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

Chapter 5: Spin Hall Effect 101
strengths V in a system with weak Rashba SOC strength α 2 = 0.5t to find the critical value
V c for the MIT. From Landauer-Bütiker calculations[SST05] it follows that at impurity
strength V = 8t we are already in the insulating regime. The typical DOS ρ typ decays
exponentially with L, as plotted in Fig.5.8 (a), which confirms this assumption. If V is
reduced the localization length ξ shows a strong increase, as shown in Fig.5.8 (b), which
in turn slows down the reduction of ρ typ , which comes along with increasing system size,
significantly in case of large localization lengths. Adding the results from the calculation of
800
0.030
600
Ρ typ
0.020
Ξ a
400
0.015
200
0.010
50 100 150 200 250
(a)
L a
0
0 2 4 6 8 10
(b)
V t
Figure 5.8: (a) Log-plot of typical DOS ρ typ at V = 8t for different system sizes L with
Rashba SOC α 2 = 0.5t at half filling, calculated with KPM. The dashed line is a linear fit
to the log-data which yields a localization length of ξ ≈ 100a. (b) Localization length ξ at
E F = 0, plotted as a function of impurity strength V.
ρ typ /ρ avr , which is plotted in Fig.5.9 as function of the inverse system size 1/L 2 for E F = 0,
we can conclude that for V 5t we are in the insulating regime. For a more precise analysis
we have to go to larger systems due to the large localization lengths.
5.3.3 SHC calculation using KPM
In this section we are going to present calculation of SHC for much larger systems
than with exact diagonalization analysis, using KPM. Also here we will use the Kubo
formalism which is why we have to reformulate Eq.(5.5) to be applicable to this iterative
method. In contrast to the calculation of the DOS, here we have to deal with a correlation
of two operators. We will follow the approach presented in the KPM review by Weiße et
al.[WWAF06].
WestartwithaKPMwhereweexpandafunctiononlyinonedimension. Givenacorrelation