first principle study of the physical properties of semiconducting...

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Materials Science in Semiconductor Processing

Materials Science in Semiconductor Processing 26 (2014) 477–490

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First principle study of the physical properties ofsemiconducting binary antimonide compounds underhydrostatic pressures

Hamdollah Salehi, Hojat Allah Badehian n, Mansoor FarbodDepartment of Physics, Shahid Chamran University, Ahvaz, Islamic Republic of Iran

a r t i c l e i n f o

Keywords:III–V compoundsFP-LAPWmBJ-GGAElastic constantsHydrostatic pressureOptical properties

x.doi.org/10.1016/j.mssp.2014.05.02001/& 2014 Elsevier Ltd. All rights reserved.

esponding author.

a b s t r a c t

First-principle calculations have been performed to investigate the structural phasetransition, electronic, elastic, thermodynamical and optical properties of III-Sb compoundsunder hydrostatic pressure up to their first order transitions pressure (Zinc Blende to RockSalt). Four different exchange–correlation functionals comprising Perdew–Burke–Ernzer-hof generalized parameterization of gradient approximation, Wu-Cohen, local densityapproximation as well as modified Becke and Johnson were used. The structural proper-ties such as phase transitions, equilibrium lattice parameters, bulk modulus and its firstpressure derivative were obtained using an optimization method. Moreover, elasticconstants, Young's modulus, shear modulus, Poisson's ratio, sound velocities for long-itudinal and shear waves, Debye average velocity, Debye temperature and Grüneisenparameters were calculated up to the first order phase transition pressure. The obtainedstructural and elastic parameters are consistent with the available experimental data. Thestatic calculations predict that Zinc Blende to Rock Salt phase transitions occur at 48.5, 9.5,5.87 and 3.15 GPa for BSb, AlSb, GaSb and InSb respectively. The optical properties of thesecompounds, such as dielectric function, refractive index and the optical band gap werealso calculated for the radiation up to 14 eV. In addition, the influence of the hydrostaticpressure on the elastic parameters, energy band structures and the refractive index ofthese compounds were investigated. The linear and quadratic pressure coefficients of thecompounds were also calculated. Results have been discussed and compared withavailable experimental and theoretical data, which show an overall good agreement withthe other studies.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

During the last century, III–V semiconductors havebeen extensively served as substances for electronic andoptoelectronic devices. These applications make them asvital elements in an advanced information society [1].Among III–V compounds, the narrow band gap antimonidebased compound semiconductors (ABCS) are broadlyregarded as the first nominee materials for the manufacturing

of the third generation infrared photon detectors and ICswith ultra-high speed and ultra-low power consumption.Their distinctive physical aspects and band gap structuresprovide a large space to design varied unique devices [2].At ambient pressure, III-Sb compounds crystallize in thezinc blende (ZB) structure. Boron antimonide and alumi-num antimonide have indirect band gap, while galliumantimonide and indium antimonide have direct ones. Eventhough a few investigations on some of the physicalproperties for this group are available, a comprehensivestudy is needed. In recent years, III-Sb compounds have

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drawn more attention owing to their possible applicationin rechargeable lithium batteries. Boron antimonide showsa strong covalent nature and a rare behavior due to thesmall core and absence of “p” electrons in boron atomcompared to other III–V substances. They make BSb as apossible material for high temperature optical and elec-tronic uses [3]. Likewise, GaSb is a noble nominee forthermo-photovoltaic cells for the structures with lowradiator temperature, as its cell technology is fairlysimple resulting in higher effectiveness than Si thermo-photovoltaic cells. III-Sb compounds, owing to their highmobility, are henceforth known as forward-thinking deviceapplications [4].

Recently, first-principle computations based on density-functional theory have developed the vital part of materialsinvestigations. The DFT (Density Functional Theory) Full-Potential linear augmented plane wave (FP-LAPW) methodhas been broadly recognized as the approach of choicefor computational solid-state researches. The calculationof various properties such as the structural, elastic, ther-modynamical, optical and electronic properties for severalcompounds has been done by the DFT method. Thecalculations supply the full recognition of materials' prop-erties and the chance to scheme new compounds forspecial uses [5].

III–V semiconductors have been broadly investigatedtheoretically and experimentally in recent decades. Even

Fig. 1. Energy versus volume curves of ZB and RS phases

though III-Sb compounds have been widely studied intheory [6–16], there is no complete study comprising allIII-Sb compounds about the effects of pressure on theirelectronic structure, elastic and optical properties. Thisstudy aims to investigate the consequence of pressure onthe structural, electronic, elastic, and the optical propertiesof the III-Sb compounds. The calculation method isFP-LAPW with various approximations. Exchange–correlationfunctionals and corresponding potential have a promi-nent role in DFT based total energy calculations. The cal-culations using local density approximation (LDA) orgeneralized gradient approximation (GGA) for exchangecorrelation functionals and corresponding potential yieldlower values of band gap energy. Thus, we used modifiedBecke and Johnson (mBJ)-GGA which is another newexchange correlation functional. This functional yieldsbetter results for the electronic properties of semiconduc-tors [17].

2. Computational approach

FP-LAPW approach within the DFT framework, wasapplied to obtain the structural, phase transition, elastic,thermodynamical, optical and electronic properties of XSb(X¼B, Al, Ga and In) compounds under hydrostatic pres-sure up to their first order phase transition pressure byusing WIEN2k package [18]. LDA, GGA (PBE: Perdew–

for III-Sb compounds with PBE-GGA approximation.

Table 1Calculated lattice constant, a0 (Å), bulk modulus, B0 (GPa), its pressure derivative, B0 , and elastic constant for III-Sb compounds in ZB phase using differentfunctionals.

Compound a (Å) B (GPa) B0 C11 (GPa) C12 (GPa) C44 (GPa)

BSbGGA-PBE 5.2809 98.9264 4.7660 182.402 54.731 117.797LDA 5.1954 115.0498 5.0570 206.238 62.008 137.664PBEsol 5.2316 108.2133 4.9070 203.981 56.247 120.575WC 5.2354 107.7334 4.8818 204.563 55.193 123.319

5.279b,5.278c,

Othertheore-ticalworks

5.191d, 5.201e , 5.145f , 5.21g , 5.252h, 5.177i, 5.278u 96b, 100c, 111d,116e , 118f, 110g,103h, 110i, 100u

4.55b, 4.40c, 4.36d,4.16e, 4.31f, 4.07g,3.62h, 4.237i, 4.40u

205.0a, 207c,223e, 236f,193.5g, 192h,207u

62.5a, 107c,62e, 62.6f,68.42g, 58.8h,47u

112.1a, 45c,122.6f,104.12g,105h, 105u

AlSbGGA-PBE 6.2230 50.4276 4.0714 81.610 35.548 53.929LDA 6.1184 56.4398 4.2531 85.005 42.238 62.450PBEsol 6.1667 54.1653 4.1540 83.984 39.424 58.190WC 6.16.69 54.4116 4.1091 85.014 37.952 57.322Exp. 6.1355j 58.2k – 87.69k, 84.4l,

89.4m43.41k, 43.2l,44.3m

40.76k,39.5l, 41.6m

Othertheore-ticalworks

6.08n, 6.230b, 6.111d, 6.090i 56n, 49b, 56d,56.1i

4.31n, 4.28b, 4.52d,4.362i

85.5a, 85.32t 41.4a, 41.51t 39.9a,39.80t

GaSbGGA-PBE 6.2145 44.6427 4.8216 73.207 31.439 49.734LDA 6.0609 56.2485 5.0813 82.124 41.728 62.363PBEsol 6.1210 51.9548 4.9121 81.550 36.756 57.955WC 6.1224 52.1292 4.8436 76.919 39.016 58.678Exp. 6.081o 56o, 56.35p – 88.39m,p

88.34q, 86.8l40.33 m,p,40.23q, 40.23l

43.16m,p,43.22q,40.7l

OtherTheore-ticalworks

5.981i, 6.219b, 6.053d 56.7i, 45b, 54d 4.662i, 4.02b, 4.26d 92.7a 38.7a 46.2a

InSbGGA-PBE 6.6240 36.9421 4.6134 54.749 28.518 39.712LDA 6.4620 46.8002 4.8882 73.082 34.154 46.619PBEsol 6.5292 43.2241 4.7305 60.947 34.516 47.745WC 6.5286 43.6707 4.7101 – – –

Exp. 66r, 69.18r,66.7m

38r,37.88s,36.5m

30r, 31.32s,30.2m

OtherTheore-ticalworks

6.346i, 6.640b, 6.456d 47.6i, 37b, 46d 4.688i, 4.43b, 4.51d 72.0a 35.4a 34.1a

a PP-LDA Ref. [6].b FP-GGA Ref. [3].c FP-GGA Ref. [7].d FP-LDA Ref. [3].e FP-LDA Ref. [7].f PP-GGA Ref. [8].g PP-GGA Ref. [9].h FP-GGA Ref.[16].i FP-GGA Ref. [5].j Exp. Ref. [29].k Exp. Ref. [30].l Exp. Ref. [31].m Exp. Ref. [32].n PP-LDA Ref. [10].o Exp. Ref. [33].p Exp. Ref. [34].q Exp. Ref. [35].r Exp. Ref. [36].s Exp. Ref. [37].t PP-LDA Ref. [38].u FP-GGA Ref. [39].

H. Salehi et al. / Materials Science in Semiconductor Processing 26 (2014) 477–490 479

Fig. 2. Enthalpies as a function of pressure for III-Sb compounds with GGA-PBE approximation.

Table 2Phase transition pressures and the difference in equilibrium volumes, ΔV0 ,for different phases of III-Sb compounds with GGA-PBE approximation.

Compound Pt(GPa)

Other calculations ΔV0

(a.u.3)

BSb 48.5 2.5a, 0.68b, 56c, 59d, 53e 34.571AlSb 9.5 8a, 8.3f, 5.3–12.5g, 8.1h, 5.6i,

7.67k,8.9o72.256

GaSb 5.9 7a, 6.2f, 6.2–7g, 7l,6.3i, 8.01k, 5.2m 75.669InSb 3.15 4.4a, 2.3n, 3.3i, 1.94k, 2.1m 103.765

a Ref. [45].b Ref. [16].c Ref. [8]d Ref. [9].e Ref. [46].f Ref. [47].g Ref. [48].h Ref. [49].i Ref. [43].k Ref. [50].l Ref. [51].m Ref. [52].n Ref. [53].o Ref. [54].


Burke–Ernzerhof [19], WC (Wu-Cohen) [20], PBEsol [21])and mBJ-GGA [22] approximations were applied to ourcalculations. Since the PBE-GGA is more efficient forestimating the physical properties [23], the GGA-PBEfunctional was used for the most of our calculations. The

wave functions cut-off were considered Kmax¼8/RMT inthe interstitial spaces, where RMT is the smallest atomicmuffin-tin sphere radius and Kmax is the largest K vector inthe plane wave extension. The valence wave functionsinside the muffin-tin spheres were expanded up tolmax¼10, whereas the charge density was Fourierexpanded up to Gmax¼14 (a.u.)�1. The self-consistentcalculations are regarded to be converged when the totalenergy of the system is steady within 10�5 Ry. Theintegrals over the Brillouin zone are done up to 300k-points in the irreducible Brillouin zone, using the Mon-khorst–Pack special k-points method [24]. The energydividing the valence state from the core state was set as�6.0 Ry. In the calculations [Kr]: 5s2 4d10 5p3, [He]: 2s2,2p1 [Ne]: 3s2 3p1,[Ar]: 4s2 3d10 4p1 and [25]: 5s2 4d10 5p1

states behave as valence electrons for Sb, B, Al, Ga, and Inrespectively.

3. Results and discussion

3.1. Structural properties

To evaluate the structural properties of III–V com-pounds containing Sb, the total energies are estimatedfor various volumes around the equilibrium cell volumeV0. The calculated total energies versus reduced andenlarged volume for these compounds with two ZincBlende (ZB) and Rock Salt (RS) phases with GGA-PBE aregiven in Fig. 1. The obtained total energy is fitted to the

Table 3Bulk modulus (B), Voigt's shear modulus GV, Reuss's shear modulus GR, shear modulus (G), Young's modulus (Y), B/G and Poisson's ratio (ν) of III-Sbcompounds with GGA-PBE approximation.

B (GPa) GV (GPa) GR (GPa) G (GPa) Y (GPa) G/B ν

BSbP¼0 97.288 96.212 88.0311 92.122 210.062 0.947 0.140p¼4.378 118.313 106.395 100.728 103.561 240.510 0.876 0.162p¼9.038 137.475 123.966 110.090 117.028 273.482 0.851 0.168p¼13.60 155.189 129.429 106.272 117.851 282.135 0.759 0.197p¼17.90 173.997 134.390 115.191 124.790 302.141 0.717 0.211Other works – – – 62.55a, 71.3b, 93.2c 221.81a, 175.8b, 212c – 0.163a, 0.234b, 0139c

AlSbP¼0 50.902 41.570 35.096 38.332 91.923 0.753 0.199p¼2.32 59.784 45.409 36.283 40.846 99.808 0.598 0.221p¼4.749 72.169 50.017 36.3133 43.165 107.969 0.647 0.251p¼7.539 80.886 55.018 33.632 44.324 112.436 0.548 0.268p¼9.675 88.012 60.124 31.397 45.761 113.715 0.520 0.278Other works 84.7d, 58.5b 0.326b

GaSbP¼0 45.361 38.194 32.033 35.113 83.734 0.775 0.192p¼1.407 51.728 42.950 36.697 39.824 95.074 0.770 0.193p¼2.796 58.549 44.906 37.551 41.228 100.173 0.704 0.214p¼4.168 64.72533 50.015 40.51981 45.2674 110.1284 0.700 0.216p¼5.498 70.671 51.797 38.831 45.314 112.004 0.641 0.236Other works 34 69.9b, 89.1d 0.295b

InSbP¼0 37.261 29.073 21.926 25.499 62.290 0.684 0.221p¼1.098 42.540 30.493 21.072 25.782 64.348 0.606 0.248p¼2.673 49.636 33.135 20.192 26.663 67.843 0.537 0.272p¼2.144 47.351 34.908 23.579 29.243 72.754 0.618 0.243p¼3.522 53.794 37.162 20.593 28.877 73.483 0.537 0.272Other works 62.1d, 48.7b 0.330b

a PP-LDA Ref. [9].b PP-HGH Ref. [6].c Ref. [62].d Ref. [48].


Murnaghan's equation of state [26] to estimate the struc-tural properties like the equilibrium lattice constant a0, thebulk modulus B and its first pressure derivative B0. Thecalculated equilibrium parameters found using LDA,WC-GGA, PBEsol and GGA-PBE approximations, are listedin Table 1, which also holds experimental data for com-parison. It may need to be marked that LDA and GGAgenerally underestimates and overestimates experimentalresults for lattice parameter respectively. The bulk mod-ulus and also its first order derivative are in good agree-ment with other results. Our results are consistent withthe general trend of the LDA and GGA approximations[27,28]. Also, it is obvious that ZB phases of these com-pounds due to lower energy, are more stable (Fig. 1).

3.2. Phase transformations

The thermodynamic stable phase at some given pres-sure and temperature, is the one with the lowest Gibbsfree energy (G) given by [40].

G¼UþPV�TS ð1Þwhere U stands for the internal energy and P, V, T and S arethe pressure, volume, temperature and entropy respec-tively. Generally, the phase equilibrium transition pressureis acquired by calculating the total energy versus volume(E–V) curves of the two phases and obtaining the commontangent, which is hard to calculate precisely. Therefore,

since DFT calculations are performed at zero temperature,we have calculated the enthalpy (H¼UþPV) of BSb, AlSb,GaSb and InSb corresponding to the ZB and RS structures.Enthalpies as a function of pressure for III-Sb compoundsby GGA-PBE approximation have been illustrated in Fig. 2.It was found that the transition pressures of BSb, AlSb,GaSb and InSb from Zb to RS phase are 48.5, 9.5, 5.9 and3.15 GPa respectively. The calculated phase transitionpressures, listed in Table 2, are consistent well with othersresults. It is found that the phase transition pressuresdecrease with increasing of the atomic radius of IIIelements. The transition pressure depends on the follow-ing factors: (a) the bulk modulus B0 (the larger B0, thelarger Pt) and (b) the difference in equilibrium volumesΔV0 for different phases (the larger ΔV0, the smaller Pt)[41]. In the sequence of compounds considered all thesefactors are dominated by volume effects which add upcoherently, therefore the transition pressures are alsodominated by volume effects, i.e., the larger the volumethe smaller the transition pressure. This behavior appearsto be typical of similar sequences of compounds [42] andwas also observed in a comparative first-principle study ofGa and Al compounds under pressure [43,44].

3.3. Elastic properties

Elastic properties play a significant role in obtaininguseful information about the anisotropic nature of binding,

Table 4Calculated mass density, ρ (kg/m3), sound velocities (m/s) for longitudinal and shear waves (VL and VS), Debye average velocity, Vm (km/s), Debyetemperature, θD (K) and Grüneisen parameter (ξ).

ρ (kg/m3) VL (m/s) Vs (m/s) Vm (m/s) θD(K) ξ

BSbP¼0 5130 6550.4 4237.6 4649.9 484.8 1.083p¼4.378 5846 6622.5 4208.9 4627.9 504.0 1.149p¼9.038 6287 6832.6 4314.4 4747.4 529.7 1.173p¼13.60 6489 6937.6 4261.6 4703.3 530.3 1.274p¼17.90 6670 7143.6 4325.4 4780.7 544.0 1.327Other works – 6170.5a, 7150g, 6072e 3911.22a, 5470g, 3932e 4302a, 1250g, 4314e 506a, 370b, 495c, 228g, 486.9e 1.158a

AlSbP¼0 3953 5079.9 3114.0 3437.5 328.6 1.282p¼2.32 4152 5245.5 3136.5 3470.9 337.2 1.373p¼4.749 4314 5483.6 3163.2 3512.0 353.5 1.503p¼7.539 4459 5603.0 3152.8 3507.9 349.0 1.592p¼9.68 4573 5708.6 3163.3 3523.9 353.9 1.646Other works 7230g 4120g 976.6g 240b, 292f,h, 380d, 170.2g,GaSbP¼0 5130 4238.9 2616.2 2885.9 300.9 1.257p¼1.41 5280 4455.7 2746.3 3029.9 318.9 1.262p¼2.80 5415 4578.6 2759.3 3051.1 323.9 1.344p¼4.17 5538 4752.4 2859.0 3161.9 338.2 1.351p¼5.50 5652 4815.9 2831.5 3138.3 337.9 1.434Other works 5614e 5790g 3110g 742.3g 266e, 266d, 143.3g, 265h

InSbP¼0 5234 3689.8 2207.2 2442.5 256.4 1.371p¼1.10 5337 3796.3 2197.9 2439.5 257.7 1.490p¼2.67 5433 3959.7 2215.3 2466.0 262.1 1.613p¼2.14 5523 3953.9 2301.0 2552.8 272.8 1.471p¼3.52 5608 4056.8 2269.2 2526.0 271.3 1.613Other works 5200g 2830g 674.6g 205d, 334g, 161b,202h

eFP-GGA Ref. [62].a PP-LDA Ref. [9].b PP-HGH Ref. [6].c Ref. [63].d Ref. [64].e Experiment Ref. [34].f Ref. [65].g Ref. [45].h Ref. [66].


structural stability and binding properties among abuttingatomic planes [55]. Hence, to investigate the stability of III-Sbcompounds in ZB phase, we calculated the elastic constantsat ambient pressure and up to their first order phasetransition pressure by the Thomas Charpin approach estab-lished recently and integrated it in the WIEN2k code [18].

Only three independent elastic constants (C11, C12 andC44) are needed to calculate cubic structures. Having theseconstants, bulk modulus, B, Young's modulus, Y, isotropicshear modulus, G, and Poisson ratio, v, which are signifi-cant parameters associated to many physical propertiessuch as internal strain, thermoelastic stress, sound velocityand fracture toughness, can be obtained readily using thefollowing expressions [56]:

B0 ¼ ðC11þ2C12Þ=3 ð2Þ

G¼ ðGV þGRÞ=2 ð3Þ

where GV stands for Voigt's shear modulus being theupper bound of G values, and GR stands for Reuss's shearmodulus for cubic crystals being the lower bound values,represented as follows:

GV ¼ ðC11�C12þ3C44Þ=5 ð4Þ

GR ¼5½ðC11�C12ÞC44�

½4C44þ3ðC11�C12Þ�ð5Þ

The terms for the Young's modulus and Poisson's ratioare as follows [56]:

Y ¼ 9GBGþ3B

ð6Þ

ν¼ ð3B�2GÞ=ð6Bþ2GÞ ð7Þ

3.4. Sound velocity, Debye temperature and Grüneisenparameter

Debye average velocity (Vm), longitudinal velocity (VL)and shear waves velocity of sound (VS) can also be attainedby elastic constants and mass density. The terms are asfollows [57]:

VL ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Bþ4G

3ρ

sð8Þ

VS ¼ffiffiffiffiGρ

sð9Þ

Fig. 3. Calculated pressure dependence of elastic constants (C11, C12 and C44) and bulk modulus (B) for III-Sb compounds in ZB phase with PBE-GGAapproximation.

Table 5Pressure derivatives, ∂C11/∂P, ∂C12/∂P, ∂C44/∂P and ∂B/∂P for III-Sb compounds.

Compound ∂C11/∂P ∂C12/∂P ∂C44/∂P ∂B/∂P

BSb 4.63a 3.91a 2.91a 4.15a

AlSb 2.86a, 2.875b 4.51a, 3.234b 2.98a 4.00a, 3.114b

GaSb 5.19a, 6.38c 4.34a, 3.97c 3.87a, 1.29c, 1.0d 4.729a, 4.78d, 4.77c

InSb 3.79 5.013a 4.15a 4.75a

a This work.b PP-LDA Ref. [38].c Ref. [71].d Ref. [72].


and

Vm ¼ 13

1

V3L

þ 2

V3S

!" #�1=3

ð10Þ

Debye temperature [57], θD, and Grüneisen parameter[58], ζ, are valuable parameters in solid state matterbecause they relate to lattice vibrations. These parametersare dependent of sound velocity and mass density, which

are evaluated using the terms given by

θD ¼ hκB

� �3n4π

Naρ

M

� �� 1=3Vm ð11Þ

ζ¼ 9ðV2L �ð4=3ÞV2

S Þ2ðV2

L þ2V2S Þ

ð12Þ

where h is the Planck's Constant and n, Na, ρ, M, and κB arethe number of atoms in the molecule, Avogadro's number,

Fig. 4. Calculated pressure dependence of sound velocities (VL, VS and Vm) in III-Sb compounds.

Table 6Lattice and elastic constants of III-Sb under pressure up to the phasetransition.

Compound a (Å) B (GPa) C11 (GPa) C12 (GPa) C44 (GPa)

BSbp¼4.378 5.2635 118.313 221.919 66.51 125.522p¼9.038 5.1375 137.477 239.503 86.461 155.596p¼13.60 5.0836 155.189 245.747 109.91 170.437p¼17.90 5.0372 173.997 276.389 122.801 172.788

AlSbp¼2.32 6.1300 59.784 89.991 44.681 60.578p¼4.749 6.0526 72.169 100.43 58.038 69.231p¼7.539 5.9864 80.886 104.906 68.876 79.686p¼9.675 5.9361 88.012 109.216 77.411 89.605GaSbp¼1.407 6.1553 51.72867 84.238 35.474 55.33p¼2.796 6.1038 58.54967 91.097 42.276 58.571p¼4.168 6.0582 64.72533 98.85 47.663 66.296p¼5.498 6.0174 70.671 101.517 55.248 70.906InSbp¼1.098 6.5813 42.54 58.482 34.569 42.852p¼2.673 6.5423 49.63633 64.039 42.435 48.024p¼ 2.144 6.5065 47.351 64.979 38.537 49.366p¼3.522 6.4734 53.79433 67.971 46.706 54.848


mass density, molecular weight and Boltzmann constantrespectively. The elastic constants of III-Sb compounds, upto their first order transition pressures, have been listed in

Table 1. The mechanical stability conditions which areC11�C1240, C1140, C4440 and C11þ↰2C1240 [59], canvalidate the calculated elastic constants. The obtainedelastic constants of XSb (X¼B, Al, Ga, In) meet the abovestability conditions exhibiting that they are elasticallystable in ZB phase. Moreover, the elastic constants alsomeet the cubic stability condition i.e. C12oB4C11. So, thevalues of the elastic constants are valid for all the com-pounds. The bulk modulus (B) and shear modulus (G) aresignificant parameters in recognizing the physical proper-ties of materials. Bulk modulus (B), Voigt's shear modulusGV, Reuss's shear modulus GR, shear modulus (G), Young'smodulus (Y), G/B and Poisson's ratio of III-Sb compoundswith GGA-PBE approximation have been calculated up tothe first order transition pressure using Eqs. (2)–(7) andare summarized in Table 3. The results are in a goodagreement with other available works. But to the best ofour knowledge, there is no available literature about theeffect of pressure on these parameters for XSb compounds.To estimate the solid ductility or brittleness, Pugh pre-sented the G/B ratio [60]; solid is brittle if the ratio is lessthan the critical value 1.75. Consequently, BSb, AlSb, GaSband InSb are brittle under ambient conditions since theirG/B values are less than 1.75 (Table 3). And also it is foundthat the brittleness of these materials decreases by pres-sure. Young's modulus (Y) is the ratio between the linear

Table 7Pressure coefficient of band gap of III-Sb compounds (E0 (P)¼E0 (0) þaPþbP2).

E0(0) a (10�2 eV/GPa) b (10�4 eV/GPa2)

BSb 1.865a, 0.740b �1.42a, �1.05b þ1.0a, 0.6b

AlSb 1.81a, 1.25b, 2.2c �2.39a, �2.44b, 10.6c, �4.2d, �1.6e þ0.7a, �0.5b, �54c, �1.0d

GaSb 0.739a, 0.215b, 0.72c þ13.36a, þ5.9b, þ13.8c,þ12e �115a, �4.0b

InSb 0.214a, 0b, 0.17c þ4.5a, 1.0b, þ16c, þ15e þ13.4a, þ11.8b

a mBJ-GGA this work.b GGA-PPE this work.c Ref. [48].d Ref. [78].e Ref. [66].

Fig. 5. Band structure profile of III-Sb compounds in mBJ-GGA approximation.


stress and strain. Higher values of Young's modulus (Y) incomparison to the bulk modulus (B) for III-compounds(Table 3) indicate that III-Sb compounds are hard to bebroken [61].

The Poisson ratio value (ν) for covalent compounds isabout ν¼0.1, while for ionic compounds an average valueof ν is about 0.25 [61]. In the present study, the value of νfor III-Sb compounds is between 0.1 and 0.25, i.e. mixedcovalent–ionic contribution should be assumed for thesecompounds. This is consistent with Aourag et al. [67] andPhillips [68] researches. Furthermore, the typical depen-dences between shear and bulk moduli are GQ1.1B andGQ0.6B for covalent and ionic compounds respectively[69]. The calculated values of G/B for III-Sb compoundsvary between 0.64 and 0.95 which also indicate that themixed covalent–ionic bonding is suitable for these com-pounds. Our findings are well consistent with Steiner, whoshowed that the nature of chemical bonds in III–V semi-conductors is of mixed covalent–ionic type [70]. Eightvalence electrons are shared between a pair of nearestatoms. Consequently, the bonding has a covalent character.Then again, since the elements of group III are moreelectropositive and elements of group V are more electro-negative, the bonding also has a partial ionic character. Butthe covalent nature is prevalent.

Afterwards, the pressure effects of the elastic moduli ofXSb compounds were studied using PBE-GGA approxima-tion. Mass density, sound velocities for longitudinal andshear waves, Debye average velocity, Debye temperature,and Grüneisen parameter of these compounds up to thefirst order phase transition pressure have been listed in

Table 4. The variations of the elastic constants and the bulkmodulus B of III-Sb compounds with respect to thevariation of pressure are shown in Fig. 3. It is obviouslyunderstood that when the pressure is enhanced, theelastic constants of these compounds increase. It is foundthat AlSb is most sensitive to pressure than others. Theresults of the pressure derivatives ∂C11/∂P, ∂C12/∂P, ∂C44/∂Pand ∂B0/∂P for III-Sb compounds are summarized inTable 5. To our knowledge, no experimental or theoreticaldata for the pressure derivative of elastic constants of BSbhas been given in the literatures. Thereupon, the calcu-lated results can be served as an estimate for futurestudies.

The velocity of sound for shear and longitudinal waves(VS and VL) and Debye average velocity (Vm) have beenestimated using Eqs. (8)–(10) respectively (Table 4). Soundvelocities, through the bulk modulus and shear modulus,depend on elastic moduli. Therefore, the greater the elasticmodulus, the greater the sound velocity. Debye tempera-ture and also Grüneisen parameter have been calculatedfor the XSb compounds using Eqs. (11) and (12) respec-tively (Table 4). Debye temperature depends on the Debyeaverage velocity (see Eq. (11)). Grüneisen parameter (ζ) isthe measure of anharmonicity of the crystal [73]. TheGrüneisen parameter also depends on elastic moduli viasound velocities (see Eq. (12)). Thus, greater elastic moduliwill lead to be a bigger Grüneisen parameter (vibrationalanharmonicity). Furthermore, we studied the pressuredependence of mass density, sound velocities and Debyetemperature listed in Table 4. The sound velocities vs.pressure up to their phase transition pressure also are

Table 8Energy band gap of III-Sb compounds in GGA-PBE and mBJ-GGA approximations.

Compound GGA-PBE mBJ-GGA Other calculation Exp.

BSb 0.741 1.190 0.75a, 0.527b, 0.844c, 0751d, 0.763e,1.334f, 0.71g, 0.56h, 0.54i, 3.096n

0.51j

p¼4.378 0.697 1.120 – –

p¼9.038 0.642 1.066 – –

p¼13.60 0.620 1.022 – –

p¼17.90 0.569 0.964 – –

AlSb 1.258 1.811 1.67n 1.6m,k,1.7l, 1.58r

p¼2.32 1.184 1.754p¼4.749 1.130 1.694p¼7.539 1.083 1.640p¼9.675 1.002 1.582

GaSb 0.220 0.734 0.729p, 0.62q, 0.8o, 0.547n 067m, 0.78l, 0.73r

p¼1.407 0.290 0.917p¼2.796 0.371 1.020p¼4.168 0.471 1.088p¼5.498 0.520 1.131

InSb 0 0.217 0.213n 0.165m, 0.23l, 0.17r

p¼1.098 0.01 0.270p¼2.144 0.07 0.373p¼2.673 0.12 0.437p¼3.522 0.17 0.533

a FP-LAPW Ref. [79].b PP-LDA Ref. [11].c PP-GGA Ref. [12].d FP-LDA Ref. [3].e FP-GGA Ref. [3].f FP-EVA Ref. [3].g FP-GGA Ref. [7].h FP-LDA Ref. [7].i FP-GGA Ref. [13].j Experiment Ref. [80].k Room temp. Experiment Ref. [74].l At 0 K. Experiment Ref. [66].m Room temp. Experiment Ref. [66].n PW-PP Ref. [5].o FP-GGA Ref. [81].p Ref. [82].q Ref. [83].r Experiment Ref. [70].


illustrated in Fig. 4. It is seen that the sound velocities,mass density and Debye temperature increase monoto-nously as pressure increases. It is known that Debyetemperature increases with pressure as it is related tomass density and Debye average velocity (Table 6).

4. Electronic properties

4.1. Band structure

Electronic properties of III-Sb compounds were inves-tigated by calculating the energy band structure. Fig. 5shows the calculated band structure of III-Sb compoundsusing mBJ-GGA approximation along some high symmetrydirections of the Brillouin zone calculated at equilibriumvolume. The band gap energies of III-Sb compounds wereimproved in comparison with the other theatrical resultsand our results are in good agreement with the experi-mental ones. The calculated energy band gaps of III-Sbcompounds in mBJ-GGA and GGA-PBE approximations arelisted in Table 8. Calculating the valence band maximum(VBM) and conduction band minimum (CBM) leads to

direct band gap compounds for GaSb and InSb, andindirect ones for BSb and AlSb. The calculated energy bandgap values of BSb, AlSb, GaSb and InSb were 1.190, 1.811,0.734 and 0.217 eV respectively by using mBJ-GGA. mBJ-GGA results in larger band gaps than those obtained fromPBE-GGA because it has orbital independent exchange–correlation potential which depends only on semilocalquantities [22]. Our results are in good agreement withthose obtained through experiments [5,66,74]. In compar-ison with the experimental data, the other functionalsunderestimate band gaps energy. The reason of the under-estimations is that the Kohn–Sham one particle equationdoes not provide the quasiparticle excitation energies[75,76]. The effect of pressure on band gap of III-Sbcompounds was also investigated up to the first orderphase transition pressure. The results are listed in Table 8.It is seen that energy band gap of BSb and AlSb decreaseswith pressure but in contrast, it increases for GaSb andInSb. As a rule, direct band gaps increase with pressureand indirect band gaps decrease with increasing pressure[77]. As it is mentioned, BSb and AlSb are indirect but GaSband InSb are direct band gap semiconductors. Moreover,

Fig. 6. Real and imaginary part of the dielectric function spectrum for a radiation up to 14 eV using PBE-GGA.


linear and quadratic pressure coefficients (E0 (p)¼E0(0)þapþbp2) of III-Sb compounds are calculated and listed inTable 7.

4.2. Optical properties

The optical properties of a solid can be described bycomplex dielectric function ε(ω). As the III-Sb compoundsin ZB phase have a cubic symmetry, we need to computeonly one dielectric tensor component for each one. ε1(ω)and ε2(ω) are the real and imaginary parts of the frequencydependent dielectric function. Significant optical functionslike the refractive index n(ω) and extinction coefficient k(ω) can be evaluated by obtaining both real and imaginaryparts of the dielectric function [84–86]as follows:

nðωÞ ¼ ε1ðωÞ2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε21ðωÞþε22ðωÞ

q2

24

351=2

ð13Þ

kðωÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε21ðωÞþε22ðωÞ

q2

�ε1ðωÞ2

24

351=2

ð14Þ

Fig. 6 illustrates the real and imaginary parts of thedielectric function spectrum of XSb compounds for aradiation up to 14 eV under hydrostatic pressure up tothe first order phase transition using PBE-GGA. It is foundthat the dielectric functions shift rightward and the staticdielectric constant ε1(0) increases by pressure. The spectralplots clearly display several peaks emanating from allowedband to band transitions. After these thresholds, ε2(ω)curves' slope rises quickly because the number of pointscontributing to ε2(ω) increases sharply. The main peaks inthe spectra are located at 4.7, 3.6, 3.5 and 3.4 eV for BSb,AlSb, GaSb and InSb with PBE-GGA approximation respec-tively. The threshold for direct optical transitions betweenthe highest valence band and the lowest conduction bandoccurs at 1.2, 1.8, 0.7 and 0.21 eV for BSb, AlSb, GaSb andInSb in mBJ-GGA approximation which correspond to theelectronic band gap obtained from the band structurecalculations. The static dielectric constants found for BSb,AlSb, GaSb and InSb in PBE-GGA are 11.19, 11.94, 19.16 and19.36 respectively. The static refractive indices (Eq. (13)) ofIII-Sb compounds in GGA-PBE and mBJ-GGA approxima-tions are given in Table 9. These values are consistent withthe refractive index measurements. It is found that PBE-GGA has better agreement with experimental results thanmBJ-GGA since mBJ-GGA is an orbital independent

Table 9Refractive indices of III-Sb compounds under pressure.

Compound GGA-PBE mBJ-GGA Other calculation

BSbP¼0 3.34 2.60 2.52a, 3.34b, 3.30c

p¼4.378 3.29 2.67 –

p¼9.038 3.25 2.69 –

p¼13.60 3.22 2.70 –

p¼17.90 3.19 2.71 –

AlSbP¼0 3.46 3.26 3.2d, 3.4e,3.20f

p¼2.32 3.40 2.86p¼4.749 3.36 2.02 .p¼7.539 3.33 1.78p¼9.675 3.32 1.77GaSbP¼0 4.38 3.64 3.80d, 3.82e, 3.85h

p¼1.407 4.43 3.35p¼2.796 4.27 2.79p¼4.168 4.08 2.39p¼5.498 3.98 2.24InSbP¼0 4.40 2.89 3.96d, 4.0e, 4.3g

p¼1.098 4.30 2.37p¼2.673 4.22 2.32p¼2.144 4.15 2.30p¼3.522 4.08 2.71

a FP-LDA Ref. [13].b FP-GGA Ref. [14].c FP-GGA Ref. [8].d Ref. [66].e Experiment Ref. [70].f Ref. [90].g Ref. [81].h Ref. [91].

Fig. 7. (αE)2–E dependence of III-Sb compounds to calculate the opticalband gap.


exchange–correlation potential which depends solely onsemilocal quantities [22]. It results in more accurate bandgaps of semiconductors and insulators than GGA and LDAand less accurate optical properties.

4.3. Optical band gap

The optical band gap Eg is a significant quantitydescribing semiconductors and dielectric compoundssince it has enormous importance in the design andmodeling of such materials [87]. The optical band gapwas obtained from the intercept of the extrapolated linearpart of the plot of (αE)1/p vs. the photon energy E withhorizontal axis. This followed from the Tauc et al. approach[88].

ðαEÞ1=p ¼ BðE�EgÞ ð15Þwhere α stands for the absorption coefficient, E is thephoton energy, B is a transition probability dependentfactor and considered to be constant in the optical fre-quency range, and the index p is related to the distributionof the density of states. Also p has distinct values such as1/2, 3/2, 2 depending on whether the transition is direct orindirect and allowed or forbidden [89]. The optical bandgap of III-Sb compounds is calculated using Tauc relation[88]. Fig. 7 displays plots of (αE)1/p versus the photon

energy E for III-Sb compounds in GGA-PBE approximation.It can be seen that the optical band gaps of BSb, AlSb, GaSband InSb are 2.1, 1.1, 1.4 and 1.35 eV respectively. Unfortu-nately, to the best of our knowledge, there is no publisheddata on this subject to compare our results with.

5. Conclusion

In summary, we have performed ab-initio calculationsof structural, elastic, electronic and optical properties ofIII-Sb compounds using the FP-LAPW method underhydrostatic pressure up to the first order phase transition.The calculations were done using various functionalsincluding PBE-GGA, PBEsol, WC and LDA as well as newapproximation mBJ-GGA. It was shown that the structuralparameters obtained after optimization are in good agree-ment with the experimental data. The first order phasetransition pressures of these compounds also were esti-mated and compared with available data. Moreover, theelastic constants, bulk modulus, shear modulus, Young'smodulus, Poisson's ratio and sound velocities for long-itudinal and shear waves up to first order phase transitionpressure were calculated. The calculated elastic constantssatisfy the stability criteria. In addition, Debye temperatureand Grüneisen parameters were calculated and the effectof pressure on these parameters were also investigated.The variations of the elastic constants and the bulkmodulus of XSb compounds with pressure were calculatedand discussed. Furthermore, the results of Poisson's ratioand G/B calculations revealed that the bonding naturebetween X-Sb is a mixture of ionic and covalent. The bandgaps of XSb compounds obtained within mBJ-GGA approx-imation, agree well with experimental result. Besides,linear and quadratic pressure coefficients of III-Sb com-pounds were calculated. The optical parameters of III-Sbcompounds under pressure were also calculated andanalyzed. The obtained optical band gaps of XSb com-pounds using Tauc method, are consistent with band gapsobtained from band gap calculations.


Acknowledgments

I would like to express my appreciation to my brotherDr. Ziaedin Badehian for his valuable and constructivesuggestions during the editing of this research work. Hiswillingness to give his time so generously has been verymuch appreciated.

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