14
1 An Overview of Spintronics
the torque is zero. Moreover, the motion of a precessing magnetic moment towards
equilibrium can be understood by including a viscous damping term. In this direction,
a dissipative term (−
∂
m
∂t
), proportional to the generalized velocity, is included with the
effective magnetic field. This dissipative term decelerates the motion of the magnetic
moment and finally brings the magnetic moment m parallel to
H
e f f
. This ultimately
provides Landau–Lifshitz–Gilbert (LLG) equation of motion:
∂
m
∂ t = γ
m ×
H
e f f
+
α
m
m × ∂
m
∂ t,
(1.25)
where α is the dimensionless phenomenological Gilbert damping constant.
The LLG equation is extensively employed to investigate and explain the
switching dynamics of small magnetic particles. For sufficiently small particles,
magnetization may be supposed to remain constant during its reversal process. In this
case, the only contributions to
H
e f f
comes from the anisotropy field, demagnetizing
field and the applied external magnetic field. On the other hand, for larger samples and
in case of inhomogeneous magnetization dynamics, the magnetic moment becomes
a function of spatial coordinates. Noteworthy, in this case, exchange interaction
also contributes to
H
e f f
. The LLG equation also provides us an opportunity to
calculate the evolution of the spin system in the atomistic limit using Langevin
dynamics. In fact, this has proved to be a powerful and efficient route to model
ultrafast magnetization processes.
A limitation of the LLG equation includes that for a very short time scale, even
shorter than the spin–orbit coupling of the order of 20 fs, the description with a
single gyromagnetic ratio fails. In this case, spin and orbital contributions must be
considered separately.
1.7 Spin-Dependent Band Gap in Ferromagnetic Materials
A spin-polarized current can be obtained by injection of unpolarized one into ferromagnetic materials. This is due to the spin-dependent band structure found in this
material. This ferromagnetism property originates from the tendency of the electron
spin to align in the same direction due to Pauli exclusion principle. Pauli stated that
two electrons with the same spin cannot be in the same position. Thus, the complete
wavefunction of the two electrons would be antisymmetric given that the two electrons interchange their positions. Therefore, the electrons will feel an additional
effective repulsion due to the Pauli principle in addition to the Coulomb one.
The difference in energy between two electron systems with a symmetric or antisymmetric spin part of the wavefunction is referred to as the exchange energy E ex .
In 1928, Heisenberg introduced a microscopic origin of the exchange energy by
1 An Overview of Spintronics
the torque is zero. Moreover, the motion of a precessing magnetic moment towards
equilibrium can be understood by including a viscous damping term. In this direction,
a dissipative term (−
∂
m
∂t
), proportional to the generalized velocity, is included with the
effective magnetic field. This dissipative term decelerates the motion of the magnetic
moment and finally brings the magnetic moment m parallel to
H
e f f
. This ultimately
provides Landau–Lifshitz–Gilbert (LLG) equation of motion:
∂
m
∂ t = γ
m ×
H
e f f
+
α
m
m × ∂
m
∂ t,
(1.25)
where α is the dimensionless phenomenological Gilbert damping constant.
The LLG equation is extensively employed to investigate and explain the
switching dynamics of small magnetic particles. For sufficiently small particles,
magnetization may be supposed to remain constant during its reversal process. In this
case, the only contributions to
H
e f f
comes from the anisotropy field, demagnetizing
field and the applied external magnetic field. On the other hand, for larger samples and
in case of inhomogeneous magnetization dynamics, the magnetic moment becomes
a function of spatial coordinates. Noteworthy, in this case, exchange interaction
also contributes to
H
e f f
. The LLG equation also provides us an opportunity to
calculate the evolution of the spin system in the atomistic limit using Langevin
dynamics. In fact, this has proved to be a powerful and efficient route to model
ultrafast magnetization processes.
A limitation of the LLG equation includes that for a very short time scale, even
shorter than the spin–orbit coupling of the order of 20 fs, the description with a
single gyromagnetic ratio fails. In this case, spin and orbital contributions must be
considered separately.
1.7 Spin-Dependent Band Gap in Ferromagnetic Materials
A spin-polarized current can be obtained by injection of unpolarized one into ferromagnetic materials. This is due to the spin-dependent band structure found in this
material. This ferromagnetism property originates from the tendency of the electron
spin to align in the same direction due to Pauli exclusion principle. Pauli stated that
two electrons with the same spin cannot be in the same position. Thus, the complete
wavefunction of the two electrons would be antisymmetric given that the two electrons interchange their positions. Therefore, the electrons will feel an additional
effective repulsion due to the Pauli principle in addition to the Coulomb one.
The difference in energy between two electron systems with a symmetric or antisymmetric spin part of the wavefunction is referred to as the exchange energy E ex .
In 1928, Heisenberg introduced a microscopic origin of the exchange energy by