PHYSICAL REVIEW B VOLUME 47, NUMBER 7 15 FEBRUARY 1993-I

Partial-ordering effects in In Gai P

R. B. Capaz* and Belita KoillerDepartamento de Fisica, Pontificia Universidade Catolica do Rio de Janeiro, Caixa Postal 88071, M)M Rio de janeiro,

Bio de Janeiro, Brazil(Received 20 October 1992)

We present a theoretical study for the band-gap energy and structural properties of partiallyordered In Gaz P alloys. Partially ordered alloys are modeled through a statistical ensemble ofsmall crystal structures defined in a 64-site periodic cell. Configurations are generated according tothe degree of ordering. The dependence of calculated properties on the long-range-order parameteris found to follow simple functional relationships. Structural anisotropies scale accurately with thesquare of the order parameter.

In Gaq P alloys have been extensively studied duringthe last few years due in great part to their adequacy asvisible-light-emitting materials. A large direct band gapand a close lattice match to GaAs at 2: = 0.5 make thisalloy an attractive alternative to Al Gai ~As in opto-electronic applications. Special attention has been givento the experimental observation of spontaneous orderinginto a metastable structure and to the consequences ofthis ordering in the electronic properties of the material,especially the order-induced band-gap narrowing.

The band-gap energy (E~) of Inc &Gas sP alloys grownon (001) GaAs substrates by metalorganic vapor-phaseepitaxy and atomic layer epitaxy7 is found to de-pend strongly on the growth conditions. According tothe growth parameters, values for Eg ranging from 1.78to 1 92 eV at 300 K (Refs. 1, 2, and 7) and from1.882 to 1.987 eV at 13 K (Ref. 6) have been mea-sured. Gomyo et al. qualitatively related this gap nar-rowing to the degree of ordering in the alloy's group-III sublattice. The reported ordered structure is the(GaP) i/(InP)i monolayer superstructure along the [ill]direction. Samples producing stronger superstructurespots in the electron-diffraction patterns also presentedlower band-gap energies. 2 Polarized photoluminescencemeasurernentss s also related ordering to the band-gapreduction and to the valence-band crystal-field splitting.

Theoretically, band-structure calculations havefocused on the (GaP) i/(InP) i [111] monolayersuperlattice, and also on a few other GaInP2-orderedcompounds. For the monolayer superlattice, Eg isfound to be lower than the average of the binary com-pounds InP and GaP at the I'-point by 0.33, 0.46,or 0.14 eV. ii Alloy calculationsi2'is have addressed thecomposition dependence Eg(z) for randomly disorderedalloys.

Kurtz, Olson, and Kibbler 4 pointed out that the cor-relation between the degree of long-range order (LRO)in Ino 5Gao 5P and changes in the band-gap properties isnot yet established in detail. They find samples with al-most "normal" band gaps but still showing a significantdegree of order in x-ray diffraction patterns. This is incontradiction with the assumption by Kanata et al. thatEg decreases linearly with the LRO parameter in these

alloys. Accurate determination of the LRO parameter isessential in the investigation of the mechanisms leadingto ordering 5 as well as to establish the effect of differ-ent growth parameters in LRO.5 In this work, partiallyordered In Gai P is studied, and a clear correlation isestablished between local structural changes, gap reduc-tion effects, and the degree of ordering.

The LRO parameter is defined in terms of site occupa-tion probabilities in the group-III sublattice. This sub-lattice is divided into two sublattices o. and P, which areoccupied, respectively, by In and Ga in the perfectly or-dered [ill] monolayer superlattice. Four site occupationprobabilities p~ give the fraction of sites in sublatticeo = ci or P which are occupied by species K = In or Ga,respectively. In terms of these, the alloy composition isz = (pi„+ p&~„)/2 = 1 —(p~& + pG )/2, and the LROparameter is defined as S = p&„—p&„——pc —pG . Thefully ordered structure corresponds to 2: = 0.5 and S = 1while, for partially ordered configurations, 0 ( 8 ( 1.

Alloys are treated within the small crystal formalismusing a 64-site cubic basic cluster, which contains eightconventional cubic cells of the diamond lattice. i7 Peri-odic boundary conditions are imposed. For the elec-tronic properties, the small crystal approach yields theexact infinite-crystal spectrum of ordered compounds atselected symmetry points. It is conceptually difFerentfrom effective potential-type alloy formalisms in which al-loy properties are obtained from a configurationally aver-aged Hamiltonian s or Green's function. i2 These approx-imations cast the problem into a zinc-blende symmetryenvironment, where local correlations and LRO cannotbe incorporated. In the present approach, Hamiltoniansassociated with different configurations accommodated ina relatively large basic cluster are determined and solvedindependently. A given alloy property is then identifiedto the average of the calculated values for this propertyover a representative set—statistical ensemble —of con-figurations. Occupational correlations within the basiccluster size are preserved. Partially ordered In Gai Palloys are simulated through a large number of smallcrystal configurations numerically generated accordingto the occupation probabilities of the 32 sites of the

47 1993 The American Physical Society

47 PARTIAL-ORDERING EFFECTS IN In Ga& „P

group-III sublattice: p&„—— (2z + S)/2 = 1 —pG,p&„——(2z —S)/2 = 1 —p~G . This defines the statis-tical ensemble from which the physical properties of analloy of composition x and LRO parameter S are calcu-lated. The generated structures provide a realistic sam-pling of 64-atom cells extracted from the macroscopicsystem, in which local fluctuations are present. For thepresent study, x = 0.5, and 400 structures are generatedfor each value of S in the range [0,1].

The binary constituents of In~ Gai ~P present a bond-length mismatch of 7.6', which results in a strained al-loy. We determine the equilibrium atomic positions foreach small crystal in the ensemble of configurations as-suming the elastic energies are described by a Keating-type valence force field model. Values of the modelparameters appropriate for In Ga~ P, namely, bond-stretching and bond-bending force constants and the un-strained bond lengths, are given in Ref. 9. The elas-tic energy was minimized using a molecular-dynamicsalgorithm, is allowing for unconstrained relaxation of alldegrees of freedom within the basic cluster. Results fromthe equilibrium configurations are summarized in Fig.1. There, the average nearest-neighbor distances alongthe [111]ordering direction (0) and along the lateral di-rections (L) are plotted versus Sz for Ga—P and In —Pbonds. o There are 128 inequivalent bonds per small crys-tal, all of which are allowed to relax towards equilibrium.Averages are thus obtained from 51200 calculated bondlengths per LRO parameter value S. The distributionof bond lengths around the average may be quantifiedby the standard deviation, which is found to vary from0.012 to 0.018 A. for difFerent bond types and degrees ofLRO.

Average values for the four types of bonds show anexcellent quadratic fit for the dependence with the orderparameter. In particular, the 0-L bond shift may bewritten as

Ab(S) = Ab, „S

with Ab „=—0.050 A for Ga—P bonds and 0.057 A. forIn—P bonds. Bond lengths for the ordered (S = 1) sys-tem are in agreement with previous calculations. s ii 0bonds are very close to the ideal unstrained length valuefor each bond type, while L bonds are shifted by Abwith respect to those. The calculated strain energy forthis configuration is U =32.4 meV/atom. As S decreasesfrom 1 to 0, average 0 and I bond lengths merge into val-ues which depend only on the species: 2.512 A. for In—Pand 2.383 A for Ga—P. This means that, instead of re-maining confined essentially to L bonds, strain is sharedamong all bonds, reducing the elastic energy. The ran-dom (S = 0) alloy has an elastic energy lower than theordered alloy by A U = 11.2+0.1 meV/atom, confirmingthat the ordered bulk alloy configuration is not thermo-dynamically stable. is Our results for the increase of Uwith S are also well described by a quadratic relation:U(S) = U(0) + AU S'.

The electronic structure of each small crystal is ob-tained in the tight-binding approximation by directlysolving for the Hamiltonian spectrum. 6 Matrix ele-

2.54I

s s II

s s sI

s s sI

s s II

s s sI

~ 2.52

A

V 2

2.4B

a

..»4

ag:G--.

'pi

0---.G--.

Q(a) In —PI s i s

II I I

I s sI

Is s

I s s s II

Is I I

2.40

2.38

V2.36

(b) Ga —P

234 —I s s s I s s s I s s s I s I s s s I—0.0 0.2 0.4 o.e 0.8 1.0

FIG. 1. Average calculated lengths of 0 bonds (solid sym-bols) and L bonds (open symbols) for (a) In—P, and (b) Ga-P bonds, vs the order paramenter squared. Statistical errorbars are at most equal to symbol size. The corresponding un-strained bond length is given by a horizontal dashed line. Dot-ted lines indicate the quadratic dependence of average bondlengths with S.

ments, taken from the 8p s* parametrization in Ref. 21,are adapted to the present system environmental disor-der. The GaP on-site elements are shifted downwardwith respect to those for InP by 0.05 eV to accountfor the band offset among these compounds. ~2 Distor-tions due to stress are incorporated locally in each near-neighbor matrix element through a (bo/b, ~) scaling,where b,~ is the calculated equilibrium distance betweenatomic sites i and j and bo is the unstrained value ofthe corresponding bond. Figure 2 gives the ensemble av-

erage values for the main energy gap as a function ofthe order parameter S. As expected, Es is a decreas-ing function of 8, but, contrary to the structural param-eters plotted in Fig. 1, a quadratic dependence alonedoes not describe the calculated trend accurately. Sym-metry implies that Eg(S) = Eg( S); therefore, p—oly-nomial corrections are restricted to even powers of S.Addition of a small quartic term yields a satisfactoryfit, indicated by the dotted line in Fig. 2. It corre-sponds to the form Eg(S) = Es(0) —AEs(S), whereAEs(S) = (0.13S —0.03 S4) eV, is the gap reduction ofordered samples soith respect to the random alloy.

The gap for the ordered structure, Es(l) = 1.87 eV, islower than the average gap of the binary compounds atthe I' point by 0.3 eV-, in good agreement with Ref. 9.Previous calculations for the random alloy gap, 2.05(Ref. 12) and 1.98 eV, is are comparable to our resultEs(0) = 1.97 eV, which is also close to the experimen-tal value 1.987 eV measured at 13 K. All measured gap

R. B. CAPAZ AND BELITA KOILLER 47

200) I I I )1 I I)

I I I)

I I I(

I I I

1.95—

'~

1.90—'~

1 85 I ( s i l s s s I ( I i I i s i I i s ( I

0.0 0.2 0.4 0.6 0.8 1.0

FIG. 2. Calculated low-temperature energy gap ofIno 5Gao. 5P vs order parameter. Each data point gives the en-semble average over 400 64-site small crystal structures. Sta-tistical error bars are indicated. The dotted line is a fourth-order polynomial fit, described in the text.

values given in Ref. 6 for ordered samples fall within therange of our calculation. Assuming our calculated E~(S)dependence, those would correspond to S between 0.5and 0.9. The order-parameter values estimated there areconsiderably smaller due to the presumed linear depen-dence of E~ with S, and also due to the value takenfrom Ref. 10 for E~(1) = 1.70 eV, which seems to be toosmall. ~~ The lowest value for E~ reported in the litera-ture is 1.868 eV (at T = 4 K) for a sample grown byatomic layer epitaxy. From our calculation, this samplewould essentially correspond to S 1.

The dependence of the gap with S obtained here ex-plains the apparent contradiction described in Ref. 14 ofsamples with almost "normal" band gaps but still show-ing order in x-ray diffraction patterns. From Fig. 2 wenote that for S ( 0.4 the gap reduction effect is almostnegligible (less than 1'%%u~), while values of S 0.4 wouldstill show superlattice diffraction spots in x-ray experi-ments.

Recently Tabor-Morris et a/. performed luminescenceand polarized extended x-ray-absorption fine-structure(EXAFS) measurements in In Gaq ~P samples. In thisway, both the gap value and 0-L bond shifts were ob-tained. For an ordered sample the measured 0-I shiftfor Ga—P bonds was Ab = (0.04 + 0.02) A.. From Eq. (1),Ab = 0.04 A. corresponds to the order parameter S = 0.9,leading to a gap narrowing AE~(0.9) = 0.086 eV at 0K, which is consistent with the experimentally measurednarrowing of 0.08 eV at 300 K.24

Our results indicate that local structural measure-ments provide essential information to establish thedegree of ordering in In Gaq P alloys. Electrondiffraction and x-ray diffraction experiments show

great quantitative uncertainties, while photolumines-cence spectra, from which Ez is usually obtained, presenta strong and anomalous temperature dependence.Therefore, diffraction spot intensities and band-gap val-

ues alone cannot give accurate estimates for the LROparameter. On the other hand, the simple quadratic de-pendence of the 0-L bond shifts with S obtained heremight be used as a complementary measure for the de-gree of ordering of these alloys. Of course, 0 Ib-ond-shiftmeasurements are also subject to uncertainties, caused inpart by the disorder Buctuation effects discussed abovewhich lead to line broadening, and by intrinsic accuracylimitations of the involved techniques. Trying to infer theabsolute value of the order parameter from these exper-iments is subject to such inaccuracies, and must also bebased on a precise knowledge of Eb ~„. The accuracy ofour bond-length calculations may be estimated by com-parison with EXAFS measurements for the deviationsof In—P and Ga—P bonds in the random x = 0.5 alloywith respect to the binary compound values. The cor-responding deviations in our calculation agree with thosewithin the reported EXAFS experimental accuracy, from0.005 to 0.01 A. . We estimate our value of Ab ~„ is alsoreliable within this range. Our calculated distribution ofwidths, about 0.02 A. , is also compatible with EXAFSexperimental linewidth values.

In conclusion, the small crystal formalism presentedhere is of general application to disordered or partiallyordered alloys, and gives direct information about thedependence of different alloy properties on statistical pa-rameters related to composition, long- or short-range or-der. Previous studies have mainly dealt with the compo-sition dependence of elastic and electronic properties ofsemiconductor random alloys. Near-neighbor distancesare usually found to follow a linear dependence with

Band gaps, however, show distinct "bowing" ef-fects, and both linear and quadratic terms in x arepresent in fits to Ez(x). In the case of order, the sim-plest functional dependence of physical quantities on anyorder parameter S for which the system has S ~ —Ssymmetry is quadratic: no linear terms are allowed. Forthe particular system considered here, this is an excellentapproximation for average near-neighbor distances, while

gap properties require higher-order corrections. The sit-uation is quite analogous to the composition dependenceof these quantities, only that expansions are performed inpowers of 2; in one case and in powers of S2 in the other.Results for the energy gap might be reined by increasingthe basic cluster size or by improving the tight-bindingparametrization, but the qualitative behavior of the gapreduction effect obtained here is not expected to change.Note that the small crystal method could, in principle,be coupled to ab initio calculations for the structuralproperties. However, solving 400 64-atom basic clusterstructures for each value of the LRO paramenter is farbeyond the current computational limitations of such ap-proaches. The agreement of our calculation with previoustheoretical results for the fully ordered structure as wellas with experimental data for partially ordered and ran-dom samples indicates that the present model accountsfor the main features of these alloys.

47 PARTIAL-ORDERING EFFECTS IN In„Ga& „P

We are grateful to M. O. Robbins and R. Osbrio foruseful conversations, and to Tabor-Morris et al. for pro-viding us their results prior to publication. This workwas partially supported by the Brazilian agencies Min-

isterio de Ciencia e Tecnologia, Conselho Nacional deDesenvolvirnento Cientffico e Tecnologico, and Fundagaode Amparo a Pesquisa do Estado do Rio de Janeiro.

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