PHYSICAL REVIEW B VOLUME 29, NUMBER 4 15 FEBRUARY 1984

Theory of the band-gap anomaly in ABC2 chalcopyrite semiconductors

J. E. Jaffe and Alex ZungerSolar Energy Research Institute, Golden, Colorado 80401

(Received 12 September 1983)

Using self-consistent band-structure methods, we analyze the remarkable anomalies (& 50%) inthe energy-band gaps of the ternary IB-IIIA-VL42 chalcopyrite semiconductors (e.g. , CuGaS2) rela-

tive to their binary zinc-blende analogs IIB-VIA (e.g., ZnS), in terms of a chemical factor AEg"'

and a structural factor EEg. We show that AE~"' is controlled by a p-d hybridization effect AEgand by a cation electronegativity effect AEg, whereas the structural contribution to the anomaly iscontrolled by the existence of bond alternation (R&c&R~c) in the ternary system, manifested bynonideal anion displacements u —4&0. All contributions are calculated self-consistently from

band-structure theory, and are in good agreement with experiment. We further show how thenonideal anion displacement and the cubic lattice constants of all ternary chalcopyrites can be ob-tained from elemental coordinates (atomic radii) without using ternary-compound experimentaldata. This establishes a relationship between the electronic anomalies and the atomic sizes in thesesystems.

I. INTRODUCTION

The ternary ABC2 chalcopyrites (A =Cu and Ag,B=A1, Ga, and In, and C=S, Se, and Te) form a largegroup of semiconducting materials with diverse optical,electrical, and structural properties. ' One of them-CuInSe2 —has recently emerged as a very promising ma-terial for photovoltaic solar-energy applications, " duepartly to the fact that it is probably the strongest absorb-ing semiconductor under sunlight' ' (cf., Fig. 1). Onecan define a binary analog to each ternary compound bytaking the cation that is situated in the Periodic Table be-tween the A and B atoms (e.g. , ZnS is the binary analog ofCuGaSq, or Zn05Cdo 5S is the binary analog of CulnSz).Despite the overall structural similarity between the ter-nary I-III-VI~ compounds and their II-VI binary analogs,the band gaps of the former compounds are substantiallysmaller than those of the latter. This can be seen in TableI, ' which depicts the band-gap anomaly AEg, de-fined as the difference between the binary gap Eg

'

and the ternary gap EEg '. In fact, it is this strong redshift of the ternary band gap that makes some of theABC2 compounds such strong absorbers of sunlight (Fig.1).

In addition to this electronic anomaly, ternary chal-copyrites have also some interesting structuralanomalies ' relative to their binary analogs (see Fig. 2for comparison of the crystal structures). First, ratherthan have a single cation, the ternary chalcopyrites, havetwo cations; starting from the A atom and translating inthe vertical direction in Fig. 2 through intervals of c/2 wefind the sequence ABAB . , whereas translating hor-izontally with an interval of a, we find the sequenceAAA . . Second, these crystals often show a tetragonaldistortion where the ratio between the lattice parametersg—:c/2a (tetragonal deformation) differ from 1 by asmuch as 12%. Third, the anions are displaced from theirzinc-blende sites. This reflects the fact that in binary AC

zinc-blende compounds each cation A has four anions C asnearest neighbors (and vice versa), whereas in a ternarychalcopyrite ABC2 each cation A and B has four anions Cas nearest neighbors, and each anion has two A and two 8cations as nearest neighbors. As a result, the anion C usu-ally adopts an equilibrium position closer to one pair ofcations than to the other, that is, unequal bond lengthsR„c&R&c (bond alternation). The nearest-neighboranion-cation bond lengths are given by

Roc ——[u +(1+g )/16]'i a

and

R~c=[(u ——,'

) +(1+g )/16]' a .

Hence, the anion displacement u ——,' =(Rzc —Rzc)/a

measures the extent of bond alternation in the system.Table II (Refs. 27—48) gives a compilation of the experi-mental data for a, rl, and u of ternary ABC2 semiconduct-ors. The structural anomalies g —1 and u —

4 relative tothe zinc-blende structure (g = 1 and u = 4 ) are seen to besignificant. Note that the anion position parameter u isoften called x of xf in the literature, and one also en-counters notations such as sr =4u —1 and @=2(1—q).

We have recently completed a detailed series of self-consistent band-structure calculations for the six Cu chal-copyrites (A=Cu, B=A1, Ga, and In, and C=S and Se)in their equilibrium crystal structure, in which we havestudied the trends in the one-electron energies and chemi-cal bonding along this series. In the present study, wefocus on the relationships between the band-gap anomaly(Table I) and the structural anomalies relative to binarycompounds (Table II). While on the simplest phenomeno-logical level one could dispose of the issue by taking theviewpoint that both the band-gap and the structuralanomalies in the ternary compounds relative to theirbinary analogs ultimately arise from the differences be-tween the A and 8 atoms and the zinc-blende cation, our

29 1882 1984 The American Physical Society

THEORY OF THE BAND-GAP ANOMALY IN ASC2 CHALCOPYRITE. . . 1883

10 2000

10'— 1200

800

IV

Cg

'aCO

LA

V

COCOC5

Ls

I

iaafsss

Wgf

II

I

1.0 1.5 2.0 2.5

400

FIG. l. Absorption spectrum (Ref. 12) of CuInSe2 compared with that of other photovoltaic semiconductors (Ref. 13). For com-parison, we give also the air mass (AM) 1.S solar-emission spectrum (Ref. 14). a-SiHo ~6 is amorphous hydrogenated Si and x-Si iscrystalline Si.

aim here will be to analyze this statement in terms ofwell-defined chemical constructs (hybridization, electro-negativities, and bond-length mismatch) with the hopethat such an analysis would also provide predictive in-sights for controlling the material properties of such com-pounds. We will use CuInSe2 as our prototype system. InSec. II we will briefly describe the theoretical tools usedto compute self-consistently the electronic band structureof these materials. We will discuss the general features ofthe electronic bands, charge densities, and bonding for the

equilibrium crystal structure used later in our discussionof structurally and chemically driven electronic anomalies.%'e will show that the band-gap anomaly can be analyzedin terms of (i) a p-d hybridization effect (Sec. III), (ii) acation-electronegativity effect (Sec. IV), and (iii) astructural effect (Sec. V). In Sec. VI we will discuss thesethree effects and their additivity. Having established thatthe anion displacements control the structural part of theband-gap anomaly, in Sec. VII we will analyze the elemen-tal factors (size mismatch between atomic radii) that con-trol the anion displacernents. This will establish a relationbetween the (semiclassical) structural coordinates (atomicradii) and the structurally driven electronic anomaly. InSec. VIII we provide predictions of structural parameters

Cb)4L 4L

~a,~% 4EIII))ii

Ca)

~ 0 OZnS CuoaS,

FIG. 2. Crystal structure of (a) chalcopyrite and (1) zinc-blende lattices.

and band gaps for ternary chalcopyrites that have not yetbccn obscrvcd while Scc. IX constitutes a summary.

1884 J. E. JAFFE AND ALEX ZUNGER

TABLE I. Band gaps of ternary semiconductors E~' and the

difference AEg =Eg Eg (band-gap anomaly) with respect tothe binary analogs. E~"' values are compiled from Refs. 8 and15—22, and correspond to room temperature, except as noted.The binary band gaps Eg

' are from Ref. 23 taken at the corre-sponding temperatures. Uncertain values are denoted by an as-terisk.

Ternary

CuA1S2CuGaS2CuInS2

TeDiaryband gapE,"' (eV)

3 49'243153

Binaryanalog

Mgo sZno sSZnS

Zno. sCdo. sS

Band-gapanomaly

AE, (eV)

2.411.371.64

CuA1Se2CuGaSe~CuInSe2

2.67'1.68'1.04

Mgp sZno sSeZnSe

Zno &Cdo qSe

1.471.001.29

CuAlTegCuGaTe2CuInTeq

2.06'

0.96~—1.06"

Mgo. sZnp. sTeZnTe

Zno, Cdo qTe

1.441.060.98—0.88

AgA1S2AgGaS2AgInS&

3.13'2.51'—2.73'1.87'

Zilo gCdo 5S

CdS0.62—0.440.66

AgA1Se2

AgGaSe2AgInSe&

2.55'1.83'1.24"

Zno gCdo 5SeCdSe

0.500.61

AgA1Te& 2.27"*

AgGaTe2 1.1g—1.326" Znp sCdp sTeAgIn Te2 0.96 —1.04"* CdTe

'Reference 15; single crystal.Reference 16; single crystal.

'Reference 17; single crystal.Reference 18; 77 K, single crystal.

"Reference 19; single crystal.'Reference 19; single crystal.Reference 20; thin-film sample.

~Reference 8, p. 336; temperature and sample type not"Reference 21; single crystal.'Reference 22; single crystal.'Reference 3, p. 118;single crystal."Reference 7, p. 165; 77 K, sample type not given.

0.84—0.620.62—0.54

given.

II. CALCULATING THE BAND STRUCTURE

= &,„,(r )+ &c,„i[p(r )]+V„[p(r )],

A. Computational strategy

We calculate the electronic structure of ternary chal-

copy rites in the local-density approach using thepotential-variational-mixed-basis (PVMB) methoddescribed before. ' In this approach, the effective one-

body Kohn-Sham (KS) potential is given as

5E„,[p(r )]

fr —r'i 5p(r )

where V,„,(r ) is the external potential (interpreted in ourall-electron approach as the electron-nuclear potential},

p(r ) is the ground-state charge density, Vc,„, is the in-

terelectronic Coulomb repulsion, and V„, is the exchange-correlation potential for which we have used bothCeperley's form, ' ' as given by Perdew and Zunger,and the Slater form, but with an exchange coefficient ofa= 1.1 (see below). The main characteristics of the non-

relativistic PVMB method are as follows. (1) The single-

particle wave functions gj(r ) are generated through thepotential gradient approach in which the conventionalwave-function gradient variational principle BE/BQ=O isreplaced by the equivalent (but computationally moretractable) condition BE/Bp; =0, where E is the total ener-

gy and the p are variational parameters of the generatingpotential U(I@I;r }. (2) The wave functions are ex-panded in a mixed-basis set consisting of coordinate-spacecompressed atom orbitals and plane waves. (3) No shapeapproximations (muffin tin or other) are applied to the po-tential; the Hamiltonian matrix is generated from the po-tential in Eq. (2) and the prescribed basis set, essentiallywith no approximations except for convergence criteriawhich are monitored to achieve a prescribed tolerance ofprecision. (4} The residual minimization direct inversionin the iterative subspace (RM DIIS) method is used fordiagonalizing the Hamiltonian matrix; this method is farmore efficient than the conventional Hausholder-Choleskiapproach, provides arbitrarily precise eigensolutions, andpermits efficient handling of large matrices (about800X 800 in our case) without requiring their storage. (5)The charge density is computed by sampling the wavefunctions at special points in the irreducible Brillouin zone

(in the present study, a single k point). (6) Acceleratedself-consistency is obtained by using the Newton-RaphsonJacobian update method to a tolerance of 1—3 mRy.Convergence studies with respect to all internal computa-tional parameters were carried out; ' we select conver-gence tolerances that produce a precision of -0.15 eV inthe band energies in a region of +10 eV around the Fermienergy E~.

B. Band structure

Figure 3 displays the band structure of CulnSei as cal-culated with (a} Ceperley's correlation and (b) the Slaterexchange. Figure 4 depicts the density of states produced

by both approaches. Whereas Ceperley's correlation pro-duces a one-electron spectrum that correlates well withthe photoemission data, it fails to reproduce the observedband gap [—1 eV(Ref. 18}]. The chemical trends in theelectronic charge densities of six CuBC2 chalcopyrites(B=A1, Ga, and In, and C=S and Se) using Ceperley'scorrelation have been discussed previously. Since, how-

ever, in the present study, we are interested in studying thechanges in the electronic band gaps induced by structuralvariations, the misrepresentation of the band gap byCeperley's correlation constitutes a serious handicap. Weseek, therefore, a simple empirical adjustment that willcorrect the band gaps of those chalcopyrites while approx-imately preserving the derivatives Bhe/BA, of the one-electron energy gaps he with structural parameters A,

THEORY OP THE BAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . .

I-III-VI2 Chalcopyritesc/aa (A)

5.31(2)5.334(1)5.3336(5)

e(A)10.4210.444(2)10,444(2)

1.9611.9581.958

272829

0.270.275(2)0.268(4)

5.6065.602(2)

0.260.269(5)

CuAlSe2 1.9451.954

10.9010.944(5)

0.263CuAlTe2

CuGaS2

5.964 11.78

1.9581.9481.958761.9586

0.250.275(5)0.2539(4)0.272(5)

5.3495.356(1)5.34741(7)5.351(1)

10.4710.435(5)10.47429(6)10.480(5)

1.9601.9651.96623

10.9911.03(1)11.0036(2)

0.250.250(1)0.2431(2)

5.6075.614(1)5.5963(1)

0.263

0.200.214(7)0.2295(4)

2.0052.0132.015 82

11.0611.12(2)11.13295(22)

5.5175.523(4)5.52279(7)

272833

0.220.224(3)0.235(5)

2.0012.0082.009 69

11.5511.616(5)11,620

5.7735.784(1)5.782

0.2315.580

0.2320.231.99511.63

0.300.290(3)

1.8021.772

10.2610.135(10)

5.6955.720(1)

10.7510.70(1)

5.9565.986(1)

AgAlTe2

AgGaS2 0.280.304(6)0.2908(4)

1.7861.7891.78968

10,2610.295(6)10.3036(2)

5.7435.754(2)5.757 22(3)

1.8231.793

10.8810.73(1)

5.9735.985(2)

AgGaTe2

AgInS2 1.9201.910

0.250.250(1)

5.8165.872(1)

11.1711.214(3)

TABLE II. Compilation of the experimental structural parameters a and e (lattice constants, measured in A), u (anion displace-ment), and 2q:—c/a (tetragonal deformation) for the ternary ABC2 system. The numbers in parentheses are uncertainties in the lastdigits. All values are taken at or near room temperature. The spread in values is often due to slight nonstoiehiometry. Values of c/aare quoted directly from the experimental work even if they are slightly inconsistent vnth the values of c and a given in the samework. The calculated values u~, are from the "CTB plus q =g„, rule, "using Pauling radii, and are described in Sec. VII A. In mostcases, only those references are included that contain a measured value for u; the values of u given in Ref. 27 appear doubtful sincethey agree poorly vnth other measurements as well as with theory.

J. E. JAFFE AND ALEX ZUNGER

CGIDpound

AgInSe2

a (A)

6.0906.109(1)6.1038(15)

11.6711.717(5)11.7118(47)

1.9161.9191.91877

TABLE II. (Continued. )

I-III-VI2 Chalcopyritese(A) c/a

0.250.250(1)0.258 45(16)

Ref.

AgInTe~

Transition-metal-containing chalcopyrites10.32 1.91 0.2710.434 1.969

836

ZnS1Pp

ZnSiAsp

5.399(1)5.399

5.60(1)5.611

10.435(2)10.436

10.88(1)10.885

II-IV-V2 Pnictides1.932 771.933

1.941.940

0.2691(4)0.271

0.265 75(12)0.269

ZnGeP2 5.4655.46(1)5.4665A6(1)

10.70010.71(1)10.72210.758

1.9581.961.9611.97(l)

0.2670.25816(44)0.264

5.6725.672

11.15311.151

1.967(2)1.966

0.2640.250

838

ZnSnP2

ZnSnAs2 5.8515.851(1)5.852(1)

11.70211.70211.705(1)

2.002.002.00

0.2310.23'0.239

6.275& 6.273

12.55012.546

4338

CdSiAs2

CdGePp

CdGeAs2

5.680(l)5.678

5,9432(1)5.942(2)5.945

10A31(3)10.430

10.881

10.775

11.2163(3)11.224411.212

1.836441.837

0.2967(2)0.302

0.2785(2)0.285(5)

10.280

3236

48,38

38

6.0936.094

11.93611.920

1.961.956

0.2610.262

838

10.115

'Qur extrapolation from the author's x-ray data.

(anion displacements, tetragonal distortions, etc.), as ob-tained with the Ceperley correlation. %"e find empiricallythat scaling the exchange potential (with an exchangecoefficient a= —', ) by 1.65 (i.e., an exchange coefficient ofa= l. 1 j satisfies these conditions to within a good approx-

38

imation: This single adjustable parameter produces bandgapa of all six chalcopyrites within -0.2 eV of experi-ment [e.g., for CuInSe2 the calculated gap is 0.985 eV,compared with the observed value of 0.98—1.04 eV (Refs.12 and 18)], while the structural derivatives are within

THEORY OP THE BAND-GAP ANOMALY IN ASC2 CHALCOPYRITE. . . 1887

-4 ~I

Cusd

-8

In-Se Bond

CulnSezCeperley

Correlation

ln-Se Bond

Guin Se2Slater

Exchange

~ MS IMa~ g~~ IN+

~gy cv Q'1a ~ ~~ ION

~~~0%

CulnSezCeperley

Cor.relationd Orbitals

Frozen

T I"

FIG. 3. Self-consistent band structure of CuInSe2 calculated at the observed crystal structure a=10.9303 a.u. , and u=0.224 andq =1.004, using (a) the Ceperley correlation (Refs. 51 and 52), (b) Slater-type exchange with o,'= 1.1, and (c) Ceperley's correlation butfreezing the Cu d orbitals. The shaded areas denote the principal band gap.

10—15% of those obtained with Ceperley's correlation.While the need for such an empirical adjustment reflectsthe inability of the state of the art first-principles local-density calculations to predict electronic excitation ener-gies in complex condensed systems with sufficient accura-cy, ' we believe that the choice of an adjustment thatpreserves the derivatives of the energies is adequate for thelimited purpose of studying structurally induced changesin the electronic properties.

C. Electronic structure of the undeformed cryital

Here we describe the band structure and the electroniccharge distribution in CulnSe2 as obtained in the scaled-exchange calculation at the observed crystal structure (theresults for the Ceperley exchange-correlation weredescribed before26' ). The electronic band structureshows a few major subbands below Ez. First, in the re-gion of 0 to —4 eV below the valence-band maximum(VBM), the upper valence band consists of Se p and Cu dstates. Figure 5(a) depicts the electronic charge density inthis subband, where the contours around the Cu —Se con-tact are shaded to highhght the formation of the bondingCu d—Sep contact and the nonbonding character of theIn—Se contact. Second, in the energy region Ev~M —4.3to EvBM —5.4 CV, we find the In—Se band (dashed lines inFig. 3) and the densely spaced Cu 3d bands. The charge

density of these subbands [Fig. 5(b)] shows the formationof a weak In—Se bond (where a partially covalent bondcharge is formed at the In—Se contact, but is ionically po-larized towards the Se site) and a nearly spherical (i.e.,closed-shell) charge on the Cu site. These subbands showup as a distinct peak in the density of states nearEvBM —4.5 eV [Fig. 4(b)]. Third, in the region betweenEvBM —13 to EvtIM —14 eV, we find the Se 4s subbandshowing up as a sharp peak in the density of states (Fig.4), and having an extended s-like charge distributionaround the Se site [Fig. 5(c)].

The total ground-state charge density in CulnSez isshown in Fig. 6, both as a contour plot [Fig. 6(a)) and asline plots along the Cu—Se [Fig. 6(b)] and In—Se bonds[Fig. 6(c)]. Our model describes the bonding in the systemas mixed ionic and covalent where the In site is isolatedfrom its nearest-neighbor Se sites by nodes, whereas thecharge on the Cu atom merges continuously into that ofthc ncRI'cst-Ilclgllbor Sc sites forming R partially covalclltbond charge closing around both atoms.

Shay and Kasper have pointed out that the anomalousreduction in the band gaps of ternary chalcopyrites rela-tive to their binary analogs is correlated with the existenceof d bonding in the former compounds. They found thatthe band-gap anomaly AEg correlates nearly linearly withthe percentage of d character ad deduced by comparing

80-

20—

10-IOP 5-COOP

55—45—35—

25Cl

Irt-SeboACI

UpperVB

r

I

IvaMI

I ~ iI I I s l iver I

s l l I I I I I

CulnSe2»~d +

Slater ExchangeI

I

I

I

IYBMI

l

I

UpperiVB

I } ~ I I I I I II

I $

CulnSe2Ceperley

Correlation

Distance (a.u.)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.010.011.012.00..0 I I I I I l I I 4 I

1.0-

2.0-

3.0—

4.0-

5sO

8.0—

7+6

8.0-

9.0-

10.0—

0.0

3 0

gg 4.0-

VO g, G

Cgg 6.0—

7.0—CiO

Cl

l i. t s I

-24 -20 -16 -12 -8 -4 0

Energy (eV)FIG. 4. Density of states of CuInSe2 calculated with (a)

Ceperley's correlation and (b) Slater-type exchange with +=1.1.Thc latter cxchRngc parameter was chosen to fit thc optical gRp

to cxpcrlHlcnt.

0.0

10-20-30-

l j I I

I'C

the spin-orbit splitting of the ternary and binary com-pouQds, 1.c., AEg =a&d &1th 0 3.125 cV. Tllcp havesuggested that CuA1S2, CuoaS2, CuGaSc2, aQd CuIQSe2have a nearly constant percentage of d character(ad =35%, 35%, 36%, and 34%, respectively), whereasCUIQS2, vvith a 1aI'gc baQd-gap aQoma1$ of 1.6 cV, has thclargest percentage of d character (45%) in this group.T4c11' aQalpsls appcaI's, ho%'cvcI', to bc lIlcoIHplctc 1Q thfccrespects. First, CuAIS2 (not shown in their correlation )has even a larger band-gap anomaly (-2.4 eV) thanCuInS2, leading, by the Shay-Kasper correlation, to 77%d character, far larger than the value of 35% determinedby them from the spin-orbit splitting. Second, the deduc-tion of the percentage of d character from the differencesin spin-orbit splittings of binary and ternary compounds isat best ambiguous due to the substantial differences inbond angles around the anion (determining the propor-tions of hybridization) between the two. Third, the inter-px'etation of trends in AZg in different chalcopyrites asarising solely from a chemical factor (d mixing) overlooks

40-5.0—

60-70-

8.0-

9.0-

10.0-

FIG. 5. Chalgc dcnsltlcs, ln Units of e/R. U. , of thc thlcc IDR-

Jor valcncc subbands of CUInSc2, calculated with Slatcl-type ex-change. Thc contoUI's arc logarlthlmcally spaced' thc Rlcas ln-sldc the solid cllclcs dcl1otc atolmc colcs Ivhcrc thc rapidly vary-ing charge density has been omitted, foI' clarity. (a) The uppervalence band. The 10 e/a. u. Contours around the Cu —Sebond are shaded to highlight the partially covalent bond. (b)The In —Se and Cu d subbands, The shadmg hlghhghts the for-rnation of a bond charge on the In—Se contact (ionicaHy polar-ized towards the anion) and the nearly spherical Cu closed-shellchRI gc. (c) The Sc s sUbbRIld. Thc 10 8/R. U. contoUrs RlGUIld

Se aI'e shaded to highlight the extended Se s chaIge.

THEORY OF THE BAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . .

10—&0o-

Culn882VBM

I=rozen Cu dhE = 07eV

(a)

CU

«c. I:~4v& = O'

C~00=

II%I~

VI~ &0&=

CulnSe2VBM

Dynamic Cu dhE —OOev

0 1 2 3 4 5 6 7 8 9 10 10-&=

10-30.0 2.0 4.0

Distance (a.u.)FIG. 7. Effect of freezing the Cu d orbitals on the charge

density of CuInSe2 at the VBM. (a) Frozen d orbitals, and (b)

dynamic d orbitals. AEg denotes the change in the direct band

gap due to d-orbital effects and Egc„[Iq„] denotes the changein the percentage of Cu d character at the VBM upon unfreezingthe d orbitals. Note that freezing the d orbitals reduces the bondcharge along the Cu —Se contact, diminishes the Cu d character,and opens up the band gap by 0.7 eV.

I

0 1I I I I 1 I l

2 3 4 5 6 7 8 9 10

the significance of the structural factor (substantial differ-ences in anion displacements) which partially control p-dhybridization, and hence o.d.

A quantum-mechanical description of the electronicstructure of a solid in terms of atomic orbitals peimits oneto assess UnaITlb1guousiy 'the Influence of 8 given atomIcsubshell on the global electronic structure of the com-pound. We have performed a self-consistent PVMB cal-

Position (a.tL)FIG. 6. Total valence charge density of CuInSe2 calculated

with Slater-type exchange. (a) Logarithmically spaced contour

plot with shading highlighting the formation of a covalent lobe

around the Cu-Se contact. (b) and (c) display line plots along theCu —Se and In—Se bonds, respectively.

culation of the band structure of CulnSez in its observedcrystal structure where the atomic Cu 3d orbitals arefrozen, i.e., all other electrons move in an external poten-tial V,„,(r ) [Eq. (2)] which contains, this time, the field ofthe (compressed ) Cu 3d states. Figure 3(c) depicts theband structure of CuInSe2 with frozen Cu d states, com-puted with the Ceperley correlation. We see that freezingthe d orbitals on Cu leads to a massive Cu d —Sep dehy-bridization, where the Cu d band separates from the uppervalence band, forming a narrow isolated band atEvBM —11.8 CV. The dehybridization is accompanied byan opening of the fundamental band gap by 0.73 eV. Wedenote this d-orbital hybridization contribution to theband-gap anomaly by EEs. If, instead of using the experi-mental anion displacement u=0.224, eve use the equal-bond structure with u = —,', the effect of freezing the d or-bitals (using Ceperley's correlation) is very similar:bEs 0.71 eV. The conseq——uences of the Cu d —Sep dehy-brIdlzatlon on thc bondIng chalgc dcnsltlcs can bc appIc-ciated from Fig. 7. It compares the charge density at thetop of the valence band after [Fig. 7(a)] and before [Fig.7(b)] freezing the Cu d orbitals. It is seen that the nonzerocovalent bond charge on the Cu—Se contact [Fig. 7(b}]vanishes upon freezing the Cu d orbitals [Fig. 7(a)] lead-ing, therefore, to an enhanced ionic polarization of' thebond charge. The Cu d character (fraction of d charge inthis state within an atomic sphere around Cu) at the top

J. E. JAFFE AND ALEX ZUNGER

~x~~z~zz~z~& CSIN

EI

VBM

hCpdFN'ozsn

AEIIAnion I;,(p)

Anti-~

bonding

~zzzzrzzzz~j. ' CBIN

EoAntibonding )I VBM

Anion I;, (p)&

i

EGpd { l

I

lfis(d)I

I

p Cation

i is(d)

FIG. 8. Schematic molecular-orbital diagram Used to discuss cation d-orbital effects on both the band-gap anomaly and acceptorbinding energies. (a) Deep d orbitals, and (b) shallow d orbitals. See text for details.

of the valence band (the I 4, state) is reduced by 10%upon freezing the Cu d orbitals. We find very similarchanges in the charge density at the conduction-bandminimum upon freezing the Cu d orbitals.

The d-hybridization contribution b,Es to the band-gapanomaly has a simple molecular-orbital interpretation(Fig. 8). The outer valence p orbitals on the anion form,in a cubic field, a threefold-degenerate state, I Is(p). Thefivefold-degenerate d orbitals on the cation (separated byan energy b.c~ from the anion p states) transform in a cu-bic field into a threefold-degenerate I »(d) combination{wltll 01'bltR1 lollcs pointing Rnd ovcflRpplllg wltll tllcnearest-neighbor anions) and into a twofold-degenerateI'Iq(d) combination {with lobes pointing between thenearest-neighbor anions, towards the next-nearest shell).(The I ls states can split further due to spin-orbit interac-tions into I 8 and I 7, and the doubly degenerate I 8 repre-sentation can split into I 6+I 7 in a noncubic field, but wewill not consider these splittings in our simple argumenthere. ) The states of the same symmetry I'Is(p) and I'ls(d)can interact, forming a lower bonding state [weightedmore by the lower-energy I ls(d) state] and an upper anti-bonding state [weighted more by a higher-energy I Is(p)state]. The I Iq(d) states do not form any o bonds withthe nearest-neighbor atoms (although they do, however,form weaker ir bonds). Perturbation theory7 suggests thatthe two ~tat~~, I Is(p) and I Is(d), will repel each other byan amount inversely proportional to 6e~ and directlyproportional to the p-d couphng matrix element

} {p (V

~

d ) (. In binary semiconductors such as GRAs,

wllcfc tllc catloll has ollly dccp, cofcllkc d states [of lllCuBCq compounds with frozen Cu 3d orbitals; see Fig.8(a)], the separation he~~ is large and the direct p-d orbitaloverlap is very small due to thc compactness of the cation

state. This leads to a small I'Is(p)-I ls(d) repulsion, andto an essentially pure cation d bonding states and a pureanion p antibonding state which forms the VBM. Howev-er, when the cation supports Ualence d states (e.g., Cu ortransition atoms) the energy denominator b,e& becomessmall [Fig. 8(b)] and the cation d orbitals are more diffuse,overlapping more effectively with the anion orbitals. Thisleads to a substantial upward repulsion of I Is{@) and areduction b,Es= } {p

~

V ~d) } /he~q in the band gap.The bonding combination results in a sharp Cu d—likeresonance in the valence band (Fig. 4) and the antibondingcombination forms the VBM (their separation is 3—4 CVin our calculation). Similar arguments will hold in thepresence of the noncubic chalcopyrite crystal field.Hence, b,Es reflects a p-d hybridization effect.

This analysis suggests that only the p-d hybridizationpart KEg of the total band-gap anomaly KEs will corre-late with the percentage of d character, and not the fullanomaly EEg, as suggested by Shay and Kasper. Thisrcsolvcs thc 1ncons1stcncy cncountcrcd by Shay andKasper for CuA1SI. In Sec. V we will calculate thestructural contribution hE~ to the band-gap anomaly. %cwill show that the chemical contribution, defined as thedifference between the total EEs and KEs, does in factscale linearly with the percentage of d character calculatedfrom tllc wave fllilc'tioll, Rlld tllat CuA1Sz ls 110 cxccpt1011.

Before closing our discussion on the p-d hybridization,ac polIlt to an intcI'cstlng analogy 1n 1nlpurlty physics.has been known for a long time (e.g., review in Ref. 56)that whereas III-V and IV-IV semiconductors can bemade lo%-rcs1stlv1ty p-type scmlcondUctoI's 4y SUbst1tu-tiolls of inlpllIlty RtoIIls wltll onc less valcllcc dcctfon Oil

the cation site (e.g., Zn in GRAS or Ga in Si), the same isnot true for II-VI compounds. Doping of ZnS or ZnSC by

THEORY OF THE SAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . .

ly in ABC2 CU Acceptor Energiesin ZnC and CdC

=CuGaC2CIIlnC,

0.0

C AtomFIQr. 9. Variation of (a) the band-gap anomaly in CuBC2 chalcopyrites, and (b) Cu acceptor binding energies in II-VI compounds

(Refs. 56 and 57), with the position of the anion in the Periodic Table.

Cu is thought to lead (aside from interstitials, Cu precipi-tates and various Cu sulphides) also to Zn substitutions.However, instead of leading to shallow states, this leads todeep acceptors, about 1,4 and 0.7 eV above the VBM,respectively. *

. We suggest that the p-d hybridizationcontribution to the reduction of the band gaps in ternarychalcopyrites and the anomalously deep Cu acceptors inII-VI compounds share a common physical origin: Bothresult from the existence of a small b,e~ energy separation(i.e., Cu d to anion p) which repels the antibonding stateupwards. This antibonding state is the VBM in ternarychalcopyrites and the acceptor level in Cu impurities inII-VI compounds. Indeed, the Cu acceptor energies in II-VI compounds decrease with he~ (in the sequenceZnS —+ZnSe~ZnTe the anion binding energy decreases,and consequently, the acceptor energy decreases from 1 4to 0.7 and 0.14 eV, respectively), whereas thecation has little infiuence on the acceptor energies (ZnXand CdX having very similar Cu acceptor energies). Thisis illustrated in Fig. 9. In contrast, doping II-VI materialsby Na that lacks d orbitals usually results in shallower ac-ceptors of energy, ' &0.1 eV (the material shows, howev-er, high p-type resistivity due to donor-acceptor compen-Satloll).

The virtual process of bringing a chalcopyrite com-pound into chemical analogy with a zinc-blende com-pound consists of two steps. (i) Freeze the d orbitals onthe Cu atom. The resulting contribution to the band-gapanomaly is the p-d —hybridization part

b,RS Es(ABC&, frozen d) Es(—A—BC@,active d)—discussed in the preceding section. (ii) With the use of the

frozen d structure, replace the two catlons by the averagecation. The resulting contrlbut~on to the band-gap anom-aly is referred to as cation electronegativity (CE) and isgiven by

b,ES Es ({AB)C fI—r—ozedn) Es(ABC2, fro—zen d),where (AB) denotes the average cation. For example,since the Zn atom is situated in the Periodic Table be-tween Cu and Ga, the CE contribution to the band-gapanomaly of CuGaS2 is the difference in the band gap ofZnZnS2 (ZnS in a chalcopyrite lattice) and CuGaS2, bothwith frozen d orbitals. The CE term refiects the fact thatin a chalcopyrite structure, even if the d electrons werechemically inactive, charge could separate differently onthe two dissimilar cations, thereby affecting the band gap.In ternary chalcopyrites, the CE contributions toEEs (-0.1—0.2 eV) are found to be considerably smallerthan the p-d hybridization contribution AE and thestructural contribution AEs discussed in Sec. V. In ter-nary pnictides (e.g., ZnSiPz) and in alloys of III-V com-pounds (e.g., InP-GaP), where EES-O, the CE contribu-tion is not negligible, as was shown before. s Electro-statically, this means that when the valence charge Z onthe zinc-blende cation is separated into Z~ ——Z —1 andZII ——Z+1 in the chalcopyrite structure, the valence elec-trons will attempt to screen these point-ion perturbations,resulting in a lower bond charge on A —C and a higherbond charge on 8—C. Clearly, the lattice is no longer inequilibrium with this new charge distributions. The sys-tem will respond by moving the atoms to new equilibriumposltlons, I'esultlllg 111 Rgc QRIIc. As a 1eslllt, IIlorecharge will be placed on the shorter bond (Cu—Se) andless charge will reside on the longer bond {In—Se). Thissimple electrostatic picture is explained in the light of ourdetailed calculations in the following section.

J. E. JAFFE AND ALEX ZUNGER

3.5 - CuAIS2

Q1.5-

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LU '|.0

Exptl. u

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0.0

-0.5

Tsv+ T4v

«1.00.20

I

0.2'tI

0.24 0.25 0.25I

0.28 0.29

Anion Displacement {u}FIG. 10. Variation of band cncI'glcs Rt high-symmetry points for CQIQSc2 Rnd CUA1S2 with thc RQ1on displaccIHcnt parameter Q.

The results for CuInSC2 are calculated with Slater-type exchange, whereas those for CuAlS2 (which has a nonvanishing gap even withCepexley's correlation) are calculated with Ceperley's correlation. Shaded areas denote the direct band gap; horizontal arrows pointto the experimental equihbrium u values.

The experimental data alone5 suggest that p-d hybridi-zation does not account for the full band-gap anomalysince compounds with similar d character (e.g., CuAISeqand CuGaSe2) differ substantially in their band-gap anom-aly. We suggest that this is due to a structural (S anoma-

ly relative to the binary analogs and denote the corre-sponding contribution to b,Es as EEs.

We start bg eliminating another possibility, namely thatthe existence of a tetragonal deformation 1I=c/2a&1controls Es. Calculation of the CulnSez band structure asa function of q indicates that the band gap changes by lessthan 0.05 eV when 11 is varied between ' 1.000 and 1.004and only the crystal-field splitting (separation between thetwo llppcl valcllcc-balld states 14' alld I 5q) changes wltll

1) (nearly linearly). [In ternary pnictides, e.g., ZnSiPz,where 11 is usually considerably smaller than 1 (cf. TableII), increasing q to 1.0 reduces the band gap by —10%.]

The next possibility we examine is the role of the

nonideal anion displacements u& —,'. We note at theoutset that bond alternation in itself could not explainb,Es since, for example, CuGaSez shows no bond alterna-tion but still has a substantial b,Es. In such cases we find(cf. Sec. VI) that the chemical effect alone (Sec. III) ex-plains EEs. To study the contributions of the structuraleffects, we first note that Eq. (1) shows that variation of u

compresses one anion —cation bond and dilates the otheraccol'dlilg to 11 —

& = Q/0 s wllcl'c cx =Rye —Rsc ls tllcbond-alternation parameter. We have performed self-consistent band-structure calculations for a number ofchalcopyrites as a function of u, keeping the unit-cell di-mensions constant (a=a,„~„and q=q, „~,). Figure 10displays the variations of the energies of a few-symmetrystates below' and above the VBM in CuInSe2 and CuA1S2with u. The experimental equilibrium value of u is indi-cated by the vertical arrows.

We find that the valence-band states drop in energywith increasing u while the conduction-band states risewith u. While the slope differs for the various states, the

THEORY GF THE BAND-GAP ANOMALY IN ASC2 CHALCOPYRITE. . .

3.0—

Icl. 2„0IC

LU

f.5—

0.00.20

I t I I 1

0.21 0.22 0.23 0.24 0.25

FIG. 1 I. Variation of the lowest direct(I"I*, XN, and TT*) and indirect (I N*) band gaps ofCuInSe2 with the anion displacement u.

lowest direct (I I, PfÃ, and TT*) and indirect (I N*)gaps vary with a similar slope (Fig. 11), with Bb,e/Bu inthe range of 19—22 eV. In ~articular, the first band gapof CuInSe2 increases by AFg =0.47 eV when u is variedfrom its equilibrium value of 0.224 to its ideal zinc-blendevalue, u = 4, complementing the chemical contribution

bEs =0.73 eV to a total of EEs=hEs+bEs =1.2 eV,close to the experimental band-gap anomaly of 1.3 eV. Inother words, setting u =0.25 and freezing the d orbitalsgives for CuInSe2 almost the same band gap as its binaryanalog. We shall see in Sec. VI that the same is true forCuA1S2, where KEs is much larger.

To study the mechanism of structurally induced gapvariations, we consideI in Fig. 12 the changes in thecharge densities at the top of the valence band of CuInSe2with u. The solid lines denote the charge densities foru& ~, whereas the dashed lines give, for comparison, theresults for the equal-bond arrangement u =—,

'(relative to

the anion site as origin). We see that as u increases from0.20 through the equilibrium value of 0.224 to u =0.25,the Cu —Se bond charge decreases while the In—Se bondcharge increases, weakening and strengthening the twobonds, respectively. This trend parallels the changes inthe bond lengths: Upon increasing u, the Cu—Se bondlength increases, whereas the In—Se bond length de-creases. Clearly, as the top of the valence band is weight-ed more heavily by Cu —Se contributions, the reductionin the covalency of this bond upon increasing u to its idealzinc-blende value opens up to the band gap. The aniondisplacement u controls, therefore, the balance betweenthe 1onlc and covalent contributions to the band gap, 1Q-

creasing the former at the expense of the latter (and in-creasing the total band gap) as u increases. The effect of uon the toto/ valence-band charge density is seen in Fig. 13,where the same contours around the Cu —Se bonds are

shaded to highlight the disengagement of the Cu—Se bondupon the formation of the more ionic zinc-blende-likeequal-bond structure. The conduction-band minimum(CBM) responds in an opposite way to the VBM: As u in-creases, its energy rises as more bond charge is placed onthis strongly antibonding Cu-Se state. The structurally in-duced changes in the atomic character of various bandstates can be seen in Fig. 14, depicting the breakdown ofthe total charge in a state

~i ) into its atom (y) centered

contributions Qr [i]. (Notice that these charges are calcu-lated from the Slater-exchange band structure and henceafe not expected to be as accurate as those calculated withCeperley's correlation; we use the former charges only todiscuss changes with u.) We see that the VBM (I 4„state)is weighted more heavily by Cu-Se contributions, and as uincreases from 0.20 to 0.25 the Cu content decreases andthe Se content increases, thereby Ieducing the p-d hybridi-zation. In contrast, the CBM (I ~, state) is weighted moreheav1ly by the catloll having less tightly bound valenceelectrons (In), and hence, constitutes an In-Se state. As uincreases, the In contribution to the CBM rises and the Secontribution decreases, leading to a destabihzation of thisstate and an increase in its energy.

The general trend that we find in a number of materialsis similar: The conduction band is lowered as the bondthat contributes the most to it (involving the cation withless-bound outer electrons, e.g., In s rather than Cu d) isstretched. For example, as u decreases in CuIQSe2, theIn—Se bond making up the CBM is stretched; in the ficti-tious compound GaIQP2 the In—P bond making up theCBM is stretched as u increases, etc. In both cases, theCBM is lowered in energy. On the other hand, the VBMis raising in energy as the bond contributing the most to it(involving the cation with the tighter bound outer elec-trons, e.g. , Cu d rather than In s) is compressed. For ex-ample, as u is decreased in CuInSe2, the Cu—Se bondmaking up the VBM is compressed, raising the energy ofthe VBM. %'e conclude that the structurally induced vari-ations in the band gaps are a consequence of the internalstress exerted by compression and stretching of the bondsthat contribute mostly to the wave functions of the VBMand CBM. The internal stress induced by bond alterna-tion has a significant consequence on the hybridization.In the ideal zinc-blende structure where the atoms are attetrahedral sites, the I 25 representations (e.g., the VBM)cannot mix s character, whereas the I

~ representations(e.g., the CBM) cannot mix p or d character. As theanions are displaced, this is no longer true, and the zinc-blende VBM and CBM states can admix in forming thechalcopyrite band edges. It is this s-p-d hybridization thatleads to the reduction in the band gap upon anion dis-placement. Clearly, higher-energy gaps or indirect gaps(which are already sp hybrides in the zinc-blende struc-ture) will be affected to a lesser extent.

Vfe wish to point to an interesting analogy between thestructural contribution AEs to the chalcopyrite band-gapanomaly and the optical bowing in semiconductor alloys.It is well known [e.g., reviews in Refs. 59(a)—59(c)] thatthe lowest band gaps of semiconductor alloys are usuallysmaller than the concentration- (x-) weighted average ofthe band gaps of the constituent binary semiconductors.

1894 J. E. JAFFE AND ALEX ZUNGER

10

Cu-Se Bond Weakened In-Se Bond Strengthened

1Q0:-

Qc &l4y]

u = 0.20.dcU se = 2.35 A

di„se = 2.68 AVBM

(a)

~se

10-2=

00

tOI10-3

1Q =

100:-

CO1Q-&=I KQc„[f4y] = 0%

u = 0.224.dc„se= 2.42 A

din-seVBM

ISe

C0VILI

10-2=

10-3

10-

QCu(14y] = -6.5% se I

u = 0.25.dcu-se 2.50 Adi„se = 2.50 A

VBM)Se

(c)

10-2=

10-30.0

I

2.0I I I I

4.0 6.0

f1I I I I I I I I

S.Q 1Q.Q Q.O 2.0 4.0 6.0 8.0 1Q.Q

Distance (a.u.)FIG. 12. Effect of varying the anion displacement parameter u on the charge density at the VBM in CuInSe2. (a) u =0.20, (b)

u=0.224 (equilibrium value), and (c) u=0.25 (ideal zinc-blende value). The dashed lines denote, for comparison, the bond-charge

(with respect to Se as origin) for the equal-bond ideal zinc-blende arrangement. EQ denotes the change in the Cu d character relative

to the equilibrium structure. Note that increasing u lowers the charge on the stretched Cu —Se bond (reducing its Cu d character),

and raises the charge on the compressed In—Se bond, weakening and strengthening these two bonds, respectively.

Distance (a.u.)0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.010.011.0 0-0 1.0 2.0 3.0 4.0 5.0 6-0 7.0 8.0 9.010.011.012.0

00 I I I I I I I I I I I I I I I I I I I I I I I

1.0-

2.0-

3.0

CO 4.0-

V S.o-

co 6 0-C

~ 7.0-

8.0-

9.0-

10.0-

FIG. 13. Logarithmically spaced contour plots of the total valence-band (VB) charge density of CuInSe~ for two values of the

anion displacement parameters, indicating the disengagement of the covalent Cu —Se bond charge (shaded areas) upon increasing u toform an equal-bond zinc-blende-like arrangement (u =0.25).

29 THEORY OF THE BAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . . 1895

60—

50—

40—

I I

Guin Se2

Qse[r4, ]

00~ %P

-0.2

-0.4OP

~ -0.6

I I I l

0

~~QlEQ

Qse[r„]~eIIe

Qin[r„]20 - Qc.[r4„]

-0.8

-1.0

Bond Length Mismatch u = R' - R& (a.u.)~OO

00.20

I

0.21

Qin[r4„1

I0.22

I

0.23I

0.24 0.25

This alloy band-gap reduction b,Es'"" is often expressedphenomenologically by the relation b,Es'""=bx(x —1),where the bowing parameter b & 0 reflects an upward con-cave nonlinearity in Es(x). We have pointed out that alarge fraction of b can be explained by considering the factthat such alloys do not have a single (average) cation-anion bond length, but instead, as discovered by Mikkel-son and Boyce, can be thought of as having a local chal-copyrite coordination with a bond alternation a=R„c—Rsc and anion displacement u ——,

' =a/a &0. For anequimolar composition of GaP and InP we calculate anequilibrium value of u =0.278. Using this result, we cannow calculate the structural contribution to ~g"' bycomparing the band gaps of an equal-bond compoundInGaP2 (u = —,) with that of a similar compound but withbond alternation (u =0.278). We find that upon increas-ing u from u =

4 (i.e., stretching the In—P bond), theCBM made up predominantly from In-P contributions (Inhas a smaller binding energy than Ga) is lowered, therebyreducing the band gap. The amount of this band-gaplowering (proportional to the sp mixing of the band edgesinduced by bond alternation) explains most of the ob-served optical bowing in equimolar InP-GaP alloys. 5 '

Anion Displacement (U)FIG. 14. Variation of the band charges [charge due to a given

band enclosed in spheres of atomic radii {Ref. 26)] with theanion displacement u in CuInSeq. The solid lines representcharges at the VBM (the I 4„state), whereas dashed lines denotecharges at the CBM (the I i, state). The difference betweenQc„+Qs,+Qi„and 100% is due to the charge outside atomicspheres. Values are calculated from the Slater-type —exchangeband structure and are hence only qualitative. Notice that uponincreasing u, the Cu charge diminishes at the VBM, whereas theIn charge increases at the CBM, leading to a decrease (increase)in the energies of the VBM (CBM), and hence an increase in theband gap.

FIG. 15. Comparison between the structurally induced reduc-tion in the band gaps of the fictitious InGaP2 chalcopyrite crys-tal and CuInSe2. The solid circles denote the equilibrium valuesof u (in InGaP&) and 2

—u (in CuInSe~). As the bond contribut-

ing mostly to the CBM (In—P and In —Se) is stretched (corre-sponding to an increase of u in InGaP2 and a decrease in uCuInSe2), the CBM is lowered and the band gap is reduced. Thestructurally induced band-gap has the same physical origin inboth classes of compounds, the only difference being the magni-tude of the effect. (Note that this definition of the structural pa-rameter a differs slightly from that given in the text in Sec.VII.)

The common origin of the b,Es effect in chalcopyrites andthe EEs'"~ effect in alloys is illustrated in Fig. 15. InInGaP2, an increase of u corresponds to stretching theIn—P bond, whereas in CuInSe2 an increase in —,

' —u cor-responds to a stretching of the In—Se bond. In both casesthe band gap decreases, primarily due to the lowering ofthe CBM. The difference between the band-gap anomalyin CuInSez and the aHoy bowing in InGaP2 is largelyquantitative: While in CuInSeq the gap decreases withdEs/d( —,

' —u)=18 eV, in InGaPq the derivative is farsmaller, BEs/Bu =1.5 eV, due to the smaller electronega-tivity differences of the constituent atoms. This analogysuggests to us that under careful alloy growth conditionsit may be possible to produce new (metastable) phases ofalloys A„Bi „Chaving a chalcopyrite-like ordering, if thealloyed elements (e.g. , A and B) are sufficiently different.

VI. DISCUSSION OF THE TOTAL BAND GAPANOMALY

We have shown that the band-gap anomaly can beanalyzed in te~s of a chemical factor b,Egh- and astructural factor AEg. The chemical factor consists of ap-d hybridization effect AEs and a cation electronegativi-ty effect EEg where the former is dominant in Cu chal-copyrites and the latter is dominant in Zn and Cd pnic-tides or in alloys of binary semiconductors, e.g., GaAs-InAs. The structural contribution to the band-gap anoma-ly AEg has a smaller contribution due to variations in the

J. E. JAFFE AND ALEX ZUNGER

TABLE III. Decomposition of the observed band-gap anomaly (cf. Table I) into its calculatedstructural component AEg and chemical component AE'"' . The major contribution to the latter isseen to arise from the d-hybridization effect (AEg ) calculated for the two end compounds. For compar-ison, me give the value of the experimental anion displacement (Ref. 28) and the calculated percentageof d character at the top of the valence band (using Ceperley's correlation at the observed value of theanion displacement; cf. Fig. 16).

Material

CuInSeqCuInS2CuGaSe2CuA1Se2CuGaS2CuAlS2

1.31.61.01.41.42.4

0.2240.2140.250.2690.2750.275

0,50.70.0

—0.38—0.50—0.50

gE chem gEobs gEsg g

0.700.91.01.811.872.90 2,98

« (%)

222425.1

27.531.535.2

tetragonal strain (ii+1), and a larger contribution due tobond alternation (a&0 or u& —,

' ). Denoting h=u ——,', wecan treat b. as the small parameter of the problem anddescribe the structural part of the band-gap anomaly byretaining two powers of 5, i.e., bEg —ab, +bb . Clearlya =0 when the two cations 3 and 8 are equal, since thetransformation b,~—4 then carries the lattice into itself.This suggests that the linear structural effect ab, dependsprimarily on the chemical difference between the cations.The electronegativity difference between A and 8 mayhence be thought of as a reasonable measure of the coeffi-cient a. Indeed, we find in our calculation a largestructural derivative (18—21 eV) for Cu chalcopyrites(e.g., 18.2 eV for CuInSe2 with a Cu —In electmnegativitydifference of 0.2, and 20 eV for CuA1S2 with a Cu —Alelectronegativity difference of 0.4) and a far smallerstructural derivative for the fictitious III-III-Vi chalcopy-rites (e.g., 1.5 eV for InGaPz with a small In —Ga elec-tronegativity difference). Note that the nonlinearstructural term bh need not vanish for the zinc-blendestructure (i.e., variations of the two bonds Zn, —S andZn~ —S in a Zn, Zn~S2 structure can change the band gapsymmetrically with 6). In systems with chemically simi-lar cations (i.e., alloys such as GaP-InP), this nonlinearterm (pmportional to the average electronegativity, ratherthan the electronegativity difference, and hence similar inbinary and ternary materials) is likely to be the dominanteffect. We conclude that the band-gap anomaly in Cuchalcopyrites is controlled by b,Es and bE&-ah„and thatthe optical bowing in semiconductor alloys is controlledby AEg and EEL-bh . This is consistent with thediscovery»«& of a remarkably successful linear scaling be-tween the optical bowing parameter of In-Ga-As-P alloysand the electronegativity difference of the substituted ele-ments. We see, however, that the existence of such a scal-ing need not indicate the predominance of disordereffects, ' but rather, it merely suggests thatb E' '"-=kg when structural distortions are small.b,E, in turn, exists already in ordered systems. In ter-nary pnictides it is likely that the band-gap anomaly iscontrolled by both AEg and EEg =a4+bh .

In Table III we have decomposed the observed band-gap anomalies of Cu chalcopyrltes mto a structural partAEg calculated from our u variation of the band gaps, andinto a chemical part defined here as the differenceb,Eg

' =AEs b,Eg. We note the followi—ng. (i) Bond

alternation is responsible for reducing the band gap ofCuInS2 and CuInSe2 relative to their binary analogs, whilein CuA1S2, CuAlSe2, and CuGaS2 bond alternation in-creases the gap relative to the corresponding binary ana-logs, consistent with b, =u ——, being negative for the firstgroup and positive for the second group. (ii) The chemicalcontribution leads uniformly to a reduction in the bandgaps. (iii) For the two "end compounds" for which wehave calculated the p-d hybridization component of thechemical shift, we find that AEg"' and AEg are veryclose, confirming the relative unimportance of CE effectsin these compounds. (iv) The calculated d character of thewave functions at the top of the valence band (denoted ad',we calculate it using the Ceperley correlation at the equi-librium anion displacement since this correlation producesbetter agreement with the position of the d states ob-served in photoemission) is proportional to the chemicalshift (Fig. 16). In contrast to the Shay-Kasper correlationof the total band-gap anomaly b,E& with the empiricallydetermined d character, CuAlS2 is no exception to ourrule. The reduction of the spin-orbit splitting in the ter-nary compounds relative to the binary compounds formsan (admittedly crude) spectroscopic measure (observable)to AEg.

VII. ANALYSIS OF STRUCTURAL PARAMETERSIN TERMS OF ELEMENTAL COORDINATES

Having established the influence of anion displacementson the structural part of the band-gap anomaly in ternarychalcopyrites, we turn to the question of what controls theanion displacements. We give in the first five columns ofTable II a compilation of the measured lattice parametersand anion displacements of ternary chalcopyrites andpnictides (note that u is not measured directly, but ratheris obtained in a crystallographic structural refinement thatis sensitive to the assumed atomic structure factors, aniso-troplc temperature cocfflclents~ dlspcrslon corrections~ andmethod of performing best-fit analysis). Despite the sub-stantial scatter in results for the same materials, signifi-cant material dependence of u is evident fmm Table II. Asuperficial inspection of these trends show that they donot follow any simple chemical rules. %hule a quantum-Iilechailical mlnlmlzatlon of tlM total eilergy E (a, Y/, u )would certainly be desirable for determining the predictedstructural parameters, we first wish to organize this rather

THEORY GF THE BAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . .

l I l

25 30 $5

d Character (/o', ~

FIG. 16. Correlation between the chemical contribution tothe band-gap anomaly and the percentage of d character of thetop of the valence band, calculated from the Ceperley-correlation band structures of six Cu-based chalcopyrites.

chaotic data base in terms of elemental properties to un-ravel the underlying chemical trends. This may be helpfulnot only for obtaln1ng the Q parameters for compoundsfor which they were not measured (e.g., II-IV-Sbl com-pounds), but also for being able to control u artificially bychemical means, and hence design compounds withdesired band gaps.

Considerable attention has been focused in the past onthe systematization of the values of the tetragonal distor-tion parameter g. Since these have only little importancefor controlling the band gaps, we will mention these ideasonly in passing. Folberth and Pfjster ' suggested that1 —II is related to the polarization of the A —C and 8 C—bonds, and should therefore be proportional to the differ-ence (r~/r~)„„„„,—(r~/ra);, „;„where r„and r~ are thecorresponding covalent and ionic radii. The correlationobtained was only qualitative. Phillips worked out aphenomenological fit of 2(1—II) to a linear combinationof elemental dielectric electronegativities which workedfairly well for II-IV-V2 compounds with small distortionsbut not for those with large distortions. Chelikowsky andPhillips later developed a theory of empirical atomic or-bital radii and applied it to the chalcopyrite tetragonal dis-tortion; they found an empirical formula that gave a goodfit for all the II-IV-V2 compounds except CdSnPl andCdSnAs2. The physical significance of the fitting parame-ters remains, however, elusive. A similar study eras re-ported by Shaukat and Singh, (although it can be

~ I —III —VI2~ II- IV-V,~ I —VIII —Vl,

V=1/2-1/4/2 I)ettpt-1 only

0.26

CUAl S

ZnSiP2ZnSiAS2

Znoep,

Agin Se2

AglnS&

88e2

0.24

Se2

0.30I

0.22 0.22 0.26 0.28 0.32Ucalc

FIG 17 Prcdjctjons of anjon djspjaccmcnt parameters II fl'OIn tllc sllllplc tctrajmdrsl 1lljc [Eq. (3)]. Data fol' II from Ha1111 8& III.(Ref. 27) are excluded since they conflict with all other available measurements. Multiple arrows denote different experimental deter-minations (cf. Table II). Poor correlation is obtained for 2 lnCI, AgBCI, and ZnSnCI compounds.

1898 J. E. JAFFE AND ALEX ZUNGER 29

dramatically improved; cf. Ref. 64), whereas Noolandi 'developed a consistent force-field model for tetragonaldistortions. The systematization of the anion displace-ment parameters u has received much less attention,and no work exists, to our knowledge, on the systematiza-tion of the lattice parameters a.

Abrahams and Bernstein have proposed that the bondangles at the B atom in ABC2 chalcopyrites would havethe ideal tetrahedral (tet) values, which would require thatu and g be related by

u =u„,(g) = —,' ——,

'(2g —1)' (3)

The experimental values of g were then used to predict u.The results of this correlation (including more compoundsthan in Ref. 30) are displayed in Fig. 17; the model givesgood agreement for the II-IV-Vz compounds (as noted byAbrahams and Bernstein) when the column-IV atom wasSi or Ge, but much worse results were obtained for II-Sn-Vz compounds (where the tendency towards metallicity ofSn makes it possible to accommodate nontetrahedralbonds around it) and for most I-III-VIz compounds (asseen in our Fig. 17}. Improved agreement was obtained byfitting the deviation from Eq. (3) to a linear combinationof atomic electronegativity with statistically fitted numeri-cal coefficients, ' especially if different coefficients wereused for I-III-VIz and II-IV-Vz compounds; however, thisimprovement comes at the cost of using a purely ad hocnumerical fit with little physical motivation. More re-cently, DeGil derived limiting inequalities forthe chalcopyrite structural parameters, while correlationsamong their temperature derivatives were studied by Bharand Samanta.

Our own approach to the problem is inspired byBragg's classical principle of transferability and conser-vation of elementary bonds in different compounds. Anenormous body of crystallographic studies has been direct-ed at defining elemental radii that add up to produce themeasured bond length R,j.-r;+rz. Surprisingly, however,this idea has received only a little attention for the ternaryABCz compounds. Pfister used a simple scheme basedon atomic radii to estimate u for a few II-IV-Vz com-pounds, while Chemla et al. developed a scheme com-bining atomic radii with an approach similar to that ofAbrahams and Bernstein. ' %orking to lowest order in1 —g and u —~, and assuming a relation equivalent to (3),they used bond-length considerations to predict both uand g, and then fitted the residual error in g to a simplefunction of the atomic electronegativity. As did other au-thors, ' they too found poor agreement for ZnSnVzcompounds, which they argued might not really have thechalcopyrite structure; they did not treat the I-III-VIz ma-terials.

The ternaries not only offer a large data base (Table II}but also provide a more stringent test to the hypothesisthat, to low order, the Rz c bond lengths do not dependon the B atom, etc. Figure 18 shows the correlation be-tween the observed bond lengths of ternary ABCz materi-als and the sum of Pauling's tetrahedral radii. A some-what better correlation can be obtained using theShannon-Prewitt radii; ' however, the latter are only avail-able for a few of the bonds considered in Fig. 18. We see

Roc(a 'rl u) ra —r—c=0, Roc(a g u} rtt —r—c=0 (4}

For ternary chalcopyrites,1/2

Rzc(a, q, u) = u + 1+9' a,' 1/2

R~c(a, g, u)= (u ——, ) + 1+g a.

3.0—

2.9—

2.8—oQ

co

2.6—

'02.5—

0lSQ 2.4—I

2.3I0

2.2

2.02.0 2.1 2.2

o+Cl

OQ

coEQ

UQ n o

40s

Q co

ogEO o+Cl COCV

ogo V

~ A

cicn0 g

cnC ~

NI

M

D „]E

JL

oi IA

p 04

cg ~,~ C" CA~+ieo ~ol - + CLI g4 g+ C f

IV

eQtOCO

Ol

VINQl

I t t I

2.3 2.4 2.5 2.6 2.7I

2.8 2.9

Calculated Bond Lengths (A)FIG. 18. Correlation between the observed bond distances

(vertical axis; the range indicated covers all available data forthe ternaries) and the sum of Pauling's tetrahedral radii (hor-izontal axis). Although the correlation can be improved by us-

ing the Shannon-Prewitt radii (cf. Table V}, the correlationpresented here already suggests that the global structural param-eters of these compounds are dictated by the mismatch in localbonds. The spread of the data shown reAects experimental er-rors as well as the deviations from pairwise additivity of elemen-tal radii.

that whereas the spread in Rtj for fixed i and j is ce~ainlynon-negligible (reflecting not only the experimental error,but also the significance of genuine chemical factors ofnonpairwise additivity), the overall correlation is reason-able, highlighting the success of Bragg s idea (dating backto 1920!) which we refer to as the "conservation oftetrahedral bonds" (CTB). In light of the classical notion,the discovery by Mikkelsen and Boyce of the conserva-tion of the In—As and Ga—As bond lengths in anIn„Chai „As alloy comes as no surprise. The implicationof this principle for the structural parameters a, g, and uof ABCz compounds is that these degrees of freedomwould attain ualues that minimize simultaneously thedifference between the actual anion-cation bond lengthsRzc and Rzc and the sums of elemental radii,

29 THEORY OF THE BAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . . 1899

Since Eqs. (4) can predict only two unknowns, a third con-dition is needed. This suggests two approaches.

P=&gC+&21C =(rg+&C) +(&21+&C) (7)

the solutions to Eqs. (4) are

2 4ap [p2 (2+ 2)~2]1/2

+p) [p2 (2+ )2 2]1/2

4a1 cx=—+4 a2

A. CTS plus g=g,„„trule

In the first approach, denoted "CTB plus g=11,»,rule, " we set the tetragonal distortion parameter equal toits experimental value and solve for a and u. This is basedon the fact that Eqs. (4) and (5) show only a weak depen-dence of the bond lengths on g.

Defining the bond mismatch parameter as

& =~~c &—ac =(&~ +&c) (&a+—rc)2 2— 2 2

and the mean-square bond as

=1 2+ ' 2+ 'u =—+ " 5+ ~ 5'+O(5') .

4 8 32

(9)

Recalling that the zinc-blende (ZB) lattice constant isgiven as aza ———,

'(—", )1/2R&c, we see from Eqs. (9) that the

correction for a —azB is second order in 5 (for 2)=1),whereas the correction for u ——,

' is already first order in

Figure 19 displays the correlation between the calculatedlattice parameter a [in Eq. (8)] and the observed one,whereas Fig. 20 gives a similar comparison for the aniondisplacements u. It is seen that this method gives excel-lent agreement for the lattice constant a and reasonableagreement for the anion displacement u (the latter beingsomewhat obscure by the large scatter in the experimentaldata). The agreement for u is, however, consistently betterthan that obtained by the Abrahams-Bernstein rule [Fig.17 and Eq. (3)].

The structure of Eqs. (8) suggests expansion in terms ofthe small parameter 5—=a/P. This gives

I /22+ '

~ 52+O(54)8

6.4-

~ I-Ill-Vl~ II-IV-Va I-Vll-Vl

CTB Plus"1="lexpt

6.0—oQ

CL

o 5.8-CO

CdSnAs2-AgGaSe2

gAISe

nS2CuTI 82CulnSe2AgGaS2AgAIS2

.4

5.2 5.6 5.8 6.0 6.2I

6.4I

6.6

alcaic (A I

FIG. 19. Correlation between the observed and calculated lattice constants of ternary ABC2 compounds using the "CTB plus77 pt rule. " The calculation is based on the Pauling tetrahedral radii of Table IV plus his radii of 1.40 and 1.23 A for Mg and Fe,

respectively.

5. This highlights the significance of the chemical con-tent of the distribution in anion displacement parameters(Table II) relative to the less sensitive distribution of lat-tice parameters a. It also explains, in alloys of binarysemiconductors (e.g., GaP-InP) which take up a localchalcopyrite arrangement, why the lattice constant closelyobeys Vegard's rule (a=azB ), whereas the two basic bondlengths remain unequal (u ——,

' &0).

B. CTB plus q=q„, rule

An alternative approach to the problem is to add toEqs. (4) a third condition that fixes rl, namely thetetrahedral bond constraint around atom 8 given by Eq.(3). The solution to Eqs. (4) and (3) is then

12m

2P+a —[(2P+a) —18a ]'~

8(P—a) (10)3Q

u = —,' ——,

' [2rlz —1] .

The power-series expansion in 5—=a/P yields

a =v'8P/3[1+5/4 —~ 53+0(5")],52 53

q= 1——„' 5+ — +O(5'),16 16

u = —,+—', 5+ ~5 + —,', 5 +O(5 ) .

This approach hence gives a, u, and rl without recourse toany ternary compound experimental data. The predic-tions for the lattice constant a are compared with the ex-perimental values in Fig. 21, and are seen to be of similarquality to those given by the CTB plus g =q,„~, rule (Fig.19), since the bond lengths depend much more strongly onu than on rj (i.e., the values of u and a that satisfy thebond-length —matching conditions are largely unaffectedby the procedure used to fix g). The predictions for theanion displacements u given by the CTB plus q =etc, ruleare so similar to those given by the CTB plus q =g,„p, rule(Fig. 20) that they are indistinguishable on the plot. Thepredicted g values are depicted in Fig. 22 showing only amediocre correlation with the experimental values (in par-

0.30—I - III - VI2

~ II- IV- Y2~ I - Vill - Vl

MgSiP

AgGe82

CTB Plus @=at t{ofQ=Q „pt}

gGeSe2

~o 0.26

'&X ZnSiP2&ZnSiAs2

~CuAISe2ZnGeAs2

ZnGeP2

0.24

0.22

0.22 0.24 0.26Llcalc

I

0.28 0.30

FIG. 20. Correlation between the observed and calculated anion displacement parameters of ternary ABC2 compounds using the"CTS plus g =g, pt rule. " Very similar predictions for u are produced by the "CTB plus g=q„t rule. " Multiple experimental valuesreflect the large disagreement between different authors' measurements. Values of u obtained by Hahn et aI. (Ref. 27) are excludedas ln Flg. 17.

29 THEORY OF THE BAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . . 1901

AtomAtom

TABLE IV. Comparison of the Shannon-Prewitt (SP) (Ref. 71) radii for the fourfold-coordinatedcations and anions with Pauling s tetrahedral radii (Ref. 70).

O 0 0

SP radius (A) Pauling radius (A) Pauling radius (A)

Cu'+Ag'+

0.6350.92

1.351.52

Zn2+

Cd+1.311.48

Al +Ga'+In +

0.560.580.765

1.261.261.44

Si4+Ge4+Sn4+

1.171.221.40

S2-Se2

Te

1.701.842.07

1.041.141.32

P3-AsSb

1.101.181.36

ticular for the compounds where 8=In, Tl, or Sn, wheretheir tendency towards metallicity enables them to deviatesubstantially from directional tetrahedral bonding aroundthe 8 site). Clearly, the CTB idea is not well suited fordetermining g, which is probably decided by longer-rangeelectrostatic forces (rather than by local bond-strain ef-fects); various effective electronegativity scales might

indeed be more suited for systematizing g (see Ref. 64 inparticular). Since, however, g variations are essentially in-consequential for understanding the band gaps of the ma-terials, we will not dwell on this quantity further.

Our results suggest that the anion displacement parame-ters u and the lattice parameters a are fundamentallydetermined by the internal bond-length mismatch in these

~ I - III - VI2~ II- IV- V2~ I - Vill - Vl

6.2— CTB Plus q=f~„,ZnSnSb2 ~

ulnTe2

6.0—CL

Oe 58—

ulnS2

GeP2

4glnSe2~i '.CdSnAs2AgGaSe~

i ~CuGa Te24g4 ' & iiCuAITe2CdSnP2 ICdGe4s2

ZnSnAs2~ ~CdSIAs2Cu T4Se,i 4glnS,

AgGsS2. ,4'CUInSe2

Mg SIP2% —AgAIS2 CZnSiAs2 —~~', ,' AgFeS2

CuAISe2 ZnGeACuGaSe2

5.2 5.4 5.6l

5.8 6.0O

~calc I,'A,'I

I

6.2I

6.4l

6.6

FIG. 21. Correlation between the observed and calculated lattice parameters of ternary ABC2 chalcopyrites using the "CTB plusg =q„, rule, "where no experimental data are needed as input.

J. E. JAFFE AND ALEX ZUNGER

compounds. Onc coUld ccrtalnly improve quantitativelyon these correlations by replacing Pauling's tetrahedral ra-dii by other radii determined from a larger data base ofcompounds, e.g., the Shannon-Prewitt (SP) radii. ' TableIV compares the two set of radii for the elements forwhich both are available. The SP radii are larger for theanions but smaller (in a similar amount) for the cations,relative to Pauling's radii. However, the structure of theCTB equations indicate that this difference is trivial sincea transformaf 1011 Tg ~Pg +l Rnd Pc~Pc —l 1caves tllcequations invariant. Table V shows that pairwise sums ofthe SP radii are, in most cases, closer to the experimentalbond lcIlgtll than sllIIls of Pauling 1'Rdl1. Table VI con1-pares the predicted parameters a, u, and 1l from the CTBplus q=qtc, rule using both the SP and Pauling's radii.Clearly, the SP radii do somewhat better than Pauling'sradii. We have repeated these calculations with theBragg-Slater and Van Vechten —Phillips radii, withworse results than with the Pauling radii.

VIII. SOME PREDICTIONS

The rules that we have provided can be used to deter-mine fairly precisely the values of a and u and the bandgap for compounds for which these have not been mea-sured or for systems where the experimental scatter is toolarge. We show some of these predictions in Table VII.More importantly to our present work, these rules suggestthat the amon displacernents can be controlled empiricallyby alloying certain elements into given compounds, there-

by controlling the optical gaps, e.g., inclusion of Ga inCuInSe2 is likely to increase the effective u, thereby open-ing the optical gap of the material. Such control of theoptical gap is needed in many applications for optoelectricdevices (e.g. , for CulnSC1 solar-cell applications, it is desir-able to increase somewhat the —1-CV gap of the material,to extract more energy per solar photon, and hence obtaina larger voltage across the solar cell ).

Table VII gives a list of some possible chalcopyrite-structure ternary semiconductors. Most of these have ap-parently never been synthesized; a few have been reportedto exist [e.g., AgT1SC2 (Ref. 8)], but without any crystallo-graphic data bclng glvcn. Wc have predicted thc stfuctul-al parameters (a, u, and q) for these 22 compounds usingthe CTB model with Pauling's tetrahedral covalent radiiplus the condition g=g„,(u); our predictions should beconsidered more reliable for a and u than for g (Ref. 64).We also give rough estimates of the lowest band gaps E&for the not-yet-synthesized compounds, based upon ourtheory of the band-gap anomaly. We expect that thelowest gaps will be direct (I 4„—+I 1, ) for the I-III-VI2compounds and the larger-molecule-weight II-IV-V1 com-pounds, while some of the lighter II-IV-V2's will havepseudodirect gaps; those most likely to be pseudodirect areso indicated in the table.

We have excluded from our list all those compoundsthat seem likely to have negative band gaps, sincetetrahedral coordination becomes unstable as one passesfrom the semiconductor to the semimetal regime. Howev-

1.02

1.00

Cul fl S2Guin Se&

CuTIS&GuV Se2

0.96

0.94

0.92

0.90

0.88 0.90 0.92 0.94 0.96 0.98peale

1.00 1.02 1.04

FIG. 22. Correlation between the observed and calculated tetragonal distortion parameters using the "CTB plus g=gt„rule"(where no data &om the ternary compounds is used as input). Multiple experimental values are included only for the few cases wheredisagreements are substantial. The overa11 correlation is rather mediocre (see discussion in text).

THEORY OF THE BAND-GAP ANOMALY IN ABC2 CHALCOPYRITE. . .

TABLE V. Comparison of the experimental bond lengths (calculated from data of Table II) in ter-nary semiconductors to the sum of the Shannon-Prewitt (SP) (Ref. 71) radii and Pauling s (P) (Ref. 70}radii. Results given in A; an asterisk denotes the approach that produces the better agreement with ex-periments.

Expt.

2.3742.312—2.3802.288—2.334

Compound

CuA182CuGaS2CuInS2

2.335 2.39'

2.4712.387—2.4172.424—2.459

CuA1Se2CuGaSeqCuInSeq

2.2192.257

2.30

2.224—2.2882.235—2.276

2.30

2.400—2.4172.416

2.465—2.5172.505

2.48

2.559—2.5982.609—2.638

2.5302.556—2.605

2.505

AgAlS2AgGaS2AgInS2

cr, the chSlcoppAtc stDlctUI'c sh001d bc $1'cssoI18blc pl'c-diction for the remaining compounds in view of the ex-istence of zinc-blende-structure compounds containingthese same elements in the same or in different combina-tions. An example is MgoeP2, which has been syn-thesized ' in a cation-disordered zinc-blende phase with

a=5.652 A, in good agreement with our prediction; itshould be possible to obtain this compound as a chalcopy-rite by extremely slow cooling from near its melting point.As another example, one can argue for the existence ofHg-containing chalcopyrites from the existence of a zinc-blende phase of HgSe. On the other hand, ternary nitrides

flSPCompound Expt. Expt.

TABLE VI. Comparison of predicted a, u, and q parameters from the "CTB plus g=g„, rule, "using the Shannon-Prewitt (SP)radii and the Pauling (P) radii. An asterisk denotes the one that produces better agreement with experiment.

Q u (A)SP I' Expt. SP P

0.27SO0.26900.2S39—0.27500.2431—0.25000.2140—0.22950.2240—0.2350

0.2641'0.2636*0.2641

0.2583' 0.2636

0.2291' 0.2359

0.2303 0.2364

5.33405.60205.3474—5.3565.5963—5.61405.5228—5.52305.7730—5.7840

5.46395.69505.4639

5.6827 5.6950

5.7833 5.7988

0.97900.97700.9740—0.97940.9800—0.98311.0025—1.00791.0005—1.0048

0.97620.97750.9825

0.9835'

1.0427

0.97210.97320.9721

0.9732

1.0287'

1.0275

J. E. JAFFE AND ALEX ZUNGER

Compound

ZnSiS12ZnGeSb2CdSiSb2CdGeSb2

6.0776.1116.3446.383

0.2700.2630.2910.285

0.9610.9750.9210.933

0.90.50.80.2

TABLE VII. Predicted structural parameters and estimatedband gaps for 22 possible chalcopyrite-structure semiconductors.The symbol PD in the band-gap column indicates compoundsmost likely to have pseudodirect los@est gaps.

a (A) Q '9 Eg (eV)

tion of Eqs. (9) and (11) shows that changes in the param-eter u are dominated by changes in the term that is linearin the small ratio 5, and it is easily sho~n that 5 dependsIDuch morc strongly on thc cation rad11 Pg and P'g than onthe anion radius rc. %e take these radii to haveconcentration-dependent effective values in the alloy, anddef1ne

Q =4R~cR~c/P

MgGePpMgSnP2MgSiAs2

MgGeAs2MgSnAs2

5.6565.7745.8045.8415.958

0.2770.2500.2840.2760,250

0.9471.0000.9350.9491.000

2.1 (PD)1.82.0 (PI3)1.61.2

85—=(~ac —B~c)Q =(ra —r~)QBrc

6e22 1

6.2586.374

0.2810.2750.250

HgSiP2HgGeP2HgSnP2HgSiAs2HgGeAs2

5.7405.7805.9095.9265.966

0.2960.2880.2620.2940.287

6.2995.8826.1136.529

0.2330.2570.2570.257

0.313

'Experimental value, Ref. 8, p. 336.

0.9390.9521.000

0.9130.9270.9770.9160.929

1.0340.9860.9860.987

0.883

1.40.90.6

1.61.20.80.70.2

0.91.10.7,0.72'0.6

8.2 (PD)

The difference rz rz is —about 8—30 times smaller in ab-solute magnitude than either bond length; thus, the anionradius has relatively little influence on u. (This can alsobe seen by examining our predicted values of u for seriessuch as ABS2, ABSez, and ABTe2. ) Now we have seenthat u controls the structural part b,Eg of the band-gapanomaly, and this sUggcsts that alloying thc anions w111

not substantially affect b,Eg. On the other hand, since thecation radii strongly affect u and hence AEg, alloyingwith either the 3 or 8 atom should have a strong structur-al effect on the band gap, especially if the size mismatchbetween alloyed (same-column) elements is large. If eitherthe concentration dependence u (x) of the anion positionsor the variation Eg(u) of the gap is nonlinear, therewill be a structurally induced bowing (Eg[u (x)]~xEs[u (1)]+(1 x)Eg[u (0)—] when 0&x & 1) that is notpresent when only the anions are alloyed.

like ZnGCN2 are known to have a structure related towurtzite, which is not surprising since their binary analogs(e.g., GaN) have the wurtzite structure. But thetetrahedrally coordinated form of boron nitride (BN) ap-pears also in the zinc-blcnde structure, which suggests thatBCCN2 may exist in a chalcopyrite structure. This ternaryanalog of diamond might have properties approximatingthose of diamond itself.

The connection we have demonstrated between anionsublattice distortion and the band-gap anomaly may helpto answer some questions about alloys of chalcopyritecompounds. For example, it is has been show that whenone has solid solutions of two different anions in thechalcopyrite lattice [e.g., CulnSzSez~&

~(Refs. 76 and 77),

CuGaSq„Se2~ ~ „~ (Ref. 78), and ZnSiP2„As2~ ~ „~ (Ref.79)], then the band gap is always almost exactly linear inthe conlposition parameter x, i.e., there is little or no opti-cal bowing in these alloys. On the other hand, when thereis alloying on the B cation [as in CuGa„In, „Se„(Ref.80)] or on the A cation [as in Ag„Cu, „InSe2 (Ref. 77)],then substantial optical bowing is usually (though not al-ways ') found. We suggest that the difference betweenthese two cases is partly duc to the behavior of the aniondlsplaceIDcnt para111etcr with alloying, whc1 c wc assumethat the structure of the alloy is approximately describedby a concentration-dependent u(x). Specifically, inspec-

IX. SUMMARY

We have proposed a theory for the band-gap anomaly,structural deformations, and the relationship betweenthem for ternary ABC2 chalcopyrites. We find that theband-gap anomaly can be analyzed in terms of a chemicalfactor EEg"' and a structural factor bEg, where

AE, =DE,'"' +DE,'.The chemical contribution to the band-gap anomaly con-sists of a d-p hybridization part b,E and a cation elec-

CEtroncgatIV&ty part ~Eg

gE chem gEd+ gE CEg g g

AEs reflects the raising of the VBM due to the levelrepulsion between the cation d orbitals and the anion p or-bitals, both forming I ~5-like representatives. This effectis small in II-VI and III-V compounds where thecation d orbitals (Zn, Cd, Ga, and In) are considerablydeeper than the anion p orbitals, but is significant in Cu-based chalcopyri. tes Rnd in Cu-based binary compounds(e.g. , CuCl), and also, to a lesser extent, in the correspond-ing Ag compounds. The p-d hybridization effect is alsorcspons1ble for thc occUrrcIlce of RnoITialous1$ deep CU ac-ceptor states in II-VI compounds, Ielative to the shallowacceptor states in III-V and column-IV semiconductors.The CE factor b,Eg reflects the ability of charge to

THEORY OF THE BAND-GAP ANOMALY IN ABC, CHALCOP YRITE. . .

separate on the two cation sublattices in ternary- corn-pounds. It is a significant factor for III-V alloys (e.g. ,GaP-InP) but small in ternary chalcopyrites relative to theother factors.

The structural contribution EEg to the band-gap anom-aly is controlled by bond alternation (i.e., R~c&RIIc, oru —

4 &0) and has only a small contribution from tetrago-Ilal dIstortIons (7/=c/20+1). Wc sl1ow tllat boIld Rltc111R-

tion can have either a positive or a negative contributionto EES and that a large fraction of the optical bowingphenomena in conventional binary alloys (e.g., InP-GaP)Is R coIlscqllcIIcc of bond RltcAlatlon (I.c., tllc breakdownof the virtual-crystal approximation). We have analyzedin detail the rearrangements in charge distribution due tobond alternation and identified the charge-polarization ef-fects associated with it.

We find that DES has a term linear in the anion distor-tion u —

~ with a linear coefficient that depends on theelectronegativity difference between the catioris (i.e., onethat vanishes for zinc-blende compounds) and a termquadratic in u —

4 with a coefficient that depends on theaverage CE (i.e., similar for ternary and the analogous

binary compounds). We suggest that most of b,ES forchalcopyrites arises from the linear term, whereas most ofthe structurally induced optical bowing in alloys of binarysemiconductors arises from the quadratic term.

FlnaHy, we show that the obselved dlstrlbutlon of lat-tice parameters Q and anion displacement parameters Q lnternary ABC& semiconductors can be systematizedthrough a semiclassical model of conservation oftetrahedral bonds. This model also provides predictionsfor a, u, and Es for compounds that have not been syn-thesized yet. The present model of the band-gap anomalyand the structure anomalies i.n ternary semiconductors arebeing used in our laboratory for guiding the design of newternary compounds and alloys with desired band gaps andlattice constants for photovoltaic applications.

ACKNO% LEDGMENTS

This work was supported in part by the Division of Ma-terials Science, Office of Energy Research, U.S. Depart-ment of Energy, under Grant No. DE-AC02-77-CH00178.

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