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

92 Chapter 5: Spin Hall Effect
result is presented in Fig.5.3. For small V one would expect the Van Hove singularity at
half filling. Due to finite Rashba SOC it is split to finite energies E = ±(2t− √ 4+α 2 2 t),
indicated in Fig.5.1 for the energy below half filling. If the impurity strength is increased,
a preformed impurity band is created. Using exact diagonalization 1 and applying the Kubo
formalism, Eq.5.22, we calculate the SHC. Exemplarily we show the result for σ SH (E) at
V = −2.8t in Fig.5.4. The SHC is strongly reduced but shows an additional maximum at
energy where the preformed impurity band is located, as can be seen in comparison with
the DOS, Fig.5.4(b).
Toanalyze thereduction ofSHCfor agiven fillingn, wevary theimpuritystrength
up to V = −5t and keep the concentration constant at 10%. Similar to the results in the
case of block distribution of impurity strength, we see a monotone suppression at all fillings,
even in the preformed impurity band, Fig.5.5.
5.3.2 Kernel Polynomial Method
The numerical calculations presented in the previous section, which were based
on exact diagonalization using LAPACK[LAP] routines, are limited to small system sizes.
This leads to finite size effects like oscillations in the SHC, e.g. Fig.5.4(b) above half filling.
For further calculations concerning the role of the impurity band it is necessary to do a
finite size scaling analysis and consider system-sizes beyond L = 64 which makes an exact
treatment on current hardware impossible: for a D-dimensional matrix such a calculation
requires memory of the order of D 2 , and the LAPACK routine scales as D 3 .
Another problem is the adjustment of the cutoff η, see Eq.(5.22), which has to be taken
with care as analyzed e.g. by Nomura et al., Ref.[NSSM05].
Toovercome thelimitation onsmall systems, therearedifferent numerical order-Dmethods.
Oneprocedureisthetimeevolution projection methoddevelopedbyTanakaandItoh[TI98],
which was already used to calculate SHC[MMF08, MM07]. However, the algorithm requires
both the choice of a sufficient number of time steps and an adjustment of cutoff η. A more
effective method, which uses Chebyshev expansion based on Kernel Polynomial Method,
will be presented in the following.
1 using LAPACK[LAP] and OpenMP[OMP] in C++