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

94 Chapter 5: Spin Hall Effect
0.10
V⩵2.8
Σ SH e 8Π
0.05
0.00
0.05
0.10
0.15
0.20
Ρ
0.15
0.10
0.05
2⋆Σ SH e 8Π
4 2 0 2 4
(a)
E
0.00
10 5 0 5 10
(b)
E
Figure 5.4: (a) SHC, as a function of Fermi energy in units of t, in presence of impurities
of binary type calculated using exact diagonalization. The impurity strength is V = −2.8t
with a concentration of 10%. The system size is L 2 = 32 2 , and the SOC is Rashba type
with α 2 = 1.2t with cutoff η = 0.06. (b) Comparison of a) with DOS (blue curve).
KPM in a Nutshell
The Kernel Polynomial Method (KPM) was first proposed by Silver et al.[SR94]
to calculate DOS of large systems. It is a method to expand integrable functions defined
on a finite interval f : [a,b] −→ R in terms of Chebyshev polynomials of the first,
T n (x) = cos(narccos(x)), (5.31)
or second kind
i.e., we can write for instance
f(x) =
U n (x) = sin((n+1)arccos(x)) ,
sin(arccos(x))
[
1
π √ µ 0 +2
1−x 2
]
∞∑
µ n T n (x) , (5.32)
if we assume that the function f has been rescaled to ˜f : [−1,1] −→ R and can be expanded
using the polynomials of the first kind, which are (as the one of the second kind) defined
n=1
on the interval [−1,1]. If so, the coefficients are given by
µ n =
∫ 1
−1
dx ˜f(x)T n (x). (5.33)