Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Chapter 5: <strong>Sp<strong>in</strong></strong> Hall Effect 105<br />
F<strong>in</strong>ally, wereconstructσ SH (E F )withthecalculatedmatrixelementdensityfunctionj M (x,y)<br />
us<strong>in</strong>g a f<strong>in</strong>ite number <strong>of</strong> moments,<br />
σ SH (ẼF,M) = e V<br />
∫ 1 ∫ 1<br />
−1<br />
−1<br />
dxdy f(x)−f(y)<br />
(y −x) 2 j M (x,y). (5.69)<br />
We set η to zero because now the divergent terms are damped by the fact that we use only<br />
a f<strong>in</strong>ite number <strong>of</strong> expansion terms M, which correspond to η as shown <strong>in</strong> Sec.5.3.2. As a<br />
presentation <strong>of</strong> the KPM we chose the same system as used for Fig.5.10. The Lorentzian<br />
function scale parameter was chosen as η = 0.07 which is why we choose M = 200 moments<br />
for the expansion, accord<strong>in</strong>g to Fig.5.11 (b). The result<strong>in</strong>g SHC σ SH (E F ,M) is plotted <strong>in</strong><br />
Fig.5.11 (red curve) and for comparison the analytical solution (blue curve). The slight<br />
asymmetry is due to an asymmetric choice <strong>of</strong> discrete k-values <strong>in</strong> the arguments <strong>of</strong> the<br />
trigonometric functions. This asymmetry disappears for larger systems.<br />
1.0<br />
0.5<br />
ΣsHe 8Π<br />
0.0<br />
0.5<br />
1.0<br />
4 2 0 2 4<br />
Figure 5.11: SHC σ SH (E F ,M) calculated with KPM for a system with 70×70 sites, pure<br />
Rashba SOC <strong>of</strong> strength α 2 = 1t and M = 200 moments (blue curve). For comparison the<br />
analytical solution is plotted (red curve), correspond<strong>in</strong>g to Fig.5.10.<br />
E F