AlAs superlattices

~

Solid State Communications, Printed in Great Britain.

LATTICE

Vol.64,No.6, pp.867-870,

DYNAMICS

1987.

0038-1098/87 $3.00 + .00 Pergamon Journals Ltd.

OF GaAs/A1As SUPERLATTICES

E . RICHTER AND D. STRAUCH

Institut f(irTheoretische Physik, UniversitKt Regensburg, D-8400 Regensburg, F R G Received July 31, 1987, by M. Cardona W e report on the results of three-dimensional shell model calculations of GaAs/AIAs superlatticephonon frequencies and eigenvectors. Their dependence on direction and magnitude of the wavevector is investigated. A m o n g others we find the occurrence of acoustical (Lamband Love-like) guided and optical (Fuchs-Kliewer-like) interface phonons for wavevectors perpendicular to the layer normal; for wavevectors parallel to the normal there are folded and confined phonons. For the folded phonons the ionic displacements can be described by continuously matched sine and cosine functions. A n effectivewavevector can be attributed to the confined phonons. With this wavevector R a m a n data in superlatticesare reevaluated and compared with recent high-accuracy neutron scattering data for bulk GaAs.

W e have investigated the latticedynamics of GaAs/A]As superlattice structures. In order to generate realisticphonon properties we have used the valence overlap shell model for G a A s } AIAs was described by the same model with only the mass of the cation being changed, while all other model parameters were kept the same. This should be a good approximation since the elasticconstants of AIAs are very similar to those of GaAs. The lattice constants of both materials are also very similar. Hence no changes of the model parameters at or near the interface have been made either. In the following we discuss some representative results for the phonon eigenfrequencies and eigenvectors for wavevectors of various directions and magnitudes in superlattices composed of layers with various thickness, even though the emphasis in this paper will be restricted to a (5/5) GaAs/AIAs superlattice. Figure 1 shows the dispersion curves for a selected set of wavevectors for a (5/5) GaAs/AlAs superlattice. The rightand left-hand portions of the figure are for wavevectors parallel (region C) and perpendicular (region A), respectively, to the layer normal, and the middle portion (region B) is for a wavevector of very small magnitude and varying direction, interpolating between the r points of the outer two portions of the figure. At the Brillouin zone center, the modes polarised parallel to the layer normal have AI or B2 symmetry and the modes polarised perpendicular to it have (two-fold) E symmetry. 2 Where the bands of the two bulk substances do not overlap confinement of the phonons with wavevector parallel to the layer normal occurs. Apart from the effectson the eigenvectors (which justify such a notion) this results in nearly dispersionless branches (region C) and in our case occurs for all optical modes and for the high-frequency folded AIAs-like T A branches. For overlapping bands one has the so-called folded phonons. (Because of our model assumptions the L A modes in G a A s and AIAs lie in exactly the same band, in good agreement with experimental data.s) The values of the lowest frequency gaps at and near r agree with results from the continuum approximation. 4

As can bee seen in Figure 1 the limiting frequencies of

B=- and E-like modes for wavevectors with directions perpendicular and parallelto the layer normal (and in the limit

Symmetry :

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Figure I. Spacial and directional phonon dispersion curves for a (5/5) GaAs/A.1As superlattice. F-Z (portion C) is the direction parallel to the layer normal, and r - M (portion A) perpendicular to it (along a cubic axis). The middle portion (B) of the figureis for very small magnitude of the wavevector and directions between parallel to the normal (d = 0 °) and perpendicular to it (~ = 90 °) (directionaldispersion). 867

868

LATTICE DYNAMICS OF GaAs/AIAs SUPERLATTICES

of almost zero magnitude) do not coincide (retardation neglected). Connected with these modes are macroscopic fields, which account for the directional dispersion in the uniaxial crystal, i.e., for the change in frequency with the direction of the wavevector (middle portion B of Figure 1). This is qualitatively known from the continuum approximation, s,s If one writes the displacement operator ff(]~[~j) of an atom at position R~ in a mode with wavevector ~'and branch index j as

Vol. 64, No. 6

the respective atoms and by - 1 for the As sublattice,

¢(fi~I¢#)=

(-1)%~(fi.l#'j)M~-

(2)

Then the corresponding expressions for AlAs- (GaAs-) like LO modes can be written to a good approximation in the form of standing waves in the AlAs (GaAs) layer, ~cos qjz, v~(z [03") ~ t sinqjz,

j=1,3,5,... j = 2, 4 , 6 , . . .

(B~modes) (A1 modes) (3)

with an effective wavevector

(1) (with the creation operator a+), then for the LA modes with wavevector parallel to the layer normal the dependence of the vibrational amplitudes ~" (actually their component along the normal) on the atom position is given to a good approximation by sine (A1) or cosine (B2) functions in each layer with logarithmic continuity at the boundaries. At higher frequencies deviations from such a behaviour occur. The displacement pattern for some branches is shown in Figure 2a. For the TA modes with wavevector parallel to the normal there is confinement to the AlAs layer for the higher (AlAslike) frequencies, and one finds larger amplitudes in the GaAs than in the AlAs layer for frequencies at the top of the bulkGaAs TA phonon band. The displacement pattern of the atoms in a confined optical mode becomes particularly simple if one multiplies the displacements ~'(_~[~'j) of equation (1) by the mass M~ of

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~; do do ~'o ~'o G'o ;, ~; ;, ;, ;, do Figure 2. Displacement pattern v of foldes L A modes (2a) and reduced displacements v I of confined L O modes (2b) of a (5/5) superlattice with wavevector parallel to the layer normal. The symbols mark the values of v and v' at the position of the G a or AI atoms (circles)and As atoms (triangles) (up to a c o m m o n factor), cf. Equ. (1), the lines through the symbols (for a given j) are guides to the eye.

2r - j qJ -- (n + 1)a

(4)

(for n monolayers of AlAs (GaAs)), and by an exponentially decaying wave in the other layer. Here j numbers the A1As(GaAs-) like branches from top to bottom, and z is the position of an atomic layer, measured from the middle between two interfaces. (With increasing values of j , the denominator in (4) decreases down to (n+0.8)a. In an electrostatiealcontinuum approximation, neglecting spacial dispersion of the optical bulk phonon bands, n + 1 is replaced by n in the expression for the effectivewavevector). Equ. (3) is in quantitative agreement with linear chain calculationsv's and the three-dimensionai treatment of Ref. 9. Expression (4) bears implications for the evaluation of experimental phonon spectra, see below. Figure 2b shows the resulting pattern of v~ for a (5/5) superstructure. For the T O (E symmetry) modes the results are somewhat more complicated because of the two-fold degeneracy for wavevectors parallel to the layer normal, 2 but the modes can also be described by the effectivewavevector of Equ. (4) (with j = 1, 2, 3,...). As the direction of the wavevector is changed from parallel to perpendicular to the layer normal the frequencies and the eigenvectors of the optical modes are changed; the dependence of the frequencies on direction is shown in the middle portion B of Figure 1; the frequencies of the B2 modes decrease, while every second branch of the two-fold degenerate E modes splits. The frequency of the mode with the greater amplitude component in the sagittalplane is shifted upward; there is no complete decoupling of the twofold E modes in those polarised purely parallel and perpendicular to the sagittal plane. For wavevector directions between parallel and perpendicular to the normal, i.e. for 0 < 0 < 90 °, the macroscopic field causes a mixing of the polarisation of purely B~ and E modes for the shifted ones to produce transverse and longitudinal modes, polarized mainly parallelto the sagittal plane. (In these cases the symmetry assignment in Figure 1 refers to the main polarisation.) At the same time the eigenvectors change from those of excitations confined to one layer (Figure 2b) to those of collectiveexcitations of the whole superlattice structure, as is shown in Figure 3a; the existing, but very small component of g perpendicular to the sagittal plane (of more acoustic character) is not shown. For structures in which the frequency of the AIAs-like B2 interface phonon lies below the frequency of the confined L O phonons (as in the (5/5) superlattice),i.e. at thickness ratios dAIAs/dGaAs __> 1, the most striking effect occurs at the AIAs-like B2 and E modes with the highest frequency (j = 1

LATTICE DYNAMICS OF GaAs/AIAs

Vol. 64, No. 6 AlAs-like

Interface

Guided Acousticcl

Modes, ~ i

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869

SUPERLATTICES

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Figure 3. Pattern of optical interface-mode displacements v' (Figure 3a) and guided (or interface like) acoustical-mode displacements v (Figure 3b) for various values of the wavevector perpendicular to the normal. The upper and lower part show the component of v' parallel to the normal and parallel to the wavevector, respectively. Symbols are as in Figure 2. in expression (3)). The displacement patterns of these new interface modes are very similar to those of the Fuchs-Kliewer interface excitations in an electrostatic continuum approach, s in which the component of the vibrational amplitude ~(zlqj) parallel to sagittal plane can be described by cosh qz and sinh qz functions, while for interface modes lying in the optical frequency bands of the bulk materials, in particular for GaAs-like interface phonons, mixing of the B2 mode with the lower-lying confined modes occurs, l° With increasing magnitude of the wavevector perpendicular to the layer normal the interface character becomes increasingly pronounced as can also be seen from Figure 3a. At the same time the other modes (with j > 1) are somewhat affected, too. In the acoustical frequency regime modes can be found for wavevectors perpendicular to the layer normal which are interface acoustic (or guided) waves in the sense that in one layer their amplitude decays exponentially with increasing distance from the interfaces, while they have the character of standing waves in the other. In the low frequency region two types of such modes can be distinguished (Figure 3b): Modes of the first type are polarized in the sagittal plane and correspond to the generalized Lamb waves (as defined for the case of excitations of an elastic plate on an elastic substrate12),

and the other modes are polarized perpendicular to it and correspond to Love waves. Equation (4) leads to a reevaluation 13 of the experimental Raman spectra. To each optical branch with frequency wj can be assigned an effective wavevector qi- A plot of room temperature data ls,le of wl of the first GaAs-like LO mode (of superlattiees with different GaAs-layer thickness of up to four monolayers) versus qi is shown in Figure 4b together and in agreement with experimental neutron scattering data. iv The similarly reevaluted low-temperature data of Sood et al.lS.14 (LO and TO modes to 6th order of a (7/7) superlattice) are compared with recent low-temperature highprecision neutron scattering data is in Figure 4a. It is not yet clear wether the discrepancy between the Raman and neutron data is due to too much simplifying harmonic model assumptions, imperfections, anharmonic effects, or other causes. A more complete account of this work is in preparation.

Acknowledgements - We have enjoyed discussions with G. Kanellis, A. Fasolino, M. Cardona, and M. Staines. The calculations and drawings were carried out using the university computer facilities.

870

LATTICE DYNAMICS

Temperature

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Vol. 64, No. 6

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Figure 4. (a) Superlattice frequencies from low-temperature Raman scattering spectra 14 versus effective wavevector and low temperature neutron scattering data le of bulk GaAs. (b) Room-temperature superlattice Raman is,is and neutron scattering data. 1~ The plot contains also the calculated bulk dispersion and calculated frequencies of a (5/5) superlattice. References

aK. Kunc and H. Bilz, Solid State Commun. 19, 1027 (1976) 2j. Sapriel, J.C. Michel, J.C. Tolddano, R. Vacher, J. Kervarec and A. Regreny, Phys. Rev. B 28, 2007 (1983) SB. Monemar, Phys. Rev. B 10, 676 (1974) 4B. Jusserand, F. Alexandre, J. Dubart and D. Paquet, Phys. Rev. B 33, 2897 (1986) SR.E. Camley and D.L. Mills, Phys. Rev. B 29, 1695 (1984) SA.K. Sood, J. Mendndez, M. Cardona and K. Ploog, Phys. Rev. Left. 54, 2115 (1985) 7B. Jusserand and D. Paquet, Phys. Rev. Lett. 56, 1752 (1086) sE. Molinari, A. Fasolino and K. Kunc, Phys. Rev. Lett. 56, 1751 (1986) 9E. Molinari, A. Fasolino and K. Kunc, Superlattices and Microstrucstures 2, 397, (1986) x°Strauch D. and Richter E., to be published lXM. Babiker, J. Phys. C 19, 683 (1986). In this paper the

phonon bands are assumed to be overlapping; this restriction is not valid for the optical modes in GaAs/A1As. 12A.A. Maradudin, FestkSrperprobleme/Advances in Solid State Physics XXI (Edited by J. Treusch), p. 25, Vieweg, Dortmund (1981) lSA.K. Sood, J. Mendndes, M. Cardona and K. Ploog, Phys. Rev. Lett. 56, 1753 (1986) 14A.K. Sood, J. Mendndez, M. Cardona and K. Ploog, Phys. Rev. Lett. 54, 2111 (1985) 15A. Ishibashi, M. Itabashi, Y. Mori, K. Kaneko, S. Kawado and N. Watanabe, Phys. Rev. B 33, 2887 (1986) XeM. Nakayama, K. Kubota, H. Kato, S. Chika and N. Sano, Solid State Commun. 53,493 (1985) XTG. Dolling and J.L.T. Waugh, Lattice Dynamics (Edited by R.F. Wallis), p. 19, Pergamon, London (1965); J.L.T. Waugh and G. Dolling, Phys. Rev. 132, 2410, (1963) lSStranch D. and Darner B., to be published