300

0 50 100 150 200 250 300 350 400

T [K]

θ [K

]

MgSiN2

AlNβ -Si3N4

Fig. 8-2: The Debye temperature (θ ) versus the absolute temperature

(T ) for MgSiN2, AlN and β -Si3N4 as obtained from heat

capacity data.


183

observed. When extrapolated to 0 K the Debye temperature is approximately equal

to the value obtained from elastic constants [12] (Table 8-2) as expected from the

literature [32]. This value equals the Debye temperature at 0 K related to the

acoustic modes only whereas the high temperature limit of this Debye temperature

is needed. With increasing temperature the Debye temperature first decreases.

Subsequently a minimum is observed and the Debye temperature starts to increase

with temperature approaching a constant value at high temperatures.

As at low temperatures only the acoustic vibration modes are excited these

determine the heat capacity. So the low temperature region of the Debye

temperature versus temperature plot is determined by the acoustic phonons

(decrease of θ ). As the temperature increases the optic phonons are also excited

resulting in an increase of the Debye temperature. Therefore the minimum in the

Debye temperature versus temperature plot θm is a more realistic estimate for the

high temperature limit of the Debye temperature related to the acoustic phonons

θ∞A than θ0, resulting in a lower value for θ∞

A (Table 8-2).

Table 8-2: The Debye temperature evaluated from elastic

constants θ E, the minimum Debye temperature

θm and the high temperature limit of the Debye

temperature θ∞C evaluated from heat capacity

measurements for MgSiN2, AlN and β -Si3N4.

Material θ E (=θ0) θm (≈θ∞A) θ∞

C

[K] [K] [K]

MgSiN2

AlNβ-Si3N4

900940955

741818837

996 989 1200

As no better estimates of (γ A)2 can be made, it was assumed that for all

temperatures (γ A)2 equals the square value of the thermodynamic Grüneisen

parameter at the Debye temperature γθ2 with θ = θm.

Chapter 8.

184

Using the exact value of B(γ ) instead of the constant B-value of 3.04 107

results in some increase of the resulting thermal conductivity (Table 8-3), whereas

the use of θm instead of the higher θ0 results in a significant decrease of the

resulting thermal conductivity (Table 8-3).

Table 8-3: The most appropriate Debye temperature θ = θm, the Grüneisen

parameter γ at T = θm, the resulting B(γ ) value, and the

theoretical and experimental thermal conductivity at T = θm,

κThe(θ ) and κExp(θ ) for MgSiN2, AlN and β -Si3N4.

Material θ (= θm) γ (= γθ) B(γ ) κThe(θ ) κExp(θ )

[K] [-] [-] [W m-1 K-1] [W m-1 K-1]

MgSiN2

AlNβ-Si3N4

741818837

0.970.930.61

3.408 107

3.415 107

3.153 107

13.947.044.7

13.462.044.5

As compared to the conventional input parameters, the agreement between

the best experimentally observed value and the calculated theoretical thermal

conductivity at T = θ = θm for MgSiN2 and β-Si3N4 considerably improves

whereas the estimate for AlN becomes worse (Table 8-3). The relatively low

predicted value for AlN is most probably caused by the fact that also optic phonons

contribute to the heat transport whereas only acoustic modes are considered. Some

experimental confirmation concerning this point can be found in Ref. 33.

Furthermore, molecular dynamic calculations indicated that AlN has some low

energy optic modes with a relatively large dispersion [28]. These modes have thus

a relative large group velocity and therefore can contribute substantially to the heat

transport. Considering the improvement of the values for MgSiN2 and β-Si3N4 it

can be concluded that the use of more appropriate input parameters results in better

estimates than the use of the conventional input parameters.


185

4. The modification of the Slack equation

The question arises whether the estimates at T = θ can be extrapolated beyond

their validity range to room temperature (300 K). As the simple T -1 dependence of

the traditional Slack equation results in questionable estimates at 300 K for

MgSiN2 and AlN (Table 8-4), a more appropriate description of the temperature

dependence is needed.

Table 8-4: The theoretical thermal conductivity (at the Debye temperature θm and 300 K) estimated

with the traditional Slack equation using more appropriate input parameters, as compared

to the highest experimentally obtained values for MgSiN2, AlN and β -Si3N4.

Slack Experimental κThe/κExp

Material κThe(θ ) κThe(300 K) κExp(θ ) κExp(300 K) θ = θm 300 K

[W m-1 K-1] [W m-1 K-1] [W m-1 K-1] [W m-1 K-1] [-] [-]

MgSiN2

AlNβ-Si3N4

144745

34128124

136245

23285122

1.080.761.00

1.480.451.02

The temperature dependence of the thermal conductivity (κ [W m-1 K-1]) is

given by the temperature dependence of thermal diffusivity (a [m2 s-1]), molar

density (ρm [mol m-3]) and heat capacity at constant volume (CV [J mol-1 K-1])

as [34]:

VCa mρκ = (3)

For T ≥ θ the T -1 dependence of the Slack equation results as in this temperature

region ρm and CV are about constant and a ∼ T -1 as will be explained below.

Combining the above mentioned equation with the classical formula for the

thermal conductivity of a phonon conductor (i.e. heat transport predominantly takes

place by lattice vibrations) VClv mtots31 ρκ = [21, 35] results in the following

expression for the thermal diffusivity a [36]:

tots31 lv a = (4)

Chapter 8.

186

in which vs [m s-1] is the average phonon velocity (i.e. essentially the velocity of

sound) and ltot [m] the total mean free path of the phonons. As the average phonon

velocity vs is almost temperature independent [21, 35], the temperature dependence

of the thermal diffusivity is mainly determined by that of the total phonon mean

free path: a ∼ ltot. So for T ≥ θ the thermal conductivity κ ~ a ∼ ltot ~ T -1 as ρm, vs

and CV are about constant, and ltot ~ T -1 [35, 37, 38]. However, for T < θ the

thermal diffusivity a and the heat capacity CV are both strong functions of the

temperature (in contrast to the density ρm) and thus both determine the temperature

dependence of the thermal conductivity.

The temperature dependence of the thermal diffusivity can be deduced by

considering that of the total phonon mean free path. The total phonon mean free

path (ltot [m]) is determined by the lattice characteristics (intrinsic properties) as

well as defects and grain boundaries present (extrinsic properties), and can be

written as [39 - 41]:

∑+=i ipptot

111lll

(5)

in which lpp [m] is the mean free path due to thermal phonon-phonon scattering and

li [m] the mean free path due to other phonon scattering mechanisms e.g. phonon-

defect scattering, phonon-grain boundary scattering, etc. For pure, defect free

single crystalline materials ltot equals lpp, and in that case the temperature

dependence of the thermal diffusivity is determined by the temperature dependence

of lpp only.

For the phonon mean free path due to thermal phonon-phonon scattering, lpp,

of pure crystalline materials it is known that approximately [35, 37]:

= 1

~exp0pp -

bTll θ (6)

in which l0 [m] is a pre-exponential factor, θ~ [K] a characteristic temperature

below which Umklapp processes start to disappear given by θ /n1/3 [11, 42], b [-] a

constant about equal to 2 [35, 37, 43] and T [K] the absolute temperature. So the


187

temperature dependence of the thermal diffusivity is given by:

1exp 31 -

bTn~a θ (7)

From the discussion of the input parameters it is clear that within the

framework of the Slack equation the Debye temperature in the above equation is

based on the high temperature limit of the acoustic phonons only (θ = θ∞A). The

value for b for describing the temperature dependence of the thermal diffusivity is

not exactly known. Based on the simple Debye theory it can be argued that b = 2

[37, 43], but larger and smaller values down to 1 are also reported [21, 35].

Leibfreid et al. [21] suggest that for a FCC lattice b = (5/3)1/2. However, the scarce

experimental results confirm the value of b ≈ 2 (2.3, 2.7 and 2.1 for solid helium,

diamond and sapphire, respectively [35]).

The heat capacity at constant volume CV is 0 J mol-1 K-1 at 0 K and with

increasing temperature it increases having a maximum value at high temperatures

of 3R per atom mole (R is the gas constant 8.314 J mol-1 K-1) (Fig. 8-3).

Theoretically the temperature dependence of CV is given by the Debye function

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50

T/θ [-]

CV [R

]

FD(T /θ )

Fig. 8-3: The heat capacity at constant volume (CV) versus the

reduced temperature (T /θ ) using the Debye function

(FD(θ /T )).

Chapter 8.

188

FD(θ /T ) [44 - 46] (Fig. 8-3). For 0.1 ≤ T /θ ≤ 0.4 the increase of FD(θ /T ) is the

largest and for T ≥ θ the heat capacity is almost constant. If the temperature is

sufficiently high (T /θ / 0.3) FD(θ /T ) describes very well the experimental heat

capacity with θ = θ∞C [K], the high temperature limit of the characteristic Debye

temperature obtained from heat capacity measurements (see Fig. 8-2).

From the temperature dependence of a, ρm and CV it may be expected that:

)/(1exp CD31

A

TF-bTn

~ ∞∞

θ

θκ (8)

in which b, θ∞A and θ∞

C are constants if the temperature is not too low (T /θ / 0.3).

For T ≥ θ, FD(θ /T ) is almost constant, and the exponential function can be

approximated by θ /bTn1/3 as for θ /bTn1/3 = x it can be written for 2x < 1 that

exp(x) - 1 = (1 + x + x2/2 +…) - 1 ≈ x, so κ ∼ T -1 as suggested by Slack's equation.

In general for bTn1/3 ≥ 2θ the temperature dependence can be calculated

accurately within 23 % (½ [exp(½) - 1]-1 × 100 % - 100%) as the thermal

diffusivity is about inversely proportional with the absolute temperature, whereas

for Tn1/3 ≥ 2θ the exact value of b is of minor importance and the maximum error

in the calculated temperature dependence equals about 15 - 20 %. Due to the large

number of atoms per primitive unit cell for MgSiN2 and Si3N4 the error made at

300 K by this assumption does most probably not exceed 15 - 20 %. However, for

AlN with n = 4 the error at 300 K can be considerable.

It is clear that the complexity of the crystal structure (n) is of importance for

the temperature dependence of the thermal conductivity below the Debye

temperature. This can be visualised by dividing the temperature dependence of κ as

function of the reduced temperature T /θ by its high temperature limit ((θ~ /b) T -1 ×

3R) assuming that b = 2 and both Debye temperatures for describing the thermal

diffusivity and the heat capacity are the same (Fig. 8-4).

For rare-gas crystals (n = 1) Julian found that κ ∼ T -1 for T ≥ θ /4 [22]. This

result is somewhat different from the one given in Fig. 8-4 which shows a faster

than T -1 dependence. From Fig. 8-4 it can be seen that for simple crystals


189

(2 ≤ n ≤ 4) with decreasing temperature the drop in the heat capacity is

compensated by the exponential increase of the thermal diffusivity resulting in a

pseudo T -1 dependence for T ≥ θ /4. For large n (n / 8) the heat capacity decreases

for T. θ whereas the thermal diffusivity is still inversely proportional with the

absolute temperature for T / 2θ /bn1/3. So, the decrease of the heat capacity is only

partially compensated by the exponential increase of the thermal diffusivity

resulting in a considerable deviation of the T -1 dependence for T /θ . 0.5 (Fig. 8-4).

Based on Julian's result De With and Groen [10] assumed that the T -1

dependence of Slack's equation is valid down to T ≅ θ /4. They used the Slack

equation to predict the intrinsic thermal conductivity of several new (oxy-)nitride

materials with large n. From the above discussion it is clear that the T -1 dependence

of the Slack formula below the Debye temperature is in general too simple and a

more general 'all temperature' formula should be used.

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

0.00 0.25 0.50 0.75 1.00 1.25T/θ [-]

F D(θ

/T)/3R

× [e

xp(θ

/2Tn

1/3 )-1

] 2Tn

1/3 /θ

[-]

n = 1 n = 2 n = 4n = 8 n = 16 n = 64

Fig. 8-4: Theoretical deviation from the T -1 temperature dependence

of the thermal conductivity versus the reduced temperature

(T /θ ) as calculated from the temperature dependence of the

thermal diffusivity and the heat capacity with b = 2 (note

that FD(θ /T )/3R × [exp(θ /2Tn1/3)-1] 2Tn1/3/θ = 1 represents

the T -1 dependence of the Slack equation).

Chapter 8.

190

For describing the absolute value of the thermal conductivity the (above

mentioned) temperature dependence of the thermal diffusivity and the heat capacity

(equation (8)) is combined with the Slack equation (equation (1)) resulting in a

modified Slack equation. It is assumed that the Slack and modified Slack equation

give the same result for T >> θ . This results in the following modified Slack

equation for the theoretical thermal conductivity:

( )

= ∞∞∞ 3

1

31

ACD

322

2A

1 - exp)(3

)(*)( bn

bTnRTF

n

MBT

θθγ

θδκ (9)

in which θ∞C [K] is the high temperature limit of the Debye temperature as

evaluated from heat capacity measurements and θ∞A [K] the high temperature limit

of the acoustic phonons. The expression before the square brackets (Slack part)

represents the thermal conductivity at T = θ∞A (= θm) for the traditional Slack

equation and between square brackets the temperature dependence of the thermal

conductivity is given. As for high temperatures (T → ∞) the θ /T temperature

dependence should result, the temperature dependence of the specific heat and the

thermal diffusivity in equation (9) are normalised by a factor 3R and bn1/3,

respectively. It is noted that at T = θ the expression between square brackets

differs from unity resulting in different estimates for the thermal conductivity at

T = θ as compared to the traditional Slack equation (see Fig. 8-4).

Table 8-5: The temperature independent Slack part, temperature dependent heat capacity and thermal

diffusivity part of the modified Slack equation, and the resulting estimated thermal

conductivity for MgSiN2, AlN and β -Si3N4 at T = θ = θm with various values for b.

Material ( )n

MB3

22m

2

θγθδγ

RTF

3 )( C

D ∞θ 31

31

m 1exp bn-bTn

θκThe(T = θm)

[W m-1 K-1] [-] [-] [W m-1 K-1]

b = 2 b = √5/3 b = 1 b = 2 b = √5/3 b = 1

MgSiN2

AlNβ-Si3N4

13.8546.9944.69

0.9150.9300.904

1.1061.1751.111

1.1711.2891.179

1.2271.3931.239

14.051.444.9

14.856.447.6

15.660.950.1


191

Using the modified Slack equation and the values for θ∞A (= θm) and θ∞

C

(Table 8-2), and b = 2, b = (5/3)1/2 and b = 1 the theoretical thermal conductivity at

T = θ = θm and 300 K was calculated (Table 8-5 and Table 8-6).

The results at T = θ = θm using the modified Slack equation (Table 8-5)

show a better agreement for AlN with the experimental values than the

conventional Slack equation (Table 8-4), whereas the estimates for MgSiN2 and

β-Si3N4 as compared to the conventional Slack equation (Table 8-4) remain about

the same being in good agreement with the experimentally observed values.

Table 8-6: The temperature independent Slack part, temperature dependent heat capacity and thermal

diffusivity part of the modified Slack equation, and the resulting estimated thermal

conductivity for MgSiN2, AlN and β -Si3N4 at T = 300 K with various values for b.

Material ( )n

MB3

22m

2

θγθδγ

RTF

3 )( C

D ∞θ 31

31

m 1exp bn-bTn

θκThe(T =300)

[W m-1 K-1] [-] [-] [W m-1 K-1]

b = 2 b = √5/3 b = 1 b = 2 b = √5/3 b = 1

MgSiN2

AlNβ-Si3N4

13.8546.9944.69

0.6090.6130.495

3.1874.3193.779

3.6985.7034.516

4.1967.2575.260

26.9124.3 83.6

31.2 164.2 99.9

35.4208.9116.4

The estimate at 300 K for MgSiN2 (27 - 35 W m-1 K-1) is in reasonable

agreement with the best experimentally observed values (23 W m-1 K-1 [20]). This

indicates that the former estimate based on the conventional Slack equation of

75 W m-1 K-1 [10] is too optimistic mainly due to the use of inappropriate input

parameters. The estimate for the thermal conductivity at 300 K of β-Si3N4

(84 - 116 W m-1 K-1) is also in favourable agreement with the best experimentally

observed value (122 W m-1 K-1 [9]). It is noted that for β-Si3N4 at room

temperature T/θ∞C = 300/1200 = 0.25 indicating that the Debye function used to

calculate the heat capacity is not totally correct (see also Fig. 8-2) resulting in an

underestimation of the heat capacity (about 5 %). For AlN the estimate

(124 - 209 W m-1 K-1) is substantially lower than the experimentally observed value

Chapter 8.

192

(285 W m-1 K-1 [8]) providing only a rough indication for the true intrinsic value.

As pointed out before, this is caused by the used assumption that only acoustic

phonons contribute to the heat conduction, resulting in an underestimation of the

thermal conductivity.

Introduction of a more realistic temperature dependence shows that AlN has

a higher intrinsic thermal conductivity than β-Si3N4 whereas the conventional

Slack equation predicted that both materials have about the same thermal

conductivity. Moreover, the calculations indicate that for applications where a high

thermal conductivity is required β-Si3N4 is a much more interesting compound

than MgSiN2. From a first approximate based on the conventional Slack equation it

was concluded that MgSiN2 might be interesting as a potential substrate material

[10]. However, this expectation is not supported by the presently discussed

refinement based on the modified Slack equation, which provides some additional

theoretical evidence for the experimentally observed limited thermal conductivity

of MgSiN2. Furthermore, by varying the value of b an impression of the reliability

of the estimate is obtained indicating that the estimate for AlN is less reliable than

the estimate for MgSiN2 and β-Si3N4.

Table 8-7: Comparison of the modified Slack estimates for b = 1 - 2 based on reliable input

parameters (κThe) and experimentally observed (κExp) thermal conductivity at T = θ 0 and

300 K for MgSiN2, AlN and β -Si3N4. Between brackets the theoretical thermal

conductivity based on the conventional input parameters using the traditional Slack

equation is presented.

Material Slack Experimental κThe/κExp

κThe(θ0) κThe(300 K) κExp(θ0) κExp(300 K) θ0 300 K

[W m-1 K-1] [W m-1 K-1] [W m-1 K-1] [W m-1 K-1] [-] [-]

MgSiN2

AlNβ-Si3N4

12 - 13 (18)45 - 52 (53)40 - 44 (54)

27 - 36 (54)124 - 209 (166) 84 - 116 (173)

10 51 38

23 285 122

1.2 - 1.30.9 - 1.01.1 - 1.2

1.2 - 1.6 0.4 - 0.7 0.7 - 1.0

Generally, the estimates using the modified Slack equation (Table 8-5 and

Table 8-6) somewhat improved as compared to the estimates obtained using the


193

traditional Slack equation (T -1 dependence) with more appropriate input parameters

(Table 8-4). However, especially the choice of reliable values for θ and γ are

important to obtain an accurate estimate of the thermal conductivity (Table 8-7).

An error of 10 % in θ results in an error of 30 - 40 % and an error of 10 % in γ

results in an error of ~ 20 % in the resulting thermal conductivity estimate. So, the

reliability of the calculated thermal conductivity is more dependent on the accuracy

of the input parameters rather than a correct description of the temperature

dependence when the temperature is not too low. However, when also optic

phonons contribute substantially to the heat transport, like in the case of AlN, the

maximum achievable thermal conductivity can be substantially underestimated

providing a conservative estimate of the maximum achievable thermal

conductivity. This provides only a rough indication of the true intrinsic thermal

conductivity. Nevertheless, the modified Slack equation results in a good

impression whether or not materials have potentially desirable thermal properties.

5. Applicability, reliability and limitations of Slack modified

From the discussion about the influence of n on the temperature dependence of the

thermal conductivity a temperature above which the modified Slack equation gives

reasonable estimates for the intrinsic thermal conductivity was estimated to be

Table 8-8: The thermal conductivity for MgSiN2, AlN and β -Si3N4 at T = θ m/n1/3 with various

values for b using the modified Slack equation.

Material θm/n1/3R

TF3

)( CD ∞θ

3

31

m 1exp1

bn-bTn

θ κThe

(T = θm/n1/3)

κExp

(T = θm/n1/3)

[K] [-] [-] [W m-1 K-1] [W m-1 K-1]

b = 2 b = 1 b = 2 b = 1MgSiN2

AlNβ-Si3N4

294 515 347

0.597 0.837 0.586

3.2692.0603.127

4.3302.7284.141

27.0 81.0 81.9

35.8 107.3 108.4

25 120 110

Chapter 8.

194

T ≅ θ∞C/3 and higher based on the specific heat part and T / θm/n1/3 for the thermal

diffusivity part. The estimates obtained at T = θm/n1/3 (/ θ∞C/3) are indeed in

reasonable agreement with the experimentally observed values (Table 8-8). For

T = 300 K the criterion of T / θm/n1/3 is not fulfilled by AlN (Table 8-8) indicating

that, besides the already discussed contribution of optic phonons to the heat

conduction, the estimate of the acoustic phonons to the heat conduction at 300 K

might be less reliable.

An important complication encountered using the (modified) Slack equation

is the estimation of γ. By using the thermodynamic Grüneisen parameter no

difference can be made between the Grüneisen parameter of acoustic and optic

phonons. Furthermore acoustic modes might have a negative gamma decreasing

the value of the thermodynamic Grüneisen parameter, however, contributing

substantially to the thermal resistance [47]. Also it was assumed that the Grüneisen

parameter is temperature independent. For temperatures near the Debye

temperature and above this assumption is allowed [23]. However, for materials

with a high Debye temperature this assumption might be incorrect at room

temperature [23, 48]. In view of the discussion about the temperature dependence

of the thermal conductivity this could be interpreted as l0, the pre-exponential

factor for describing the phonon mean free path (equation (6)), being not constant

but l0 ~ B(γ )/γ 2. Assuming that γ Α is temperature dependent and can be

approximated by the (temperature dependent) thermodynamic Grüneisen parameter

(when the temperature is not too low (T / θ∞C/3)), the maximum achievable

thermal conductivity at 300 K was calculated using the modified Slack equation.

For b = 2 a value of 33 W m-1 K-1, 215 W m-1 K-1 and 133 W m-1 K-1 is obtained for

the intrinsic thermal conductivity at 300 K of MgSiN2, AlN and β-Si3N4,

respectively. These estimates are in very good agreement with the experimentally

observed values (Table 8-7). However, this result might be a coincidence /

fortuitous.

Although, the κ ∼ n-2/3 of the Slack formula (equation (1)) works well some

discussion about this point can be found in the literature (κ ∼ n-1/3 [42] and κ ∼ n-1/2


195

[49]). Considering this point it would be interesting, as suggested previously by

Spitzer [50], to study the thermal conductivity of materials with different

modifications, having only a slightly different atomic orientation, resulting in about

the same density, Debye temperature and Grüneisen parameter, yet different

number of atoms per primitive unit cell like α-Si3N4 (n = 28) and β-Si3N4 (n = 14).

6. Conclusions

In order to obtain reliable estimates of the intrinsic thermal conductivity at T = θ

using the Slack equation the input parameters should be carefully chosen. The T -1

dependence of the Slack equation can generally not be used at T < θ and should be

modified in order to obtain a more realistic description of the temperature

dependence of the thermal conductivity. The presented modified Slack equation is

considered to describe the temperature dependence of the intrinsic thermal

conductivity for T ≅ θ /3, and higher. Furthermore, the choice of the appropriate

input parameters, especially the Debye temperature and Grüneisen parameter, is of

crucial importance.

Considering the accuracy and boundary conditions using the modified Slack

equation it can be stated that the Slack equation provides a reasonably good

impression of the thermal conductivity when T / θ /n1/3 as for this temperature the

exact value of b is of minor importance.

Estimates of the maximum thermal conductivity at 300 K resulted in a value

of 27 - 35 W m-1 K-1, 124 - 209 W m-1 K-1 and 84 - 116 W m-1 K-1 as compared to

the highest experimentally observed values of 23 W m-1 K-1, 285 W m-1 K-1 and

122 W m-1 K-1 for MgSiN2, AlN and β-Si3N4, respectively. The calculated values

for MgSiN2 and β-Si3N4 are in good agreement with the measured thermal

conductivity whereas the calculated value for AlN is considerably below the

experimentally observed thermal conductivity as the (modified) Slack equation

neglects the contribution of optic phonons to the heat conduction.

Chapter 8.

196

Although the match with the true intrinsic thermal conductivity can be

disappointing due to the contribution of optic phonons to the heat conduction, the

modified Slack equation is a useful tool for estimating the intrinsic thermal

conductivity and understanding the differences in thermal conductivity between

several materials. Considering the accuracy of the modified Slack equation at lower

temperatures it would be desirable to have an alternative method for estimating the

intrinsic thermal conductivity in order to get a better impression of the true

maximum achievable thermal conductivity of (new) potentially interesting

materials.

References





1994) 239.








Compon., Hybrids, Manuf. Technol., CHMT-7 (1984) 399.

5. T.B. Jackson, A.V. Virkar, K.L. More, R.B. Dinwiddie, Jr. and R.A. Cutler,



(1997) 1421.


197



14 (1999) 1538.



Ceram. Soc. 79 (1996) 2485.



9. N. Hirosaki, Y. Okamoto, M. Ando, F. Munakata and Y. Akimune, Thermal

Conductivity of Gas-Pressure-Sintered Silicon Nitride, J. Am. Ceram. Soc. 79

(1996) 2878.

10. G. de With and W.A. Groen, Thermal Conductivity Estimates for New (Oxy)-

nitride Ceramics, Fourth Euro Ceramics 3, Basic Science - Optimisation of

Properties and Performances by Improved Design and Microstructural

Control, Faenza, Italy, October 1995, edited by S. Meriani and V. Sergo

(Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 405.




12. Chapter 6; R.J. Bruls, H.T. Hintzen, R. Metselaar and G. de With, The

temperature dependence of the Young's modulus of MgSiN2, AlN and Si3N4,







Hausner, G.L. Messing and S. Hirano (The American Ceramic Society,

1995) 585.



Chapter 8.

198



1995) 289.

16. I.J. Davies, H. Uchida, M. Aizawa, and K. Itatani, Physical and Mechanical





published

18. J.S. Haggerty and A. Lightfoot, Oppertunities for enhancing the thermal

conductivities of SiC and Si3N4 ceramics through improved processing,

Ceram. Eng. Sci. Proc. 16, 19th Annual Conference on Composites, Advanced

Ceramics, Materials, and Structures-A, Cocoa Beach, Florida, USA, January

8 - 12, 1995, edited by J.B. Wachtman (The American Ceramic Society,

Westerville, OH, 1995) 475.




20. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modeling of the Thermal



Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 26-29, 1997,

edited by P.S. Gaal and D.E. Apostolescu (Technomic Publishing Co., Inc.,

Lancaster, 1999) 3.



22. C.L. Julian, Theory of Heat Conduction in Rare-Gas Crystals, Phys. Rev. 139

(1965) A128.

23. Chapter 7; R.J. Bruls, H.T. Hintzen, R. Metselaar, G. de With and J.C. van

Miltenburg, The temperature dependence of the Grüneisen parameter of

MgSiN2, AlN and β-Si3N4, submitted to J. Phys. Chem. Solids


199






December 1 - 5 1997, edited by F.A. Ponce, S.P. DenBaars, B.K. Meyer,

S. Nakamura and S. Strite, (Materials Research Society, 1998) 863.



74 (1975) 49.




Properties of Aluminum Nitride, Phys. Rev. B. 57 (1998) 10550.

29. R. Grün, The Crystal Structure of β-Si3N4; Structural and Stability

Considerations Between α- and β-Si3N4, Acta Cryst. B35 (1979) 800.



(1998) 7871.

31. C. Domb and L. Salter, The Zero Point Energy and Θ Values of Crystals, Phil.

Mag. 43 (1952) 1083.

32. P. Debye, Zur Theorie der spezifischen Wärmen, Ann. Phys. 39 (1912) 789.

33. Chapter 9; R.J. Bruls, H.T. Hintzen, R. Metselaar, A new Estimation Method

for the Thermal Diffusivity/Conductivity of Non-Metallic Compounds: A case

study for MgSiN2, AlN and β-Si3N4 ceramics, to be published.

34. R. Berman, Thermal Conduction in Solids (Clarendon Press, Oxford,

1976), p. 7.

35. R. Berman, The Thermal Conductivity of Dielectric Solids at Low

Temperatures, Advances in Phys. 2 (1953) 103.

Chapter 8.

200



Ed. 91 (1989) 170.

37. P. Debye, Zustandsgleichung und Quantenhypothese mit einem Anhang über

Wärmeleitung, in: Vorträge über die Kinetische Theorie der Materie und der

Elektrizität (Teubner, Berlin, 1914), pp. 19 - 60.

38. R. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen, Ann. Phys.

3 (1929) 1055.

39. F.R. Chavat and W.D. Kingery, Thermal Conductivity: XIII, Effect of


40 (1957) 306.

40. P.G. Klemens, The thermal conductivity of dielectric solids at low

temperatures, Proc. Roy. Soc. (London), A208 (1951) 108.



Sci. 28 (1993) 3709.

42. M. Roufosse and P.G. Klemens, Thermal Conductivity of Complex Dielectric

Crystals, Phys. Rev. B. 7 (1973) 5379.

43. J.R. Drabble and H.J. Goldsmid, Thermal Conduction in Semiconductors,

International Series of Monographs on Semiconductors 4, edited by H.K.

Henisch (Pergamon Press, Oxford, 1961), p. 141.


45. See for numerical calculation W.M. Rogers and R.L. Powell, Tables of

Transport Integrals, Natl. Bur. Stand. Circ. 595 (1958) 1.

46. See for example C. Kittel, Introduction to Solid State Physics, fifth edition

(John Wiley & Sons, Inc., New York, 1976), pp. 136 - 140.

47. W.B. Daniels, The anomalous thermal expansion of germanium, silicon and

compounds crystallizing in the zinc blende structure, International Conference

on the Physics of Semiconductors, Exeter, UK, July 1962, edited by A.C.

Stickland (London Institute of Physics, 1962) 482.


201

48. G.A. Slack and I.C. Huseby, Thermal Grüneisen parameters of CdAl2O4,

β-Si3N4, and other phenacite-type compounds, J. Appl. Phys. 53 (1982) 6817.

49. A. Missenard, Conductivité Thermique des Solides, Liquides, Gaz et leurs

Mélanges, ch.1.II, Eyrolles, Paris, 1965.

50. D.P. Spitzer, Lattice Thermal Conductivity of Semi-Conductors: a Chemical

Bond Approach, J. Phys. Chem. Solids 31 (1970) 19.

Chapter 8.

202

203

Chapter 9.

A new method for estimation of the intrinsic thermal

conductivity

A case study for MgSiN2, AlN and ββββ-Si3N4

Abstract

A new method for estimating the maximum achievable thermal conductivity of

non-electrically conducting materials is presented. The method is based on

temperature dependent thermal diffusivity data using a linear extrapolation method

enabling to distinguish between phonon-phonon and phonon-defect scattering. The

thermal conductivity estimated in this way for MgSiN2, AlN and β-Si3N4 ceramics

at 300 K equals 26 - 28 W m-1 K-1, 178 - 200 W m-1 K-1 and 79 - 94 W m-1 K-1,

respectively in favourable agreement with the highest experimental values of

23 W m-1 K-1, 246 - 266 W m-1 K-1 and 106 - 122 W m-1 K-1. The difference

between the estimated and experimentally observed value for AlN can be

understood in view of optic phonons that are substantially contributing to the heat

conduction. The reliability, accuracy and limitations of this method are discussed.

1. Introduction

Several ceramic materials have been investigated intensively for substrate

applications [1] because of their potentially high thermal conductivity in

combination with a high electrical resistivity. Especially AlN has drawn a lot of

attention [2 - 4], but recently also the nitride materials β-Si3N4 [5 - 7] and MgSiN2

[8, 9] are considered to be potentially interesting. In a previous paper [10] the

Chapter 9.

204

intrinsic thermal conductivity of MgSiN2, AlN and Si3N4 was (theoretically)

estimated based on a modification of Slack’s equation for non-metallic materials

[11]. Based on the results obtained it can be concluded that this equation only

provides a rough indication of the maximum achievable thermal conductivity, and

that a more accurate and simpler estimation method would be useful.

Another (experimental) method reported in the literature to estimate the

maximum achievable thermal conductivity is by linear extrapolation of the

measured inverse thermal conductivity (thermal resistivity) values [12] versus the

absolute temperature. Usually, it is assumed that the slope is determined by the

lattice characteristics (intrinsic properties) and the intercept at 0 K by defects

(impurities, grain boundaries, etc.) [12 - 14]. It will be shown that this last

assumption is only partially correct. So, also this method is not generally

applicable. However, by combining some of the concepts of both approaches a new

estimation method was developed.

In this chapter a new method will be described for the estimation of the

maximum achievable thermal conductivity of non-metallic crystals (i.e. heat

transport takes place by lattice vibrations) based on temperature dependent thermal

diffusivity data. With this method the maximum achievable thermal conductivity of

MgSiN2, AlN and β-Si3N4 was calculated at 300, 600 and 900 K. The results were

compared with experimental values, values obtained using the (modified) Slack

theory and other (theoretical) estimates. Some preliminary results considering

MgSiN2 have already been reported elsewhere [15].

2. The temperature dependence of the thermal diffusivity and

conductivity

The thermal conductivity (κ [W m-1 K-1]) of a material can be calculated using [16]:

VCa mρκ = (1)

in which a [m2 s-1] is the thermal diffusivity, ρm [mol m-3] the molar density and CV

[J mol-1 K-1] the heat capacity at constant volume. The density is only a weak

A new method for estimation of the intrinsic thermal conductivity

205

function of the temperature, so the temperature dependence of the thermal

conductivity is determined by that of the thermal diffusivity and the heat capacity.

For a phonon conductor (i.e. heat transport predominantly takes place by

lattice vibrations) the thermal diffusivity a equals [16 - 19]:

tots31 lv a = (2)

in which vs [m s-1] is the average phonon velocity (i.e. essentially the velocity of

sound) and ltot [m] the total mean free path of the phonons. The average phonon

velocity vs is almost temperature independent [20], so that a ∼ ltot. If secondary

phases are not taken into account then the total phonon mean free path is

determined by the lattice characteristics (intrinsic properties) as well as defects and

grain boundaries present in the material (extrinsic properties), and can be written as

[12, 21 - 23]:

∑+++=x xgbpdpptot

11111lllll

(3)

in which lpp [m] is the mean free path due to thermal phonon-phonon scattering, lpd

[m] the mean free path due to phonon-defect (vacancies, impurities, isotopes)

scattering, lgb [m] the mean free path due to phonon-grain boundary scattering and

lx [m] the mean free path due to other scattering mechanisms induced by e.g.

stacking faults, dislocations, etc.

For the temperature dependence of the phonon mean free path due to thermal

phonon-phonon scattering, lpp, of pure crystalline materials it is known that

approximately [17, 24]:

= 1

~exp0pp -

bTll θ with

31

~

n

θθ = (4)

in which l0 [m] is a pre-exponential factor, θ~ [K] a characteristic temperature

(so-called reduced Debye temperature) below which Umklapp processes start to

disappear [11, 25], b [-] a constant ≈ 2 [17, 18, 24, 26], T [K] the absolute

Chapter 9.

206

temperature, θ [K] the Debye temperature and n [-] the number of atoms per

primitive unit cell.

For most materials only the first three terms of equation 3 are considered to

be of importance [12, 22, 27]. However, for the present discussion it is sufficient to

assume that lx is temperature independent. The temperature dependence of phonon-

defect scattering lpd has been studied by Klemens [21, 28] and Ambegaokar [29]. It

was shown that this term for low defect concentrations is (almost) temperature

independent. The phonon-grain boundary scattering term lgb is temperature

independent if the influence of the thermal expansion is neglected. So, the

temperature dependence of ltot is dominated by the lpp term, whereas the other terms

can be assumed to be negligibly temperature dependent [12, 21, 30]. This implies

that in general at low temperature ltot is determined by temperature independent

extrinsic scattering processes (at defects and grain boundaries), whereas at high

temperatures it is determined by the temperature dependent intrinsic phonon-

phonon scattering process.

If the temperature is sufficiently high (T > θ~ /b) we can write more generally

(i.e. including all above mentioned phonon scattering mechanisms) for the inverse

of the thermal diffusivity:

a1 ∼

tot

1l

∼ B

-bT

A +

1

~exp θ

=

BbTbT

-AbT +

+

+

...

~

121~

211~

2θθ

θ≈ (5)

B-bTA +

21

~θ = ( )A-BTbA

21~ +

θ

The constant A in equation (5) is related to the temperature dependent phonon-

phonon scattering processes (intrinsic lattice diffusivity) and B to the temperature

independent phonon scattering processes (impurities, defects, grain boundaries,

etc.). It is obvious that equation (5) shows a linear relation between a -1 and T:


207

''1 BTAa +=− (for T / θ~ /b) (6)

in which the slope A' (= bA/θ~ ) [m-2 s K-1] is determined by the intrinsic lattice

characteristics (phonon-phonon scattering mechanisms), and the intercept

B' (= B - ½ A) [m-2 s] by the impurities and microstructure (B: temperature

independent scattering processes) as well as the intrinsic lattice characteristics (A).

From equation (6) it can be concluded that for pure defect free single crystalline

materials (B = 0) a plot of the inverse of the thermal diffusivity versus the absolute

temperature for measurements at T > θ~ /b extrapolated to 0 K should result in a

straight line with (negative) intercept -½ A and which intercepts the temperature

axis at T = θ~ /2b (= ½ A/A').

If the temperature is sufficiently high so that the heat capacity is temperature

independent (T / θ [31]) then κ ~ a (~ ltot) and the well known linear relation for

the thermal resistivity results [12 - 14]:

κ -1 = A''T + B'' (for T / θ ) (7)

This equation is often interpreted as being A'' (~ bA/θ~ ) determined by the intrinsic

lattice diffusivity, which is correct, and the intercept value B'' (~ (B - ½ A)) as being

only determined by the microstructure and impurities, which is incorrect. This

results in the erroneous conclusion that for a pure defect free single crystalline

material (for T > θ ) the thermal resistivity versus the absolute temperature plot

gives a straight line through the origin [5, 13, 14, 32] as B = 0 instead of B'' = 0.

It is noted that at very high temperatures (T ≥ 2θ ), where the phonon mean

free path is limited by the inter-atomic distances, equations (5) and (6) are no

longer valid [33] because they predict a decrease of the phonon mean free path to

zero. For most materials n > 1 so that θ~ < θ . Considering the above discussion it

is clear that the linear temperature dependence for the inverse thermal diffusivity

a -1 can be observed at much lower temperatures (T > θ~ /b) than for the thermal

resistivity κ -1 as for T . θ the heat capacity is still temperature dependent.

Furthermore the thermal diffusivity is directly related to the total phonon mean free

Chapter 9.

208

path which has to be maximised in order to optimise the thermal conductivity. So

for identifying the dominant scattering mechanisms it is much more interesting to

study the temperature dependence of the thermal diffusivity rather than that of the


So, temperature dependent thermal diffusivity measurements when

performed in a suitable temperature region (θ~ /b (= θ /bn1/3 ) ≤ T ≤ 2θ ) can be a

powerful tool in understanding and optimising the thermal conductivity of

promising materials.

3. Experimental

For MgSiN2 the thermal diffusivity a as a function of the temperature T

(300 - 900 K) was measured on small ceramic samples (∅ 11 mm × 1 mm) cut

from several large fully dense ceramic pellets processed under different conditions

(for details see [34]) using the photo/laser flash method [35] (laser flash equipment,

Compotherm Messtechnik GmbH). The method used to prepare the ceramic pellets

is described elsewhere [34, 36, 37]. By carefully grinding and polishing, samples

with a uniform thickness and a low roughness were obtained. Samples varying in

microstructure, oxygen content and processed with and without additive were

investigated (Table 9-1). The accuracy of the measurement was estimated to be

within 5%. Some samples were coated with a thin layer of gold and/or carbon

before measuring the thermal diffusivity. The thin gold layer prevents direct

transmission of the laser beam and aids the energy transfer to the sample. Carbon

was used to increase the absorptivity of the front surface, and the emissivity of the

back surface. These additional layers reduce the measured thermal diffusivity only

slightly. A gold layer is always coated with a carbon layer because the gold layer

reflects the laser flash. The radiative heat losses were minimised by measuring the

samples in vacuum. The molar density ρm and heat capacity at constant volume CV

required for calculating the thermal conductivity were obtained from the literature

(density [38] and heat capacity [38, 39] assuming that CV = Cp resulting in a

maximum relative error of approximately 10 % [40]).


209

Table 9-1: Preparation characteristics as reported for several MgSiN2 ceramic samples [34].

Sample Densification method andreaction conditions

Additives Oxygencontent

Grainsize

[wt. %] [µm]

RB02

RB11

RB13

RB32

RB34

RB37

hot-pressing1823 K, N2, 75 MPa, 2 hhot-pressing1823 K, N2, 75 MPa, 2 hreaction hot-pressing1873 K, N2, 75 MPa, 2 hreaction hot-pressing1873 K, N2, 75 MPa, 2 hreaction hot-pressing1973 K, N2, 75 MPa, 2 hreaction hot-pressing1873 K, N2, 75 MPa, 2 h

None

None

None

4.2 wt. % Mg3N2

none

6.0 wt. % Y2O3

3.8

1.8

1.0

1.0

1.0

—

—

—

~ 0.5

—

~ 1.5

—

In the literature many temperature dependent thermal diffusivity/

conductivity data are reported for several AlN [4, 32, 41 - 47] and Si3N4 [6,

48 - 52] ceramics having different thermal properties. When necessary, the thermal

diffusivity as a function of the temperature was calculated from the temperature

dependence of the thermal conductivity, the density and heat capacity reported in

the corresponding reference or literature [38].

4. Results for MgSiN2, AlN and ββββ-Si3N4

4.1. The temperature dependence of the thermal diffusivity a

As expected, the thermal diffusivity for the MgSiN2 samples processed in different

ways decreases for higher temperatures (Fig. 9-1). The same is true for the AlN

(Table 9-2) and (β-)Si3N4 (Table 9-3) samples prepared in different ways (for

details concerning the processing see the corresponding references). For all three

materials the observed thermal diffusivity/conductivity at 300 K varied over a

Chapter 9.

210

Table 9-2: Preparation characteristics as reported in the literature for several AlN ceramic materials.

Sample Densification method and

reaction conditions

Additives

Single crystal W-201 [41] Sublimation-recondensation2523 K, 95 % N2/ 5 % H2 [53]

none

Shapal [42, 54] Not reported

AlN without additive [4] Hot-pressing, 2123 K, 10 min.annealing, 2123 K, 100 min.

none

BP research AlN [32, 54] Not reportedShapal SH-04 [32, 54] Not reportedShapal SH-15Super ShapalToshiba TAN-170 [32, 54] Not reportedCarborundum AlN [32, 54] Not reported

AlN [43] Pressureless sintering2023 K, N2, 10 h

4 wt. % Y2O3

B(N2) [44] Pressureless sintering2133 K, N2, 1 h

1 wt. % Y2O3

H(N2) 3 wt. % Y2O3

G(N2) 10 wt. % Y2O3

C1 [44, 45] Pressureless sintering2098 K, N2, 1 h

3 wt. % Y2O3 + 0 wt. % CaO

I1 [44] 3 wt. % Y2O3 + 1 wt. % CaOB1 [44, 45] 3 wt. % Y2O3 + 2 wt. % CaOH(N2) [44, 46] Pressureless sintering

2133 K, N2, 1 h3 wt. % Y2O3 + 0 wt. % SiO2

N(N2) [44] 3 wt. % Y2O3 + 0.3 wt. % SiO2

O(N2) [44, 46] 3 wt. % Y2O3 + 1 wt. % SiO2

Q(N2) [44, 46] 3 wt. % Y2O3 + 2 wt. % SiO2

S(N2) [44, 46] 3 wt. % Y2O3 + 5 wt. % SiO2

relatively broad range (MgSiN2: 16 - 23 W m-1 K-1 (Table 9-4); AlN:

24 - 285 W m-1 K-1 (Table 9-5); β-Si3N4: 14 - 122 W m-1 K-1 (Table 9-6)),

indicating large differences in impurity content and microstructure for the different

samples. The difference in thermal diffusivity between the samples is less


211

Table 9-3: Preparation characteristics and resulting β -fraction as reported in the literature for several

β -Si3N4 and α/β -Si3N4 composite ceramic materials.

Sample Densification methodand reaction conditions

Additives β-fraction

SN5 [51] gas-pressure sintered 2473 K, 30 MPa (N2), 4 h

0.5 mol % Y2O3 + 0.5 mol %Nd2O3

100

A [48] high pressure hot-pressing2173 K, 3 GPa, 1 h

None 100

B 4 wt. % MgO 100D 4 wt. % Al2O3 100C high pressure hot-pressing

2073 K, 3 GPa, 1 h4 wt. % Y2O3 100

[49] capsule-HIPped1973 K, 60 MPa (Ar), 1 h

3 mol % Y2O3 + 3 mol % Al2O3 ~ 25

2 mol % Y2O3 + 4 mol % Al2O3 ~ 34 4 mol % Y2O3 + 2 mol % Al2O3 ~ 67-100 capsule-HIPped

2023 K, 60 MPa (Ar), 1 h3 mol % Y2O3 + 3 mol % Al2O3 100

- 90 capsule-HIPped1973 K, 60 MPa (Ar), 1 h

3 mol % Y2O3 + 3 mol % Al2O3 ~ 90

- 34 capsule-HIPped1823 K, 60 MPa (Ar), 1 h

3 mol % Y2O3 + 3 mol % Al2O3 ~ 34

+ 6/0 capsule-HIPped2023 K, 60 MPa (Ar), 1 h

6 mol % Y2O3 + 0 mol % Al2O3 100

+ 5/1 5 mol % Y2O3 + 1 mol % Al2O3 100+ 4/2 4 mol % Y2O3 + 2 mol % Al2O3 100+ 3/3 3 mol % Y2O3 + 3 mol % Al2O3 100+ 2/4 2 mol % Y2O3 + 4 mol % Al2O3 100+ 1/5 1 mol % Y2O3 + 5 mol % Al2O3 100+ 0/6 0 mol % Y2O3 + 6 mol % Al2O3 100Tape cast [6] hot-pressed, 2073 K,

40 MPa, 2 h and5 wt. % Y2O3 + 5 vol. %rod-like β -Si3N4 seeds

—

Subsequently HIPped,2773 K, 200 MPa (N2), 2 h

pronounced at higher temperatures (see e.g. Fig. 9-1) as then intrinsic phonon

scattering processes are dominating the thermal diffusivity/conductivity (because

∑++>x xgbpdpp

1111llll

so A'T > B').

Chapter 9.

212

4.2. Inverse thermal diffusivity a -1 versus temperature T plots

For all three compounds the inverse of the thermal diffusivity plotted against the

absolute temperature can be described with a linear fit (Fig. 9-2 - Fig. 9-8) resulting

2.0E-06

4.0E-06

6.0E-06

8.0E-06

1.0E-05

0 200 400 600 800 1000

T [K]

a [m

2 s-1]


Fig. 9-1: The thermal diffusivity (a ) plotted versus the absolute

temperature (T ) for several MgSiN2 samples.

0.0E+00

0.5E+05

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

4.0E+05

4.5E+05

0 200 400 600 800 1000

T [K]

a-1 [s

m-2

]

RB02 RB11

RB13 RB32

RB34 RB37

Fig. 9-2: The inverse thermal diffusivity (a -1) versus the absolute

temperature (T ) plot for MgSiN2 ceramic samples

processed in different ways.


213

in a good description of the temperature dependence (R > 0.99) (Table 9-4 -

Table 9-6). The indicated uncertainties for the slope and the intercept correspond

with the 95% confidence interval.

For all MgSiN2 samples about the same slope A' of 400 - 430 m-2 s K-1 is

observed (Table 9-4 and Fig. 9-2) indicating that the lattice characteristics are not

influenced by the processing conditions used. On the contrary, the intercept B'

shows a relative large variation, as the samples differ in impurity content and grain

size [34]. As expected, the samples with the highest purity and grain size have in

general the lowest intercept value.

Table 9-4: Slope and intercept values (with R-value) of linearly fitted inverse

thermal diffusivity (a -1) versus the absolute temperature (T ), and

measured room temperature (~ 300 K) thermal conductivity κ300 for

MgSiN2 ceramic samples proceed in different ways.

Sample Slope A' Intercept B' R-value κ300

[m-2 s K-1] [m-2 s] [-] [W m-1 K-1]


424.6 ± 7.5 409.8 ± 8.0 411.8 ± 9.5 394.5 ± 6.6 402.8 ± 4.4 437.2 ± 5.1

5.3 ± 4.6 103

27.7 ± 4.9 103

4.0 ± 5.8 103

- 1.7 ± 4.0 103

-13.6 ± 2.7 103

-19.7 ± 3.1 103

0.99920.99900.99870.99930.99970.9997

191620212322

Mean 413.5 ± 14.0 — — —

The AlN ceramics processed with several additives have a typical slope

value of 80 - 90 m-2 s K-1 (Table 9-5). These values are somewhat smaller than the

value observed for the hot-pressed AlN sample without sintering additive

(104.8 ± 3.0 m-2 s K-1). Also for the best heat conducting sample (single crystal

W-201 [41]) a somewhat larger slope is observed (100.0 ± 1.0 m-2 s K-1) as

compared to the typical value (Fig. 9-3). This observation is related to the fact that

this sample is a single crystal for which the thermal conductivity was determined

Chapter 9.

214

along the c-axis resulting for the a -1 versus T plot in the anisotropic slope value of

the c-axis.

Table 9-5: Slope and intercept values with R-value of linearly fitted inverse thermal diffusivity (a -1)

versus the absolute temperature (T ) and measured room temperature (~ 300 K) thermal

conductivity κ300 for AlN ceramic samples processed in different ways.


[m-2 s K-1] [m-2 s] [-] [W m-1 K-1]

single crystal W-201 [41] 100.0 ± 1.0 - 22.79 ± 0.86 103 0.9999 285Shapal [41, 54] 83.5 ± 1.5 - 8.08 ± 1.10 103 0.9986 141 AlN without additive [4] 104.8 ± 3.0 3.38 ± 1.75 103 0.9988 70BP research AlN [32, 54] 81.6 ± 1.0 - 14.33 ± 0.45 103 0.9996 228Shapal SH-04 [32, 54] 85.6 ± 1.6 - 11.66 ± 0.77 103 0.9995 167Shapal SH-15 91.9 ± 1.5 - 14.62 ± 0.77 103 0.9995 144Super Shapal 84.4 ± 1.9 - 14.79 ± 0.94 103 0.9987 212Toshiba TAN-170 [32, 54] 85.0 ± 1.5 - 11.69 ± 0.69 103 0.9995 170Carborundum AlN [32, 54] 82.4 ± 1.1 - 13.81 ± 0.51 103 0.9995 212

AlN [43] (4 wt. % Y2O3) 89.3 ± 9.0 - 16.11 ± 4.21 103 0.9850 208B(N2) [44] (1 wt. % Y2O3) 87.0 ± 1.9 - 4.72 ± 1.05 103 0.9990 119H(N2) (3 wt. % Y2O3) 91.6 ± 2.4 - 13.41 ± 1.35 103 0.9986 159G(N2) (10 wt. % Y2O3) 94.3 ± 1.1 - 12.74 ± 0.58 103 0.9998 148C1 [44, 45] (0 wt. % CaO) 89.2 ± 1.0 - 10.50 ± 0.56 103 0.9997 144I1 [44] (1 wt. % CaO) 91.5 ± 0.5 - 9.11 ± 0.30 103 0.9999 129B1 [44, 45] (2 wt. % CaO) 100.0 ± 1.0 - 10.97 ± 0.52 103 0.9998 124H(N2) [44, 46] (0 wt. % SiO2) 91.6 ± 2.4 - 13.41 ± 1.35 103 0.9986 159N(N2) [44] (0.3 wt. % SiO2) 88.5 ± 0.6 - 8.19 ± 0.32 103 0.9999 129O(N2) [44, 46] (1 wt. % SiO2) 96.0 ± 1.5 - 6.62 ± 0.83 103 0.9995 106Q(N2) [44, 46] (2 wt. % SiO2) 148.9 ± 5.3 9.10 ± 2.93 103 0.9935 46S(N2) [44, 46] (5 wt. % SiO2) 200.4 ± 11.4 45.83 ± 6.30 103 0.9975 24

The resulting slope value is not much influenced by the addition of small

amounts of Y2O3 (≤ 10 wt. %) (Fig. 9-4) and CaO (≤ 2 wt. % together with 3 wt. %

Y2O3 (Table 9-5)), whereas the slope changes drastically for larger amounts SiO2


215

addition (≥ 2 wt. % together with 3 wt. % Y2O3) (Fig. 9-5). From these

observations it can be concluded that Y2O3 and CaO additions mainly influence the

defect chemistry and microstructure of the AlN ceramics (phonon-defect and

phonon-grain boundary scattering), whereas SiO2 addition also results in a change

of the lattice characteristics (phonon-phonon scattering). In complete agreement

with this conclusion, De Baranda et al. [46] reported that for an SiO2 addition of

2 wt. % together with 3 wt. % Y2O3 and above sialon polytypoids with an AlN like

structure are formed, resulting in the formation of a different lattice and thus a

different slope value (Table 9-5 and Fig. 9-5).

The intercept value B' is the smallest for the (almost) defect free single

crystal and largest for hot-pressed ceramics processed without additives containing

many defects due to the oxygen impurities dissolved into the AlN lattice (Fig. 9-3

and Table 9-5). By suitable processing (typical sample) the defect concentration in

the AlN lattice is reduced resulting in a decrease of the intercept approaching the

value for the (almost) defect free single crystal.

0.0E+00

0.2E+05

0.4E+05

0.6E+05

0.8E+05

1.0E+05

0 200 400 600 800 1000 1200T [K]

a-1 [s

m-2

]

without additivesShapal, non-irradiatedsingle crystal W-201

Fig. 9-3: The inverse thermal diffusivity (a -1) versus temperature (T )

plot for AlN samples without additive (×), a typical sample

( ) and a single crystal ().

Chapter 9.

216

With increasing Y2O3 addition the intercept value B' first decreases and

subsequently increases again (Table 9-5 and Fig. 9-4) in agreement with other

observations [55, 56] that with increasing Y2O3 addition the thermal conductivity

first increases (till about 4 - 6 wt. % addition [56]) and subsequently decreases.

This indicates that (as expected) Y2O3 is an effective sintering aid for sintering of

0.0E+00

0.2E+05

0.4E+05

0.6E+05

0.8E+05

1.0E+05

0 100 200 300 400 500 600 700 800T [K]

a-1 [s

m-2

]

AlN without additives1 wt. % Y2O3

3 wt. % Y2O3

AlN 4 wt. % Y2O3

10 wt. % Y2O3


plot for AlN ceramics, processed with different amounts of

Y2O3 as a sintering additive (data from several references).

0.0E+00

0.5E+05

1.0E+05

1.5E+05

2.0E+05

2.5E+05

0 100 200 300 400 500 600 700 800 T [K]

a-1 [s

m-2

]

3 wt. % Y2O3 + 0 wt. % SiO2

3 wt. % Y2O3 + 0.3 wt. % SiO2

3 wt. % Y2O3 + 1 wt.% SiO2

3 wt. % Y2O3 + 2 wt. % SiO2

3 wt. % Y2O3 + 5 wt. % SiO2


plot for AlN ceramics, processed with 3 wt. % Y2O3 and

different amounts of SiO2 as sintering additives.


217

AlN by reducing the defect concentration (Al vacancies) in the AlN lattice. For

higher dopant levels the thermal conductivity decreases as the thermal conductivity

of yttrium aluminates (and Y2O3) is much lower than that for AlN [4, 55] resulting

in an increase of the observed slope value too.

The lowest slope value A' for the isotropic β-Si3N4 samples equals

110 - 130 m-2 s K-1 (Table 9-6). The slope observed for the sample with the highest

thermal diffusivity/conductivity (SN5) equals (129.1 ± 2.9 m-2 s K-1). The addition

of MgO and Y2O3 has only a limited influence on the slope, whereas in contrast the

addition of Al2O3 has a strong effect (Fig. 9-6). The reason for this different

behaviour is that the Al2O3 addition can dissolve into the β-Si3N4 lattice resulting

in the formation of a β-sialon (Si6-zAlzOzN8-z), whereas Y2O3 and MgO can only

react with SiO2 on the surface of the Si3N4 grains to form a separate secondary

phase. The relatively large scattering in the data points for the samples A to D,

especially at higher temperatures (Fig. 9-6), can be partially ascribed to the

inaccuracy introduced when obtaining the data from a plot of ref. 48.

For a lower β content of the α /β-Si3N4 composite ceramics (Table 9-3) the

observed slope increases (Table 9-6). This observation can be explained in view of

0.0E+00

0.5E+05

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

3.5E+05

4.0E+05

0 500 1000 1500 2000T [K]

a-1 [s

m-2

]

D 4wt% Al2O3

C 4wt% Y2O3

B 4wt% MgOA no additivesSN5 β -Si3N4


plot for β -Si3N4 ceramics processed in different ways.

Chapter 9.

218

the difference between the crystal structure of the α- and β-modifications of Si3N4.

As the α-modification is more complex than the β-modification (α-Si3N4: n = 28;

β-Si3N4: n = 14) it is expected for α-Si3N4 to have a higher value for the slope A'

(= bA/θ~ = bAn1/3/θ ~ n, assuming that b, A, and θ are about the same for both

modifications) and thus a lower intrinsic thermal diffusivity/conductivity than

β-Si3N4.

Table 9-6: Slope and intercept values with R-value of linearly fitted inverse thermal diffusivity (a -1)

versus the absolute temperature (T ) and measured room temperature (~ 300 K) thermal

conductivity κ300 for several (β -)Si3N4 ceramic samples.


[m-2 s K-1] [m-2 s] [-] [W m-1 K-1]

SN5 [51] 129.1 ± 2.9 - 22.84 ± 3.14 103 0.9985 122A [48] (without additive) 125.7 ± 6.3 42.57 ± 4.49 103 0.9838 30B (4 wt. % MgO) 143.6 ± 5.1 38.46 ± 3.51 103 0.9910 29D (4 wt. % Al2O3) 251.3 ± 19.3 106.38 ± 13.67 103 0.9636 14C (4 wt. % Y2O3) 159.0 ± 9.5 64.24 ± 6.71 103 0.9910 22

[49] 210.6 ± 4.3 20.36 ± 3.18 103 0.9990 28 199.8 ± 6.6 31.35 ± 4.82 103 0.9967 26 146.2 ± 5.4 48.01 ± 3.91 103 0.9960 24-100 143.5 ± 4.4 38.24 ± 4.02 103 0.9986 28- 90 143.8 ± 6.6 53.44 ± 4.82 103 0.9937 22- 34 169.2 ± 10.4 83.86 ± 7.56 103 0.9889 16+ 6/0 112.1 ± 1.3 - 3.77 ± 1.20 103 0.9998 73+ 5/1 116.0 ± 1.2 6.40 ± 1.10 103 0.9998 53+ 4/2 127.7 ± 3.6 26.94 ± 3.30 103 0.9988 35+ 3/3 143.5 ± 4.4 38.24 ± 4.02 103 0.9986 28+ 2/4 150.9 ± 4.7 50.57 ± 4.34 103 0.9985 23+ 1/5 137.5 ± 7.4 67.91 ± 6.71 103 0.9957 21+ 0/6 146.2 ± 6.1 92.43 ± 5.53 103 0.9974 17

tape-casting direction [6] 84.5 ± 2.1 -8.96 ± 1.52 103 0.9954 155Stacking direction 187.0 ± 4.8 -22.29 ± 3.10 103 0.9960 70


219

A nice illustration of the influence of the type and amount of additive on the

slope and intercept values can be obtained from the data of Watari [49] who

studied the influence of in total 6 mol % Y2O3 and Al2O3 addition on the thermal

conductivity of β-Si3N4 (Table 9-6 and Fig. 9-7). It can be concluded that Y2O3

without Al2O3 is an effective additive for increasing the thermal conductivity of

β-Si3N4 because it does not dissolve in the lattice (slope A' ≈ constant

≈ 110 m-2 s K-1) and decreases the intercept B' (< 0), whereas with increasing

Al2O3/Y2O3 ratio a sialon is formed resulting in a change of the lattice

characteristics (increase of the slope A' (= bAn1/3/θ ) due to lowering θ as a

consequence of Si-N → Al-O replacement) and defect concentration (increase of

the intercept B' due to Al on Si site and O on N site acting as scattering centres for

phonons) (Fig. 9-7 and Table 9-6).

Recently it was demonstrated that the thermal diffusivity/conductivity of

β-Si3N4 is strongly anisotropic [57]. This observation is fully supported by the a -1

versus T plot of thermal conductivity data of a tape-cast sample (Fig. 9-8) obtained

from Ref. 6 showing two different slope values, the one in the casting direction

(predominantly along c-axis) below the typically observed value and the one in the

stacking direction (predominantly along a-axis) above the typically observed slope

0.0E+00

0.5E+05

1.0E+05

1.5E+05

2.0E+05

2.5E+05

3.0E+05

0 500 1000 1500T [K]

a-1 [s

m-2

]

6 mol % Y2O3 / 0 mol % Al2O3

4 mol % Y2O3 / 2 mol % Al2O3

2 mol % Y2O3 / 4 mol % Al2O3

0 mol % Y2O3 / 6 mol % Al2O3


plot for β -Si3N4 ceramics using mixtures of Y2O3 and Al2O3

as sintering additives.

Chapter 9.

220

value (Table 9-6). It is worth noting that the average slope value of the tape-cast

sample equals 133 m-2 s K-1 (= 3(1/84.5 + 2/187)-1) which is very close to the value

observed (129 m-2 s K-1) for the best isotropic sample SN5 (Table 9-6).

5. Discussion

From the results of the a -1 versus T plots it is clear that these plots can be very

useful for optimisation of the thermal diffusivity/conductivity. The data of a

material processed in different ways can be used to study the influence of several

additives. Increase of the slope indicates that the additive dissolves into the lattice,

whereas a decrease in slope or intercept indicates that the additive improves the


5.1. Interpretation of the fitting parameters

In general the observed slopes A' for the three materials have a typical constant

value (MgSiN2: 400 - 430 m-2 s K-1 (Table 9-4); AlN: 80 - 90 m-2 s K-1 (Table 9-5);

0.0E+00

0.5E+05

1.0E+05

1.5E+05

2.0E+05

2.5E+05

0 250 500 750 1000 1250T [K]

a-1 [s

m-2

]

Tape-casting directionStacking directionisotropic β -Si3N4 (SN5)


plot for β -Si3N4 ceramics along the casting and stacking

direction as compared to an isotropic sample.


221

and β-Si3N4: 110 - 130 m-2 s K-1 (Table 9-6)) and deviations from this constant

value can be explained in view of the lattice characteristics. For the samples with

the same lattice characteristics (constant A') but with different impurity content and

microstructure a relatively large variation in the intercept value B' can be observed.

Considering the large variation in thermal conductivity observed for the samples

with a approximately constant slope value for the inverse thermal diffusivity versus

absolute temperature plot, it can be concluded that all phonon scattering processes,

except the intrinsic phonon-phonon scattering, are indeed (almost) temperature

independent. As expected from the theory moreover also negative intercept values

are found. This indicates that the presented theoretical concept has a sound basis.

5.2. Thermal conductivity estimates for MgSiN2, AlN and β-Si3N4

In order to estimate the maximum achievable theoretical thermal diffusivity/

conductivity (B = 0), besides the slope A' (= bA/θ~ ) the (theoretical) intercept with

the a -1-axis (= -½ A) or the T-axis (= θ~ /2b) should be known. For the present

discussion the intercept with the T-axis was used as this value is only dependent on

θ~ and b. For estimation of this intercept with the T-axis at T = θ~ /2b (= θ /2bn1/3)

reliable values for θ and b are needed.

The Debye temperature can be evaluated from elastic constants or heat

capacity data near 0 K [11] resulting in θ0. Recently the θ0 data obtained from

elastic constants for MgSiN2, AlN and Si3N4, have been reported [58] (Table 9-7).

The number of atoms per primitive unit cell n can be obtained from

crystallographic data (Table 9-7). This results in a reduced Debye temperature ( 0~θ )

of 357, 592 and 396 K for MgSiN2, AlN and Si3N4, respectively. As expected, AlN

with the highest 0~θ shows the lowest slope value A' (Table 9-7). The value of b for

describing the temperature dependence of the thermal diffusivity is not exactly

known. Based on the simple Debye theory it can be argued that b = 2 [17, 18, 24,

26]. This results in a theoretically calculated intercept of 89, 148 and 99 K for pure,

defect free MgSiN2, AlN and Si3N4 ceramics, respectively (Table 9-7).

Chapter 9.

222

Table 9-7: The measured slope A' (= bA/θ~ ), the Debye temperature θ0, the

number of atoms per primitive unit cell n, the resulting reduced

Debye temperature 0~θ (= θ0/n1/3) and the calculated intercept

(= 0~θ /2b with b = 2) for MgSiN2, AlN and β -Si3N4.

Material slope A' θ0 [58] n 0~θ intercept

[m-2 s K-1] [K] [-] [K] [K]

MgSiN2

AlNβ-Si3N4

4.0 - 4.3 102

0.8 - 0.9 102

1.1 - 1.3 102

900940955

16 [59] 4 [60] 14 [61]

357592396

8914899

For T ≥ θ~ /2 (T ≥ 179 K, 296 K and 198 K for MgSiN2, AlN and β-Si3N4,

respectively) and T ≤ 2θ the maximum achievable thermal diffusivity of MgSiN2,

AlN and β-Si3N4 can be estimated by using the linear extrapolation method

of temperature dependent thermal diffusivity measurements.

Using the theoretical intercept with the T-axis (Table 9-7) and the

experimental slope values for non-optimised (badly-conducting) fully dense

MgSiN2 (409.8 m-2 s K-1 for sample RB11 (Table 9-4)), AlN (104.8 m-2 s K-1 for

sample AlN without additive [4] (Table 9-5)) and β-Si3N4 (125.7 m-2 s K-1 for

sample A [48] (without additive) (Table 9-6)) ceramics using no additives during

sintering (so the influence on A' is limited) a conservative estimate of the maximum

achievable (intrinsic) thermal diffusivity was made. From these data the

corresponding thermal conductivity was calculated using the heat capacity and the

density data of the corresponding materials. This resulted in the prediction that the

room temperature thermal conductivity can be at least improved from 16 to

28 W m-1 K-1 for MgSiN2, from 70 to 153 W m-1 K-1 for AlN and 30 to

82 W m-1 K-1 for β-Si3N4. Considering the quality of the used samples reasonable

estimates as compared to the highest experimental values of MgSiN2 (23 W m-1 K-1

[15]), AlN (246 W m-1 K-1 [4] - 266 W m-1 K-1 [62]) and β-Si3N4 (106 W m-1 K-1

[57] - 122 W m-1 K-1 [51]) for isotropic materials are obtained, indicating that this


223

method is in general very powerful in providing an indication of the maximum

achievable thermal conductivity of optimised ceramics.

Taking the typical experimentally observed slope A' and the theoretically

calculated intercept (Table 9-7), the maximum achievable thermal diffusivity at

300, 600 and 900 K was calculated (Table 9-8) for MgSiN2, AlN and Si3N4. From

these data the corresponding maximum achievable thermal conductivity was

calculated using the heat capacity and the density data of the corresponding

materials and compared with the highest reported experimental values as a function

of the temperature for MgSiN2 (300 - 900 K, RB34, see Fig. 9-1 and Table 9-4)

[15], AlN (0.4 - 1800 K, (almost) pure single crystal along the c-axis) [41] and

β-Si3N4 (300 - 1700 K) [51].

Table 9-8: The estimates for the maximum achievable thermal diffusivity aThe usig the data

of Table 9-7 for MgSiN2, AlN and β -Si3N4 (at 300, 600 and 900 K) and resulting

thermal conductivity κ The (obtained from the molar density ρm and heat capacity

Cp), compared with corresponding highest experimentally observed thermal

conductivity κ Exp.

Material a ρm Cp = CV κThe κExp

[m2 s-1] [mol m-3] [J mol-1 K-1] [W m-1 K-1] [W m-1 K-1]

300 KMgSiN2

AlNβ-Si3N4

1.10 - 1.18 10-5

7.31 - 8.22 10-5

3.83 - 4.52 10-5

3.90 104

7.94 104

2.29 104

61.7 30.6 90.6

26 - 28 178 - 200 79 - 94

23246 - 285106 - 122

600 KMgSiN2

AlNβ-Si3N4

0.46 - 0.49 10-5

2.46 - 2.77 10-5

1.54 - 1.81 10-5

3.88 104

7.91 104

2.28 104

88.1 44.0 144.5

16 - 17 86 - 96 51 - 60

159663

900 KMgSiN2

AlNβ-Si3N4

0.29 - 0.31 10-5

1.48 - 1.66 10-5

0.96 - 1.13 10-5

3.86 104

7.87 104

2.27 104

95.6 47.7 157.0

11 - 11 56 - 62 34 - 40

115541

Chapter 9.

224

For MgSiN2 ceramics the highest experimentally obtained thermal

conductivity at 300 K does not exceed 25 W m-1 K-1 despite the fact that already

considerable effort has been made to improve the thermal conductivity [9, 34, 36,

37, 63, 64]. As the predicted value of 26 - 28 W m-1 K-1 is close to this value it can

be concluded that the highest experimentally observed value is close to the intrinsic

one. From Table 9-8 and Fig. 9-2 it is obvious that a further reduction of the defect

concentration in the MgSiN2 lattice will not result in a significant increase of the

thermal diffusivity/conductivity, because for the best samples the intercept with the

T-axis (Fig. 9-2) is already very close to the theoretical value of 89 K (Table 9-7).

The estimates for β-Si3N4 are in reasonable agreement with the highest

experimentally observed values indicating that the thermal conductivity cannot be

significantly increased. The measured value of 122 W m-1 K-1 at 300 K [51] is

somewhat higher than the expected 106 W m-1 K-1 estimated from the measured

thermal conductivity along the c-axis (180 W m-1 K-1) and a-axis (69 W m-1 K-1)

for a single grain [57].

However, the estimate at 300 K (178 - 200 W m-1 K-1) for the thermal

conductivity of AlN is significantly lower than the observed experimental values

for isotropic materials (246 - 266 W m-1 K-1). Evidently the intercept with the

T-axis is underestimated. This is caused by underestimation of the reduced Debye

temperature as also optic phonons contribute to the heat conduction [10] whereas

only acoustic phonons are considered. This underestimation of the reduced Debye

temperature θ~ also results in a too low value for the maximum achievable thermal

conductivity as estimated with the Slack equation [10]. When using the a -1 versus

T method, both the slope A' (= bA/θ~ ) and the (theoretical) intercept with the T-axis

(= θ~ /2b) are related to θ~ . However, especially at high temperatures the influence

of θ~ on the estimate is limited as a -1 = A'T + B' ≈ A'T and A' is determined

experimentally (see Table 9-8: 600 and 900 K estimates).

In general the theoretical estimates are in good agreement (within 20 %) with

the best experimentally observed values (Table 9-8), unless also optic phonons

contribute substantially to the heat conduction (like in the case of AlN), resulting in


225

an underestimation of the intercept with the T-axis (= θ~ /2b). However, at higher

temperatures (T / 3 × 041 ~θ ) the exact value of the intercept with the T-axis

becomes less important, and therefore also the influence of optic phonons

contributing to the heat conduction, as a -1 = A'T + B' ≈ A'T (high T ) resulting in a

better agreement between the estimated and the experimentally observed thermal

conductivity. At high temperatures the accuracy of the estimate is consequently

determined by the error in the slope A'. Furthermore, these calculations directly

indicate that the thermal conductivity of AlN and β-Si3N4 is relatively high

whereas that of MgSiN2 is limited. So, for applications where a high thermal

conductivity is required β-Si3N4 (and AlN) is a more interesting compound than

MgSiN2.

5.3. Comparison with other estimates

If the here presented estimated value of the maximum achievable thermal

conductivity of MgSiN2 ceramics at 300 K of 26 - 28 W m-1 K-1 is compared with

previous estimates which have been made for MgSiN2 ceramics at 300 K

(Table 9-9), it is noticed that the presented value is the lowest of all. Especially the

difference with the first estimated values using the theory of Slack is considerable.

During time these values were adjusted downwards, as more accurate input

parameters became available [38, 39, 58].

The estimate of the thermal conductivity for AlN (178 - 200 W m-1 K-1) is

much lower than that of 319 W m-1 K-1 at 300 K [41]. In the meantime this value of

319 W m-1 K-1 has been widely interpreted as being the true intrinsic value for AlN.

However, this value was obtained by correcting the measured thermal conductivity

for defect scattering by oxygen impurities using the experimental value of

285 W m-1 K-1 measured on a single crystal providing the thermal conductivity

along the c-axis [41]. This axis has the lowest thermal expansion coefficient [65]

and therefore it is expected to show the highest thermal conductivity, as it is

empirically known that in general the direction with the lowest thermal expansion

Chapter 9.

226

Table 9-9: Theoretical estimates of the thermal conductivity for MgSiN2, AlN and β -Si3N4 ceramics

at 300 K. For comparison the highest measured values (κ Exp) are also given (* probably

based on n-2/3 dependence of the Slack equation [11] and the intrinsic estimate of about

300 W m-1 K-1 for AlN).

Estimated Value

[W m-1 K-1]

Reference Estimation Method Based On

MgSiN2 κExp = 23 W m-1 K-1 [15]

time

→

26 - 2827 - 35

3426 ± 437 ± 1335 - 5040 - 70

75120

this work[10][10][15][15][63][63][66][8]

a -1 versus Tmodified Slack equationstandard Slack equation

thermal diffusivity measurementsmodified Slack equation

defect scatteringSlack equationSlack equationnot specified*

AlN κExp = 266 W m-1 K-1 [62]

time

→

178 - 200124 - 209

128319320

this work[10][10][41][2]

a -1 versus Tmodified Slack equationstandard Slack equation

defect scattering

scaling factor 3δθM

β-Si3N4 κExp = 106 W m-1 K-1 [57]

time

→

79 - 9484 - 116

124177

200 - 320

this work[10][10][68][5]

a -1 versus Tmodified Slack formulastandard Slack formula

two-phase composite modelSlack equation

coefficient has the highest thermal conductivity [67]. Also the observed slope value

in the a -1 versus T plot for this sample (Fig. 9-3) differed from the typically

observed value confirming the anisotropic behaviour of the thermal conductivity.

Therefore, the present author has the feeling that the estimate of 319 W m-1 K-1 [41]


227

related to one direction is too high for isotropic AlN, and the intrinsic thermal

conductivity for isotropic AlN equals about the highest experimentally observed

thermal conductivity of 246 - 266 W m-1 K-1 [4, 62] for isotropic samples. So, the

here presented estimated value of 178 - 200 W m-1 K-1 is lower as compared to the

true intrinsic value of about 246 - 266 W m-1 K-1 for isotropic material [4, 62].

However, this estimate of 178 - 200 W m-1 K-1 (at 300 K) is in much better

agreement with the highest experimentally observed value than the estimates

obtained using the Slack formula or the modified Slack formula (Table 9-9).

The present estimate of 79 - 94 W m-1 K-1 for the intrinsic thermal

conductivity of β-Si3N4 is significantly lower than the first reported estimate of

200 - 320 W m-1 K-1 [5], whereas it is in good agreement with previous estimates

(Table 9-9) based on the Slack equation, the modified Slack equation, and the

experimentally measured value on a single grain (106 W m-1 K-1 [57]), indicating

that the present estimate is only slightly lower than the intrinsic value of β-Si3N4.

The measured thermal conductivity along the c-axis of 180 W m-1 K-1 is in good

agreement with the value estimated using a two-phase composite model resulting

in a value of 177 W m-1 K-1 [68] along the c-axis. This provides some further

confidence that the intrinsic value equals about 106 W m-1 K-1 [57] indicating that

the value of 122 W m-1 K-1 [51] was measured on a somewhat anisotropic sample.

5.4. Limitations, accuracy and reliability

It should be noted that the new estimation method based on equation (5) was

obtained by approximating an already simple description of the (temperature

dependence of the) thermal diffusivity (equation (4)) of a pure phonon conductor.

Furthermore the value of b = 2 in equation (4), which was used to calculate the

intercept with the T-axis (= θ~ /2b), may differ somewhat from 2 and vary from

substance to substance [17]. Leibfreid et al. [20] suggest that for a FCC lattice

b = √5/3. However, some scarce experimental results confirm the value of b ≈ 2

(2.3, 2.7 and 2.1 for solid helium, diamond and sapphire, respectively [18]). Also

the choice of θ = θ0 is somewhat arbitrary. In general the Debye temperature θ can

Chapter 9.

228

be obtained from elastic constants or heat capacity data [31, 58] resulting in θ E and

θ C, respectively. Assuming that the acoustic phonons are the major heat carriers

the high temperature limit of the Debye temperature based on the acoustic phonons

θ∞A is needed [11] to evaluate the reduced Debye temperature. In a previous paper

[10] a more appropriate estimate for θ∞A based on heat capacity data was presented

resulting in somewhat lower values for θ . However, if also optic phonons

contribute to the heat conduction, which is to some extend always the case for

n > 1, this estimate for θ is too low. So, for practical use the choice of θ = θ0

seems to be a good compromise as it can be easily obtained from elastic constants,

and in combination with the assumption that b = 2 seems to results in reasonable

estimates for the intercept with the T-axis.

The presented estimation method seems to be more reliable than the

theoretical Slack equation [11] or the modification of this equation [10]

(Table 9-9). This can be explained in view of the influence of the accuracy of the

reduced Debye temperature on the resulting estimate. The (more complicated)

Slack equation is very sensitive for relatively small deviations in the input

parameters [10], whereas the here presented method is relatively easy and less

sensitive for small deviations in slope and intercept. In general, the Slack equation

is especially useful when no samples for thermal diffusivity/conductivity

measurements are available whereas the a -1 versus T plots give a more accurate

indication of the maximum achievable thermal conductivity, and moreover can be

used to guide the optimisation of the processing in order to obtain the desired


6. Conclusions

A new simple method for estimating the maximum achievable (intrinsic) thermal

conductivity of non-metallic compounds was presented based on temperature

dependent thermal diffusivity measurements. Its strength is that non-optimised

samples can be used to provide a good impression of the intrinsic thermal


229

conductivity. It was successfully applied to MgSiN2 and β-Si3N4 providing some

evidence for its general applicability. For AlN too low estimates were obtained due

to the fact that optic phonons, which are not considered when using this method,

contribute substantially to the heat conduction. However, in general the estimates

are accurate within 20 % and become more accurate with increasing temperature,

independent of the fact whether or not optic phonons contribute substantially to the

heat conduction. Furthermore, the method is a useful tool for optimising the

processing as it enables discrimination between the lattice characteristics, defects

and microstructure.

References


Electroceramics IV 2, Aachen (Germany), September 5 - 7 1994, edited by R.

Waser, S. Hoffmann, D. Bonnenberg and Ch. Hoffmann (Augustinus






Compon., Hybrids, Manuf. Technol., CHMT-7 (1984) 399 - 404.

4. T.B. Jackson, A.V. Virkar, K.L. More, R.B. Dinwiddie, Jr. and R.A. Cutler,



(1997) 1421.

5. J.S. Haggerty and A. Lightfoot, Oppertunities for enhancing the thermal

conductivities of SiC and Si3N4 ceramics through improved processing,

Ceram. Eng. Sci. Proc. 16, 19th Annual Conference on Composites, Advanced

Ceramics, Materials, and Structures-A, Cocoa Beach, Florida, USA, January

8 - 12 1995, edited by J.B. Wachtman (The American Ceramic Society,

Westerville, OH, 1995) 475.

Chapter 9.

230



14 (1999) 1538.



Ceram. Soc. 79 (1996) 2485.





1994) 239.




10. Chapter 8; R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, Room

Temperature Estimates of the Thermal Conductivity of MgSiN2, AlN and

β-Si3N4 using a Modified Slack Equation, to be published.




12. F.R. Chavat and W.D. Kingery, Thermal Conductivity: XIII, Effect of


40 (1957) 306.

13. D.-M. Liu and B.-W. Lin, Thermal Conductivity in Hot-Pressed Silicon

Carbide, Ceram. Int., 22 (1996) 407.

14. W.D. Kingery, Thermal Conductivity: XII, Temperature Dependence of

Conductivity for Single-Phase Ceramics J. Am. Ceram. Soc. 38 (1955) 251.

15. R.J. Bruls, H.T. Hintzen and R. Metselaar, Modeling of the Thermal



Expansion Symposium, Pittsburgh, Pennsylvania, USA, October 26 - 29,


231

1997, edited by P.S. Gaal and D.E. Apostolescu (Technomic Publishing Co.,

Inc., Lancaster, 1999) 3.

16. R. Berman, Thermal Conduction in Solids, Clarendon Press, Oxford (1976).

17. P. Debye, Zustandsgleichung und Quantenhypotheses mit einem Anhang über

Wärmeleitung in: Vorträge über die Kinetische Theorie der Materie und der

Elektrizität (Teubner, Berlin, 1914), pp. 19 - 60.

18. R. Berman, The Thermal Conductivity of Dielectric Solids at Low

Temperatures, Advances in Phys. 2 (1953) 103.

19. P.G. Klemens, Theory of the Thermal Conductivity of Solids, Thermal

Conductivity 1, edited by R.P. Tye (Academic Press, London, 1969).



21. P.G. Klemens, The thermal conductivity of dielectric solids at low

temperatures, Proc. Roy. Soc. (London), A208 (1951) 108.

22. P.G. Klemens, Thermal Conductivity and Lattice Vibrational Modes, Solid

State Phys. 7, edited by F. Seitz and D. Turnbull (Academic Press Inc., New

York, 1958) 1.



Sci. 28 (1993) 3709.

24. R. Peierls, Zur kinetischen Theorie der Wärmeleitung in Kristallen, Ann. Phys.

3 (1929) 1055.

25. M. Roufosse and P.G. Klemens, Thermal Conductivity of Complex Dielectric

Crystals, Phys. Rev. B. 7 (1973) 5379.

26. J.R. Drabble and H.J. Goldsmid, in International Series of Monographs on

Semiconductors 4, Thermal Conduction in Semiconductors, edited by

H.K. Henisch (Pergamon Press, Oxford, 1961), p. 141.

27. A.K. Collins, M.A. Pickering and R.L. Taylor, Grain size dependence of the

thermal conductivity of polycrystalline chemical vapor deposited β-SiC at low

temperatures, J. Appl. Phys. 68 (1990) 6510.

Chapter 9.

232

28. P.G. Klemens, Thermal Resistance due to Point Defects at High

Temperatures, Phys.Rev. 119 (1960) 507.

29. V. Ambegaokar, Thermal Resistance due to Isotopes at High Temperatures,

Phys. Rev. 114 (1959) 488.

30. P.G. Klemens, Thermal Conductivity of Lattice Vibrational Modes, Solid

State Phys. 7, edited by F. Seitz and D. Turnbull (Academic Press Inc.,

Publishers, New York, 1958) 1.


32. R.C. Enck and R.D. Harris, Temperature Dependence of Thermal

Conductivity of Electronic Ceramics by an Improved Flash Diffusivity

Technique, Mat. Res. Soc. Proc. 167, Advanced Electronic Packaging

Materials, Boston, Massachusetts, USA, November 27 - 29 1989, edited by

A.T. Barfknecht, J.P. Partridge, C.J. Chen and C.-Y. Li (Materials Research

Society, Pittsburgh, 1990) 235.

33. M.C. Roufosse and P.G. Klemens, Lattice Thermal Conductivity of Minerals

at High Temperatures, J. Geophys. Res. 79 (1974) 703.



published.

35. W.J. Parker, R.J. Jenkins, C.P. Butler and G.L. Abbott, Flash Method of

Determining Thermal Diffusivity, Heat Capacity, and Thermal Conductivity,

J. Appl. Phys. 32 (1961) 1679.


Preparation and Densification of MgSiN2, Ceram. Trans. 51 Int. Conf. Cer.


Hausner, G.L. Messing and S. Hirano (The American Ceramic Society,

1995) 585.




233

38. Chapter 7; R.J. Bruls, H.T. Hintzen, G. de With, R. Metselaar and J.C. van

Miltenburg, The temperature dependence of the Grüneisen parameter of

MgSiN2, AlN and β-Si3N4, submitted to J. Phys. Chem. Solids.



(1998) 7871.

40. R.A. Swalin, Thermodynamics of Solids, second edition (John Wiley & Sons,

Inc., New York, 1972) 82.



42. M. Rohde and B. Schulz, Thermal Conductivity of Irradiated and

Non-irradiated Fusion Ceramics, 21st International Thermal Conductivity

Conference and 9th International Thermal Expansion Symposium (ITCC 21

and ITES 9), 1990, edited by C.J. Cremers and H.A. Fine (Plenum Publishing,

New York, 1990) 509.

43. T.B. Jackson, K.Y. Donaldson and D.P.H. Hasselman, Temperature

Dependence of the Thermal Diffusivity/Conductivity of Aluminum Nitride, J.

Am. Ceram. Soc. 73 (1990) 2511.

44. P.S. de Baranda, Thermal Conductivity of Aluminum Nitride, Ph. D. Thesis,

Rutgers University, Piscataway, NJ, USA (1991) pp. 1 - 294.

45. P.S. de Baranda, A.K. Knudsen and E. Ruh, Effect of CaO on the Thermal


46. P.S. de Baranda, A.K. Knudsen and E. Ruh, Effect of Silica on the Thermal


47. A. Witek, M. Bockowski, A. Presz, M. Wróblewski, S. Krukowski, W.

Wlosinski and K. Jablonski, Synthesis of oxygen-free aluminum nitride

ceramics, J. Mater. Sci. 33 (1998) 3321.

48. K. Tsukuma, M. Shimada and M. Koizumi, Thermal Conductivity and

Microhardness of Si3N4 with and Without Additives, Am. Ceram. Soc. Bull. 60

(1981) 910.

Chapter 9.

234



Ed. 97 (1989) 170.

50. V.E. Peletskii, The Investigation of Thermal Conductivity of Silicon Nitride,

High Temp. 31 (1993) 668 (translated from Teplofizika Vysokikh Temperatur

31 (1993) 727).

51. N. Hirosaki, Y. Okamoto, M. Ando, F. Munakata and Y. Akimunde, Effect of

Grain Growth on the Thermal Conductivity of Silicon Nitride, J. Ceram. Soc.

Jpn. Int. Ed. 104 (1996) 50.

52. K. Watari, M.E. Brito, M. Toriyama, K. Ishizaki, S. Cao and K. Mori,

Thermal conductivity of Y2O3-doped Si3N4 ceramics at 4 to 1000 K, J. Mater.

Sci. Lett. 18 (1999) 865.

53. G.A. Slack and T.F. McNelly, AlN Single Crystals, J. Crystal Growth 42

(1977) 560.

54. R.H. Bogaard, personal communication, providing the raw data for ref. 32

and 42.

55. A.V. Virkar, T.B. Jackson and R.A. Cutler, Thermodynamic and Kinetic

Effects of Oxygen Removal on the Thermal Conductivity of Aluminum

Nitride, J. Am. Ceram. Soc. 72 (1989) 2031.

56. H. Buhr, G. Müller, H. Wiggers, F. Aldinger, P. Foley and A. Roosen, Phase

Composition, Oxygen Content, and Thermal Conductivity of AlN(Y2O3)

Ceramics, J. Am. Ceram. Soc. 74 (1991) 718.




58. Chapter 6; R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The

Temperature Dependence of the Young's Modulus of MgSiN2, AlN and Si3N4,





235


Properties of Aluminum Nitride, Phys. Rev. B. 57 (1998) 10550.

61. R. Grün, The Crystal Structure of β-Si3N4; Structural and Stability

Considerations Between α- and β-Si3N4, Acta Cryst. B35 (1979) 800.

62. M. Okamoto, H. Arakawa, M. Oohashi and S. Ogihara, Effect of

Microstructure on Thermal Conductivity of AlN Ceramics, J. Ceram. Soc.

Jpn. Inter. Ed. 97 (1989) 1486.


Ceramics, Fourth Euro Ceramics 2, Faenza (Italy), October 1995, edited by

C. Galassi (Gruppo editoriale Faenza editrice S.p.A., Faenza, 1995) 289.

64. I.J. Davies, H. Uchida, M. Aizawa, and K. Itatani, Physical and Mechanical




Soc. Symp. Proc. 482, Nitride Semiconductors, edited by F.A. Ponce,

S.P. DenBaars, B.K. Meyer, S. Nakamura and S. Strite (Materials Research

Society, Warrendale, Pennsylvania, 1998) 863.


(Oxy)-nitride Ceramics, Fourth Euro Ceramics 3, Basic Science -

Optimisation of Properties and Performances by Improved Design and



67. W.D. Kingery, The Thermal Conductivity of Ceramic Dielectrics, Progress in

Ceramic Science 2, edited by J.E. Burke (Pergamon Press Ltd., Oxford, 1962),

pp. 182 - 235.



Ceram. Soc. 79 (1996) 2485.

Chapter 9.

236

237

Chapter 10.

Conclusions

In Chapter 1 the importance of non-metallic materials showing a high thermal

conduction by phonons and the need for new materials with desirable thermal

properties is pointed out. It is concluded that (1) for substrate applications, where a

high thermal conductivity is needed, the traditional oxide materials are replaced by

nitride materials, and (2) the relatively new material MgSiN2 might be potentially

interesting (based on an estimated thermal conductivity of 120 W m-1 K-1 at room

temperature reported in the literature).

In chapters 2 and 3 it is shown that the processing of MgSiN2 can be

optimised by using pure starting materials and suitable reaction conditions.

MgSiN2 powders and ceramics with an oxygen content far below 1 wt. % are

obtained as compared to previously synthesised materials containing about 4 wt. %

oxygen. However, the thermal conductivity at room temperature of the resulting

ceramics did not exceed 25 W m-1 K-1.

Therefore, also the theoretical thermal conductivity of MgSiN2, AlN and

β-Si3N4 was considered. It turned out that the theoretical thermal conductivity

calculated using Slack's formula is relatively sensitive for small variations in the

Debye temperature θ and Grüneisen parameter γ which are needed as input

parameters. The Debye temperature was calculated using either heat capacity data

(chapter 5 and 8) or elastic constants (chapter 6), whereas for the evaluation of the

Grüneisen parameter (chapter 7) both are needed in combination with thermal

expansion data (chapter 3, 4 and 7). The elastic constants are almost temperature

independent and decrease almost linearly for T > 450 K whereas the thermal

expansion coefficient and heat capacity show an S-shaped increase as a function of

Chapter 10.

238

the absolute temperature approaching a constant value at higher temperatures

(T > θ ≈ 1000 K). From these data the Debye temperature near 0 K (chapter 6) and

the Grüneisen parameter as a function of the absolute temperature (chapter 7) for

MgSiN2, AlN and β-Si3N4 were obtained. The Debye temperatures at 0 K (θ 0 ) of

MgSiN2, AlN and β-Si3N4 are about the same (θ 0 ≈ 900 - 950 K). The Grüneisen

parameter increases as a function of the temperature approaching a constant value

(γ ≈ 1.0 for AlN and the structurally related MgSiN2, and γ ≈ 0.63 for β-Si3N4) at

high temperatures (T/θ ≥ 0.8).

Using Slack's equation (chapter 8), reasonable estimates for the thermal

conductivity at the Debye temperature as compared to experimental values (within

20 %) were obtained. For extending the validity of the Slack equation below the

Debye temperature, the equation was modified resulting in a more realistic

description of the temperature dependence. This justified the theoretical calculation

of the thermal conductivity of MgSiN2, AlN and β-Si3N4 down to the more

interesting room temperature region. The resulting estimates for the maximum

achievable thermal conductivity were in rough agreement with the highest

experimental values providing a reasonable indication for the usefulness of the

Slack equation. From the calculations it became clear that at room temperature AlN

has a high thermal conductivity (>> 100 W m-1 K-1), MgSiN2 a low thermal

conductivity (<< 100 W m-1 K-1) and β-Si3N4 a thermal conductivity in between

(~ 100 W m-1 K-1).

Considering the limitations of the Slack equation a new method for

estimating the maximum achievable thermal conductivity is proposed, based on

temperature dependent thermal diffusivity measurements on non-optimised

samples using a linear extrapolation method (chapter 9). The estimates obtained

with this method at room temperature for MgSiN2 (26 - 28 W m-1 K-1), AlN

(178 - 200 W m-1 K-1) and β-Si3N4 (79 - 94 W m-1 K-1) are in favourable agreement

with the best experimental values (23 W m-1 K-1, 266 W m-1 K-1 and 106 W m-1 K-1,

respectively) indicating the general applicability of this relatively simple method.

Furthermore this method is a strong tool for guiding the optimisation of the

Conclusions

239

material with respect to the desired thermal conductivity and can be used on

non-optimised samples in order to get an impression of the maximum achievable

value.

Finally it can be concluded that, in contrast to the expectations at the

beginning of this work, the thermal conductivity at room temperature of MgSiN2

ceramics is limited to 25 - 30 W m-1 K-1, as shown both experimentally and

theoretically, reducing its potential for applications. Moreover, as a spin-off the

described theoretical approach resulted in a new generally applicable method for

estimation of the maximum achievable thermal conductivity of non-metallic

materials. This method not only provides a good indication of the potential thermal

conductivity of new non-optimised interesting materials, reducing the time and

effort normally needed to obtain a reliable indication of the maximum achievable

thermal conductivity, but can be also used for guiding the optimisation of it.

Chapter 10.

240

241

List of symbols

Lower-case symbols

a [m2 s-1] : thermal diffusivity

a, b, c [Å] : lattice parameters

b [m] : width

cp [J kg-1 K-1] : specific heat capacity at constant pressure

d [Å] : d-value, interplanar spacing

d [µm] : particle size

f [s-1] : flexural frequency

h [J s] : Planck's constant (6.626 10-34 J s)

h [m] : height

hkl [-] : Miller indices

k [J K-1] : Boltzmann's constant (1.381 10-23 J K-1)

k [eV K-1] : Boltzmann's constant (8.62 10-5 eV K-1)

l [m] : length

l [m] : phonon mean free path

lgb [m] : phonon mean free path due to grain boundary scattering

lpd [m] : phonon mean free path due to phonon-defect scattering

lpp [m] : phonon mean free path due to phonon-phonon scattering

ltot [m] : total phonon mean free path

m [kg] : (sample) mass

n [-] : number of atoms per primitive unit cell

p [Pa] : pressure

q [J] : energy

s, st [-] : (total) number of atoms per formula unit / molecule

List of symbols

242

sa [-] : number of anions per formula unit / molecule

t [s, h, years] : time

vl [m s-1] : longitudinal sound velocity

vs [m s-1] : sound velocity

vt [m s-1] : transverse sound velocity

wi : 1/yi, weight factor

wRp [-] : Σ wi (yi(obs) - yi(calc))2/ Σ wi (yi(obs))21/2, weighted

R-pattern

x, y, z [-] : position along the x, y and z direction

yi(calc) : calculated intensity at the ith step

yi(obs) : observed (gross) intensity at the ith step

Upper-case symbols

Cp [J mol-1 K-1] : heat capacity at constant pressure

Cpo [J mol-1 K-1] : heat capacity at standard pressure

CV [J mol-1 K-1] : heat capacity at constant volume

E [GPa] : Young's modulus

FD(θ /T) [-] : Debye function

G [J mol-1 K-1] : Gibbs energy

GTo [J mol-1 K-1] : Gibbs energy function at standard pressure

GTo - H0

o [J mol-1 K-1] : energy function at standard pressure

H [J mol-1 K-1] : enthalpy

HK [GPa] : Knoop hardness

HV [GPa] : Vickers hardness

H0o [J mol-1 K-1] : standard formation enthalpy

HTo - H0

o [J mol-1 K-1] : enthalpy function at standard pressure

I/I0 [%] : relative intensity

M [kg mol-1] : mole mass

M [kg mol-1] : mean atomic mass

List of symbols

243

NA [mol-1] : Avogadro's number (6.022 1023 mol-1)

[N] [wt. %] : nitrogen content

[O] [wt. %] : oxygen content

Q [J] : energy

Qv [J, eV] : energy required for the formation of a vacancy

R [J mol-1 K-1] : gas constant (8.314 J mol-1 K-1)

R [-] : statistical R-value

Rp [-] : Σyi(obs) - yi(calc)/ Σ yi(obs), R-pattern

S [J mol-1] : entropy

STo (- S0

o) [J mol-1] : entropy function at standard pressure

T [K] : absolute temperature

T [°C] : temperature

W [Å3] : volume per anion bond

V [m3] : volume

V0 [m3 mol-1] : molar volume at 0 K

Vm [m3 mol-1] : molar volume

V [Å3] : volume of a unit cell

Z [-] : number of formula units per unit cell

Greek symbols

α [K-1] : thermal expansion coefficient

αa, αb, αc [K-1] : thermal expansion coefficient along the a-, b- and c-axis

αlat [K-1] : linear lattice thermal expansion coefficient

αlin [K-1] : linear thermal expansion coefficient

βS [Pa-1] : adiabatic compressibility

βT [Pa-1] : isothermal compressibility

χ 2 [-] : wRp/Rp, chi-square, goodness of fit

δ [m, Å] : cube root of the average volume per atom

List of symbols

244

δ 3 [m3, Å3] : average volume per atom (volume of a unit cell divided

by the number of atoms per unit cell)

δ [-] : Anderson-Grüneisen parameter

γ [-] : Grüneisen parameter

γθ [-] : Grüneisen parameter at the Debye temperature

γ∞ [K] : high temperature limit of the Grüneisen parameter

η [-] : number of bonds (per anion)

κ [W m-1 K-1] : thermal conductivity

ν [-] : Poisson's ratio

ρ [kg m-3] : density

ρm [mol m-3] : molar density

θ [K] : Debye temperature, characteristic temperature

θ0 [K] : Debye temperature at 0 K

θ∞ [K] : high temperature limit of the Debye temperature

2θ [°] : diffraction angle

245

Summary

The objective of this work was to investigate, understand and optimise the thermal

conductivity of MgSiN2 ceramics. In order to obtain a high thermal conductivity

the impurity content and especially the oxygen content in the MgSiN2 lattice was

considered to be of crucial importance. Therefore this work first concentrated on

the optimisation of the synthesis of pure MgSiN2 powder and ceramics by suitable

processing. Although, originally a high thermal conductivity (~ 120 W m-1 K-1) was

theoretically expected for MgSiN2 this value could by far not be confirmed

experimentally. Therefore, the theoretical method to predict the maximum

achievable thermal conductivity (Slack's theory) was reconsidered. This resulted in

an improved theory of Slack and moreover, the development of a new prediction

method based on temperature dependent thermal diffusivity measurements. This

was done with the intention to avoid putting a lot of time and effort in process

optimisation of materials for which less interesting thermal properties can be

expected. The improved prediction methods were also applied to the commercial

materials AlN and β-Si3N4 in order to check the general validity of the used

methods. So, an experimental as well as a theoretical approach is described in this

thesis.

The first chapters of the thesis deal with the preparation of MgSiN2 powders

and ceramics. By suitable processing it is possible to control the oxygen content of

MgSiN2 powders and ceramics. As a consequence very pure, oxygen poor

materials (<< 1 wt. %) could be obtained. However, the thermal conductivity at

room temperature of the resulting ceramics did not exceed 25 W m-1 K-1.

The middle part of the thesis deals with the properties of MgSiN2, AlN and

β-Si3N4. Most data for AlN and β-Si3N4 could be obtained from the literature. For

MgSiN2 the specific heat, thermal expansion coefficient and Young's modulus

Summary

246

were experimentally determined as a function of the temperature. These data were

used to evaluate the Debye temperature θ near 0 K and Grüneisen parameter γ,

which are important input parameters for theoretical modelling of the thermal

conductivity. The Debye temperatures are relatively temperature independent. The

Debye temperatures at 0 K (θ 0 ) of MgSiN2, AlN and β-Si3N4 are about the same

(θ 0 ≈ 900 - 950 K) and with increasing temperature the Debye temperature first

decreases and subsequently increases approaching a constant value (θ ≈ 1000 K for

MgSiN2 and AlN, and θ ≈ 1200 K for β-Si3N4) at intermediate temperatures

(T/θ / 0.3). The Grüneisen parameter increases as a function of the temperature

approaching a constant value (γ ≈ 1.0 for MgSiN2 and AlN, and γ ≈ 0.63 for

β-Si3N4) at high temperatures (T/θ ≥ 0.8).

The last chapters of the thesis deal with the theory of Slack and a new

method for predicting the maximum achievable thermal conductivity of non-

metallic solids. The assumptions made in Slack's theory are briefly discussed and

some improvements concerning the temperature dependence are presented,

resulting in a modified Slack theory. The second estimation method proposed in

this thesis is based on temperature dependent thermal diffusivity measurements on

non-optimised samples (a -1 versus T method). This method has the advantage that

it directly provides a minimum value for the maximum thermal conductivity. The

validity and limitations of these methods are discussed using MgSiN2, AlN and

β-Si3N4 as model compounds. The Slack equation provides a rough indication of

the maximum achievable thermal conductivity whereas the a -1 versus T method

provides more reliable estimates. From both estimation methods and also from the

experimental results it can be concluded that, in contrast to the expectations at the

beginning of this work, the thermal conductivity at room temperature of MgSiN2

ceramics is limited to 25 - 30 W m-1 K-1 reducing its potential for applications.

247

Samenvatting

De hoofddoelstelling van dit promotieonderzoek was het bestuderen, begrijpen en

vervolgens optimaliseren van de warmtegeleidbaarheid van het relatief nieuwe

keramische materiaal MgSiN2. Voor het verkrijgen van een hoge

warmtegeleidbaarheid werd de reductie van de concentratie verontreinigingen, met

name zuurstof, van groot belang geacht. Het eerste gedeelte van het onderzoek

concentreerde zich derhalve op de synthese van zuiver MgSiN2 poeder en

keramiek. Aangezien de na optimalisatie gemeten warmtegeleidbaarheid

(< 25 W m-1 K-1 bij kamertemperatuur) veel lager was dan de (hoge) theoretisch

voorspelde waarde (120 W m-1 K-1) werd de gebruikte methode voor de schatting

van de maximaal haalbare warmtegeleidbaarheid (Slack formule) opnieuw in detail

bekeken. Dit heeft geleid tot een verbeterde versie van de Slack formule, en

bovendien in een geheel nieuwe schattingsmethode gebaseerd op thermische

diffusiviteitsmetingen als functie van de temperatuur. Deze algemeen bruikbare

methode werd mede ontwikkeld met het oog op toekomstige vraagstellingen met

betrekking tot de thermische eigenschappen van keramische materialen. Hierdoor

kan vroegtijdig, zonder al te veel tijd en moeite te spenderen aan

procesoptimalisatie, worden onderkend welke materialen potentieel de gewenste

thermische eigenschappen bezitten. Om de algemene geldigheid van beide

schattingsmethoden te toetsen werd ook de warmtegeleidbaarheid van de

commercieel interessante materialen AlN en β -Si3N4 berekend. Dus, zowel een

experimentele als een theoretische aanpak is in dit proefschrift beschreven.

De eerste hoofdstukken beschrijven de synthese van zuiver MgSiN2 poeder

(hoofstuk 2) en keramiek (hoofdstuk 3). Met name geschikte processing maakt het

mogelijk om het zuurstofgehalte in MgSiN2 poeder en keramiek te beheersen. Dit

resulteerde in zeer zuiver, zuurstofarm materiaal (<< 1 gewichts % zuurstof).

Samenvatting

248

Desondanks bleef de warmtegeleidbaarheid van de keramiek beperkt tot

25 W m-1 K-1.

In de volgende hoofstukken worden enkele thermische en mechanische

eigenschappen van MgSiN2, AlN en β -Si3N4 behandeld, die nodig zijn om de

theoretische warmtegeleidbaarheid te berekenen. De meeste gegevens voor AlN en

β -Si3N4 waren reeds in de literatuur gerapporteerd. De thermische

expansiecoëfficiënt (hoofstuk 3 en 4), soortelijke warmte (hoofdstuk 5) en de

elasticiteitsmodulus (hoofdstuk 6) van MgSiN2 werden gemeten als functie van de

temperatuur. Deze gegevens werden gebruikt om de Debye temperatuur θ bij 0 K,

θ 0 (hoofdstuk 6) en de Grüneisen parameter γ (hoofdstuk 7) van MgSiN2, AlN and

β -Si3N4 te bepalen. Deze twee grootheden zijn belangrijke parameters voor de

theoretische modellering van de warmtegeleidbaarheid. Alle drie de materialen

blijken een relatief hoge Debye temperatuur te hebben (θ 0 ≈ 900 - 950 K). De

Grüneisen parameter vertoonde een (relatief sterke) temperatuursafhankelijkheid.

Met stijgende temperatuur nam de Grüneisen parameter eerst toe om vervolgens bij

hogere temperatuur (T/θ ≥ 0.8 met θ ≈ 1000 K) constant (γ ≈ 1.0 voor MgSiN2 en

AlN, en γ ≈ 0.63 voor β -Si3N4) te worden.

De laatste twee hoofdstukken van dit proefschrift behandelen de aanpassing

van de Slack formule (hoofdstuk 8) en een nieuwe methode om de maximale

warmtegeleidbaarheid af te schatten (hoofdstuk 9). De toepasbaarheid en

beperkingen van beide methoden worden besproken aan de hand van de resultaten

voor MgSiN2, AlN en β -Si3N4. De Slack formule is een relatief simpele manier om

de maximale warmtegeleidbaarheid van niet-metallische (keramische) materialen

ruwweg te kunnen schatten. De aannames en beperkingen van deze formule

worden kort besproken en enkele eenvoudige verbeteringen worden

geïntroduceerd, welke resulteren in een gemodificeerde Slack formule. Hierdoor

wordt het temperatuurgebied waarbinnen deze formule normaal toepasbaar is

(T ≥ θ ) substantieel uitgebreid naar het praktisch interessante lagere

kamertemperatuurgebied. De nieuwe schattingsmethode is gebaseerd op

thermische diffusiviteitsmetingen aan niet-geoptimaliseerde preparaten als functie

Samenvatting

249

van de temperatuur. Deze eenvoudige methode heeft als voordeel dat vrij

gemakkelijk een minimale waarde voor de maximale warmtegeleidbaarheid kan

worden verkregen. De methode is betrouwbaarder dan de traditionele Slack

formule. Bovendien kan deze methode gebruikt worden om de optimalisatie van de

warmtegeleidbaarheid te sturen. Beide schattingsmethoden bevestigen de

experimentele resultaten dat de warmtegeleidbaarheid bij kamertemperatuur van

MgSiN2 gelimiteerd is tot 25 - 30 W m-1 K-1. Dit reduceert de potentiële

mogelijkheden van MgSiN2 keramiek voor toepassingen waarbij een hoge

warmtegeleidbaarheid van belang is.

Samenvatting

250

251

Nawoord

Het nawoord van het proefschrift wordt altijd gebruikt om de mensen met wie je

hebt samengewerkt tijdens je promotieonderzoek te bedanken. Het is dan ook in

het algemeen zowat het allerlaatste wat er tijdens het tot stand komen van een

proefschrift door de promovendus geschreven wordt. Vandaar dat het gehalte aan

standaardzinnen in de meeste nawoorden zo hoog is. Je bent al lang blij dat 'het'

erop zit. Hiermee doe je m.i. de mensen met wie je met zoveel plezier hebt

samengewerkt toch wel een beetje te kort. Met deze korte (ongebruikelijke)

inleiding op het nawoord probeer ik dan ook enigszins af te wijken van de

standaard door niet direct met de deur in huis te vallen.

Net zoals iedere promovendus ben ik dank verschuldigd aan heel veel

mensen. Het tot stand komen van dit proefschrift was alleen mogelijk door de

steun en hulp van deze mensen. Alleen had ik het nooit voor elkaar gekregen! Als

allereerste zou ik die mensen willen bedanken die ik onverhoopt vergeet te

noemen. Verder zal ik min of meer een chronologische volgorde van het verloop

van het onderzoek aanhouden bij het bedanken van de diverse personen:

Pim Groen (Philips Research), Bert de With en Bert Hintzen voor de eerste

kennismaking met het boeiende onderwerp. Mijn eerste promotor Ruud Metselaar,

die na mijn afstuderen zoveel vertrouwen in mijn werk had, dat hij mij een

promotieplaats aanbood en met veel interesse mijn werk gevolgd heeft. Hierbij

dien ik direct ook mijn directe begeleider Bert Hintzen te noemen die altijd

enthousiast, kritisch en behulpzaam was.

Direct betrokken bij mijn onderzoek waren voor korte of langere tijd Henk

Eekhof, Agnieszka Kudyba-Jansen, Peter Gerharts en Tarek Gueddas. Bedankt

voor de experimentele en wetenschappelijke bijdrage aan mijn werk. Veel

technische en/of wetenschappelijke steun binnen de groep heb ik ontvangen van

Nawoord

252

Henk van der Weijden (ovens, glove-box), Gerrit Bezemer (ovens, TGA/DTA,

glove-box), Hans de Jonge Baas (röntgendiffractie), Toon Rooijakkers (SEM

preparaatbereiding), Hans Heijligers (SEM), Marco Hendriks (SEM, meting

elasticiteitsconstanten m.b.v. puls-echo methode), Gerben Boon en Dick Klepper

(TEM), Joost van Krevel (reflectie metingen), Paul van der Varst (meting Young's

modulus m.b.v. impuls-excitatie methode) en Anneke Delsing (glove-box,

thermische diffusiviteit). Mijn dank hiervoor. Verder alle medewerkers,

promovendi en studenten van de capaciteitsgroep Vastestof- en Materiaalchemie

(SVM) die ik de afgelopen jaren gekend heb. Waarbij ik met name mijn

kamergenoten Henk, Robert, Stephan, Joost en Maru wil bedanken voor alle

discussies en gezelligheid.

Ook buiten de groep ben ik veel mensen dank verschuldigd: Joost van Eijk

(TNO/TPD), Han van der Heijde (TNO/TPD), Hans-Joachim Sölter (Compotherm

GmbH, Duitsland), Pim Groen (Philips Research, Aken), Theo Kappen (Philips

Lighting, Eindhoven), Harrie van Hal (Philips Research, Eindhoven), Kees van

Malsen (Universiteit van Amsterdam), Kees van Miltenburg (Universiteit Utrecht),

Anil Virkar (University of Utah, USA), Dale Niesz (Rutgers University, USA),

Chun Loong (Intense Pulse Neutron Source, Argonne, USA), Simine Short

(Intense Pulse Neutron Source, Argonne, USA), Ron Bogaard (Purdue University,

USA), Jos van Wolput (TU Eindhoven), Koji Watari (National Industrial Research

Institute of Nagoya (NIRIN), Nagoya, Japan) en Naoto Hirosaki (National Institute

for Research in Inorganic Materials (NIRIM), Japan).

Voor de technische ondersteuning, het verbeteren van de diverse

experimentele opstellingen en de bewerking van de keramische preparaten ben ik

veel dank verschuldigd aan de mensen van de Faculteitswerkplaats en de

Gemeenschappelijke Technische Dienst.

Verder wil ik de leescommissie (bestaande uit: Ruud Metselaar, Kiyoshi

Itatani (Sophia University, Tokyo, Japan), Bert Hintzen, Roger Marchand

(Université de Rennes I, France) en Bert de With) hartelijk bedanken voor de vele

nuttige aanwijzingen, suggesties, tips en discussies over het onderwerp.

特に，私の副査である板谷清司助教授（上智大学，日本）には，学位審査

Nawoord

253

委員会のメンバーとしてMgSiN2粉体およびそのセラミックスのプロセスに

関連した多くの貴重なご質問およびご助言を頂きましてお礼を申し上げま

す。(Especially, my second promotor Prof. K. Itatani I would like to thank for

being a member of my Ph. D. committee, his comments on the manuscript as well

as his many motivating critical questions and remarks concerning the processing of

the MgSiN2 powder and ceramics). Je voudrais remercier le Professeur Marchand

d'avoir accepté d'être membre de mon jury de thèse et d'avoir pris le temps de

discuter et commenter ma thèse. Voor de natuurkundige inbreng en de daaruit

voortvloeiende nuttige discussies en begripsvorming ben ik prof. G. de With grote

dank verschuldigd.

Shell Nederland B.V. wil ik bedanken voor de financiële steun (Shell

reisdonatie) die het deelnemen aan een congres in de V.S. en het bezoeken van

diverse Amerikaanse universiteiten en instituten mogelijk maakte.

De nodige ontspanning en gezelligheid vond ik ondermeer in de muziek.

Vooral het lidmaatschap van het Eindhovens Studenten Muziek Gezelschap

Quadrivium heb ik altijd als ontzettend leuk, inspirerend en vooral gezellig

ervaren. Ook de vrienden buiten de muziek wil ik bedanken voor al hun steun.

Vooral als het een keer niet mee zat dan stonden ze altijd met raad en daad voor me

klaar.

Mijn ouders wil ik bedanken voor hun betrokkenheid, steun en vertrouwen.

Mijn "broertje" Dominique wil ik bedanken voor alle steun maar vooral voor zijn

gevoel voor humor en Karin voor de getoonde interesse en betrokkenheid. Mijn

vriendin Marianne ben ik heel veel dank verschuldigd voor alle steun, liefde,

geduld, het "er zijn", en het me op sleeptouw nemen altijd wanneer dat nodig was.

254

Curriculum Vitae

Richard Joseph Bruls werd geboren op 26 mei 1972 te Sittard. Na het behalen van

zijn Atheneum B diploma in 1990 aan het Serviam Lyceum Scholengemeenschap

te Sittard, studeerde hij Scheikundige Technologie aan de Technische Universiteit

Eindhoven. Na het behalen van zijn ingenieurstitel door het afronden van zijn

afstudeeropdracht getiteld "Investigation of the Thermal Diffusivity/Conductivity

of Hot-Pressed MgSiN2 Ceramics" in februari 1996 bij de vakgroep Vastestof- en

Materiaal Chemie, werd in aansluiting daarop onder begeleiding van prof.dr. R.

Metselaar en dr. H.T. Hintzen een promotie onderzoek gestart op hetzelfde

onderwerp. De resultaten van dit onderzoek zijn beschreven in dit proefschrift.

Richard Joseph Bruls was born in Sittard (The Netherlands) on May 26th 1972. In

1990, after completing the secondary school at the Serviam Lyceum

Scholengemeenschap in Sittard, he started his study Chemical Engineering at the

Eindhoven University of Technology. In February 1996 he finished his graduation

work entitled "Investigation of the Thermal Diffusivity/Conductivity of

Hot-Pressed MgSiN2 Ceramics" at the Laboratory of Solid State and Materials

Chemistry and obtained his Masters Degree. Subsequently, he started his Ph.D.

study in the same field under supervision of Prof. R. Metselaar and Dr. H.T.

Hintzen. The results of this investigation are described in this thesis.

255

List of publications

H.T. Hintzen, R. Bruls, A. Kudyba, W.A. Groen and R. Metselaar, Powder

Preparation and Densification of MgSiN2, Ceramic Transactions, Vol. 51, Ceramic

Processing Science and Technology (Friedrichshafen, Germany, September 11 - 14,

1994), editors H. Hausner, G.L. Messing and S. Hirano, 1995, pp. 585 - 589.

H.T. Hintzen, R.J. Bruls and R. Metselaar, The Thermal Conductivity of MgSiN2

Ceramics, Fourth Euro Ceramics (Faenza, Italy, October 2 - 6, 1995), Vol. 2, Basic

Science - Developments in Processing of Advanced Ceramics - Part II, editor

C. Galassi, 1995, pp. 289 - 294.

H.T. Hintzen, R. Bruls and R. Metselaar, Thermal Conductivity of MgSiN2

Ceramics, The American Ceramic Society 98th Annual Meeting Abstracts,

(Indianapolis, IN, USA, April 14 - 17, 1996), p. 249.

R.J. Bruls, H.T. Hintzen and R. Metselaar, Modeling of the Thermal

Diffusivity/Conductivity of MgSiN2 Ceramics, Thermal Conductivity 24, Thermal

Expansion 12 (Pittsburgh, USA, October 26 - 29, 1997), editors P.S. Gaal and D.E.

Apostolescu, 1999, pp. 3 - 14.

R.J. Bruls, H.T. Hintzen, R. Metselaar and J.C. van Miltenburg, Heat Capacity of

MgSiN2 between 8 and 800 K, J. Phys. Chem. B, Vol. 102, 1998, pp. 7871 - 7876.

C.M. Fang, R.A. de Groot, R.J. Bruls, H.T. Hintzen and G. de With, Ab initio Band

Structure Calculations of Mg3N2 and MgSiN2, J. Phys.: Condens. Matter, Vol. 11,

1999, pp. 4833 - 4842.

List of publications

256

R.J. Bruls, H.T. Hintzen and R. Metselaar, Preparation and Characterisation of

MgSiN2 Powders, J. Mater. Sci., Vol. 34, 1999, pp. 4519 - 4531.

R. Metselaar and R. Bruls, The Thermal Conductivity of MgSiN2 in Comparison

with AlN and Si3N4, The American Ceramic Society 102nd Annual Meeting

Abstracts, (St. Louis, Missouri, USA, April 30 - May 3, 2000), p. 271.

R.J. Bruls, H.T. Hintzen, R. Metselaar and C.-K. Loong, Anisotropic Thermal

Expansion of MgSiN2 from 10 to 300 K as Measured by Neutron Diffraction, J.

Chem. Phys. Solids, Vol. 61, 2000, pp. 1285 - 1293.

R.J. Bruls, A.A. Kudyba-Jansen, H.T. Hintzen and R. Metselaar, Preparation,

Characterisation and Properties of MgSiN2 Ceramics, to be published.

R.J. Bruls, H.T. Hintzen, G. de With, R. Metselaar and J.C. van Miltenburg,

Thermodynamic Grüneisen Parameter of MgSiN2, AlN and β-Si3N4, submitted to

J. Chem. Phys. Solids.

R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, The Temperature

Dependence of the Young's Modulus of MgSiN2, AlN and Si3N4, accepted for

publication in J. Eur. Ceram. Soc.

R.J. Bruls, H.T. Hintzen, G. de With and R. Metselaar, Estimates of the Maximum

Achievable Thermal Conductivity of MgSiN2, AlN and β-Si3N4 using a Modified

Slack Equation, to be published.

R.J. Bruls, H.T. Hintzen and R. Metselaar, A New Estimation Method for the

Intrinsic Thermal Diffusivity/Conductivity of Non-Metallic Compounds: A case

study for MgSiN2, AlN and β-Si3N4 ceramics, to be published.

The thermal conductivity of magnesium silicon nitride ... · The Thermal Conductivity of Magnesium...

Documents

Transcript of The thermal conductivity of magnesium silicon nitride ... · The Thermal Conductivity of Magnesium...