Electronic structure and density of states of the random ...7948 RITA MAGRI, SVERRE FRQYEN, AND ALEX ZUNGER tuations in different properties survive averaging. The various theoretical

PHYSICAL REVIEW B VOLUME 44, NUMBER 15 15 OCTOBER 1991-I

Electronic structure and density of states of the random Ala 5Gao 5As, GaAso ~PO 5,and Gao 5lno 5As semiconductor alloys

Rita Magri, Sverre Froyen, and Alex ZungerSolar Energy Research Institute, Golden, Colorado 80401

(Received 8 February 1991)

The electronic density of states (DOS), charge densities, equilibrium bond lengths, and optical bowingof the direct band gaps are calculated for three perfectly random semiconductor alloys within the first-principles pseudopotential method using the concept of "special quasirandom structures" (SQS's). TheSQS's are periodic structures with moderately large unit cells whose sites are occupied by atoms in a waydesigned to reproduce the structural features of the infinite, perfectly random substitutional alloys. Inavoiding averaging over atoms (as in the virtual-crystal approximation) or over atomic environments (asin the site-coherent-potential approximation), this approach is capable of revealing the multisite natureof chemical disorder, as well as atomic-relaxation effects. We show how the existence of different localenvironments about chemically identical sites leads to splittings and fine structures in the density ofstates, and how atomic relaxations are induced by such nonsymmetric environments and lead tosignificant modifications in these DOS features. The calculated alloy bond lengths and optical-bowingcoefficients are found to be in good agreement with experiment. Relaxation-induced splittings in theDOS are offered as predictions for future photoemission studies.

I. INTRODUCTION

Isovalent zinc-blende AC and BC semiconductors tendto form stable and continuous substitutional solid solu-tions A& „B C when grown above the miscibility gaptemperature. ' X-ray diffraction, ' nuclear-magnetic-resonance chemical-shift measurements, Raman scatter-ing, and extended x-ray-absorption fine-structure(EXAFS) experiments show that while low-temperature vapor-phase growth often leads to substan-tial deviations from randomness (e.g., to short-range clus-tering ' or even to long-range ordering' ), liquid-phase ormelt-grown alloys tend to organize microscopically innear-random atomic arrangements. ' There, the occu-pations of the "mixed sublattice" (A and B) follow closelyBernoulli's random (R) statistics. For example, thedifferent C-centered nearest-neighbor atomic tetrahedraA 4 A 3B, A 2B2, AB 3, and B4 appear at alloy concentra-tion x in the relative ratios given by '

Pz"'(x)=(„)x"(1—x) ", where n is the number of 3atoms in the A„B4 „cluster. Similarly, the twelveA B&2 arrangements of the next-nearest neighborsabout A occur in the relative proportionsPz' '=(' )x (1—x)' . Indeed, the existence of suchdistributions of "local environments" about sites is thehallmark of disordered alloys, distinguishing them fromordered intersemiconductor compounds (e.g., short-period superlattices, where only a small number of clustertypes exist), or from the pure binary AC and BC constitu-ents (where only A4 or B~ clusters occur). These non-periodic distributions of local environments lead tochanges in the physical properties (alloy band gaps,effective masses, density of states, bond lengths) and con-sequently to the technologically attractive possibility oftuning these properties with x to meet desired

specifications.Understanding the fashion in which such nonperiodic

cluster distributions control optical and structural alloyproperties has been the focus of much of the theoreticalresearch both of metal"' and of semiconductor' alloys.As translational invariance is lost, Bloch-periodic func-tions such as band structures are replaced by statisticalconstructs including density of states (DOS), local DOS(LDOS), spectral functions, and average bond lengths.We know that at a composition x there areN!/(xN)![(1 —x)N]! ways of distributing the 2 and Batoms on the N available lattice sites; each suchconfiguration is distinguished from others by havingdifferent local environments. Even though the occupa-tions of the diff'erent sites in a configuration may be (e.g. ,by construction) perfectly random, the properties of thealready occupied sites need not follow random statistics.For example, a C atom coordinated locally by four Aatoms could have a different charge transfer or local bondlengths than a C atom coordinated locally by, say four Batoms. ' The fact that such chemically identical atomsmay, in general, have different local environments (i.e., bestructurally inequivalent), does not of course affect theproperties that depend only on the global alloy composi-tion (e.g., the molecular weight of A, „B C). Yet, phys-ical properties that involve local coupling between atoms(e.g. , charge transfer, bond relaxation) may exhibit finite(nonrandom) correlations even in nominally (occupation-wise) random alloys. For example, the configurationalaverage of the product of charges on two sites (Q;Qi)(which determines electrostatic energies) is not'(Q; ) (Q. ) =0, even though the occupation variables of aperfectly random alloy are uncorrelated, i.e.,(S;SJ.) = (SJ.) (S; ). Since macroscopic observables cor-respond to averages over all configurations, the centralquestion here is the extent to which environmental fluc-

1991 The American Physical Society

7948 RITA MAGRI, SVERRE FRQYEN, AND ALEX ZUNGER

tuations in different properties survive averaging. Thevarious theoretical approaches applied previously to de-scribe the electronic structure of random alloys differ intheir treatment of this issue.

The virtual-crystal approximation' (VCA) averagesthe potentials of all atoms in the alloy, hence, all Auctua-tions are lost. The failure of the VCA in accounting fordistinct A-like or 8-like spectral features is well docu-mented for both semiconductor' ' and metal" alloys.The site-coherent-potential approximation' (SCPA) re-tains the separate identifies of the 3 and 8 sublattices butaverage over the environments of each site. Since it as-sumes that random occupations of sites also implies thatthe physical properties of the sites are uncorrelated (e.g.,that (Q, Q ) = (Q; ) (g ) =0), all C atoms (and separate-ly, all 2 or B atoms) are assumed to scatter electronsequivalently. While this approximation captures some"chemical disorder" effects due to the different potentialson A and B (see applications of the CPA in Refs. 18—20to 3

&B„Csemiconductor alloys), it misses distinct en-

vironmental effects' ' ' associated with the existence of adifferent distribution of A (and separately B) atoms. Themolecular CPA (Refs. 19 and 22) (MCPA) retains alonger correlation length over which structural andchemical characteristics are retained. This often leads toadditional structure' in the DOS. The supercell ap-proach considers a crystallographic unit cell whosesites are occupied by atoms in a manner aimed at repro-ducing approximately a random alloy. The electronicstructure is then solved (either with periodic or withfree-cluster ' boundary conditions) using any of theavailable electronic-structure methods. Correlation be-tween the properties of different sites are permitted (andare limited in extent by the supercell size). A distributionof various local environments is naturally included. Thisresolves even more DOS features than the MCPA; exam-ples include large (X—10 atoms/cell) randomly occu-pied supercells treated by tight-binding recursionmethods ' and the 16-atom chalcopyrite model ofzinc-blende alloys treated by first-principles density-functional methods. Even more detailed representa-tions of alloy environmental effects were obtained by per-forming a set of different supercell calculations, eachrepresenting a distinct (repeated) cluster, and superposingthose according to the statistical probabilities in whichthe underlying clusters appear in an infinite random al-loy. Such approaches have been applied to study spin-orbit splitting and band-gap bowing jn II-VI and III-Valloys.

The computational complexity of the SCPA, '

MCPA, ' ' and large-supercell approaches has lim-ited their application for semiconductor alloys tosimplified electronic models, mostly empirical tight-binding methods. On the other hand, while small super-cell approaches were applied in the context of more so-phisticated first-principles electronic structuremethods ' ' ' ' ' ' the small number of atomsused to model a random alloy (and their high-symmetryarrangements) could misrepresent the correct statisticalstructure.

We have recently developed ' ' a systematic method

of constructing small supercells whose sites are occupiedby atoms in a manner designed to reproduce approxi-mately the structure of an infinite random alloy. Theatomic structure of an infinite, perfectly random alloywith a given Bravais lattice (e.g. , fcc) and composition xis characterized by its many-body correlation functions '

( IIk (x) ). We consider "clusters" consisting of katoms separated by up to mth-neighbor distances. Wethen search for a single ordered, X-atom supercell whosecorrelation functions IIk (X) have minimal deviationsfrom (IIk (x)) up to a given order in (k, m). Interest-ingly, such "special quasirandom structures" (SQS's) withonly eight 3 and 8 atoms per cell already reproducerather accurately "' ' the first few (physically mostsignificant) random correlation functions( IIk ) =(2x —1)". Here, the constraint of matching thefirst few correlation functions of the infinite, perfectlyrandom alloy requires that some chemically identical sitesbe crystallographically inequivalent; any other occupa-tion scheme in a small supercell results, by necessity, in aworse description of the real random alloy. In this paperwe use a self-consistent pseudopotential description of theproperties of Alo 5Gao 5As, Ino 5Gao 5As, and GaPO ~Aso 5

random alloys modeled by the SQS8. The general prop-erties of such SQS's (Refs. 30 and 31) and their applica-tions to II-VI semiconductor alloys and sometransition-metal alloys were described previously.

II. METHOD OF CALCULATION

The SQS8 is a (AC), /(BC)2/( AC)3/(BC)2 superlatticein the [113] direction with four 3, four B, and eight Catoms per unit cell. Detailed information about the lat-tice vectors and the atomic positions inside the unit cellfor this structure were given elsewhere. ' The SQS8has two variants, obtained by switching the A and 8atoms (denoted as SQS8,and SQS8b). For completely re-laxed structures we find that the formation enthalphies ofthese variants differ by as little as 1 meV/atom pair whilethe minimum direct band gaps at I differ by only 0.01eV. As noted above, the atomic sites within the SQS unitcell are structurally inequivalent. Considering first thenearest neighbors to the common atom C, we see differentcoordinating atoms: in SQS8, we have the C-centeredclusters A4, 338, A282, and 383 with the relative fre-quency 1:1:3:3,whereas in the complementary SQS8b wehave the clusters B4, AB3, A2B2, and A3B (with thesame frequencies). Notice therefore that the occurrenceof the A„B& „clusters with n =4, 3, 2, 1, and 0 aroundthe C atoms follows the random probability at composi-tion x =

—,' (ratios of 1:4:6:4:1)if we consider SQS8, and

SQS8b together. Considering next second neighbors, theA and the 8 atoms again have different environments.For example, in SQS8, there are two inequivalent Batoms: 8' ', surrounded in the second shell by five 3 andseven 8 atoms and 8' ', surrounded by seven A and four8 atoms. Again the average number of atoms of oppositetype to that at the origin follows closely the average valuein the random alloy.

We studied the electronic structure of the SQS usingthe first-principles, self-consistent pseudopotential ap-

ELECTRONIC STRUCTURE AND DENSITY OF STATES OF. . . 7949

proach with nonlocal pseudopotentials constructed bythe approach of Kerker. Exchange and correlationeffects were treated within the local-density approxima-tions using the Ceperley and Alder potential asparametrized by Perdew and Zunger. The wave func-tions were expanded in a basis set with about 150 planewaves per atom (kinetic energy cutoff of 15 Ry).Brillouin-zone integrations were performed by Fourierquadrature using a set of 21 k points equivalent to 29 fccspecial points. For the DOS calculation the eigenvaluesare obtained directly at a set of 73 k points (equivalent to85 zinc-blende k points), including high-symmetry points.We have checked the convergence of the results as afunction of the number of k points and find that a largerset consisting of 146 k points gives the same DOS peakpositions to within 0.05 eV. The eigenvalues were thenlinearly interpolated on a denser k-point mesh using thetetrahedron interpolation scheme. For the local DOScalculations each energy was weighted with a weightequal to the projection of the corresponding wave func-tion in a Wigner-Seitz sphere around an atom. TheWigner-Seitz radii depend on the equilibrium volume andare 3.23, 3.29, and 3.41 a.u. for GaAso ~PO 5,Alo 5Gao ~As, and Gao 5Ino 5As, respectively. SinceLDOS results depend on these radii, they are quoted inwhat follows only in "arbitrary units. "

The equihbrium atomic positions for the lattice-mismatched systems Gao 5Ino 5As and GaAs05P05 weredetermined by minimizing the total energy with respectto the cell-internal structural parameters, retaining thecubic unit cell dimensions. The calculated equilibriumlattice constants were within 0.2%%uo of the average of thecalculated values for the two zinc-blende constituents(Vegard's rule). A fully relaxed configuration was ob-tained through an iterative process using first-principlescalculated forces and an approximate force constant ma-trix based on Keating's valence force-field model. Thefinal (zero-force) geometry does not depend on the forceconstants used; only the convergence rate of the iterative

relaxation is. Because of the structural inequivalence ofatoms of the same chemical identity, the relaxed SQS8manifests a distribution of bond lengths for each (e.g. ,A -C or B-C) atom pair. Table I compares the calculatedaverage nearest-neighbor bond lengths d in SQS8 withthe virtual-lattice average dvL~=xd„c+(1 —x)s c andwith the values d in the binary constituents. The alloyvalues d deviate substantially from the virtual-latticeaverage values and are closer to the binary equilibriumbond lengths. As seen in Table I, the magnitudes of thesedeviations agree well with those found in EXAFS mea-surements on these systems. '

III. GENERIC FEATURES IN THE DOSOF III-V ZINC-BLENDE COMPOUNDS

We start our discussion of the DOS by establishing thebasic features of the DOS of a pure III-V zinc-blendecompound; this will serve below as a benchmark for as-sessing alloy effects on the states of a pure compound.An analogous discussion for II-VI compounds was givenin Ref. 32.

We use A1As as an example for III-V compounds. Fig-ure 1 gives the calculated total DOS of A1As. It exhibitsthree distinct valence-band peaks denoted P1, P3, and P4and two conduction-band peaks denoted P5 and P6. Fornotational compatibility with Ref. 32 we have omittedP2: this state corresponds to the metal d states; in II-VIsemiconductors they are positioned energetically be-tween P1 and P3, whereas in III-V semiconductors theylie deeper than ' P1 and hence will not be consideredhere.

Figure 2 depicts the electronic charge densities in ener-

gy regions corresponding to these peaks, while Fig. 3gives the site and angular momentum decomposedLDOS. From these data we can characterize the locali-zation and hybridization of each generic DOS peak asfollows.

(i) P 1 is an isolated band around E„11eV consi-sting of

TABLE I. Calculated bond lengths for the binary zinc-blende constituents (db;„„„), the x =~

virtual-lattice average (VLA) dv« =2

d & z+ 2 d& z, and the average (with fluctuation denoted as +)over the distribution of bond 1engths in SQS8, and SQSgq (ds&ss). The quantity d —d gives the pre-dicted change in bond length in the x =

2 alloy relative to the pure constituents. Since the local density

method used here makes a small error in db;„„„,we calculate the predicted alloy bond length by sub-tracting our LDA-calculated d —d value from the measured db;„„„(2.448, 2.623, and 2.360 A in CxaAs,

InAs, and GaP, respectively). This gives our predicted alloy bond lengths dp„d that are compared tothe EXAFS-measured values given in the last column.

Bond

Ga—AsIn—As

Cxa—AsGa—P

d;„„„(A)

2.4332.616

2.4332.341

0

dvz. A {A)

2.5252.525

2.3872.387

dsgs, (A)

Cheap 5Inp 5AS2.449+0.0152.595+0.011

GaAsp 5Pp 5

2.419+0.0062.354+0.006

d' —d

—0.0160.021

0.014—0.013

pred

2.4642.602

2.4342.373

Expt.

2.46'2.608b

2.43'2.38'

'Reference 9(b).Reference 7.

'Reference 9(a).

7950 RITA MAGRI, SVERRE FROYEN, AND ALEX ZUNGER

I Ii

I I I Ii

I I I[

I I I I(

I I

~ 25—E

CIC4

lXe 20

15

M

Qw)0—~~fh

Ch

eO

P1

P3

P4

Al AsTotal DOS

P6

P5

I l I I I I I I I I I

-10 -5 0Energy (eV)

FIG. 1. Total density of states of AlAs at the cubic latticeconstant a =5.619 A, labeling the various "generic" peaks dis-cussed in the text (Sec. III).

an atom-centered anion s valence state. It contains asmall admixture of states with s, p, and d character aboutthe cation sites.

(ii) P3 is an sp -like bonding valence state evolvinglargely from cation s and anion p orbitals. It is spatiallylocalized on the anion-cation bond with a peak chargedensity polarized towards the anion site.

(iii) P4 is a spatially extended anion p, d and cation p, dbonding valence state. Its peak charge density is near theanion site along the anion-cation [111]bond direction.

(iv) P5 is the antibonding of sp -like counterpart of theP3 state. It is composed from cation s, p, d and anion s,

F ~IG. 2. Contour plots of the electronic charge densities inthe (110) plane for the main peaks in the DOS of AlAs (Fig. 1).(a) Total valence pseudo-charge-density, (b) P1, (c) P3, (d) P4,(e) P5, and (f) P6. The peaks P1—P4 represent valence bandswhereas P5 and P6 are conduction bands. The contour steps in(a)—(f) are, respectively, 2, 1, 0.4, 1, 0.5, and 0.5 (in units of elec-trons per unit cell volume).

p, d orbitals and is spatially localized on the "antibond"[1 1 1] direction with a node along the bond.

(v) P6 is a spatially extended antibonding conductionstate made of cation p, d and anion p, d orbitals. Like P5but unlike P3, its charge density lobes point away fromthe nearest-neighbor bonds.

The DOS features discussed above for AlAs appearsimilarly in all other III-V semiconductors. Table II

TABLE II. Peak positions (indicated by arrows in Fi . 1) in thGP f d'F

ig. ) in e density of states of the binary compounds GaAs, AlAs InAsa, or dN'erent values of the lattice arameters. Wep e use the calculated lattice constants for the binary constituents and for the cor-

responding alloys. Energies are given in eV relative to the valence-band maximum

SystemP1 P3

Peak energies (eV)P4 P5 P6

GaAs, a =ac;,A,GaAs, a =a(- i A

GaAs, a =aG,A,p

—10.4—10.2—10.6

—6.5—6.1—6.8

—3.7—3.3—40

—2.6—2.3—2.8

2.52.32.6

4.5444.5

5.65.45.6

7.67.27.8

AlAs, a =aA, A, =a~,A,—10.0 —5.1 —3.3 —2.2 3.3 4.5 5.6

IQAs, a =ay„A,InAs, a =a&,&„A,

—10.1—10.5

—5.6—6.1

—3.1—3.7

—2.2—2.6

2.42.6

4.34.4

5.35.5

6.97.4

GaP, a =ac.,p

GaP, a =a~ A p

—9.6—9.4

—6.4—6.2

—3.8—3.5

—2.6—2.5

2.92.7

4.54.5

5.75.5

7.87.6

7951RE A» DEN SITY OF STATES OFCTR~NIC STRUC

As Gap, and InAs for aivest eph llo di-of unit cell volumes pertinen onumber o uni

he s stematic trends in a

p

t e lat'

h the pertinent atom'h trends corret e late witA endixergies (see the ppe

ETHOD OF ANALYSISIV. MOF ALLOY-INDUCED D

re of the DOS features int tdh d 'bpure I po

to analyze alloyingcturalh 6' t of t t

met o ooto discern t e e cSince we wis

'al erturbat1ons,

+ t:S h 1 1

se of the1tbt thoofa A1 xconstituents (usua yAC and BC cons i

ted average denoted V), in s'

onentconcen tration-weig t

s the volume o ef th smaller compmismatched alloys t e v

lar er comp o ent cpntractshile that of the a g14 then couple

exPands w 1 ermation Potentia sh d'ostatic deforma1

as shpwn 1nsuW see that the boo lng

mailer bindingFi . 4. We see

~

u on volume expume contraction

ansion into sma4

pon4(d)] whereas volumenergies [Fig .; s 4(a) and

states in oto larger bindi gp

energies Flg'moves bond g l, the &ritibonding4( )

n p5 and4O ) and 4(c)]. Colume expansion Ã'g 'os a .

e cpntrac 1

t tes drop in energyt'on increases t4(d)] hjie volum

g t are weil u nder-

fprmationS.d shifts for the a py

he next steps studied here.

calcu-eds 1

ll pirtuaI-crystafrom the cons

ects.' In t e nextituentswe construct a vir

d at the alloy s ca c-irtual latticecu-~cthat are already P P

t t o) A. direct1 ted mPlar volumeof the bands Pf a vpseudpPoten

ith concentratipn-w g 'th

-bl nde structure w'

s the DO»nzinc- en

onstituents givesare

of the deformed co " All bond lengthstials p

1 roximation. "e t at their

" irtual-crysta apPb nd angles are eP4 and all bon

ation the zinc-vajl.ues. Sinceh DOS has precisely

ende com-blende symmetry

in the pure zinc-plogical features as'

deducible by com-same opo

gC-&C band splittingf gC and gC

ppnents.anng e

hence e imine correspond g

'nated. » this Pnds in»g 4der" (resulting fr

compounh "structural disor errpximatipn. , bpth

I ' I ' I ' I ' I ' I I16:.:(a) Al, s

12:-

I ' I ' iI ' I ' ii '1

' I

(e) As, s

8 .--

lAI~C

II40

o~~N

I"U

(5Vo

o: W, , l.2 . i ' I '

I ' i

'- (b) AI, pi II ' I

I I0:, , i

20- '

:(d)Al, tot15.10 .-

I

6— (0) AI, d

I ' i II

: '-(f) As, p

I I ' I ' I

(g) As, d

I ' I ' I ' I::(h) As, tot

~ .

I ' i 'II

20 .—thEo 15CO

C4~ &0-K

itA

ItD ii5-0$ i

th Ii

tO

: (c)20-

th~ &5-tDO

o 10I-

II

iIIIIIiII

II

0

GaAsa (Ga2AsP)

——-- a (GaAs)

GaAsa (GalnAs2)

---- a (GaAs)IiIiIiIiIiIiIiIiiiI,I

I

I

I

II

I

IiI!I

I

I

iiII

I

I

I

IIIIII

I ~ il

i ' t ' I II 'I

'I '

I

GaPa (Ga2AsP)

———-a (GaP)

gl

IIIII '

i I

i iI

I

iI

I

iI

I

InAsa (GalnAs2)

---- a (InAs)iI

iIi

II II

IlI i

~Iii i I

/I

4 8-12 -8

Energy (eV)

4 80

0-12 -8 -4 0

i

4 8 12 -8 -4Energy (eV)

0 4 8

l d local densityan ular momeentum reso vein Sec. IIIis information is used inc Aof states in cu

'

't plots of Fig. ochar e density paDOSb enericor itao l nature of each g

ifts in the total densi-d h drostatic shifts inlines

give the DO aar semiconductors, w

the corresponding ad o bre iven in

0 evolume. e

h de dilation, on

ing states sucthe valence-band mThe zero of energy

'is taken at


dissimilar bond lengths and angles) and "chemical disor-der" are excluded by construction. Note that we includethe VCA step only as a benchmark against which otherresults can be compared, not as a viable alloy theory.

(iii) Chemical dis-order sects: We next calculate theDOS in the SQS8 model at the volume V, keeping howev-er all bond lengths and angles equal to those in the virtuallattice ("unrelaxed SQS"). Like in the VCA calculationsof step (ii), structural disorder is excluded, but unlike it,chemical disorder effects are included: distinct A and Batoms are retained with no averaging. Note that unlikethe SCPA, this model retains the chemical fluctuationsarising from the existence of different A and, separately,different B sites. A recent test compared the results ofthis model in the context of a tight-binding description tothose obtained in the SCPA (where all A atoms are aver-aged and so are all B atoms) for a case where structuraldisorder is negligible (the lattice-matched Alo 5Gao 5Assystem). It was found that the SCPA captures well chem-ical disorder effects in the spectral ranges where the alloyperturbation is reasonably small (in the I „and X„re-gions) but that noticeable differences exist when the per-turbation is larger (e.g., in the X3, spectral region). Inthe later cases, a multisite description of chemical disor-der effects (underlying the SQS) is important.

(iv) Structural disorder -effects: In the last step we con-trast the DOS of the unrelaxed SQS model of step (iii)with that obtained from the SQS whose atomic positionsare relaxed to obtain a minimum in the total energy ofthe cubicly shaped cell. The nearest-neighbor A -C andB-C bond lengths (Table I) are no longer equal, as insteps (i)—(iii). Relative to the virtual lattice average, theshorter of the two (AC and BC) bonds become (slightly)shorter, whereas the longer of the two becomes longer(Table I). Such relaxations couple to the electronic statesthrough phononlike deformation potentials, causing ad-ditional shifts and splittings.

Following the analysis above, we will calculate the to-tal DOS of (i) the binary constituents at their equilibriumvolumes as well as at the alloy volume, (ii) the VCA den-sity of states, (iii) the unrelaxed SQS density of states and,for lattice-mismatched alloys, (iv) the fully relaxed SQSdensity of states. This will provide a description ofvolume deformation as well as chemical and positionaldisorder effects on the global density of states.

In addition to the total DOS, we are interested to ex-plore how "environmental effects" lead to distinctfeatures in the electronic structure. To this end we willproject the DOS onto distinct clusters that exist in theSQS. Each "common atom" C in A, „B„Cis surround-ed, in successive shells by the coordination spheres( A 4 „B„),(C,2), ( A, 2 B ), for the first, second, andthird neighbors, respectively. To delineate the effects ofthe first coordination sphere, we calculate the five C-centered cluster LDOS corresponding to the clustersA4 „B„(0~n~4), where the contribution of furthershells (e.g. , A, 2 B ) is averaged out. This is obtainedby averaging the LDOS on the structurally inequivalentC atoms in SQS8, and SQS8b corresponding to the samecluster n. Obviously, the higher the number ofconfigurations of further shells corresponding to a given

V RESULTS FOR Alp 5Gap 5As

We first apply our method to the lattice-matchedA1As-GaAs system which manifests only chemical disor-der. Figure 5 depicts the DOS of (a) the zinc-blende con-stituents, (b) the virtual crystal and (c) the SQS structure.

-(e)20—

15—

GSAs---- AIAs

I'

I' j

IEQCO

C4

Kth

lh

VlIM

0~~M

IU

CO

I-

10-I

I

0

:(b)20

15—

10—

5-0

Alp 5Gap 5AsVCA

P4

P6

I

Alp 5GBp 5AsSQS8

: (c) pi20- B

15- P~A p6AP68 =

P5 '

P6C

5- P40-I J g I s I s I ~ I s I a I i I a

-12 -10 -8 -6 -4 -2 0 2 4

PSA p38

Energy(eV}

FIG. 5. Total DOS of the AlAs-CzaAs system. (a) Binarycomponents, (b) VCA, and (c) average of SQS8, and SQS8b.

cluster n on which it is possible to average, the more ac-curate is the "average LDOS" obtained. In SQS8 thenumber of such configurations is limited by the small sizeof the unit cell. This procedure gives the C-centerednearest-neighbor LDOS denoted Dc ( A ~ „B„). Aweighted sum g„w„Dc(A4 „B„) over these fiveLDOS's gives the "average C-atom LDOS," similar towhat can be expected from the SCPA. Similarly, furthershell effects are calculated by keeping a fixed A4 „B„cluster and examining Dc"'(A i2 B ) due to variationsin the occupations of the third and further cation shells.To examine second-neighbor effects we consider the firstcoordination shells about A: (C4), (Ai2 B ), (Ci2),(Ab B ), . . . and calculate D~(A, 2 B ) where fur-ther shells are again averaged out. AveragingD~(A, 2 B ) over the possible configurations Im I inthe second shell gives the "effective" A-atom LDOS. Wewill see that in some cases a splitting in the tota/ DOS istraceable to distinct features in the cluster I.DOS. Thisthen permits the interpretation of alloy DOS features interms of such "environmental effects. "


A. General features

Figure 5(a) and Table II show that when the density ofstates of GaAs and AIAs are aligned using a commonzero of energy (the valence-band maxima), the As-like Plstructures and the cationlike P3 structures occur at dis-tinctly different energies in the two constituents. Indeed,the substantial difference in the atomic s orbital energiesof Al and Ga result in a difference of about 1.4 eV inthe P3 energies in GaAs and A1As [Fig. 5(a)]. In theVCA [Fig. 5(b)] each As atom is surrounded by four iden-tical "effective" cations, so this approximation is obvious-ly incapable of revealing any "split band" behavior anddepicts a single peak for each of the characteristic zinc-blende DOS structures. In contrast, in the SQS modeleach As atom has around it some Al and some Ga atoms,so a fraction of the splitting seen in the pure constituents[Fig. 5(a)] is retained in the alloy. Figure 5(c) indeedshows fine structure and splittings, most pronounced inthe P1, P3, and P6 regions. The nature of these splitbands can be judged by inspection of the electronic

charge density calculated from the wave functions ofeach split component. Figure 6 shows the charge densi-ties of the P3& and P3& peaks, revealing that the deeperP3& state represents primarily (but not entirely) Ga—Asbonds, whereas the lower binding energy P 3z peakrepresents mostly Al—As bonds. Since each of these sub-peaks contains also "minority character" (e.g., Al charac-ter in P3&), the splitting in the alloy (0.7 eV} is smallerthan that in the pure constituents (-1 eV, see the Appen-dix).

While the Al and Ga atoms differ substantially in theirs energies, they have almost identical p energies. Thus,the P4 and P5 DOS structures do not exhibit a strongsplit-band behavior. In particular, P4 displays only finestructure, while P5 shows only a broadening. In the twobinary compounds the P5 peaks are broader and closer toeach other (about 0.8 eV) than the corresponding P3peaks. This gives rise in the SQS8 structure to only onevery broad structure since, in this case, the two com-ponents are not completely resolved. The P6 structure,showing two main features in the virtual crystal, exhibitsa further splitting of the lower binding-energy com-ponent, giving rise to two distinct P6~ and P6C peaks,more like in the binary AlAs. Analysis of the charge den-sities analogous to that shown in Fig. 6 reveals that thefine structure of the P1 As, s peak is related to the contri-bution from the different Al„Ga4 „,0~ n ~4 local envi-ronments about As, affecting the energies of the As, s lev-els. It is hence natural to analyze the DOS in terms ofsuch clusters.

B. Analysis in terms of the 6rst coordinationshell about As

FIG. 6. Electronic charge density contours in the (110) planefor Ala 5Gao, As SQS8 in the P3 region. (a) P3„at E„—6.5 eVto E„—5.8 eV; (b) P3~ at E, —5.8 eV to E,—4. 1 eV. Thesesplit peaks in the DOS are seen in Fig. 5(c). Note that P3„ ispredominantly an As-Cxa state, while P3 is predominantly anAs-Al state. The contour step is 3 (in units of electrons per unitcell volume).

To isolate the effects of the nearest coordination of theAs-centered LDOS, we have calculated the DOS of allAs-centered clusters and averaged over the next-nearest-neighbor variants present in the SQS8 structure, produc-ing the five density-of-states curves Dz, (A1~ „Ga„)asso-ciated with the five A14 „Ga„,0~n ~4 clusters. Theseare depicted in Figs. 7(a)—7(e). Focusing on the P3 re-gion, where the largest variations are apparent, we see alarge P3& intensity in the LDOS corresponding to theAl~ cluster [Fig. 7(a)]. The intensity of this P3Ji com-ponent diminishes as the number of Al atoms in eachcluster decreases in the series A14~ A13Ga& ~A12Ga2~A1,Ga3~Ga~ [Figs. 7(a)—7(e), respectively]. Similarly,the intensity of the larger binding-energy componentP3„ is small in the A14 cluster; it increases as the Ga con-tent of the cluster increases until in the Ga4 cluster [Fig.7(e)] this peak is dominant. Clearly, the intensities of thevarious substructures in the P3 region reAect local envi-ronmental effects about the As site even though the alloyconsidered here has a fixed global composition x = —,'. Su-perposition of all five LDOS of Figs. 7(a)—7(e} with therandom probability ratios 1:4:6:4:1for n =0, 1,2, 3,4, re-spectively, produces [Fig. 7(f)] the average contributionof the anion to the DOS.

7954 FROYEN, AND ALEX ZUNGERRITA MAGRI, SVERRE FR

C. Ana ysxs ol ' f further-neighbor effects

ted above, each of the five first-neighbor clus-

tons of the nexta a ears in the

d b the occupa ionsg. While we have average eneighbor sites.

h r-nei hbor effects on thefocus on such further-neiguter odd-shell neighbor effects abouDOS. Third and furth

are a arent when one xes athe As at the origin are app) and exam-i hbor cluster (A14 „Ga„gat a iven n

nces in the LDOS's due to various occupa-hofurther shells. igutions in the fu

= 1) Al Gai nearest-Ga ) for fixed (n =n

'd

'h th' d-shell occupationsneighbor cluster and y'dvar ingt e ir-

alld (b)], ill =6 Fig. cThe common c arac er

f t i that the P3„peakAs 12—m Ga ) density of states is

shape and intensi y o- ei hbor cation coor-

arison of Fig. 8(a) with Fig. 8(b)dination shells. Comparison o ig.

showst ee ecsh ff t due to different occup ations of the coor-dination she yelis be ond the third.

b t Al can be discerned byGa for various m values de-

d-nei hbor effects about cancalculating Dzi(Al, z Ga or va

' e-. To focus on the secon -neig

further coordination shells. npations of the furs of the second coor-

he second shell is a decrease inatoms in the1-like P3 peaks and an in-

in' ' t the expense of that of I'3z.

the Ga-like and the A - i ecrease inin the P3 intensity at t e expB

th second she11 increases,As the number of Af Al atoms in t e seco1 cal Al density ofGa ) approaches the ocaDAt(Ali2 ~ Ga~

3(d)]. As before, we findinar A1As [Fig.

11 h """d"'blf two Al-centere c us er

ordination can sti s owth oo d'differences if the occupaations o ur er

shells are different.

D. Comparison with experiment

1.5

1.0

DAS (Al4) DAs (AI1Ga3)

P3

Al AsEffects of first coordination shell in Gao 5I

'I ' II

' II'

I

. (a) -

{d)

eA1Ll. "have recently measured t e 23- -ra em' '

1 Ga As rejecting the- -ra emission (SXE) in Al„a&what follows we compare our reAl-centered LDOS. In w at o ow,

1' 'ble dependence onith their data.

ener shows neg igi e

i . 9) coordination shells. The( d 11 P3) doshows, indeed, that this peak terme

shift with Al concentration.0.0

5 . (b)

m1.5-

F) 1.00'o) 0.5CC0Cl

0.0O0

4 I ~ i

I

DAs (AI3Ga)

P3A

I II

I t ' ['I ' t 'I

(c)(n)Z WnDAs

n=0DA (Al2Ga2)

1.5—

1.0—

0.5—

0.0-14 -10

j-2 0 -14

Energy (eV)

-10 -6 -2 0

I It

I'

II'

I'

I' II

'I '

IJ

' I 'I

(e)DAs (Ga4)

1.0—C

J5m 0.5—OlI

0.0M0

1.5—OlCl~ 1.0-0

0.5—

(a)

(c)

I

DA {AI7Gag)(~)

P38

I

(AlsGa6)(1)

DAs (AI5Ga7)(~)

a Al AsEffect of Third Coordination Shell in Gao 5I

(1) - - (b)DAs(Al7Ga5)

osition of the total DOS of Alo 5Gao 5As intopos-centered contributiolons D s ( A14 —n an

i hbor cation environmen s pth Al ot ti-f P3~ increases as eNote that the amphtude o

de of P3~ increasesster whereas the amplitu e ocreases in each clus er, wRecall that all ofentration in each c uster. ecwith the Ga concen ra

'

nd to a xed (x =— g o= 2) 1 bal composition inthese results correspond o ae As LDOS"llo . Part (f) shows the "average sa homogeneous alloy. ar

nDAs A 4—nobtained by superposing „„,Adom tetrahedral probabilities w„.

0.0-2-8

Energy (eV)

the D~, (A1,Ga) LDOS [Fig. 7(b)]FIG. &. Decomposition of the DAs 3'sin from different occupations ointo contributions arising rom i

nd (b) have thea shell. Note that parts (a an'd d to hll ththrou h the thir coorsame occupations g

small differences in their DOSOS reAect e ec sshells.


(ii) The calculated P3 peak shows a deep, Ga-like com-ponent (P3&) and a lower binding-energy Al-like com-ponent (P3~). As the number N~&' of Al atoms in thesecond shell about Al increases, the Ga-like peak P3„narrows (Fig. 9) but the intensity stays nearly constant(although not shown, the same trend is found in the Ga-centered LDOS). On the other hand, the Al-like P3~peak shows a pronounced increase in intensity (and somenarrowing) as N~~' increases. While we have not calculat-ed the LDOS at other compositions, we can speculatethat the trends with increasing the Al composition x willparallel those of increasing X~& at a fixed x. While theSXE spectra do not resolve P3~ from P3~, they clearlyshow that as the Al concentration increases, the deeperbinding energy part of P3 [termed P2 in Ref. 47(a)] nar-rows asymmetrically, while the higher binding-energy

Effects of Second Coordination Shell in Ga& 5AIO 5As

part stays at a similar width but the peak intensity in-creases significantly. Hence, as xA& increases, there is atransfer of spectral intensity from P3& to P3&, as seen inour calculation (Fig. 9). This "nonrigid" behavior of P3is the hallmark of this spectral region, found also in otheralloys (Sec. VI below). The same trends were seen in a re-cent tight-binding CPA calculation by Hass. '"' Themeasured position of the peak intensity of P3 at x =0.5

(extrapolated between x =0.4 and x =0.6) is E„b —5.8

eV, close to our result (average of P3„and P3~) ofE„b —( 5.8+0. 1) eV.

(iii) Our P4 peak [denoted P 1 in Ref. 47(a)] shows onlyweak dependence on N~&', in agreement with the fact thatits width and intensity do not vary much with x A&.

(iv) Comparison of our results for the pure constituents[Fig. 5(a)] with those of the x =0.5 alloy [Fig. 5(c)] andthe foregoing analysis of the effects of X&&' suggest thatwhile the global composition x changes the energy posi-tion of the various peaks, the shape of the peaks is deter-mined primarily by the local environment (N~& and No, )

in successive shells.

I

. (a)1.5—

I'

I'

I'

I'

I'

I' I

DAI (AI2 Ga10)

1.0—

0.5—

I ' I ' I ' I ' I ' I I ' I ' I I

.:(e) AIAs

I 'I

' I ' I ' I ' I ' I ' I ' I ' I

(b) GaAs

M~~C

L0$

MED

lA

O

~~thcx$tO

O

0.0- (b)

15—

1.0—

0.5—

0.0

- (c)1.5—

l.0—

0.5—

0.0

- (d)1.5—

I I

I1

I~

II

II

DAI (AI5Ga7}

I 'I

'I

DAI (AI7Ga5}

II

I II

II

II

DAI (AlaGa4)

5-M p.O I ' I

20 ( )

Kv)

P) 10thoO

QI

(e)

15-

I ' I ' I ' I ' I ' 1 ' I ' I

LuzoniteAl& Ga3As4

I' I '

\'

I'

I' t '

I' I

CuAu-IAI2Ga2As

- (d) LuzoniteAI3Ga& As4

I ' I ' I ' I ' I '&

' ~

(f) chalcopyriteAI2Ga2As4

1.0—Q-12 -8 -4 0 4 8 -12 -8 -4 0 4

Energy (eY)05-0.0

-14I

-12 -10 -8 -6 -4

Energy (eY)

0

FIG. 9. Decomposition of the Al-centered LDOS into contri-butions arising from different occupations of the secondAl» Ga shell: (a) (A12Galo)' (b) (A15Ga7); (c} (A17Gas); (d)(AlsGa4).

FIG. 10. Total DOS of ordered periodic Al„Ga As„+structures: (a) and (b) give the results for the binary zinc-blendecompounds, (c) gives the DOS of A1Ga3As4 in the luzonitestructure, (d) gives the DOS the A13GaAs4 in the luzonite struc-ture, (e} shows the DOS in the CuAu-I —like structure (an(A1As)&(GaAs)& superlattice with layer alternation along the[001] direction), (f) shows the DOS in the chalcopyrite structure(an (A1As)2(GaAs)2 superlattice with layer alternation along the[201]direction).


E. Comparison with the method of superpositionof periodic structures

II

m .(a)E 20-C5

C4

~ 10-MED

CO~peal

0-th

-(b)Pe 20V0~~" 10-

:P

CS

O 0-12

pie

-8

P3A

Alp. 5Gap 5As4Z W„D„(ordered)

n=p

Alp. 5Gap. 5As:SQS8

6A P6B

-4 0Energy (eV)

FIG. 11. Comparison of the total DOS of Alo 5Gao 5As as ob-tained (a) from averaging SQS8, and SQS8„[identical to Fig.5(c)] with (b) the model of "superposition of periodic struc-tures" [Ref. 29(c)] where the density of states of the five orderedstructures Al& „Ga„As4 (0~ n ~4) given in Fig. 10 are super-posed with random probability weights. [This is distinct fromaveraging LDOS shown in Fig. 7(f).]

It has been suggested previously ' that the electronicdensity of states of a random alloy could be modeled bysuperposing those of a number of ordered structures eachcontaining a distinct repeated cluster. For example, theA4 or B4 clusters appear as a repeated motive in thezinc-blende structures A C and BC, respectively, theAzB~ cluster appears in the (001) monolayer superlattice( AC), l(BC ), (The CuAu-like structure, or CA), and the3 3B and AB 3 clusters appear in the "Luzonite" struc-tures A 3BC4 and AB3 C4 (denoted as L 1 and L 3, respec-tively). To examine this approach we have calculated theDOS of the five periodic crystals 3C, 33BC4, 32Bz C4,AB3C~, and BC (Fig. 10) and superposed them using asweights the Bernoulli probabilities appropriate to x =

—,'.The resulting DOS [Fig. 11(a)] compares reasonably wellto that obtained in the more accurate SQS8 model [Fig.11(b)]. In particular, both models show the splitting ofP3 structure and the broadening of P5 peak. However,we notice from Fig. 10 that the long-range order can sub-stantially modify the "cluster" extracted from these or-dered structures. While the total DOS obtained by themethod of superposition of periodic clusters is rathersimilar to what is obtained from the SQS, the partial DOScan difFer. First, note that ordered structures of the sameglobal composition (e.g., CA and CH) differ in their coor-dination arrangements. For example, relative to an 3atom at the origin, the CA structure has 4, 6, 8, 12, 8, 8,and 16 3 atoms in shells 1 through 7, respectively,whereas the CH structure has 4, 4, 16, 4, 8, 0, and 32 3

-:(a)20—

10-

GaAs---- InAs

MEOl5

C4

K40I40CD

6$VJ

O

~~lO

'vC5

oI-

0 -s~ I I

-(~)20-

10-

P1

P3

P4

- (c)20-

i,I

sI

'I

10-

-(d)20—

I'

I'

I'

I'

I'

I

I

Gap 5lnp 5AsVGA

Gap 5lnp 5As-SQS8

unrelaxed

I I I I ~ I F

Gap 5lnp 5As-SQSSrelaxed

10-

0 -12II a I a I I

-4 0Energy (eV)

FIG. 12. Total DOS of the GaAs-InAs system. (a) Binaries0

at the unrelaxed alloy equilibrium lattice constant aUR =5.80 A,(b) VCA, at the same lattice constant, (c) unrelaxed SQS8 (aver-aged over variants a and b), and (d) relaxed SQS8 (see Table IIfor the relaxed bond lengths). The calculated equilibrium latticeconstants in SQS8 for fixed and relaxed cell-internal parametersare 5.80 and 5.82 A, respectively. Note in (c) and (d) therelaxation-independent splitting in the P1 region absent in theVCA and the relaxation-induced splitting in the P3 region.

atoms in these respective shells. The DOS of these twocrystals are indeed somewhat di6'erent: CH has a struc-ture between P3& and P3~ that CA lacks and the inten-sity ratios of the various P1 subpeaks are diferent.Clearly, the success of the method of superposition ofperiodic structures depends on a judicious choice ofstructures. Second, note that the DOS associated to theclusters in the simple periodic structures (Fig. 10) differfrom those extracted from the SQS (Fig. 7). For example,the DOS of the AIGa3 Luzonite cluster [Fig. 10(c)] differsfrom that in the SQS [Fig. 7(d)] in that P3 „ofthe formeris slightly split. Also, the DOS of the A13Ga Luzonitecluster [Fig. 10(d)] differs slightly from that of Fig. 7(b) inthe relative P3 ~:P3~ intensities.

Finally, note that using a single simple (not a SQS)periodic structure to describe the random alloy of the


same composition can lead to significant errors unless theperiodic structure is selected carefully. This has beenpointed out by Boguslawski and Baldereschi who usedperiodic Ga~ „In„As& crystals to describe, for n = 1 (L 1),n =2 (CA), and n =3 (L3) the Ga In, „As alloy at com-positions x = 4, —,', and 4, respectively. The fact that thisis not a very good approximation can be also seen bycomparing the n =2 (CA) density of states of Fig. 10(e)with the corresponding SQS results of Fig. 5(c).

VI. RESULTS FOR Gao 5Ino 5As and GaAso &Po z

(e)I I

I / I i ~/

I i II I I

GaAs————-GaP

Comparison of the density of states of the lattice-mismatched Gao 5Ino 5As and GaAso 5P0 5 systems intheir unrelaxed and relaxed structures permits analysis ofstructural disorder effects. Because of the different envi-ronment of each atom, structural relaxations lead to a

distribution of bond lengths and angles even in a perfectlyrandom alloy. This feature is included in our calculationby permitting the atoms in the SQS8 to relax into theirminimum energy positions. Since all nearest-neighborA„84 „clusters are present in SQS8, + SQS8b, theeffects of relaxation on the DOS should mimic realistical-ly the situation in the random alloy. The density of statesof Gao 51no sAs and GaAso sPo s SQS8 are shown in Figs.12 and 13 where they are compared with the DOS of thebinary constituents and of the virtual crystal. In whatfollows we analyze the behavior of each feature in theDOS.

A. The P1 region

The P 1 (anion, s) region in the SQS of Ino 5Gao sAs ex-hibits a fine structure with many subpeaks. This behavioris enhanced in the mixed anion system GaAso ~P0 5,where two different anion s orbitals contribute to thisstructure.

Since the wave functions associated with the P1 statesare largely atom-centered states with nearly spherical

10 , &I

/"I

II

I

0-I

'I

'I

'I

'I

'I

'I

Effects of first coordination shell in Ga As& 5 Pp 5 (unrelaxed)I

' I '1

' I '1

'I

'I I ' I ' I ' I ' I ' i ' I

(a) : (d)DGg (P4)

~ IoQa(p4s3)

eP3

20 — (b)

(c)

10-

0

P1o 10C5

CV

K 0 I s II ~

CO

I'

I

9)0CO

Q

aAsp. 5 0.5VGA

P4

GaAsp 5 Pp 5SQSS

unrelaxed

p6P1A

0.5 — P1cN

a - P&p

- (b)N

1.5—

M1.0-

N

=:,"~-=li"I ' I '

I

O . (c)

1.5—

I I ~ I

I 'I

' I

loo, (P2As, )

l

(f)

P1BP1co P&A

I

I' I '

I' I

'I

'I

ion (As4)

P1c

4~ WnDGa

!n=O

20—

10-

0-

I'

I'

I 'I

'I

' I ' I 'I

'I

GaAsp. 5 Pp.5SQSS

relaxed

1.0-

-10-14

B P)

Pip0 I

-2 0 -14 -10

Energy (eV)

-2 0

-12 -4 0Energy (eV)

FIG. 13. Total DOS of the GaAs-GaP system. (a) Binaries atthe alloy lattice constant a=5.51 A, (b) VCA, {c) unrelaxedSQS8 (averaged over the variants a and b), and (d) relaxed SQS8(see Table I for the relaxed bond lengths) all at the same latticeconstant. Note in (c) and (d) the broadening and splitting in theP1 region and that the P3 region remains unsplit in the alloy,unlike the case in GaAs-InAs (Fig. 12).

FIG. 14. Decomposition of the total DOS of GaAso 5PO, intoGa-centered contributions DG, (As4 „P„) associated with thefive (0& n ~4) first-neighbor anion environments [parts (a)—(e)].Note in the P1 region how the amplitude of the different sub-peaks reAects the relative P/As content in each first-neighborcluster: P1 & and P1& are reduced in the seriesP4~P3As~P~As2~PAs3~As4 whereas P1~ and P1D in-crease as the As content increases in each cluster. Part (f) showsthe "average Ga LDOS" obtained by superposingDG, {P4 „As„) with the random tetrahedral probabilities.


symmetry about the anion sites [see Fig. 2(a)], the energylevels in this region are not perturbated severely by relax-ing the bond lengths and angles at a fixed volume. Likein Alo 5Gao 5As, the splitting and broadening in the P1region of Ino 5Gao 5As can be traced back to the existenceof structurally inequivalent As atoms. In GaAso 5PO 5 theP1 region shows a greater complexity, because of themixing between the P and As s levels. Moreover, itshows some changes on passing from the unrelaxed to therelaxed structure. We have analyzed the contribution ofthe five nearest-neighbor clusters on the local density ofstates on Ga atoms. The results are displayed for unre-laxed GaAs05P05 in Fig. 14. The amplitudes of thedi6'erent P1 subpeaks change following the cluster se-quence P4 —+P3As2~PAs3~As4. The amplitude of thepeak P1~ at about E, —9.9 eV decreases as the P contentof the cluster diminishes. Examination of the charge den-sity in this energy region [shown in Fig. 15(a)), revealsthat this feature is associated with states having s-likesymmetry around the P atoms, just like in pure GaP. Inthis energy region there is no mixing between P and As sorbitals. On the other hand, the feature P1D at aboutE„—12.3 eV (Fig. 14) grows in going from the cluster P4to the cluster As4. Inspection of the associated chargedensity in Fig. 15(d) shows that this structure is relatedmostly to s orbitals on the As atoms. The regions Pl&

and P1& between E, —10 eV and E, —11.8 eV show themost pronounced changes on passing from one cluster toanother (Fig. 14). The charge densities of these struc-tures are plotted in Figs. 15(b) and 15(c), respectively.We see from Fig. 15(b) that the wave functions associatedwith the region P1& are mostly centered around thephosphorus atoms, with a considerable probability alsoaround the As atoms. From Fig. 15(c) we deduce thatthe Plc state, extending from about E, —11 eV to higherbinding energies, is As,s like with mixing of amplitude onthe P atoms. From this analysis we can divide the P1peak into three regions: (i) an As, s like region (P 1D and alarge part of P lc) from about E„—11 eV to E, 12—.8 eV,(ii) a P,s like region (Pl„) from E, —9.5 eV to aboutE, —10 eV, and (iii) an intermediate region where thewave functions are extended on both As and P atoms.This is confirmed by the comparison with the DOS of re-laxed GaAso 5Po s, Fig. 13(d), where we see that the larg-est change in the shape of P1 with respect to the unre-

Ga Get

QA.

GaAsp 5 Pp 5

(c

FIG. 15. Electronic charge density contours in the (110)plane for the unrelaxed GaAso, PO, SQS8 in the Pl region. (a)

P1& at E, —9.5 eV to E, —10.0 eV; (b) P1& at E, —10.0 eV toE„—10.9 eV; (c) P1C at E, —10.9 eV to E, —12.0 eV; (d) P1& atE, —12.0 eV to E, —13.0 eV. P1~ ad P1& are predominantlyP-centered states, while P1& and P1D are predominantly As-centered states. The contour steps for parts (a)—(d) are, respec-tively, 2.5, 2.5, 5.1 (in units of electrons per unit cell volume).

FIG. 16. Electronic charge density contours in the (110)plane for the relaxed Ino, Ciao, As SQS8 in the P3 region. (a)P3& from E, —6.9 eV to E, —5.9 eV; (b) P3& from E, —5.9 eVto E„—4.0 eV. P3„ is mostly centered on the Ga—As bondswhile P3& on the In—As bond. The contour steps for parts (a)and (b) are, respectively, 4 and 2 (in units of electrons per unitcell volume).


laxed case occurs just in this energy region. These mixedstates are t'he most sensitive to the structural relaxation.We notice, moreover a change in the energy onset of P1 ~peak with relaxation, reQecting the shift of the P1 peak inGaP to higher binding energies with volume compressing(bond shortening).

Equal BondLengths

- &oIn-As - ~ge~

Ga-As @a~

Relaxed

P5B

P5A

(c}

Equal BondLengths

Ga-P

Ga-As Longer-

Relaxed

P5B

P5A

B. The P3 and P5 regions

Unlike the atom-centered P1 states, the bonding andantibonding P3 and P5 states have directional character,and are hence expected to respond more strongly toconstant-volume bond relaxations. Figure 16 depicts thecharge density in the P3 region of Gao 5Ino 5As, andshows that the deeper binding-energy P3~ component isconstructed predominantly from Ga-As states, whereasthe lower binding energy P3& component has moreweight on the In—As bond. The behavior of the P3peaks both in Ino 5Gao 5As and in GaAso sPO 5 is particu-larly interesting, in that the P3 energies of the purebinary constituents at the alloy's equilibrium volume arenearly degenerate in GaAs and InAs [Fig. 12(a) andTable II], whereas in GaAs and GaP the P3 energy sepa-ration is significant [Fig. 13(a) and Table II]. However,we see from Figs. 12 and 13 that alloy relaxation inducesa splitting in the P3 region of Gao 5Ino 5As, whereas inGaAso 5Po 5 the alloy environment acts to remove the ini-tial P3 splitting. Similar effects were noticed and ana-lyzed in II-VI alloys.

This behavior is caused by two eff'ects. (i) While inA1As and GaAs each As is coordinated, respectively, byA14 and Ga4, leading to a maximal difference in the posi-tions of the respective I'3 peaks (Appendix), in the alloythere are also mixed Al-Ga clusters that fil1 in additionalDOS features between the A14 and Ga4 peaks. Indeed,Fig. 16 shows that in Ino 5Gao 5As the P3~ peak involvesIn—As minority character whereas the P3~ peak in-volves Ga—As minority character. The same is true inGaAso gPO 5 where the 0.6-eV P3 splitting in the binaries(Table II) is reduced [Fig. 13(c)] by hybridization. (ii)Bond relaxation further modifies these states. To under-stand this difFerent behavior we have schematically plot-ted in Fig. 17 the shifts in the sp bonding (P3) and anti-bonding (P5) states due to relaxations. Previous calcula-tions ' ' have shown that bonding (antibonding) statesare displaced to larger (smaller) binding energies whenthe bond lengths are shortened. Our calculation showed(Table I) that upon relaxation of Ino ~Gao 5As, the In—Asbond becomes longer, while the Ga—As bond becomesshorter relative to the unrelaxed virtual lattice average.These changes are denoted in Fig. 17 by the arrows. As aconsequence, the In—As bonding P3& component shiftsto lower binding energies while the Ga—As P3„com-ponent shifts to deeper binding energies [Fig. 17(b)].Thus, relaxation increases the P3 ~ -P3& separation inInn ~Gao 5As, as observed in Figs. 12(c) and 12(d). Theopposite is true for the antibonding P5 peak inIn& 5Gao 5As, where relaxation tends to bring the peakscloser [Fig. 17(a)]. In GaAso &PO 5 relaxation elongatesthe Ga—As bond while the Ga—P bond becomes short-

~e& W~o& ~

In-As

Ga-AsO~

P3B

P3A

(d) Ga-P

Ga-As

P3B

P3A

FIG. 17. Schematic plot of the effects of bond relaxation onthe energies of the bonding P3 and antibonding P5 states in theCxao, lno, As [parts (a) and (b)] and GaAso, PO, [parts (c) and(d)] alloys. "Longer" and "shorter" refer to changes in the bondlengths relative to the equal A -C and 8-C bond lengths in theunrelaxed alloys (denoted VLA in Table I).

Like in Alo &Gao 5As, the P4 region shows a fine struc-ture in the SQS model. These features are very similar inboth the common-anion and the common-cation alloysand change relatively little with relaxation. The P6 re-gion shows broadening for both systems. In GaAso 5Po 5

we can notice a splitting of the lower binding-energycomponent as already seen in Alo ~Gao 5As.

VII. COMPARISON WITH PREVIOUSDOS CALCULATIONS

Lempert and co-workers' have calculated the DOS ofIn05Gao5As in three difFerent alloy models: (a) VCA,where chemical and structural disorder are both neglect-ed, (b) site CPA, where only the chemical disorder hasbeen introduced, and (c) MCPA, where both kinds of dis-order were taken into account. Comparing SCPA withVCA (Fig. 18) they found that the introduction of chemi-cal disorder broadens the peaks P3 and P5, leaving thestructures P1 and P4 in the valence band substantiallyunchanged. Similar results are obtained here in passingfrom the VCA [Fig. 12(b)] to the unrelaxed SQS [Fig.12(c)], except that we find distinctly new fine structures inP1 and P4. Comparing the MCPA to the SCPA, Lem-

er. Hence, the higher binding energy Ga-As P3& statesshift upon relaxation to a lower binding energy, while theGa-P component P3~ shifts to higher binding energy[Fig. 17(d)]. Thus, relaxation reduces the energy-levelseparation between the two P3 components inGaAso 5Po s. The opposite is true for the P5 peak [Fig.17(c)]. Since the splittings are rather small inGaAso 5PO 5, the changes with relaxation are not apparentin Fig. 13.

C. The P4 and P6 regions

RITA MAGRI, SVERRE FROYEN, AND ALEX ZUNGER

pert and co-workers find that the introduction of thestructural disorder tends to increase the broadeningthroughout the upper valence bands. They observed thepresence of two bumps in the I'3 peak, analogous to P3~and P3s found here [compare Fig. 12(c) with Fig. 12(d)],except that the sphtting found here is considerably larger.Another feature related to the introduction of thestructural relaxation in their model is the decrease ofbroadening of the P5 peak relative to the site-CPA modelanalogous to the present results. Hence, the MCPA cal-culations capture the overall shape (but not the details)found in SQS models.

All other existing calculations on III-V semiconductoralloy DOS are restricted to the VCA and SCPA. Themost detailed are the CPA calculation of Chen and Sher'on GaPo 5Aso 5 and on Gao 5Ino. sAs. For Gao. 5Ino. sAs(where the scattering strength is largest), they find [Fig.19(a)] that the CPA density of states represents a merebroadening of the valence-band structures with respect tothe VCA. In convict with Lempert and co-workers'their P1 and P4 peaks show CPA-induced broadening.The fine structure in Pl and P4 (Fig. 12) and relaxationeffects are all missed by this calculation. The results forCraPo 5Aso 5 [Fig. 19(b)] show essentially the samevalence-band features in the scaled VCA and CPA mod-els. Only for the first conduction bands does the CPAshow a slight broadening of the peaks. Our results onGaPo 5Aso 5 system (Fig. 13) show instead a pronouncedbroadening of the structures, in particular P3, P5, andI'6 on passing from the VCA model to the unrelaxedSQS8. In particular the Pl structure is the most affectedby the introduction of the chemical disorder. For the

lattice-Inatched Al Gaj As system, the present resultsare very similar to the CPA results of Hass.

VIII. BOWING OF THE DIRECTOPTICAL BAND GAPS

We have calculated the direct band gaps at I in theSQS8 model for the three systems. Table III gives the en-ergies at the top of the valence band and the energy of theconduction state of I &, character. The crystal-field split-ting at the top of the valence band is enhanced by relaxa-tion. Table IV gives the calculated optical bowingcoe%cient for the three systems together with theirdecompositions in chemical and structural contributions.The bowing coe%cient is defined as four times thedifference between the average band gap of the twobinary constituents and the band gap (relative to theaverage of the crystal-field components at VBM) of theSQS alloy. The "volume deformation" is defined as thecontribution due to the compression and/or dilation ofthe binary constituents into the alloy's volume V. The"charge exchange" is defined as the change in going fromthe binaries at V to the unrelaxed SQS, while the"structural relaxation" is the change in passing from theunrelaxed to the relaxed SQS. The contributions fromvolume deformation (VD) and structural relaxation (SR)are present only for lattice-mismatched systems. We no-tice that the chemical exchange (CE) contribution issmall and negative for both lattice-mismatched systems,while it is positive and larger for Ala 5Ciao 5As having thegreatest mismatch in atomic orbital energies. This isconsistent with the observation that the effect of compo-

~ 3.50)O)

0

~ 3.5M

0(g) 0Q)

C3X

~ 3.5P)

P1

I I I

(b) S-CPA

(c) MCPA

P3P5

7I0 I I I [ Il i

f I

(a) VCA Inane GaosAs

NEOC5

CV

KfA47

F)co0O

(a) Gao sino s As

P4

P1

(b)4- ~GaAszsP~s

~

P1

P3

P4P6

-12 -10 -8 -6 -4 -2 0.0 2.0 4.0Energy (eV)

-12.0 -8.0 -4.0 0.0Energy (eV)

4.0

FIG. 18. Total DOS of the Ino ~Cxao 5As alloy in the (a) VCA,(b} site CPA, and (c) MCPA from Ref. 19.

FIG. 19. (a) Total DOS of the Gao 5Ino 5As alloy in the CPAfrom Ref. 18 [Chen and Sher, Phys. Rev. 8 17, 4726 (1978)]; (b)total DOS of the CxaAso 5PO, alloy in the CPA from Ref. 18[Chen and Sher, Phys. Rev. 8 23, 5360 (1981)].


sitional disorder is larger in lattice-matched systems.Structural relaxation contributes in a different way to thebowing of GaAso 5Po ~ and Gao 5Ino ~As. To explain thisdifferent behavior we notice that the top of the valenceband in Gao &Ino &As has more In than Ga character (seealso the Appendix), while the bottom of the conductionband has more Ga than In character. Structural relaxa-tion increases the In—As bond lengths thereby raisingthe energy of the bonding valence-band maximum(VBM), while the reduction in the Ga—As bond lengthsraises the energy of the antibonding states at theconduction-band minimum (CBM). The net effect is aslight increase of the direct band gap with respect to theunrelaxed configuration. In GaAso 5Po ~, on the otherhand, both the VBM and the lowest conduction state atI have more As than P character. Since structural relax-ation increases the Ga—As bond lengths, this reduces theband gap giving a positive contribution to the band bow-ing. In all cases the total bowing b =bvD+bcE+bsR ispositive, leading to a reduction of the direct band gap.The reduction is biggest for Gao &Ino 5As that presentsthe largest lattice mismatch. Neglect of relaxation leadsto an underestimation of the bowing coefficient of about42% in GaAso 5PO 5 and to its overestimation of about18% in Gao &Ino 5As.

In the case of Gao 5Ino 5As and Alo 5Gao ~As randomalloys there is a substantial spread in the experimentaloptical bowing parameters. The reported values rangebetween 0.32 and 0.61 eV for Gao ~lno ~As (Ref. 50) andbetween 0 and 0.37 eV for Alo 5Gao ~As. ' ForGao 5Alo 5As the most recent determination of the directgap bowing by Bosio et al. gives the value (0.22+0.06)eV in very good agreement with our calculated value of

0.20 eV. For the GaInAs alloy, lattice matched to InP(0.44&x (0.49), Goetz et al. give the value b =0.475eV, close to our calculated value of 0.40 eV. Also forGaAso 5Po z there is a good agreement between the vari-ous experimental values b,„,=0.17—0.21 eV (Ref. 54)and our calculated total optical band bowing of 0.19 eV.

The spread of calculated values is even bigger. Thebowing calculated here for Gao sino ~As using the SQSSmodel in the pseudopotential approach (0.40 eV) is invery good agreement with our previously calculatedlinearized-augmented-plane-wave value for SQS4 of 0.43eV. ' ' Lempert and co-workers' have found that theintroduction of the structural disorder in Gao ~Ino ~Asthrough the MCPA model reduces the optical bowing.They give the following values: bvcA =0.24 eV,bcpA=0. 32 eV, and bMcpA=0. 26 eV. Baldereschi andMaschke, using the pseudopotential method andsecond-order perturbation theory on GaP As, , foundb=0.27 eV. Lee et al. , using a modified empiricaltight-binding method, calculated b =0.74 eV forGa, In As. Both studies conclude that. structural re-laxation (not included in their model) has almost a negli-gible effect on the direct band gap bowing, the main con-tribution coming from the compositional disorder. Thisis in disagreement with our findings, where the structuraleffects are very important (Table IV), mostly for theGaAso 5Pc s alloy, and the compositional disorder (CE)has little effect in lattice-mismatched systems.

We conclude with a note on the different bvcA valuesobtained here for the three alloys. In the literature onefinds different values for the direct gap optical bowing ofthe GaAs& „virtual crystal. Ting and Chang, using anempirical k.p method to calculate the band structure,

TABLE III. Calculated energies (in eV) of the I »„and I"&, electronic states relative to the top of the valence band for the unre-laxed (UR) and the relaxed {R)SQS8 and the virtual crystal. For SQS8 we give the crystal-field components of the I &,„state. The en-

ergy levels are compared with the average of the energies of the corresponding states in the binary compounds at their equilibriumlattice constant a,q and with those at the alloy lattice constant a.

Electroniclevel

I )5

Average energy ofbinaries at aeq

0.000

1.443

0.000

1.433

0.000

0.261

Average energy ofbinaries at a

GaASO. 5PO. s

0.000

1.412Alo ~Gao 5As

0.000

1.433Gao. sIno. sAs

0.000

0.133

SQS8 energyUR at a

—0.009~ —0.006

0.000

1.410

—0.014—0.005

0.000

1.376

—0.005' —0.004

0.000

0.140

SQS8 energyRata

—0.061

)—0.049

0.000

1 ~ 358

—0.080~ —0.069

0.000

0.116

VCA energyat a

0.000

1.367

0.000

1.436

0.000

0.2560'For GaAso 5Po 5 we use the alloy value a =5.51 A.

0For Gao 5Alo 5As we use the alloy value a =5.619 A.

0'For Gao o5Ino 5As we use the alloy value a =5.818 A.


TABLE IV. Decomposition of the direct gap optical bowingcoefficient b for SQS8 (using the crystal-field average of thestates at the (VBM) into "volume deformation" (VD), "chargeexchange" (CE), and "structural relaxation" (SR) contributions.The corresponding optical bowing coeKcient of the virtual crys-tal (bvcA) and experimental values are given for comparison.Note that the SQS model gives bowing effects due to onlyshort-range interactions.

&vr&CE

&sR

b

bexpt

~vcA

GaAsp 5Pp 5

+0.12—0.01+0.08+0.19

0.17-0.21'+0.30

Crap, Alp, As

0.00+0.20

0.00+0.20

0.0—0.37—0.01

Gap 5Inp 5As

+0.51—0.04—0.07+0.40

0.32—0.61'+0.02

'Reference 54.References 51 and 52.

'References 50 and 53.

find a negatiue bowing (b ——0.56 eV). The virtual-crystal approximation of Chen and Sher, who used atight-binding model for the wave functions and empiricalpseudopotentials, gives a positive bowing coefficientb =+0.10 eV. As indicated by Bernard and Zungerdifferent VCA calculations within empirical methods canlead to a large spread in predictions even though the un-derlying fits to the band structures of the pure constitu-ents are equally good. These values are seen to stronglydepend on the theoretical method. As recognized by oth-ers' ' the VCA is not a viable approach for estimatingalloy bowing.

IX. SUMMARY

We have calculated the electronic properties of therandom III-V semiconductor alloys Alp 5Gap 5As,Gap 5Inp 5As, and GaAsp 5Pp 5 focusing on the effects ofchemical and structural disorder. In a random alloythere exists a distribution of many distinct localconfigurations. Chemically identical atoms contributedifferently to the alloy properties if their "local environ-ments" are different. In this work we have analyzed howthe electronic properties of semiconductor alloys dependon the average atomic arrangements around sites. Theuse of the special quasirandom structures approach per-mits a study of the interactions between these differentatomic local environments and the global properties ofthe alloy in that each atom retains its identity.

Aligning the peaks of the density of states (Table II) oftwo semiconductors on an absolute energy scale (Appen-dix) reveals differences in the positions of correspondingDOS features. Compressing and dilating these semicon-ductors to the alloy volume V further shifts these peaksdue to hydrostatic effects (Fig. 4), producing the "un-mixed" states of the constituents. Creating the unrelaxedrandom alloy at V presents chemically identical atomswith different local environments. This leads to coupling

of the "unmixed" states of the constituents, most ap-parent by the appearance of splittings in the Pl anion sstates [Figs. 5(c), 12(c), 13(c)] and fine structure in thebonding P4 and antibonding P5 bands. These featurescan be traced to a superposition of the various nearest-neighbor local environments about the common atom(Figs. 7 and 14), as well as to second-shell (Fig. 9) andeven third-shell (Fig. 8) effects. The SQS construct pro-vides a direct way for inspecting such "superpositioneffects" and depicting the electronic charge density ofvarious states in real space (Figs. 6, 15, and 16). Relaxa-tion of the atomic positions leads to "bond alternations"whereby the shorter of the two (A -C and 8-C) binarybonds become yet shorter relative to the virtual latticeaverage, whereas the longer of the two becomes yetlonger in quantitative agreement with EXAFS measure-ments (Table I). Such nonhydrostatic deformations cou-ple to the electronic states (Fig. 17), enhancing (reducing)the P3~ P3~ spl-itting in Ino ~Gao 5As (GaAso 5PO ~) andreducing (enhancing) the P5 „P5~ s-plittings inIno~Gan ~As (GaAs05P05). These structural relaxationscontribute significantly to the bowing of the direct bandgaps in lattice-mismatched alloys (Table IV).

Our results are qualitatively similar to those obtainedby the molecular CPA, where distinct (nearest-neighbor' ) environmental effects are retained, and to theanalysis of Gonis, Butler, and Stocks " ' of environmen-tal efFects in metallic alloys. Since, however, the single-site CPA averages out at the outset all environmentsabout single sites, the multisite nature of chemical disor-der is lost, as are all relaxation effects. These effects areshown to be significant for size-mismatched semiconduc-tor alloys. While they can be captured by the MCPA, thecomplexity of this method has so far hindered applica-tions in the context of self-consistent first-principles ap-proaches. We find here, however, that the method of"superposition of periodic structures" ' (that can bereadily implemented in the context of first-principles ap-proaches) captures closely (Figs. 10 and 11) most of thedetails of chemical and relaxation disorder effects. Theresults of the SQS method await experimental testing.

ACKNO%'I. KDGMKNTS

We thank S.-H. %'ei for helpful discussions and K. C.Hass for valuable comments on the manuscript. Thiswork was supported by the Office of Energy Research,Basic Energy Science (OER-BES), Division of MaterialsResearch, under Grant No. DE-AC02-77-CH00178.

APPENDIX; TRENDS IN DOS PEAKENERGIES IN III-V COMPOUNDS

These trends can be understood qualitatively in termsof the pertinent atomic orbital energies by realizing thatthese are given on an absolute energy scale, while those ofTable II need to be shifted by the relative band offsets tobe on an absolute energy scale.

(i) Table II shows that relative to the respectivevalence-band maxima, P3(GaAs) is 1.4 eV deeper thanP3(A1As), whereas P 1(GaAs) is about 0.4 eV deeper than


Pl(AlAs). Notice that P3 constitute mainly cation s andanion p states while Pl is mainly anion s states (with asmall amount of cation s). On an absolute energy scalethe VBM of A1As is 0.4 0.5 eV deeper than that ofGaAs. Shifting the AlAs DOS to deeper binding energiesby this amount shows that P3(GaAs) is —1 eV deeperthan P3(A1As) whereas the two Pl states are approxi-mately degenerate. This follows the trends in the atomicorbital energies: the Ga,s orbital is —1.2 eV deeper thanAl, s.

(ii) Relative to the VBM, Pl(GaAs) is 0.8 eV deeperthan Pl(GaP), when both constituents are taken at theirequilibrium lattice constants, a, . On an absolute scale,the VBM of GaP is -0.4 eV deeper than that of GaAs(Ref. 45) (both at the alloy intermediate lattice parametera,&&,„), so on this scale Pl(GaAs) is only -0.4 eV deeperthan Pl(GaP). This is consistent with the fact that theatomic s orbital energy of As is -0.7 eV deeper thanthat of P and that GaP has shorter bond length, so itsbonding states have lower energies. P3(GaAs) is -0.1

eV deeper than P3(GaP) when both are measured withrespect to their valence-band maxima at a, . On an abso-lute scale the difference is 0.3 eV, GaP being now deeper.Indeed, the P p atomic orbital energy is -0.3 eV deeper

than that of As. At a,~&,„,however, P3(GaAs) deepens by0.3 eV and P3(GaP) becomes shallower by 0.2 eV, sothese hydrostatic effects reverse again the P3 energy or-der.

(iii) Relative to the VBM, P3(GaAs) is 0.9 eV deeperthan P3(InAs) while Pl(GaAs) is 0.3 eV deeper thanP 1(InAs) when both constituents are taken at theira, values. The band offset between GaAs/InAs is only-0.2 eV, so on an absolute scale the P3 difference is in-creased to 1.1 eV whereas the P1 difference is about 0.5eV. Indeed, the Ga s orbital energy is 0.7 eV deeper thanthat of In, and GaAs having shorter bond lengths, haslower bonding energies. However, at a Qpy the P3 energydifference in the solid is diminished to near zero, whereasthe P1 difference is 0.3 eV, with InAs having lower ener-

gy. This rejects the fact that dilation of GaAs reducesthe binding energies of its P1 and P3 states, whereascompression of InAs increases the binding energies of itsbonding states (Fig. 4 and Table II). This reverses thetrends seen at a, .

In summary, we see that the trends in the bonding-state energy levels can be explained semiquantitatively onthe basis of their initial atomic values and a tight-bindingdescription of the binary compounds.

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" The LDA-claculated semirelativistic s orbital energies of Al,Ga, In, P, and As are —7.91, —9.25, —8.56, —14.09, and—14.77 eV, respectively. The p orbital energies are —2.86,—2.82, —2.78, —5.68, and —5.42, respectively.

~See S.-H. Wei and A. Zunger, Phys. Rev. Lett. 59, 144 (1987},and references therein.

45R. G. Dandrea and A. Zunger, Appl. Phys. Lett. 57, 1031(1990). Neglecting spin-orbit effects, the I 5, offset in theGaAs-GaP system is 0.52 eV and the I 3, offset is 0.12 eV, sotheir 2:1 average used here is 0.39 eV.G. Martinez, in Handbook on Semiconductors, edited by M.Balkanski (North-Holland Amsterdam, 1980), Vol. 2, p. 181;A. Blacha, H. Presting, and M. Cardona, Phys. Status Solidi8 126, 11 (1984).

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Electronic structure and density of states of the random ...7948 RITA MAGRI, SVERRE FRQYEN, AND ALEX ZUNGER tuations in different properties survive averaging. The various theoretical