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Chapter 3 Transport in a Two-Dimensional Disordered Electron Liquid with Isospin Degrees of Freedom
Published in Sergey Kravchenko, Strongly Correlated Electrons in Two Dimensions, 2017
The essence of the phenomenon of Anderson localization is quantum interference that can fully suppress diffusion of a quantum particle in a random potential (Anderson, 1958). The wave function of a particle with a given energy can be localized or delocalized in space. It leads to a possibility for existence of quantum phase transition with changing particle energy or parameters of the random potential. This quantum phase transition is usually termed as Anderson transition. Instead of quantum mechanical problem for a single particle one can consider a system of noninteracting electrons. Then the Anderson transition can occur under change of the chemical potential or dimensionless parameter kFl where kF denotes the Fermi momentum and l stands for the elastic mean free path. From the point of view of transport properties the phase with delocalized states is a metal whereas the phase with localized states is an insulator.
Weak Localization and Electron Dephasing in Disordered Conductors I: Metallic Limit
Published in Andrei D. Zaikin, Dmitry S. Golubev, Dissipative Quantum Mechanics of Nanostructures, 2019
Andrei D. Zaikin, Dmitry S. Golubev
In practice, however, Anderson localization turns out to be a pretty fragile phenomenon because it requires complete phase coherence of the electron wave functions. While the electron coherence is not damaged by elastic scattering (e.g., on impurities or otherwise), it can be destroyed by processes such as electron–electron and electron–phonon interactions, spin-flip electron scattering on magnetic impurities, and so on. As a result, quantum coherence of electrons propagating in disordered conductors can be maintained only during some finite time τφ and within some finite length Lφ=Dτφ, where D = vFℓ/3 is the electron diffusion coefficient. The so-called dephasing time τφ and dephasing length Lφ were already encountered before in Chapters 3 and 8. These parameters will also play a crucial role in our subsequent considerations. For instance, we will demonstrate that in generic metallic conductors of a reduced dimension (and described by the condition (14.1)), electron–electron interactions restrict the electron dephasing length Lφ to always be parametrically shorter than the localization length ξloc. Under this condition, strong (Anderson) localization cannot be observed and turns into weak localization (WL), which will be the main subject of both this and the next chapters.
Effect of Magnetic Field on the Transport Phenomena in Quantum Nanostructures
Published in Jyoti Prasad Banerjee, Suranjana Banerjee, Physics of Semiconductors and Nanostructures, 2019
Jyoti Prasad Banerjee, Suranjana Banerjee
According to Anderson localization, all states in a bulk 3-D sample are localized for sufficiently disordered material. Electron scattering with point defects like impurities, structural disorder, and interface roughness in less disordered material leads to partial localization of electron states. The broadening of delta function shape of DOS function, shown in Figure 8.14, is due to partial localization of electron states. The peaks of DOS curve are observed at and around the energy E=(n+12)ℏωc. The extended states that propagate through the sample lie at the center of each LL. The extended Bloch states correspond to mobile electrons and localized non-conducting state. The electron states lying near the tails of LLs are localized and bound, which do not contribute in conduction. In other words, if the Fermi level is within the localized states, electrons cannot participate in conduction and carry current. The location of Fermi level between LLs is no longer required for observation of IQHE. Rather, observation of IQHE depends on the location of Fermi level within the localized states. The finite width of resistance plateaus can be understood from the concept of localization of states in the tail of broadened LLs. The origin of IQHE remained unresolved because the expected value of Hall resistance can be obtained provided all electrons participate in conduction. Contrary to our expectation, only the electrons in the localized states participate in conduction. This fallacy can be resolved, subject to the criterion that the extended states compensate for the localized states and the extended states carry an additional current over that in a defect-free sample. On the other hand, if the Fermi level is within the extended states, the electrons participate in conduction. Theoretical analysis shows that constant value of resistance can be maintained, if the extended states carry more current to compensate for the non-localized states. This mysterious behavior can be explained from a rigorous theoretical analysis. In this analysis, the electrons occupying the extended states carry more current. This is due to the fact that the electrons occupying the extended states are accelerated by the scattering potential arising from disorder with an increase of their average drift velocity.
Statistical analysis of wave localization and delocalization in one-dimensional randomly disordered phononic crystals with finite cells
Published in Waves in Random and Complex Media, 2022
Ke Ma, Ruo-Xi Liu, Feng Wu, Jia Xu
In practical applications, due to manufacturing errors during machining, random disorder in phononic crystals is inevitable. This can lead to several new phenomena such as the well-known Anderson localization in disordered systems [34]. When the random disorder is considered, the band structure is always random, which leads to a problem of how to study the random behavior of the wave propagation in the randomly disordered phononic crystal (RDPC). The most conveniently and widely used method is the localization factor [35], first introduced by Li et al. The localization factor, which is often combined with the transfer matrix method, can characterize the average exponential rate of growth or decay of the wave amplitudes [36]. One advantage of the localization factor is that it can be computed by using an infinite RDPC generated from any a single sample. With the help of the localization factor, it has been observed in many studies [37,38] that, for the infinite phononic crystal, the larger the degree of disorder, the stronger the wave localization. It must be noted that all these works focused on the infinite RDPCs, while there is much less research on finite cells. Obviously, the localization factor of the RDPC with finite cells should also be random, which means that the statistical method may be necessary.