Electrons in incommensurate crystals: Spectrum and localization

Abstract

Electronic properties of incommensurate crystals are studied on a one-dimensional Kronig-Penney model with sinusoidally modulated atomic positions. Spectra and wave functions have been calculated numerically as a function of the modulation wave vector. Information for the incommensurate case is obtained by considering series of commensurate approximants. It is shown that the spectrum of this model has a hierarchical and recursive nature connected with the continued-fraction expansion of the modulation wave vector. In the present model there are parts of the spectrum where the energy eigenvalues occur in bands with a finite number of numerically nonvanishing gaps (bandlike spectrum) and parts where new gaps open up at each step of the continued-fraction expansion (Cantor-type or discrete spectrum). Numerical evidence is presented that the electron states are extended in the bandlike regions and localized in the discrete regions. The characterization of the electron density is made more transparent if one considers the generalized density in (two-dimensional) "superspace." The localized electronic states violate the Fr\"ohlich assumption for the translational freedom of the modulation wave. This has consequences for the electrical conductivity in this model.