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

104 Chapter 5: Spin Hall Effect
(a)
(b)
Figure 5.10: a)+b) Matrix element density function j(x,y) for a clean system with 70×70
sites and pure Rashba SOC of strength α 2 = 1t using the analytical solution Eq.(5.20).
As already mentioned, we can use the great benefit to replace the trace by an average over
a number R ≪ D = 2×L 2 (the two is due to spin degree of freedom) of random vectors,
Eq.(5.39). Having the µ mn calculated, we can reconstruct j(x,y) according to Eq.(5.32),
∞∑ µ mn h mn T m (x)T n (y)
j(x,y) =
π 2√ (1−x 2 )(1−y 2 ) , (5.65)
m,n=0
where we added normalization functions
h mn =
4
(1+δ m,0 )(1+δ n,0 ) . (5.66)
On a computer we have to replace the infinite sum with a finite one where only M momenta
are taken into account. To cure the mentioned appearance of Gibbs oscillations we use
a convolution with a Jackson kernel, i.e. the coefficients µ mn are replaced by µ mn →
µ mn g m (M)g n (M), with kernel-damping factors[WWAF06]
g n (M) =
(M −n+1)cos(
πn
M+1
)
+sin(
πn
M+1
M +1
The function j(x,y) for finite M is therefore given by
j M (x,y) =
M−1
∑
m,n=0
) )
cot(
π
M+1
. (5.67)
µ mn h mn g m (M)g n (M)T m (x)T n (y)
π 2√ . (5.68)
(1−x 2 )(1−y 2 )