44
2 Basic Elements of Spintronics
spin polarization is already an eigenstate [discussed in Chap. 1] and therefore
stable. As can be understood, such situation is a bit rare.
(ii) Most of the time, electron’s spin points along some arbitrary direction in
space, and hence they are not either parallel or antiparallel to that effective magnetic field. As a result, such spin will undergo the familiar Larmor
precession about the effective magnetic field, B eff with an angular frequency
Ω e f f = gμ B B e f f
, where g is the Landé g-factor of the medium. Time
evolution of the spin, undergoing Larmor precession (which follows from timedependent Pauli Equation) is described by the equation
d s
dt
=
Ω e f f ×
S, which
present us a picture where the initial spin will not be stable and begin to change
with time when the electrons interact with the effective magnetic field.
Furthermore,
B e f f depends on the velocity of electrons
ν or wavevector
k (will be
discussed later in detail). Since different electrons suffer scattering differently, hence
electron’s velocity or wavevector changes randomly. This in turn leads to randomization of
B e f f . Consequently, the axis of precession of electron’s spin changes
its direction and the frequency of Larmor precession also changes. Since scattering
events are random, such changes occur randomly in time, thereby resulting in the
orientation of electron’s spin changes randomly and gradually with time. This leads
to spin relaxation. On the other hand, if the spin changes suddenly and discretely in
time, then such phenomenon is referred to as ‘spin flip’, which corresponds to an
‘up-spin’ state becoming a ‘down-spin’ state and vice versa. As discussed in Chap. 1,
‘up-spin’ and ‘down-spin’ states would not have any coupling between them since
they are mutually orthogonal and the corresponding matrix element connecting these
two states should be zero. It is the ‘scatterer’ that couples and causes transitions from
one state to another, thereby resulting in a spin flip. Noteworthy, the scatterer must
have an internal magnetic field of some sort.
2.5.2 What Is Spin–Orbit Interaction?
First, we should have knowledge on ‘spin–orbit interactions’ before going into details
of spin relaxation mechanisms. The spin–orbit interaction or coupling is the interaction of a particle’s spin with its motion. In order to understand the interaction process,
let us take a simple example of an electron orbiting around a nucleus in an atom.
In such case, ‘spin–orbit interaction’ is basically an electromagnetic interaction of
spin magnetic moment of electron with the magnetic field, which is produced due
to the orbiting negatively charged electron around the positively charged nucleus,
hence have coined the name spin–orbit interaction. A well-known consequence is the
shifting of the atomic energy levels of the electrons, thus causing splitting in atomic
spectral lines. Similar effect can also be found in case of protons and neutrons moving
inside the nucleus. In the arena of spintronics and many spin-based devices, spin–
orbit interaction is one of the major mechanisms that determine the spin relaxation
process; therefore, it is indeed necessary to learn the spin–orbit interaction in detail.
2 Basic Elements of Spintronics
spin polarization is already an eigenstate [discussed in Chap. 1] and therefore
stable. As can be understood, such situation is a bit rare.
(ii) Most of the time, electron’s spin points along some arbitrary direction in
space, and hence they are not either parallel or antiparallel to that effective magnetic field. As a result, such spin will undergo the familiar Larmor
precession about the effective magnetic field, B eff with an angular frequency
Ω e f f = gμ B B e f f
, where g is the Landé g-factor of the medium. Time
evolution of the spin, undergoing Larmor precession (which follows from timedependent Pauli Equation) is described by the equation
d s
dt
=
Ω e f f ×
S, which
present us a picture where the initial spin will not be stable and begin to change
with time when the electrons interact with the effective magnetic field.
Furthermore,
B e f f depends on the velocity of electrons
ν or wavevector
k (will be
discussed later in detail). Since different electrons suffer scattering differently, hence
electron’s velocity or wavevector changes randomly. This in turn leads to randomization of
B e f f . Consequently, the axis of precession of electron’s spin changes
its direction and the frequency of Larmor precession also changes. Since scattering
events are random, such changes occur randomly in time, thereby resulting in the
orientation of electron’s spin changes randomly and gradually with time. This leads
to spin relaxation. On the other hand, if the spin changes suddenly and discretely in
time, then such phenomenon is referred to as ‘spin flip’, which corresponds to an
‘up-spin’ state becoming a ‘down-spin’ state and vice versa. As discussed in Chap. 1,
‘up-spin’ and ‘down-spin’ states would not have any coupling between them since
they are mutually orthogonal and the corresponding matrix element connecting these
two states should be zero. It is the ‘scatterer’ that couples and causes transitions from
one state to another, thereby resulting in a spin flip. Noteworthy, the scatterer must
have an internal magnetic field of some sort.
2.5.2 What Is Spin–Orbit Interaction?
First, we should have knowledge on ‘spin–orbit interactions’ before going into details
of spin relaxation mechanisms. The spin–orbit interaction or coupling is the interaction of a particle’s spin with its motion. In order to understand the interaction process,
let us take a simple example of an electron orbiting around a nucleus in an atom.
In such case, ‘spin–orbit interaction’ is basically an electromagnetic interaction of
spin magnetic moment of electron with the magnetic field, which is produced due
to the orbiting negatively charged electron around the positively charged nucleus,
hence have coined the name spin–orbit interaction. A well-known consequence is the
shifting of the atomic energy levels of the electrons, thus causing splitting in atomic
spectral lines. Similar effect can also be found in case of protons and neutrons moving
inside the nucleus. In the arena of spintronics and many spin-based devices, spin–
orbit interaction is one of the major mechanisms that determine the spin relaxation
process; therefore, it is indeed necessary to learn the spin–orbit interaction in detail.