Itinerant Spin Dynamics in Structures of ... - Jacobs University

Chapter 5: Spin Hall Effect 99
the case where time-reversal and spin-rotational symmetry are preserved, i.e. unitary and
orthogonal universality class are present, the scaling function[AALR79]
β(g) = dln(g)
dln(L) , (5.45)
where g is the dimensionless conductivity, L the side length of our system, scales like
β(g) ≈ − 1 g
(5.46)
for large g, which means that all macroscopic systems are insulators. Here we used the
scaling parameter g = E Th /∆ LS , where E Th = D e /L 2 is the Thouless energy and ∆ LS =
1/(ρ EF L 2 ) the typical energy level spacing. However, Hikami et al. could show in Ref.
[HLN80] that if the universality class changes from orthogonal to symplectic, a MIT can
appear in a 2D system. Because SOC breaks spin rotation symmetry one can show that
SO interaction can enhance the localization length ξ drastically.[AT92, KKA10, SST05]
Recalling results from transfer matrix calculations of the Anderson model in 2D with SOC,
the critical disorder strength V c for the MIT is for strong Rashba SOC α 2 = 1t given by
V c ≈ 6.3t[SST05, And89] and for weaker SOC α 2 = 0.1t given by V c ≈ 4.6t[SST05]. As a
first application of the KPM we use the typical DOS,
ρ typ (E) = exp[〈〈log(ρ i (E))〉〉], (5.47)
in comparison to the local DOS
ρ i (E) = 1 D
D−1
∑
k=0
|〈i|k〉| 2 δ(E −E k ), (5.48)
as an indicator for the metallic or insulating regime. In contrast to the arithmetic mean of
ρ i , ρ avr (E) = 〈〈ρ i (E)〉〉, the geometric mean is suppressed until it vanished for V > V c : The
impurity is added to the Hamiltonian by adding the term H imp = ∑ iσ ǫ ic † iσ c iσ where the
ǫ i are uniformly distributed between [−V/2,V/2]. In the following we consider the Fermi
energy to be at half-filling. In the metallic regime we have ρ = 1/(L 2 ∆ B ) at all sites,
therefore we expect exp[〈〈log(ρ i (E))〉〉]/〈ρ〉 = 1.
In contrast, being deeply in the localized regime the states decay exponentially on the