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

Chapter 5: Spin Hall Effect 105
Finally, wereconstructσ SH (E F )withthecalculatedmatrixelementdensityfunctionj M (x,y)
using a finite number of moments,
σ SH (ẼF,M) = e V
∫ 1 ∫ 1
−1
−1
dxdy f(x)−f(y)
(y −x) 2 j M (x,y). (5.69)
We set η to zero because now the divergent terms are damped by the fact that we use only
a finite number of expansion terms M, which correspond to η as shown in Sec.5.3.2. As a
presentation of the KPM we chose the same system as used for Fig.5.10. The Lorentzian
function scale parameter was chosen as η = 0.07 which is why we choose M = 200 moments
for the expansion, according to Fig.5.11 (b). The resulting SHC σ SH (E F ,M) is plotted in
Fig.5.11 (red curve) and for comparison the analytical solution (blue curve). The slight
asymmetry is due to an asymmetric choice of discrete k-values in the arguments of the
trigonometric functions. This asymmetry disappears for larger systems.
1.0
0.5
ΣsHe 8Π
0.0
0.5
1.0
4 2 0 2 4
Figure 5.11: SHC σ SH (E F ,M) calculated with KPM for a system with 70×70 sites, pure
Rashba SOC of strength α 2 = 1t and M = 200 moments (blue curve). For comparison the
analytical solution is plotted (red curve), corresponding to Fig.5.10.
E F