Physica B 406 (2011) 3951–3955

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Physica B

0921-45

doi:10.1

$The

(110743

Researc� Corr

E-m

journal homepage: www.elsevier.com/locate/physb

First-principles calculations on temperature-dependent elastic constantsof rare-earth intermetallic compounds: YAg and YCu$

Rui Wang �, Shaofeng Wang, Xiaozhi Wu, Yin Yao

Institute for Structure and Function and Department of Physics, Chongqing University, Chongqing 400044, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 25 April 2011

Received in revised form

19 July 2011

Accepted 20 July 2011Available online 27 July 2011

Keywords:

Rare-earth intermetallics

Temperature-dependent elastic constants

First-principles calculations

26/$ - see front matter & 2011 Elsevier B.V. A

016/j.physb.2011.07.034

work is supported by the National Natural

13) and Project No. CDJXS11102211 supp

h Funds for the Central Universities of China

esponding author. Tel.: þ86 13527528737.

ail address: rcwang@cqu.edu.cn (R. Wang).

a b s t r a c t

We present the temperature-dependent elastic constants of two ductile rare-earth intermetallic

compounds YAg and YCu with CsCl-type B2 structure by using a first-principles approach. The elastic

moduli as a function of temperature are predicted from the combination of static volume-dependent

elastic constants obtained by the first-principles total-energy method with density-functional theory

and the thermal expansion obtained by the first-principles phonon calculations with density-functional

perturbation theory. The comparison between our calculated results and the available experimental

data for Ag and Cu provides good agreements. In the calculated temperature 0–1000 K, the elastic

constants of YAg and YCu follow a normal behavior with temperature that those decrease with

increasing temperature, and satisfy the stability conditions for B2 structures. The Cauchy pressure for

YAg and YCu as a function of temperature is also discussed, and our results mean that YAg and YCu

become more ductile while increasing temperature.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

The rare-earth intermetallic compounds typically possess highstrength and stiffness, low specific weight corrosion resistance,and hot strength superior to ordinary metals [1,2]. However,their room temperature, brittleness and poor fracture toughnessseverely restrict their applications. Recently, a large family ofrare-earth intermetallics have been discovered to be ductile andtough at room temperature [3], in some cases exceeding 20%ductility in tension. These alloys have the B2-type structure inspace group Pm3m (number 221) and chemical formula RM,where R denotes a rare-earth element, and M denotes a latetransition metal or an early p-element. YAg and YCu are two mosttypical ductile rare-earth intermetallic compounds, and manytheoretical and experimental works [4–13] have been done tounderstand their mechanism of high ductility at room tempera-ture, such as elasticity, defect properties, phase stability, disloca-tions, stacking fault, electronic structure and density of states(DOS), etc. However, the properties and mechanism have notbeen understood completely hitherto.

To get a better understanding of the anomalous ductility of RMintermetallics, more fundamental investigations of temperature-dependent properties such as thermal expansion, heat capacity,

ll rights reserved.

Science Foundation of China

orted by the Fundamental

.

and thermoelasticity, etc., are obviously required. In general,elastic properties of a solid are very important because they areclosely associated with various fundamental solid-state proper-ties such as interatomic potentials, equation of state, and phononspectra. The temperature dependence of the elastic constantsof a material is important for predicting and understanding themechanical strength, stability, and phase transitions of a material[14]. So far as we know, the temperature-dependent moduli of thenovel B2-type structures YAg and YCu have not been reported inthe literature. The first-principles methods based on density-functional theory (DFT) can compute the single-crystal elasticmoduli at zero-temperature accurately, but treating the corre-sponding temperature dependence of elastic constants is still aformidable challenge [15]. More recently, Wang et al. [16,17]proposed a simple quasistatic approach to determine the elasticconstants at finite temperatures by using first-principles calcula-tions, and the excellent agreement between the calculated resultsand experimental data is found. Their approach is based on thefact that the temperature dependence of elastic constants mainlyresults from volume expansion with increasing temperature[14,18,19].

In the present work, we apply the first-principles quasistaticapproach to investigate the temperature-dependent elasticconstants for the novel intermetallics YAg and YCu. The staticvolume-dependent elastic constants are obtained by using the DFTtotal-energy calculations combined with the method of homoge-neous deformation. A first-principles quasiharmonic approach [20]is applied to predict the phonon free energy and volume versustemperature by using density-functional perturbation theory (DFPT).

R. Wang et al. / Physica B 406 (2011) 3951–39553952

To benchmark the reliability results of the presented method, wehave also calculated the temperature-dependent elastic constantsfor Ag and Cu, and the comparison between our predicted resultsand the available experimental data provides good agreements. Inaddition, the ductility of YAg and YCu as a function of temperaturehas been investigated by using the temperature-dependent elasticmodulus.

2. Theory

In this section, we present the necessary theory for thecalculation of the temperature dependence of cubic crystal elasticmoduli in B2 structures. For programming the calculation ofthe elastic constants, we consider a procedure of homogeneousdeformation of a bulk-crystal. With the displacement given asui ¼ xi�ai (i¼1, 2, 3), between the initial configuration ai and thestrained configuration xi, the deformation applied to the crystal isdescribed as the deformation gradient matrix Jij ¼ @xi=@aj. Then,we may define the Lagrangian strain Zij ¼

12ðJikJkj�dijÞ [21]. Thus,

the isothermal elastic constants CijklT are defined as the derivative

with respect to the Lagrangian strain on the Helmholtz freeenergy equation (8) at constant temperature [22],

CTijkl ¼

1

V

@2F

@Zij@Zkl

�����T

Z0

, ð1Þ

where V¼V(T) is the equilibrium volume at temperature T, and Z0indicates that all other stains are held fixed. This is the generalway to compute the finite temperature elastic constants inprinciple. However, this procedure is quite cumbersome at pre-sent since it involves calculation of the second derivatives of Fvib.For cubic materials, there are three independent elastic constantsCT

11, CT12, and CT

44 (in Voigt notation) which describe the elasticbehavior completely. A more convenient set for computations areCT

44 and two linear combinations BT and mT . The bulk modulus

BT ¼ ðCT11þ2CT

12Þ=3, ð2Þ

is the resistance to deformation by a uniform hydrostatic pres-sure; the shear modulus

mT ¼ ðCT11�CT

12Þ=2, ð3Þ

is the resistance to shear deformation across the (1 1 0) plane inthe /1 1 0S direction, and C44 is the resistance to shear deforma-tion across the (1 0 0) plane in the /0 1 0S direction. The bulkmodulus B is determined from the Vinet equation of state [23].The shear modulus m is calculated from volume-conservingorthorhombic strain:

ZðdÞ ¼d 0 0

0 d 0

0 0 ð1þdÞ�2�1

0B@

1CA, ð4Þ

and the Helmholtz free energy related to this strain is

FðV ,dÞ ¼ FðV ,0Þþ6mT Vd2þOðd3

Þ, ð5Þ

where F(V,0) is the free energy of the unstrained structure.We use a volume-conserving tetragonal strain to determine CT

44,

ZðdÞ ¼0 d 0

d 0 0

0 0 d�2=ð1�d�2Þ

0B@

1CA, ð6Þ

which leads to the free energy change

FðV ,dÞ ¼ FðV ,0Þþ2CT44Vd2

þOðd4Þ: ð7Þ

In the usual way and following Moriarty et al. [24], theHelmholtz free energy F for a crystal at temperature T and volume

V can be separated as

FðV ,TÞ ¼ E0ðVÞþFvibðV ,TÞþFelðV ,TÞ, ð8Þ

where E0ðVÞ is the zero-temperature total energy of the electronicground state, FvibðV ,TÞ is the vibrational free energy which comesfrom the phonon contribution, and where Fel represents thethermal electronic contribution to free energy from finite tem-perature. Here V is taken as the volume per unit cell and F is thefree energy per unit cell. The total hydrostatic pressure in thecrystal can be calculated by solving

PðV ,TÞ ¼�@FðV ,TÞ

@V

����T

, ð9Þ

with corresponding relations for its three components. Otherthermodynamic functions can be obtained from F(V,T) similarly,e.g., entropy S¼�ð@F=@TÞV , internal E¼FþTS, and the Gibbs freeenergy G¼FþPV.

In usual way, the vibrational free energy FvibðV ,TÞ includesboth quasiharmonic and anharmonic components. The anharmo-nic effects which represent the phonons interaction and electron–phonon coupling are so weak and can be neglected [15]. In thiswork, we only consider the remaining quasiharmonic phonon freeenergy which can be written as

FvibðV ,TÞ ¼Xql

1

2‘oqlþkBT lnð1�e�‘oql=kBT Þ

� �, ð10Þ

where kB represents the Boltzmann constant, ‘ is the reducedPlanck constant, and oql represents the frequency of the l thphonon branch at wave vector q.

With finite temperature DFT calculations [25], the thermalelectronic contribution to free energy Fel is obtained from theenergy and entropy contribution and given by

Fel ¼ Eel�TSel, ð11Þ

with the bare electronic entropy Sel

SelðV ,TÞ ¼�kB

Z 10

nðe,VÞ½f ðeÞ ln f ðeÞþð1�f ðeÞÞlnð1�f ðeÞÞ� de, ð12Þ

and the thermal electronic energy Eel

EelðV ,TÞ ¼

Z 10

nðe,VÞf ðeÞe de�Z eF

0nðe,VÞe de, ð13Þ

where nðe,VÞ, f ðeÞ, and eF represent the electronic density ofstate (DOS), the Fermi–Dirac distribution, and the Fermi energy,respectively.

In present work, we employ a first-principles quasistaticapproach, which was developed by Wang et al. [16,17], to calculatethe temperature-dependent elastic constants Cij(T). Following thequasistatic approximation, the change of elastic properties atelevated temperatures is mainly caused by volume change due tothermal expansion, and the contributions of Fvib and Fel to thesesecond derivations can be neglected. In fact, the contributions dueto the kinetic energy and the fluctuation of microscopic stresstensor are small and can be ignored reasonably [26]. The thermalexpansion and the equilibrium volume V at T are obtained by usingthe quasiharmonic approximation in our calculation. A similarprocedure was employed by Kadas et al. [27], though theycalculated thermal expansion by using Debye-type model. Weobtain the temperature dependence of isothermal elastic constantsCT

ij ðTÞ by the application of the following three-step procedure. Thefirst step is calculating the thermal expansion and the equilibriumvolume V(T) at T by using the first-principles quasiharmonicapproach. In this procedure, one can compute volume dependenceof phonon density of states at 0 K in a set of volume points, and thepredicted equilibrium volume V(T) at Ta 0 K (or inversely T(V)relation) is determined from fitting to the Vinet equation of state

R. Wang et al. / Physica B 406 (2011) 3951–3955 3953

[23] by minimizing free energy F with respect to V, whileisothermal bulk modulus BT as a function of temperature T is alsoobtained. In the second step, we obtain the volume-dependentelastic constants CT

ij ðVÞ at T¼0 K as the second derivatives of theHelmholtz free energies with respect to strain tensor by using theenergy–strain relation based on Eqs. (2)–(7). In the third step, thecalculated elastic constants from the second step at the volumeV(T) are approximated as those at finite temperatures, i.e.,CT

ij ðTÞ ¼ CTij ðTðVÞÞ. There is ample experimental evidence [28,19,29]

to support this approximation in which the temperature depen-dence of elastic constants are solely caused by thermal expansion.

In order to compare with experimental data, the isothermalelastic moduli Cij

T must also be transformed to the adiabatic elasticmoduli Cij

S by the following relation [30]:

CSij ¼ CT

ijþTV

CVlilj, ð14Þ

where CV is the specific heat at constant volume and

li ¼X

k

akCTik, ð15Þ

with the linear thermal expansion tensor ak. For cubic crystals,Eq. (14) simplifies to

CS44 ¼ CT

44, ð16Þ

CS11 ¼ CT

11þD, ð17Þ

CS12 ¼ CT

12þD, ð18Þ

where [15]

D¼TV

CVa2BT 2

, ð19Þ

with the volume thermal expansion coefficient a¼ V�1@V=T9p. Itis worth noted that the correction D increases with temperature.

Table 1

The present calculated lattice constants a (A), isentropic elastic constants CijS (GPa),

and Cauchy pressure PCS(GPa) for YAg and YCu at T¼0 and 300 K compared to

previous computed results and experimental data. Note that all the previous

calculated results are T¼0 K values, while the experiments are room temperature

values about T¼300 K.

Present calculation Previous calculations Experiment

YAg

a 3.645a, 3.656b 3.634c, 3.627d 3.619e

CS11

98.9a, 95.7b 105.0c, 102.5d 102.4e

CS12

52.4a, 51.4b 50.0c, 56.5d 54.0e

CS44

35.5a, 32.7b 37.0c, 37.8d 37.2e

PCS 16.9a, 18.7b 13.0c, 18.7d 16.8e

YCu

a 3.484a, 3.492b 3.477c, 3.472d 3.477e

CS11

117.4a, 114.6b 116.0c, 113.6d 113.4e

CS12

45.9a, 45.6b 47.0c, 48.4d 48.4e

CS44

36.3a, 33.4b 35.0c, 36.8d 32.3e

PCS 9.6a, 12.2b 12.0c, 11.6d 15.9e

a This work at T¼0 K.b This work at T¼300 K.c Ref. [4] obtained from first-principles calculations at T¼0 K.d Ref. [7] obtained from first-principles calculations at T¼0 K.e Ref. [4] obtained from experiments at T¼300 K.

3. Computational details

In present work, the static energy and the thermal electroniccontribution to the Helmholtz free energy were computed by usingthe first-principles calculations in the framework of the density-functional theory (DFT). We employed the plane-wave basis pro-jector augmented wave (PAW) method [31,32] within the general-ized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof(PBE) [33,34] exchange–correlation functional as implemented inthe VASP code [35–37]. The radial cutoffs of the PAW potentials ofAg, Cu, and Y were 1.50, 1.31 and 1.81 A, respectively. The 4d and 5selectrons for Ag, the 3d and 4s electrons for Cu, and the 4s, 4p, 4dand 5s electrons for Y were treated as valence and the remainingelectrons were kept frozen. The Brillouin zones (BZ) of the unit cellsare represented by Monkhorst–Pack special k-point scheme [38].Since high accuracy is needed to evaluate the elastic constants, theconvergence of strain energies with respect to the Brillouin zoneintegration was carefully checked by repeating the calculations for21�21�21 and 25�25�25 grid meshes at equilibrium volume at0 K, and we found at most 0.5 GPa difference for both C44 and m.Hence, we used 21�21�21 in the full Brillouin zone giving726 irreducible k-points. In addition, we used a high plane-waveenergy cutoff of 600 eV which is sufficient to calculate the elasticmoduli accurately. The thermal electronic energies and entropiesare evaluated by using one-dimensional integrations from the self-consistent DFT calculations of electronic DOS using FD smearing asshown in Eqs. (12) and (13). Calculations of the static strain–energywere performed by the tetrahedron method with Blochl correc-tions [39], which give a good account for total energy.

The vibrational free energy was obtained from the first-principles phonon calculations by using PHONOPY [20,40,41]which can support VASP interface to calculate force constantsdirectly in the framework of density-functional perturbationtheory (DFPT) [42]. Phonon calculations were performed by thesupercell approach. Since the chosen supercell size stronglyinfluences on the thermal properties, we compare the vibrationalfree energies of 3�3�3 supercell with those of 5�5�5 super-cell at 300 and 1000 K, and find that the energy fluctuationsbetween 3�3�3 and 5�5�5 supercells are less than 0.01%.Hence, we chose the 3�3�3 supercell with 54 atoms tocalculate phonon dispersions. We carried out DFPT calculationson this 54 atoms supercell using PBE-GGA exchange-correlationseffects and 7�7�7 k-point grid meshes for BZ integrations. Inorder to deal with the possible convergence problems for metals,a smearing technique is employed by using the Methfessel–Paxton scheme [43], with a smearing with of 0.05 eV.

We have calculated Helmholtz free energy [the right-hand sideof Eq. (8)] at temperature points with a step of 1 K from 0 to1000 K at 15 volume points. At each temperature point, theequilibrium volume V and isothermal bulk moduli BT are obtainedby minimizing free-energy with respect to V from fitting theintegral form of the Vinet equation of state (EOS) [23]. Then, thevolume thermal expansion coefficient was obtained by numericaldifferentiation for @V=@T.

4. Results and discussion

Table 1 gives our calculated lattice constants and isentropicelastic constants Cij

S at T¼0 and 300 K at ambient pressure(P¼ 0 GPa) in comparison with the results from the previouscalculations [4,7] (T¼0 K) and experimental data at ambient condi-tions [4] (P¼ 0 GPa, T¼300 K). In all cases, the comparison is quiteagreeable. At T¼0 K our calculated results show a very goodagreement with the other theoretical data. For YCu, the presentresults at 300 K are very closed to the experimental values obtainedby Morris et al. [4] at ambient conditions. The largest discrepancybetween theory and experiment is the value for CS

44, where thedifference is approximately 10% for YAg.

20

40

60

80

100YAg

C11S

C12S

C44S

ntro

pic

elas

tic c

onst

ants

Cij [

GP

a]S

R. Wang et al. / Physica B 406 (2011) 3951–39553954

For the temperature dependence of the isentropic elasticmoduli Cij

S, we present in Figs. 1–4 our calculation in the range0–1000 K at ambient pressure. Figs. 1 and 2 show our findings forbenchmark metals Ag and Cu respectively, accompanied byavailable experimental data taken from ultrasonic measurements[44–46]. Figs. 3 and 4 give our prediction for the unknown valuesof temperature-dependent Cij

S for YAg and YCu, respectively.Through the calculated values of isentropic elastic constants forAg and Cu in comparison with the values measured by experi-ments, we get a useful test of the accuracy of the method and theprecision of our calculations of temperature dependence of elasticconstants for YAg and YCu. The same DFT method had also beenemployed to calculate the elastic constants of Cu as a function of

0 200 400 600 80020

40

60

80

100

120

140

Temperature [K]

AgC11

S

C12S

C44S

Isen

tropi

c el

astic

con

stan

ts C

ij [G

Pa]

S

Fig. 1. For single-crystal Ag, we present calculated isentropic elastic constants and

corresponding experimental data. The solid, dashed, and dashed-dotted curves

denote the present values of CS11, CS

12 and CS44, respectively. The open symbols

denote the corresponding values of ultrasonic measurements by Neighbours and

Alers [44].

0 200 400 600 800 10000

20

40

60

80

100

120

140

160

180

Temperature [K]

CuC11S

C12S

C44S

Isen

tropi

c el

astic

con

stan

ts C

ij [G

pa]

S

Fig. 2. For single-crystal Cu, we present calculated isentropic elastic constants and

corresponding experimental data. The solid, dashed, and dashed-dotted curves

denote the present values of CS11, CS

12 and CS44, respectively. The open symbols

denote the corresponding values of ultrasonic measurements by Overton et al.

[45], and the filled symbols display those from the measurements of Chang and

Himmel [46].

0 200 400 600 800 10000

Temperature [K]

Ise

Fig. 3. The predicted isentropic elastic constants as a function of temperature for

YAg. The solid, dashed, and dashed-dotted curves denote the present values of

CS11, CS

12 and CS44, respectively. Note that C11 4C12, C11 40, and C44 40 at all

temperatures, which is consistent with stability of B2 lattice.

0 200 400 600 800 10000

20

40

60

80

100

120

Temperature [K]

YCuC11

S

C12S

C44S

Isen

tropi

c el

astic

con

stan

ts C

ij [G

Pa]

K

Fig. 4. The predicted isentropic elastic constants as a function of temperature for

YCu. The solid, dashed, and dashed-dotted curves denote the present values of

CS11, CS

12 and CS44, respectively. Note that C114C12, C1140, and C4440 at all

temperatures, which is consistent with stability of B2 lattice.

temperature by Wang et al. [16], and our results show nodiscrepancy from their values. The overall observation is that allthe calculated values of CS

11, CS12, and CS

44 decrease with increasingtemperature, since thermal expansion may soften the elasticmoduli at high T. At higher temperature, we also find that trendof Cij

S approach linearity. Considering our calculated resultsfor both YAg and YCu, we find the values of CS

11 decrease to thelargest extant in the whole temperature range 0–1000 K, andthose of CS

12 decrease least. CS11, CS

12, and CS44 for YAg decrease by

12.8, 7.3, and 9.6 GPa, and those for YCu decrease by 13.9, 5.9, and10.3 GPa, respectively. The requirement of mechanical stability ina cubic crystal leads to the following restrictions on the elasticconstants, C11�C1240, C1140, and C4440[47]. Our results of theelastic constants as shown in Figs. 3 and 4 satisfy these stabilityconditions in the range 0–1000 K.

Next we discuss the temperature dependence of Cauchy pressurewhich was suggested that it could be used to describe the angularcharacter of atomic bonding in metals and compounds by Pettifor

0 200 400 600 800 10000

5

10

15

20

25

Temperature [K]

Cau

chy

pres

sure

[GP

a]

Fig. 5. The predicted isentropic Cauchy pressure defined by PSC ¼ CS

12�CS44 as

a function of temperature. The solid and dashed curves denote the values for

YAg and YCu, respectively.

R. Wang et al. / Physica B 406 (2011) 3951–3955 3955

[48]. The isentropic Cauchy pressure PCS is defined as

PSC ¼ CS

12�CS44: ð20Þ

If the value of Cauchy pressure is more positive and bigger, thebonding of material is more metallic in character. In contrast,negative Cauchy pressure requires an directional character andlow mobility in the bonding. Generally, for ductile materials suchas metals Ag and Cu, the Cauchy pressures have positive values,while for brittle semiconductors such as Si, the Cauchy pressure isnegative. The YAg and YCu are a new class of highly ordered andductile intermetallics, especially for YAg in some cases exceeding20% ductility in tension [4]. In Table 1, the present results of Cauchypressure T¼0 and 300 K have positive values for YAg and YCu, andagree well with the previous theoretical results at T¼0 K and theexperimental results at room temperature (about 300 K). Fig. 5illustrates that the Cauchy pressure for YAg and YCu as a function oftemperature. In the range of temperature 0–1000 K, the Cauchypressure of YAg is always greater than that of YCu, and those forboth YAg and YCu increase with elevating temperature. Our calcu-lated results demonstrate that high temperature may soften thedirectional character, and YAg and YCu will become more ductilewhile the temperature is increased.

5. Conclusions

In this work, we present the temperature-dependent elasticconstants of two ductile rare-earth intermetallic compounds YAgand YCu with CsCl-type B2 structure by using a first-principlesquasistatic approach, in which the static elastic constants as afunction of volume are determined by the first-principles DFTtotal-energy calculations within the framework of the method ofhomogeneous deformation and the volume–temperature relationdetermined by first-principles phonon calculations based on DFPTwith quasiharmonic approach. To benchmark the reliabilityresults of the presented method, the comparison between ourcalculated results for Ag and Cu with available experimentaldata has been performed and shows good agreements. In the

whole temperature range 0–1000 K, the elastic constants follow anormal behavior with temperature that those decrease withincreasing temperature, and satisfy the stability conditions forB2 structures. The values of CS

11 decrease to the largest extant, andthose of CS

12 decrease least. The temperature dependence ofCauchy pressure is also discussed. The Cauchy pressures for bothYAg and YCu increase with elevating temperature, and that of YAgis always higher than that of YCu. Our results mean that increas-ing temperature may improve ductility of YAg and YCu.

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