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

Chapter 5: Spin Hall Effect 85
current in the latter is defined by[MM05]
where
I s p,µ = 1
4π
∑
p≠q,ν
Tr[Γ µ pG R Γ ν qG A ](V p −V q ), (5.3)
Γ µ p = i(Σ µ p −(Σ µ p) † ), (5.4)
with the retarded self energy Σ µ p due to coupling of lead p, which has voltage V p , and
sample for spin channel µ and advanced and retarded Green’s function G A/R . However,
in an experiment the SHE can be measured. This measurement has to be connected to
the theoretical description using linear response to a transverse electrical field with the
frequency ω which yields, Appendix B, Eq.(B.16),
σ µν (ω) = i V
∑
m,n
(f(E m )−f(E n ))〈m|j ν |n〉〈n|j µ |m〉
E n −E m E n −E m +ω +iη . (5.5)
To calculate the spin Hall conductivity (SHC) the correlation function consists, in contrast
to charge conductivity, of the charge current and a current which contains a spin operator.
In the simplified model we use the spin current which is given by the anticommutator of
the spin and the group velocity v = i[H,r],
J z = 4 {σ z,v}. (5.6)
This spin current does not differ from the one defined in Sec.2.3.[ESL05, BNnM04] Note
that this quantity is dissipationless, because the spin current is even under the time-reversal
operation.
5.2 SHE without Impurities: Exact Calculation
We consider theHamiltonian for alattice which provides linear Rashba, Eq.(2.15),
and linear and cubic Dresselhaus SOC, Eq.(2.14), as introduced in Sec.2.3.3. The confinement
to generate the 2D electron gas is in [001] direction. In order to do calculations
numerically, one needs to define a tight binding model on a discrete lattice of finite lattice
spacing a. It has the following characterization:
• square lattice,