Some elementary questions in the theory of quasiperiodic heterostructures

Abstract

The characterization of the spectrum of eigenstates of quasiperiodic heterostructures is discussed by focusing on three questions. Arguments are advanced to justify the often indiscriminate use of different approximants in the calculation of the eigenvalue spectra. It is stressed that the calculation of the fractal dimension may be rather inaccurate if the high eigenvalue range is not included, even if physically the interest is limited to the low range. The question of self-similarity is critically examined and found to have a very limited range of validity in practice. The unique properties of the Rudin-Shapiro sequence are also stressed.

Properties of elastic waves in quasiregular structures with planar defects

“…We consider finite systems with stress free bounding surfaces. It has been seen [16,18,26] that the characteristics of the spectrum in periodic or finite realizations of quasiregular systems are not essentially modified, beyond the appearance of localized states in the primary and secondary gaps in the finite case.…”

Self-Similarity in Semiconductors: Electronic and Optical Properties

“…(1; 2)) and numerous references can be found in two recent reviews. (3; 4) In these kind of systems, the question of self-similarity was deeply examined but found to have a very limited range of validity in actual practice (5). In addition, the fractal character (6) of the spectrum of elementary excitations is rigorously proved (7; 8) and confirmed in many numerical calculations (see for example Refs.…”

“…For the calculation of the eigenvalues we use the boundary conditions, i.e., our potential is infinite in z = l qw and z = −l qw , the transcendental equation by calculation of eigenvalues is M 12 (0, −l qw )=0, for odd states and M 22 (0, −l qw )=0 for even states, where M ij are the elements ij of the matrix M(l qw , −l qw ). Once we have the eigenvalues, the eigenfunctions are calculated in the usual manner, using the transference matrix method [5]. This system is actually a set of coupled quantum wells, therefore, we expect to observe self-similarity for basic states corresponding different wells as for excited states.…”

Transmittance and Fractality in a Cantor-Like Multibarrier System

“…The reader is addressed to references [10,12] in order to face some practical aspects and tricks on this matter. …”