ARTICLE IN PRESS

Physica B 405 (2010) 1663–1676

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

Univers

Tel./fax:

E-m

(D. Vars

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

Pressure induced structural phase transition and elastic properties in BSb,AlSb, GaSb and InSb compounds

Dinesh Varshney a,b,�, G. Joshi a, Meenu Varshney c, Swarna Shriya a

a School of Physics, Vigyan Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore 452001, Indiab School of Instrumentation, USIC Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore 452001, Indiac Department of Physics, M.B. Khalsa College, Indore 452002, India

a r t i c l e i n f o

Article history:

Received 21 November 2009

Received in revised form

19 December 2009

Accepted 26 December 2009

Keywords:

Alloys

Computational techniques

Ductility

Mechanical properties

Phase transitions

26/$ - see front matter & 2009 Elsevier B.V. A

016/j.physb.2009.12.064

esponding author at: School of Physics, V

ity, Khandwa Road Campus, Indore 452001, I

+91 7312465689.

ail addresses: vdinesh33@rediffmail.com, vdin

hney).

a b s t r a c t

By formulating an effective interionic interaction potential that incorporates the long-range Coulomb,

the covalency effects, the charge transfer caused by the deformation of the electron shells of the

overlapping ions, the Hafemeister and Flygare type short-range overlap repulsion extended upto the

second neighbour ions and the van der Waals (vdW) interaction, the pressure dependent elastic and

thermodynamical properties of the III–V antimonide semiconductors as YSb (Y=B, Al, Ga, and In) are

investigated. Estimated values of phase transition pressure of YSb antimonides are consistent with the

available data on the phase transition pressures. The ratio RS/B of S (Voigt averaged shear modulus) over

B (bulk modulus), elastic wave velocity, average wave velocity and Debye temperature as functions of

pressure is calculated. From the ratio RS/B it is inferred that YSb (Y=Al, Ga, and In) are ductile and BSb is

brittle in zinc blende (B3) phase. To our knowledge this is the first quantitative theoretical prediction of

the ductile (brittle) nature of YSb antimonides and still awaits experimental confirmations.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

The structural phase transformations, as well as metallizationunder high pressure in III–V binary compounds as YSb (Y=B, Al,Ga, and In) is of considerable interest and has been studied in therecent past due to its technological importance in high tempera-ture electronic and optical devices [1,2]. These binary compoundsform a closed-shell ionic system crystallized in the ZnS-type (B3)structure with four-fold coordination at ambient conditions. TheBoron compounds display rather peculiar behaviour whencompared to other III–V compounds and appears to originatefrom the small core size and the absence of p electrons in the Bcore. In antimonides as BSb, the B attractive potential competeswith the Sb potential for the valence charge, a situation of smallheteropolarity and consequently small ionicity. The electronega-tivity of the anions is only slightly smaller than that of the cation,and possesses strong covalent character, and hence the inverserole between the cation and the anion in terms of charge transferis important in these materials.

There have been a number of first-principles theoreticalstudies of the structural and elastic properties of antimonides.

ll rights reserved.

igyan Bhawan, Devi Ahilya

ndia.

esh33@gmail.com

The elastic and bonding properties of the Boron compounds havebeen calculated using full potential linearized augmented planewave (FP-LAPW) method. The zinc-blende boron compounds BSbtransforms from B3-B1 at 0.68 GPa [2]. The ground state and thehigh-pressure properties of BSb following the plane-wave basissets and pseudopotential approximation (PW–PP) within localdensity approximation (LDA) have been calculated by Zaoui andresearchers and identified the structural transformation (B3-B1)at about 2.16 GPa [2].

Although there have been several theoretical studies of the YSb(Y=B, Al, Ga, and In), little experimental work has been devoted tothese compounds. This might be due to difficulties in measuringthe elastic properties by growing larger bulk epitaxial films asinformation on the elastic constants of the substrate and theepitaxial film is indispensable. Experimental transition pressuresfor AlSb (GaSb) from zincblende to rocksalt transition at about 8.3(6.2) GPa with volume collapses 20 (17.1) % [3] is reported. Thestructural transition in Al(Ga) antimonides with a transitionpressure in the range 5.3–12.5 (6.2–7) GPa is also documented [4].Using angle-dispersive X-ray (ADX) powder diffraction methodNelmes and researchers found the transition of AlSb fromzincblende to Cmcm at about 8.1 GPa. It is believed that GaSbtransform from zincblende to b-tin structure [5]. However, theADX pattern strictly rules out for the d-b-tin structure in III–V andII–VI systems. Later on, angle-dispersive X-ray (ADX) powderdiffraction experiments performed in diamond anvil cell alsoidentifies absence of b-tin-like structure but a disordered

ARTICLE IN PRESS

D. Varshney et al. / Physica B 405 (2010) 1663–16761664

orthorhombic structure with space group Imma stabilized undernon-hydrostatic conditions [6]. Previously, the volumetric mea-surement has shown the structural phase transition of InSb fromzincblende to b-tin at about 2.3 GPa [7].

Prior to high-pressure ADX measurements, the first-principlesab initio calculations using pseudopotential total-energy approachshows that the transition pressure for AlSb [zincblende torocksalt], GaSb [zincblende to b-tin-like] and InSb [zincblendeto b-tin-like] occur at 5.6, 6.3 and 3.3 GPa with volume collapse19.1%, 18.6% and 20.7% [8]. Following ab initio ionic pseudopo-tential calculations, Chelikowsky finds the phase transformationfrom zincblende to b-tin-like as well Rocksalt structure for II–VIand III–V semiconducting compounds. For the zincblende toRocksalt structural transition, the phase transition pressures areobtained as 10.17 [10.06 (14.09)] GPa for AlSb [GaSb (InSb)],respectively [9].

Following LDA pseudopotential calculations, Ozolins andZunger have explain the absence of b-tin structure in terms ofthe dynamical phonon instabilities in most of the compoundsemiconductors. From equation of state, the phase transitionpressure for both zincblende to b-tin-like as well Rocksaltstructure are calculated. The phase transition pressure withzincblende to rocksalt (zincblende to b-tin) for GaSb and InSbare about 5.2 (4.7) and 2.1 (1.6) GPa, respectively [10]. It is thusestablished that in on raising the pressure in most of the binarysemiconducting compounds, one expects a sequence of ZB-NaCl-b-Sn-like structural phase transitions. Furthermore, thedeclined transition pressure trend in III–V compounds is attrib-uted to an increase in the atomic numbers of the constituentelements. However, BSb is exceptional, because of its anomalousnature. The band gap is lower in high-Z compounds viz, 0.17 eV inInSb. The gap, which stabilizes the zinc-blende structure at lowpressures, is also responsible for the structural instability athigher pressures [11].

The high-pressure ADX and first-principles ab initio calcula-tions have thus motivated us to investigate the pressuredependent elastic properties in YSb (Y=B, Al, Ga, and In)semiconductors. Furthermore, the covalent bonding characteristicof these compound semiconductors is important and is an acidtest for many body interactions which can predict at least theextent of pressure one should generate to observe a structuralphase transition and other related properties. The modelling oflattice models in binary III–V semiconducting compounds is acomplicated task and, in many instances, must be guided byexperimental evidence of the low degree of freedom in order toobtain a correct minimal model, which will capture the observedeffect and will make useful predictions.

The first-principles ab initio calculations are complex andrequire significant effort; therefore phenomenological latticemodels [12–19] have been developed, to compute properties ofmaterials. The advantage of phenomenological model is theirsimplicity, applicability for a broad class of materials and areuseful to seek a systematic trend but less accurate. The reliabilityof results obtained from first principle depends upon suitablemodelling of interatomic interactions. These interactions areusually result of fits various data at 0 K, room temperature ortemperature at which experiment was carried out. However, it isnot clear whether simulations performed at other temperaturesstill reproduce the data accurately. Comparing theoretical andexperimental elastic constants and other properties at varioustemperatures can serve as a measure of reliability and usefulnessof lattice dynamical models.

The present investigations on elastic and structural phasetransition of YSb (Y=B, Al, Ga, and In) semiconductors based onphenomenological lattice models are aimed at understanding thecovalent bonding characteristics in terms of charge transfer

between the cation and the anion as well the impact of covalency.Besides these important effects, we have included the vdWinteraction effects [16–18]. The overlap repulsion is expressed interms of the Hafemeister and Flygare [14] type interactionsextended up to the second neighbour ions. The estimation of thevan der Waals coefficients (vdW) has been made following theSlater–Kirkwood variational method [19] with an idea that boththe ions are polarizable.

We introduce the assumptions for the formulation of aneffective interionic interaction potential (EIoIP) between a pair ofions and support them by physical arguments. The phasetransition pressures, associated volume collapses, the secondorder elastic constants, stability of crystal structures, pressurederivatives of elastic constants, sound velocity and Debyetemperature of YSb (Y=B, Al, Ga, and In) semiconductors are thencomputed. Theoretical results are compared and discussed inSection 3 with the existing first principles, experimental, andpredicted data with concluding remarks presented in Section 4.

2. Theory and method of computation

The application of pressure causes an increase in the overlap ofadjacent ions in a crystal and hence charge transfer takes placebetween the overlapping electron shells. The transferred chargesinteract with all others of the lattice via Coulomb’s law and giverise to many-body interactions, of which the most significant iscovalency and charge transfer effect leading to three-bodyinteractions. It is well known that three body interactions comefrom the covalent bonding in tetrahedral semiconductors. Theincreased effect due to covalent nature and of charge transfer,thus obtained, lead to an obvious necessity of their inclusion inthe elastic properties and high-pressure study of antimonidessemiconductors.

In general, the III–V binary compounds transform from theirinitial B3 to B1 structure under pressure and the thermodynami-cal properties are described by the appropriate thermodynamicalpotential relevant to the given ensemble. Either variable pressureor temperature acting on the systems is altered, the free energychanges smoothly and continuously. A phase transition is said tohave occurred if the change is discontinuous or continuous butwith a change in crystal symmetry. The stability of a particularstructure is decided by the minima of Gibbs’s free energy,G=U+PV–TS, U being the internal energy, which at 0 K corre-sponds to the cohesive energy, S is the vibrational entropy atabsolute temperature T, pressure P and volume V. The Gibbs freeenergies of the different possible arrangements of atoms varyunder compression, and at some stage it becomes favourable forthe material to change the type of atomic arrangement.

In the present calculations, we assume zero temperature i.e.,the frozen ionic degrees of freedom. Although, the experimentalresults are obtained at ambient temperature inferring a certainsmall temperature dependence of the transition pressures in therange of low temperatures. Henceforth, it is safe to consider thelattice model calculations that are representative of the resultsthat would be obtained under the actual experimental conditions.At zero temperature, the thermodynamically stable phase at agiven pressure P is the one with lowest enthalpy, and thethermodynamical potential is the Helmholtz free energy (H).

We begin by expressing the Gibbs’s free energies:

GB3ðrÞ ¼UB3ðrÞþPVB3 ð1Þ

GB1ðr0Þ ¼UB1ðr

0ÞþPVB1 ð2Þ

at T=0 K for ZnS (B3, real) phase and NaCl (B1, hypothetical) phasebecome equal at the phase-transition pressure P. Here, VB3

ARTICLE IN PRESS

D. Varshney et al. / Physica B 405 (2010) 1663–1676 1665

(=3.08r3) (VB1 (=2r03)) as the unit cell volume and r (r0) being thenearest-neighbour distance for B3 (B1) phase. The notationsUB3(r) and UB1(r0) denote cohesive energies for B3 and B1 phasesand are expressed as

UB3 ¼ ð�aMZe2=rÞ½Zþ2nf ðrÞ��X

ij

Cr�6ij �

Xij

Dijr�8

þnbbijexp½ðriþrj�rijÞ=r�þðn0b=2Þ½biiexpðð2ri�krijÞ=rÞþbjjexpðð2rj�krijÞ=rÞ� ð3Þ

UB1 ¼ ð�a0MZe2=r0Þ½Zþ2mf ðr0Þ��X

ij

Cr0�6ij �

Xij

Dr0�8ij

þmbbijexp½ðriþrj�r0ijÞ=r�þðm0b=2Þ½biiexpðð2ri�k0r0iiÞ=rÞ

þbjjexpðð2rj�k0r0jjÞ=rÞ� ð4Þ

Here the first term is the potential energies due to long-rangeCoulomb, and the second term is due to the charge transfercaused by the deformation of the electron shells of the over-lapping ions and the covalency effects. The III–V semiconductingcompounds contain covalent bonds so that some electrons aredistributed over the region between neighbouring atoms; in sucha situation, the ionic charge for each atom cannot be determineduniquely and hence the calculation of the Madelung energy ismodified by incorporating the covalency effects [20]. We thusneed to incorporate the effective charge arose due to thepolarization of a spherical shaped dielectric in displacingthe constituent positive ions. The third and fourth terms are theshort-range (SR) vdW attraction energies due to dipole–dipoleand dipole–quadrupole interaction. The C and D are the overallvdW coefficients, which are evaluated from the variationalapproach [19].

The vdW coefficients due to dipole–dipole and dipole–quadrupole interactions are calculated from the Slater andKirkwood variational approach and are

cij ¼3

2

e‘ffiffiffiffiffiffiffimep aiaj

ai

Ni

� �1=2

þaj

Nj

� �1=2" #�1

ð5Þ

dij ¼27

8

‘ 2

meaiaj

ai

Ni

� �1=2

þaj

Nj

� �1=2" #2

ai

Ni

� �þ

20

3

aiaj

NiNj

� �1=2

þaj

Nj

� �" #�1

ð6Þ

cii ¼3

2

e‘ffiffiffiffiffiffiffimep aiai

ai

Ni

� �1=2

þai

Ni

� �1=2" #�1

ð7Þ

dii ¼27

8

‘ 2

meaiai

ai

Ni

� �1=2

þai

Ni

� �1=2" #2

ai

Ni

� �þ

20

3

aiai

NiNi

� �1=2

þai

Ni

� �" #�1

ð8Þ

cjj ¼3

2

e‘ffiffiffiffiffiffiffimep ajaj

aj

Nj

� �1=2

þaj

Nj

� �1=2" #�1

ð9Þ

djj ¼27

8

‘ 2

meajaj

aj

Nj

� �1=2

þaj

Nj

� �1=2" #2

aj

Nj

� �þ

20

3

ajaj

NjNj

� �1=2

þaj

Nj

� �" #�1

ð10Þ

where m, e and Z are the electron mass, charge and valence of theconstituent metallic element respectively, (ai, aj) are the polariz-abilities of ith and jth ion, respectively; Ni and Nj are the effectivenumber of electrons responsible for polarization. The overall vdWcoefficients C and D are then expressed in terms of cij and dij, by aappropriate lattice sums (Sij and Tij) the values of which are takenfrom [18] such that and the expression is given by

C ¼ cijSijþciiSiiþcjjSjj ð11Þ

D¼ dijTijþdiiTiiþdjjTjj ð12Þ

The last two terms are SR repulsive energy due to the overlaprepulsion between ij, ii and jj ions. aM (aM

0) are the Madelungconstants for B3 (B1) phases. bij are the Pauling coefficient definedas bij=1+(Zi/ni)+(Zj/nj) with Zi(Zj) and ni(nj) as the valence and thenumber of electrons in the outermost orbit. Ze is the static chargeof the ion, k (k0) is being the structure factor for B3 (B1) structures,and b (r) are the hardness (range) parameters. r(r0) are the nearestneighbour ion separations for B3 (B1) structures.

The second term in Eqs. (3 and 4) is an algebraic sum of three-body force parameters f(r) and the force parameter arises due tocovalent nature i.e., f(r)= fcte(r)+ fcov(r). The charge transfer forceparameter fcte(r) is expressed as [12,13,15]

fcteðrÞ ¼ f0 expð�r=rÞ ð13Þ

Because the III–V semiconducting compounds are partially ionicand partially covalent in bonding, the attractive forces due tocovalency are important that modifies the effective charge. Thepolarization effects originates from changes in covalency due toelectric fields and the covalency term is expressed as [20]

fcovðrÞ ¼4e2V2

sps

r0E3g

ð14Þ

Vsps is being the transfer matrix element between the outermost p

orbital of anion and the lowest excited of s state of cation, Eg is thetransfer energy of electron from anion to cation. The effectivecharge eS* of the host crystal is related with the number ofelectrons transferred to the unoccupied orbitals of a cation fromits surrounding anions at the nearest neighbour and is nc=1�eS

*/e.Furthermore, nc=12ffiV2

sps=E2g and the transfer matrix element

Vsps and the transfer energy Eg is related to effective charge eS*

following [20]

V2sps

E2g

¼1�e�s

12ð15Þ

The transfer energy Eg is further expressed as

Eg ¼ E�Iþð2a�1Þe2

rð16Þ

Here E is being the electron affinity for Sb i.e. for the non-metalatoms and I is the ionization potential of the constituent metalatom as B, Al, Ga, and In.

The optical static dielectric constant es and the high frequencydielectric constant eN are intimately related to Szigeti effectivecharge eS

* (=Ze)* [15] as follows:

e�2s ¼9mo2

TOðes�e1Þ4pNkðe1þ2Þ2

ð17Þ

and

e�2s

e2¼

9Vmo2TOðes�e1Þ

4pe2ðe1þ2Þ2ð18Þ

Here, m is the reduced mass, Nk is the number of atoms presentper unit cell volume i.e. Nk=1/V, oTO is the long wavelengthtransverse optical phonon frequency. For partially ionic andpartially covalent crystals as YSb (Y=B, Al, Ga, and In) semi-conductors, eS

* deviates from Ze in view of the effects arose due tocovalency.

In the absence of any barrier, a transition occurs when thethermodynamical potential relevant to the given ensemble of thelower-pressure phase equals that for some other structure, whichbecomes the stable phase above this coexistence pressure. As anext step, we have investigated the pressure variations of thesecond-order elastic constants (SOECs) and those of the thirdorder elastic constants (TOECs). The relevant expressions for theSOECs, TOECs and the pressure derivatives of SOECs are given inAppendix. We have emphasized the effects of charge transfer

ARTICLE IN PRESS

D. Varshney et al. / Physica B 405 (2010) 1663–16761666

mechanism, we have computed the phase transition pressures,the associated volume collapses, the mechanical and thermodynamical behaviour of BSb, AlSb, GaSb and InSb compounds inthe next section.

Table 2Input crystal data and model parameters for BSb, AlSb, GaSb and InSb compounds.

Compounds Input parameters Model parameters

ri (A) rj

(A)

a0 (A) BT

(GPa)

f(10�3) b(10�12 erg) r(10�1 A)

BSb 0.108 1.16 5.12

[21]

103 [2] 21 24.92 4.26

AlSb 1.38 1.16 6.135

[22]

58 [22] 4.485 2.496 3.89

GaSb 1.73 1.16 6.118

[22]

57 [22] 5.179 1.16 4.23

InSb 0.98 1.16 6.478

[22]

48.3

[22]

9.495 12.05 3.77

3. Results and discussion

The high-pressure experiments and the theoretical calcula-tions infers that new crystal phases appear in materials under theinfluence of applied pressure and the relative stability of twocrystal structures requires an extremely accurate predictions. Asdiscussed in previous section the effective interionic interactionpotential with charge transfer and covalency effect has beenapplied to investigate the structural phase transitions and elasticproperties in YSb (Y=B, Al, Ga, and In) semiconductors. The phasetransition pressure is thus determined by calculating the Gibbsfree energy G=U+PV–TS for the two phases. The Gibbs free energybecomes equal to the enthalpy H=U+PV at T=0 K.

The high-pressure experiments results huge pressure thatcauses a reduction of the material volume and the temperaturevariations will normally produce much smaller changes in therelative stabilities of different phases. The Gibbs free energy atzero temperature, which is the enthalpy H is important in highpressure studies. At T=0 K, the thermodynamically stable phase atpressure P is the one with the lowest enthalpy and the zero-temperature theory results in consistent agreement with experi-ment, however, the effects of finite temperature may besignificant. We have undertaken such structural and elasticproperties in an ordered way.

The values of thermodynamical potential G or H have beencomputed using the values of the three material dependentparameters, namely, hardness, range and force parameter (b, rand f(r)), which have been evaluated from the equilibriumcondition:

dUðrÞ

dr

��������r ¼ r0

¼ 0 ð19Þ

and the bulk modulus (BT):

d2UðrÞ

dr2

��������r ¼ r0

¼ ð9kr0Þ�1BT ð20Þ

For the actual calculations, we have used the experimental valuesof lattice constant (2a), Bulk modulus (BT) and the second orderelastic constants. The values of the overall vdW coefficient C and D

involved in Eqs. (3) and (4) have been evaluated from the well-known Slater–Kirkwood variational method [19] and are listed inTable 1. We consider that the BSb, AlSb, GaSb and InSbcompounds to be partially ionic and covalent to describe theirstructural and elastic properties in a systematic manner. We shalluse the experimental values of lattice constant (2a), Bulk modulus(BT), ionic (Ze) and effective charge (es*).

For the computation purpose, we have deduced the values ofmaterial parameters range (r), hardness (b) and force parameters(f(r)) from the knowledge of equilibrium distance [21,22] and the

Table 1The values of van der Waals coefficients of BSb, AlSb, GaSb and InSb compounds. cij (i, j=

van der Waals coefficients (C, D).

Compounds cii cij cjj C

BSb 186.93 89.65 45.43 478

AlSb 24.87 33.46 45.43 172

GaSb 65.11 54.29 45.43 278

InSb 1155 199.33 45.43 1325

bulk modulus [1,22] following the equilibrium conditions aregiven in Table 2. The values of electronic polarizabilities of YSb(Y=B, Al, Ga, and In) semiconductors have been directly takenfrom the experimental data [23–25] using the additivity rule andLorentz factor (4p/3). While estimating the effective charge eS

*, thevalues of optical dielectric constant es and the high frequencydielectric constant eN, and the long wave length transverseoptical phonon frequency oTO are taken from [4,10] to have thecovalency contribution.

While estimating, the structural phase transition of the III–Vsemiconductors, we first minimize the Gibbs’s free energies GB3(r)and GB1(r0) for the equilibrium interatomic spacing (r0) and ðr00Þ.Fig. 1 illustrates the Gibbs’s free energies for B3 and B1 phaseshave been plotted as functions of pressure (P) by using theformulated EIoIP for YSb (Y=B, Al, Ga, and In) semiconductors. Thelattice model’s ability to predict realistic cohesive energy,associated volume collapses, the bulk modulus, its derivativewith pressure, the relative stability of crystal structures, andtransition pressures are exemplified in various plots. Let ussummarize the results of the plots. At zero pressure, the Gibb’sfree energy for B3 crystal phase is more negative therefore it isthermodynamically and mechanically stable, while the B1 is not.As pressure increases, beyond the phase transition pressure (Pt),the Gibb’s free energy for B1 system becomes more negativethan B3 phase, so B1 will be more mechanically andthermodynamically stable. The binary semiconductor as BSb,AlSb, GaSb and InSb infers the zinc-blende structure at low andmoderate pressures and a crystallographic transition from B3-B1occurs.

Within this scheme, the phase-transition pressures (Pt) thusobtained are listed in Table 3 and are compared with highpressure ADX experimental [3–7] and other first principles ab

initio calculations [1,2,8–10]. The calculations reveal that thetransition pressure decreases from AlSb to InSb and is attributedto an increase in the atomic numbers of the constituent elements.However, the minor deviation in phase-transition pressures (Pt)values is mainly due to the fact that the present calculations doesnot considered the zero point contribution. We admit that the

1, 2) [in units of 10�60 erg cm6], dij (i, j=1,2) [in units of 10�76 erg cm8] and overall

dii dij djj D

.92 173.2 69.48 26.28 308.65

.48 11.78 17.69 26.28 76.96

.53 42.46 33.47 26.28 145.21

1963 261.25 26.28 1317

ARTICLE IN PRESS

-2900

-2800

-2700

-2600

-2500b

Pt

AlSb G3 G1

Gib

b's

free

eneg

y(K

J/m

ole)

-2900

-2800

-2700

-2600

-2500c

Pt

GaSb G3 G1

Gib

b's

free

eneg

y(K

J/m

ole)

0 6 12 15-2645

-2530

-2415

-2300

-2185

-2070

P (GPa)

d

Pt

InSb G3 G1

Gib

b's

free

eneg

y(K

J/m

ole)

-3630

-3575

-3520

-3465

-3410

-3355 a

Pt

BSb G3 G1

Gib

b's

free

eneg

y(K

J/m

ole)

3 9

Fig. 1. Variation of Gibbs free energy with pressure.

D. Varshney et al. / Physica B 405 (2010) 1663–1676 1667

transition pressures are obtained from enthalpies of two phases atT=0 and zero point contribution will contribute to the transitionpressures. The consistency between experimental data and latticemodel calculation is attributed to the formulated effectiveinterionic potential with partial ionic and covalent bondingcharacteristics, which considers the various interactions as wellusage of material parameters from the reported experimentaldata.

The values of the relative volumes V(P)/V(0) associatedwith various compressions have been computed following the

Murnaghan equation of state [26]:

V

V0¼ 1þ

B0

B0P

� ��1=B0

; ð21Þ

with V0 being the cell volume at ambient conditions.Deduced values of the pressure-dependent radii r(P) for both

the structures (B3 and B1) have been used to compute the valuesof V(P)/V(0) and are plotted them against the pressure (P) asillustrated in Fig. 2 for YSb (Y=B, Al, Ga, and In) semiconductors. Itis noticed from the plot that the present model calculations have

ARTICLE IN PRESS

D. Varshney et al. / Physica B 405 (2010) 1663–16761668

predicted correctly the relative stability of competitive crystalstructures, as the values of DG are positive. The magnitudes of thediscontinuity in volume collapse [�DV (Pt)/V (0)] at the transitionpressure are obtained from the phase diagram and the values arelisted in Table 3. We note that the volume discontinuity DV/V atB3-B1 transition is lower than that of high pressure ADXexperimental [3,27] and other ab initio first principlescalculations [8]. The developed EIoIP considers only the overlaprepulsive interactions significant up to the second nearestneighbours and does not incorporate the thermal fluctuations,results in a poor prediction of the volume change associated withthe phase transition. Although the transition pressures may bewell reproduced up to certain extent, the idea is to incorporateboth the issues such that a balance of Pt and DV/V must yield theequilibrium of the systems under investigations.

For the high-pressure elastic behaviour of these compounds,we have computed the second-order elastic constants (SOECs)and their variation with pressure as illustrated in Fig. 3 for YSb(Y=B, Al, Ga, and In) semiconductors. We note that C44 decreasewith increasing pressure, away from zero at the phase transitionpressures. However, the values of C11 and C12 increase linearlywith pressure and in accordance with the first-order character ofthe transition for these compounds. In passing we refer to theBorn criterion for a lattice to be in the mechanically stable statesis that the elastic energy density must be a positive definitequadratic function of strain. This requires that the principalminors (alternatively the eigenvalues) of the elastic constantmatrix should all be positive.

The stability of a cubic crystal is expressed in terms of elasticconstants [28]:

BT ¼ ðC11þ2C12Þ=340 ð22Þ

Table 3Calculated transition pressures from B3-B1 and volume collapse of BSb, AlSb,

GaSb and InSb.

Compounds Transition pressure (GPa) Volume collapse (%)

Present Others Present Others

BSb 2.5 0.68 [2] 2.16 [2] 5.3

AlSb 8 8.3 [3] 5.3–12.5 [4] 12.5 20 [3]

8.1 [5] 5.6 [8] 7.67 [9] 19.1 [8]

GaSb 7 6.2 [3] 6.2–7 [4] 7 [6] 8.9 17.1 [3]

6.3 [8] 8.01 [9] 5.2 [10] 18.6 [8]

InSb 4.4 2.3 [7] 3.3 [8] 1.94 [9] 16.3 20.7 [8]

2.1 [10] 18.5–19.7 [27]

0 6

0.68

0.72

0.76

0.80

0.84

0.88

0.92

0.96

1.00

Vol

ume

Col

laps

e

P3

Fig. 2. Variation of volume

C4440 ð23Þ

and

CS ¼ ðC11�C12Þ=240: ð24Þ

Here, Cij are the conventional elastic constants.The calculated values of bulk modulus (BT), shear moduli (C44)

and tetragonal moduli (CS) well satisfied the above elastic stabilitycriteria for BSb, AlSb, GaSb and InSb compounds. Deduced valuesof second order elastic constants, bulk modulus (BT) andtetragonal moduli (CS) are mentioned in Table 4 and are alsocompared with various experimental [4,22,29–31] and othertheoretical works [1,8,21,32–35]. The variation of bulk moduluswith pressure is shown in Fig. 4. The figures show linear increasein bulk modulus with increase in pressure even after transitionpressure, reveals that the structure becomes more harder andmore strengthens. We must refer to Vukcevich [36], whoproposed a high-pressure stability criterion for ionic crystals,combining mechanical stability with minimum energy conditions.In accordance, the stable phase of the crystal is one in which theshear elastic constant C44 is nonzero (for mechanical stability) andwhich has the lowest potential energy among the mechanicallystable lattices.

The elastic constant C44 is a very small quantity, as thecalculated value of [(4r0/e2) C44–0.556Zm

2] is found to be anegative quantity so that (A2–B2) is negative and infers that theseterms belong to an attractive interaction and possibly arise due tothe van der Waals energy. Precisely, the van der Waals energyconverges quickly, but the overlap repulsion converges muchmore quickly. The above fact illustrates that the second neighbourforces are entirely due to the van der Waals interaction and thefirst neighbour forces are the results of the overlap repulsion andthe van der Waals attraction between the nearest neighbours.

Also, at high-pressures the short-range forces for thesecompounds increase significantly, which, in turn, is responsiblefor change in the coordination number and phase transformation.Other than deriving the equation of states correctly from a modelapproach and then to analyse the variation of short-range forces,at present we have no direct means to understand the interatomicforces at high pressure. We now turn to analyse the anharmonicproperties of YSb (Y=B, Al, Ga, and In) semiconductors bycomputing the third order elastic constants (TOECs) and thepressure derivatives of SOECs at zero pressure. These pressurederivatives are related to the third-order constants [4,37]:

dB

dP¼�

C111þ6C112þ2C123

9B

� �ð25Þ

9 12 15

BSbAlSbGaSbInSb

(GPa)

collapse with pressure.

ARTICLE IN PRESS

0.8

1.2

1.6

2.0

2.4

2.8 AlSb

B1B3

0.69

0.92

1.15

1.38

1.61

1.84

2.07

2.30GaSb

B1

B3

30 96 12 15

0.3

0.6

0.9

1.2

1.5

1.8

P (GPa)

InSb

B1

B3

0.0

0.4

0.8

1.2

1.6

2.0

2.4 BSb

B1

B3C11C12C44

C11C12C44

Cij(

1011

Nm

-2)

Cij(

1011

Nm

-2)

Cij(

1011

Nm

-2)

Cij(

1011

Nm

-2)

C11C12C44

C11C12C44

Fig. 3. Variation of second order elastic constants with pressure for BSb, AlSb, GaSb and InSb compounds.

D. Varshney et al. / Physica B 405 (2010) 1663–1676 1669

dCS

dP¼

�1

2ðC11þ2C12Þð3C11þ3C12þC111�C123Þ ð26Þ

dC44

dP¼

�1

ðC11þ2C12ÞðC44þC11þ2C12þC144þ2C166Þ ð27Þ

The values of the pressure derivatives of SOECs (dBT/dP, dC44/dP and dCS/dP) for YSb (Y=B, Al, Ga, and In) semiconductors andare listed in Table 5 and are compared theoretical works[4,8,21,34,35]. It inferred from the table that our results deviatefrom reported data for dBT/dP and might be due to overlook ofmany body contribution and the estimation of elastic propertiesat zero K. The variation of TOECs with pressure is shown in Fig. 5.It can be seen that the variation of third order elastic constantswith pressure points to the fact that the values of C111, C112, C123,C166, C456 are negative while that of C144 is positive after aparticular pressure as obtained from the effective interionicpotential at zero pressure (please see Table 5). With theseresults, we can conveniently comment that the developed EIoIP

properly describes the high pressure and elastic behaviour of BSb,AlSb, GaSb and InSb semiconducting alloys.

The average wave velocity vm has been approximatelycalculated from

vm ¼1

3

2

v3t

þ1

v3l

!" #�1=3

ð28Þ

where vl and vt are the longitudinal and the transverse elasticwave velocity respectively, which are obtained from Navier’sequation in the following forms:

vl ¼3Bþ4S

3r

� �1=2

ð29Þ

vt ¼S

r

� �1=2

ð30Þ

where S is the Voigt averaged shear modulus, B is the bulkmodulus, and r is the density. The pressure dependence of the

ARTICLE IN PRESS

Table 4Second order elastic constants (C11, C12 and C44), bulk modulus (BT) and tetragonal moduli (CS) (all are in 1010 N m�2) for BSb, AlSb, GaSb and InSb compounds in B3 phase.

Compounds C11 C12 C44 BT CS

BSb

Present 16.94 1.76 25.4 6.82 7.58

Others 19.2 [1] 5.85 [1] 10.5 [1] 10.3 [1]

20.5 [33] 6.25 [33] 11.21 [33] 10.8, 10.6 [32,35]

22.3 [34] 6.2 [34] 14 [34] 11.5 [21]

AlSb

Present 15.83 11.23 10.66 12.76 2.29

Others 8.77 [4] 4.34 [4] 4.08 [4] 5.82 [4] 2.21 [4]

8.94 [29,31] 4.43 [29,31] 4.16 [29,31] 5.43 [8]

8.55 [33] 4.14 [33] 3.99 [33] 5.8 [22]

5.7 [32]

GaSb

Present 12.66 9.7 7.37 10.7 1.467

Others 8.84 [4,30,31] 4.03 [4,30,31] 4.32 [4,30,31] 5.63 [4] 2.41 [4]

9.27 [33] 3.87 [33] 4.62 [33] 5.57 [8]

5.7 [22]

5.8 [32]

InSb

Present 10.32 7.9 6.3 8.78 1.178

Others 6.61 [4] 3.53 [4] 3.03 [4] 4.56 [4] 1.54 [4]

6.67 [29,31] 3.65 [29,31] 3.02 [29,31] 4.7 [8,32]

7.2 [33] 3.54 [33] 3.41 [33] 4.83 [22]

0 6 12 15

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4BSbAlSbGaSbInSb

Bul

k M

odul

us (1

011N

m-2

)

P (GPa)93

Fig. 4. Variation of Bulk modulus with pressure.

Table 5The values of pressure derivatives of SOECs (dBT/dP, dC44/dP and dCS/dP) and third order elastic constants (C111, C112, C123, C144, C166 and C456) (in units of 1011 N m�2) and

Debye temperature in (K). (Others values are given in parenthesis).

Quantities AlSb GaSb InSb BSb

dBT/dP 3.8 (4.55 [4], 3.71 (4.78 [4], 4.25 (4.89 [4], 3.54 (5.28 [21],

4.01 [8]) 3.83 [8]) 5.21 [8]) 4.03 [35], 4.16 [34])

dC44/dP 1.01 0.32 (1 [4]) 0.02 (0.533 [4]) 4.135

dCS/dP �0.66 �0.42 �0.89 1.27

C111 �4.34 �4.68 (�4.75 [4]) �2.91 (�3.56 [4]) �5.78

C112 �6.17 �4.9 (�3.08 [4]) �4.44 (�2.66 [4]) �4.52

C123 �1.23 �0.699 (�0.44 [4]) �2.06 (�1 [4]) 5.22

C144 0.39 1.26 (0.5 [4]) 0.51 (0.16 [4]) 2.65

C166 �4.63 �3 (�2.16 [4]) �1.98 (�1.39 [4]) �8.1

C456 �1.48 �2.28 (�0.25 [4]) �2.59 (�0.004 [4]) �0.25

D. Varshney et al. / Physica B 405 (2010) 1663–16761670

sound velocity is documented in Fig. 6. It is noticed that vl

increases in both phases while vt decreases in B3 phaseand increases in B1 phase. The values of longitudinal, transverse

and average sound velocities are given in Table 6 at zerotemperature and pressure. Furthermore, the Debye temperature(yD) is estimated from the knowledge of elastic constants. We

ARTICLE IN PRESS

-10

-8

-6

-4

-2

0

2

4 AlSb

-8

-6

-4

-2

0

2

4GaSb

30 96 12 15

-7.0

-5.6

-4.2

-2.8

-1.4

0.0

1.4

2.8

P (GPa)

InSb

Cijk

(1011

Nm

-2)

Cijk

(1011

Nm

-2)

Cijk

(1011

Nm

-2)

Cijk

(1011

Nm

-2)

-10

-8

-6

-4

-2

0

2

4

6

BSb

C 111 C 112C 123 C 144C 166 C 456

C 111 C 112C 123 C 144C 166 C 456

C 111 C 112C 123 C 144C 166 C 456

C 111 C 112C 123 C 144C 166 C 456

Fig. 5. Variation of third order elastic constants with pressure for BSb, AlSb, GaSb and InSb compounds.

D. Varshney et al. / Physica B 405 (2010) 1663–1676 1671

define yD as

y3D ¼

3:15

8ph

kB

� �3 r

M

� 3=2 C11�C12

2

� �1=2 C11þC12þ2C44

2

� �1=2

C1=244 ;

ð31Þ

Here, M is the acoustic mass of the compound; h and kB are thePlanck and Boltzmann constants, respectively.

The Debye temperature as function of pressure is plotted inFig. 7 and compared with experimental results [4] in Table 6. Thevariation of our results is because of the ignorance of zero pointcontribution. It is inferred from the figure that yD decrease withincreasing in pressure before transition pressure and after phasetransition pressure Debye temperature increases. To explain thevariation of yD with the pressure, we attempt to analyse our

results in the framework of dynamics of lattice with pressure. Thechange of the force constants induced by pressure increases yD.The pressure dependence of yD after phase transition pressuresuggests that increasing of Debye temperature drives the systemeffectively toward the hardening of lattice with increasingpressure. However, we do not claim the process to be rigorous,but a consistent agreement following EIoIP is obtained on Debyetemperature as those revealed from experiments. The Debyetemperature is a function of temperature and varies fromtechnique to technique and depends on the sample quality witha standard deviation of about 15 K. One can approximate thisresult motivates the definition of an ‘average’ elastic constant as

C ¼8p

3:15

� �2=3 kB

h

� �2 M

r

� �y2

D ð32Þ

ARTICLE IN PRESS

3000

4500

6000

7500

Vt

Vl

B1B3

BSb

Elas

tic w

ave

velo

city

(m s

ec-1

)

2400

3600

4800

6000

Vt

Vl

B1B3

GaSb

Elas

tic w

ave

velo

city

(mse

c-1)

0 6 9 12 15

2200

3300

4400

5500

Vt

Vl

P (GPa)

B1B3

InSb

Elas

tic w

ave

velo

city

(mse

c-1)

3200

4800

6400

8000

Vt

Vl

B1B3

AlSb

Elas

tic w

ave

velo

city

(mse

c-1)

3

Fig. 6. Pressure dependence of elastic wave velocity vl and vt.

Table 6Calculated longitudinal, transverse and average elastic wave velocity of anti-

monides in m/sec.

Compounds vl vt vm YD

BSb 7150 5470 1250 228.35

370 [4]

AlSb 7230 4120 976.59 170.21

240 [4]

GaSb 5790 3110 742.29 143.34

161 [4]

InSb 5200 2830 674.64 334.91

D. Varshney et al. / Physica B 405 (2010) 1663–16761672

which in turn is calculated from the Debye temperature allow usto correlate Cauchy discrepancy in elastic constant following

C� ¼C12�C44

C12þC44; ð33Þ

at zero pressure. Fig. 8 show variation of ‘average’ elastic constant(C) with Cauchy discrepancy (C*). We have observed negativeCauchy discrepancy for BSb compound is because of the state ofthe BSb compound is intermediate between covalency andionicity and positive Cauchy discrepancy for AlSb, GaSb, InSb. Itis worth to mention that the III–V, II–VI and diluted magneticsemiconductors with either zinc blende to NaCl structure (B3-B1structural phase transition) or NaCl to CsCl structure (B1-B2structural phase transition) [38–41] and most of the body centredcubic transition metals shows a positive Cauchy deviation C*.

The mechanical properties as ductility and brittleness of III–Vcompounds are important and can be known from second orderelastic constants. We refer to Pugh [42] who suggested a simplerelationship, empirically linking the plastic properties of materialswith their elastic moduli. Another thermodynamical property asthe Voigt averaged shear modulus G representing the resistance toplastic deformation, while the bulk modulus B represents theresistance to fracture. It is worth commenting that the high B/G

ARTICLE IN PRESS

0 6 12 150

40

80

120

160

200

240

280

320

BSbAlSbGaSbInSb

ΘD(K

)

P (GPa)3 9

Fig. 7. Variation of Debye temperature (yD) with pressure.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.24

6

8

10

12

14

16

18

20BSb

InSb

GaSb

AlSb

C (1

010N

m-2

)

C∗

Fig. 8. Shows ‘average’ elastic constant (C) as a function of Cauchy discrepancy (C*) for BSb, AlSb, GaSb and InSb compounds.

D. Varshney et al. / Physica B 405 (2010) 1663–1676 1673

ratio may then be associated with ductility whereas a low valuewould correspond to a more brittle nature. The critical valuewhich separates ductile and brittle materials is around 1.75, i.e. ifB/G 41.75 the material behaves in a ductile manner; otherwisethe material behaves in a brittle manner. The model calculationsyield the ratio B/G as 1.75 [AlSb], 2.13 [GaSb], 2.04 [InSb] and 0.37[BSb]. Thus, the semiconducting AlSb, GaSb and InSb are ductilematerials, while to that BSb is brittle in nature. We may add arecent work on Lutetium chalcogenides LuX (X=S, Se, and Te)which on the basis of Pugh [42] relationship are classified asbrittle materials [43].

4. Conclusion

At ambient pressures the semiconductors based on zincblendeIII–V compounds, such as BSb, AlSb, GaSb and InSb crystallizes inB3 structure. As pressure is raised, one expects that thesesemiconductors show the zincblende (ZB)-rock salt (RS) struc-tural phase transitions. To determine the most stable structure atfinite pressure and temperature, the free energy G=E+PV�TS

should be used. Since the reported phase transformations for YSb

(Y=B, Al, Ga, and In) semiconductors are temperature indepen-dent, we neglect the last term and therefore calculated thepressure induced elastic properties with stable zincblendestructures following the lattice models. It is worth to considerthe lattice model calculation results as representative of theresults that would be obtained under the actual experimentalconditions.

The computational methods for the determination of cohesive,structural and vibrational properties of semiconductors underpressure are routinely being performed by means of firstprinciples ab initio calculations. Despite the rapid developmentof computational techniques, the nature of interatomic forces isnot well understood about these materials and lattice dynamicalmodels are important in interpreting and covered the chemicaltrends in the structural stabilities. While estimating the elasticproperties and structural phase transition, the phenomenologicallattice model does not yield a highly accurate results for eachspecific material, but they are less time consuming and still can beuseful in determining a systematic trend for YSb (Y=B, Al, Ga, andIn) semiconductors.

The realistic description of structural and mechanical proper-ties of YSb (Y=B, Al, Ga, and In) semiconducting compounds need

ARTICLE IN PRESS

D. Varshney et al. / Physica B 405 (2010) 1663–16761674

to take in to account various interactive forces when the lattice isstrained and a balance of them to attend the stable structuredepending upon the ionic or covalent nature. We thereforecalculate the pressure induced elastic properties of antimonidessemiconducting compounds with ZnS (B3)-NaCl (B1) structuralphase transition by formulating an effective interionic interactionpotential. To further simplify the phenomenological latticemechanical model calculations, the temperature has been set tozero. The entropy of the crystal is therefore ignored, keeping inmind that the contribution of temperature to free energy is smallfor the experimental data is considered.

The lattice model calculations yield the phase transitionpressure of 2.5, 8.0, 7.0 and 4.4 GPa for BSb, AlSb, GaSb andInSb semiconductors, respectively consistent with the high-pressure angle-dispersive X-ray and first principles ab initio

calculations. The volume discontinuity in pressure volume phasediagram identifies the structural phase transition from ZnS toNaCl type structure. The model’s ability to predict realisticcohesive energy, associated volume collapses, the bulk modulus,its derivative with pressure, the relative stability of crystalstructures, and transition pressures exemplified in terms of thescreening of the effective Coulomb potential through modifiedionic charge.

The lattice model calculations also support the validity of theBorn criterion. The SOECs: C11 and C12 increase with increase inpressure up to the phase transition pressure that supports high-pressure structural stability of BSb, AlSb, GaSb and InSbcompounds. Also, C44 decreases with the increase of pressureand does not tend to zero at the phase transition pressures andthis feature is in accordance with the first-order character of thetransition. The consistency of the results obtained from chargetransfer arose from the electron–shell deformation when thenearest-neighbour ions overlap is enhanced under pressure andthe covalency effects. Our results are quite consistent with knowntheoretical and experimental works. However, the deviationsmight be ascribed to the extension of the covalent and zero pointmotion effects

The pressure dependence of yD in BSb, AlSb, GaSb and InSbinfer the increase in Debye temperature and is attributed to thefact the application of pressure drives the system effectivelytoward the hardening of lattice with increasing pressure. B/G ratioshows that the semiconducting AlSb, GaSb and InSb are ductilematerials, while to that BSb is brittle in nature.

In conclusion, the proper incorporation of the charge transferand covalent effects in the effective interionic interactionpotential and the realistic physical parameters based on theexperimental observations will allow us to reveal a consistenthigh-pressure structural and mechanical behaviour of YSb (Y=B,Al, Ga, and In) semiconducting compounds. Deviations appearingmight be ascribed to the extension of the zero point motioneffects.

Acknowledgements

Financial support from DRDO, New Delhi is gratefully acknowl-edged.

Appendix A

The expressions for the SOECs and TOECs for AIIIBV with ZnSstructure are

C11 ¼ L½0:2477ZðZþ8f ðr0ÞÞþ13 ðA1þ2B1Þþ

12ðA2þB2Þþ5:8243Zaf 0ðr0Þ� ðA:1Þ

C12 ¼ L½�2:6458ZðZþ8f ðr0ÞÞþ13 ðA1�4B1Þþ

14ðA2�5B2Þþ5:8243Zaf 0ðr0Þ� ðA:2Þ

C44 ¼ L½�0:123ZðZþ8f ðr0ÞÞþ13 ðA1þ2B1Þþ

14ðA2þ3B2Þ

�13rð�7:53912ZðZþ8f ðr0ÞÞþA1�B1Þ� ðA:3Þ

C111 ¼ L0:5184ZðZþ8f ðr0ÞÞþ

1

9ðC1�6B1�3A1Þþ

1

4ðC2�B2�3A2Þ

�2ðB1þB1Þ�9:9326Zaf 0ðr0Þþ2:522Za2f00

ðr0Þ

264

375ðA:4Þ

C112 ¼ L0:3828ZðZþ8f ðr0ÞÞþ

1

9ðC1þ3B1�3A1Þþ

1

8ðC2þ3B2�3A2Þ�

11:6482Zaf 0ðr0Þþ2:522Za2f00

ðr0Þ

264

375 ðA:5Þ

C123 ¼ L6:1585ZðZþ8f ðr0ÞÞþ

1

9ðC1þ3B1�3A1Þ�12:5060Zaf 0ðr0Þþ

2:5220Za2f00

ðr0Þ

264

375 ðA:6Þ

C144 ¼ L

6:1585ZðZþ8f ðr0ÞÞþ1

9ðC1þ3B1�3A1Þ�4:1681Zaf 0ðr0Þþ

0:8407Za2f00

ðr0Þþ

rð�3:3507ZðZþ8f ðr0ÞÞ�2

9C1þ13:5486Zaf 0ðr0Þ�1:681Za2f

00

ðr0ÞÞþ

r2

�1:5637ZðZþ8f ðr0ÞÞþ2

3ðA1�B1Þþ

1

9C1�5:3138Zaf 0ðr0Þþ2:9350Za2f

00

ðr0Þ

0BB@

1CCA

2666666666666664

3777777777777775ðA:7Þ

C166 ¼ L

�2:1392ZðZþ8f ðr0ÞÞþ1

9ðC1�6B1�3A1Þþ

1

8ðC2�5B2�3A2Þ�

ðB1þB2Þ�4:1681Zaf 0ðr0Þþ0:8407Za2f00

ðr0Þþ

r�8:3768ZðZþ8f ðr0ÞÞþ

2

3ðA1�B1Þ�

2

9C1þ13:5486Zaf 0ðr0Þ�

1:681Za2f00

ðr0Þ

0B@

1CA

þr2ð2:3527ZðZþ8f ðr0ÞÞþ

1

9C1�5:3138Zaf 0ðr0Þþ2:9350Za2f

00

ðr0ÞÞ

26666666666664

37777777777775ðA:8Þ

C456 ¼ L

4:897ZðZþ8f ðr0ÞÞþ1

9ðC1�6B1�3A1Þ�B2þ

r �5:0261ZðZþ8f ðr0ÞÞ�1

9C1

� �þ

r2 7:0580ZðZþ8f ðr0ÞÞþ

1

3C1

� �þ

r3�4:8008ZðZþ8f ðr0ÞÞþ

1

3ðA1�B1Þ�

1

9

1

9C1

� �

26666666666664

37777777777775

ðA:9Þ

Using the equilibrium condition:

B1þB2 ¼�1:261Z½Zþ8f ðrÞ� ðA:10Þ

Various symbols appear in the above expressions are associatedwith the crystal energy and have the following form

A1 ¼ Aij ¼ L0d2

dr2VSR

ij ðrÞ

� �r ¼ r0

ðA:11Þ

A2 ¼ Aii ¼ Ajj ¼ L0d2

dr2VSR

ii ðrÞþd2

dr2VSR

ij ðrÞ

� �r ¼ r0

ðA:12Þ

B1 ¼ Bij ¼L0

a

d

drVSR

ij ðrÞ

� �r ¼ r0

ðA:13Þ

B2 ¼ Bii ¼ Bjj ¼L0

a

d

drVSR

ii ðrÞþd

drVSR

jj ðrÞ

� �r ¼ r0

ðA:14Þ

C1 ¼ Cij ¼ L0ad3

dr3VSR

ij ðrÞ

� �r ¼ r0

C2 ¼ Cii ¼ Cjj ¼ L0ad3

dr3VSR

ii ðrÞþd3

dr3VSR

jj ðrÞ

� �r ¼ r0

ðA:15Þ

ARTICLE IN PRESS

D. Varshney et al. / Physica B 405 (2010) 1663–1676 1675

In addition, the pressure derivatives of second order elasticconstants are expressed as

3OdBT

dp¼�½20:1788ZðZþ8f ðr0ÞÞ�3ðA1þA2Þþ4ðB1þB2Þ

þ3ðC1þC2Þ�104:8433Zaf 0ðr0Þþ22:7008Za2f00

ðr0Þ� ðA:16Þ

2Odsdp¼�½�11:5756ZðZþ8f ðr0ÞÞþ2ðA1�2B1Þþ

32 A2�

72 B2þ

14C2þ37:5220Zaf 0ðr0Þ

ðA:17Þ

and

OdC44

dp¼�

½þ4:9667Zaf 0ðr0Þþ2:522Za2f00

ðr0Þ�

þr17:5913ZðZþ8f ðr0ÞÞþA1�B2�

2

3C1þ40:6461Zaf 0ðr0Þ�

5:044Za2f00

ðr0Þ

264

375

þr2 3:1416ZðZþ8f ðr0Þþ2

3ðA1�B1Þþ

C1

3�15:9412Zaf 0ðr0Þþ

8:8052Za2f00

ðr0Þ

264

375

8>>>>>>>>>>>><>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;ðA:18Þ

r ¼�7:5391ZðZþ8f ðr0ÞÞþðA1�B1Þ

�3:141ZðZþ8f ðr0ÞÞþðA1þ2B1Þþ21:765Zaf 0ðr0Þ

� �; ðA:19Þ

BT ¼13ðC11þ2C12Þ ðA:20Þ

Similarly the expressions for the SOECs for AIIIBV with NaCl phase

C11 ¼ L½�5:112ZðZþ12f ðr0ÞÞþA1þ12ðA2þB2Þþ9:3204Zaf 0ðr0Þ�

ðA:21Þ

C12 ¼ L½0:226ZðZþ12f ðr0ÞÞ�B1þ14ðA2�5B2Þþ9:3204Zaf 0ðr0Þ�

ðA:22Þ

C44 ¼ L½2:556ZðZþ12f ðr0ÞÞþB1þ14ðA2�5B2Þ� ðA:23Þ

We must mention that the elastic constant C11 is a measure ofresistance to deformation by a stress applied on (1 0 0) plane withpolarization in the direction /1 0 0S, and the C44 refers to themeasurement of resistance to deformation with respect to ashearing stress applied across the (1 0 0) plane with polarizationin the /0 1 0S direction. Henceforth, the elastic constant C11

represents elasticity in length and a longitudinal strain produces achange in C11. No doubt, the elastic constants C12 and C44 areintimately related to the elasticity in shape, which is a shearconstant. However, a transverse strain causes a change in shapewithout a change in volume. Thus, the second order elasticconstants as C12 and C44 are less sensitive of pressure as comparedto C11.

Various symbols appear in the Eqs. (A.21–A.23) are associatedwith the crystal energy and have the following form:

A1 ¼ Aij ¼ 2L0d2

dr2VSR

ij ðrÞ

� �r ¼ r0

ðA:24Þ

A2 ¼ Aii ¼ Ajj ¼ 4L0d2

dr2VSR

ii ðrÞþd2

dr2VSR

jj ðrÞ

� �r ¼ r0

ðA:25Þ

B1 ¼ Bij ¼ 2L0d

drVSR

ij ðrÞ

� �r ¼ r0

ðA:26Þ

B2 ¼ Bii ¼ Bjj ¼ 4L0d

drVSR

ii ðrÞþd

drVSR

jj ðrÞ

� �r ¼ r0

ðA:27Þ

For both ZnS phase and NaCl phase L=(e2/4a4) and L0=(4a3/e2),and in terms of the short-range energy

VSrij ðrÞ ¼ bbij exp

riþrj�rij

r

� ��

cij

r6ij

�dij

r8ij

ðA:28Þ

The short-range interaction (SR) energy is expressed in terms ofthe overlap repulsion (first term) and the vdW d�d and d�q

attractions (second and third terms), respectively.

References

[1] R.M. Wentzccovitch, K.J. Chang, M.L. Cohen, Phys. Rev. B 34 (1986) 1071;R.M. Wentzcovitch, M.L. Cohen, J. Phys. C: Solid State Phys. 19 (1986) 6791;R.M. Wentzcovitch, M.L. Cohen, P.K. Lam, Phys. Rev. B 1986 36 (1987) 6058.

[2] F.El.H. Hassan, H. Akbarzadeh, M. Zoaeter, J. Phys. Condens. Matter 16 (2004)293;A. Zaoui, M. Ferhat, Phys. Status Solidi (b) 225 (2001) 15;A.El.H. Zaoui, F. Hassan, J. Phys. Condens. Matter 13 (2001) 253.

[3] S.C. Yu, I.L. Spain, E.F. Skelton, Solid State Commun. 25 (1978) 49.[4] S. Adachi, Properties of Group—IV, II–V and II–VI semiconductors, Wiley, UK,

2005.[5] R.J. Nelmes, M.I. McMahon, S.A. Belmonte, Phys. Rev. Lett. 79 (1997) 3668.[6] M. Mezouar, H. Libotte, S. Deputier, T.L. Bihan, D. Hausermann, Phys. Status

Solidi (b) 211 (1999) 395.[7] A. Jayaraman, R.C. Newton, G.C. Kennedy, Nature 191 (1961) 1290.[8] S.B. Zhang, M.L. Cohen, Phys. Rev. B 35 (1987) 7604.[9] J.R. Chelikowsky, Phys. Rev. B 35 (1987) 1174.

[10] V. Ozolins, A. Zunger, Phys. Rev. Lett. 82 (1999) 767.[11] S.M. Sze, Physics of Semiconductor Devices, second ed., Wiley, New York,

1981 p. 849.[12] P.O. Lowdin, Adv. Phys. 5 (1956) 1.[13] S.O. Lundqvist, Ark. Fys. 12 (1957) 263.[14] D.W. Hafemeister, W.H. Flygare, J. Chem. Phys. Soc. 43 (1965) 795.[15] R.K. Singh, Phys. Rep. 85 (1982) 259.[16] M.L. Huggins, Y. Sakamoto, J. Phys. Soc. Japan 12 (1957) 241.[17] H.B. Huntington, Solid State Phys 7 (1958) 214.[18] M.P. Tosi, Solid State Phys 16 (1964) 1.[19] J.C. Slater, J.G. Kirkwood, Phys. Rev. 37 (1931) 682.[20] K. Motida, J. Phys. Soc. Japan 49 (1980) 213;

K. Motida, J. Phys. Soc. Japan 55 (1986) 1636.[21] M. Ferhat, B. Bouhafs, A. Zaoui, H. Aourag, J. Phys. Condens. Matter 10 (1998)

7995.[22] R.W.G. Wyckoff, Crystal Structures, Interscience, New York, 1963.[23] N.W. Grimes, R.W. Grimes, J. Phys. Condens. Matter 9 (1997) 6737.[24] R.C. Weast, D.R. Lide, CRC Handbook of Chemistry and Physics E, 70, 1990.[25] S.C. Tidrow, A. Tauber, W.D. Wilber, R.D. Finnegan, D.W. Eckart, W.C. Drach,

Appl. Superconductivity 7 (1997) 1769.[26] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 3 (1944) 244.[27] R.E. Hanneman, M.D. Banus, H.C. Gatps, J. Phys. Chem. Solids 25 (1964) 293.[28] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford, Clarendon,

1956.[29] Londolt-Bornstein, In: K.H. Hellewege (Ed.), Numerical Data and Functional

Relationships in Science and Technology, New Series, Springer, Berlin, 1966,Group 3, Band 5.

[30] H.J. McSkimin, A. Jayaraman, P. Andreatch, T.B. Bateman, J. Appl. Phys. 39(1968) 4127.

[31] R. Weil, W.O. Groves, J. Appl. Phys. 39 (1968) 4049.[32] M.L. Cohen, Phys. Rev. B 32 (1985) 7988.[33] S.O. Wang, H.O. Ye, Phys. Status Solidi (b) 240 (2003) 45.[34] H. Meradji, S. Drablia, S. Ghemid, H. Belkhir, B. Bouhafs, A. Tadjer, Phys. Status

Solidi (b) 241 (2004) 2881.[35] B. Bouhafs, H. Aourag, M. Ferhat, M. Certier, J. Phys. Condens. Matter 11

(1999) 5781.[36] M.R. Vukcevich, Phys Status Solidi B 54 (1970) 435.[37] M. Gluyas, BRIT. J. Appl. Phys. 18 (1967) 913.[38] D. Varshney, P. Sharma, N. Kaurav, R.K. Singh, Phase Trans. 77 (2004) 1075;

D. Varshney, N. Kaurav, P. Sharma, S. Shah, R.K. Singh, Phys. Status Solidi B241 (2004) 3179;D. Varshney, N. Kaurav, P. Sharma, S. Shah, R.K. Singh, Phys. Status Solidi B241 (2004) 3374;D. Varshney, N. Kaurav, R. Kinge, S. Shah, R.K. Singh, High Pressure Res. 77(2005) 1075.

[39] D. Varshney, P. Sharma, N. Kaurav, S. Shah, R.K. Singh, J. Phys. Soc. Japan 74(2005) 382;D. Varshney, N. Kaurav, R. Kinge, R.K. Singh, J. Phys. Condens. Matter 19 (236)(2007) 204.

[40] D. Varshney, N. Kaurav, U. Sharma, R.K. Singh, J. Phys. Chem. Solids 69 (2008)60;D. Varshney, N. Kaurav, R. Kinge, R.K. Singh, J. Phys. Condens. Matter 20(2008) 075204;D. Varshney, V. Rathore, N. Kaurav, R.K. Singh, Int. J. Modern Phys. B 22 (2008)2749;D. Varshney, N. Kaurav, R. Kinge, R.K. Singh, Comput. Mater. Sci. 41 (2008)529;D. Varshney, N. Kaurav, U. Sharma, R.K. Singh, J. Alloys Compounds 448(2008) 250;D. Varshney, N. Kaurav, R. Kinge, Phase Trans. 81 (2008) 1;D. Varshney, N. Kaurav, U. Sharma, Phase Trans. 81 (2008) 525.

ARTICLE IN PRESS

D. Varshney et al. / Physica B 405 (2010) 1663–16761676

[41] D. Varshney, G. Joshi, European Phys. J. B. 70 (2009) 523;D. Varshney, G. Joshi, N. Kaurav, R.K. Singh, J. Phys. Chem. Solids 70 (2009)451;D. Varshney, V. Rathore, R. Kinge, R.K. Singh, J. Alloys Compounds 484 (2009)239;

D. Varshney, V. Rathore, R. Kinge. R.K. Singh, Int. J. Modern Phys. B (2009) InPress.

[42] F. Pugh, Phil. Mag. 45 (1954) 823.[43] T. Seddik, F. Semari, R. Khenata, A. Bouhemadou, B. Amrani, Phys. B 405

(2009) 394.