Superlattices and Microstructure.% Vol. 7, No. 2, 1985 165 PHONON DISPERSION RELATIONS IN SEMICONDUCTOR SUPERLATTICES IN THE ADIABATIC BOND-CHARGE MODEL Sung-kit Yip and Yia-Chung Chang Department of Physics and Materials Research Laboratory University of Illinois at Urbana-Champaign Urbana, Illinois 61801 (Received 13 August 1984 by J. D. Dow) The phonon dispersion in semiconductor superlattices is studied in the adiabatic bond-charge model. The complex phonon dispersion relations of the constituent bulk materials are obtained via the eigenvalue method in a zeroth-order calculation, which includes only short range forces and Coulomb interactions between ions and bond charges in the same and neighboring layers. The eigenmode displacements of the superlattice are obtained by matching the eigenvectors associated with complex phonon branches at the interfaces. The effect of the remaining Coulomb interactions are then included in the first-order approximation. The results for the superlattice phonon frequencies compare favorably with the existing experimental data. L. Introduction Phonon properties of superlattices was first studied by Rytovl in a continuum model and later by Barker et al2 in a simple linear chain model with short-range interactions. As noted by Merlin et al3 the long-range Coulomb interaction plays a crucial role in the phonon- properties of superlattices. In this paper, we shall present a more realistic calculation which takes into account both the presence of a lattice and the long range Coulomb interaction. We adopt the adiabatic bond charge model (BCM)4-6 for describing the phonon dispersion in the bulk semiconductor and then solve the superlattice problem using the idea of complex band structure7p8 (complex phonon dispersion relation). Due to the difficulty of finding the complex phonon dispersion relation in a model with long-range interactions, we shall treat the problem by a perturbation method. We first solve the problem in a zeroth-order model, in which the Coulomb interaction between any two layers of ions are truncated at a short distance, and with the remaining Coulomb interaction treated as a perturbation. We find that the dispersion curves for the bulk obtained this way are extremely close to those obtained by a full calculation. In the zeroth order model we can proceed in an analogous way to the electronic “complex band structure” calculation8 to obtain the complex phonon dispersion and the associated eigenvectors of the constit’uent semiconductors and then match them at the boundaries to obtain the superlattice phonon dispersion relation. Finally we add back the remaining long-range Coulomb interaction using a first order perturbation theory. 0749-6036/85/020165+07 $02.00/O II. Adiabatic bond charge model (BCM) We shall refer the reader to the original papers for the details of BCM. Here we only remind the reader that in BCM, we imagine that the lattice contains bond charges (BC) lying on the bonds between cations and anions, dividing it in the ratio of 5:3. The cations and the anions have charge 22 and BC, -2. Table I shows the parameters used in BCM for seven III-V compound semiconductors. The first four rows are adopted from Ref. 4. The InP parameters are obtained by fitting to the Table 1. BCM parameters for some III-V semiconductors, given in units of e2 /va, where v, is the unit cell volume. Ion-ion Ion-BC BC-ion-BC $;_i,3 $;,3 f&3 B1 B2 221, GaAs 6.16 2.36 16.05 5.36 a.24 0.187 GaP 6.04 2.40 17.91 5.20 10.00 0.203 GaSb 6.77 2.37 13.10 6.28 7.08 0.160 InSb 7.47 2.32 14.09 4.56 6.24 0.172 InP 7.16 2.95 21.62 3.43 a.37 0.249 InAs 7.31 2.64 17.86 3.99 7.30 0.210 AlAs 5.80 2.27 15.48 5.79 a.54 0.180 0 1985 Academic Press Inc. (London) Limited