Anomalous properties of the Kronig-Penney model with compositional and structural disorder

We study the localization properties of the eigenstates in the KronigPenney model with weak compositional and structural disorder. The main result is an expression for the localization length that is valid for any kind of selfand inter-correlations of the two types of disorder. We show that the interplay between compositional and structural disorder can result in anomalous localization. Pacs numbers: 73.20.Jc, 73.20.Fz, 71.23.An Recently, much attention was paid to low-dimensional disordered models with long-range correlations in random potentials. Apart from the theoretical aspects, the interest on this issue has increased significantly due to the possibility of constructing random potentials with specific correlations which result in a strong enhancement or reduction of the localization length [1, 2, 3, 4, 5, 6]. These new effects allow for the fabrication of electron and optic/electromagnetic devices with desired anomalous transport properties. As was shown analytically [2, 3, 4] and confirmed experimentally [5, 6], 1 one can arrange prescribed windows of energy with perfect transmission (or reflection) of scattering waves. One of the most important models, both from the theoretical and experimental point of view, is the Kronig-Penney (KP) model, which was introduced long ago to analyze electronic states in crystals [7]. Since the ’80s, this model has attracted considerable attention because it provides a convenient description of superlattices (see, e.g., [8] and references therein). Modifications of the standard Kronig-Penney model have been suggested for a study of the physics of random and quasi-periodic systems with various applications, see, e.g., [9]. Recently, the Kronig-Penney model has been used to discuss the possibility of selective transmission in waveguides (see [5, 6] and references therein). In this paper we study the KP model with two types of weak disorder. Disorder of the first kind, or “compositional”, is due to small variations in strength of the delta-shaped barriers. In addition, the spacings between the barriers can be also randomly perturbed (the so-called “structural” disorder). Our interest lies in the interplay of these two kinds of disorder which can exhibit both self-correlations and mutal correlations. Our goal is to derive a formula for the localization length, and to analyze it. The stationary Schrödinger equation for the eigenstates ψ(x) has the form − h̄ 2 2m ψ(x) + ∞