Itinerant Spin Dynamics in Structures of ... - Jacobs University
Itinerant Spin Dynamics in Structures of ... - Jacobs University Itinerant Spin Dynamics in Structures of ... - Jacobs University
94 Chapter 5: Spin Hall Effect 0.10 V⩵2.8 Σ SH e 8Π 0.05 0.00 0.05 0.10 0.15 0.20 Ρ 0.15 0.10 0.05 2⋆Σ SH e 8Π 4 2 0 2 4 (a) E 0.00 10 5 0 5 10 (b) E Figure 5.4: (a) SHC, as a function of Fermi energy in units of t, in presence of impurities of binary type calculated using exact diagonalization. The impurity strength is V = −2.8t with a concentration of 10%. The system size is L 2 = 32 2 , and the SOC is Rashba type with α 2 = 1.2t with cutoff η = 0.06. (b) Comparison of a) with DOS (blue curve). KPM in a Nutshell The Kernel Polynomial Method (KPM) was first proposed by Silver et al.[SR94] to calculate DOS of large systems. It is a method to expand integrable functions defined on a finite interval f : [a,b] −→ R in terms of Chebyshev polynomials of the first, T n (x) = cos(narccos(x)), (5.31) or second kind i.e., we can write for instance f(x) = U n (x) = sin((n+1)arccos(x)) , sin(arccos(x)) [ 1 π √ µ 0 +2 1−x 2 ] ∞∑ µ n T n (x) , (5.32) if we assume that the function f has been rescaled to ˜f : [−1,1] −→ R and can be expanded using the polynomials of the first kind, which are (as the one of the second kind) defined n=1 on the interval [−1,1]. If so, the coefficients are given by µ n = ∫ 1 −1 dx ˜f(x)T n (x). (5.33)
Chapter 5: Spin Hall Effect 95 0.4 0.0 V Σ sH e 8Π 0.3 0.2 0.5 1.0 1.5 0.1 2.0 0.0 0.02 0.04 0.06 0.08 0.10 0.12 n 2.5 (a) Σ sH e 8Π 0.06 0.05 0.04 0.03 0.02 0.01 3.0 3.5 4.0 4.5 V 0.00 0.02 0.04 0.06 0.08 0.10 0.12 n 5.0 (b) Figure 5.5: SHC σ SH as function of filling n in presence of impurities of binary type calculated using exact diagonalization. The impurity concentration is set to 10% for all plots and the average is performed over 200 impurity configurations. The system size is L 2 = 32 2 , and the SOC is Rashba type with α 2 = 1.2t with cutoff η = 0.06 (a) V = −0.2t... −2.8t (b) V = −2.8t...−5t.
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Chapter 5: <strong>Sp<strong>in</strong></strong> Hall Effect 95<br />
0.4<br />
0.0<br />
V<br />
Σ sH e 8Π<br />
0.3<br />
0.2<br />
0.5<br />
1.0<br />
1.5<br />
0.1<br />
2.0<br />
0.0<br />
0.02 0.04 0.06 0.08 0.10 0.12<br />
n<br />
2.5<br />
(a)<br />
Σ sH e 8Π<br />
0.06<br />
0.05<br />
0.04<br />
0.03<br />
0.02<br />
0.01<br />
3.0<br />
3.5<br />
4.0<br />
4.5<br />
V<br />
0.00<br />
0.02 0.04 0.06 0.08 0.10 0.12<br />
n<br />
5.0<br />
(b)<br />
Figure 5.5: SHC σ SH as function <strong>of</strong> fill<strong>in</strong>g n <strong>in</strong> presence <strong>of</strong> impurities <strong>of</strong> b<strong>in</strong>ary type calculated<br />
us<strong>in</strong>g exact diagonalization. The impurity concentration is set to 10% for all plots and<br />
the average is performed over 200 impurity configurations. The system size is L 2 = 32 2 ,<br />
and the SOC is Rashba type with α 2 = 1.2t with cut<strong>of</strong>f η = 0.06 (a) V = −0.2t... −2.8t<br />
(b) V = −2.8t...−5t.
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