2.5 Spin Relaxation
49
one, as already mentioned, is the familiar hyperfine interaction originating due to the
interaction between carrier and nuclear spins.
2.6 Elliott–Yafet Mechanism
2.6.1 What Is the Prerequisite Condition for Elliott–Yafet
Mechanism?
Elliott–Yafet spin relaxation mechanism in real crystals occurs owing to the presence
of spin–orbit interaction in the solid (Pramanik et al. 2006). In fact, this mechanism
arises owing to the fact that in a real crystal, Bloch states, i.e., the solutions of the
Schr ¨
odinger equation in the periodic lattice potential, are not pure spin eigenstates
(|↑ or |↓), but an admixture of both
u k
r
= a k
r
|↑ + b k
r
|↓
(2.51)
This means that electron’s spin in a crystal does not really have one of two fixed
spin polarizations, i.e., spin-up or spin-down with a unique axis defined by spin
quantization. Rather, the electron’s spins are orientated either in pseudo-up or down
directions. More subtle point is that the degree of admixture of up-spin or downspin electrons, determined by the quantities a k and b k , is a function of the electronic
wavevector k. This in turn implies that for electrons in a crystal, its spin orientation
depends on the electron’s wavevector k. This is evident from the schematic demonstration (Fig. 2.13), where for an arbitrary band, energy has been plotted as a function
of the wavevector. Such plot explores that the spin orientations of electrons (shown
by arrow) are different in different wavevector states. However, corresponding to
each wavevector state there are two possible mutually antiparallel spin orientations.
Fig. 2.13 Energy dispersion
relation showing the spin
polarizations at different
wavevector states
k1
k
k 2
49
one, as already mentioned, is the familiar hyperfine interaction originating due to the
interaction between carrier and nuclear spins.
2.6 Elliott–Yafet Mechanism
2.6.1 What Is the Prerequisite Condition for Elliott–Yafet
Mechanism?
Elliott–Yafet spin relaxation mechanism in real crystals occurs owing to the presence
of spin–orbit interaction in the solid (Pramanik et al. 2006). In fact, this mechanism
arises owing to the fact that in a real crystal, Bloch states, i.e., the solutions of the
Schr ¨
odinger equation in the periodic lattice potential, are not pure spin eigenstates
(|↑ or |↓), but an admixture of both
u k
r
= a k
r
|↑ + b k
r
|↓
(2.51)
This means that electron’s spin in a crystal does not really have one of two fixed
spin polarizations, i.e., spin-up or spin-down with a unique axis defined by spin
quantization. Rather, the electron’s spins are orientated either in pseudo-up or down
directions. More subtle point is that the degree of admixture of up-spin or downspin electrons, determined by the quantities a k and b k , is a function of the electronic
wavevector k. This in turn implies that for electrons in a crystal, its spin orientation
depends on the electron’s wavevector k. This is evident from the schematic demonstration (Fig. 2.13), where for an arbitrary band, energy has been plotted as a function
of the wavevector. Such plot explores that the spin orientations of electrons (shown
by arrow) are different in different wavevector states. However, corresponding to
each wavevector state there are two possible mutually antiparallel spin orientations.
Fig. 2.13 Energy dispersion
relation showing the spin
polarizations at different
wavevector states
k1
k
k 2