The phenomenon of the Anderson localization of waves in elastic systems is studied. We analyze this phenomenon in two different sets of systems: disordered linear chains of harmonic oscillators and disordered rods which oscillate with torsional waves. The first set is analyzed numerically whereas the second one is studied both experimentally and theoretically. In particular, we discuss the localization properties of the waves as a function of the frequency. In doing that we have used the inverse participation ratio, which is related to the localization length. We find that the normal modes localize exponentially according to the Anderson theory. In the elastic systems, the localization length decreases with frequency. This behavior is in contrast with what happens in analogous quantum mechanical systems, for which the localization length grows with energy. This difference is explained by means of the properties of the reflection coefficient of a single scatterer in each case.
We present numerical calculations of electronic structure and transport in the Penrose approximants. The electronic structure of perfect approximants shows a spiky density of states and a tendency to localization that is more pronounced in the middle of the band. Near the band edges the behavior is more similar to that of free electrons. These calculations of band structure and in particular the band scaling suggest an anomalous quantum diffusion when compared to normal ballistic crystals. This is confirmed by a numerical calculation of quantum diffusion which shows a crossover from normal ballistic propagation at long times to anomalous, possibly insulator-like, behavior at short times. The time scale t^*(E) for this crossover is computed for several approximants and is detailed. The consequences for electronic conductivity are discussed in the context of the relaxation time approximation. The metallic-like or non-metallic-like behavior of the conductivity is dictated by the comparison between the scattering time due to defects and the time scale t^*(E).
The degree of electronic localization in disordered one-dimensional systems is discussed. The model is simplified to a set of Diracδ-like functions used for the potential in the Schrödinger equation and calculations are carried out for the ground state. The disorder of topological character is introduced by the random shifts of the potential peaks. For comparison, we also discuss two aperiodic systems of the potential peaks: Thue-Morse and Fibonacci sequences. The localization, both in the momentum and the real space, is analyzed for different disorder strengths and sizes of the system. We calculate the localization length, and additionally we express the localization effects in terms of the inverse participation function and also by means of the Husimi quasi-classical distribution function in the phase space of the electron (position, momentum) coordinate system. We present the influence of disorder generated by the random and aperiodic sequences of potential on the energy spectrum.
The temperature dependences of electric conductivity σ, the Hall coefficient R of p-Ag_{2}Te in 4.2-200 K temperature interval for acceptor concentration N_{a} ≤ 6 × 10^{16} cm^{-3} were investigated. The minimum σ(T) was observed for all samples in ≈50÷80 K temperature interval. It was observed that the depth of minimum is increased with N_{a} decrease. It was shown that the part resonance scattering of electrons in minimum of σ(T) and maximum of |α_{n} (T)| region is 16-18%.
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