Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices
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B.3 Nonmagnetic/Ferromagnetic Interface
The two layers are connected at \(z=0\) using the BCs from the Magnetoelectronic circuit theory to determine the remaining unknown coefficients \(\bm {B}\) and \(C\). To simplify the derivation, \(\bm {B}\) is decomposed into
a longitudinal and two transverse components: \(\bm {B} = B_l(\bm {m}\cdot \bm {\hat {y}})\bm {m} + B_d\bm {m}\times (\bm {m}\times \bm {\hat {y}}) + B_f\bm {m}\times \bm {\hat {y}}\).
The transverse components of the spin current at the NM side of the interface are then determined by the following BC:
\(\seteqnumber{0}{B.}{10}\)
\begin{equation}
[I-\bm {m}\otimes \bm {m}]\bm {j^N_{sz}}(0) = -2\operatorname {Re}\{G_{\uparrow \downarrow }\}\bm {m}\times (\bm {m}\times \bm {\mu _s^N}(0)) - 2\operatorname {Im}\{G_{\uparrow \downarrow
}\}\bm {m}\times \bm {\mu _s^N}(0).
\end{equation}
Inserting the expressions for the current and spin potential yields
\(\seteqnumber{0}{B.}{11}\)
\begin{multline}
-2\frac {\sigma ^N}{\lambda ^N_{sf}}\sinh \left (d_{N}/\lambda ^N_{sf}\right )(B_d\bm {m}\times (\bm {m}\times \bm {\hat {y}}) + B_f\bm {m}\times \bm {\hat {y}}) + \alpha _\mathrm {SH}\sigma
^N E_x\left [1-e^{d_{N}/\lambda ^N_{sf}}\right ]\bm {m}\times (\bm {m}\times \bm {\hat {y}}) = \\ +4\operatorname {Re}\{G_{\uparrow \downarrow }\}\cosh \left (d_{N}/\lambda ^N_{sf}\right
)(B_d\bm {m}\times (\bm {m}\times \bm {\hat {y}}) + B_f\bm {m}\times \bm {\hat {y}}) +2\operatorname {Re}\{G_{\uparrow \downarrow }\}\alpha _\mathrm {SH}E_x\lambda _{sf}e^{d_{N}/\lambda
^N_{sf}}\bm {m}\times (\bm {m}\times \bm {\hat {y}}) \\ +4\operatorname {Im}\{G_{\uparrow \downarrow }\}\cosh \left (d_{N}\lambda ^N_{sf}\right )(B_d\bm {m}\times \bm {\hat {y}}-B_f\bm
{m}\times (\bm {m}\times \bm {\hat {y}})) + 2\operatorname {Im}\{G_{\uparrow \downarrow }\}\alpha _\mathrm {SH}E_x\lambda _{sf}e^{d_{N}/\lambda ^N_{sf}}\bm {m}\times \bm {\hat {y}}.
\end{multline}
As the \(\bm {m}\times \bm {\hat {y}}\) and \(\bm {m}\times (\bm {m}\times \bm {\hat {y}})\) terms are mutually orthogonal, they can be treated separately, yielding the following set of equations:
\(\seteqnumber{0}{B.}{12}\)
\begin{equation}
\left (\tanh ^2\left (\xi ^N \right )+\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\right )B_d -\operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}B_f = + \left [\frac
{\alpha _\mathrm {SH}E_x\lambda ^N_{sf}}{2\sinh \left (\xi ^N \right )}\right ]\left (\tanh ^2\left (\xi ^N \right )(1-e^{\xi ^N }) -\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow
}\}\tanh \left (\xi ^N \right )e^{\xi ^N }\right ),
\end{equation}
\(\seteqnumber{0}{B.}{13}\)
\begin{equation}
\left (\tanh ^2\left (\xi ^N \right )+\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\right )B_f +\operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}B_d = -\left [\frac {\alpha
_\mathrm {SH}E_x\lambda ^N_{sf}}{2\sinh \left (\xi ^N \right )}\right ]\operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}\tanh \left (\xi ^N \right )e^{\xi ^N }.
\end{equation}
Here the scaled mixing conductance \(\tilde {G}^N_{\uparrow \downarrow } = 2(\lambda ^N_{sf}/\sigma ^N)\tanh \left (\xi ^N \right ) G_{\uparrow \downarrow }\), and the scaled thickness \(\xi ^N =
d_N/\lambda _{sf}^N \) was introduced. Solving the set of equations yields
\(\seteqnumber{0}{B.}{14}\)
\begin{align}
B_d & = \left [\frac {-\alpha _\mathrm {SH}E_x\lambda ^N_{sf}}{2\sinh \left (\xi \right )}\right ]\Bigg [\left (e^{\xi ^N}-1\right ) + \left (\frac {(1-e^{-\xi ^N})^2}{1+e^{-2\xi
^N}}\right )\frac {\vert \tilde {G}^N_{\uparrow \downarrow }\vert ^2 +\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\tanh ^2\left (\xi ^N \right )}{\left (\tanh ^2(\xi ^N
)+\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\right )^2 +\operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}^2}\Bigg ], \\ B_f & = \left [\frac {-\alpha _\mathrm
{SH}E_x\lambda ^N_{sf}}{2\sinh \left (\xi ^N \right )}\right ]\left (\frac {(1-e^{-\xi ^N })^2}{1+e^{-2\xi ^N }}\right )\frac {\operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}\tanh
^2(\xi ^N )}{\left (\tanh ^2(\xi ^N )+\operatorname {Re}\{\tilde {G}^N_{\uparrow \downarrow }\}\right )^2 + \operatorname {Im}\{\tilde {G}^N_{\uparrow \downarrow }\}^2}.
\end{align}
The longitudinal spin current at the interface is continuous:
\(\seteqnumber{0}{B.}{16}\)
\begin{equation}
\bm {j^F_{sz}(0)}\cdot \bm {m} =\bm {j^N_{sz}(0)}\cdot \bm {m}.
\end{equation}
Inserting the expressions for the spin currents yields
\(\seteqnumber{0}{B.}{17}\)
\begin{equation}
-2\frac {\sigma ^F(1-\beta _D\beta _\sigma )}{\lambda _{sdl}}C\sinh \left (-\xi _F\right ) = -2\frac {\sigma ^N}{\lambda ^N_{sf}}\sinh \left (\xi ^N\right )B_l(\bm {\hat {y}}\cdot \bm {m}) -
\alpha _\mathrm {SH}\sigma ^N E_x\left [1-\exp \left (\xi ^N\right )\right ](\bm {\hat {y}}\cdot \bm {m}),
\end{equation}
where \(\xi _F = d_{F}/\lambda _{sdl}\). Solving for \(C\) yields
\(\seteqnumber{0}{B.}{18}\)
\begin{equation}
C = (1-\beta _D\beta _\sigma )^{-1}\frac {\sigma ^N\lambda _{sdl}}{\sigma ^F\lambda ^N_{sf} }\frac {\sinh \left (\xi ^N\right )}{\sinh \left (-\xi ^F\right )}B_l(\bm {\hat {y}}\cdot \bm {m}) +
\frac {\alpha _\mathrm {SH}}{2}E_x (1-\beta _D\beta _\sigma )^{-1}\lambda _{sdl}\frac {\sigma ^N}{\sigma ^F}\frac {\left [1-\exp \left (\xi ^N\right )\right ]}{\sinh \left (-\xi ^F\right
)}(\bm {\hat {y}}\cdot \bm {m}).
\end{equation}
The spin accumulation in the FM layer is then determined by \(B_l\), thus one obtains
\(\seteqnumber{0}{B.}{19}\)
\begin{equation}
\bm {V^F_s}(0) = 2(1-\beta _D\beta _\sigma )^{-1}\frac {\sigma ^N\lambda _{sdl}}{\sigma ^F\lambda ^N_{sf} }\sinh \left (\xi ^N\right )\coth \left (-\xi ^F\right )B_l(\bm {\hat {y}}\cdot \bm
{m})\bm {m} + \alpha _\mathrm {SH}E_x(1-\beta _D\beta _\sigma )^{-1}\lambda _{sdl}\frac {\sigma ^N}{\sigma ^F}\left [1-\exp \left (\xi ^N\right )\right ]\coth \left (-\xi ^F\right )(\bm {\hat
{y}}\cdot \bm {m})\bm {m}.
\end{equation}
The charge currents at either side of the interface are given by
\(\seteqnumber{0}{B.}{20}\)
\begin{equation}
j^N_{cz}(0) = j^F_{cz}(0) = G_+\Delta V_c - G_-\Delta \bm {V_s}\cdot \bm {m} = 0,
\end{equation}
thus the potential drop across the interface is given by
\(\seteqnumber{0}{B.}{21}\)
\begin{equation}
\Delta V_c = \frac {G_-}{G_+}\Delta \bm {V_s}\cdot \bm {m}.
\end{equation}
The longitudinal spin current at the interface can now be expressed solely in terms of the spin potential drop:
\(\seteqnumber{0}{B.}{22}\)
\begin{equation}
\bm {j^N_{sz}(0)}\cdot \bm {m} = -G_-\Delta V_c + G_+\Delta \bm {V_s}\cdot \bm {m} = \left [\frac {G_+^2-G_-^2}{G_+}\right ](\Delta \bm {V_s}\cdot \bm {m}).
\end{equation}
Inserting the expressions for the spin current and potentials yields
\(\seteqnumber{0}{B.}{23}\)
\begin{multline}
-2\frac {\sigma ^N}{\lambda ^N_{sf}}\sinh \left (\xi ^N\right )B_l(\bm {\hat {y}}\cdot \bm {m}) - \alpha _\mathrm {SH}\sigma ^N E_x\left [1-e^{\xi ^N}\right ](\bm {\hat {y}}\cdot \bm {m}) =
\left [\frac {G_+^2-G_-^2}{G_+}\right ]\bigg [2(1-\beta _D\beta _\sigma )^{-1}\frac {\sigma ^N}{\lambda ^N_{sf}}\frac {\lambda ^F_{sdl}}{\sigma ^F}\sinh \left (\xi ^N\right )\coth \left (-\xi
^F\right )B_l(\bm {\hat {y}}\cdot \bm {m}) \\ + \alpha _\mathrm {SH}E_x(1-\beta _D\beta _\sigma )^{-1}\lambda _{sdl}\frac {\sigma ^N}{\sigma ^F}\left (1-e^{\xi ^N}\right )\coth \left (-\xi
^F\right )(\bm {\hat {y}}\cdot \bm {m}) - 2\cosh \left (\xi ^N\right )B_l(\bm {\hat {y}}\cdot \bm {m}) + \alpha _\mathrm {SH}E_x\lambda _{sf}e^{\xi ^N}(\bm {\hat {y}}\cdot \bm {m} )\bigg ].
\end{multline}
After rearranging the terms, one obtains
\(\seteqnumber{0}{B.}{24}\)
\begin{multline}
\bigg [1 + G_{\|}\bigg ((1-\beta _D\beta _\sigma )^{-1}\frac {\lambda ^F_{sdl}}{\sigma ^F}\coth \left (-\xi ^F\right ) - \frac {\lambda ^N_{sf}}{\sigma ^N}\coth \left (\xi ^N\right )\bigg
)\bigg ]B_l = \\ \left [\frac {-\alpha _\mathrm {SH}\lambda _{sf}^N E_x}{2\sinh \left (\xi ^N\right )}\right ]\left (1-e^{xi^N}\right )\bigg [- G_{\|}\frac {\lambda _{sf}^N }{\sigma ^N}\frac
{1-e^{-\xi ^N}}{1-e^{-2\xi ^N}}e^{-\xi ^N} 1 + G_{\|}\bigg ((1-\beta _D\beta _\sigma )^{-1}\frac {\lambda _{sdl} }{\sigma ^F}\coth \left (-\xi ^F\right )- \frac {\lambda _{sf}^N }{\sigma
^N}\coth \left (\xi ^N\right )\bigg ) \bigg ],
\end{multline}
where the longitudinal conductance \(G_{\|} =(G_+^2-G_-^2)/G_+ \) was introduced.
Solving for the longitudinal coefficient yields
\(\seteqnumber{0}{B.}{25}\)
\begin{equation}
B_l = \left [\frac {-\alpha _\mathrm {SH}\lambda _{sf}^N E_x}{2\sinh \left (\xi ^N\right )}\right ]\bigg [\left (1-e^{\xi ^N}\right )+ \frac {(1-e^{-\xi ^N})^2}{1-e^{-2\xi ^N}}\frac {\tilde
{G_{\|}}^N\tanh (\xi ^N)}{2 - \tilde {G_{\|}}^F - \tilde {G_{\|}}^N}\bigg ],
\end{equation}
where the scaled conductances \(\tilde {G_{\|}}^N = 2(\lambda ^N_{sf}/\sigma ^N)\coth \left (\xi ^N \right ) G_{\|}\) and \(\tilde {G_{\|}}^F = 2(1-\beta _D\beta _\sigma )^{-1}(\lambda
^F_{sdl}/\sigma ^F)\coth \left (\xi ^F \right ) G_{\|}\) were introduced.
