Modeling Spin-Orbit Torques
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Chapter 3 Micromagnetic Modeling
Micromagnetic simulations are a powerful tool for studying magnetic materials at the mesoscopic scale, where atomic details can be neglected while complex magnetic structures, such as skyrmions and domain walls, are still
resolved. When extended to include spin and charge transport, these simulations can provide valuable insights for advancing next-generation spintronic technologies. In this chapter, an overview of the Landau-Lifschitz Gilbert
(LLG) equation, describing the magnetization dynamics in FM materials, is presented. Furthermore, the effective field contributions considered in this work are introduced. Finally, the current-induced torques arising from
spin-polarized currents are discussed.
3.1 The Landau-Lifshitz-Gilbert (LLG)
At the beginning of the 20th century, Bohr and van Leeuwen discovered independently that statistical and classical mechanics cannot account for any magnetization in a material, suggesting that material magnetism is a purely
quantum mechanical effect [48, 49]. In 1928, a theory of ferromagnetism was developed by Heisenberg based on quantum mechanical exchange interactions between the magnetic moments of the atoms in a material [50]. It became
apparent that quantum mechanical descriptions of ferromagnetism could be combined with Maxwell’s equations to describe the mesoscopic behavior of magnetic materials. In 1935, Landau and Lifshitz introduced a continuous
description of the motion of the magnetization in a ferromagnet, which only accounted for weak damping [51]. Strong damping was later addressed in 1955 by Gilbert, who introduced a phenomenological damping term, resulting in
the LLG equation [52]. In this section, the evolution from the simple equation of rotational motion of a single magnetic moment in a magnetic field to the LLG equation, which describes the spatially resolved motion of the
magnetization due to an effective magnetic field, is presented.
A magnetic dipole moment \(\boldsymbol {\mu }\) is considered. The relation of the dipole moment of a particle or system to its angular momentum is given by \(\boldsymbol {\mu } = \gamma \bm {L}\), where \(\gamma \)
is the gyromagnetic ratio and \(\bm {L}\) is the angular momentum. In the presence of a uniform external magnetic field \(\bm {H}\), the magnetic moment experiences a torque:
\(\seteqnumber{0}{3.}{0}\)
\begin{equation}
\label {eq:magmom_torque} \frac {\bm {\tau }}{\mu _0} = \boldsymbol {\mu } \times \bm {H},
\end{equation}
where \(\mu _0\) is the permeability of vacuum. By its definition, the torque is the rate of change of the angular momentum:
\(\seteqnumber{0}{3.}{1}\)
\begin{equation}
\label {eq:def_of_torque} \bm {\tau } = \frac {\partial \bm {L}}{\partial t} = \frac {1}{\gamma }\frac {\partial \boldsymbol {\mu }}{\partial t}.
\end{equation}
Inserting Eq. (3.2 ) into Eq. (3.1 ) yields
\(\seteqnumber{0}{3.}{2}\)
\begin{equation}
\label {eq:eom_magmom} \frac {\partial \boldsymbol {\mu }}{\partial t} = \gamma \mu _0 \boldsymbol {\mu } \times \bm {H},
\end{equation}
which describes the precessional motion of the magnetic moment about the external field, a phenomenon known as Larmor precession.
Equation (3.3 ) describes only the rotational motion of a single magnetic moment; however, in micromagnetics, it is of interest to describe the
collective behavior of numerous magnetic moments in a material, which interact with each other through exchange and dipole-dipole interactions. This is achieved by considering the magnetization as a continuous vector field \(\bm
{M}(\bm {r},t)\). Assuming a uniform density of spins and a constant temperature, the magnitude of the magnetization at each point is given by the saturation magnetization \(M_s\) of the material. The normalized
magnetization field is then given by
\(\seteqnumber{0}{3.}{3}\)
\begin{equation}
\label {eq:def_of_magnetization} \bm {m}(\bm {r},t ) = \frac {\bm {M}(\bm {r},t)}{M_s}, \text { with } \| \bm {m}(\bm {r},t) \| = 1.
\end{equation}
The equation of motion for the magnetization is then obtained by replacing the magnetic moment in Eq. (3.3 ) with the normalized magnetization,
and the uniform external magnetic field with a position-dependent effective magnetic field \(\bm {H}(\bm {r},t)\), resulting in
\(\seteqnumber{0}{3.}{4}\)
\begin{equation}
\label {eq:eom_magnetization} \frac {\partial \bm {m}(\bm {r},t )}{\partial t} = -\gamma _0 \bm {m}(\bm {r},t ) \times \bm {H}(\bm {r},t),
\end{equation}
where \(\gamma _0 = -\mu _0\gamma \) is a rescaled gyromagnetic ratio. Equation (3.5 ) describes the precessional motion of the
magnetization field due to the effective field at position \(\bm {r}\) and time \(t\). The effective field describes not only external fields but also internal fields resulting from exchange interactions, crystalline anisotropy, electrical
currents, and demagnetizing effects.
In 1935, Landau and Lifshitz introduced a phenomenological damping term to account for the dissipative magnetization dynamics observed in experiments, resulting in the Landau-Lifshitz equation [51]:
\(\seteqnumber{0}{3.}{5}\)
\begin{equation}
\label {eq:LL_eq} \frac {\partial \bm {m}}{\partial t} = -\gamma _0 \bm {m} \times \bm {H} - \gamma _0\lambda \bm {m} \times \left (\bm {m}\times \bm {H}\right ).
\end{equation}
From here on, the position and time dependence of the magnetization and effective field are omitted for brevity. The new term describes the damping of the precessional motion of the magnetization vector towards the direction of
the effective field. The damping is characterized by the material-dependent damping constant \(\lambda \), which is determined from experiments. The direction of each term and the resulting dynamics are illustrated in
Fig. 3.1 .
(a) Precession.
(b) Damping.
(c) Both.
Figure 3.1: The path of a single magnetic spin (red arrow) subjected to an effective magnetic field \(\bm {H_\mathrm {eff}}\) (black arrow) as described by the LLG equation. The field term causes precession around the
effective field (a), while the damping term causes the spin to relax towards the effective field (b). The combination of both effects results in a spiral path towards the effective field (c).
To account for materials with strong damping, in 1955, Gilbert used an Euler-Lagrange formulation where dissipative processes were described by the Rayleigh dissipation function, and derived a phenomenological damping term
expressed in terms of the time derivative of the magnetization [52]:
\(\seteqnumber{0}{3.}{6}\)
\begin{equation}
\frac {\partial \bm {m} }{\partial t} = -\gamma _0 \bm {m} \times \left [ \bm {H} -\xi \frac {\partial \bm {m} }{\partial t} \right ],
\end{equation}
which is often rewritten as
\(\seteqnumber{0}{3.}{7}\)
\begin{equation}
\label {eq:LLG_eq} \frac {\partial \bm {m}}{\partial t} = -\gamma _0 \bm {m} \times \bm {H} + \alpha \bm {m} \times \frac {\partial \bm {m}}{\partial t},
\end{equation}
where \(\alpha = \gamma _0\xi \geq 0\) is the dimensionless Gilbert damping constant. Equation (3.8 ) is known as the LLG equation, and is the
most widely used equation in micromagnetics to describe the dynamics of the magnetization in a FM material.
With a few algebraic manipulations, the LLG equation can be rewritten into an explicit form:
\(\seteqnumber{0}{3.}{8}\)
\begin{equation}
\label {eq:LLG_explicit} \frac {\partial \bm {m}}{\partial t} = -\frac {\gamma _0}{1+\alpha ^2}\bm {m} \times \bm {H} - \frac {\alpha \gamma _0}{1+\alpha ^2}\bm {m} \times \left (\bm {m}
\times \bm {H}\right ).
\end{equation}
In certain cases, this form of the LLG equation is more convenient to use, as it separates the precessional and damping terms.
