Computational Materials Science 68 (2013) 379–383

Contents lists available at SciVerse ScienceDirect

Computational Materials Science

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

Ab initio investigation of the elastic and piezoelectric properties of lithium basedChalcogenides LiMX2 (M = Ga,In; X = S,Se)

Brahim Lagoun a,b,1, Tayeb Bentria a,⇑, Bachir Bentria a,2

a Laboratoire de Sciences Fondamentales, Université Amar Telidji de Laghouat, BP 37G, Laghouat 03000, Algeriab School of Physics, Faculty of Science, Université A.B. Belkaid-Tlemcen, BP 119, Tlemcen 13000, Algeria

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 August 2012Received in revised form 4 November 2012Accepted 5 November 2012Available online 12 December 2012

Keywords:PiezoelectricElasticDFTDebye temperatureTernary Chalcogenides

0927-0256/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.commatsci.2012.11.010

⇑ Corresponding author. Tel.: +213 792 33 68 39; fE-mail addresses: lag17_brahim@yahoo.fr (B. Lagou

(T. Bentria), bentria_b@yahoo.co.uk (B. Bentria).1 Tel.: +213 553 83 72 86.2 Tel.: +213 553 35 47 05.

In this paper we report theoretical ab initio study of the elastic and piezoelectric properties of LiMX2

(M = Ga,In; X = S, Se) by means of density functional theory (DFT). Other relevant quantities such as shearmodulus, Young’s modulus, sound velocity, Poisson’s ratio, anisotropy factors, and Debye temperaturehave been calculated. The calculated Debye temperatures for LiInS2 and LiInSe2 are very close to exper-imental counterparts. Moreover, we present for the first time the investigation of the piezoelectric prop-erties of LiGaS2, LiGaSe2 and LiInSe2 using the density-functional perturbation theory (DFPT). Thecalculated piezoelectric tensors for LiInS2 are in good agreement with the experimental data. These com-pounds that have reached crystal growth maturity show good piezoelectric property. For example, thecalculated d15 and d33 piezoelectric coefficients of LiGaSe2 are 5.02pN/C and 3.17pN/C respectively.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The investigation of elastic and piezoelectric properties of newand/or more efficient crystals is motivated by the growing demandof piezoelectric crystals. These materials are used in a wide rangeof devices such as pressure sensors, ultrasonic transducers formedical and sonar imaging and various types of micro-positionersand actuators. This diversity in use stimulates researchers to inves-tigate the properties of new potential piezoelectric crystals. Ourattention is turned toward the noncentrosymmetric Li-containingternary Chalcogenides LiMX2 (M = Ga,In; X = S,Se). These ortho-rhombic crystals are known as good infrared nonlinear opticalcrystals with unique optical properties such as wide band gap, highenough thermal conductivity, low anisotropy of linear thermalexpansion coefficient and wide opportunities of phase-matchingconditions [1]. These properties lead to a range of laser applicationssuch as second harmonic generation (SHG) and optical parametricoscillators in the mid-IR domain [1,2].

The maturity of Li based Chalcogenides took a longer time thanthe traditional counterparts such as AgGaS2(Se2) ZnGaP2. This wasdue to the high chemical reactivity of Li+ ion that interacted withthe container walls during the processes of synthesis and growth[3]. In recent years, much progress in LiMX2 crystal growth has

ll rights reserved.

ax: +213 29 93 21 45.n), tayeb.bentria@gmail.com

been achieved. Single crystals of these compounds are grown tocentimeter size and high optical quality in many laboratoriesaround the world using the Bridgman–Stockbarger technique [4–6]. LiGaS2 and LiInS2 single crystals have reached commercialmaturity and are used in mid-IR nonlinear optics for frequencyconversion [4].

Recently Wang et al. measured the piezoelectric, elastic anddielectric properties of LiInS2. Their results show very large piezo-electric constants (d24 = �13.3 pC/N; d33 = 9.6 pC/N) [3] and smalldependence of the elastic confidents S44 and S66 with temperature,which is advantageous for device applications. To the best of ourknowledge no experimental or computational investigations havebeen carried out on the piezoelectric properties of LiGaS2, LiGaSe2

and LiInSe2. Furthermore, the elastic properties of LiInSe2 havenever been calculated or measured so far. Therefore, further theo-retical investigations of the elastic and piezoelectric properties arerequired to explore more these important non-centrosymetriccrystals.

This paper is organized as follow. After the introduction, Sec-tion 2 gives the computational details. Section 3 presents the cal-culated results and thorough discussions on the structural, elasticand piezoelectric properties of our materials. A brief summaryand conclusion are presented in Section 4.

2. Computational details

The elastic properties were calculated by means of pseudopo-tential plane wave method within the density functional theory

380 B. Lagoun et al. / Computational Materials Science 68 (2013) 379–383

(DFT) frameworks as implemented in CASTEP package [7]. Perdew–Burke–Ernzerhof generalized gradient approximation (GGA-PBE)was used for the exchange–correlation potentials [8]. Ultrasoftpseudopotentials (USP) [9] were used with the following valenceelectronic configurations Li:2s1, In:4d105s25p1, Ga:4s24p1, Se:4s24-p4 and S:3s23p4 and a plane wave cutoff energy of 600 eV. A4 � 4 � 4 Monkhorst Pack mesh grid was used for k-points sam-pling of the first Brillouin zone. The calculations were assured ahigh level convergence of the total-energy difference within5 � 10�6 eV/atom and maximum Hellmann–Feynman force within0.1 meV/Å.

The piezoelectric coefficients were calculated within the frame-work of modern theory of polarization and density-functional per-turbation theory DFPT as implemented in the open-source ABINITcode package [10,11]. This theory has been shown to give success-ful description of the dielectric and piezoelectric properties of awide range of materials in which electronic correlation is not toostrong, see Zheng et al. [12,13]. Calculations were carried out usingTroullier–Martins norm-conserving pseudopotentials [14]. The ex-change and correlation effects were treated within the local-density approximation (LDA) [15] Perdew–Wang 92 functionalparameterizations [16]. A convergence study was made forplane-wave energy cutoff and Monkhorst–Pack grid of Brillouinzone k-point sampling to insure a 10�6 Ha energy tolerance whichis enough for second order derivation of total energy with respectto electric fields and Cartesian strains. Due to the close unit cellvolumes and electrons numbers in LiXM2 (X = Ga,In, M = S,Se), thisconvergence study resulted in 60 Hartree for cutoff energy and a8 � 8 � 8 mesh grid for all four compounds. The ground statestructure was fully optimized by minimizing the total energy andHellmann–Feynman forces. Owing to the conventional underesti-mation of the equilibrium lattice constants by local-densityapproximation pseudopotentials PP-LDA and also the sensitivityof piezoelectricity to cell volume [17], the optimization steps pro-ceeded until the maximum force components on any atom are lessthan 10�6 Hartree/Bohr.

3. Results and discussion

3.1. Structural properties and mechanical stability

Since all physical properties are related to the total energy, theoptimized structure ‘‘lattice constants and internal parameter’’ thatminimize the total energy for each approximation was carefullyperformed. Table 1 presents the calculated cell parameters of therelaxed LiMX2 structures using GGA (CASTEP) and LDA (ABINIT).

Table 1The equilibrium structural parameters of the orthorhombic chalcopyrite LiMX2 calculatedfrom literature.

Method a b c V

LiInS2

LDA 6.772 7.846 6.360 338.0GGA 7.000 8.180 6.070 347.6Experimental 6.823d 8.266d 6.524d 367.9GGA-NCP 7.053a 8.265a 6.626a 386.2

LiGaS2

LDA 6.692 8.055 6.409 345.5GGA 6.600 7.940 6.310 330.7Experimental 6.519d 7.872d 6.238d 320.1GGA-PBE 6.550a 7.854a 6.272a 322.7

a Data taken from Ref. [20].b Data taken from Ref. [22].c Data taken from Ref. [21].d Data taken from Ref. [1].

These values are used in subsequent calculations of physical prop-erties. For comparison, the experimental and theoretical valuesavailable in the literature are also included in Table 1.

The lattice parameters calculated by means of GGA, are in aver-age 1.5% larger than experimental values whereas the calculatedcounterparts using LDA are in average 1.8% smaller. This goodagreement with experimental values confirm also the overestima-tion of GGA and underestimation of LDA trends of cell volume.

The mechanical stability of these crystals depends on the posi-tive definiteness of stiffness matrix and the Born stability criteria.For the Orthorhombic case, these criteria are [18,19]

ðC11 þ C22 � 2C12Þ > 0; ðC11 þ C33 � 2C13Þ > 0;ðC22 þ C33 � 2C23Þ > 0; C11 > 0;C22 > 0;C33 > 0; C44 > 0; C55 > 0; C66 > 0;ðC11 þ C22 þ C33 þ 2C12 þ 2C13 þ 2C23Þ > 0:

ð1Þ

The calculated elastic constants for all four structures, pre-sented in the next section, do obey the above conditions whichindicates that these orthorhombic structures are mechanicallystable.

3.2. Elastic properties

Elastic properties of materials are very important because theyare related to various fundamental properties such as equation ofstate and phonon spectra. They are also thermodynamically linkedto specific heat, thermal expansion, Debye temperature and melt-ing point.

Nine independent strain coefficients (Cij) are necessary to calcu-late the elastic constants of Orthorhombic Pna21 compounds. Theresults of the calculated elastic constant and bulk modulus (inGPa) are presented in Table 2 together with available experimentaland the previously calculated values. Our results for LiGaS2 andLiGaSe2 are in good agreement with those calculated by Ma et al.[20] taking in consideration the difference between ultrasoft andnorm-conserving pseudopotentiels. The only known experimentalelastic constants for our compounds are those of LiInS2 by Wanget al. [5]. In their work, the authors present negative values forC12, C13 and C23 with no further explanation for this situation.Our results show close agreement with C33, C44 and C66 valuesand total disagreement with negatives C12, C13 and C23.

There are unfortunately no available theoretical or experimen-tal studies of the elastic properties for LiInSe2 to compare with.However, due to the close nature of our compounds and the good

using LDA and GGA approximation and experimental and theoretical result obtained

Method a b c V

LiInSe2

LDA 7.146 8.267 6.743 398.4GGA 7.270 8.590 6.880 429.7Experimental 7.192d 8.412d 6.793d 411.0GGA-NCP 7.085b 8.251b 6.751b 394.7GGA-USP 7.274a 8.491a 6.980a 431.1LiGaSe2

LDA 6.692 8.055 6.409 345.4GGA 6.900 8.240 6.640 377.5Experimental 6.829d 8.239d 6.538d 367.9GGA(LAPW + LO) 6.911c 8.392c 6.622c 384.1LDA CA-PZ 6.792c 8.094c 6.490c 356.7

Table 2The calculated bulk modulus B0 (GPa), the elastic constants (GPa) for orthorhombic LiMX2, and result available in literature.

C11 C22 C33 C44 C55 C66 C12 C13 C23 B0

LiInSe2

Ultrasoft GGA 50.3 44.0 58.2 12.6 11.3 13.6 25.8 19.6 19.5 31.2Experimental/theory N/A N/A N/A N/A N/A N/A N/A N/A N/A N/A

LiGaS2

Ultrasoft GGA 81.3 68.3 89.8 22.4 17.3 23.9 38.0 27.6 29.9 47.6NCP-GGA a 77.7 61.8 85.6 22.3 17.8 25.0 31.7 16.6 18.9 39.6

LiInS2

Ultrasoft GGA 70.8 64.4 66.0 18.3 16.6 14.2 45.8 40.4 41.1 50.5Experimental b 50.1 44 61.7 �11.7 �3.5 �9.6 20.8 19.2 36.4 N/A

LiGaSe2

Ultrasoft GGA 80.6 71.1 87.7 23.1 17.5 23.9 41.1 26.5 29.3 48.0Ultrasoft LDA a 85.3 66.1 83.7 23.4 19.4 25.7 25.2 16.4 14.3 38.0

a Data taken from Ref. [21].b Data taken from Ref. [5].

B. Lagoun et al. / Computational Materials Science 68 (2013) 379–383 381

value of the Debye temperature see Table 2, the present values canserve as a reference for future investigations.

It is known that the elastic constant C44 is the most importantparameter indirectly governing the indentation hardness of amaterial. Large C44 means strong resistance to the monoclinic sheardistortion in the (100) plane, and the C66 is related to the resis-tance to the shear in the h110i direction [23]. We notice that ourcompounds present low C44 and C66 coefficient compared to quartz(C44 = 57, C66 = 40 GPa) [24], and are also lower than those of thenon Li containing Chalcogenides CuGaS2, CuInS2, AgGaS2 whichvary between 30 and 43 GPa [25,26].

For the polycrystalline case which has been synthetized mainlyfor LiInS2 case [27], the bulk modulus (B) and the shear modulus(G) may be determined. There are two approximation methods tocalculate the polycrystalline modulus, namely, the Voigt method[28] and the Reuss method [29]. For specific case of an orthorhom-bic lattice, the Reuss shear modulus (GR) and the Voigt shear mod-ulus (GV) are given by

GR ¼4

15ðS11 þ S22 þ S33Þ �

415ðS12 þ S13 þ S23Þ þ

315ðS44 þ S55

þ S66Þ; ð2Þ

and

GV ¼1

15ðC11 þ C22 þ C33Þ �

115ðC12 þ C13 þ C23Þ þ

115ðC44

þ C55 þ C66Þ; ð3Þ

The Reuss bulk modulus (BR) and the Voigt bulk modulus (BV)are defined as:

BR ¼ ðS11 þ S22 þ S33Þ þ 2ðS12 þ S13 þ S23Þ: ð4Þ

and

BV ¼19ðC11 þ C22 þ C33Þ þ

29ðC12 þ C13 þ C23Þ: ð5Þ

In Eqs. (2) and (4), Sij are the elastic compliance constants. Tak-ing the Hill empirical average [30], the elastic module of the poly-crystalline material for shear G and the bulk B modulus are

G ¼ ðGR þ GV Þ2

: ð6Þ

B ¼ ðBR þ BV Þ2

: ð7Þ

The Young’s modulus (E) and Poisson’s ratio (v) for an isotropicmaterial are given by E = 9B � G/(3B + G) and v = (3B–2G)/2(3B + G), respectively [31,32]. Using the above relations we calcu-

lated the bulk modulus, shear modulus, Young’s modulus, andPoisson’s ratio for all LiMX2 (M = Ga,In; X = S, Se) compoundswhich are presented in Table 3.

It is known that the isotropic shear modulus and bulk modulusare measures of the hardness of a solid. The bulk modulus is a mea-sure of resistance to the volume change by an applied pressure,whereas the shear modulus is a measure of resistance to thereversible deformation upon shear stress [23]. Therefore, the iso-tropic shear modulus is more pertinent to hardness and a largershear modulus is mainly due to a larger C44. In general, the valueof the shear modulus is an indication of the directional bonding be-tween atoms. The degree of Cauchy violation (C44/C12 – 1) indi-cates the deviation from a two-body central force model [23].That is, the closer to 1 the C44/C12 ratio is, the stronger the ioniccharacter between two atoms. In this work, the C44/C12 ratio forLiGaSe2, LiGaS2 LiInSe2 are 0.56, 0.59, 0.49 respectively. This showsthat these crystals exhibit to a certain extent an ionic character. Acombination of covalent and ionic character for LiGaS2 and LiGaSe2

is demonstrated in reference [2]. LiInS2 exhibit remarkable lowerCauchy violation (0.40) which agrees with the remarque of Maet al. that the bonding characteristics in our compound cannot bepurely ionic but must exhibit a large covalent part [20].

According to the criterions in Ref [23], a material is brittle (duc-tile) if the B/G ratio is less (greater) than 1.75. In fact the value ofB/G is greater than 1.75 for all four crystals. Hence, our materialsshould behave in a ductile manner.

The Young’s modulus is defined as the ratio of stress to strain,and used to provide a measure of the stiffness of a material. Thematerial is stiffer if the value of the Young’s modulus is higher.In a comparative way, the value of the Young’s modulus for LiGaS2

and LiGaSe2 are higher than LiInS2, LiInSe2, so the Gallium contain-ing materials are stiffer.

The value of the Poisson’s ratio is indicative of the degree ofdirectionality of the covalent bonds. This value is small (v = 0.1)for covalent materials, whereas for ionic materials a typical valueof v is 0.25 [23]. The calculated Poisson’s ratios are all above0.25. Therefore, the ionic contribution to the interatomic bondingfor these compounds is dominant.

Many low-symmetry crystals exhibit a high degree of elasticanisotropy [23]. The shear anisotropic factors in different crystal-lographic planes provide a measure of the degree of anisotropyof atomic bonding in different planes. The shear anisotropic fac-tors for the (100), (010), and (001) planes, are respectivelygiven by

A1 ¼4C44

ðC11 þ C33 � 2C13Þ; ð8Þ

Table 3Reuss, Voigt and Hill values for shear modulus (G) and the bulk modulus (B), Young’s modulus (E), and Poisson’s ratio (v) are in Hill’s approximation. All moduli values are in GPa.Longitudinal, transverse, and the average elastic wave velocity (VL, Vt and Vm) in m/s, the Debye temperature is in Kelvin.

GR GV GH BR BV BH EH VH VL Vt Vm hD hD,exp

LiGaS2 21.4 22.3 21.8 47.6 47.8 47.7 56.8 0.301 10426 5559 6210 425 N/ALiInS2 14.2 14.7 14.5 50.5 50.7 50.6 39.6 0.369 8871 4038 4552 306 310 � 50a

LiGaSe2 21.4 22.4 21.9 48.0 48.1 48.1 57.0 0.302 8664 4611 5152 337 N/ALiInSe2 12.8 13.3 13.1 31.2 31.4 31.3 34.4 0.317 6720 3481 3897 244 265 � 10b 240 � 50a

a Data taken from Ref. [37].b Data taken from Ref. [38].

Table 5

382 B. Lagoun et al. / Computational Materials Science 68 (2013) 379–383

A2 ¼4C55

ðC22 þ C33 � 2C23Þ; ð9Þ

A3 ¼4C66

ðC11 þ C22 � 2C12Þ; ð10Þ

A value of unity means that the crystal exhibits isotropic prop-erties, while values other than unity represent varying degrees ofanisotropy. Therefore, it can be seen from Table 4 that LiMX2 crys-tals exhibit low anisotropy in the (010) plane. Another way ofmeasuring the elastic anisotropy is to give the percentages ofanisotropy in the compression (Acomp) and shear (Ashear) [21,22]which are defined as:

Acomp ¼Bv � BR

Bv þ BR� 100: ð11Þ

Ashear ¼Gv � GR

Gv þ GR� 100: ð12Þ

For crystals, these values can range from zero (total isotropy) to100% (maximum anisotropy). The percentages of anisotropy Acomp

and Ashear are calculated and presented in Table 4. These valuesdemonstrate that the anisotropy in shear is at least four timeshigher than in compression.

The Debye temperature (hD) is an important fundamentalparameter closely related to many physical properties such asspecific heat and the melting temperature. It is the highest tem-perature that can be achieved due to a single normal vibration[33]. At low temperature the vibrational excitations arise solelyfrom acoustic vibrations. Hence, at this condition hD calculatedfrom elastic constants are the same as that determined by thespecific heat measurements. Based on the elastic constants, theDebye temperature is calculated using the following commonrelation [34].

hD ¼hkb

3n4pVa

� �1=3

Vm; ð13Þ

where h is Planck’s constant, kb is Boltzmann’s constant, n is thenumber of atoms per unit formula, Va is the unit cell volume andVm is the average sound velocity which is given by [35]:

Vm ¼13

2V3

t

þ 1V3

l

!" #�13

: ð14Þ

Table 4The shear anisotropic factors A1 A2 A3 for (100), (010), and (001) planes andpercentages of anisotropy in the compression (Acomp) and shear (Ashear).

A1 A2 A3 Acomp Ashear

LiGaS2 0.77 0.70 1.30 0.245 2.113LiInS2 1.30 1.38 1.31 0.184 1.823LiGaSe2 0.80 0.70 1.38 0.075 2.347LiInSe2 0.73 0.71 1.27 0.260 1.980

where Vl and Vt are the longitudinal and transverse elastic wavevelocities, respectively. They can be obtained from Navier’s equa-tions [36]

Vl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Bþ 4G

3q

s; Vt ¼

ffiffiffiffiffiffiGq:

sð15Þ

The values of Vm, Vl, Vt and hD calculated using the above formu-las are presented in Table 3 together with available experimentalvalues of Debye temperature. For LiInS2 and LiInSe2, the calculatedhD is, within the experimental error, in good agreement with theexperimental value of refs. [37,38]. This result validates somehowthe value of the present calculated elastic constants for LiInSe2 andLiInS2 since hD is derived mainly from the elastic constants. As faras LiGaS2 and LiGaSe2 are concerned, and to the best of our knowl-edge, no experimental or theoretical work has been performed sofar on their Debye temperature.

3.3. Piezoelectric properties

In piezoelectric materials, it is the elastic deformations of thelattice under stress that cause the electric polarization by shiftingthe centroids of positive and negative charges and vice versa[39]. A material must be formed as a single crystal and have anon-centrosymmetric structure to be truly piezoelectric. Our com-pounds belong to the non-centrosymmetric mm2 class for whichthe piezoelectric tensor exhibits five nonzero components, namelyd15, d24, d31, d32, d33.

Abinit package calculates the piezoelectric coefficients dij bycalculating the derivatives of the total energy with respect to elec-tric fields, and Cartesian strains [11]. The ANADDB module of ABI-NIT calculates the relaxed piezoelectric coefficients eij. The dij

coefficients are obtained from the relation dja ¼ SðeÞjk eka where SðeÞjk

are the compliance constants. Table 5 regroups the calculated pie-zoelectric coefficient dij for LiMX2 (M = Ga,In; X = S,Se) obtained bythe PP-LDA approximation. For comparison, the available experi-mental piezoelectric coefficients are also presented in Table 5.

To the best of our knowledge, there are no experimental or the-oretical piezoelectric data for LiGaS2, LiGaSe2 or LiInSe2. The calcu-lated dij coefficients presented in Table 5 show that our threecompounds have high piezoelectric coefficients in comparison

The calculated relaxed Piezoelectric coefficient dij (pC/N), and experimental resultavailable in literature.

d15 d24 d31 d32 d33

LiGaS2 PP-LDA 4.78 4.04 1.43 1.38 �2.63LiInS2 PP-LDA �12.34 �11.39 �5.94 �9.76 16.52

Experimental a �10.20a �13.30a �4.60a �6.30a 9.60a

Experimental b �5.60b �2.70b 7.50b

LiGaSe2 PP-LDA 5.41 4.55 1.58 1.61 �3.23LiInSe2 PP-LDA 6.76 5.82 1.75 2.97 �4.66

a Data taken from Ref. [5].b Data taken from Ref. [40].

B. Lagoun et al. / Computational Materials Science 68 (2013) 379–383 383

with quartz (d14 = 0.727 pC/N, d11 = �2.310 pC/N). These data con-firm previous remarks that the good piezoelectric properties ofLiInS2 have been attributed not only to the polarized In–S andLi–S bonds, but more importantly to the constructive addition ofthese polarizations while applying external force or electric fieldon the crystal [5]. Moreover these results show that LiInS2 hasthe highest piezoelectric coefficients in these Li basedChalcogenides.

For the four compounds of the Li-Chalcogenides family, the onlyknown experimental values of the piezoelectric coefficient dij arethose of LiInS2 [5,4] Table 5. There is a clear discrepancy betweenthe two experimental results reported in [5,40] especially for d32

coefficient. Wang et al. [5] explain this difference by the fact thatthe piezoelectric coefficients are very sensitive to crystal quality,the design of the samples and data processing method. Our calcu-lated dij are in reasonable agreement with the experimental valuesfor the LiInS2 case.

4. Conclusion

In summary, we have calculated the structural, elastic and pie-zoelectric properties of the LiMX2 compounds by mean of densityfunctional theory. The calculated elastic constants are inagreement with the available theoretical result and do obey themechanical stability conditions. The relevant mechanical and ther-mo-dynamical properties such as shear modulus, Young’s modu-lus, Poisson’s ratio, Debye temperature, and shear anisotropicfactors are calculated and discussed. Since there are no experimen-tal data available for the elastic constant of LiGaS2 LiGaSe2 LiInSe2

we believe that due to the good agreement of the Debye tempera-ture with the experimental result, the theoretical elastic constantscan serve as reference for further investigation.

The piezoelectric tensor dij have been calculated using plane-wave pseudopotentials method with local-density approximation(LDA) for exchange and correlation effects. These results show thatour three compounds exhibit high piezoelectric coefficient in com-parison with quartz and confirm that LiInS2 has very important dij

coefficients. We conclude that besides having good mid-IR nonlin-ear optical properties, all four Li Chalcogenides presented in thiswork are promising candidates for piezoelectric applications.

Acknowledgements

The authors wish to thank the Laboratoire de sciences Fonda-mentales, Laghouat University, Algeria for providing adequateworking conditions to accomplish this work. We also wish to thank

the Algerian Research Network for providing machine time on theircomputer grid.

References

[1] L. Isaenko et al., J. Cryst. Growth 275 (2005) 217–223.[2] L. Isaenko, I. Vasilyeva, A. Yelisseyev, S. Lobanov, V. Malakhov, L. Dovlitova, J.-J.

Zondy, I. Kavun, J. Cryst. Growth 218 (2000) 313–322.[3] V.V. Badikov et al., Opt. Mater. 23 (2003) 575–581.[4] L.I. Isaenko, I.G. Vasilyeva, J. Cryst. Growth 310 (2008) 1954.[5] Shanpeng Wang et al., J. Cryst. Growth, 2, 2011 (in press).[6] Nonlinear optical crystals (LiGaS2), (LiGaSe2), (LiGaTe2), <http://www.ascut-

bridge.de/product.php?pid=11>.[7] S.J. Clark, M.D. Segall, C.J.P.J. Hasnip, M.J. Probert, K. Refson, M.C. Payne,

Zeitschriftfuer 220 (5–6) (2005) 567–570.[8] B. Hammer, L.B. Hansen, J.K. Norskov, Phys. Rev. B 59 (1999) 7413–7421.[9] D. Vanderbilt, Phys. Rev. B 41 (1990) 7892–7895.

[10] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L.Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, A. Roy, M. Mikami, Ph.Ghosez, J.-Y. Raty, D.C. Allan, Comput. Mater. Sci. 25 (2002) 478.

[11] D.R. Hamann, X. Wu, K.M. Rabe, D. Vanderbilt, Phys. Rev. B71 (2005) 035117.[12] Y. Zheng et al., J. Phys: Conference Series 29 (2006) 61–64.[13] Y. Zeng et al., Comput. Mater. Sci. 56 (2012) 169–171.[14] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993.[15] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566.[16] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244.[17] X. Gonze, C. Lee, Phys. Rev. B 55 (1997) 10355.[18] Wallace D C, Thermodynamics of Crystals, Wiley, NewYork, 1972. ISBN 0-486-

40212.[19] F. Birch, Phys. Rev. 71 (1947) 809, http://dx.doi.org/10.1103/PhysRev.71.809.[20] T.-h. Ma et al., Computational Materials Science 47 (2009) 99–105.[21] T.-h. Ma et al., Physica B 405 (2010) 363–368.[22] T. Ma et al., J. Alloys Compd. 509 (2011) 9733–9741.[23] A. Yildirim, H. Deligoz, H.E. Chin, Phys. B 21 (3) (2012) 037101.[24] Heyliger et al J. Acoust. Soc. Am., Vol. 114, No. 2, August 2003.[25] H. Hai-Jun et al., Phys. Scr. 82 (2010) 055601.[26] G. Mikhail, P. Brik, Status Solidi C 8 (9) (2011) 2582–2584.[27] S.P. Wang, X.T. Tao, C.M. Dong, J. Liu, M.H. Jiang, J. Synth. Cryst. 35 (2006)

1167–1171.[28] W. Voigh, Lehrbook der kristallphysi, 1928. Leipzig: Teubner.[29] A.Z. Reuss, Angew. Math. Mech. 9 (1929) 49.[30] R. Hill, Proc. Phys. Soc. Lond. A 65 (1952) 349.[31] K.B. Panda, K.S.R. Chandran, Acta Mater. 54 (2006) 1641.[32] P. Ravindran, L. Fast, P.A. Korzhavyi, B. Johansson, J. Wills, O. Eriksson, J. Appl.

Phys. 84 (1998) 4891.[33] T.L. Hill, An Introduction to Statistical Thermodynamics, Dover, New York,

1986.[34] I. Johnston, G. Keeler, R. Rollins, S. Spicklemire, Solids State Physics

Simulations, Wiley, New York, 1996.[35] O.L. Anderson, J. Phys. Chem. Solids 24 (1963) 909.[36] E. Schreiber, O.L. Anderson, N. Soga, Elastic Constants and Their

Measurements, McGraw-Hill, New York, 1973.[37] G. Kühn, E. Pirl, H. Neumann, E. Nowak, Cryst. Res. Technol. 22 (1987) 265–

269.[38] E. Gmelin, W.H. onle, Thermochim. Acta 269/270 (1995) 575–590.[39] Michel Dupeux aide-Mémoire science des Matériaux Dunod, Paris, 2004, ISBN

2 10 005458 9.[40] S. Fossier, et al, Opt.Soc. Am. J. B 21 (11) (2004) 1981–2007.