STATISTICS OF THE LYAPUNOV EXPONENT IN 1D RANDOM PERIODIC-ON-AVERAGE SYSTEMS

In this paper we numerically study localization properties of band gap states in a one-dimensional periodic-onaverage random system (PARS). These kinds of system were extensively studied in the past in the context of electron localization, where they were known as Kronig-Penny-like models (see, for example, Ref. [1–3] and references therein). Classical wave versions of 1-D PARS also recently attracted a considerable attention [4–8]. Most of these studies focused upon localization properties of states from pass (conduction) bands of the respective initial periodic systems, or states at band edges of the original spectrum. They were found to behave similarly to the one-dimensional Anderson model, demonstrating single-parameter scaling (SPS) and universality [9]. Disorder, however, not only localizes states in the conduction bands of 1-D systems, it also gives rise to localized states inside band gaps of the original spectrum. This is well known in the physics of disordered semiconductors, where a vast literature on properties of localized states arising within forbidden gaps of semiconductors exists (see, for example, book [10]). In the case of one-dimensional models, however, these states have been studied surprisingly little. Particularly, statistical properties of the Lyapunov exponent (the inverse localization length), λ, for these states have not been studied at all. At the same time, it turns out that the variance, var(λ), of the Lyapunov exponent contains important information about spectral properties of these systems. From the frequency dependence of the variance, we find that the band gap states can be divided into two groups with qualitatively different localization properties separated by a sharp boundary. This means that though all states in 1-D systems are localized, there might be two qualitatively different regimes of localization. The first regime corresponds to the band and band edge states, and has regular Anderson behavior (if disorder is locally weak). The second regime, associated with the gap states, does not obey SPS and is not universal. The regular tight-binding Anderson model also demonstrates violation of SPS, when disorder becomes locally strong [11,12]. It should be empasized, therefore, that in our case the absence of SPS is caused not by the strength of disorder, but by the different nature of the gap states. Studying how var(λ) and the Lyapunov exponent (LE) itself depend upon the degree of disorder (rms fluctuations of a random parameter, σ) we find that there exists a critical value, σcr, at which the boundary between the groups of states with different localization properties disappears. At σ > σcr all the states have the regular Anderson-like behavior. It is interesting to note that in this situation SPS is restored when disorder becomes stronger, contrary to what one would expect in the Anderson model. In the paper we deal with the classical wave version of PARS and consider localization properties of scalar waves in a 1-D superlattice composed of two alternating layers A and B with dielectric constants eA and eB, respectively. The results, however, can also be applyed to Kronig-Penny-like models of electron localization. We introduce disorder in the system forcing the thickness dB of the B layers to change randomly assuming that dB is drawn independently from a uniform distribution. The structure of the described model is periodic on average with the spatial period equal to d = dA + h dBi and with random positions of the boundaries between different layers. The states of the model are characterized by a dimensionless wave number k = (ω/c)d, where ω is the frequency and c is the vacuum speed of light. We study the model by means of the transfer-matrix method. The state of the system is described by the vector un with components representing the wave field, En, and its spatial derivative, E ′ . The evolution of the vector un is