Correlations in phonon-transmission spectra in periodic and quasiperiodic superlattices.

Abstract

We study the origin of correlations in phonon-transmission spectra in Fibonacci and periodic su­ perlattices (SL's), which have recently been recognized by numerical calculations and then confirmed by phonon-imaging experiments. The structure factors describing the interference effects of phonons reflected from the interfaces of both periodic and Fibonacci SL's are calculated. It is found that the wave number Qm.n (indexed by two integers m and n) for which the largest reflection occurs in a Fibonacci SL is identical to the wave number Qm + n for which (m + n )th Bragg reflection of phonons occurs in the corresponding periodic SL. We also show more generally that major phonon-transmission dips in the Fibonacci SL occur at frequencies close to each transmission dip in the periodic SL. Two-dimensional maps for the Bragg and Bragg-like reflections of phonons in these systems indicate that the correlations are applicable, irrespective of propagation directions and modes participating in the reflection processes. Since the discovery of quasicrystals,1 very significant advance has been made in the study of systems with quasiperiodic order.2 In particular, the spectral property of one-dimensional quasiperiodic (Fibonacci) lattices has been one of the topics studied extensively.3 The interest stems partly from the fact that the problem represents, in some sense, an intermediate case between periodic and random lattices. Undoubtedly, a major impetus toward reexamining the Fibonacci lattices was given by Merlin et al.4 In their pioneering work they fabricated a semi­ conducting layered structure based on the Fibonacci se­ quence, known as a Fibonacci superlattice (SL). The measurements of x-ray scattering revealed that this artificial heterostructure indeed exhibited properties characteristic of quasiperiodic systems.4,5 Raman experi­ ments have been performed on similar Fibonacci SL'S.6-9 Recently, the phonon-transmission spectra in a Fi­ bonacci SL have also been examined both experimental­ lylO and theoretically. 11 A striking similarity has been found between the Fibonacci and periodic SL's when the major transmission dips in the former are compared with the transmission dips in the latter. (The distribution of frequency gaps in the phonon dispersion relations also ex­ hibits the same similarity.) It has been predicted numeri­ cally on the basis of the transfer-matrix method for both the fre~uency and angular dependences of the transmis­ sion.IO, 1 The predicted angular dependences of transmis­ sion spectra in these SL's are in reasonable agreement with the results of phonon-imaging experiment. 10 The purpose of the present work is to clarify the origin of these correlations found in the phonon spectra of the periodic and Fibonacci SL's. This will be made by study­ ing SL structure factors which describe the interference effects of phonons reflected at the interfaces of constitu­ ent layers. We shall find that the wave numbers of pho­ nons for which the largest reflections occur in the Fi­ bonacci SL coincide with the wave numbers satisfying the Bragg condition in the associated periodic SL. This ob­ servation leads more generally to the result that major 39 reflections in Fibonacci SL closely correlate with Bragg reflections in the periodic SL in both their magnitudes