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

84 Chapter 5: Spin Hall Effect
DOS, their average distance and the disorder strength [SBK + 10].
Another way for efficiently injecting spin currents into semiconductors is to make
use of the spin Hall effect (SHE). This effect was first proposed by D’yakonov and Perel
[DP71a] and describes in today’s terminology the extrinsic SHE which requires spin dependent
impurity scattering. It was experimentally confirmed by the angle-resolved optical
detection of spin polarization at the edges of a two-dimensional layer [WKSJ05, KMGA04].
The theory of the SHE has been developed in the last 10 years, as reviewed in Refs.
[Sch06, ERH07].
The spin transport not only occurs due to the spin precession in the bulk, but is affected
by the scattering from nonmagnetic impurities, which can depend on the direction of the
spin itself due to the so called skew scattering and the side jump mechanism[Sch06]. In the
first part of the following chapter, we will outline the formalism to calculate analytically
the SHE which arises even in the absence of impurities, the so called intrinsic SHE, which
is due to the bulk SOC. For the clean case we will include both, Rashba and Dresselhaus
SOC. In the second part we focus on numerical methods: The first attempt is application of
exact diagonalization which has of course strong limitations concerning the system size. To
overcome this limitations, we apply the Kernel Polynomial Method (KPM) in the last part
of this thesis. It is first applied to treat the metal-insulator transition (MIT) in a symplectic
system, finally we calculate the SHE using the KPM.
5.1.1 About the Definition of Spin Current
We have learned in Sec.2.3.4 that in presence of the spin-orbit interaction, the
spin current components are not conserved[CSS + 04, SZXN06] even if we assume no spin
relaxation: In the continuity equation
∂s z
∂t +D e∇J spin = T s − 1
(ˆτ s ) ij
s j (5.1)
= τ〈∇v F (B SO (k)×S) i
〉− 1
(ˆτ s ) ij
s j . (5.2)
anadditionaltorquetermT s appearsincontrasttoEq.2.12besidesthetermwhichdescribes
the spin relaxation rate 1/τ s . It follows that Noether’s theorem is not applicable to define
the spin current. This is a reason why the comparison between results done with Kubo
formalism and calculations using Landauer-Büttiker approach is not trivial, where the spin