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

90 Chapter 5: Spin Hall Effect
as an effective magnetic field, which however does not break time reversal symmetry, this
is very similar to the case where a Peierls substitution is applied,[Pei33] i.e. where a Peierls
phase is added to the electron whenever it hops in the direction of finite vector field.
The sign changes according to the sign of α 2 2 − ˜α2 1 and exhibits the value e/(8π) for small
filling with the condition that both bands, E ± , are filled, independent of the strength of
SOC. This can bee seen by expanding the spectrum around the Γ point which yields[She04]
σ SH =
∫
e 2π
16m e π 2
= e
8π
with k x = kcos(ϕ) and k x = ksin(ϕ).
0
dϕ (α2 2 − ˜α2 1 )cos2 (ϕ)(k + −k − )
(α 2 2 + ˜α2 1 −2α , (5.25)
2˜α 1 sin(2ϕ)) 3 2
α 2 2 − ˜α2 1
|α 2 2 − (5.26)
˜α2 1 |,
Looking at the results from the calculation on a clean lattice, Fig.(5.2) (b), it can be seen
that the value e/(8π) decreases with increasing SOC strength in the case of pure Rashba
SOC. This can be understood by noticing that the value of the SHC[SCN + 04]
σ SH =
e
16m e πα 2
(k F+ −k F− ) (5.27)
is diminished when we add corrections to the parabolic assumption: On the lattice we have
( )
2
(k F+ −k F− ) = arccos
1+ ( α 2
) 2
−1
(5.28)
2t
= 2m e α 2 − 2 3 (m eα 2 ) 3 +O(m e α 2 ) 4 (5.29)
and therefore the diminishment is given by
σ SH = e
8π − e
24π (m eα 2 ) 2 . (5.30)
5.3 Numerical Analysis of SHE
5.3.1 Exact Diagonalization
For linear Rashba coupling, the value σ SH = e 2 /(8π), as presented in the previous
section, has been obtained both by analytical calculations in the continuum model, and
by numerical calculations of the tight binding model[Sch06]. However, in the presence
of nonmagnetic impurities, the DC spin Hall conductance is diminished to exactly zero,