ABSTRACT

Before we review the spin dynamics of conduction electrons and holes in semiconductors and metals, let us rst reconsider the spin dynamics of a localized spin, as governed by the Bloch equations.

28.1 Introduction ...........................................................................................................................28-1 28.2 Spin Dynamics .......................................................................................................................28-1

Dynamics of a Localized Spin • Spin Dynamics of Itinerant Electrons 28.3 Spin Relaxation Mechanisms .............................................................................................. 28-6

28.5 Experimental Results on Spin Relaxation Rate in Semiconductor Quantum Wires ..................................................................................28-13 Optical Measurements • Transport Measurements

28.6 Critical Discussion and Future Perspective ....................................................................28-15 28.7 Summary ...............................................................................................................................28-15 Symbols .............................................................................................................................................28-16 Acknowledgments ...........................................................................................................................28-16 References .........................................................................................................................................28-16

A localized spin sˆ, like a nuclear spin, or the spin of a magnetic impurity in a solid, precesses in an external magnetic eld B due to the Zeeman interaction with Hamiltonian Hz = −γgsˆB, where γg is the corresponding gyromagnetic ratio of the nuclear spin or magnetic impurity spin, respectively, which we will set equal to one, unless needed explicitly. is spin dynamics is governed by the Bloch equation of a localized spin,

∂ γ ×gˆ ˆ = .ts s B (28.1)

is equation is identical to the Heisenberg equation ∂tsˆ = −i[sˆ, Hz] for the quantum mechanical spin operator sˆ of an S = 1/2-spin, interacting with the external magnetic eld B due to the Zeeman interaction with Hamiltonian Hz. e solution of the Bloch equation for a magnetic eld pointing in the z-direction is sˆz(t) = sˆz(0), while the x-and y-components of the spin are precessing with frequency 𝛚0 = γgB around the z-axis, sˆx(t) = sˆx(0) cos ω0t + sˆy(0) sin ω0t, sˆy(t) = − sˆx(0) sin ω0t + sˆy(0) cos ω0t. Since a localized spin interacts with its environment by exchange interaction and magnetic dipole interaction, the precession will dephase a er a time τ2, and the z-component of the spin relaxes to its equilibrium value sz0 within a relaxation time τ1. is modi es the Bloch equations to the phenomenological equations:

1ˆ ˆ ˆ ˆ( B B )

1ˆ ˆ ˆ ˆ( B B )

1ˆ ˆ ˆ ˆ( B B ) ( ).