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

88 Chapter 5: Spin Hall Effect
with c iα c † iα c iα|0〉 = c iα |0〉 and c iα c † jβ c jβ = c † jβ c jβc iα
= 1 i
v = 1 i
∑
ij
αβ
ψ ∗ n(i,α)[(r i −r j )H αβ
ij ]ψ m(j,β), (5.15)
∑
(r i −r j )H αβ
ij , (5.16)
ij
αβ
where we used i,j and k as site indices and the other for spin. Using the definition of the
spin current, Eq.(5.6), we get accordingly
〈n|J z |m〉 = e ∑
(r i −r j )ψ n (i,α){σ z ,H} αβ
ij
4i
ψ n(j,β). (5.17)
ij
αβ
Going into momentum space we are left with
v y = ∑ k y
2tsin(k y )½+(α 2 σ x + ˜α 1 σ y )cos(k y ), (5.18)
and for the spin current
J z x = ∑ k x
tsin(k x )σ z . (5.19)
The last Eq. follows from {σ z ,H D } = {σ z ,H R } = 0. To evaluate the rhs of Eq.(5.5) we
need the matrix elements of both operators in the eigenvalue basis, which yields the pure
imaginary result
〈∓|Jx|±〉〈±|v z y |∓〉 = ±it(α 2 2 − ˜α 2 1) sin(k x) 2 cos(k y )
. (5.20)
∆(k)
Assumingzerotemperaturetocalculateσ z xy ,Eq.(5.5)finallysimplifiesto[NSSM05,MMF08]
σ z xy ≡ σ SH = − ie
V
= 2 e V
∑
m,n
∑
f(E m )−f(E n )〈m|Jx z |n〉〈n|v y |m〉
E m −E n (E m −E n )+iη , (5.21)
I(〈m|Jx z |n〉〈n|v y |m〉)
(E n −E m ) 2 +η 2 . (5.22)
E m