As the spin hot spot condition is characterized by a strong increase of the mixing
of the up- and down-spin states, the equivalent subband splitting at the spin hot
spots is purely determined by the effective spin-orbit term, which is in turn linear
with , where
with
=(
One can estimate the spin expectations which depend on the spin injection orientation. The spin Pauli matrices for the two [001] valleys can be written as,
Now one can express the expectation values of the spin projection on the
coordinate axes, denoted as (
![]() | (4.7) |
The total spin expectation can be represented as,
![]() | (4.8) |
When spin is injected along =
=0 but
=1. It
is spotted that at spin hot spots, the
value also drops to zero. When spin is
injected along
,
=
=0 but
=1. On the
contrary at the spin hot spots [153, 154],
thus making = sin
arctan
. When spin is injected along an
arbitrary direction on the
and hence
![]() | (4.11) |
Figure 4.6 and Figure 4.7 portray the values for over a range of (
always remains to one, but its value drops to zero
when the spin is injected along
= sin
arctan
at the spin hot spots. When
gradually increases at the spin hot
spots. This indicates that the spin relaxation rate (lifetime) is expected
to decrease (increase), when the injection orientation is drawn towards
in-plane.
The effective spin-orbit field SOF is given by and lies on the