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

102 Chapter 5: Spin Hall Effect
1.00
0.70
0.50
Ρ typ
Ρaver
0.30
0.20
0.15
0.10
0.00000 0.00005 0.00010 0.00015 0.00020
1
L 2
Figure 5.9: Finite size analysis of the typical DOS in relation to the averaged one, ρ typ /ρ avr ,
here plotted for E F = 0: The system size has been changed with L = 70, 140, 200, 280.
The impurity strength for the different curves is given by V/t = 1,3,4,5,6,8 (monotone
from top to bottom).
function
〈
∣ 〉
〈A;B〉 ω = 0
∣ A 1 ∣∣∣
ω +iǫ−H B 0 , (5.52)
the imaginary part of this function yields
− 1 π I[〈A;B〉] ω =
D−1
∑
k=0
〈0|A|k〉〈k|B|0〉δ(ω −E k ) (5.53)
assuming that 〈0|A|k〉〈k|B|0〉 is real. To apply KPM, we have to rewrite this expression
in terms of a trace. This can be done similar to a local DOS calculation according to
Eq.(5.37), which gives the coefficients
µ n = 1 D
D−1
∑
k=0
∫ 1
|〈i|k〉| 2 T n (Ẽk) (5.54)
D−1
∑
=
−1d˜ω 1 〈i|½|k〉〈k|½|i〉δ(˜ω −Ẽk)T n (˜ω) (5.55)
D
k=0
= 1 〈
∣
i∣T n (
D
˜H)
〉
∣
∣i . (5.56)