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

Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 11
2.3.3 Spin Orbit Interaction in Semiconductors
SOI in semiconductors is closely related to breaking of symmetries which lift spin
degeneracy: In the case without magnetic field B we start with a twofold degeneracy: Time
inversion symmetry, E ↑ (k) = E ↓ (−k) and space inversion symmetry, E ↑ (k) = E ↑ (−k). As
a consequence we have E ↑ (k) = E ↓ (k). In the following we show how the degeneracy is
lifted in semiconductor devices in case of B = 0.
While silicon and germanium have in their diamond structure an inversion symmetry
around every midpoint on each line connecting nearest neighbor atoms, this is not
the case for III-V-semiconductors like GaAs, InAs, InSb, or ZnS. These have a zinc-blende
structure which can be obtained from a diamond structure with neighbored sites occupied
by the two different elements. Therefore the inversion symmetry is broken, which results
in spin-orbit coupling. This can be understood by noticing that pairs like Ga-As are local
dipoles whose electric field is responsible for SOC if inversion symmetry is broken 2 . Similarly,
that symmetry is broken in II-VI-semiconductors. This bulk inversion asymmetry
(BIA) coupling, or often so called Dresselhaus-coupling, is anisotropic, as given by [Dre55]
[
H D = γ D σx k x (ky 2 −k2 z )+σ yk y (kz 2 −k2 x )+σ zk z (kx 2 −k2 y )] , (2.13)
where γ D is the Dresselhaus-spin-orbit coefficient. Band structure calculations yield the
following values: γ D = 27.6 eVÅ(GaAs), = 27.2 eVÅ(InAs), = 760.1 eVÅ(InSb) [Win03].
Some values extracted in experiments are listed in Tab.A. Confinement in quantum wells
with width a z on the order of the Fermi wave length λ F yields accordingly a spin-orbit
interaction where the momentum in growth direction is of the order of 1/a z . Because
of the anisotropy of the Dresselhaus term, the spin-orbit interaction depends strongly on
the growth direction of the quantum well. Grown in [001] direction, one gets, taking the
expectation value of Eq.(2.13) in the direction normal to the plane, noting that 〈k z 〉 =
〈kz〉 3 = 0, [Dre55]
H D[001] = α 1 (−σ x k x +σ y k y )+γ D (σ x k x ky 2 −σ y k y kx). 2 (2.14)
where α 1 = γ D 〈kz〉 2 is the linear Dresselhaus parameter. Thus, inserting an electron with
momentum along the x-direction, with its spin initially polarized in z-direction, it will
2 Starting from an extended Kane model where p-like higher energy bands are included one gets a Hamiltonian
with matrix elements which are only nonzero if the crystal has no center of inversion.[FMAE + 07]