Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices
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Chapter 5 Interfacial Spin-Orbit Coupling
The original formulation of the BCs from MCT only considered the case of a negligible SOC [65]. However, in SOT systems, NM HMs with strong SOC are used to generate spin currents through the SHE, thereby manipulating the
magnetization of adjacent FM layers. The NM/FM interface in these systems can therefore potentially have a strong interfacial SOC, which can modify the BCs from the MCT. Early approaches to account for SOC at the interface
of NM/FM systems have mainly focused on 2D REE models, which treat the interface as a 2D electron gas that becomes spin-polarized under an applied electric field [45]. However, NM/FM bilayers require a 3D description to
capture the effects arising from the electrons incident from the bulk on either side of the interface scattering off the spin-orbit field.
Haney et al. proposed a simple Hamiltonian to model interfacial SOC at NM/FM interfaces [70]. The model generalizes the Rashba SOC to three dimensions by treating the interface as a spin- and momentum-dependent
delta function potential barrier. The spin-quantization axis is aligned with the effective interface field, combining the interfacial magnetic exchange and Rashba spin-orbit fields. The resulting scattering matrices for the plane-wave
solutions of the Schrödinger equation were used as BCs for the non-equilibrium density functions in a spin generalized Boltzmann equation. Later, Amin and Stiles derived BCs for the drift-diffusion equations based on the
previous Boltzmann formalism [75, 76], which introduced modifications to the BCs from the MCT. The torques calculated using the drift-diffusion approach agreed well with those obtained using the more rigorous Boltzmann
equation approach. In 2017, Kim et al. presented a perturbation theory approach, which treats the delta-function SOC as a perturbation of the scattering matrices at the interface [77]. The approach was shown to be
flexible and could be used to describe the SOC at various interfaces.
In this chapter, BCs for a NM (\(z < 0\))/FM (\(z>0\)) interface at \(z=0\) with SOC, based on the previous works, are derived. First, the BCs from the MCT are extended by considering a drift term in the
non-equilibrium distribution function, which captures the deformation of the Fermi surface by an applied electric field. This term is often ignored as it yields a vanishing contribution; however, it will be demonstrated that in the
presence of interfacial SOC, this term can give rise to a finite contribution to the currents at the interface. Next, BCs for the interfacial SOC are derived using two different approaches. First, the perturbation theory approach is
used, which extends the BCs from the MCT and is valid for interfaces with weak SOC. Finally, BCs based on the delta function scattering potential barrier are derived, which are valid for interfaces with strong SOC.
5.1 The Nonequilibrium Drift Term
The BCs from the MCT were rigorously derived by applying the Keldysh NEGF to a Stoner Hamiltonian [69]; thus, they provide a good starting point for deriving BCs that consider interfacial SOC at NM/FM interfaces. The
\(2\times 2\) current density along \(z\) at the NM (\(0^-\)) side of the interface reads
\(\seteqnumber{0}{5.}{0}\)
\begin{equation}
\label {eq:interface_current_from_MCT} \hat {j}_z(0^-) = -\frac {e}{h A}\sum _{\bm {k_\|}} \int d\epsilon \left [ \hat {f}^N_{\bm {k}} - \hat {r}_{\bm {k}}\hat {f}^N_{\bm {k}}(\hat {r}_{\bm
{k}})^\dagger - \hat {t}_{\bm {k}}^{\prime }\hat {f}^F_{\bm {k}}(\hat {t}^{\prime }_{\bm {k}})^\dagger \right ],
\end{equation}
where \(\hat {f}^N_{\bm {k}}\) and \(\hat {f}^F_{\bm {k}}\) are the distribution functions in the NM and FM layers for states with wave vector \(\bm {k}\), respectively. The summation is over the in-plane wave vectors
\(\bm {k_\|}\) and \(A\) is the area of the interface.
In equilibrium, the probability that the state with momentum \(\bm {k}\) is occupied by an electron is described by the Fermi-Dirac distribution; the distribution functions in equilibrium are diagonal in spin space:
\(\seteqnumber{0}{5.}{1}\)
\begin{equation}
\hat {f}_{\bm {k},eq}^{N/F} = \hat {1} \frac {1}{\exp ((\epsilon _{\bm {k}} - \epsilon _F)/k_BT) + 1},
\end{equation}
where \(\epsilon _F = \hbar ^2k_F^2/2m_e\) is the Fermi energy, \(k_F\) is the Fermi wave number, \(\epsilon _{\bm {k}}\) is the energy of the state with momentum \(\bm {k}\).
When an electric field is applied, the distribution of occupied states is shifted. In the linear response regime, the out-of-equilibrium distribution in spin space is given by
\(\seteqnumber{0}{5.}{2}\)
\begin{equation}
\hat {f}_{\bm {k}}^{N/F} = \hat {f}_{\bm {k},eq}^{N/F} + \frac {\partial \hat {f}_{\bm {k},eq}^{N/F}}{\partial \epsilon _{\bm {k}}}\hat {g}^{N/F}_{\bm {k}},
\end{equation}
where \(\hat {g}^{N/F}_{\bm {k}}\) is the non-equilibrium distribution function, which describes the perturbation of the Fermi-Dirac distribution function.
The non-equilibrium distribution function is given by the isotropic in momentum non-equilibrium electrochemical potential and an anisotropic "drift" term describing the deformation of the Fermi surface by the applied field [78, 65]:
\(\seteqnumber{0}{5.}{3}\)
\begin{equation}
\label {eq:noneq_distr_func} \hat {g}^{N/F}_{\bm {k}} = \hat {\mu }^{N/F}+\hat {\gamma }^{N/F}_{\bm {k}}.
\end{equation}
The anisotropic term is often ignored, as it vanishes when averaging over the Fermi surface. However, if the interface scattering matrices in Eq. (5.1) are uneven in transverse momentum, this is no longer the case, and the anisotropic term must be taken into account.
In a steady state, the shift of the Fermi surface due to an applied electric field is given by \(\Delta \bm {k} = e\bm {E}\tau /\hbar \) [60], where \(\tau \) is the momentum relaxation time. Ignoring terms second order in
\(\tau \), the deformation of the Fermi surface by an applied electric field in the FM and NM layer can be expressed by
\(\seteqnumber{0}{5.}{4}\)
\begin{equation}
\hat {\gamma }^{N/F}_{\bm {k}} = \frac {e\hbar }{m_e}(\bm {k}\cdot \bm {E})\hat {\tau }^{N/F},
\end{equation}
where \(\hat {\tau }^{F} = \tau ^\uparrow \hat {p}^\uparrow +\tau ^\downarrow \hat {p}^\downarrow \) and \(\hat {\tau }^N=\tau ^N\hat {1}\) is the momentum relaxation time of each layer, respectively. In
the FM layer one can define a polarization in terms of the majority and minority relaxation times: \(\beta _\tau = (\tau ^{\uparrow } - \tau ^{\downarrow })/(\tau ^{\uparrow } + \tau ^{\downarrow })\), and an
average momentum relaxation time: \(\tau ^F = (\tau ^{\uparrow } + \tau ^{\downarrow })/2\).
Using the unitarity relation of the scattering matrices: \(\hat {1} - \hat {r}_{\bm {k}}(\hat {r}_{\bm {k}})^\dagger = \hat {t}^\prime _{\bm {k}}(\hat {t}^\prime _{\bm {k}})^\dagger \), it can be shown that
only the non-equilibrium part of the distribution function contributes to the current density. By treating both sides of the interface as degenerate metals [60]: \(\partial \hat {f}_{\bm {k},eq}^{N/F}/\partial \epsilon
_{\bm {k}} \approx \hat {1}\delta (\epsilon _{\bm {k}}-\epsilon _F)\). After integrating over the energy, the current density at the interface reads
\(\seteqnumber{0}{5.}{5}\)
\begin{multline}
\label {eq:interface_current_with_drift} \hat {j}_z(0^-) = \frac {-e}{2\pi \hbar A}\sum _{\bm {k_\|} \in \mathrm {FS}} \left [ (\hat {\mu }^N + \hat {\gamma }_{\bm {k}}^N) - \hat {r}_{\bm
{k}}(\hat {\mu }^N+\hat {\gamma }_{\bm {k}}^N)(\hat {r}_{\bm {k}})^\dagger - \hat {t}^{\prime }(\hat {\mu }^F+\hat {\gamma }_{\bm {k}}^F)(\hat {t}^{\prime }_{\bm {k}})^\dagger \right ] \\ =
-\frac {e}{2\pi \hbar A}\sum _{\bm {k_\|} \in \mathrm {FS}} \left [ \hat {\mu }^N - \hat {r}_{\bm {k}}\hat {\mu }^N(\hat {r}_{\bm {k}})^\dagger - \hat {t}^{\prime }_{\bm {k}}\hat {\mu }^F(\hat
{t}^{\prime }_{\bm {k}})^\dagger \right ] +\frac {e^2}{2\pi m_e A}\sum _{\bm {k_\|} \in \mathrm {FS}}(\bm {k}\cdot \bm {E})\left [\hat {t}^{\prime }_{\bm {k}}(\hat {\tau }^F - \hat {\tau
}^N)(\hat {t}^{\prime }_{\bm {k}})^\dagger \right ],
\end{multline}
where \(\hat {\tau }^N - \hat {r}_{\bm {k}}\hat {\tau }^N(\hat {r})^\dagger = \hat {t}^{\prime }_{\bm {k}}\hat {\tau }^N(\hat {t}^{\prime })^\dagger \) was used. The summation over the in-plane wave
vectors is limited to wave vectors on the Fermi surface, which is denoted by \(\bm {k_\|} \in \mathrm {FS}\). Since \((\bm {k}\cdot \bm {E})\) is odd in \(\bm {k_\|}\), if \(\hat {t}^{\prime }_{\bm {k}}\) is even in
\(\bm {k_\|}\) the last term vanishes after averaging over the Fermi surface, and the typical expression for the current at the interface is recovered.
