Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University 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++
Chapter 5: Spin Hall Effect 93 V⩵0.2 V⩵2.8 0.20 0.15 Ρ 0.15 0.10 0.05 Ρ 0.10 0.05 0.00 10 5 0 5 10 E 0.00 10 5 0 5 10 E V⩵ 3 V⩵ 3.6 0.15 0.15 Ρ 0.10 Ρ 0.10 0.05 0.05 0.00 10 5 0 5 10 E 0.00 10 5 0 5 10 E V⩵4 V⩵5 0.15 0.15 0.10 0.10 Ρ Ρ 0.05 0.05 0.00 10 5 0 5 10 E 0.00 10 5 0 5 10 E Figure 5.3: DOS, as a function of Fermi energy in units of t, in presence of impurities of binary type with a concentration of 10% calculated using exact diagonalization. The system size is L 2 = 32 2 , and the SOC is Rashba type with α 2 = 1.2t with cutoff η = 0.02t.
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Chapter 5: <strong>Sp<strong>in</strong></strong> Hall Effect 93<br />
V⩵0.2<br />
V⩵2.8<br />
0.20<br />
0.15<br />
Ρ<br />
0.15<br />
0.10<br />
0.05<br />
Ρ<br />
0.10<br />
0.05<br />
0.00<br />
10 5 0 5 10<br />
E<br />
0.00<br />
10 5 0 5 10<br />
E<br />
V⩵ 3<br />
V⩵ 3.6<br />
0.15<br />
0.15<br />
Ρ<br />
0.10<br />
Ρ<br />
0.10<br />
0.05<br />
0.05<br />
0.00<br />
10 5 0 5 10<br />
E<br />
0.00<br />
10 5 0 5 10<br />
E<br />
V⩵4<br />
V⩵5<br />
0.15<br />
0.15<br />
0.10<br />
0.10<br />
Ρ<br />
Ρ<br />
0.05<br />
0.05<br />
0.00<br />
10 5 0 5 10<br />
E<br />
0.00<br />
10 5 0 5 10<br />
E<br />
Figure 5.3: DOS, as a function <strong>of</strong> Fermi energy <strong>in</strong> units <strong>of</strong> t, <strong>in</strong> presence <strong>of</strong> impurities <strong>of</strong><br />
b<strong>in</strong>ary type with a concentration <strong>of</strong> 10% calculated us<strong>in</strong>g exact diagonalization. The system<br />
size is L 2 = 32 2 , and the SOC is Rashba type with α 2 = 1.2t with cut<strong>of</strong>f η = 0.02t.
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