PHYSICAL REVIE% B VOLUME 41, NUMBER 6 15 FEBRUARY 199G-II

Phonon spectra of ultrathin GaAslAlAs superlattices: An ab initio calculation

Stefano BaroniScuola Internazionale Superiore di Studi Avanzati (SISSA), Strada Costiera I l, 1-340l4 TriesteI, taly

Paolo GiannozziInstitut Romand de Recherche Numerique en Physique des Materiaux (IRRMA),

PHB-Ecublens, CH-1015 Lausanne, Switzerland

Elisa MolinariConsiglio Nazionale delle Ricerche, Istituto di Aeustica "O.M. Corbino" Via Cassia 1216, I-00189 Roma, Italy

(Received 24 October 1989)

Phonon spectra of ultrathin (GaAs), (A1As)„(001) superlattices are studied theoretically using

linear-response density-functional techniques. Results are presented for n 1,2,3 superlattices,along with prototype superce11 calculations aimed at simulating a completely disordered (alloy) aswell as some partially disordered superlattices. Besides interfacial disorder, which modifies theeffective confinement length of low-order longitudinal-optic phonons, we find that —in the ul-

trathin regime —some degree of cationic mixing must also affect inner planes in order to explainexperimental findings.

Since the development of epitaxial techniques for grow-ing artificial semiconductor heterostructures, phonon Ra-man spectroscopy stood out as one of the most importanttools for their characterization. ' Much of the experimen-tal work performed so far in this field concerns GaAs/AIAs (001) superlattices (SL's). For thick samples, thestudy of folded acoustical and confined optical modes hasbeen used to obtain information on the period, thicknessof the individual components, and on the abruptness of in-terfaces. In ultrathin SL's, the interpretation of Ramanmeasurements, particularly in the longitudinal-optical(LO) region, is still controversial. Available spectra showthat, both in the GaAs- and A1As-like energy ranges, thefrequencies of the highest LO modes coLO, almost coincidewith the LO modes in the corresponding alloy for themonolayer SL, whereas they smoothly tend to the bulklimit toLo(I ) for increasing thickness of the slab.

On the theoretical side, previous investigations based onmodels at various levels of sophistication predicted for(GaAs)„(A1As)„a smooth dependence of toLo, (n) uponthickness for n ~ 2, and an abrupt variation between n 1

and n 2. The discrepancy between theory and experi-ment, together with the similarities between the spectrumof the superlattice and that of the corresponding alloy, hasbeen interpreted as the evidence that spectra of monolayerSL's are heavily affected by disorder, while they are notfor thicker slabs. The question was still open whether thisdiscrepancy should be attributed to inadequacies oftheoretical models or to disorder effects in monolayerSL's.

%e have undertaken a series of 6rst-principles calcula-tions of the phonon spectra of (GaAs), (A1As)„(001) forn 1,2,3. Preliminary results of this research have beenreported in Ref. 3, to which we refer for the technica1 de-tails of our calculation. %e remind the reader that thedynamical matrix is obtained within state-of-the-artdensity-functional linear-response theory, using norm-

conserving pseudopotentials, and large plane-wave basissets. The macroscopic electric fields arising from long-wavelength longitudinal phonons are dealt with ab initio,without adjusting the ionic effective charges to any empir-ical data. Our previous experiencei indicates that pseudo-potentials from pubhshed tables, though appropriate forreproducing phonon dispersions of bulk materials, are notas accurate for superlattices. 6 For this reason, we havegenerated new norm-conserving pseudopotentials for Ga,Al, and As, which are able to correctly predict the ob-served lattice mismatch between GaAs and AlAs, andalso to describe even more accurately the phonon disper-sions of the two bulks.

In Fig. 1 we report our calculated phonon dispersions

400

~ 300

~ POO

100

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xrFIG. &. Calculated phonon dispersions in GaAs (left) and

AlAs (right) along (001) (cm '). Experimental points aremarked by triangles and circles for longitudinal modes and

squares and diamonds for transverse modes. Data for GaAs arefrom Ref. 8(a) (circles and diamonds), and Ref. 8(b) (trianglesand squares); for AlAs from Ref. 9(a) (circles and diamonds)and Ref. 9(b) (triangles and squares).

3870

PHONON SPECTRA OF. . . 3871

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FIG. 2. Frequencies of the GaAs-like and A1As-like eLp,modes (cm '}of (GaAs)„(AlAs)„, as a function of n. Note thebreak in the energy scale: The dashed band separates the GaAsand A1As energy ranges. Left-hand side: theory, present work;diamonds, "ordered structures;" triangles, alloy from a 16-atomsupercell calculation; squares, chalcopyrite structure. Right-hand side: experiment; diamonds, from Ref. 13(a); squares,from Ref. 13(b); triangles, from Ref. 13(c); circles, from Ref.13(d).

for GaAs and A1As. For GaAs, the agreement with ex-periment is excellent, particularl with respect to recentneutron-diffraction experiments which are believed tobe very accurate. Notice that both the LO-TO splitting atzone center (and hence effective charges) and the flatnessof the TA branch near the zone boundary are accuratelypredicted by the present calculation. ' For A1As, this isthe first ab initio calculation of a full phonon dispersion. "Our results agree well with experimental data, which are,however, very scarce for this material. For later discus-sion it is important to notice that our calculations predictand the few available experiments confirm that the LObranch of A1As along the 6 direction is very flat (thewidth of the dispersion being five times smaller than theLO-TO splitting); in GaAs —on the contrary —the widthof the LO dispersion is twice as large as the LO-TO split-ting.

We focus on the highest-lying longitudinal frequenciesrcILo, which are the most intense Raman modes observablein the usual backscattering configuration. A more com-plete analysis of the full zone-center spectrum will be re-ported elsewhere. ' On the left-hand side of Fig. 2 wedisplay our results for aILo, (II) for ordered SL's, as a func-tion of the slab thickness n. For n ~ 2, coL~, roughly fol-low the trend predicted by confinement arguments, ' i.e.,oILo, (II) =«Ih"[q, Ir/(n+I)]. For n 1, this is nolonger true; however, mL~, still falls within the bulk bandwidth, as suggested by the previous argument. In particu-lar, the very small sensitivity of A1As-like LO~ modes toslab thickness is due to the flatness of the bulk LQ disper-sion. On the right-hand side, we report the correspondingavailable experimental data. It appears that —at variancewith our theoretical results for ideal SL's—both GaAs-and AlAs-like LO~ modes show a smooth thickness depen-dence for 1 ~ n ~ 3. Furthermore, A1As- and GaAs-like

coLO, for n 1 are very close to the LO modes in the alloyof equivalent composition. In the GaAs-like region, ex-perimental data lie below our predictions for n 2, 3, andslightly above for n l. In the AlAs region, they fall con-sistently below theoretical results and out of the bulk bandwidth.

In order to understand whether such discrepancies aredetermined by disorder effects, we have calculated phononfrequencies of selected supercell systems aimed at simu-lating cationic intermixing. In particular, a bulk Ga05-A105As alloy has been simulated with a 16-atom fcc su-percell where four Al and four Ga ions occupy the cationicsites at random. '5 Results for a more ordered stacking offully intermixed cationic planes are obtained from calcu-lations on a GaA1As2 chalcopyrite structure. The(GaAs)2(AlAs)2 SL with interfacial disorder is simulatedby a Ga-L-Al-X stacking of cationic planes, where the Xplane is a mixed (Ga-Al) plane (no virtual-crystal approx-imation is made for Xplanes: i.e., we consider planes withtwo cations per unit 2D cell).

Let us consider first the fully disordered situation. TheLOI modes of both our model alloy (oI""")and the chal-copyrite (rII'"" ) fall slightly above our monolayer SLresult in the GaAs-like range (Iu""" 267~2 cmoI'"" 270 cm '), and substantially lower in the A1Asrange (oI""" 386 ~ 2 cm ', oI'"" 381 cm '). Theseresults are in good agreement with the observed alloy fre-quencies, and also with the observed monolayer SL data.The agreement between our calculations and experimentsboth for the individual bulks and for the alloy makes usconfident that the present first-principles results are accu-rate and reliable enough to allow predictive statements onthe SL spectra. We conclude that the origin of thediscrepancy between our results and experiments has to besearched in the difference between actual samples and thestructures studied theoretically, i.e., in disorder effects.

The mode which is most sensitive to disorder is theAIAs-like rather than the GaAs-like one. In fact, whileconfinement effects in ideally ordered (001)-grown SL'smay only shift the modes within the frequency range ofthe corresponding bulk (001) dispersions, disorder effectsmove A1As-like modes out of such range. This reflectstwo specific features of AIAs: (i) the anisotropy of its op-tical phonons, i.e., the fact that the dispersion along (001)is much flatter than along other directions;' (ii) the im-portance of macroscopic polarization, which is responsiblefor the large LO-TO splitting at I". The two main effectsof disorder on LO frequencies are: (i) the mixing of stateswith wave vectors of arbitrary orientations, and (ii) thereduction of effective polarization produced by alloying. '

Due to the above features of the bulk dispersion, botheffects contribute to shifting A1As-like modes out of the(001) LO dispersion. In GaAs, on the other hand, thebulk dispersion is more isotropic' and much larger thanthe LO-TO splitting: For SL GaAs-like modes it is thendificult to distinguish the effects of disorder from those ofconfinement, because they both lead to a frequency shiftwithin the bulk-allowed range. %'e conclude that theA1As-like eLp, frequency would be most suited for char-acterization purposes, as it is almost unaffected byconfinement effects, and its departure from the value of

3872 BARONI, GIANNOZZI, AND MOLINARI

the bulk tuLo(I ) is a direct measure of disorder effects.We have seen that —for monolayer GaAs/A1As SL's

—a substantial degree of cationic intermixing can explainthe discrepancies between our calculations for orderedstructures and experiments. For thicker SL's, one can dis-tinguish the situation where disorder only affects the in-terface planes, and the one where it extends to all theinner planes. In the case of (GaAs)2(A1As)2, we havesimulated the former situation by studying the phononspectrum of the GaAs-L-As-A1As-LAs SL, as mentionedabove. Our results indicate that the A1As-like LO1 fre-quency practically coincides with the corresponding valuefor the ordered SL (403 and 402 cm ' for the orderedand disordered cases, respectively), whereas the GaAs-like mode is lowered by 8 cm ', in better (but still notvery good) agreement with experiments. Inspection ofphonon eigenvectors shows that ion displacements ofA1As-like and —to a minor extent —GaAs-like LO1modes are well confined within the corresponding pure-cation plane. It turns out that this kind of interfacial dis-order affects the I.O& frequencies only through amodification of the relevant confinement length, shiftingthem closer to the LO~ modes of the n 1 SL. As theAlAs-like LO~ frequencies for n 1 and n 2 are veryclose, such a shift is very small in this case. We conclude,therefore, that no pure-cation plane can exist in (GaAs)2-(A1As)z samples in order to shift the A1As LO1 frequencyto the observed value. This picture is confirmed by inspec-tion of the displacement patterns corresponding to LOAIAs modes of lower frequency. These are confined in-stead to mixed cationic planes, thus indicating that a rela-tion exists between the composition of the purest cationicplane and the frequency of the highest LO mode. Unfor-tunately this relation is not simple: In fact—as the disper-sion of the AlAs LO branch in the alloy is probably largerthan in the bulk —the dependence of mLo, upon composi-tion is complicated by confinement effects. Our con-clusions are also consistent with the analysis of GaAs-likeLO modes. In this case, however, the large dispersion ofthe bulk LO band makes a rather large shift of the LO~mode possible even in the presence of pure Ga planes. Itis this dif5culty of disentangling disorder and confinementeffects which makes GaAs-like modes less suited for char-acterization purposes than A1As ones.

We come now to a comparison of the present resultswith those obtained by a linear-chain model similar to

that used in previous theoretical investigations. Startingfrom our calculated interplanar force constants for bulkGaAs, we have calculated phonon frequencies of bulkA1As and of GaAs/AlAs SL's. We find that these resultsdiffer from those of full self-consistent calculations for aquasirigid shift of the LO bands of, at most, a few cmNote that such an accuracy has been achieved simply byreplacing the relevant masses without adjusting anyeffective charges. This indicates that the difference be-tween our results for ideal SL's and previous ones is notdue to important chemical effects occurring at the inter-face, but rather to an improved quality of the present bulkphonon dispersions. This fact gives confidence that inter-atomic force-constants models based on first-principlescalculations may prove to be an accurate tool for studyinglarge unit-cell systems aimed at simulating alloys ofdifferent compositions and thick disordered SL's. Work isin progress along these lines.

In conclusion, our calculations definitely rule out anysignificant role of interface charge rearrangement in thephonon spectra of even ultrathin GaAs/A1As SL's. In-stead, our first-principles study of madel-disordered sys-tems indicates that a substantial degree of cationic inter-mixing extending beyond the interface planes is necessaryto account for the discrepancies between observed spectraand calculations made for ordered ultrathin SL's. Finally,we provide the first ab initio phonon dispersion of bulkA1As. The Aatness of the LO branch suggests that A1As-like LO& modes of ultrathin SL's are very sensitive to thepresence and spatial extension of cationic intermixing.Therefore, an experimental effort aimed at monitoringA1As-like features would be most useful.

The authors are grateful to A. Fasolino and J.Menendez for very useful discussions. This work has beenpartially supported by the Italian Ministry of Educationthrough the collaboration between SISSA and CINECASupercomputing Center and by the Italian Consiglio Na-zionale delle Ricerche through Progretto FinalizzatoSistemi Informatici e Calcolo Parallelo, under Grants No.89.00006.69 and No. 89.00011.69. S.B. acknowledgesfinancial support from the European Research Office ofthe U.S. Army, under Grant No. DAJA45-89-C-0025,and P.G. acknowledges support from the Swiss NationalScience Foundation, under Grant No. 20-5446.87.

'For a recent review, see B. Jusserand and M. Cardona, in LightScattering in Solids V, edited by M. Cardona and G.Giintherodt (Springer-Verlag, Berlin, 1989),p. 49.

2See A. Fasolino and E. Molinari, in Proceedings of the FourthInternational Conference on Modulated SemiconductorStructures, Ann Arbor, 1988 [Surf. Sci. (to be published)),and references quoted therein.

S. Baroni, P. Giannozzi, and E. Molinari, in Proceedings of theIVinereenth International Conference on the Physics of Semiconductors, Warsa~, 1988, edited by W. Zawadzki (Instituteof Physics, Polish Academy of Sciences, Warsaw, 1989), p.795.

S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 58,

1861 (1987).5G. B. Bachelet, D. R. Hamann, and M. Schluter, Phys. Rev. B

26, 4199 (1982).The reason is that phonon frequencies are rather sensitive to

the lattice parameter at which they are calculated. In orderto obtain accurate phonon frequencies, one has to calculatethem at the theoretical equilibrium volume. The mismatchbetween GaAs and AlAs predicted by the potentials of Ref. 5is larger than 1%, instead of —0.1% as observed experirnen-tally. It then turns out that a very accurate description ofboth GaAs and AlAs-like frequencies in a mixed crystal canhardly be obtained with these potentials.

7The predicted values of the lattice constants are 10.60 and

PHONON SPECTRA OF. . . 3873

10.61 a.u. for GaAs and AlAs, respectively. This latticematching between GaAs and AlAs has been obtained by im-

proving the choice of the reference configuration for the GaI 2 potential.

s(a) G. Dolling and J. L. T. Waugh, in Lattice Dynamics (Per-gamon, London, 1965), p. 19; (b) B. Dorner and D. Strauch(unpublished), as quoted in E. Richter and D S.trauch, SolidState Commun. 64, 867 (1987).

9(a) A. Onton, in Proceedings of the Tenth InternationalConference on the Physics of Semiconductors, Cambridge,MA, 1970, edited by S. P. Keller, J. C. Hensel, and F. Stern(U.S. AEC, Springfield, VA, 1970), p. 107; (b) B. Monemar,Phys. Rev. B 8, 5711 (1973).

' Our results for GaAs are qualitatively similar to those of K.Kunc and R. M. Martin, Phys. Rev. Lett. 48, 406 (1982).The improved quantitative agreement with experiment ismainly due to the use of norm-conserving pseudopotentials.The only previous ab initio calculation on phonons in AlAs areby K. J. Chang and M. L. Cohen, in Proceedings of theSeventeenth International Conference on the Physics ofSemiconductors, San Francisco, 1984, edited by J. D. Chadiand W. A. Harrison (Springer-Verlag, New York, 1985), p.

1151. This calculation only provided results for TO (I ) andXmodes.

' S. Baroni, P. Giannozzi, and E. Molinari (unpublished).'3(a) M. Cardona, T. Suemoto, N. E. Christensen, T. Isu, and

K. Ploog, Phys Rev. B 36, 5906 (1987); (b) A. Ishibashi, M.Itabashi, Y. Mori, K. Kawado, and N. Watanabe, Phys. Rev.B 33, 2887 (1986); (c) M. Nakayama, K. Kubota, and N.Sano, Solid State Commun. 53, 493 (1985); (d) T. Toriyama,N. Kobayashi, and Y. Horikoshi, Jpn. J. Appl. Phys. 25, 1895(i986).

'4B. Jusserand and D. Paquet, Phys. Rev. Lett. 56, 1752 (1986).'5Tests made within the bond-charge model confirm that the

relevant results do not depend crucially on the chosen ionicarrangement and the size of the cell [L. Colombo (unpub-lished) J.

1sP. Pavone, S. Baroni, S. de Gironcoli, and P. Giannozzi (un-published).

' This is so because —in a two-mode alloy such as GaAlAs —theGa(Al) ions do not participate in the vibration of A1As-(GaAs-) like modes, and therefore give no contribution to thecorresponding polarization.