Phonon spectra of ultrathin GaAslAlAs superlattices : An ab initio calculation

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) as well as some partially disordered superlattices. Besides interfacial disorder, which modifies the effective confinement length of low-order longitudinal-optic phonons, we find that —in the ultrathin regime —some degree of cationic mixing must also affect inner planes in order to explain experimental findings.

Since the development of epitaxial techniques for grow- ing artificial semiconductor heterostructures, phonon Raman spectroscopy stood out as one of the most important tools 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, the study of folded acoustical and confined optical modes has been used to obtain information on the period, thickness of the individual components, and on the abruptness of in- terfaces.In ultrathin SL's, the interpretation of Raman measurements, particularly in the longitudinal-optical (LO) region, is still controversial.Available spectra show that, both in the GaAsand A1As-like energy ranges, the frequencies of the highest LO modes coLO, almost coincide with the LO modes in the corresponding alloy for the monolayer SL, whereas they smoothly tend to the bulk limit toLo(I ) for increasing thickness of the slab.
On the theoretical side, previous investigations based on models at various levels of sophistication predicted for (GaAs)"(A1As)"a smooth dependence of toLo, (n) upon thickness 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 spectrum of the superlattice and that of the corresponding alloy, has been interpreted as the evidence that spectra of monolayer SL's are heavily affected by disorder, while they are not for thicker slabs.The question was still open whether this discrepancy should be attributed to inadequacies of theoretical models or to disorder effects in monolayer SL's.%e have undertaken a series of 6rst-principles calcula- tions of the phonon spectra of (GaAs), (A1As)"(001) for n 1,2, 3. Preliminary results of this research have been reported in Ref. 3, to which we refer for the technica1 de- tails of our calculation.%e remind the reader that the dynamical matrix is obtained within state-of-the-art density-functional linear-response theory, using norm-conserving pseudopotentials, and large plane-wave basis sets.The macroscopic electric fields arising from long- wavelength longitudinal phonons are dealt with ab initio, without adjusting the ionic effective charges to any empirical data.Our previous experiencei indicates that pseudopotentials from pubhshed tables, though appropriate for reproducing phonon dispersions of bulk materials, are not as accurate for superlattices. 6For this reason, we have generated new norm-conserving pseudopotentials for Ga, Al, and As, which are able to correctly predict the ob- served lattice mismatch between GaAs and AlAs, and also to describe even more accurately the phonon disper- sions of the two bulks.
In Fig. 1   for GaAs and A1As.For GaAs, the agreement with ex- periment is excellent, particularl with respect to recent neutron-diffraction experiments which are believed to be very accurate.Notice that both the LO-TO splitting at zone center (and hence effective charges) and the flatness of the TA branch near the zone boundary are accurately predicted by the present calculation.' For A1As, this is the 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 predict and the few available experiments confirm that the LO branch of A1As along the 6 direction is very flat (the width of the dispersion being five times smaller than the LO-TO splitting); in GaAson the contrarythe width of the LO dispersion is twice as large as the LO-TO split- ting.
We focus on the highest-lying longitudinal frequencies rcILo, which are the most intense Raman modes observable in 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 we display our results for aILo, (II) for ordered SL's, as a func- In order to understand whether such discrepancies are determined by disorder effects, we have calculated phonon frequencies 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 cationic sites at random.'5 Results for a more ordered stacking of fully intermixed cationic planes are obtained from calculations on a GaA1As2 chalcopyrite structure.The (GaAs)2(AlAs)2 SL with interfacial disorder is simulated by a Ga-L-Al-X stacking of cationic planes, where the X plane is a mixed (Ga-Al) plane (no virtual-crystal approximation is made for Xplanes: i.e. , we consider planes with two cations per unit 2D cell).
Let us consider first the fully disordered situation.The LOI modes of both our model alloy (oI""") and the chal- copyrite (rII'"" ) fall slightly above our monolayer SL result in the GaAs-like range (Iu""" 267~2 cm oI'"" 270 cm '), and substantially lower in the A1As range (oI""" 386 ~2 cm ', oI'"" 381 cm ').These results are in good agreement with the observed alloy frequencies, and also with the observed monolayer SL data.
The agreement between our calculations and experiments both for the individual bulks and for the alloy makes us confident that the present first-principles results are accurate and reliable enough to allow predictive statements on the SL spectra.We conclude that the origin of the discrepancy between our results and experiments has to be searched in the difference between actual samples and the structures studied theoretically, i.e. , in disorder effects.
The mode which is most sensitive to disorder is the AIAs-like rather than the GaAs-like one.In fact, while confinement effects in ideally ordered (001)-grown SL's may only shift the modes within the frequency range of the corresponding bulk (001) dispersions, disorder effects move A1As-like modes out of such range.This reflects two 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 responsible for the large LO-TO splitting at I".The two main effects of disorder on LO frequencies are: (i) the mixing of states with wave vectors of arbitrary orientations, and (ii) the reduction of effective polarization produced by alloying.'  Due to the above features of the bulk dispersion, both effects contribute to shifting A1As-like modes out of the (001) LO dispersion.In GaAs, on the other hand, the bulk dispersion is more isotropic' and much larger than the LO-TO splitting: For SL GaAs-like modes it is then dificult to distinguish the effects of disorder from those of confinement, because they both lead to a frequency shift within the bulk-allowed range.%'e conclude that the A1As-like eLp, frequency would be most suited for char- acterization purposes, as it is almost unaffected by confinement effects, and its departure from the value of the bulk tuLo(I ) is a direct measure of disorder effects.
We have seen thatfor monolayer GaAs/A1As SL's a substantial degree of cationic intermixing can explain the discrepancies between our calculations for ordered structures 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 the inner planes.In the case of (GaAs)2(A1As)2, we have simulated the former situation by studying the phonon spectrum of the GaAs-L-As-A1As-LAs SL, as mentioned above.Our results indicate that the A1As-like LO1 frequency practically coincides with the corresponding value for the ordered SL (403 and 402 cm ' for the ordered and disordered cases, respectively), whereas the GaAs- like mode is lowered by 8 cm ', in better (but still not very good) agreement with experiments.
Inspection of phonon eigenvectors shows that ion displacements of A1As-like andto a minor extent -GaAs-like LO1 modes 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 a modification of the relevant confinement length, shifting them closer to the LO~modes of the n 1 SL.As the AlAs-like LO~frequencies for n 1 and n 2 are very close, 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 frequency to the observed value.This picture is confirmed by inspection of the displacement patterns corresponding to LO AIAs 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 cationic plane and the frequency of the highest LO mode.Unfor- tunately this relation is not simple: In factas the disper- sion of the AlAs LO branch in the alloy is probably larger than in the bulkthe dependence of mLo, upon composi- tion is complicated by confinement effects.Our con- clusions are also consistent with the analysis of GaAs-like LO modes.In this case, however, the large dispersion of the bulk LO band makes a rather large shift of the LOm ode possible even in the presence of pure Ga planes.It is this dif5culty of disentangling disorder and confinement effects which makes GaAs-like modes less suited for characterization purposes than A1As ones.
We come now to a comparison of the present results with those obtained by a linear-chain model similar to that used in previous theoretical investigations.
Starting from our calculated interplanar force constants for bulk GaAs, we have calculated phonon frequencies of bulk A1As and of GaAs/AlAs SL's.We find that these results differ from those of full self-consistent calculations for a quasirigid shift of the LO bands of, at most, a few cm Note that such an accuracy has been achieved simply by replacing the relevant masses without adjusting any effective charges.This indicates that the difference be- tween our results for ideal SL's and previous ones is not due to important chemical effects occurring at the inter- face, but rather to an improved quality of the present bulk phonon dispersions.This fact gives confidence that inter- atomic force-constants models based on first-principles calculations may prove to be an accurate tool for studying large unit-cell systems aimed at simulating alloys of different compositions and thick disordered SL's.Work is in progress along these lines.
In conclusion, our calculations definitely rule out any significant role of interface charge rearrangement in the phonon 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 necessary to account for the discrepancies between observed spectra and calculations made for ordered ultrathin SL's.Finally, we provide the first ab initio phonon dispersion of bulk A1As.The Aatness of the LO branch suggests that A1As- like LO& modes of ultrathin SL's are very sensitive to the presence and spatial extension of cationic intermixing.
Therefore, an experimental effort aimed at monitoring A1As-like features would be most useful.
The authors are grateful to A. Fasolino and J.   (1982).
The reason is that phonon frequencies are rather sensitive to the lattice parameter at which they are calculated.In order to obtain accurate phonon frequencies, one has to calculate them at the theoretical equilibrium volume.The mismatch between GaAs and AlAs predicted by the potentials of Ref. 5 is larger than 1%, instead of -0.1% as observed experirnen- tally.It then turns out that a very accurate description of both GaAs and AlAs-like frequencies in a mixed crystal can hardly be obtained with these potentials.
7The predicted values of the lattice constants are 10.60 and FIG. &.Calculated phonon dispersions in GaAs (left) andAlAs (right) along (001) (cm ').Experimental points are marked by triangles and circles for longitudinal modes and squares and diamonds for transverse modes.Data for GaAs are from Ref. 8(a) (circles and diamonds), and Ref. 8(b) (triangles and squares); for AlAs from Ref. 9(a) (circles and diamonds) and Ref. 9(b) (triangles and squares).
Menendez for very useful discussions.This work has been partially supported by the Italian Ministry of Education through the collaboration between SISSA and CINECA Supercomputing Center and by the Italian Consiglio Nazionale delle Ricerche through Progretto Finalizzato Sistemi Informatici e Calcolo Parallelo, under Grants No. 89.00006.69and No. 89.00011.69.S.B. acknowledges financial support from the European Research Office of the U.S. Army, under Grant No. DAJA45-89-C-0025, and P.G. acknowledges support from the Swiss National Science Foundation, under Grant No. 20-5446.87.