8/2/2019 Jaffe and Zunger Anion Displacement

1/4

PHYSICAL REVIEW B VOLUME 27, NUMBER 8 15 APRIL 1983Anion displacements and the band-gap anomaly in ternary ABC2

chalcopyrite semiconductorsJ. E. Jaffe and Alex Zunger

Solar Energy Research Institute, Golden, Colorado 80401(Received 14 January 1983)

Using a first-principles all-electron band-structure approach, we show that the anomalous(& 50'lo) reduction in the band gaps of the A B "C2 chalcopyrite semiconductors relative totheir II-VI isoelectronic analogs results both from a pure structural effect (the anion displace-ments reflecting the mismatch of classical elemental radii) and from a purely electronic effect(p-d repulsion in the valence band), with a small coupling between the two factors.

The ternary chalcopyrite semiconductors A'B"'C2 '(isoelectronic analogs of the II-VI binaries) andA "B'vC2v (isoelectronic analogs of the III-V binaries)form a group of nearly 40 different species, spanninga wide range of structural, chemical, and electronicproperties. ' ' Their crystal structure is closely relatedto that of the zinc-blende binary analogs' but withnonideal tetragonal strain parameter rt =c/2a A 1and anion displacement parameter u = 4 +n/a A 41 2 1reflecting the unequal cation-anion bond lengths R&cand Rqc, where 0.R~cR~c. We find that thechemical trends in the magnitudes of the 36 observedcubic lattice constants' aven for the active delectron materials CuB"'C2"'an be understoodquantitatively on the basis of a simple zinc-blendestructural model. Specifically, assuming that thecation-anion bond lengths R& equal the sum ofPauling's tetrahedral radii d, +dj (derived solelyfrom data on the binary compounds) and that inanalogy with zinc-blende materials the bond anglesaround the Batom remain tetrahedral, 5 we find thatthe distribution of all observed a values (Refs. 1and references therein) is closely reproduced (Fig. 1).Remarkably, however, despite this close structuralresemblance to zinc-blende semiconductors, the ob-served optical band gaps of the AiBIC2vi semiconductors are more than 50'/0 smaller than in theirbinary analogs6 (Table I). In fact, it is thisanomalous band-gap downshift hE~ that makes someof the ternary chalcopyrites (e.g. , CulnSeq) amongthe strongest known absorbers in the solar spectrum7(e.g. , linear absorption coefficient at hv =1.5 eV of10, 10, and =10 cm ' for CuInSe2, GaAs, and Si,respectively). This anomaly is largely responsible forthe recent emergence of ternary chalcopyrites as newand promising solar cell materials. Shay and Kasperhave observed that this band-gap anomaly correlateswith the independently observed anomalous reduc-tion in the spin-orbit splitting of the valence-bandmaximum. They suggested that both anomalies arerelated to the observed d character in the upper halfof the valence band found in the ternaries, but not in

i I

~ I- III- VI2+ II - IV- V2~ I - Vill - VI26.46.26.0

CLIe 5.85.6

5.4

CdSnPZnSnAss~Cu TISeaiAgGPS~~Ptculnse,MgglPIk xcgGep,Z Sip dSIPsAgFeSs

ZnGeAssSea

5.$ I I I5.4 5.6 5.8 6.0 6.2 6.4 6.6caic( )

FIG. 1. Correlation between the observed (Refs. 1)and calculated (using the conservation of tetrahedral bondmodel) cubic lattice constants of ternary chalcopyrites.

the binaries. Available band-structure calculations(e.g. , Ref. 10) have ignored the d levels and henceshed no light on the electronic origin of the Shay-Kasper correlation, nor did the calculations show anyband-gap anomaly.

' We have applied the recentlydeveloped all-electron first-principles band-structuremethod" to CuInSe2. We show that the band-gapanomaly has two physically independent sources withcomparable contributions structural source due tothe existence of a nonideal anion displacement u ~ 4(i.e., bond alternation, Rqc &Rsc) which is dictatedby the mismatch in the classical atomic radii, and anelectronic source due to the p-d level repulsion asso-ciated with the noble-atom d orbitals. Having identi-fied a simple structural parameter u that sensitivelycontrols the optical band gap of these materials, wewill further show that the distribution of the ob-served anion displacement parameters u can be

27 5176 1983 The American Physical Society

8/2/2019 Jaffe and Zunger Anion Displacement

2/4

27 ANION DISPLACEMENTS AND THE BAND-GAP ANOMALY IN. . . 5177TABLE I. Observed lowest band gaps of the Cu-based ternary chalcopyrites (Ref. 1) and theamount hE~ by which they are lowered from the values of the binary analogs (Ref. 6 ).

TernaryBand gap(eV) Binary Band-gap anomalyaE (eV)

CuA1S2CuGaS2CuInS2CuA1Se2CuGaSe2CuInSe2CuGa Te2CuIn Te2

3.492.431.532.711.681.041.241.06

Mgp 5Znp 5SZnSZnp 5Cdp 5SMgp 5Znp 5SeZnSeZno sCde sSeZnTeZnp 5Cdp 5Te

2.41.41.61.41.01.31.31.1

predicted solely from Pauling's tetrahedral radii4 orfrom the quantum-mechanical orbital radii' thatscale linearly with them, "by assuming the conserva-tion (and hence transferability) of elemental atomicradii in compounds. This establishes a unique rela-tionship between the structural and optical propertiesof ternary chalcopyrite semiconductors.We use the newly developed first-principles andself-consistent mixed-basis potential-variation(MBPV) band-structure method" recently usedwithin the density functional formalism to obtain thefirst theoretical analysis of the observed valence-bandx-ray photoemission spectra" of CuInSe2. Unlikepseudopotential methods, ' the all-electron MBPVapproach can treat self-consistently all core andvalence orbitals on equal footing, and does not biasthe calculation by using the crystal potential of thebinary analogs to describe the properties of ternarysemiconductors. ' Furthermore, it offers a means foraccurately assessing the influence of selected atomicsubshells on the global electronic structure of thecompound by solving for the self-consistent electron-ic bands in the field of selectively frozen atomic orbi-tals. To elucidate quantitatively the microscopic ori-gin of the Shay-Kasper correlation, we have solvedthe self-consistent band structure of CuInSe2 in thefield of frozen Cu 3d orbitals. We use a mixed basisconsisting of 130 numerical "compressed atom" orbi-tals" plus 380 plane waves; self-consistency is at-tained to a tolerance level of 1 mRy. Comparisonwith the dynamic all-electron results of Ref. 11 showsthat upon freezing, the noble-atom 3d bands separatefrom the main valence band and are lowered by 8.7eV. This virtually eliminates the p-d hybridization,leading to a reduce repulsiveness between the(upper) p-like and the (lower) d-like I'is-derivedvalence-band states as the energy denominator in theinteraction matrix element

(I'is(p) I P II'is(d) )&(er&,er(gj

increases. This leads to a lowering of the valence-band maximum and to an increase of 0.7 eV in theband gap, both for the experimental and anion-distorted configuration (u =0.224), and for the idealzinc-blende configuration (u = 4 ). Hence, p-d hy-bridization accounts for about half of the band-gapanomaly.Examination of the experimental data ' ' alreadysuggests that p-d hybridization is insufficient to fullyexplain the band-gap anomaly. For instance, whereasCuInSe2 and CuGaSe2 have nearly identical d charac-ters, ' the former compound has a band gap that is40% smaller (Table I). We suggest that the structuralanomaly of ternary chalcopyrites reflected in bondalternation and hence in a displaced arrangement ofanions (e.g. , u =0.250 for CuGaSe& but 0.224 inCuInSei) is directly related to the band-gap anomaly.To understand how the structural anomaly is dictatedby the properties of the atoms, we hypothesize thatthe independent crystal parameters u, q, and a willattain such values that the two anion-cation bondlengths Rqc(u, q, a) and Rsc(u, q,a ) will equal therespective sums of Pauling's elemental tetrahedral ra-dii d~ +d~ and d~ + d~ and that the bond anglearound the Batom will be tetrahedral, implyingg- [-, +8( 2 u)~]' t. This simple rule of conserva-tion of tetrahedral bonds (CTB) in different com-pounds, underlying Bragg's classical notion oftransferability of elemental radii in molecules andcrystals, '4 provides direct predictions for a (Fig. I), u(Fig. 2; notice the large scatter of the experimentaldata), and q with no adjustable parameters and, incontrast with previous work, ' without the use of datafrom the ternaries. Its success enables one to predicta, u, and q for compounds for which they are notknown' and to systematize the known structuralparameters in terms of simple and known classicalelemental radii. In turn, the classical radii are linear-ly related to the quantum-mechanical orbital radii"reflecting the shell structure of the atoms. This

8/2/2019 Jaffe and Zunger Anion Displacement

3/4

5178 J.E. JAFFE AND ALEX ZUNGER 27

0.30 4.0

0.283.530

g 0.26 2.50.24

IP 2.04lcUJ

0.221.5

0.22 0.24 0.30.26 0.28UcalcFIG. 2. Correlation between the observed (Refs. 1)and calculated (using the CTB model) anion displacement

parameters for the ternary chalcopyrites. The uncertain datafor the telurides is omitted. Multiple arrows denote variousexperimental determinations.

1.00.50.0 0.20 I I I I I0.21 0.22 0.23 0.24 0.25

Anion Displacement umethod is also consistent with the behavior of chal-copyrite lattice constants with alloying; for example,in the system CuInS2~i ~Se2, if we take

damon =xdsa+ (1 &)dsthen the variations in a i,(x) obtained by the CTBmethod are within &0.1% of the observed (Vegard-type) variations. '6By mapping the all-electron band structure as afunction of the structural parameters, we next inves-tigate how atomic radii, encoded in u, g, and a (Figs.1 and 2) control the electronic properties. '7 We findthat whereas variations in the tetragonal strainparameter rt merely lead to small (=10 t eV)changes in the crystal-field splitting and the bandgaps, the anion displacement parameter sensitivelycontrols all of the band gaps in the system (Fig. 3).Even a small increase in u from its equilibrium valueof 0 224 (Rcn-sa =2 42 A~ Rin-sa =2 60 A) to itsideal zinc-blende value u =0.25 (Rcs,=Ris,=2.51 L) leads to a substantial ionic polarization ofthe bonds (more charge on Se and less on Cu) andconsequently to a dramatic increase in the band gaps(0.5 eV for CulnSei) with a rate 8'/8u rangingfrom 17.5 eV (the I'I' gap) to 25.6 eV (the TT'gap). We have verified that this conclusion is not af-fected by the choice of the exchange-correlation po-tential' by performing a similar calculation forCuAISi, using Ceperley's correlation" (this materialshows a large I I' band gap even with this correla-tion). We find 8Es/8u -17.4 eV, confirming thecontrolling influence of u on the band gaps ofA 'BIC2vi compounds.

FIG. 3. Dependence of the direct (I'I' ) and indirectband gaps of CuInSe2 on the anion displacement parameteru. The equilibrium value is u =0.224 whereas u =0.25 isthe ideal (equal bond) zinc-blende value. The vertical ar-rows denote the lowering in the direct gap due to unfreezingthe Cu 3d orbitals.We conclude that the band-gap anomaly in ternarychalcopyrites is produced both by a purely electronicfactorhe repulsive p-d interactions in the valenceband associated with the A-atom d orbitals, and by apure structural factorhe anion distortions, reflect-

ing the classical atomic size mismatch in the A-C andB-Cbonds. By commuting the operations of freezingthe d orbitals and displacing the anions we find thatthe coupling between these two factors is small((0.1 eV). This analysis clarifies the hitherto unex-plained difference in band gaps between chalcopyriteswith equal d character9 but different sizes of the 8atom (e.g., CuInSe2 vs CuGaSeq), or materials withequal anion displacements but different d character(e.g., CuAISeq and ZnSiPi, both with u =0.269).Indeed, recent experiments on CuInSe2-CuInS2 al-loys' have indicated that despite a substantial differ-ence in the d character of the valence bands in thesetwo materials, 9 the alloy band gap follows a preciselinear variation with composition. In contrast, similarexperiments on CuInSe2-CuGaSe2 alloys show con-siderable positive bowing (the gap of the alloy islower than the composition weighted average of thegaps of the constituent semiconductors), despite anearly identical d character in the two systems. This

8/2/2019 Jaffe and Zunger Anion Displacement

4/4

27 ANION DISPLACEMENTS AND THE BAND-GAP ANOMALY IN . ~ . 5179is consistent with the controlling role of the aniondisplacements: We calculate the same equilibrium ufor CuInSe2 and CuInS2 ( u =0.236), but a larger uvalue for CuGaSe2 (0.264), leading to

Es(CuGaSe2) Es(CuInSe2) = )au = 0.5 eVs,aQ

and positive bowing, in good agreement with experi-ment. Our separation of the band-gap anomaly intostructural and chemical factors may hence provide in-sight and guidelines for designing and constructingternary chalcopyrites with tailored optical properties,as needed in device applications.

'J. L. Shay and J. H. Wernick, Ternary Chalcopyrite Semicon-ductors (Pergamon, Oxford, 1975).A. Miller, A. MacKinnon, and D. Weaire, in Solid StatePhysics, edited by H. Ehrenreich, F. Seitz, and D. Turn-bull (Academic, New York, 1981),Vol. 36, p. 119.B.R. Pamplin, Prog. Crystal Growth Charact. 1, 331(1979).4L. Pauling, The Nature of the Chemical Bond (CornellUniversity Press, Ithaca, New York, 1967), p. 246.5S. C. Abrahams and J. L. Bernstein, J. Chem. Phys. 55,796 (1971) We find in our model similar predictions fora if we do not assume tetrahedral angles around the Batom but use the observed q instead of the Abrahams-Bernstein value.6W. H. Strehlow and E. L. Cook, J. Phys. Chem. Ref. Data2, 163 (1973).~C. Rincon, J. Gonzalez, and G. S. Perez, Phys. Status Solidi(b) 108, K19 (1981).SR. A. Mickelsen and W. S. Chen, in Proceedings of theFifteenth IEEE Photovoltaic Specialists Coqference-1981(IEEE, New York, 1981), p. 880.9J. L. Shay and H. M. Kasper, Phys. Rev. Lett. 29, 1162(1972). Figure 1 of this reference shows a linear relationbetween 4E~ and the percentage d character, as deter-mined from the spin-orbit splitting. It does not includeCuA1S2 (LEE~=.5 eV). This correlation would havepredicted 5% d character for CuA1S2, in contrast withexperimental value (Ref. 1) of 35%. This suggests thatthe d character could not be the only factor controllinghE~, as indeed confirmed by the present study.' A. S. Poplavnoi and Y. I. Polygalov, Izv. Akad. Nauk.SSSR, Neorg. Mater. 7, 1706, 1711 (1971) [Inorg. Mater.

(USSR) 7 1527, 1531 (1971)];C. V. de Alvarez and M.L. Cohen, Phys. Rev. Lett. 30, 979 (1973).P. Bendt and A. Zunger, Phys. Rev. B 26, 3114 (1982).' A. Zunger, Phys. Rev. Lett. 44, 582 (1980);Phys. Rev. B22, 5839 (1980).' J. C. Rife, R. N. Dexter, P. M. Bridenbaugh, and B.W.Veal, Phys. Rev. B 16, 4491 (1977).' W. L. Bragg, Philos. Mag. 40, 169 (1920).' For example, in CdSiAs2 we have a~,-5.926 A inreasonable agreement with ae, =5.885 A, while ua, isunknown and we predict u~~, =0.294. Similarly, wepredict u =0.291, 0.285, 0.262, and 0.261 for CdSiSb2,CdGeSb2, CdSnP2, and CdSnAs2, respectively.G. H. Chapman, J. Shewchun, J. J. Loferski, B.K. Gar-side, and R. Beaulieu, Appl. Phys. Lett. 34, 735 (1979).'7Since the most accurately known interelectronic correlationpotential (due to Ceperley, cf. J. P. Perdew and A.Zunger, Phys. Rev. B 23, 5048 (1981) used in our bandstructure (Ref. 11) underestimates the band gaps of semi-conductors and insulators (see Perdew and Zunger), it isnot well suited to the present purpose. We therefore ad-just the exchange-correlation coefficient to 1.1, yieldingfor CuInSe2 the observed band gap at the equilibriumcrystal structure (u =0.224, rt= 1.004, a =5.784 A). Thiscoefficient is used with no further adjustments in the stud-ies of the structural variations of the gaps.' See J. P. Perdew and A. Zunger, Ref. 17 and referencestherein.'9I. V. Bodner, B.V. Korzun, and A. I. Lukomskii, Phys.Status Solidi (b) 105, K143 (1981).I.V. Bodner, A. P. Bologa, and B.V. Korzun, Phys. StatusSolidi (b) 109, K31 (1982).