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

Chapter 1
The Rashba effect
Effects of spin-orbit (SO) interaction are well-known from atomic physics. The
relativistic nature of this coupling can be understood by a low-velocity approximation
to the Dirac equation [16]. This approach yields, among other fine-structure
corrections to the non-relativistic Schrödinger equation, the Pauli SO term
H SO = −
4m 2 σ · (p
× ∇V (r) ) , (1.1)
0 c2
where is Planck’s constant, m 0 the bare mass of the electron, c the velocity of
light, and σ the vector of Pauli matrices. V (r) is the electrostatic potential in
which the electron propagates with momentum p. In atomic physics V (r) is the
Coulomb potential of the atomic core.
In semiconductor physics, the spectral properties of electrons which move in
a periodic crystal are characterised by energy bands E n (k). Here also, effects
of SO coupling emerge in the band structure. A prominent example is the energy
splitting of the topmost valence band in GaAs. This splitting can be determined up
to high precision in band structure calculations [17, 18]. The microscopic origin
of the energy splitting in such calculations is again given by Eq. (1.1).
In the following, we consider the effects of SO coupling in two-dimensional
(2D) electron systems such as quantum wells (QWs) which can be tailored experimentally
e.g. in semiconductor heterostructures [3]. Strong confining potentials at
the interface of the heterostructure result in quantised energy levels of the electron
for one spatial direction whereas it is free to move in the other two spatial directions
[19]. Here, we focus on the introduction of the Rashba effect [11, 12] as a
model for the dominating SO coupling in a certain class of 2D systems. We will
need this model in the subsequent chapters to understand SO effects in electron
systems with further reduced dimension (quasi-one-dimensional ‘quantum wires’
in chapter 1 and quasi-zero-dimensional ‘quantum dots’ in chapter 2).
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