Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Abstract

This paper investigates a simple one-dimensional model of incommensurate "harmonic crystal" in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional space. It is shown that both approaches lead to essentially the same results; that is, the lower spectrum is splitted between "Cantor-like zones" and "impurity bands" to which correspond critical and extended eigenstates, respectively. These "new bands" seem to emerge inside the band gaps of the unperturbed problem when certain conditions are met and display a parabolic nature. Numerical tests are extensively performed on both steady and time-dependent problems.Key words: Quasi-periodic potential, impurity band, quasi-Bloch wave, quantum chaos, spectral algorithm, deformed crystal, Charge-Density Wave (CDW). 1991 MSC: 81Q05, 81Q20 IntroductionThis article is a step further in the numerical study of electronic motion in a one-dimensional model of "harmonic crystal", that is to say, a modeling in solid-state physics going beyond the rigid periodic lattice picture. Indeed, it consists in assuming that, within the BornOppenheimer approximation (the motion of nucleons is much slower than the electron's ones and thus decouples adiabatically), ionic cores are linked together by springs and vibrate like classical coupled oscillators; consult the book [1] and the recent paper [14] to which we are about to borrow notation.In [14], the following rescaled problem describing the motion of a free electron inside a 1-D lattice of atoms which display single-mode vibrations has been investigated within the * Corresponding Author Email address: l.gosse@ba.iac.cnr.it (Laurent Gosse). Preprint submitted to Elsevier ScienceWKB semi-classical framework [5,19] for the following Schrödinger equation,and under the fundamental assumption that k and Ω(k) = | sin(kπ)| are rational and commensurate. This is perhaps the simplest non-trivial model available which describes the effects of phonons onto electronic conduction and is sometimes referred to as the "modulated crystal" [1,9,25,26,32]. Since the potential in (1) isn't of the form cos(x − 1 2 at 2 ), the unitary transform propsed in [35] and leading to Bloch oscillations doesn't apply here (see also [33] for an AC-type modulation). A particular feature pointed out in [39] and called lattice tracking which refers to the dragging effect of heavy nucleons onto the much lighter electron was easily observed in the numerical results.The commensurability hypothesis is of critical importance in this global picture in order to keep on applying the classic Bloch decomposition for (1); we assume the reader familiar with this theory and refer only to [1,5,15,19] for details. The first step in this line of thinking is the derivation of "energy bands", which are dispersion relations for electrons around certain levels of excitation. Solids are clas...

The numerical spectrum of a one-dimensional Schrödinger operator with two competing periodic potentials

“…The new bands look like being k-periodic hence the original Brillouin zone is now too narrow to include them; see also [11] for a similar observation with a different type of perturbation. In [11], we observed that impurity bands appear to be parabolic: a plausible interpretation is that electrons perceive somehow the average of the corresponding very oscillating potential (like the so-called Jellium metal model [6]) and thus display a parabolic dispersion relation. The effect of the disordered medium boils down to the effective mass only.…”

“…It looks also like a smooth process as has been shown in [2] using the concepts of "resonance tongues" and "instability pockets". The upper part of the spectrum is rather messy, so we didn't display it; see however [11].…”

“…Indeed, periodicity is a special case of short-range order, leading generally to an absolutely continuous spectrum (the well-known "band structure") and extended eigenstates; quasi-periodicity, though not being completely disordered, displays only long-range order, thus allowing for singular Cantor-type spectrum and critical (very peaked) eigenstates, some of which can be seen in [11]. What we aim at here are the variations in the spectrum of (1.1) as k and ε vary; for very low k's, very few impurities show up in the medium hence based on common belief, the original Mathieu band structure should be visible modulo very small changes, see §3.1.…”

Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients II

“…This governs the quantum wave function of an independent electron within a crystalline lattice perturbed by some impurities whose dissemination induces long-range order only, which is rendered by means of the quasi-periodic potential q. In the paper [21], a simple onedimensional model of an incommensurable "harmonic crystal" is studied in terms of the spectrum of the corresponding Schrödinger equation.…”

Localized wavefunctions in quantum systems with multiwell potentials

“…This numerical evidence leads to the conjecture that the wavefunction (and other quantities related to it) might not be continuous functions of the parameter α. Related work has also been presented in [6,7]. In the context of free-electron lasers (which are periodic systems), related work has been presented in [8].…”