2.5 Spin Relaxation
47
where S is the spin angular momentum vector, μ B is the Bohr magneton and g s ≈ 2
denotes the electron spin g-factor. It is obvious from Eq. 2.4 that μ is antiparallel to
the spin angular momentum.
In effect, spin–orbit interaction energy is made up of two parts. The Larmor part
is related to the interaction, occurring between the magnetic moment of electron and
the magnetic field produced by the nucleus. The Larmor interaction energy is
H L = −μ · B
(2.48)
By using the expressions of the magnetic moment and the magnetic field, we
obtain from Eq. 2.4
H L =
2μ B
m e ec 2
1
r
∂U (r )
∂r
L.S
(2.49)
The second part is related to Thomas precession for the electron’s curved trajectory
(we would not go into details of this). Actually, the electron in a rotating orbit is
constantly accelerating because the direction of the velocity is changing with time,
even though the magnitude is not changing. Therefore, it is not enough to transform
the laboratory frame to the rest frame using the electron’s instantaneous velocity.
This requires an additional correction factor ½, also known as Thomas correction
factor, to be introduced when the above fact is taken into account.
Thus, the net spin–orbit interaction energy takes the form
H ≡ H L + H T =
μ B
m e ec 2
1
r
∂U (r )
∂r
(L.S)
(2.50)
The net effect of Thomas precession comes from the reduction of the Larmor interaction energy by factor ½, which is familiar as Thomas half. It is obvious that the
above energy depends on the scalar product
L.
S, where
L and
S are the orbital and
the spin quantum number, respectively. Hence, the above equation reflects the interaction or coupling between the magnetic moment and the orbital angular momentum
associated with an electron in the atom and thereby denotes spin–orbit interaction.
2.5.3 Spin Relaxation Process
Let us return back to our discussion on the spin relaxation process. In spintronics, after
spin injection from the ferromagnetic contact, the spin-polarized carriers flow within
the paramagnetic semiconductor under the application of a bias voltage, which in turn
produces a transport-driving electric field inside the semiconductor. During the travel
of spin-polarized carriers, spins undergo different interactions (spin–orbit, hyperfine,
carrier–carrier interactions etc.) with their environments and their initial orientations
get altered to different extent considerably. As a result, the spin polarization of the
47
where S is the spin angular momentum vector, μ B is the Bohr magneton and g s ≈ 2
denotes the electron spin g-factor. It is obvious from Eq. 2.4 that μ is antiparallel to
the spin angular momentum.
In effect, spin–orbit interaction energy is made up of two parts. The Larmor part
is related to the interaction, occurring between the magnetic moment of electron and
the magnetic field produced by the nucleus. The Larmor interaction energy is
H L = −μ · B
(2.48)
By using the expressions of the magnetic moment and the magnetic field, we
obtain from Eq. 2.4
H L =
2μ B
m e ec 2
1
r
∂U (r )
∂r
L.S
(2.49)
The second part is related to Thomas precession for the electron’s curved trajectory
(we would not go into details of this). Actually, the electron in a rotating orbit is
constantly accelerating because the direction of the velocity is changing with time,
even though the magnitude is not changing. Therefore, it is not enough to transform
the laboratory frame to the rest frame using the electron’s instantaneous velocity.
This requires an additional correction factor ½, also known as Thomas correction
factor, to be introduced when the above fact is taken into account.
Thus, the net spin–orbit interaction energy takes the form
H ≡ H L + H T =
μ B
m e ec 2
1
r
∂U (r )
∂r
(L.S)
(2.50)
The net effect of Thomas precession comes from the reduction of the Larmor interaction energy by factor ½, which is familiar as Thomas half. It is obvious that the
above energy depends on the scalar product
L.
S, where
L and
S are the orbital and
the spin quantum number, respectively. Hence, the above equation reflects the interaction or coupling between the magnetic moment and the orbital angular momentum
associated with an electron in the atom and thereby denotes spin–orbit interaction.
2.5.3 Spin Relaxation Process
Let us return back to our discussion on the spin relaxation process. In spintronics, after
spin injection from the ferromagnetic contact, the spin-polarized carriers flow within
the paramagnetic semiconductor under the application of a bias voltage, which in turn
produces a transport-driving electric field inside the semiconductor. During the travel
of spin-polarized carriers, spins undergo different interactions (spin–orbit, hyperfine,
carrier–carrier interactions etc.) with their environments and their initial orientations
get altered to different extent considerably. As a result, the spin polarization of the