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Concept
Anderson localization
Summary
In condensed matter physics, Anderson localization (also known as strong localization) is the absence of diffusion of waves in a disordered medium. This phenomenon is named after the American physicist P. W. Anderson, who was the first to suggest that electron localization is possible in a lattice potential, provided that the degree of randomness (disorder) in the lattice is sufficiently large, as can be realized for example in a semiconductor with impurities or defects.
Anderson localization is a general wave phenomenon that applies to the transport of electromagnetic waves, acoustic waves, quantum waves, spin waves, etc. This phenomenon is to be distinguished from weak localization, which is the precursor effect of Anderson localization (see below), and from Mott localization, named after Sir Nevill Mott, where the transition from metallic to insulating behaviour is not due to disorder, but to a strong mutual Coulomb repulsion of electrons.
In the original Anderson tight-binding model, the evolution of the wave function ψ on the d-dimensional lattice Zd is given by the Schrödinger equation
where the Hamiltonian H is given by
with Ej random and independent, and potential V(r) falling off faster than r−3 at infinity. For example, one may take Ej uniformly distributed in [−W, +W], and
Starting with ψ0 localised at the origin, one is interested in how fast the probability distribution diffuses. Anderson's analysis shows the following:
if d is 1 or 2 and W is arbitrary, or if d ≥ 3 and W/ħ is sufficiently large, then the probability distribution remains localized:
uniformly in t. This phenomenon is called Anderson localization.
if d ≥ 3 and W/ħ is small,
where D is the diffusion constant.
The phenomenon of Anderson localization, particularly that of weak localization, finds its origin in the wave interference between multiple-scattering paths. In the strong scattering limit, the severe interferences can completely halt the waves inside the disordered medium.
For non-interacting electrons, a highly successful approach was put forward in 1979 by Abrahams et al.
Official source
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