# Regensburg 2004 – wissenschaftliches Programm

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# HL: Halbleiterphysik

## HL 53: Elektronentheorie

### HL 53.1: Vortrag

### Freitag, 12. März 2004, 12:00–12:15, H13

**A renormalization approach for the Anderson model in ***d*=1** and ***d*=2** at the band edge: Scaling of the localization length** — •Stefanie Russ — Institut für Theoretische Physik III, Universität Giessen, D-35392 Giessen, Germany

A renormalization procedure for the electronic wave functions
of the one- and two-dimensional Anderson model with diagonal disorder is
developed that applies close to the band edges. This theory leads in *d*=1
to a scaling form of the localization length
λ∼ ⟨є^{2}⟩^{−1/3}
*f*((2−| *E*|)/⟨є^{2}⟩^{2/3}), in agreement with
former
works. In *d*=2 a similar scaling
form applies for the localization volume *V* (participation ratio), i.e.
*V*^{1/2} ∼⟨є^{2}⟩^{−1/2}
*g*((4−| *E*|)/⟨є^{2}⟩). Here,
⟨є^{2}⟩
is the variance of the site potentials, *E* is the energy and
*f*(*x*) and *g*(*x*) are the scaling functions (*f*(0)=*g*(0)=1).

λ and *V* close to the band edges are
studied by numerical simulations and confirm this scaling ansatz.
The scaling functions *f*(*x*) and *g*(*x*) show a crossover
between the two regimes *x*≪ 1 and *x* ≫ 1. It is shown that these two
regimes refer to
Λ ≫ λ_{0} and Λ ≪ λ_{0}, respectively where
Λ
is the wavelength, λ_{0} in *d*=1 is the localization length at the
band
edge and λ_{0} in *d*=2 is the effective localization length
*V*^{1/2} at
the band edge.