Spin-orbit coupling and electron-phonon scattering - Fachbereich ...

2.1 Rashba effect and magnetic field in semiconductor quantum wires 19
SOI, the transversal wavefunction component satisfies
for k fixed with the Hamiltonian
H(k)
ω 0
=Ω
H(k) φ k (x) = E k φ k (x), (2.4)
(
a † k a k + 1 2
⎛
)
+ 1 (kl 0 ) 2
2 Ω 2
+ 1 ξ 1 kl 0 + ξ 2 (a k + a † k ) ⎞
⎝ ξ
2 3 (a k − a † k ) ⎠ · σ, (2.5)
δ
the abbreviations
Ω =
√
ω 2 0 + √
ω2 ( ) 4
c l0
= 1 + , (2.6)
ω 0 l B
ξ 1 = l 0
l SO
1
Ω , (2.7)
ξ 2 = 1 √
2
l 0
l SO
(
l0
l B
) 2
1
√
Ω
, (2.8)
ξ 3 =
and the dimensionless Zeeman splitting
δ = 1 2
i √
2
l 0
l SO
√
Ω, (2.9)
(
l0
l B
) 2
m
m 0
g, (2.10)
(m 0 is the bare mass of the electron).
This representation of the Hamiltonian corresponds to expressing the transverse
wavefunction in terms of oscillator eigenstates such that a † k a k gives the subband
index of the electron which propagate with longitudinal momentum k. The
magnetic field leads to the lateral shift of the wavefunction and the renormalisation
of the oscillator frequency Ω. Moreover, the effective mass in the kinetic
energy of the longitudinal propagation is changed. The last term in Eq. (2.5) describes
how the SOI couples the electron’s orbital degree of freedom to its spin.
Due to the operators a † k and a k the subbands corresponding to one spin branch
are coupled to the same and nearest neighbouring subbands of opposite spin, see
Fig. 2.2a.