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

Chapter 2: Spin Dynamics: Overview and Analysis of 2D Systems 21
2.4.3 Elliott-Yafet Spin Relaxation
Because of the spin-orbit interaction the conduction electron wave functions are
not eigenstates of the electron spin, but have an admixture of both spin up and spin down
wave functions. Thus, a nonmagnetic impurity potential V can change the electron spin,
by changing their momentum due to the spin-orbit coupling. This results in another source
of spin relaxation which is stronger, the more often the electrons are scattered, and is
thus proportional to the momentum scattering rate 1/τ[Ell54, Yaf63]. For degenerate III-V
semiconductors one finds[Cha75, PT84]
1
τ s
∼
∆ 2 SO
(E G +∆ SO ) 2 E 2 k
E 2 G
1
τ(k) , (2.32)
where E G is the gap between the valence and the conduction band of the semiconductor,
E k the energy of the conduction electron, and ∆ SO is the spin-orbit splitting of the valence
band. Thus, the Elliott-Yafet spin relaxation (EYS) can be distinguished, being proportional
to 1/τ, and thereby to the resistivity, in contrast to the DP spin scattering rate, Eq.
(2.31), which is proportional to the conductivity. Since the EYS decays in proportion to
the inverse of the band gap, it is negligible in large band gap semiconductors like Si and
GaAs. The scattering rate 1/τ is again the sum of the impurity scattering rate [Ell54], the
electron-phonon scattering rate [Yaf63, GF97], and electron-electron interaction [Bog80], so
that all these scattering processes result in EY spin relaxation. In degenerate semiconductors
and in metals, the electron-electron scattering rate is given by the Fermi liquid inelastic
electron scattering rate 1/τ ee ∼ T 2 /E F . The electron-phonon scattering time 1/τ ep ∼ T 5
decays fasterwith temperature. Thus, at low temperatures theElliott-Yafet spinrelaxation,
Eq.(2.32), is dominated by elastic impurity scattering τ 0 . In non-degenerate semiconductors,
where the Fermi energy is below the conduction band edge, one finds 1/τ s ∼ T 2 /τ(T).
2.4.4 Spin Relaxation due to Spin-Orbit Interaction with Impurities
The spin-orbit interaction, as defined in Eq.(2.6), arises whenever there is a gradient
in an electrostatic potential. Thus, the impurity potential gives rise to the spin-orbit
interaction
V SO = 1
2m 2 c2∇V ×k s. (2.33)