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

100 Chapter 5: Spin Hall Effect
localization length ξ, |ψ(x)| 2 ∼ exp(−x/ξ), which leads to
ρ exp[
1
∫ ( ( )]
L
typ πL
=
2 0 drr∫ 2π
0
dϕ ln exp − r ξ
)ρ 0
ρ
∫ [
(5.49)
avr 1 L
πL 2 0 drr∫ 2π
0
dϕ exp − r ξ
]ρ 0
]
L 2
=−
=−
L
ξ
exp[
1
3
(
2ξ L+ξ −ξexp[
L
ξ
with u = L/ξ. From the last equation we can conclude that
]) (5.50)
exp [ 1
3 u] u 2
2(u+1−exp[u]) , (5.51)
ρ
• this value vanishes in the thermodynamic limit, lim typ
L→∞
ρ avr
= 0,
• the fraction is a monotone function of the system length L (in contrast to e.g. the 1d
case).
The explained difference between the averaged and typical DOS is plotted in Fig.5.7: In (a)
the averaged DOS is plotted for different impurity potentials V. The band edges are shifted
to larger energies with larger V. In contrast, in (b) the typical DOS is plotted for different
system sizes at impurity strength V = 8t. The product of local densities leads to a strong
reduction with the size. This reduction is also significant at the band edges. Knowing the
0.25
0.20
0.03
0.15
Ρaver
Ρ typ
0.02
0.10
0.05
0.01
0.00
5 0 5
0.00
6 4 2 0 2 4 6
E
E
(a)
(b)
Figure 5.7: (a) Averaged DOS ρ avr calculated with KPM (30 impurity configurations) for
system size of L 2 = 280 2 with Rashba SOC α 2 = 0.5t, for different impurity strengths:
V = 1t(black), V = 4t(red), V = 6t(green), V = 8t(blue) (b) Monotoneous reduction of
typical DOS ρ typ with system size L = 70, 140, 200, 280, for impurity strength V = 8t.
behavior of the typical DOS ρ typ , we carried out a finite size analysis for different impurity