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

8 Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems
2.2 Dynamics of a Localized Spin
A localized spin ŝ, like a nuclear spin, or the spin of a magnetic impurity in a solid,
precesses in an external magnetic field B due to the Zeeman interaction with Hamiltonian
H Z = −γ g ŝB, where γ g is the corresponding gyromagnetic ratio of the nuclear spin or
magnetic impurityspin, respectively, whichwewill set equal toone, unlessneededexplicitly.
This spin dynamics is governed by the Bloch equation of a localized spin,
∂ t ŝ = γ g ŝ×B. (2.4)
This equation is identical to the Heisenberg equation ∂ t ŝ = −i[ŝ,H Z ] for the quantum
mechanical spin operator ŝ of an S = 1/2-spin, interacting with the external magnetic
field B due to the Zeeman interaction with Hamiltonian H Z . The solution of the Bloch
equation for a magnetic field pointing in the z-direction is ŝ z (t) = ŝ z (0), while the x- and
y- components of the spin are precessing with frequency ω 0 = γ g B around the z-axis,
ŝ x (t) = ŝ x (0)cosω 0 t+ŝ y (0)sinω 0 t, ŝ y (t) = −ŝ x (0)sinω 0 t+ŝ y (0)cosω 0 t. Since a localized
spininteracts withits environment by exchange interaction and magnetic dipoleinteraction,
the precession will dephase after a time τ 2 , and the z-component of the spin relaxes to its
equilibrium value s z0 within a relaxation time τ 1 . This modifies the Bloch equations to the
phenomenological equations,
∂ t ŝ x = γ g (ŝ y B z −ŝ z B y )− 1 τ 2
ŝ x
∂ t ŝ y = γ g (ŝ z B x −ŝ x B z )− 1 τ 2
ŝ y
∂ t ŝ z = γ g (ŝ x B y −ŝ y B x )− 1 τ 1
(ŝ z −s z0 ). (2.5)
2.3 Spin Dynamics of Itinerant Electrons
2.3.1 Ballistic Spin Dynamics
Starting from Dirac equation we have seen that one obtains in addition to the
Zeeman term a term which couples the spin s with the momentum p of the electrons, the
spin-orbit coupling
H SO = − µ B
2m e0 c 2ŝ p×E = −ŝB SO(p), (2.6)