Quantum diffusion and localization in disordered systems

M. Kaveh

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2 Scopus citations


We describe quantum diffusion of the electrons in a disordered system by requiring that ψ(r, t)2 obeys a diffusion equation, where ψ(r, t) is the time-dependent wavefunction. It is found that, regardless of the weakness of the disorder, this requirement leads to electron eigenstates which consist of a power-law component for dimension d> 1 and a logarithmic correction for d = 1, in addition to an extended function. For d≤2, this is correct only below a certain length scale. As a result, even for kFl>>1, the conductivity σ is reduced from the Boltzmann conductivity σB in agreement with diagrammatic calculations. By expanding the eigenstate in terms of 1/rnit is shown that the 1 /rd−1 term is responsible for the reduction of σ in the weak-disorder limit. It is demonstrated that in three dimensions one can extrapolate the formula for the conductivity down to the Anderson transition to obtain σ =g2σB[1−(C/g2kF2l2) (1 − l/L)] where g is the reduction in the density of states due to disorder and C is a dimensionless constant of order unity which depends on some cut-off length.

Original languageEnglish
Pages (from-to)453-464
Number of pages12
JournalPhilosophical Magazine B: Physics of Condensed Matter; Statistical Mechanics, Electronic, Optical and Magnetic Properties
Issue number4
StatePublished - Apr 1985


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