The absence of localization in one-dimensional disordered harmonic chains

Abstract

We study one-dimensional harmonic chains in which clusters of two or three defect atoms are embedded randomly. The disorder in the systems appean in the masses of the atoms. Reflectionless modes are obtained by studying different kinds of correlation among the m e s . The localization behaviour of the modes around these special frequencies is examined analytically a s well as numerically. To discem the nature of the modes at and around those frequencies, density of states, bandwidth scaling and site Green functions are sNdied. If the special frequencies lie within the common band of the constituent atoms and at zero the modes are extended ak and around them. However, the modes are critical when the special frequency appears at the upper band edge of the host system. The number of non-scatted modes is estimated for all cases. It isf i for the dimer problem. For the trimer problem with degenerate resonances appearing inside the constituent band it is -N3/'. If Ule degenerate resonances of the trimer appear at W O freqency the number of non-scattered modes is -NSl6.

Delocalization in harmonic chains with long-range correlated random masses

“…Once the initial values for u 0 and u 1 are known, the value of u n can be obtained by repeated iterations along the chain, as described by the product of transfer matrices The localization length of each vibrational mode is taken as the inverse of the Lyapunov exponent γ defined by [13,14,25] …”

“…Strong fluctuations in the DOS are related to the presence of localized states, whereas smooth a DOS is usually connected with the emergence of delocalized states [5,14]. In Fig.…”

Vibrational modes in aperiodic one-dimensional harmonic chains

“…However, there are a few low-frequency modes not localized, whose number is of the order of ͱ N. 18,19 It was shown that correlations in the mass distribution produce a new set of nonscattered modes in this system. 20 Nonscattered modes have also been found in disordered harmonic chains with dimeric correlations in the spring constants. 21 A large amount of work has been done in recent decades to understand localization behavior in randomly disordered chains.…”

Absence of reflection as a function of the coupling constant

“…Consider solutions of (2) in the situation where the coupling constant λ is a random variable. As was first observed in the context of the random dimer model [6,7], it is possible for the localization length to diverge (i.e., Lyapunov exponent vanish) at a fixed energy. In particular, this happens at positive energies when the reflection coefficient associated to the single site potential vanishes almost surely for a particular choice of random coupling constant.…”