Eur. Phys. J. B (2013) 86: 469DOI: 10.1140/epjb/e2013-40736-4

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

The structural, electronic and optical propertiesof CuGa(SexS1−x)2 compounds from first-principle calculations

Ke-Sheng Shen1, Zhao-Yong Jiao1,a, Xian-Zhou Zhang1,b, and Xiao-Fen Huang2

1 College of Physics and Electronic Engineering, Henan Normal University, Xinxiang, Henan 453007, P.R. China2 College of Physics and Electronic Engineering, Sichuan normal university, Chengdu, Sichuan 610068, P.R. China

Received 5 August 2013 / Received in final form 24 September 2013Published online 13 November 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. The structural, electronic and optical properties of the CuGa(SexS1−x)2 alloy system have beenperformed systematic within generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE)implemented in the Cambridge serial total energy package (CASTEP) code. We calculate the lattice param-eters and axial ratio, which agree with the experimental values quite well. The anion position parameters uare also predicted using the model of Abrahams and Bernstein and the results seem to be trustworthy ascompared to the experimental and theoretical values. The total and part density of states are discussedwhich follow the common rule of the conventional semiconductors. The static dielectric tenser and refractiveindex are summarized compared with available experimental and theoretical values. Also the spectra of thedielectric functions, refractive index, reflectance, absorption coefficient and real parts of photoconductivityare discussed in details.

1 Introduction

The I-III-VI2 ternary compounds are isoelectronic withthe zinc-blende II-VI semiconductors. For instance,CuGaX2 (X =S and Se) are the corresponding analogs ofthe binary compounds ZnX and can be generated throughcross-substituting pairs of group I and III atoms for thegroup II atoms, with the octet rule still obeyed [1]. Dueto the added chemical and structural freedom, these chal-copyrite compounds and their alloys possess some inter-esting properties and play an important role in optoelec-tronic devices, solar cells, and light emitting diodes or innonlinear optics [2–5]. A typical representative of this fam-ily is CuInSe2, which has a direct bandgap of 1.04 eV andis widely used in high-efficiency thin-film solar cells dueto its high absorption coefficient under sunlight and goodchemical stability. Furthermore, Cu(In, Ga)Se2 is one ofthe most successful thin-film PV materials with demon-strated power conversion efficiencies (PCEs) of nearly 20%and commercial use [6].

Recently, the (Cu, Ag)GaX2 alloy systems becomeknown to us as they have direct band gaps between 1.68 eVand 2.65 eV, which is in the range satisfactory for appli-cations in solid state lighting [7] and high-efficiency tan-dem solar cells [8,9]. It is reported that the substitutionof III or VI elements by others can produce more abun-dant band gap and some related works are listed as fol-lows. The band gap of the solid solution AlCu(SxSe1−x)2

a e-mail: zhy_jiao@htu.cnb e-mail: xz-zhang@htu.cn

was found to vary nonlinearly with composition byBodnar [10]. The solid solution CuInxGa1−xTe2 [11] andCuInSe2xTe2−2x [12] have also been investigated. More-over, the structural and electronic properties of these com-pounds CuGa(SexS1−x)2 have been studied by Chen andGong [13] and its ternary parents CuGaS2 and CuGaSe2

are well investigated by Aguilera et al. [14], Xu et al. [15],Parlak and Eryigit [16], respectively.

Although there is a number of literature [17–19] re-lated to these compounds, their optical properties havenot been systematically studied up to now. It is foundexperimentally that Cu2ZnSn(S, Se)4 alloy has a higherefficiency as absorber when Se was introduced, and thereis still no accurate knowledge to account for this phenom-ena. This inspires us to investigate whether the similarquaternary alloy CuGa(S, Se)2 also has a possibility byvarying the amount of Se or improve the band-gap en-ergy just like Zhan et al. [20] reported that a significantimprovement is observed in AlCu(Se1−xTex)2 by the mix-ture of 20% Se and 80% Te. In this paper, we carry outthis task by calculating the structural, electronic and op-tical properties of the quaternary chalcogenides in first-principle calculations.

2 Computational details

All first-principle calculations in present article were per-formed in the framework of density functional theory(DFT) [21], as implemented in the CASTEP code [22].The electron-ion potential is described by means of the

Page 2 of 8 Eur. Phys. J. B (2013) 86: 469

Table 1. The lattice parameters a, axial ratio η, anion position parameters u of CuGa(SexS1−x)2.

Compositiona η u

x Present Exp. Cal. Present Exp. Cal. Present Exp. Cal.

0 5.363 5.349a 5.370e 0.992 0.979a 0.991e 0.243 0.25a 0.2491e

5.33b 5.407f 0.98b 0.995f 0.275b 0.244f

0.25 5.422 5.445e 0.994 0.2420.5 5.489 5.52e 0.994 0.2420.75 5.551 5.595e 0.995 0.2421 5.616 5.614c 5.670e 0.995 0.982c 0.993e 0.242 0.25c 0.2443e

5.614d 5.624f 0.98d 1.00f 0.259d 0.242f

Exp.a reference [2]; Exp.b reference [28]; Exp.c reference [29]; Exp.d reference [30]; Cal.e reference [13]; Cal.f reference [31].

Fig. 1. A unit cell of CuGaS2 crystal with the chalcopyritestructure.

Vanderbilt ultrasoft pseudo-potential [23]. The exchange-correlation functional with generalized gradient approxi-mation (GGA) of Perdew-Burke-Ernzerhof (PBE) [24] isused to solve the Kohn-Sham equations with a kineticenergy cutoff of 800 eV. The Brillouin-zone is sampledwith a 8 × 8 × 4 k-point mesh using the Monkhorst-pack scheme [25]. The structural relaxation was conductedby using the Broyden-Fletcher-Goldfarb-Shanno (BFGS)minimization [26]. The geometries are considered to beconverged only when the convergence criteria are fit-ted: energy, maximum force, maximum displacement, andmaximum stress are 5.0 × 10−7 eV/atom, 0.03 eV/nm,0.05 GPa and 10−4 nm, respectively. The electronic config-urations in the ground state are taken as Cu: 3d104s1, Ga:3d104s24p1, S: 3s23p4, Se: 4s24p4 in all the calculations.

3 Results and discussion

3.1 Structural and electronic properties

At room temperature, the ternary parents of these com-pounds CuGaS2 and CuGaSe2 have a tetragonal chalcopy-rite structure with the space group I-42d (No. 122) andeight atoms per unit cell. Each atom in this structure isfour-fold coordinated, like in the zinc-blende or diamondcrystal structures [27]. The crystal structure of CuGaS2

is depicted in Figure 1 as a representative of this family

Fig. 2. Relaxed lattice parameter of the CuGa(SexS1−x)2compounds as a function of composition (x).

to save space. A comparison of the zero pressure latticeparameters a, axial ratio η = c/2a and anion position pa-rameters u with experimental and other calculation valuesare given in Table 1. Moreover, the variation of the lat-tice parameters is plotted in Figure 2, which produces avisual expression, and a, c change approximately linearlywith a slope 0.25 and 0.52, respectively. That is to say, theresults of the alloys obey Vegard’s rule [32] with bowingparameter ba is zero, or

a(x) = xa(A) + (1 − x)a(B). (1)

It is obvious our calculated lattice parameters increasemonotonously with the increasing concentration x, whichagree well with the experimental values and seem to bemore accurate than the listed calculation values. Theground state properties in the pressure-free cases are ob-tained by calculating the total energy for a number offixed η values, and for each η, u is determined by mini-mizing the force on the atoms. Also, all chalcopyrite CIGSunit cells present an internal distortion with respect tothe standard zinc-blende lattice, since the existence ofdifferent cations results in different bonding lengths be-tween cations and anions. The anion position parameteru not only measures this distortion of the (S, Se) sub-lattice but also has a large influence on the band gap of

Eur. Phys. J. B (2013) 86: 469 Page 3 of 8

Table 2. Calculated GGA band gaps (in eV) compared to experimental and other theoretical values.

CompositionEg Scissor/eV

x Present Exp.a Cal.b Cal.c(LDA)

0 2.257 2.43 0.6932 0.92 1.60.25 2.109 1.60.5 1.964 1.60.75 1.8 1.61 1.618 1.68 0.0291 0.2 1.6

Exp.a reference [41]; Cal.b reference [13]; Cal.c reference [42].

CIGS chalcopyrites [14,33–36]. Therefore, it is meaningfulto estimate u. To predict the anion position parameter uwe performed the model of Abrahams and Bernstein [37]which estimates u by:

u = 0.5 − (c2/32a2 − 1/16)1/2. (2)

However, this equation does not take into accountthe sublattice distortion Δu with an average error of0.03 A [38]. Then for predicting the sublattice distortionin chalcopyrite semiconductors, a modified equation wasused:

Δu = 0.0024YI + 0.0280YIII − 0.0248YVI, (3)

where YI, YIII, and YVI are the Phillips’ electronegativityvalues for the atoms of the I, III and VI groups [39]. Inthe present system CuGa(SexS1−x)2, we take

YVI = xY Se + (1 − x)YS. (4)

The Phillips’ electronegativity values for copper, gallium,selenium and sulfur are 0.79, 1.13, 1.79 and 1.81, respec-tively. Unfortunately, no related experimental data of uare available for comparison for all the considered com-pounds except the ternary parents CuGaS2 and CuGaSe2.

It has been shown by previous experimental studiesand theoretical calculations that many physical proper-ties (including band gaps, mixing enthalpies, etc.) P ofsemiconductor alloy AxB1−x as a function of x follow thequadratic rules as follows:

P (x) = xP (A) + (1 − x)P (B) − bPx(1 − x), (5)

where bP is the so-called bowing parameter. Chen andGong [13] have calculated the electronic band gaps ofCuGa(SexS1−x)2 using the special quasirandom structures(SQS) approach [40]. They assume that the GGA bandgap error is linear with composition x, then together withthe calculated band gap values of pure ternary compoundsa bowing parameter of 0.07 is obtained. Our calculatedband gaps for these compounds compared with availableexperimental and other theoretical values are illustrated inTable 2. The underestimation of band gaps is well knownin GGA as an inherent feature due to not taking intoaccount the quasiparticle self-energy correctly, then theso-called scissor operator corrections was applied for allthe considered compounds with a value of 1.6 eV. Despite

Fig. 3. The total and atom-projected density of states forCuGa(SexS1−x)2 (composition x = 0.75).

the large GGA error in the calculations, which would notbe assumed to be linear with composition x at present,different bowing parameters are obtained. Our bowingparameters of the band gap at x = 0.25, 0.5, 0.75 are−0.0627, −0.106, −0.1187, respectively. It is obvious theresults have an upward shift from the linear average whileChen and Gong [13] declared there is a downward shift dueto the level repulsion between chalcopyrite energy levelsin the alloy. Similarly, Zhan et al. [20] showed the trendof the calculated band gaps for CuAl(TexSe1−x)2 whosesituation is as same as ours in general. From the view offundamental research and practical application, it is vi-tal to obtain an accurate knowledge of the band gaps ofCuGa(SexS1−x)2, so additional experiments are necessary.

Considering all the compounds have almost the samedensity of states due to their similar electronic structures,we choose one (at x = 0.75) as a representative as depictedin Figure 3. With the concentration x increasing the totaldensity of states (TDOS) and atom-projected density ofstates (PDOS) of Cu and Ga are almost the same, thatis to say the amount of Se has almost no influence onthe DOS of Cu and Ga. As expected, the contributionof Se to the TDOS is proportional to the concentrationwhile that of S is on the contrast. It can be seen clearlythat the uppermost valence band has two sharp peaks atabout −3.36 eV and −1.72 eV, respectively, which aremainly attributed to the strong overlapping of Cu-d, S-p,Se-p states. Also the states of Se centralize in a higher

Page 4 of 8 Eur. Phys. J. B (2013) 86: 469

Table 3. The static dielectric tenser and refractive index of CuGa(SexS1−x)2.

x0 0.25 0.5 0.75 1

Present Exp.f Cal. Present Present Present Present Exp.g Cal.

ε1(0)7.66a 8.905b 8.55a 9.353a 10.594a 12.4a

5.487c 6.88d 5.865c 6.232c 6.639c 7.17c 7.98e

ε1‖(0)7.55a 8.839b 8.383a 9.326a 10.526a 12.25a

5.362c 6.817c 5.747c 6.127c 6.535c 7.019c

ε1⊥(0)7.77a 8.883b 8.634a 9.368a 10.63a 12.47a

5.551c 6.859d 5.925c 6.286c 6.692c 7.245c

n0(0)2.775a 2.98b 2.923a 3.076a 3.254a 3.51a

2.342c 2.437 2.62d 2.422c 2.496c 2.576c 2.676c 2.692 2.82e

n‖(0)2.748a 2.973b 2.895a 3.054a 3.244a 3.50a

2.315c 2.412 2.611d 2.397c 2.475c 2.556c 2.649c

n⊥(0)2.788a 2.984b 2.938a 3.087a 3.26a 3.53a

2.356c 2.427 2.623d 2.434c 2.507c 2.586c 2.69c

Presenta, Cal.b without use the correction scissors; Presentc, Cal.d,e with use the correction scissors; Exp.f reference [45];Exp.g reference [46]; Cal.b,d reference [47]; Cal.e reference [30].

energy level than that of S in the region −2.5 to −5.5 eVdespite the magnitude of their contribution. The S-s andSe-s states make up the valence band between about −12.3and −14.5 eV while the lowest valence band is mainly com-posed of Ga-d states mixed with some s states of both Sand Se, approximately −14.5 to −16.3 eV. Moreover, thelowest conduction band is mainly composed of s, p statesof both S and Se, along with small admixture of Ga-sstates attributing to its lower orbital energy.

3.2 Optical properties

The complex dielectric function ε(ω) = ε1 + iε2 can beused to measure the linear response of the system to anexternal electromagnetic field with a small wave vector.The imaginary components ε2 could be calculated fromthe momentum matrix elements connecting the occupiedand unoccupied electronic states and is given by [43]

ε2(ω) =2e2πΩε0

∑

k,c,υ

∫|ψc

k < ur > ψvk |2δ(Ec

k − Evk − E),

(6)where the integral is over the first Brillouin zone, ω isthe frequency, e is the electronic charge, Ω ∝ m2ω2, m isthe electron effective mass, ψc

k and ψvk are the conduction

and valence band wave functions at k, ε0 is the dielectricfunction of free space. In addition, the imaginary compo-nent ε2 is associated with the interband transitions, andthe intraband transitions are neglected because the intra-band transitions are considered only in metallic materi-als. The real components ε1(ω) can be obtained from theimaginary components ε2(ω) using the Kramers-Kroningrelation [44]:

ε1(ω) = 1 +2πP

∫ω′ε2(ω′)ω′2 − ω2

dω′. (7)

All of the other optical properties can be directly calcu-lated from ε1(ω) and ε2(ω) [43] such as the absorption

coefficient α(ω), refractive index n(ω) etc. Furthermore,scientists found there are some other effects affecting thespectra profoundly including local field effects, excitoniceffects etc. Our calculated spectra using KS GGA eigen-values do not include local field effects but are verifiedbasically identical to the ones with local field effects andGGA wave functions are expected to be too delocalized.As to the excitonic effects, one has to admit they playan important role in the optical properties of the consid-ered materials due to electron-electron and electron-holeinteractions. The intensity of the absorption spectrum justabove the onset always appears underestimated, which iscaused by the neglect of excitonic effects. However, ex-citonic effects mainly affect the band gaps but not theshapes of the spectra. The long-range contribution (LRC)kernel using scCOHSEX + G0W0 eigenvalues [14] worksparticularly well in the case of a continuum exciton, butthe GW correction, which is too computationally demand-ing at present for large supercells made of atoms withmany valence electrons, is basically equivalent to the useof a scissor operator within GGA during the energy rangeof interest for absorption. In the present work, all the spec-tra are plotted only with the incident radiation polarizedperpendicular to the tetragonal c-axis of the crystals al-though they have anisotropic properties. One of the rea-sons is that the behaviors in the two directions are rathersimilar for the studied compounds with few differences indetails, and the other is that our focus is on the compar-ison of the composition dependence of behaviors but notthe anisotropic properties. However, the data both per-pendicular and parallel to the c-axis for all the discussedcompounds have been shown in Table 3 and we displaythe corresponding data both with and without the cor-rection scissors. Figures 4a and 4b display the calculatedreal components of the dielectric functions at x = 0, 1 andx = 0.25, 0.5, 0.75, respectively. Combining Figure 4 withTable 3 we can get that the performance of the real compo-nents of the dielectric constant ε1(ω) is ordinary becauseit increases monotonously along with the substitution of S

Eur. Phys. J. B (2013) 86: 469 Page 5 of 8

Fig. 4. Real and imaginary parts of the dielectric function for CuGa(SexS1−x)2 crystals.

by Se, and a smaller energy gap yields a larger ε1(0) value,which can be explained by the Penn model [48]

ε1(0) ≈ 1 + (�ωp/Eg)2. (8)

Furthermore, the line shapes of ε1(ω) are almost the samebecause of their similar electronic structures and are char-acterized by two sharp peaks, approximately 1.5−3.5 eV.One can observe that the two peaks have a red shiftalong with the substitution of S by Se and the sec-ond peak is higher than the first one at x = 0, butthe discrepancy is reducing until x = 1, the first peakalmost surpasses the second. As expected, the electrontransition from the hybridization orbitals of S 3p, Se 4pand Cu 3d to Ga 4p states leads to the high peaks forCuGa(SexS1−x)2. Our calculated dielectric functions ofCuGaS2 and CuGaSe2 are highly consistent with that ofthe other calculation [31,47] curves, however, the relatedexperimental data for CuGa(SexS1−x)2 are insufficient forcomparison at present. Figures 4c and 4d depict the imag-inary components ε2(ω) for all the considered compounds,and the behaviors are also rather similar.

Figure 5 illustrates the refractive indices n(ω) alongwith the optical reflectivity spectra R (ω), both of whichare characterized by two peaks during the area 1.5−3.5 eV.The peaks for refractive index are much sharper thanthat for reflectivity during the area, especially the sec-ond peak. Furthermore, the peaks mentioned above in-crease monotonously along with the substitution of Sby Se, but they are still relatively low to a certain extent.Calculated refractive indices both perpendicular and par-allel to the c-axis are summarized in Table 3 compared

with available experimental data for all the discussed com-pounds. To have more insight into the optical propertiesof CuGa(SexS1−x)2, the calculated absorption coefficientand the real components of photoconductivity of the crys-tals are plotted in Figure 6 with using the correction scis-sors. The absorption coefficient is a basic way to measurethe fraction of energy lost by the electromagnetic wavewhen it passes through a unit thickness of the materialand photoconductivity parameters related to the photo-electric conversion efficiency are mainly used to measurethe change caused by the illumination. The correspond-ing absorption spectra are produced from the differentinter-band transitions and can be calculated by:

l(ω) = 2ω{

[ε21(ω) + ε22(ω)]1/2 − ε1(ω)2

}1/2

. (9)

Note that the spectra of the absorption coefficient andthe real components of photoconductivity are rather sim-ilar with some differences in the middle energy region andcharacterized by two peaks in the low area, which notonly have a red shift as well as other optical spectra butalso increase more remarkably due to the substitution of Sby Se. For the two kinds of spectra, the first peaks amongthe area 0.7−1.18 eV are relatively low and unconspicu-ous while the second peaks among 1.9−2.4 eV are highand distinct, both of which indicate the strong absorp-tion energy region. To be used in photovoltaics, a mate-rial should satisfy some requirements: to be a direct-gapsemiconductor first, and then have a band gap preferablylarge enough (close to 1.4 eV), the small reflectance and

Page 6 of 8 Eur. Phys. J. B (2013) 86: 469

Fig. 5. Dispersion of the refractive index and reflectivity for CuGa(SexS1−x)2 crystals with scissor correction.

Fig. 6. Dispersion of the absorption coefficient (α) and real parts of photoconductivity (σ1) of CuGa(SexS1−x)2 crystalswith scissor correction.

large absorption coefficient are also necessary to absorbthe essential part of the visible light spectrum [49]. As in-dicated above, the band gaps of CuGa(SexS1−x)2 signifi-cantly shift towards the ideal 1.4 eV for single junction so-lar cells [50] (according to the Shockley-Queisser limit [51])monotonously along with the substitution of S by Se, andCuGaSe2 seems to perform best considering the band gapand large absorption coefficient. However, the alloys at

concentration x = 0.25, 0.5, 0.75, which possess not onlysmaller refractive indices and reflectance but also suitabledirect band gaps and relatively strong absorption ability,may also perform as well as the (Cu, Ag)GaX2 alloy sys-tem in solid state lighting and high-efficiency tandem solarcells or better as a window layer for photovoltaic applica-tions than CuGaS2. Predictably, more applications will befound due to the relatively large variation in properties.

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4 Conclusions

We have studied the structural, electronic and op-tical properties of the quaternary chalcogenidesCuGa(SexS1−x)2 using first-principle calculations tomake an exploration as already mentioned in the intro-duction. It is a pity that the substitution of S by Se seemsnot give rise to a significant improvement of the band gapwhich shifts towards the ideal 1.4 eV for single junctionsolar cells with the increasing concentration at present.Despite the large GGA error in the calculations, whichwould not be assumed to be linear with composition x,different bowing parameters are obtained. Our calculatedlattice parameters and axial ratio agree with the experi-mental values quite well and seem to be more accurate ascompared to other calculation values. We also predict theanion position parameters u using the model of Abrahamsand Bernstein and the results seem to be trustworthy ascompared to the experimental and theoretical data, whichare only available for the ternary parents CuGaS2 andCuGaSe2. Due to the similar electronic structures, thealloys have almost the same density of states followingthe common rule of the conventional semiconductors.The amount of Se has almost no influence on the DOS ofCu and Ga. Moreover, the spectra of the real componentsof the dielectric functions are characterized by two sharppeaks in the region 0−2.0 eV, and the two peaks have ared shift along with the substitution of S by Se and thesecond peak is higher than the first one at x = 0, but thediscrepancy is reducing until x = 1, the first peak almostsurpasses the second. As to other optical properties, onecan observe that the behaviors of the spectra are rathersimilar with some differences in details, especially themiddle energy region, and increase monotonously duringthe low energy region along with the substitution of Sby Se. Calculated static dielectric tenser and refractiveindices both perpendicular and parallel to the c-axisare summarized for all the discussed compounds whilethe data with correction scissors are also, however, theexperimental and theoretical values are available onlyfor the ternary parents CuGaS2 and CuGaSe2. AlthoughCuGaSe2 seems to perform best considering the band gapand large absorption coefficient, the alloys at concentra-tion x = 0.25, 0.5, 0.75, which possess not only smallerrefractive indices and reflectance but also suitable directband gaps and relatively strong absorption ability, mayalso perform as well as the (Cu, Ag)GaX2 alloy system insolid state lighting and high-efficiency tandem solar cellsor better as a window layer for photovoltaic applicationsthan CuGaS2.

This work was supported in part by the Foundation of NSFC(No. 11304085), the Foundation for Key Program of Ministryof Education, China (Grant No. 212104), the Natural ScienceFoundation of Henan Province, China (No. 132300410017).

References

1. C.H.L. Goodman, J. Phys. Chem. Solids 6, 305 (1958)

2. J.L. Shay, J.H. Wernick, Ternary Chalcopyrite Semicon-ductors: Growth, Electronic Properties and Applications(Pergamon Press, Oxford, 1975), Vol. 1, p. 244

3. M.C. Ohmer, R. Randey, B.H. Bairamov, Mater. Res. Bull.23, 16 (1998)

4. R.W. Birkmire, E. Eser, Annu. Rev. Mater. Sci. 27, 625(1997)

5. T. Xu, C.-Y. Zhang, G.-M. Wei, J. Li, X.-H. Meng, B.Tian, Eur. Phys. J. B 55, 323 (2007)

6. H. Katagiri, K. Jimbo, W.S. Maw, K. Oishi, M. Yamazaki,H. Araki, A. Takeuchi, Thin Solid Films 517, 2455 (2009)

7. S.F. Chichibu, T. Ohmori, N. Shibata, T. Koyama, T.Onuma, J. Phys. Chem. Solids 66, 1868 (2005)

8. T.J. Coutts, K.A. Emery, J.S. Ward, Prog. Photovoltaics10, 195 (2002)

9. T.J. Coutts, J.S. Ward, D.L. Young, K.A. Emery, T.A.Gessert, R. Noufi, Prog. Photovoltaics 11, 359 (2003)

10. I.V. Bodnar, Inorg. Mater. 38, 647 (2002)11. V.Yu. Rud, Yu.V. Rud, R.N. Bekimbetov, V.F. Gremenok,

I.A. Viktorov, I.V. Bodnar, D.D. Krivolap, Semiconductor33, 757 (1999)

12. I.V. Bodnar, V.S. Gurin, A.P. Molochko, N.P. Solovei, P.V.Prokoshin, K.V. Yumashev, J. Appl. Spectrosc. 67, 482(2000)

13. S. Chen, X.G. Gong, Phys. Rev. B 75, 205209 (2007)14. I. Aguilera, J. Vidal, P. Wahn’on, L. Reining, S. Botti,

Phys. Rev. B 84, 085145 (2011)15. B. Xu, X. Li, Z. Qin, C. Long, D. Yang, J. Sun, L. Yi,

Physica B 406, 946 (2011)16. C. Parlak, R. Eryigit, Phys. Rev. B 73, 245217 (2006)17. M. Quintero, C. Rincon, P. Grima, J. Appl. Phys. 65, 2739

(1989)18. J. Gonzalez, E. Calderon, F. Capet, L. Aioouy, J.C.

Chervin, Jpn J. Appl. Phys. 32, 462 (1993)19. A.A. Lavrent’ev, B.V. Gabrel’yan, I.Ya. Nikiforov, Phys.

Solid State 43, 1438 (2001)20. Y.Z. Zhan, M.J. Pang, H.Z. Wang, Y. Du, Curr. Appl.

Phys. 12, 373 (2012)21. P. Hohenberg, W. Kohn, Phys. Rev. B 136, 864 (1964)22. M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard,

P.J. Hasnip, S.J. Clark, M.C. Payne, J. Phys.: Condens.Matter 14, 2717 (2002)

23. D. Vanderbilt, Phys. Rev. B 41, 7892 (1990)24. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77,

3865 (1996)25. H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13, 5188 (1976)26. B.G. Pfrommer, M. Cote, S.G. Louie, M.L. Cohen, J.

Comput. Phys. 131, 233 (1997)27. T. Sakuntala, A.K. Arora, Phys. Rev. B 53, 15667 (1996)28. P.W. Yu, Y.S. Park, S.P. Faile, J.E. Ehret, Appl. Phys.

Lett. 26, 717 (1975)29. L. Garbato, F. Ledda, R. Rucci, Prog. Cryst. Growth

Charact. 15, 1 (1987)30. S. Shirakata, S. Chichibu, S. Isomura, Jpn J. Appl. Phys.

36, 7160 (1997)31. P. Nayebi, K. Mirabbaszadeh, M. Shamshirsaz, Physica B

416, 55 (2013)32. L. Vegard, Z. Phys. 5, 17 (1921)33. J.E. Jaffe, A. Zunger, Phys. Rev. B 29, 1882 (1984)34. J.E. Jaffe, A. Zunger, Phys. Rev. B 27, 5176 (1983)35. F.D. Jiang, J.Y. Feng, Semicond. Sci. Technol. 23, 025001

(2008)

Page 8 of 8 Eur. Phys. J. B (2013) 86: 469

36. J. Vidal, S. Botti, P. Olsson, J.-F. Guillemoles, L. Reining,Phys. Rev. Lett. 104, 056401 (2010)

37. S.C. Abrahams, J.L. Bernstein, J. Chem. Phys. 55, 796(1971)

38. S.C. Abrahams, J.L. Bernstein, J. Chem. Phys. 59, 5415(1973)

39. J.C. Philips, Covalent Bonding in Crystals (Chicago Univ.Press, Chicago, 1969)

40. A. Zunger, S.-H. Wei, L.G. Ferreira, J.E. Bernard, Phys.Rev. Lett. 65, 353 (1990)

41. Semiconductors: Data Handbook, 3rd edn., edited by O.Madelung (Springer, Berlin, 2004)

42. S.N. Rashkeev, W.R.L. Lambrecht, Phys. Rev. B 63,165212 (2001)

43. S. Saha, T.P. Sinha, Phys. Rev. B 62, 8828 (2000)

44. G. Wang, S. Wu, Z.H. Geng, S.Y. Wang, L.Y. Chen, Y.Jia, Opt. Commun. 283, 4307 (2010)

45. G.D. Boyd, H. Kasper, J.H. McFee, IEEE J. QuantumElectron. 7, 563 (1971)

46. G.D. Boyd, H. Kasper, J.H. McFee, F.G. Storz, IEEE J.Quantum Electron. 8, 900 (1972)

47. S. Laksari, A. Chahed, N. Abbouni, O. Benhelal, B. Abbar,Comput. Mater. Sci. 38, 223 (2006)

48. D.R. Penn, Phys. Rev. 128, 2093 (1962)49. R.H. Bube, Photovoltaic Materials (Imperial College

Press, London, 1998), Vol. 1, pp. 1–3350. M.A. Green, K. Emery, Y. Hisikawa, W. Warta, Prog.

Photovoltaics 15, 425 (2007)51. W. Shockley, H.J. Queisser, J. Appl. Phys. 32, 510 (1961)