Thermal conductivity of polycrystalline ZnS, ZnSe, and CdTe in the

temperature range 4-400 K

S.M. Luguev, N.V. Lugueva, and A.B. Batdalov

Institute of Physics, Dagestan Scientific Center Russian Academy of Sciences,

ul. Jaragskogo 94, Makhachkala 367003 Dagestan, Russia

E-mail: luguev.if@mail.ru

The thermal conductivity of ZnS, ZnSe, CdTe was experimentally studied in the

temperature range 4-400 K. It was found that in the ZnS samples with micron

grain sizes in comparison with the single crystals and large-grained polycrystals

the thermal conductivity are essential reduced and its low-temperature maximum

shifts towards higher temperature region and its height decreases. In the low

temperature range, the textured samples exhibit the anisotropy of the thermal

conductivity which is caused by scattering by dislocations oriented along the

crystals growth direction. The anisotropy of the thermal conductivity is also

persisted after deformation and recrystallization of these samples. It was

established the correlation between the phonon density of states and the features

of the trmperature dependences of the thermal resistance of the investigated

samples.

KEY WORDS: ZnS; ZnSe; CdTe; thermal conductivity; anisotropy; defect;

dislocation.

1. INTRODUCTION

Zinc sulfide, zinc selenide, and cadmium telluride are used in optical instrument

production owing to their in the IR-spectral range. The service conditions of the

structural optics elements under varying thermal duty and heavy thermal loads

require the knowledge of the thermal conductivity coefficient data of given optical

materials. In the commercial production optical ZnS, ZnSe, and CdTe-based

materials are usually prepared by the recrystallization compaction of a finely

disperse powder or through vapor deposition of raw materials. The materials

produced by the vapor deposition exhibit a pronounced texture. In recent years for

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reduction of the negative effect of texture this materials have been subjected to

pressure deformation and recrystallizasion. Because the thermal conductivity is a

structure-sensitive parameter it is important study the peculiarities of heat transfer

in the materials having the different production histories and accordingly the

different defect structure. Data are available on the thermal conductivity of single

crystals of ZnS and ZnSe [1] and single crystals and polycrystals of CdTe [1-5]

obtained by crystallization from the melt. In recent years the series paper [6-10]

with the data on the thermal conductivity of these materials were published. The

present paper reports on the results of investigations into the thermal conductivity

of the ZnS, ZnSe, CdTe having different production histories in wide temperature

range including liquid helium interval.

2. EXPERIMENTAL

The thermal conductivity coefficient κ was measured by a steady-state absolute

method in the temperature range 4-400 K. In the temperature range 4-100 K the

measurements were performed using a setup 1 according to the method described

in [11]. The schematic diagram of measuring cell of this setup is presented in

Fig.1.

Fig. 1. Diagram of measuring cell of setup 1.

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The investigated sample (1) soldering to copper rod (2) is placed to the vacuum

chamber (3). The copper rod has a direct contact with the thermostatic fluid (liquid

helium). The coiled on the copper rod heater (4) is used for the change of the

temperature at the measurements of the temperature dependence of the thermal

conductivity coefficient. The temperature gradient is created in sample by heater

(5). Carbon resistance thermometers (6) at 4-40 K and copper-constantan

thermocouple (7) soldering direct to samples served as temperature probe. All

soldering to the sample (of thermometers, heater, measuring probe) are carried out

by pure indium. At 80-400 K the measurements were performed using setup 2

similar to the type A setup described in [12] where the schematic diagram device

and design formulas are given. The scheme of arrangement of samples, gradient

heater, refrigerator, thermocouples is shown in Fig. 2.

The heater (1) was placed between two identical samples (2). The heater was 0.3

mm in thickness and its section was equal to the sample section. The samples have

an effective thermal contact with device case serving as refrigerator (3) and

locating directly to thermostating medium. The heat flow from the heater produced

temperature difference which was measured by particularly graduated copper-

constantan thermocouples (4) (copper and constantan wire diameter was 0.1 mm).

The thermocouples were soldered to silver pin which tightly fixed into thin parallel

holes (0.3 mm in diameter) perforated in the samples. At that pins have the

temperature of given section of the sample. For improvement of the thermal

Fig. 2. Diagram of measuring cell of setup 2.

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contact the pin with the sample the holes were lubricated by special black-lead

lubrication.

In the investigated temperature interval the loss of heat are possible as a result

of radiation from the lateral surfaces of samples. The sample lateral surfaces were

blacked and it permitted it consider that the emissivity of this surfaces is close to

emissivity of absolute black body. Hence the heat losses by radiation were

estimated. During the measurements the inside of the vacuum chambers was kept

at a pressure lower than 10-5 Torr to eliminate the effect of heat losses by

convection from the sample surface. The relative error in the thermal conductivity

measurements for setup 1 and 2 was within 2-3 % depending on temperature range.

Samples for investigation were prepared from blocks of ZnS, ZnSe, and CdTe

of different production histories. The samples prepared by deposition from the

vapor phase onto a preheated substrate were characterized by a texture with grains

extended in the direction of growth. The diameter of crystallites in the samples was

1-3 mm. Some samples of ZnSe and CdTe polycrystals prepared by deposition

from the vapor phase were subjected to deformation by pressure in a direction

parallel to that of crystallite growth. As a result of deformation and

recrystallization, these samples lost the texture, and the average grain size in the

samples was 1 mm. In the investigation of these samples, κ was measured both

with a heat flux along the axis of polycrystal growth, κ||, and with a heat flux

normal to this direction, κ┴.

We further investigated single crystals of ZnS and samples prepared by

vacuum recrystallization pressing of finely divided powders. ZnS samples prepared

by pressing exhibited a relative density of 0.998 of the single crystal density and

consisted of grains 1 to 2 µm in size. According to the data of X-ray phase

analysis, the sample of all investigated materials had a cubic structure.

3. RESULTS AND DISCUSSION

The results of the investigation of the temperature dependence of κ|| of the

ZnS, ZnSe , CdTe polycrystals are given in Fig. 3. It is known that the thermal

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conductivity of defect-free crystals can be represented by Leibfried-Schlemann

formula [13]

TManAp 2

32/1

γθκ = , (1)

where A is coefficient depending on the lattice structure and type of the chemical

bond; n is number of atoms per unit cell; a3 is the mean values occupied by one

atom in crystal; M is the mean atomic mass; θ and γ are, respectively, the mean

values of the Debye temperature and the Grüneisen constant for all phonon

branches contributing to the heat transfer. Also in Fig. 3 there are presented the

data calculated by formula (1) for ZnS. The entering to formula (1) values of the

parameter θ and γ is borrowed from [14-16].

Fig. 3. The temperature dependences of the thermal conductivity coefficient of polycristals:

experimental data for (1) ZnS, (2) ZnSe, (3) CdTe, (4) calculated data for ZnS

One can see in Fig.3 that the thermal conductivity of investigated polycrystals

regularly decreases in the ZnS, ZnSe, CdTe series, which is associated with the

decrease in the Debye temperature and the increase in the anharmonicity of crystal

lattice vibrations with increasing mean atomic weight and interatomic spacing

from ZnS to CdTe. The comparison of the experimentally obtained values of κ for

ZnS with the values calculating by formula (1) show that the calculating values are

rather higher than experimentally obtained values. The similar pattern is observed

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for the samples of ZnSe and CdTe. It is shown in [6-10] that in the investigated

temperature interval the heat transfer in ZnS, ZnSe ,CdTe largely proceeds owing

to the crystal lattice vibrations (by phonons). The contribution of the photon

component to the heat transfer even in the high-temperature range is within the

experimental error. In the investigated temperature range the heat transfer by

phonons in ZnS, ZnSe, CdTe polycrystals is restricted by the phonon scattering by

phonons and defects. The thermal conductivity in single crystals of these

compounds is limited by the mechanisms associated with phonon-phonon

processes and phonon scattering by point (zero-dimensional) and linear (one-

dimensional) defects existing in real crystals. In polycrystalline samples the heat

transfer can be also by two-dimensional defects (grain boundaries), and in the

samples prepared by recrysallizaton compacting by bulk defects (pores) in

addition. In polycrystalline samples, grain boundaries and surface layers of grains

always contain a large amount defect which distort the crystal lattice, strongly

scatter phonons. All listed factors define quantity and temperature dependence of

the thermal conductivity coefficient.

The prepared by vapor deposition ZnS, ZnSe, CdTe polycrystals exhibit

anisotropy in their thermal conductivity relative to the crystal growth direction.

The experimental data on the thermal conductivity of ZnSe samples with various

technological histories and heat-flow directions with respect to the texture of the

samples are shown in Fig.4. The curve 1 is data κ|| obtained for the samples in

which the heat-flow direction was coincident with the crystal growth direction. The

curve 2 is data κ┴ obtained for the sample in which these directions were normal.

At 80 K data for the samples 1 and 2 differ by 14 %. The thermal conductivity

coefficients of the samples additionally subjected to deformation and

recrystallization (samples 3, 4 in Fig. 4) also exhibit the anisotropy depending on

the heat-flow direction throw the sample.

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Fig. 4. The temperature dependences of the thermal conductivity coefficient of polycrystalline

ZnSe: (1) κ׀׀ and (2) κ┴ of the textured samples, (3) κ׀׀ and (4) κ┴ the samples subjecting deformation and recrystallization. In the inset: the temperature dependences of anisotropy

coefficient: (1) textured samples, (2) these samples subjecting deformation and recrystallization.

The inset in Fig. 4 shows the temperature dependence of the anisotropy

coefficient (κ|| - κ┴.)/κ┴. for the textured polycrystalline ZnSe and samples

subjecting deformation and recrystallization. The investigated samples has a cubic

structure and the anisotropy of κ is not related to the anisotropy of the elastic

properties of crystals. The estimation of free path of phonons in the ZnSe

polycrystals [6] is showed that in the investigated samples with crystallite size ~ 2

mm the scattering by boundaries can affect on κ only at very low temperatures,

and at T > 80 K the observed anisotropy does not determine by the boundary

scattering in textured samples. The examination of the phonon scattering by

dislocations provides the explanation of the anisotropy of κ in the samples

prepared by vapor deposition. It is known [17] that the phonon scattering by

dislocations results in an additional thermal resistance Wd which may be

represented as

Wd = k Lσ sinα, (2)

where L is the dislocation length, σ is the dislocation scattering cross section for

phonons, α is the angle between the dislocation axis and the temperature gradient

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direction, k is a constant depending on the phonon frequency. It follows from

Eq. (2) that the thermal resistance due to the phonon scattering by dislocations

increases with increasing angle between the dislocation axis and the temperature

gradient direction. Since the dislocations in investigated samples of A2B6

compounds exhibit a preferred orientation in the direction of polycrystal growth

[18], they scatter phonons most strongly when the heat flow in normal to this

direction. As a result κ|| exceeds κ┴, and the presence of the anisotropy of the

thermal conductivity coefficients of the ZnS, ZnSe, CdTe polycrystals prepared by

deposition from the vapor phase may be associated with the scattering of phonons

by oriented dislocations.

After deformation under pressure , the ZnS, ZnSe, CdTe samples prepared by

deposition from the vapor phase consisted of smaller crystallites without definite

directionality. The anisotropy has been observed in these samples also, and it is

evidence that preferred orientation of dislocations is preserved after deformation as

well. A decrease in the κ was observed in these samples because the deformation

resulted in the emergence of an additional number of defects (dislocations, grain

boundaries) which scatter phonons.

As already it was mentioned, the phonon scattering by grain boundaries can

appreciable influence on κ of polycrystals at low temperature. In the zinc sulfide

samples, with a grain size of 1 µm it was observed the contribution of the phonon

scattering by grain boundaries at T < 130 K [8]. For a determination of a influence

pattern of the grain size on the thermal conductivity of ZnS the temperature

interval of the κ measurements was extended to the liquid helium temperatures.

The measurements were carried out on the single crystal and the polycrystalline

samples with mean grain size of 1-2 mm and 1-2 µm. The experimental results are

shown in Fig. 5. At can be seen from the Fig. 5 the data for the single crystal and

polycrystal with a grain size of 1-2 mm are coincident range T > 70 K, and at

lower temperatures it is observed a essential divergence.

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Fig. 5. The temperature dependences of the thermal conductivity coefficient of ZnS: (1) single

crystal, (2) polycrystal prepared by deposition from the vapor phase, (3) polycrystal prepared by hot pressing.

The thermal conductivity of the samples with a grain size of 1-2 µm is

substantially less in all investigated temperature intervals. The most divergence the

data of κ this samples with the data for the single crystal and large-grained

polycrystal is observed in the area and below of thermal conductivity maximum.

The lower values of κ of this polycristals are determinate on the contribution in

their thermal resistance of the phonon scattering by defects in the grain layers near

the surface. The concentration of these defects in the samples with smaller grain

size is significantly higher than in large-grained samples, and accordingly their

contribution to the thermal resistance is more considerable. In the samples with a

grain size of 1 µm, the contribution of the phonon-phonon scattering and phonon

scattering by grain boundaries to the limitation of mean free path of phonons are

comparable to each other in magnitude already at 130 K [6,8]. A further decrease

in the temperature leads to an increase in the relative contribution of the phonon

scattering by grain boundaries to the limitation of both the mean free path of

phonons and the thermal conductivity. This explains the observed removal of the

thermal conductivity maximum to high- temperature range.

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Fig. 6. The temperature dependences of the thermal resistance of the ZnS : the polycrystalline

samples with a grain size of (1) 1-2 µm, (2) 1-2 mm; (3) single crystal. In top: the phonon density of state for ZnS [14].

Fig. 7. The temperature dependence of the thermal resistance of the polycrystalline ZnSe. In top: the phonon density of state for ZnSe [14].

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Fig. 8. The temperature dependence of the thermal resistance of the polycrystalline CdTe. In top: the phonon density of state for CdTe [19].

In treating the temperature dependences of the thermal resistance of the lattice

of investigated samples (Fig. 6-8), irrespective of the production history of the

samples, it was found that the function W (T) has a kink in medium-temperature

range. It is known [17] that temperature dependence of the thermal resistance of

real crystals may be represented as

W = BT + C, (3)

where B ~ γ2/ θ3 [13], and C is constant which are defined the defects containing in

the specific samples (their number and scattering cross section for phonons). The

γ2/ θ3 ratio can change when the contribution from different phonon branches to the

heat transfer vary.

On top of the Fig. 6-8 for the investigated materials the phonon density of

states from [14-19] are presented. At T > 200 K in ZnS the relative number of the

phonons corresponding to the acoustic longitudinal (LA) branch begins to increase.

An increase in the contribution of theirs phonons to heat transfer leads to a

variation in the slope W (T) curve, which, as experiment shows, takes place in the

temperature region of 200-280 K. In ZnS the γ2/ θ3 ratio for the acoustic transverse

(TA) branch is three times lower than that for LA branch [8]. Hence the change in

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the slope of the dependence W (T) for ZnS in the high-temperature range can be

explained by the increase in the contribution of longitudinal acoustic phonons to

the heat transfer. The optical phonons in ZnS begin to become excite only in high-

temperature range of the measurements. The energy gap between acoustic and

optical branches is considerable and in investigated temperature range optical

phonons do not influence on W (T).

In the zinc selenide the number of phonons corresponding to the LA phonon

branch increases gradually from 160 to 280 K. The slope of the W (T) curve of the

investigated ZnSe samples in this temperature range does not vary since in ZnSe

γ2/ θ3 value differs little for the TA and LA phonon branches and the increase of

the fraction of LA phonons in heat transfer at T > 160 K has no effect on the

behavior of W (T) dependence. The excitation of the optical phonons begins in

the temperature region T > 280 K. Because their dispersion is low, the optical

phonons do not transfer heat. The energy gap between the acoustic and optical

phonon branches in ZnSe is insignificant, and the scattering of acoustic phonons by

optical ones turns out to be possible. This scattering causes an additional increase

in the thermal resistance of the lattice and a variation of the slope of W (T) in ZnSe

at T > 270 K.

For the cadmium telluride in the phonon spectrum there is only a small

dispersion in the optical branches [19, 20]. Consequently, the group velocity of the

optical phonons is rather low, and their contribution to heat transfer is

insignificant. Nearly all of the heat is transferred by acoustic phonons. According

to calculations [21], the contributions of LA and TA phonons to the thermal

conductivity of CdTe are roughly equal above 80 K and γ2/ θ3 ratio remains

constant in this temperature range The slope W(T) begins to change just above 180

K, where the excitation of optical phonon branches occurs. As in ZnSe the energy

gap between acoustic and optical branches in CdTe is not very large, and acoustic

phonons can be scattering by optical phonons. This process increases the thermal

resistance of the CdTe lattice and, accordingly, the slope of W (T) in the

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temperature range of optical-phonon excitation. At temperatures, where all of the

optical phonons a excited, W is again a linear function of temperature.

4. CONCLUSION

As a results of the experimental investigation, the absolute values and

temperature dependences of the thermal conductivity coefficient of the ZnS, ZnSe,

CdTe crystals were determined. It is established that in the samples with a micron

grain size at low temperatures the phonon scattering by grain boundaries is

essential for the limitation of heat transfer as well in all investigated temprature

range the phonon scattering by defects in the grain layers near the surface. In the

textured samples the scattering of phonons by oriented dislocation is observed. The

observed change in the slope of the temperature dependences of the thermal

resistance of ZnS, ZnSe, and CdTe above the Debye temperature is caused by a

features of theirs phonon spectrum.

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