Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices
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Chapter 4 Charge and Spin Transport
The electron transport in multilayer systems involving FM regions, such as spin valves, is naturally strongly affected by the spin of the electrons. Consequently, the description of the electron transport in these systems requires a
treatment of the spin degree of freedom. In this chapter, a drift-diffusion description of the charge and spin transport in FM systems is presented.
4.1 Charge and Spin Currents in a Ferromagnet
The charge and spin current densities in a FM layer, for an arbitrary flow direction, can be defined in terms of the majority and minority electron currents \(j^\uparrow \) and \(j^\downarrow \), respectively. The charge and
spin current densities are defined by the sum and difference of the two currents, i.e.
\(\seteqnumber{0}{4.}{0}\)
\begin{equation}
j_c = j^\uparrow + j^\downarrow ,
\end{equation}
and
\(\seteqnumber{0}{4.}{1}\)
\begin{equation}
j_s = j^\uparrow - j^\downarrow ,
\end{equation}
respectively. The spin current density is here in units of charge current density, and can be converted to units of angular momentum current density by multiplication with \(\hbar /(2e)\).
To generalize the currents for any direction of the magnetization, the currents can be described in terms of spin projection matrices along the magnetization direction \(\bm {m}\). The current densities can then be compactly given
as a \(2\times 2\) matrix in Pauli spin space by the sum
\(\seteqnumber{0}{4.}{2}\)
\begin{equation}
\hat {j} = \sum _s \hat {p}^s j^s,
\end{equation}
where \(s\in \{\uparrow ,\downarrow \}\), and \(\hat {p}^{\uparrow /\downarrow } = (\hat {1} \mp \bm {\hat {\sigma }}\cdot \bm {m})/2\) is the spin projection matrix for majority/minority electrons, \(\hat
{1}\) is the \(2\times 2\) identity matrix, \(\bm {\hat {\sigma }}\) is the vector of Pauli matrices. In this and the next chapter, a hat \(\hat {\,}\) denotes a \(2\times 2\) matrix in spin space. This spin generalized
current density can be expanded into a basis of the unit matrix and the vector of Pauli matrices, yielding a charge and a spin component:
\(\seteqnumber{0}{4.}{3}\)
\begin{equation}
\hat {j} = \frac {1}{2}\left (j_{c} \hat {1} + \bm {\hat {\sigma }} \cdot \bm {j_{s}}\right ),
\end{equation}
where \(\bm {j_s} = (j_x,j_y,j_z)\) is the spin polarization current density describing both the magnitude of the spin current density and its polarization direction. The current densities in real space are then obtained from
taking the trace of \(\hat {j}\) over the spin components:
\(\seteqnumber{1}{4.5}{0}\)
\begin{align}
j_{c} = \text {Tr}[\hat {j}], \\ \bm {j_{s}} = \text {Tr}[\bm {\hat {\sigma }}\hat {j}].
\end{align}
For a single FM layer, the spin polarization current density is given by \(\bm {j_s} = -\bm {m}(j^\uparrow - j^\downarrow )\). However, in a multilayer system consisting of several FM layers with non-collinear magnetization
directions, the spin currents polarized in one FM layer can flow into an adjacent NM layer, or another FM layer with a different magnetization direction. In this case, the spin current polarization can deviate from the local
magnetization direction, which yields a spin torque on the magnetization. Therefore, a comprehensive spin-generalized transport framework is required for multilayer systems to account for spatially varying spin-polarized currents
and spin torques.
