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

98 Chapter 5: Spin Hall Effect
This leads us to the following condition:
σ ! > ∆ (5.41)
πt
M > D B
N
with the band width D B , and the number of states N.
(5.42)
M < πtN
D B
, (5.43)
To get an impression of the relation between both the η cutoff in exact diagonalization and
the finite number of moments in the KPM, we fix the system size, apply Rashba SOC, and
calculate the DOS using the eigenvalues E i calculated with exact diagonalization,
ρ η (E) = 1 ∑
[ ]
1
I . (5.44)
π E −E λ +iη
λ
Now ρ can be calculated using KPM, and M is adjusted to fit best to ρ η (E). The relation
between M and η is plotted in Fig.(5.6). Over a large interval of M we have η ∼ 1/M.
Only when the oscillations are too strong the differences between the Lorentz kernel, i.e.
using Eq.(5.44), and the Jackson kernel appear.
0.5
0.10
Ρ
0.4
0.3
0.2
Η
0.08
0.06
0.04
0.02
0.1
0.0
4 2 0 2 4
E
0.00
0.002 0.003 0.004 0.005 0.006
1
M
(a)
(b)
Figure 5.6: (a) DOS of a system of size L 2 = 40 2 with Rashba SOC, α 2 = 0.8t calculated
with exact diagonalization with cutoff η = 0.0215 (blue) and KPM with M = 500 moments.
(b) Relation between number of moments M and cutoff η.
First application: Metal-Insulator Transition
To determine if a 2D system has a metal-insulator transition (MIT) it is important
to analyze its symmetries: It is well known from the scaling theory of localization that in