PHYSICAL REVIEW B VOLUME 44, NUMBER 20 15 NOVEMBER 1991-II

Spontaneous surface-induced long-range order in Gao 5Ino 5P alloys

James E. Bernard, S. Froyen, and Alex ZungerSolar Energy Research Institute, Golden, Colorado 80401

(Received 25 April 1991)

Previous total-energy calculations for bulk Gao 5Ino 5P alloys have demonstrated that the lowest-

energy configuration at T=O corresponds to phase separation into GaP+InP, followed by the orderedGaInP2 chalcopyrite phase as the next lowest state; the (111)-ordered CuPt-like superstructure is predict-ed to lie at a much higher energy. Yet, vapor-phase crystal growth has shown CuPt-like long-range or-dering in relatively thick Gao 5Ino 5P films grown on a lattice-matched (001) GaAs substrate. We presenthere first-principles local-density total-energy calculations for Ga05Ino 5P/GaAs(001) in various two-dimensional structures, each having a free surface. For one-monolayer coverage, we find electronicallydriven surface reconstructions, consisting not only of the previously known cation dimerization, but alsoof buckling and tilting of the surface dimers. These considerably stabilize the CuPt-like surface topologyover all other forms of surface order, including phase separation. Furthermore, a Ga/In layer covered

by three monolayers still exhibits a significant energy preference (relative to kTg, where Tg =900 K is thegrowth temperature) for the CuPt structure. If complete atomic mobility were to exist irrespective ofhow deeply buried the atoms are, we would then expect that the surface-stable CuPt ordering would ex-ist in the near-surface regions, whereas deeper layers would revert to the bulk-stable structures. Since,however, surface atomic mobilities are far larger than bulk mobilities, it is possible that surface-stabilized structures will be frozen in and consequently ordering will propagate into macroscopic film di-

mensions. In light of our results, we describe several possible ways that surface effects could lead tolong-range CuPt-like ordering.

I. INTRODUCTION: CAN ORDEREDINTERSEMICONDUCTOR BULK COMPOUNDS

BE STABLET

Isovalent pseudobinary semiconductor alloysB C are widely used in electronic devices, ' princi-

pally because they allow a continuous range of materialsparameters, tunable by changing the composition x.Their utility in such applications rests largely on the pos-sibility of achieving a single, homogeneous solid phase.Thermodynamically, however, such disordered single-phase bulk alloys are expected to be stable only at rela-tively high temperatures. This can be seen by consid-ering the excess enthalpy AHb„&k of phase n in bulk formdefined as the energy of a with respect to the energy ofequivalent amounts of the constituents AC and BC attheir bulk equilibrium lattice parameters a~c and a~c:

bHb„I„=E(a,a ) —(1 x)E(AC, a„c)—xE(BC,a~c) . —

Fitting the measured liquidus and solidus curves hasshown ' that all isovalent III-V and II-VI semiconductoralloys have AHb'„, 'k )0 for the a=disordered (D) phase.Hence such disordered alloys are expected to be thermo-dynamically stable only above a "miscibility gap" (MG)temperature TMG, where the negative entropy term—TMzhS overwhelms AHb„I'k. Classic theories ex-plained why EHb„j'k )0 by postulating that all AC-BC in-teractions in such systems are fundamentally repulsive,due to the increase in elastic energy upon mixing constit-

uents of dissimilar atomic sizes. Since the present paperdeals with a surface-induced mechanism for ordering inIII-V alloys, we first summarize in this section thecurrent understanding of bulk ordering mechanisms, sothat the need for alternative mechanisms becomes ap-parent.

First-principles total-energy calculations ' haveshown that AHb„&k consists of three contributions.

(i) A non-negative energy term associated with volumedeformation of the constituents into the alloy's volume.This uolume deformation -energy vanishes for size-matched binary components (e.g. , A1As-GaAs andCdTe-HgTe); otherwise, it scales with (b,a ), as correctlysurmised by the classic elastic models. This termtends to drive phase separation for alloys with b,a %0.

(ii) A strain relief term -rejecting energy-loweringatomic relaxations that reduce the strain associated withpacking difFerent bond arrangements onto a fixed lat-tice. ' ' ' Like the volume-deformation energy, thisterm is small for nearly size-matched components, butunlike it, the strain-relief term depends sensitively on theinterfacial symmetry of the atomic arrangements.Among the pseudobinary adamantine 3

&B C systems,

the chalcopyrite structure (Fig. 1) is best able to accom-modate dissimilar tetrahedral arrangements, ' whereasamong binary adamantine 2, „B phases, the zinc-blende and the rhombohedral' RH1 structures have thelowest (in fact, zero) strain energies. Since the random al-loy manifests a statistical distribution of (both stable andunstable) local atomic arrangements, it is overall lessable to accommodate strain than these special orderedstructures. Strain relief can therefore drive selective or-

11 178 1991 The American Physical Society

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . . 11 179

Chalcopyrite CuAu- i CuPt

OAQB

C

FIG. 1. Some of the observed ordered 3D structures ofadamantine pseudobinary ABC2 semiconductors.

dering of the random alloy: for certain strain-minimizingordered structures 0 one can have AHbulk ~Hbulk

(0) (D)

gardless of the sign of the two individual formationenthalpies. The sum of the volume-deformation andstrain-relief energies (total strain) can be zero only in to-pologically unconstrained" binary structures (e.g., zincblende and RH1), which possess sufficient structural de-grees of freedom to accommodate all of the bonding con-straints. All pseudobinary adamantine structures areconstrained, so for h a %0 their total strain energy is posi-tive. '

(iii) The charge excha-nge energy reflects chemical in-teractions attendant, e.g., upon charge transfer betweenthe constituents. ' ' ' For size-matched systems thiscan be conveniently modeled by the Madelung energy ofa fixed lattice, ' showing that among pseudobinaryadamantine structures this term is lowest for theA C +BC (phase-separated) system, followed by the(111)-oriented ( AC)~(BC)~ superlattices and the randomalloy. For size-mismatched systems, charge, in general, istransferred from the constituent with the smaller to thatwith the larger lattice constant. ' This lowers the ener-gy' if the direction of charge Qow is toward the constitu-ent with the larger electronegativity, i.e., if the smaller ofthe two constituents has the smaller electronegativity.

The results of quantitative first-principles total-energycalculations for a number of IV-IV, III-V, and II-VI in-tersemiconductor compounds in various structures'can be summarized in light of the above analysis, thus ad-dressing the question posed in the title of this section, asfollows.

(i) In size matched -pseudobinary systems (e.g., AIAs-GaAs or CdTe-HgTe), the constituent-strain and strain-relief energies vanish separately for all structures. Hence,the stability order is determined by the short-range chern-ical interactions, which are weakly repulsive. The lowestenergy then corresponds to phase separation, followed bythe (111)superstructures and by the random alloys as thenext lowest energy phases. We hence do not expect, onthe basis of thermodynamics, any ordered bulk intersem-iconductor compounds in this class of materials.

(ii) In size mismatche-d binary systems there are twocases. First, in Si& „Ge the stability sequence is phaseseparation, followed by the random alloy, then the rhom-

bohedral RH1 structure. Here, phase separation intoSi+Ge is favored both by vanishing strain and charge-exchange energies, both of which are positive in the com-bined SiGe system. The "topologically unconstrained"RH1 (Ref. lg) and zinc-blende structures have vanishingtotal strain energies but unfavorable charge-exchange en-ergies. The second case is exemplified by SiC. Like SiGe,SiC too has vanishing total strain energy in the topologi-cally unconstrained structures. " However, strong chargetransfer stabilizes its ordered phases (e.g., zinc-blende andits polytypes) over phase separation. " Hence, there arestable compounds for bulk SiC but not for Sioe.

(iii) In size mis-matched pseudobinary systems, such asGa, In P or GaAs, Sb, the stability sequence isphase separation, followed by the chalcopyrite structureand the random alloy; the (111) superstructures (Fig. 1)have the highest energies in this sequence. Again, phaseseparation is favored by vanishing strain and charge-exchange energies, while chalcopyrite is favored over theremaining structures by the favorable strain-relief energy.In the special case of the chalcopyrite phases of A1InP2and A1InAs2, a coInbination of favorable strain relief andsignificant charge transfer from the smaller Al atom tothe larger and more electronegative In atom leads to theprediction' that the CH phase of these two compoundsis absolutely stable, so AHb„&~' & O. Hence, size-misrnatched pseudobinary semiconductors are character-ized by metastable chalcopyrite bulk ordering for all butA1InP2 and A1InAsz for which stable chalcopyrite bulkordering is predicted; all other ordered structures of thisclass are expected to be unstable.

II. INTKRSKMICONDUCTOR COMPOUNDSIN THIN FILMS

All of the above expectations correspond to bulk equi-librium conditions appropriate to cases where the grow-ing alloy is free to attain its three-dimensional minimum-energy crystal arrangement. ' Vapor-phase crystalgrowth techniques such as organometallic vapor-phaseepitaxy (OMVPE) or molecular-beam epitaxy (MBE) in-volve thin three-dimensional films that are often growncoherently on a fixed substrate. Recently, such OMVPEand MBE growth experiments have revealed spontaneouslong-range order in a number of isovalent pseudo-binary and binary semiconductor alloys. Theseordered structures (see Fig. 1) can be described as short-period superlattices ( AC) (BC) whose atomic layers areoriented in the direction G (not necessarily the growthdirection). The CuAu-I-like structure (CA) has p =q =1and G = [001], the chalcopyrite (CH) structure hasp =q =2 and G= [201] and the CuPt-like (CP) structurehas p =q =1 and G=[111].Observations of CA order-ing in AIGaAs2/GaAs (110) were reported in Ref. 20,while CH ordering in InGaAsz/InP (110) was describedin Ref. 31. A similar CH ordering was seen inGa2AsSb/InP (100) in Ref. 32, while the same materialwas reported also to order in the CP structure in Ref. 33.Indeed, CuPt ordering seems to be the most prevalentform: It has been seen also in In2AsSb/InSb (100),

11 180 JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

A1InP2/GaAs (100), GalnAs~/InP (100) andGaInAsP, 3 InA1As2/InP (100), Ga2AsP on a (100) al-loy substrate and A1GaInP/GaAs (100).Rhombohedral-type ordering was seen in Si, Ge„.

The prototype examples for ordering in pseudobinaryB C alloys that we wish to discuss here are the CP

ordering of GaInP2/GaAs (100) and the CA ordering ofAIGaAsz/GaAs (110). Our foregoing discussion on thestability of various bulk intersemiconductor phases servesto explain why these particular forms of spontaneouslong-range order are especially interesting.

(i) These isovalent alloys are known ' to have positivemixing enthalpies AHb„&I, &0 when disordered, so or-dered phases can represent stable equilibrium only in theunusual case where special structures 0 have EHb„&k &0despite AHb„&'& & 0. Initially, a nu~ber of calculations onA1GaAs2 indicated that AHb„&&' &0 for the observedCA structure. However, these were later shown tobe either incomplete or incorrect, so that in facthH' ' )0 (actually, even' b,H' ') b,H' ' is true).Calculations on GaInP2 also show' thatEHb„&&' & AHb„&'k &0. Hence, the observed ordering can-not represent bulk equilibrium.

(ii) Even if phase separation were slowed down by bulkkinetic limitations, calculations of AHb„&k and our forego-ing discussions show that the next lowest-energy state ofsize-mismatched alloys is the chalcopyrite structure (ob-served in GaAso 5Sbo 5), not the CP structure seen inGaInP2, whereas for the lattice-matched Alp 5Gap 5As al-loy, the next states are the (111)-ordered structures andthe random alloy, not the (100)-ordered CA structure.Therefore, even if the ordering were metastable (in thesense that an excited configuration is grown, then"frozen") bulk energetics indicates that this would not re-sult in the structures that are actually observed forGap 5Inp 5P and Alp 5Gap 5AS.

(iii) In the case of lattice-mismatched systems, coherentepitaxial growth could stabilize the homogeneous phaseby raising the energies of the constituents. Here, oneconsiders the case where the growing film is sufficientlythin so that the homogeneous phase e, as well as itsdecomposition products AC and BC, are all coherentlymatched in the parallel direction a

~~

to the substrate's lat-tice constant a, (all other structural parameters relax sub-ject to this constraint). In this case the relevant excessenthalpy'"' is not that given by Eq. (1) but rather the en-ergy relative to constituents that are strained coherently:

H~ ~ (a'all a ) (1 x)E(AC, all a, )

xE(aC, a„~ =a, ) —.Note therefore that if a, is chosen to match the bulkequilibrium lattice constant of the homogeneous phase a(but a~cAa~cXa ), the existence of a substrate poses noefFect on o, but raises the energies of AC and BC. Thiscould then expose the previously unstable (EHb„Iq &0)homogeneous phase a as epitaxially stable, i.e.,AH,' & 0. Calculations have shown that suchepitaxial-stabilization effects can suppress the bulkmiscibility-gap temperature (indeed, the bulk-insoluble

GaP-GaSb system has been observed to exhibit extend-ed solubility when grown epitaxially). However, suchcoherent epitaxial effects do not alter the relative stabili-ties' of the CH, the disordered, and the CP phasesof Gao 5Ino sP/GaAs. For the lattice-matchedAlo 5Gao sAs/GaAs system coherent epitaxy has no efFectsince aac=aac =a

Hence, our current theoretical understanding suggeststhat neither bulk thermodynamics nor coherent epitaxialeffects can explain the ordered CP structure seen inGaInP2 or the CA structure seen in A1GaAs2. Such ob-servations are nevertheless of great practical interest, asthe optical properties of such "ordered alloys" are sub-stantially different from those of the disordered alloys ofthe same composition. This opens the possibility oftuning alloy properties (through selection of growth pa-rameters controlling ordering) even at fixed composition.Before abandoning a thermodynamic reasoning for order-ing in favor of some "kinetic factors, " we examine nextthe role that surface energetics might play.

III. THE POSSIBILITY OF SURFACE-INDUCEDINTERSEMICONDUCTOR ORDERED COMPOUNDS

There are experimental indications ' ' ' suggestingthat the CP-type ordering could be induced at the freesurface during growth. If the [111]ordering were of bulkorigin one would expect all four I ill] variants to bepresent (these are equivalent in the bulk by symmetry);only two, however (the [111]and [111]denoted as CP~;the other two are CP„), are seen. ' ' Furthermore, agiven type of ordering is frequently observed only forgrowth on a given substrate orientation. We will there-fore consider the excess energies of various epitaxiallycoherent phases a, each having a free surface:

bH', „,'z E(a,„,r, a~~=a—,—)

—(1 x)E( AC,„,&,—

a~~ =a, )

—xE(aC,„„,a„=a, ) .

Like in the three-dimensional (3D) structures underlyingEq. (2), here too, coherent epitaxy will tend to suppressphase separation. However, unlike the 30 case, the ex-istence of a free surface permits both new, strain-relievingrelaxations and electronically driven reconstructions. Wewill therefore investigate the possibility that the lowest-energy atomic topology at the free surface of an alloy isqualitatively difFerent from that of the three-dimensionalbulk. While surface structures without bulk counterpartsoften have been observed in meta1 alloys, ' these persistonly for a few monolayers. The far slower bulk diffusionin semiconductors could, however, cause the surface-stable topology to be retained metastably after the surfacehas been covered by growth of subsequent layers. Thiswould lead to the unusual effect of surface-induced order-ing that propagates to macroscopic length scales. Wefocus here on Ga05In05P on the (001) GaAs substrateand contrast AH,'„,'f for this alloy with the correspondingvalues for Alo 5Gao sAs, which does not order on (001)substrates [it does on the (110) surface, which is notconsidered here].

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . .

Previous theories ' ' have considered elasticallydriven surface re-laxation (not reconstruction) effects as apossible driving force for ordering. We find (Sec. V),however, that the resulting energy differences betweenthe various competing topologies are too small to accountfor ordering at a (typical) growth temperature (T ) of 900K. Another model discussed ordering in terms of adhoc "growth rules, " which were designed to produce theobserved CP structure but lack any other justification.We have performed first-principles total-energy calcula-tions for P-terminated and Ga/In-terminated GaInP2 bi-layers on a GaAs substrate, permitting the cation layer totake up different topologies, ranging from phase separat-ed to various ordered structures. The presence of such afree surface leads to surface reconstructions. Our calcu-lations show that for the cation-terminated surface suchelectronically driven surface reconstructions (i.e., dimeri-zation, buckling, and tilting, see Sec. V) stabilize the observed CPti structure of GaInP2 (but not AlGadsg overother structures by 84 meV per surface atom, so this typeof surface ordering is predicted to be thermodynamicallypreferred at Tg. We have next considered structuralpreferences for "buried" Ga/In layers. While coverage iscertain to undo the reconstruction of buried layers, it ispossible that the reconstruction of the top (surface) layerwould exert (chemical or elastic) effects a few layersbelow the surface, thus stabilizing there a certain atomic"topology" over the others. In systems with buriedGa/In layers the Ga/In layers below the top two layershave a bulklike environment, so we use the computation-ally simpler valence-force-field (VFF) method (whoseparameters are fit to first-principles results) to calculatethe relative stability of such "buried" layers. We consid-er various arrangements of 2, 4, or 6 monolayers (refer-ring to each cation and anion layer as a monolayer) ofGaInP on a GaAs substrate of at least 11 monolayers,with the deepest Ga/In layer denoted as the hth subsur-face layer. The top two monolayers (surface h =0 andfirst subsurface h =1) are kept fixed at the geometrydetermined from first-principles calculations, with allremaining atomic positions being relaxed to minimize theVFF energy We find . that CPti is selectively stabilizedalso for h =3. Our calculations hence show that surfacereconstruction not only favors atomic arrangements thatare unstable in the bulk, but that it also results in energydifferences large enough to produce order at T . Ofcourse, the chemical or elastic influence of the top surfacecan penetrate only a few layers into the film. Hence, eventhough we find that the CPz structure is energeticallypreferred at the third subsurface layer, this structurewould not propagate much deeper if atoms could diffusefreely inside the bulk (recall that the CH, not the CPstructure is the stablest in bulk form). However, if atom-ic mobility decreases sufFiciently rapidly below the grow-ing surface, the CPz structure formed at the surfacewould be frozen in and perpetuated throughout the film.In light of our results, we propose in Sec. VIII severalpossible scenarios by which the full three-dimensionalCP~ structure could be formed near the surface andfrozen in by subsequent growth. We next discuss the

meaning of such constraints on atomic mobilities in deeplayers.

IV. METHOD OF CALCULATION

A. Basic premises and approximations

Bulk thermodynamic models assume that sufhcientatomic mobility exists during growth so that atoms canalways explore and find their global equilibrium posi-tions; the microscopic structure of the crystal then doesnot depend on the way it was grown. While this is oftenthe case in melt and solution growth, low-temperaturevapor growth techniques are characterized by large sur-face mobilities but by small mobilities for atoms in thesolidified, buried layers. Our basic premise here will bethat at the free surface, atoms will take up only energy-minimizing positions, but that when buried by subsequentlayers the topology (but not the reconstruction) estab-lished at the free surface is frozen in, even though it nolonger represents the lowest energy of the bulk. The term"topology" requires some explanation: we knowthat, e.g., cation-terminated (001) surfaces of zinc-blendesemiconductors exhibit dimerization and possibly otherforms of reconstruction. For an alloy surface we need todistinguish between such structural degrees of freedom(which we denote as "geometry") and the overall surfacepattern or connectivity of the alloy atoms (which wedenote as "topology" ). We hence assume that subsequentcoverage of the surface freezes in the surface-stable topol-ogy (although coverage must, of course, undo surface-driven relaxation and dimerization). The number of ad-layers h, required to effectively halt atomic diffusion isunknown, and will be treated as a parameter. In bulkequilibrium h, —+ ~.

If h, is a single monolayer, one expects that theenergy-minimizing Ga/In topology of the cation-terminated surface would be perpetuated during growth,since each new Ga/In layer would adopt this topologyand each buried layer would be frozen in this topology.We will hence determine first the energy-minimizingstructure of the cation-terminated surface (Sec. V). If h,is larger, one expects that the structure and topology ofthe top layer could still influence the topology a few lay-ers below, e.g., through subsurface strain. Conversely,ordered buried layers could influence the placement ofthe surface reconstructions, an effect that could producethe correct layer phasing necessary to build up a truethree-dimensional CP~ structure. We will therefore cal-culate next AH', „,'f for various Ga/In layers that are 1, 2,3, 4, and 5 monolayers below the surface (Secs. VI andVII). Finally, if h, is even larger, one expects to recoverthe result of three-dimensional epitaxial calculations.

The present approach differs from Monte Carlogrowth-simulation models, ' which explore the effecton the free surface of kinetic variables such as activationbarriers for adsorption, migration, and desorption of indi-vidual atoms. Instead, we assume a complete (fiat) sur-face and explore its equilibrium structure and energy.Our approach is also distinct from molecular dynamics

11 182 JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

simulations, which study the time evolution of the ki-netic pathway, or from stochastic growth models, ' whichsearch for a set of empirical "growth rules" producing(through atomistic simulations) a desired atomic orderwithout minimizing any energy functional. Hence, ourbasic approach consists of exploring first the extent towhich surface equilibrium efFects [through Eq. (3)] canexplain ordering, falling back on kinetic arguments onlywhen the former approach fails to explain the observa-tions.

We describe the (001) surface structure of the alloy us-

ing a (2X2) surface unit cell. This cell is large enough todescribe the topologies of several common bulk struc-tures (including CP, CH, and CA) yet simple enough tobe treated using the first-principles pseudopotentialmethod. It differs from the observed and calculated(2X4) structure of GaAs (001), which is believed to be apartially covered surface.

There is evidence that surface steps are important forthe selective growth of one of the two subvariants ofCP& (either [ill] or [111])and may be responsible forperiodic lateral phase separation observed, e.g., inAl, Ga As/GaAs (100). It has also been speculatedthat steps may be necessary for the correct CPz stackingof the layers during growth. Our results show, however,that the CP~ structure can be achieved without the pres-ence of steps, although step motion and other kinetic fac-tors could determine the relative proportions of the twosubvariants of CPz.

B. Structure of supercells

The total excess energy b,HI„,)f of Eq. (3) is calculatedin a repeated slab geometry. Each structure has a fewmonolayers of GaAs representing the substrate and a fewlattice-matched GaInP layers on top of it. Above themwe retain a few empty layers; this structure is then re-peated periodically. Our basic computational strategy is

to compare the energies of such supercells, varying (i) theatomic topology of the Ga/In layer(s), and (ii) the depth h

of the deepest Ga/In layer below the surface (h =0denotes the surface layer). We hence fix the topology ofthe unit cell (i.e., the connectivity of atoms) but permitrelaxations and reconstructions that lead to a minimumenergy structure for each topology.

Considering first the energies of a single Ga/In surfacelayer on a (001) substrate we ask whether bH', „,)f of Eq.(3) can explain (i) the preference of ordering (CA, CP,CH) over phase separation in GaInP; (ii) the preferencefor CP over other structures (CA, CH, etc); (iii) thepreference of CP~ over CP~; and (iv) the absence of or-der in the lattice-matched A10~Gao ~As/GaAs (001) al-loy. These questions can be phrased in terms of the rela-tive stability of the five prototype (001) alloy bilayersshown in Fig. 2. By stacking these layers in differentways, several three-dimensional structures are obtained(Table I). These questions are addressed by calculatingthe appropriate hH,'„,'f using the first-principles pseudo-potential method. Second, we ask whether an energeticpreference persists between structures a —e if such layersare buried h monolayers below the surface.

C. Pseudoyotential calculations

Two types of slabs were used. The first, which wasused for most of the cation-terminated surfaces, consistedof three atomic substrate layers (Ga-As-Ga) covered oneach side by single, cation-terminated bilayers of GaInP2(or AIGaAsz) in the topologies a —e (of Fig. 2) andseparated by four empty layers. The bottom and top lay-ers are kept identical to prevent spurious charge transfer.The second type was a single-surface slab with aGao ~Ino ~P top surface and a GaAs bottom surface ter-minated by half a monolayer of Ga. The bottom sur-face is semiconducting; this prevents spurious charge

TABLE I. 3D structures characterized by stacking of the (001) bilayers shown in Fig. 2. The struc-tures are identified both by a superlattice notation (the direction Cx and the repeat period 2n) and by thename used in the text. Each layer is shifted laterally as indicated in parentheses (in units of the zinc-blende lattice constant). Several other structures are degenerate with those tabulated: [110] n =1 isidentical to [001] n = 1 CA; [201] n =2 CH is degenerate with [021] n =2 CH, and [010] CA is degen-erate with [100] CA. The [102] n =2 and [012] n =2 CH structures have (001) layers that cannot berepresented by the 2X2 patterns in Fig. 2. Notice that the d layer occurs only in the observed CP&phase and in the [110]n =2 superlattice. The two are distinguished only by the third layer stacking.

StructureName

Bilayer number2 3

(binary)

(binary)

[021]

[100][001][111][110]

[111][110]

CH

CACA

CP

CP~

a (0,0)

b(0, 0)

e(0,0)

e(0,0)a(0, 0)c(0,0)

c(0,0)

d(0, 0)

d(0, 0)

a( 2,0)& ( —,',o)

e ( —,', o)

e ( —,', 0)b(-,',0)c( —',0)

c ( —',0)

d( -', 0)

d(-,',0)

a (0,0)

b(0, 0)

e(0,0)a(0, 0)

c(0,0)

d(0, 0)

a(-,', 0)

b ( —', 0)

e(0, —,'

)

e ( —,', 0)b ( —', 0)

c(0, ——')c( —',0)

d(0, —')d( —',0)

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . . ll 183

(a)

0 .. 00::

(c)

VNC;; 'I

::: 0 0:::''. 0.. 0

(d)

/

.. '6 '. 0:::, 0 0.'

. 0. '0

(e)

,.''0 '.. 0

''. 0 .'~0

0 Ga surface

Q In surface

~ P subsurface

[010]

= [100]

FIG. 2. Surface atomic arrangements for the 2X2 surfacecell. See Table I for the description of the bulk structures thatcan be obtained by different stacking of these surface unit cells.

transfer between opposite surfaces. The single-surfaceslab is computationally more expensive but it allows re-laxations five layers below the surface and was used tocheck oux results. We use the first-principles local-density pseudopotential method with the exchangecorrelation potential of Perdew and Zunger to calculatethe total energies of the slabs; quantum-mechanicalforces are used to find equilibrium geometries. In orderto reduce the computational cost, we generated "soft"pseudopotentials using the method of Vanderbilt. Thisallows us to achieve basis-set convergence with a relative-ly small plane-wave cutoff of only 10 Ry. The resultinglattice constants of the binary constituents (GaP, InP,GaAs, and A1As) are about 4% smaller than experiment;the deviation, however, is uniform so lattice constantmatches and/or mismatches are preserved. We use a 2Dprojection of the 10 face-centered-cubic special k pointsto perform the surface Brillouin-zone integrals.

We have performed several tests in order to check thatinteractions between the two surfaces of our seven-layerslab did not bias our results (the geometry was separatelyrelaxed in every case): (i) We added four additional layersof GaAs to the substrate and compared the unrecon-structed surfaces (a +b) and e. The energy differencechanged by less than 3 meV per surface atom. (ii) For thefully relaxed geometries, we shifted the upper half of theslab by half the lattice constant of the surface unit cell.Energies changed by less than 6 meV. (iii) We fixed theatomic positions in the central layers of the slab in theirideal zinc-blende positions. Fixing one layer increasedthe energy of the e surface by 7 me, and fixing three lay-ers raised the energy by 17 meV. (iv) Finally, we com-pared our results from the normal slab, with those usinga single-surface slab. The energy difference between thefully relaxed d and e surfaces changed by less than 4meV. This last test allowed surface-induced relaxationsto propagate five layers into the bulk. Based on thesetests, we estimate that energy differences between the

various surfaces are accurate to better than 10 meV persurface atom.

D. Valence-force-Seld calculations

Calculations for supercells representing thick (up to 17monolayers) systems are done using the valence-force-field method. The total elastic energy of such supercellstructures is minimized subject to the constraint that thetop two GaInP2 layers are fixed at the geometry found inself-consistent pseudopotential calculations. Hence, un-like Boguslawski, "who used an unreconstructed sur-face, we explore here the energetic consequences of sur-face reconstruction on structural selectivity a few layersbelow the surface. We use as input to the VFF equilibri-urn bond lengths and elastic parameters deterxnineddirectly from the pseudopotential calculations for zinc-blende GaP, InP, and GaAs and InAs. These differsomewhat from the empirical values tabulated, e.g., inRef. 68. The structural optimization is carried out bydirectly relaxing the coordinates of the (nonfrozen) atomsusing a conjugate-gradient method. Since the top twomonolayers are frozen in the pseudopotential-determinedgeometry, we omit from the energy calculation all bondslying within those layers, as well as all bond angles in-volving exclusively atoms within those layers.

V. PSEUDOPOTENTIAI. STUDIES OF THECATION- TERMINATED SURFACES

A. Relative energies of various surface topologiesof cation-terminated GaInP2/GaAs (001)

1. Relaxed but unreconstructed surfaces

Calculated energies for relaxed but unreconstructedsurfaces are given in the first row of Table II. In agree-ment with the calculations (but not the conclusions) ofBoguslawski, "all energy differences between unrecon-structed surfaces (alloyed or phase separated) are consid-erably smaller than kT, where T -900 K is the growthtemperature. Similarly, the model of Matsumura,Kuwano, and Oki, ' ' which assumes an unreconstructed1 X 1 surface unit cell, gives, in light of our calculated en-ergies, the wrong ground-state structure and negligibleorder-disorder transition temperature. Clearly, surfaceeffects without reconstruction do not give rise to anysignificant structural preference. Suzuki, Gomyo, andIijima suggested that surface-relaxation effects associat-ed with incorporation of a large atom (In) create lateralstrains that encourage subsequent acceptance of only asmall atom (Ga) next to it. We find, however, that elasticsize effects are more easily accoxnrnodated at the free sur-face (where atoms can relax freely in the perpendiculardirection) than in the bulk, in that the energy differencesindicated in the first line of Table II are smaller thanthose appropriate for the corresponding bulk systems(-40 meV/atom pair). ' Hence, strain relief alone in-duces no structural preferences at the surface. Electroni-cally we find that all unreconstructed surfaces are metals.

11 184 JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

TABLE II. Surface energies for the varoius GaInP2/GaAs (001) reconstruction modes discussed inthe text. The energies are in meV per surface atom relative to the unreconstructed a +b (phase separat-ed) surface (surfaces a and b have their own separate zero of energy). We show separately the results ofthe dimerzied+ buckled+ tilted (DBT) surface with Ga up and with In up.

Surfacegeometry

Surface typea+b C

UnreconstructedDimerizedDimerized+ buckledDBT-Ga upDBT-In up

0—785—732—836—836

0—366—448—564—564

0—575—590—701—701

—684—684

—692—623—799

2—620—602—715—705

2. The reconstructed surface

The situation changes significantly when reconstruc-tions are permitted (Figs. 3 and 4). Surface dimerizationresults in heteropolar (Ga-In) dimers on surfaces c and e(Fig. 4) but in homopolar (Ga-Ga or In-In) dimers on sur-faces a, b, and d (Fig. 3). Relative to the undimerizedsurfaces, dimerization lowers the energy by an average of600 meV per surface atom (Table II, line 2). In additionto dimerization we And in all cases two other energy-lowering reconstructions within the 2X2 surface unit

cell. First, dimers relax perpendicularly to the surfacecreating [110]dimer rows of alternating high and low di-mers [e.g., see Fig. 4(c)]. We will refer to this as "buck-ling. " This motion does not alter the symmetry of the dsurface, where the two dimers are already distinct before

Chalcopyrite like GalnP2/GaAs (001)

(cation terminated)

(a) Relaxed

CuPt - like GalnP&/GaAs (001)

(cation terminated)

(a) Relaxed(b) Dimerized (D)

(b) Reconstructed (DBT)

(c) Dimerized + Buckled (DB)

(c) ionized

(d) Dimerized + Buckled + Tilted (DBT)

FIG. 3. Side view of atomic geometries for the cation ter-minated d (CP) surface of GaInP2. The surface atoms are Ga(white), In (grey), and P (black) on top of a (001) substrate GaAslayer (white). (a) Relaxed but undimerized, (b) fully reconstruct-ed with buckled and tilted dimers, and (c) ionized (Sec. V D).

FIG. 4. Side view of atomic geometries for the various recon-struction modes described in the text for the cation terminated e(CH) surface of GaInP2. The atoms for Ga (white), In (grey),and P (black) on top of a (001) substrate GaAs layer (white). (a)Relaxed but undimerized, (b) dimerized, (c) dimerized and buck-led, and (d) fully reconstructed.

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . . 11 185

buckling, but it breaks the symmetry of the other sur-faces. Buckling lowers the energy of the b surface (TableII, line 3), but raises the energy for surfaces a and e (sothese surfaces will not buckle). Second the high dimertilts in the [110] direction becoming nonhorizontal,whereas the low dimer remains virtually horizontal [e.g. ,see Fig. 4(d)]. This tilt is natural for surfaces with hetero-polar dimers (c and e), but constitutes a symmetry break-ing for the other surfaces [see Fig. 3(b)]. Since the foursurface sites are inequivalent in the final geometry, thereare two different ways of distributing the two Ga and thetwo In atoms in each of the topologies, c, d, and e. Wewill characterize this by the type of atom (Ga or In) occu-pying the site on the high dimer that tilts upwards. Thethird and fourth lines of Table II show that the d surfacestrongly prefers having the larger In atom on the high di-mer, whereas the c and e surfaces show a slight prefer-ence for the smaller (but more electronegative) Ga atomto be tilted up. Tilting leads to a uniform energy lower-ing of 100 meV but does not affect the relative stability ofthe surfaces ' (Table II, lines 3 and 4). This insensitivityof the relative surface energies to tilting of the high dimersuggests that the low dimer might be responsible for theincreased stability of the d surface over the others. Notefrom Table I that among the a +b, c, d, and e surfaces, dis the only one where the low dimer contains only smallatoms (Ga-Ga). Therefore, it is best able to relax into theelectronically optimal (see below) geometry where it be-comes nearly coplanar with the P atoms.

As shown in Table II, the reconstructions (dimeriza-tion, buckling, and tilting) considerably lower the energyof all the surfaces and, most significantly, make the sur-face corresponding to the observed CP& ordering (d) thelowest in energy by 84 meV per surface atom. Thus, sur-face reconstruction not only fauors atomic arrangementsthat are unstable in the bulk, but it also results in energy

differences large enough to produce order at T .Note that in GaInP2/GaAs the CH surface e (ob-

served in Ga2AsSb) is the next lowest-energy structureafter CP~ and that the CP ~ structure (never observed) isthe highest-energy member in this series. Within the pre-cision of our calculation we hence find that surface recon-struction of a cation layer changes the energy orderCH C&A (&a+b) &CPz =CP& found in the bulk, ' 'yielding instead CPs & CH & (a +b) & CP „.

Og ale, Thomsen, and Madhukar noted in theirMonte Carlo simulation of the growth of (001)Alo 33Gao 67As that if one assumes some simple form ofsurface reconstruction this can produce ordering in thefollowing adsorbed layer through modification of kineticparameters. We have shown here that electronic effectsat the surface can lead to a thermodynamic (as opposed tokinetic) preference for ordering. Kondow et al. sug-gested that CP~ ordering in GalnPz/GaAs (100) is kinet-ically driven, since at its growth temperature the bulkdisordered alloy or phase separation should have been ob-served. However, it has been shown before ' thatcoherence of a film with its substrate can remove phaseseparation from contention even without the presence ofa surface or kinetic effects.

B. Electronic structure: A negative-U system

We find that all of the reconstructions in GaP, InP,and GaInP2, are electronically driven. Unreconstructedcation surfaces have two broken bonds per surface atom,each containing —,

' electron. When two surface atomsdimerize, two of their three electrons occupy the bondingorbital [Figs. 5(c) and 5(d)]. This leaves one unpairedelectron per dimer; hence the unbuckled, dimerized(2 X 2) surface is metallic. Energy can, however, belowered by breaking the symmetry through buckling andtilting. Of the four atoms in a dimer pair, we find thatthree relax toward planar sp -like configurations and thefourth atom moves up into a pyramidal s p -likeconfiguration with bond angles approximating 90'. Thisallows the fourth atom to bind the two unpaired electrons

(a)

+As~

(b)

p„bonding(unoccupied

18.5C&

Ga Ga

~A, ~+A, QA9

I I

Dangling bond

In

QA~s

p bonding

QA

(d)&~c.

p bonding

Fi3'A '9~S

FIG. 5. Contour plots of selected surface states for the d sur-face at the I point in the Brillouin zone in two cuts (throughthe low Ga-Ga dimer, left, and the high In-In dimer, right). (a)shows a Ga-Ga p bonding state (lowest unoccupied state), (b)shows an In dangling-bond state (highest occupied state), and (c)and (d) show the deeper Ga-Ga and In-In p bonding states.

JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

in a "dangling-bond" (lone-pair) state [Fig. 5(b)]. Hence,the high atom acts as an "anion, " receiving electronsfrom the low atoms that behave therefore as "cations. "The lowest unoccupied state is a bonding p„state on thelow dimer [Fig. 5(a)]. Note therefore that a pair of di-mers forms a negative-U system where two neutral (hor-izontal) dimers are unstable with respect to dispropor-tionation into a positively charged low (L) dimer and anegatively charged high (H) dimer. For example,

11.00

) (0.

11.12

I34) (0.

1.05 1.21

Cation terminated

GalnP2 lGaAs (001)

00

U(CPB )

(Ga —Ga) +(In —In): (Ga —Ga)L +(In —In)H .(4)

The energy gain involved (difference between the fifth andthe second lines of Table II) is as large as U= —200meV/surface atom.

0.0

5) (0.72) 1.05

(c-1)94

11.06

1) (0

(c-2)

1.06I

26) (0.

00

.10

C. Structure of relaxed and reconstructedGaInp2 surfaces

1.13

8) (0

92'

06

1.13

Figures 6(a) —6(e) tabulate the geometries of fully re-laxed and reconstructed cation surfaces for the neutralGalnP2/GaAs (001) system. In addition to the bond-angle changes already discussed, several features arenoteworthy: (i) The cation-anion bond lengths are mainlydetermined by the dimer type (high or low). The bondlengths for the higp dimers are 6% and 11% longer thanthe bond lengths for the low dimer, whereas thedifference between Ga-P and In-P bond lengths for agiven dimer type is only 6%. (ii) The cation-cation dimerbond lengths are comparable to the cation-anion bondlengths and are longer for the high dimer and for dimerscontaining In. (iii) For a given dimer type (high or low),and occupation (by Ga or In) the geometry is highlytransferable. For example, the low dirner on the a sur-face is almost identical to the low dirner on the d surface.Similarly, compare high b with high d and high c withhigh e. Even the low dimer on the c surface resemblesthe low dimer on the e surface after 180' rotation. (iv)The low dimer, preferring coplanar bonding arrangementwith its P neighbors, moves downward and the P sub-layer atoms relax towards the high dimers in order to fa-cilitate this. Low dimers with two small Ga atoms areable to achieve a more optimal bonding arrangementthan dimers containing In.

0.0

11.11

I

5) (0.

8) (0

11.06

I3) (0.

1.13

7) (0

(d*)

11.00

a G36) (0.

7 11.10

n I52) (0.

00

03

00

04

0.0

(d-2)

11.00

) (0

1.21

0) (0

(e-2)

11.06

) (0.

1.15

0) (0

(e*)

11.06

) (0.

0' 11.06

a I

40) (0.

.06

12

93

.06

12

D. Structural consequences of ionizing the dangling bonds

The foregoing analysis suggests that if the two elec-trons in the dangling-bond state on the high dimer [Fig.5(b)] could be removed, e.g. , by heavy p doping or by ex-citation, both dimers would prefer planar sp -like ar-rangements and the energy advantage of the d surfaceshould vanish. To examine this, we have performed a to-tal energy minimization for the doubly ionized surfaces.This showed that the total energy di6'erence between thed* and e* surfaces (we denote ionized structures by anasterisk) is indeed reduced from 84 to I meV per surfaceatom; hence, the In dangling-bond electrons are indeedresponsible for the buckling of CP~. A side view of theresulting d* surface is shown in Fig. 3(c) and the dimer

FIG. 6. Schematic top view of the 2X2 unit cell depictingfully relaxed and reconstructed geometries (dimerized, buckled,and tilted) of the five prototype cation-terminated GaInP& sur-faces shown in Fig. 2. For the c, d, and e surfaces we show thetwo variants with Ga up (denoted 1) and In up (denoted 2).Parts (d*) and (e*) give the geometries for the ionized d and esurfaces. In addition to bond lengths and bond angles, theheights (in parentheses) of the Ga (white) and In (grey) atomsover the P (black) subsurface layer, and the approximate direc-tion and magnitude of the horizontal displacement of the Patoms from their ideal zinc-blende positions are given. Indivi-dual vertical displacements of the P atoms are all small.Lengths are given in units of the average of the bond lengths ofbulk GaP and InP. In these units, the bulk bond lengths of GaPand InP are 0.96 and 1.04, respectively.

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . . 11 187

geometries are tabulated in Fig. 6 [see (d') and (e*)]. Wewould expect the ionized surfaces to have dimergeometries similar to the low dimer geometries for theneutral surfaces. Comparing the two, we see that thebond lengths are closely preserved but that dimers on theionized surfaces sit higher above the P layer than the cor-responding low dimers on the neutral surfaces. This isbecause the P atoms in the former case are unable to re-lax. The vanishing (d*) vs (e*) energy difference uponionization is a possible explanation for the fact that p-

doped samples show dramatically reduced ordering; italso opens the possibility for undoing the reconstruction(hence, ordering) by optical excitations during growth.This argument also shows that ordering depends on cov-erage: if the cation surface coverage 0 is less than —,

'monolayer, all high dimer dangling-bond electrons cantransfer to the P dangling bonds with a consequent loss ofstructural selectivity. For —,

' &O&1 the transfer is par-tial. We therefore expect that the contribution to order-ing from reconstructed cation surfaces will be reduced forsurfaces with —,

' & 0 & 1 and vanish for 0 & —,'.

0.

0.

0.

0.0

(a)

0.99

) (0.

5

0.1.03

) (0

0.0

1.05

8) (0

7' 1.08

1.04

) (0.

7

0.

1-2)

10.98

0.

1.06

7) (0

7

06

1.08

Cation terminated

A GaAspiGaAs (001)

.09

5

.09

05

E. Absence of ordering in Alo &Gao &As/GaAs (001)

Since Ala ~oao ~As does not order (or orders onlyweakly) when grown on (001) substrates, it is interestingto compare the energies of the GaInP2 surfaces to thoseof A1GaAs2. We separately considered Ga and Al as thehighest atom on the high dimer (Table III). We find thatwhen the Ga atom is in the up position, the energy islowest. Recall (Sec. V B) that the high atom accepts elec-trons from the "low" atoms. Since Ga has a deeper lyings orbital energy than Al (the calculated I.DA s orbital en-ergies for Al, Ga, and In are —7.83, —9. 17, and —8.46eV, respectively) and since the p orbital energies are simi-lar, the system prefers to have Ga, not Al, as the highatom. Most importantly, we find that the energy of sur-face d is now only 9 meV per surface atom below that ofthe e surface, so the energy difference is negligible com-pared to kT . Hence, the resulting surface topology islikely to be disordered. This is consistent with experi-ment, where strong ordering is seen only for growth onthe [110] surface. Figure 7 tabulates the resultinggeometries for these surfaces. Again the major variationis between the high and low dimers. The Ga dimers haveGa—As bond lengths about 2% longer than the Ga—Pbond lengths, but are otherwise remarkably similar to theGa dimers on the GaInP2 surfaces. The Al—As bondlengths are identical to the Ga—As bond lengths, but theAl-Al dimer length is slightly longer than the Ga—Ga di-mer length.

e-1)

11.01

A3) (0.

8

0.

(e-2)

1.01

) (0.

CuPt - like GalnP2/GaAs (001)

(anion terminated)

0.0 0.0 .08

1.07

4) (0

7' 1.05

6) (

0

05

FIG. 7. Schematic top view of the 2X2 unit cell depictingfully relaxed and reconstructed geometries (dimerized, buckled,and tilted) for the a, b, d, and e cation-terminated AlGaAs2 pro-totype surfaces. For the d and e surfaces we show the two vari-ants with Ga up (1) and Al up (2). The notation and units are asin Fig. 6.

FIG. 8. Side view of the single-surface slab used for theanion-terminated surface calculations. The surface atoms are P(black), Ga (white), and In (grey) on top of a (001) substrateGaAs layer (white).

11 188 JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

Ga

0.

(0.0.

Ga

0.

1(0.50.9

Ga

Anion terminated

Ga In

0.87 26'34)

Ga

11(0.41.0

In

0.86

0.9355)97

Ga

(0.

In

0.92

GalnP2lGaAs (001)

In

845)05

In

03

66)03

F. Implications for zinc-blende systems

The buckled and tilted surfaces described above shouldbe considered as a possible candidate geometry forcation-terminated surfaces of binary III-V semiconduc-tors. A 2X4 cell is frequently observed, ' and is be-lieved to be a partially covered, dimerized surface withone out of four dimers missing. It should be noticed,however, that a variation of our buckled surface, with adimer sequence up-up-down-down, would also result in a2X4 cell. It is possible, however, that the elastic relaxa-tions in the phosphorus sublayer (see Fig. 6) make thisvariation energetically unfavorable. Another possiblemodification would be a translation of every second di-mer chain by a/V'2 along the chain. This would lead toa (4X2) cell with alternating high-low dimers also in thedirection perpendicular to the dimer chains. We find thatthe chain-chain interaction is weak, and that this "chainsliding" modifies the energy by less than 10 meV/surfaceatom.

ln Ga In (e) Ga VI. PSEUDOPOTENTIAL STUDIESOF THE ANION- TERMINATED SURFACES

1(0.1.

In

(0.

10.85

0.93

2730)

Ga

02'.54}96

Ga

126(o.0.

Ga

0.

(0.1.

In

10.87

7' 10.93

26'35}04

02

61)96

Ga

FIG. 9. Schematic top view of the 2X2 unit cell depictingfully relaxed and reconstructed geometries (dimerized, buckled,adn tilted) for the a, b, d, and e anion terminated GaInP2 protot-pye surfaces. The notation and units are as in Fig. 6.

We next ask how surface energies are modified whenthe cation layers are covered by phosphorus. We denotethis first subsurface layer by h =1. As before, we use a2X2 unit cell and calculate the energies of the prototypetopologies (a, b, d and e on a GaAs substrate) but allcovered by a single monolayer of P that is allowed todimerize. The P-P dimer length is found to be shorterthan the Ga-In cation dimer length, leading to muchlarger subsurface relaxations (compare the cation sur-faces in Figs. 3 and 4 with the anion surface illustrated inFig. 8). For this reason we exclusively used a single-surface slab with 7—,'-monolayer thickness. The resultingdimer geometries are tabulated in Fig. 9. In addition tothe dimerization we observe a buckling of the alternatedimers. The buckling is less than that for the cation sur-

X P Y P Ga As GaAs

X=CPA100-

c 60—~~

e 40—

~) 0

CPA CQ CH

X =CPB

CPA CQ CH

X=CH

CPA CQ CH

X=PS

CPA CQ CH

FIG. 10. Strain energies of GaInP2 due tostructural changes in the second subsurfacelayer (h =2). Each panel gives the energies ofthe c, d, and e topologies in layer Y (relative tophase separation a +b), for specific topologiesin layer X. The empty and shaded bars corre-spond to translations of the pattern by a /&2in the [110]or [110]directions.

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . . 11 189

Dimertype

Al upGa up

00

Surface typed

38—35

7—26

TABLE III. Surface energies for fully reconstructed cation-terminated Alo &Ga05As/GaAs (001) system. The energies arein meV per surface atom relative to the a +b phase-separatedsurface. The dimer type (Al up or Ga up) refers to the atomfarthest from the surface.

mirror plane being vertical and containing the dimer),whereas the pseudopotential-determined geometry exhib-its a small breaking of that symmetry caused by the resid-ual interaction between the (mutually rotated) top andbottom surfaces. This symmetry breaking causes a smallspurious splitting of the energies of the CP~ and CH sub-variants at h =2, and to avoid this we have imposed thesymmetry on the surface geometries used to generate Fig.10 (only). The results show that all subsurface cation ar-

face (compare Figs. 6 and 9), and the dimers are horizon-tal. The surfaces are still metallic and therefore likely toundergo additional reconstruction. Using argumentssimilar to those we used for the cation surfaces, a semi-conducting surface might possibly be achieved in a 4 X 2cell with three high dimers followed by a low. Within the2 X 2 cell we find that the e surface has the lowest energy,nearly degenerate with d (3 meV higher) and that thephase-separated (a+b) is 33 meV higher. Thus, theh = 1 layer affords little structural selectivity.

VII. VALENCE-FORCE-FIELD CALCULATIONSOF BURIED CATION LAYERS

Having discussed the relative energies of the cation-terminated (h =0) and anion-terminated (h = 1) surfaces,we now turn to discussion of the energies of more deeplyburied (h ~ 2) cation layers. Figures 10—13 show the slabstructures and results of our VFF calculations of the sub-surface strain energies for h =2, 3, 4, and 5, respectively,where the hth layer (denoted I" in Figs. 10—13) can haveeach of the topologies shown in Fig. 2, i.e., CP~ =c,CP&=d, and CH=e or phase separated (PS). In eachcase the top two layers were frozen in thepseudopotential-determined geometry, and their strainenergy was omitted. The energies are given relative tothe phase-separated state (the average of the pure Ga[Fig. 3(a)] and pure In [Fig. 3(b)] configurations) in thehth layer, since absolute comparisons of the strain ener-gies with difFerent surfaces (denoted X in Figs. 10—13)and/or different values of h are not meaningful on ac-count of omission of surface strain and chemical energiesand the chemical-energy difference engendered whentransmuting a layer of GaAs into GaInP2 or vice versa.Note that the subsurface elastic effects of the surfacereconstruction result in a splitting of the energies of thetwo phase-reversed partners of a particular topology, andthese are depicted separately in Figs. 10—13 by the whiteand shaded areas. We will refer to these phase-reversedstructures as subvariants of CP „CP~, or CH (the term"variant" is reserved for the A and B partners of the CPstructure). The results shown include all of thepseudopotential-calculated surface geometries (X) dis-cussed above. Layer-by-layer discussion of the resultsfollows.

A. Results for h =2

The cation-terminated surfaces should have a mirror-plane symmetry operation located at each dimer (with the

h=3 P X P Y As Ga GaAs

(a) Buckled dimers

125E

100

o 75CI~c 50

25

0

-25

g -50

-75

X=CP

CPa CQ CH

X=CH

CPa CP4 CH

X=PS

CPa CQ CH

(bj Unbuckled dirners

CO

lh

0IE

CA

lPCCP

~~C5

N49CPlO

ill

lhtD

~~CO

4)K

125—

100—

0

-25—

-50—

-75—

X= CPB

CPa CQ CH

X=CH

CPa CQ CH

X=PS

CPa CQ CH

FIG. 11. Strain energies of GaInP& due to structural changesin the third subsurface layer (h =3). Each panel gives the ener-gies of the c, d, and e topologies in layer Y (relative to phase sep-aration a +b), for specific topologies in layer X. The empty andshaded bars correspond to translations of the pattern by a/&2in the [110] or [110] directions. We show the energies of thelayers underneath a P-terminated surface with buckled dimers(a) and with the buckling removed (b).

11 190 JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

X P CPB P Y P GaAs

X=CPA

30—

20—

c q0

o

-10—

-20—I f

~~-30—

cQ cg cH

X=CP X=CH

CQ CQ CH CPa CQ CH

X=PS

CPa CQ CH

FIG. 12. Strain energies of GaInP2 due tostructural changes in the fourth subsurface lay-er (h =4). Each panel gives the energies of thec, d, and e topologies in layer Y (relative tophase separation a +b), for a specific topologyin layer X. The empty and shaded bars corre-spond to translations of the topologies bya /i/2 in the [110]or [110]directions.

rangements X have similar energies, except the high-energy CPz subvariant, whose energy is considerablyhigher than the others. The low-energy CP~ subvariantis favored slightly (by about 10 meV/surface atom) overthe next lowest-energy arrangement, the phase-separatedstate. Note the very large splitting between the CPz sub-variants induced by the surface reconstruction, and therelatively weak dependence of the relative subsurfacestrain energies on the topology of the surface (X). Theformer reflects the strong influence that surface recon-

h=5 P X P CP P Y GaAs

CIth

E

Ul20—

4)

Olr~~6$ka

10—4)0$

lA

rn 0~~CII

IX

X =CPB

CPa CFI, C

X=CH

CPa CQ CH

X=PS

CPa CQ CH

FIG. 13. Strain energies of GaInp2 due to structural changesin the fifth subsurface layer (h =5). Each panel gives the ener-gies of the c, d, and e topologies in layer Y(relative to phase sep-aration a +b), for a specific topology in layer X. The empty andshaded bars correspond to translations of the pattern by a/&2in the [110]or [110]directions.

structions exert on subsurface layers, and the latterreflects the strong similarities in the reconstructions ofthe various cation-terminated surfaces. One can gain aqualitative understanding of the reason CP~ is favoredover the other ordered structures and why the two CP &

subvariants exhibit such large splitting by examination ofFig. 14(a), which shows the VFF-relaxed atomic positionsfor the CPz surface on a thick GaAs substrate. The ar-rows indicate the major relaxation directions of subsur-face atom rows in response to primarily bond-angle con-straints induced by dimerization of surface cations.From these one can see that in the second subsurface lay-er the [110]-oriented row of atoms beneath the surfacedimer row has stretched bonds relative to its parallelpartner that does not lie beneath a dimer row. Thus, ifthat layer is occupied by a mixture of Ga and In atoms,the In atoms will prefer to lie beneath the dimer row, andthe Ga atoms in the adjacent row, thus producing a layerof the CPz type. The other CP„subvariant would havethe rows reversed, and would produce the worst possibleelastic fit, thereby having much higher strain energy.Hence, dimerization of the cation surface both favorsCP z at h =2 and splits the energies of CP „ into two sub-variants in that layer.

B. Results for h =3

For h = 3 shown in Fig. 11 (which has an anion-terminated surface) the energy selectivity is muchgreater, with the CP~ layer structure being preferred byabout 35 meV/surface atom over the lowest-energy CP~subvariant, regardless of the topology of the first subsur-face layer (X). Note also the large splitting of the twosubvariants of CPz and those of CP~. One can under-stand these trends qualitatively by inspecting Figs. 14(c)and 14(d), which show the first-principles-relaxed atomicpositions for the CP~ surface on a GaAs substrate, witharrows indicating major relaxation directions of subsur-face atom rows. In Fig. 14(c) we see row relaxations in-

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . . 11 191

duced primarily by dimerization of the surface P atoms.It is clear that at h =3 one of the CPz subvariants pro-vides the best fit, with the other CPz subvariant provid-ing the worst fit. In Fig. 14(d) we see row relaxations in-duced primarily by buckling of the surface dimer rows,and there it is clear that the two CP „subvariants pro-vide, respectively, the best and worst fits. That one of theCP~ subvariants wins out overall is a consequence of thelarger relaxations induced by dimerization than by buck-ling.

Because the 2X2 cell used in the first-principles calcu-lations may not permit an accurate representation of theground-state surface reconstruction of the anion-terminated surfaces (e.g., the calculated surfaces are stillmetallic), we also show in Fig. 11(b) the subsurface strainenergies of anion-terminated surfaces with the bucklingof the frozen surface removed, i.e., with a (2X 1) surfacesymmetry. In this case the splitting of the CPz and CHsubvariants disappears, as expected. The amount bywhich the lowest CPz subvariant is favored over thenearest competitor (now phase separation) is increased to-90 meV/surface atom, comparable to the energy selec-tivity at the cation-terminated (h =0) surface. This largeenergy preference for CP~ induced at h = 3 by the dimer-ization of the P surface is similar to the mechanism pro-posed by Leooues et al. to explain one type of orderingobserved in SiGe alloys. The quantitative difference inthe degree of energy selectivity between h =2 (Fig. 10)and h =3 (Fig. 11) is primarily a result of the fact thatthe P-P dimers of the anion-terminated surface are moretightly bound than their cation counterparts, thus induc-ing greater subsurface strain. Comparison of the h =3results for buckled and unbuckled surfaces [Figs. 11(a)and 11(b), respectivelyj indicates that splitting of theCP ~ and CH variants by higher-order (than mere dimeri-

zation) reconstructions can alter the details of the picturesomewhat, but the dominant selectivity is due to the di-merization itself, and we expect that the preference forCPz is likely to survive the smaller degree of bucklinglikely to be found in a calculation using a larger surfaceunit cell provided the dimers are organized in chains.

C. Results for h =4

For h =4 shown in Fig. 12 (which has a cation-terminated surface) we find that the relative energies areonly weakly dependent on the topology of the secondsubsurface layer, so we show only those energies forwhich that layer has the lowest energy, i.e., the CP~ layerstructure. Here again CPz is favored over the otherstructures, though the next lowest-energy structure (CH)is only 7—15 meV/surface atom higher in energy, de-pending on the surface topology. Figure 14(b) shows themajor row relaxations, induced primarily by buckling ofthe surface dimer rows. One can see from these that thetwo CPz subvariants provide the best and worst fits ath =4. We are not certain as to the degree to which thepreference for the lowest of the CP~ subvariants over theCH structure would survive the enlargement of the sur-face unit cell to allow more complex higher-order recon-structions, but in any case, the preference is small enoughthat it may be largely lost at growth temperatures. Wenote that the low-energy subvariant corresponds to truethree-dimensional CuPt ordering of the surface togetherwith the second and fourth subsurface layers, whereas thehigh-energy subvariant corresponds to having an anti-phase boundary between the second and fourth subsur-face layers. In contrast, Boguslawski, "who did notconsider reconstructed surfaces, found the antiphaseboundary to give the lower energy.

(a)[«0]

(b)[110]

D. Results for h =5

(c)[«0]

h=3

(d)110

For h =5 shown in Fig. 13 (which has an anion-terminated surface) we again present only results forwhich the third subsurface (cation) layer has the lowest-energy CP~ layer structure, since there is only weakdependence of the relative energies on the state of orderof that layer. Here the stability sequence has nearly re-verted to that of the bulk, ' ' with the phase-separatedand CH states being the most stable and CP the leaststable. We also find little change in the results when thesurface buckling is removed. Of the two CPz subvariantsshown under the CPz surface, the one with the lower en-

ergy (by about 10 meV/surface atom) is the one with anantiphase boundary, whereas the higher-energy varianthas the true three-dimensional CuPt structure.

FIG. 14. Side view of atomic geometries for the cation ter-minated [(a) and (b)] and anion-terminated [(c) and (d)] GalnP2d surfaces. The surface atoms are Ga (white), In (grey), and P(black) on top of a (001) substrate GaAs layer (white). The ar-rows illustrate the relaxation of subsurface atom rows (see text).

K. Summary of subsurface calculations

In summary, we see that the surface dimerization, themost well verified and the most energetically profitable

11 192 JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

(Table II) of the surface reconstructions found in thefirst-principles calculations, induces a strong preferencefor CPz layer ordering at h =3, and weak preference forCP~ at h =2, whereas surface dimer row buckling in-duces a weak preference for CPz layer ordering at h =4.Tilting of surface dimers appears to have no major quali-tative effect on subsurface layers, though electron count-ing arguments suggest that it is closely related to thebuckling of dimer rows. A nearly bulklike stability se-quence is found by h =5. We find that surface energetics(not kinetics) explains (i) why GalnP/GaAs does notphase separate, (ii) why the ordering is of the (111) type(not CH or CA), (iii) why the (111) ordering occurs onvariant 8 (not 3).

VIII. DISCUSSION AND CONCLUSIONS

Our results show that the CPz topology has the lowestenergy both at the free (cation-terminated) h =0 surfaceand at the third h =3 subsurface layer, while past thefourth subsurface layer, the energy order of different to-pologies approaches that of the bulk crystal. Hence, ifatoms could freely move to attain the global minimumenergy, we will have a CP~ structure in the near-surfacelayers, and chalcopyrite inside the epitaxial bulk. As dis-cussed, however, in Sec. IVA, atomic mobility is largenear the surface, and is slowed down considerably indeeply buried layers. To see how this could producelong-range ordering through the film, consider the fol-lowing scenarios:

Scenario (i). Assume that the growing surface has cat-ions exposed for a sufficiently long time so that a locallycomplete (a few surface unit cells) cation-terminatedreconstructed surface can form before subsequent cover-age, and that bulk diffusion following subsequent cover-age is largely ineffective in rearranging the order estab-lished at the surface. Local completion could easily beenvisioned to occur, for example, via growth at steps.Then (Table II) the first cation layer grown on the sub-strate would prefer to order in the CP~ topology(presumably with domains of both subvariants present).When the second cation layer is deposited, and takes theCPz form, no particular preference for phasing (stackingrelationship) between that layer and the buried CP~ layeris expected, so both CP~ subvariants would form withmore or less equal probability. When the third cationlayer is deposited, the strong preference for true CPzstacking at h =4 (rather than formation of an antiphaseboundary), would drive the placement of the surface CP~layer to be appropriate within a given domain to the truethree-dimensional CP~ stacking begun there when thesecond cation layer was deposited. Thus, a mix ofdomains of the two CP~ subvariants would be formed,with the degree of order depending on the degree of sur-face CP~-layer order achieved, as well as on subsequentdisordering of buried cation layers through slower bulkdiffusion.

Scenario (ii). Assume that the growing surface doesnot have cations exposed for a sufficient time needed toform CPz at the surface, and that subsurface diffusionoccurs at a substantial rate to a depth of h =3 but be-comes insignificant thereafter. Then, each mixed-cation

layer, as it passes through a depth of h =3, would rear-range into a CP& layer structure (Fig. 11) as a result ofthe dimerization of the P-terminated surface, regardlessof the state of order of the mixed-cation layer above it ath = 1. However, in the absence of significant lifetimes forcation-terminated surfaces, we cannot invoke the CPzlayer-phasing mechanism that occurs at h =4 and needto look further for a layer-phasing mechanism that wouldbe active with P-terminated surfaces. At present weknow of no such mechanism, though it is possible thatsurface steps may exert some inhuence through a choiceof which pairs of rows of atoms dimerize or through anenhancement of the probability of formation of CP& ath =1 with fixed phase relative to the CPz layer at h =3.The latter would not require diffusion to h =3, but onlyto h = 1 (or at steps). Note that ordering was observed tooccur only in a "window" of temperatures. It is hencenatural to assume in this scenario that if the temperatureis too low, then only layers h =0, 1 are fiuid, and orderingdoes not occur since h = 1 has but a small ordering selec-tivity (Sec. VI). On the other hand, if the temperature istoo high, even layers 4 and 5 are Quid, and again orderingdoes not occur, since these layers too have small structur-al selectivity (Figs. 12 and 13). At intermediate tempera-tures, where diffusion penetrates to h =3 (but no further),ordering is maximized because of the significant energeticpreference for CP~ ordering there (Fig. 11).

Scenario (iii). A variation of the scenario above in-volves the assumption of a locally complete cation-terminated surface existing for a time too short for sur-face diffusion to create much CPz ordering at h =0, butlong enough for the reconstruction of that surface to takeplace. Assume further that there is a gradual falloff ofsubsurface diffusion with increasing depth into the film,rather than a relatively abrupt falloff for h )3. Thenwhen the only slightly ordered surface is buried to adepth of h =3, the CP~ ordering will be enhanced by the(somewhat limited) subsurface diff'usion. At this level,the initially small degree of ordering will select a surfaceP-P dimerization pattern that will, in turn, enhance theexisting CPz order at the expense of the other variant.When this layer passes to h =4, it will dictate the patternof cation-surface reconstruction such that the small de-gree of CPz order achieved there will have the correctphase with respect to the h =4 layer, so that true three-dimensional CP~ ordering results, though the imperfec-tions due to rapid coverage of the cations at h =0 and thelimited subsurface diffusion at h =3 would be substantial.

All of our results and the foregoing discussion have ap-plied to mixed-cation materials, in particular GaInPz.However, spontaneous ordering into the CuPt structurehas been observed in a number of mixed-anion III-V sys-tems as well, and it has been conclusively verified thatthe structure observed there is also CPz, rather thanCP ~. Some previous hypotheses concerning mechanismsfor surface-driven ordering correctly predict CPz or-dering in mixed-cation systems, but erroneously predictCP ~ ordering in mixed-anion systems. Our calculationsfor the P-terminated surface, as well as electron-countingarguments, would suggest a buckled dimer structure,which would tend to favor the CP~ structure at h =0.

SPONTANEOUS SURFACE-INDUCED LONG-RANGE ORDER IN. . . 11 193

Based on the relaxations seen in Fig. 14(c), we would ex-pect that Cpz is elastically preferred at h =2. Thus,while the preferred state of order of the reconstructedsurface cannot be ascertained, we can say that the subsur-face strain induced by dimerization would still promoteCP~ ordering.

ACKNOWLEDGMENT

This work was supported in part by the 0%ce of Ener-gy Research, Materials Science Division, U.S. Depart-ment of Energy, under Grant No. DE-AC02-77-CH00178.

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~8There are two distinct rhombohedral superlattice structures

for IV-IV systems: the RH1 form, in which the Si-Ge inter-faces occur within bilayers along (111) containing one layereach of Si and Ge (intrabilayer or mixed-bilayer interfaces),and the RH2 form, in which the interfaces occur between bi-layers (interbilayer or pure-bilayer interfaces). The bilayer se-quence in RH1 is (SiGe)(GeSi}. , while in RH2 it is(SiSi)(GeGe) . RH1 is a topologically unconstrainedstructure in that it has zero strain energy when no externalconstraints are applied. This can be understood geometrical-ly by realizing that the mixed bilayers that constitute the in-terfaces in RH1 have the Si—Ge bond lengths and angles ful-

ly relaxed at the RH1 lattice constant (essentially the averageof the Si and Ge lattice constants), and the bilayers are thenfree to adjust along [ill] until both the Si—Si and Oe—Cue

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' H. Nakayama and H. Fujita [in GaAs and Related Compounds1985, edited by M. Fujimoto, IOP Conf. Proc. No. 79(Hilger, Bristol, 1986), p. 289] observed ordering inIno &Ga05As/InP (001) grown by liquid-phase epitaxy. Theordered In3GaAs4 and InGa3As4 (201) structures were laterreinterpreted to include InGaAs2 in the (201) chalcopyriteform [see T. S. Kuan, W. J. Wang, and E. L. Wilkie, Appl.Phys. Lett. 51, 51 (1987)].T. S. Kuan, T. F. Kuech, W. I. Wang, and E. L. Wilkie, Phys.Rev. Lett. 54, 201 (1985).Ordering in GaInP& was discussed by A. Gomyo, T. Suzuki,and S. Iijima, Phys. Rev. Lett. 60, 2645 (1988); A. Gomyo, K.Kobayashi, S. Kawata, I. Hino, and T. Suzuki, J. Crys.Growth 77, 367 (1986); A. Gomyo, T. Suzuki, K. Kobayashi,S. Kawata, I. Hino, and T. Kusas, Appl. Phys. Lett. 50, 673(1987).

The effect of substrate orientation on ordering in GaInP2 wasdiscussed by A. Gomyo, T. Suzuki, S. Iijima, H. Hotta, H.Fujii, S. Kawata, K. Kobayshi, Y. Ueno, and I. Hino, Jpn. J.Appl. Phys. 27, L2370 (1988); A. Gomyo, S. Kawatawa, T.Suzuki, S. Iijima, and I. Hino, ibid. 28, L1728 (1989); T.Suzuki, A. Gomyo, and S. Iijima, J. Cryst. Growth 99, 69(1990).

The effect of p doping on ordering in GaInPz was discussed byT. Suzuki, A. Gomyo, I. Hino, K. Kobayashi, S. Kawata, andS. Iijima, Jpn. J. Appl. Phys. 27, L1549 (1988); F. P. Dab-kowski, P. Gavrilovic, K. Meehan, W. Stutius, J. E. Williams,M. A. Shahid, and S. Mahajan, Appl. Phys. Lett. 52, 2142(1988); S. R. Kurtz, J. M. Olson, J. P. Goral, A. Kibbler, andE. Beck, J. Electron. Mater. 19, 825 (1990).Ordering-induced changes in the band gap of GaInP2 werediscussed by T. Suzuki, A. Gomyo, S. Iijima, K. Kobayashi,S. Kawata, I. Hino, and T. Yuasa, Jpn. J. Appl. Phys. 27,2098 (1988); Y. Inone, T. Nishino, Y. Hamakawa, M. Kon-dow, and S. Minagawa, Qptoelectron. Dev. Technol. 3, 61(1988);T. Nishino, Y. Inone, Y. Hamakawa, M. Kondow, andS. Minagawa, Appl. Phys. Lett. 53, 583 (1988); S. R. Kurtz, J.

11 194 JAMES E. BERNARD, S. FROYEN, AND ALEX ZUNGER

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Ordering in Gao 7Ino 3P was discussed by M. Kondow, H.Kakibayashi, T. Tanaka, and S. Minagawa, Phys. Rev. Lett.63, 884 (1989).

2 InAuence of growth temperature on ordering in GaInP2 wasdiscussed by M. Kondow, H. Kakibayashi, S. Minagawa, Y.Inone, T. Nishino, and Y. Hamakawa, Appl. Phys. Lett. 53,2053 (1988); M. Kondow, H. Kakibayashi, and S. Minagawa,J. Cryst. Growth 88, 291 (1988); M. Kondow, S. Minagawa,Y. Inone, T. Nishino, and Y. Hamakawa, Appl. Phys. Lett.54, 1760 (1989)~

Work at Fujitsu Laboratories on GaInP& ordering includes S.Ueda, M. Takikawa, J. Komeno, and I. Umebu, Jpn. J. Appl.Phys. 26, L1824 (1987); O. Ueda, M. Takikawa, M. Takechi,J. Komeno, and I. Umebu, J. Cryst. Growth 93, 418 (1988);O. Ueda, M. Takechi, and J. Komeno, Appl. Phys. Lett. 54,2312 (1989); O. Ueda, M. Hoshino, K. Kodama, H. Yamada,and M.Ozeki, J. Cryst. Growth 99, 560 (1990); H. Okuda, C.Anayama, S. Narita, M. Kondo, T. Tanahashi, O. Ueda, andK. Nakejima, Appl. Phys. Lett. 55, 690 (1989).Models of substrate-induced ordering in GaInP& were ad-vanced by P. Bellon, J. P. Chevalier, L. Augarde, J. P. An-dre, and G. P. Martin, J. Appl. Phys. 66, 2388 (1989).S. McKernan, B. C. DeCooman, C. B. Carter, D. P. Bour, andJ. R. Shealy, J. Mater. Res. 3, 406 (1988).CH and CA ordering in InGaAs2 was discussed in Ref. 15.CH and CA ordering in Ga2AsSb/InP (001) was seen by H. R.Jen, M. J. Cherng, and G. B. Stringfellow, J. Appl. Phys. 48,1603 (1986); H. R. Jen, M. J. Cherng, M. J. Jou, and G. B.Stringfellow, in Gads and Related Compounds, 1986, editedby W. T. Lindley, IOP Conf. Ser. Proc. No. 83 (Institute ofPhysics, Bristol, 1987), p. 159; H. R. Jen, M. J. Jou, Y. T.Cherng, and G. B. Stringfellow, J. Cryst. Growth. 85, 175(1987).CP ordering in Ga2AsSb/InP (001) has been seen in OMVPEgrowth by G. B.Stringfellow, J. Cryst. Growth 98, 108 (1989);J. J. Murgatroyd, A. G. Norman, and G. R. Booker, J. Appl.Phys. 67, 2310 (1990); A. G. Norman, R. E. Mallard, I. J.Murgatroyd, G. R. Booker, A. H. Moore, and M. D. Scott,in Microscopy of Semiconducting Materials, edited by A. Cr.

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~~CP ordering was seen in A1InP2/GaAs (001) by M. Kondow,H. Kakibayashi, and S. Minagawa, Phys. Rev. B 40, 1159(1989).CP ordering and phase separation were seen in GaInAs&/InP(001) and GaInAsP by M. A. Shahid, S. Mahajan, D. E.Laughlin, and H. M. Cox, Phys. Rev. Lett. 58, 2567 (1987);M. A. Shahid and S. Mahagan, Phys. Rev. B 38, 1344 {1988);T. L. McDevitt, S. Mahajan, D. E. Laughlin, W. A. Bonner,and V. G. Keramidas, in Epitaxia/ Heterostructures, editedby D. W. Shaw, J. C. Bean, V. G. Keramidas, and P. S. Peer-cy (Material Research Society, Pittsburgh, 1990), Vol. 198, p.609.CP ordering was seen in InA1As2/InP (001) by O. Ueda, T.

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