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

86 Chapter 5: Spin Hall Effect
• localized site orbitals are of s symmetry.
Applying this assumptions the tight-binding version of the Hamiltonian is given by
H = H 0 +H R +H D,lin +H D,cubic , (5.7)
= ∑ ǫ i c † i,σ c i,σ −t ∑
c † i,σ c j,σ
i,σ 〈i,j〉,σ
∑
{c † l,m,σ
(iσ ′ y ) σσ ′c l+1,m,σ −c † l,m,σ
(iσ ′ x ) σσ ′c l,m+1,σ }
+ α 2
2a
+ α 1
2a
σ,σ ′
l,m
∑
{c † l,m,σ
(iσ ′ x ) σσ ′c l+1,m,σ −c † l,m,σ
(iσ ′ y ) σσ ′c l,m+1,σ }
σ,σ ′
l,m
⎧
⎪⎨
+ γ ∑
D
a 3 {c †
⎪
l,m,σ
(−iσ
⎩ ′ x ) σσ ′c l+1,m,σ +c † l,m,σ
(iσ ′ y ) σσ ′c l,m+1,σ }
σ,σ ′
l,m
σ,σ ′
l,m
⎫
+ 1 ∑
⎪⎬
{c †
2
l,m,σ
(i(σ ′ x −σ y )) σσ ′c l+1,m+1,σ +c † l,m,σ
(i(σ ′ x +σ y )) σσ ′c l+1,m−1,σ }
⎪⎭
+h.c.. (5.8)
where c † i,σ is the creation operator at site index i with spin σ =↑,↓ and c† l,m,σ
the creation
operator at site (index x ,index y ) = (l,m). The hopping coupling t is given by t = 1/(2m e a)
with the lattice constant a. In the following we take the cubic Dresselhaus term only as
a shift of, ˜α 1 = α 1 − 2γ D /a 2 , according to Eq.(3.44), and assume a clean system, i.e the
on-site energy is set to ǫ i = 0. Applying a Fourier transformation to Eq.(5.7) and going to
momentum space we get (we set a ≡ 1)
⎧
H = ∑ ⎪⎨
−2t(cos(k
⎪ x )+cos(k y )) δ σσ ′c †
kx,ky ⎩ } {{ }
k,σ
c ′ k,σ
σ,σ ′ E 0
+ (α 2 sin(k y )− ˜α 1 sin(k x ))c † k,σ ′ (σ x ) σσ ′c k,σ
+ (˜α 1 sin(k y )−α 2 sin(k x ))c † k,σ ′ (σ y ) σσ ′c k,σ
}
. (5.9)
The corresponding eigenvalues are
E ± (k) = E 0 (k)±∆(k) (5.10)