Modeling Spin-Orbit Torques
in Advanced Magnetoresistive Devices

4.3 Boundary Conditions

In order to solve the drift-diffusion equations (4.21) and (4.18), the BCs at the interfaces between different materials and at the external boundaries of the system need to be specified. Considering a general system \(\Omega \) consisting of multiple NM regions \(\Omega ^N_i\) and FM regions \(\Omega ^F_i\), the BCs can be divided into two categories: those at external boundaries, which specifying the electrical contacts \(\Gamma _i\) and boundaries not containing a contact \(\partial \Omega \); And those at internal boundaries, specifying the interfaces between different material layers \(\Gamma _{ij}\).

At external boundaries not containing a contact (\(\partial \Omega \)), it is typically assumed that there is no flow of currents, which is expressed through Neumann BCs:

\begin{align} \bm {j_c}\cdot \bm {n} & = 0 & \text { on } \partial \Omega , \\ \tilde {J}_s\bm {n} & = 0 & \text { on } \partial \Omega , \end{align} where \(\bm {n}\) is the normal vector of the interface. Applied to Eqs. (4.21), this yields BCs for the gradients:

\begin{align} (\nabla V)\cdot \bm {n} & = \frac {e}{\mu _B}\alpha _\mathrm {SH}\frac {D_e}{\sigma }(\nabla \times \bm {S})\cdot \bm {n} & \text { on } \partial \Omega , \\ (\nabla \bm {S})\bm {n} & = \frac {\mu _B}{e}\alpha _\mathrm {SH}\frac {\sigma }{D_e}\bm {E}\times \bm {n} & \text { on } \partial \Omega . \end{align}

At external boundaries containing a contact (\(\Gamma _i\) ), the BC for the electrical potential directly corresponds to the voltage applied to the system through the electric contacts. At the contacts, a Dirichlet condition can then be applied directly to the electrical potential, which is expressed as

\begin{equation} V = V_i \text { on } \Gamma _i. \end{equation}

Another option is to describe the applied current at the contact directly with the Neumann condition

\begin{equation} \bm {j_c}\cdot \bm {n} = j_{c}^i \text { on } \Gamma _i, \end{equation}

where \(j_{c}^i\) is the current density flowing through the contact.

For the electrical contacts, the no-flux condition for the spin current is valid if the region is NM, and its thickness exceeds the spin-flip length. In this case, the spin accumulation can be assumed to have decayed to zero at the contact; otherwise, the spin current has to be treated with the Robin BC [54]

\begin{equation} \tilde {J}_s\bm {n} + \frac {D_e}{\lambda _{sf} }\bm {S} = 0 \text { on } \Gamma ^N_i, \end{equation}

which takes into account the exponential decay of the spin accumulation.

If the contact is FM, assuming the spin accumulation is fully decayed, i.e. \((\nabla \bm {S})\bm {n} = 0\), the BC for the spin current is obtained by expressing the electric field in terms of the charge current using Eq. (4.10a) and using the Neumann condition for the charge current:

\begin{equation} \tilde {J}_s\bm {n} = -D_e(\nabla \bm {S})\bm {n} - \frac {\mu _B}{e}\beta _{\sigma }\sigma \bm {m} \otimes \left [\bm {j_{c}} - \frac {e}{\mu _B}\beta _D D_e (\nabla \bm {S})^T\bm {m}\right ]\bm {n} = - \frac {\mu _B}{e}\beta _{\sigma }\sigma \bm {m}(\bm {j_{c}}\cdot \bm {n}) \text { on } \Gamma ^F_i. \end{equation}

The internal boundaries can be treated by assuming that the electrical potential, the spin accumulation, and the currents are continuous across the interface. For the interface between material \(1\) and \(2\) these BCs are expressed as follows:

\begin{align} V^1 & = V^2 \text { on } \Gamma _{12}, \\ \bm {S}^1 & = \bm {S}^{2} \text { on } \Gamma _{12}, \\ \bm {j^1_c}\cdot \bm {n} & = \bm {j^2_c}\cdot \bm {n} \text { on } \Gamma _{12}, \\ \tilde {J}^1_s\bm {n} & = \tilde {J}^2_s\bm {n} \text { on } \Gamma _{12}. \end{align} This assumption is required for the charge current to ensure that the flux of particles is conserved. However, the assumption that the electric potential, spin accumulation, and spin current are continuous is only valid if there is no interfacial resistance or spin-flip scattering at the interface.

Semiclassical theories of spin transport, such as the drift-diffusion equations, are generally only valid when the length scales are larger than the mean free path; thus, abrupt interfaces between materials cannot be properly described. Therefore, BCs based on quantum mechanical scattering are essential for describing the interface between two materials. Since the STTs and SOTs are predominantly affected by the interfaces in MTJs and between the NM and FM layers, respectively, the BCs at these interfaces will be discussed in more detail.

4.3.1 Nonmagnetic/Ferromagnetic Interface Boundary
Conditions

In contrast to the Zhang-Levy-Fert model, where the electrical potential and spin accumulation are assumed to be continuous across NM/FM interfaces, the original Valet-Fert model allowed them to be discontinuous. The interface was treated as a fictitious, infinitesimally thin layer with resistances that described the interfacial scattering, which could be fitted to the experiments. The currents at the interface would then be determined by charge and spin potential jumps across this layer caused by the interface resistances. However, this approach does not account for non-collinear systems where there can be spin currents transverse to the magnetization incident on the interface, such as ones generated through the SHE.

An alternative method for addressing both collinear and non-collinear transport across NM/FM material interfaces was developed within the framework of magnetoelectronic circuit theory (MCT) [68]. This approach introduced quantum mechanical BCs, which assume distribution functions that are isotropic in momentum on both sides of the interface and strictly longitudinal spin accumulation and currents in the FM regions. Under these assumptions, the charge and spin currents at the interface can be succinctly described using only a handful of interface conductances that describe both collinear and non-collinear spin transport. Since the description requires few parameters to characterize the most important properties of the interface, which can be computed using ab initio methods or be extracted from experiments [65], it has gained significant popularity in the spintronics community.

A NM layer (\(N\)) with an adjacent FM layer (\(F\)) is considered. The bulk of each layer is treated as a node described by a distribution matrix in spin space \(\hat {f}^a(\epsilon )\) for electrons with energy \(\epsilon \), where \(a\in \{N,F\}\) denotes which layer the distribution describes. In equilibrium the matrix is diagonal in spin-space \(\hat {f}^a(\epsilon ) = \hat {1}f^a(\epsilon )\), where \(f^a(\epsilon )\) is the Fermi-Dirac distribution function. Out of equilibrium, assuming rapid dephasing of transverse components (i.e. \(\lambda _\phi \to 0\) ), the distribution function in the FM layer can be limited to the form \(\hat {f}^F = \hat {1}f_0^F +\bm {\hat {\sigma }}\cdot \bm {m}f_s^F\), while in the NM layer, \(\hat {f}^N\) is an arbitrary Hermitian matrix.

The interface is treated as a contact between the nodes. The charge and spin currents through the contacts are then, in the spirit of the Landauer-Büttiker formalism, related to the distribution matrices of the adjacent nodes through microscopic scattering matrices. The resulting energy resolved \(2\times 2\) current at the NM side of the interface reads [69]

\begin{equation} \hat {i} = -\frac {e}{h}\sum _{nm} \left [ \delta _{nm}\hat {f}^N - \hat {r}_{nm}\hat {f}^N(\hat {r}_{nm})^\dagger - \hat {t}^{\prime }_{nm}\hat {f}^F(\hat {t}^{\prime }_{nm})^\dagger \right ], \end{equation}

where \((\hat {r}_{nm}(\epsilon ))_{ss^\prime }\) and \((\hat {t}^{\prime }_{nm}(\epsilon ))_{ss^\prime }\) are the reflection and transmission matrices, respectively, which describe the electrons approaching from the NM (FM) layer in transverse momentum mode \(m\) with spin \(s^\prime \) being reflected (transmitted) to mode \(n\) with spin \(s\). The energy dependence has been omitted for brevity. From here on \(A^\dagger \) denotes the Hermitian transpose of a matrix \(A\). It should be noted that current conservation requires the following unitarity relation [60]: \(\hat {r}(\hat {r})^\dagger + \hat {t}^{\prime }(\hat {t}^{\prime })^\dagger = \hat {1}\).

The scattering matrices can be expressed in terms of a sum over the majority/minority spin projection matrices and scattering coefficients:

\begin{align} \label {eq:scattering_matricies} \hat {r}_{nm} & = \sum _s \hat {p}^s r^{s}_{nm}, \\ \hat {t}_{nm} & = \sum _s \hat {p}^s t^{s}_{nm}. \end{align} Here, \(r^{s}_{nm}\) and \(t^{s}_{nm}\) are the reflection and transmission coefficients for majority (\(s=\uparrow \)) and minority (\(s=\downarrow \)) electrons, respectively . Inserting Eqs. (4.30a) into the expression for the current and using the unitarity relation for the scattering matrices gives

\begin{equation} \label {eq:energy_resolved_MCT_current} e\hat {i} = \mathcal {G}_{\uparrow \uparrow }\hat {p}^\uparrow (\hat {f}^{F}-\hat {f^N})\hat {p}^\uparrow + \mathcal {G}_{\downarrow \downarrow }\hat {p}^\downarrow (\hat {f}^{F}-\hat {f^N})\hat {p}^\downarrow - \mathcal {G}_{\uparrow \downarrow }\hat {p}^\uparrow \hat {f^N}\hat {p}^\downarrow - (\mathcal {G}_{\uparrow \downarrow })^*\hat {p}^\downarrow \hat {f^N}\hat {p}^\uparrow , \end{equation}

where the spin-dependent majority and minority conductances \(\mathcal {G}^{\uparrow \uparrow }\) and \(\mathcal {G}^{\downarrow \downarrow }\), respectively, read

\begin{equation} \mathcal {G}_{ss} = \frac {e^2}{h}\sum _{nm} \left [ \delta _{nm} - \left \vert r_{nm}^s \right \vert ^2\right ] = \frac {e^2}{h}\sum _{nm} \left \vert t_{nm}^s \right \vert ^2. \end{equation}

The superscript \(^*\) denotes the complex conjugate of a complex quantity. The non-collinear transport is captured by the complex spin-mixing conductance \(\mathcal {G}_{\uparrow \downarrow }\), which is given by

\begin{equation} \mathcal {G}_{\uparrow \downarrow } = \frac {e^2}{h}\sum _{nm} \left [ \delta _{nm} - r_{nm}^\uparrow (r_{nm}^{\downarrow })^*\right ]. \end{equation}

The total current is obtained by integrating Eq. (4.31) over the energy, \(\hat {I}=\int d\epsilon \hat {i}(\epsilon )\). Assuming both sides of the interface behave as spin-dependent reservoirs (voltage sources), the currents can be expressed in terms of the electrochemical potential matrix \(\hat {\mu }^a = \int d\epsilon \hat {f^a}(\epsilon ) = \mu ^a_c\hat {1} - \bm {\hat {\sigma }}\cdot \bm {\mu ^a_s}\), where \(\mu ^a_c\) is the electrochemical potential and \(\bm {\mu ^a_s}\) is a vector quantity referred to as the spin chemical potential which is here defined to be parallel to the spin magnetic moment accumulation \(\bm {S}\) (opposite to the spin direction). The currents in real space are then obtained by taking the trace over the spin components:

\begin{align} eI_c = e\operatorname {Tr}[\hat {I}] & = (\mathcal {G}_{\uparrow \uparrow } + \mathcal {G}_{\downarrow \downarrow })\Delta \mu _c + (\mathcal {G}_{\uparrow \uparrow } - \mathcal {G}_{\downarrow \downarrow })\Delta \bm {\mu _s}\cdot \bm {m}, \\ -e\bm {I_s} = -e\operatorname {Tr}[\bm {\sigma }\hat {I}] & = (\mathcal {G}_{\uparrow \uparrow } - \mathcal {G}_{\downarrow \downarrow })\Delta \mu _c \bm {m} + (\mathcal {G}_{\uparrow \uparrow } + \mathcal {G}_{\downarrow \downarrow })(\Delta \bm {\mu _s}\cdot \bm {m})\bm {m} -2\operatorname {Re}\{\mathcal {G}_{\uparrow \downarrow }\}\boldsymbol {m}\times (\bm {\mu _s^N}\times \boldsymbol {m}) + 2\operatorname {Im}\{\mathcal {G}_{\uparrow \downarrow }\}\bm {\mu _s^N}\times \boldsymbol {m}, \end{align} where \(\Delta \mu _c =\mu _c^F - \mu _c^N\) and \(\Delta \bm {\mu _s} =\bm {\mu _s^F} - \bm {\mu _s ^N}\). The sign in front of the spin current changes the direction of the spin current to be aligned with the magnetic moment of the electrons. The expression of the currents on the FM side of the interface is the same, except for the two transverse terms described by the spin-mixing conductance, vanishing as the transverse spin current is assumed to be instantaneously destroyed. The spin torque acting on the magnetization is then localized at the interface and is completely described by these two terms.

Equation (4.34) can be modified to a form compatible with the Zhang-Levy-Fert model by assuming that the expressions for the currents hold in the microscopic limit. The local current densities at the interface are then expressed in terms of local distributions: \(\mu _c(\bm {r})\), \(\bm {\mu _s(\bm {r})}\), \(\bm {m}(\bm {r})\). The resulting equations can then be expressed as [70]

\begin{align} \bm {j_c}\cdot \bm {n} & = (G_{\uparrow \uparrow } + G_{\downarrow \downarrow })\Delta V_c + (G_{\uparrow \uparrow } - G_{\downarrow \downarrow })\Delta \bm {V_s}\cdot \bm {m}, \\ \label {eq:haney_spin} \tilde {J}_s\bm {n} & = \frac {\mu _B}{e}\left [(G_{\uparrow \uparrow } - G_{\downarrow \downarrow })\Delta V_c \bm {m} + (G_{\uparrow \uparrow } + G_{\downarrow \downarrow })(\Delta \bm {V_s}\cdot \bm {m})\bm {m} 2\operatorname {Re}\{G_{\uparrow \downarrow }\}\boldsymbol {m}\times (\Delta \bm {V_s}\times \boldsymbol {m}) - 2\operatorname {Im}\{G_{\uparrow \downarrow }\}\Delta \bm {V_s}\times \boldsymbol {m}\right ]. \end{align} Here, \(V_i = \mu _i/e\) for \(i\in \{x,y,z,c\}\) are spin and charge electrochemical potentials in units of voltage, and \(G_{ss^\prime }\) are the interface conductances \(\mathcal {G}_{ss^\prime }\) divided by the area of the interface. The position dependence is omitted for brevity. The relation between the spin chemical potential and the spin accumulation is given by \(\bm {V_s} = (e/\mu _B)(D_e/\sigma )\bm {S}\). The spin-mixing terms in Eq. (4.35b) only agree with the original formulation of the MCT when \(\bm {\mu _s^F} \| \bm {m}\), otherwise a transmission spin-mixing conductance is required to describe the transverse components on the FM side of the interface [71]. However, when \(\lambda _\phi \) is sufficiently small, Eqs. (4.35) are a good approximation.

The formulation can be further simplified by introducing a rank 4 spin-charge interface conductance tensor, yielding a spin generalized Ohm’s law for the interface:

\begin{equation} \mathfrak {j} = \check {G} \Delta \mathfrak {v}, \end{equation}

where \(\mathfrak {j} = [(e/\mu _B)\bm {J_{sz}}, j_{cz}]\) and \(\mathfrak {v} = [\bm {V_s},V_c]\), and \(\check {\,} \) denotes the rank 4 tensor. For a magnetization along the \(z\) direction, the interface conductance tensor can be expressed as

\begin{equation} \check {G} = G_+ \begin{pmatrix} a & -b & 0 & 0 \\ b & a & 0 & 0 \\ 0 & 0 & 1 & P \\ 0 & 0 & P & 1 \end {pmatrix}, \end{equation}

where \(G_+ = G_{\uparrow \uparrow } + G_{\downarrow \downarrow }\) is the interface charge conductance, \(P = (G_{\uparrow \uparrow } - G_{\downarrow \downarrow })/G_+ \) is the interface polarization parameter, \(a = 2\operatorname {Re}\{G_{\uparrow \downarrow }\}/G_+\) and \(b = 2\operatorname {Im}\{G_{\uparrow \downarrow }\}/G_+\) are scaled real and imaginary parts of the spin-mixing conductance, respectively.

4.3.2 Magnetic Tunneling Junctions

As the TB in a MTJ is an insulator, the charge and spin drift-diffusion formalism cannot be applied directly. Instead, the transport across it must be treated with BCs that describe the quantum mechanical tunneling process. Furthermore, the BCs must take into account the dependence on the relative angle between the magnetizations of the two FM layers. The non-equilibrium Green functions (NEGF) approach has often been applied to calculate the tunneling currents through the MTJ. Calculating the tunneling currents using the NEGF approach for a three-dimensional system, while considering the time evolution of the magnetization, is computationally demanding; consequently, more simplified approaches that still capture the core physics are required.

Camsari et al. proposed an approach for computing the currents through a MTJ based on the MCT [72], which showed good agreement with both experimental results and NEGF calculation in the ballistic limit. Similar to Juliere’s treatment for tunneling between two collinear FM films [20], the approach treats the MTJ as a product of two FM/NM interfaces. The tunneling of non-equilibrium charge and spin accumulation from one side to the other is then captured through a conductance matrix. Assuming this approach holds in the microscopic limit, the currents at either side of the MTJ are described by

\begin{equation} \begin{pmatrix} \mathfrak {j}_1 \\ \mathfrak {j}_2 \end {pmatrix} = \begin{pmatrix} \check {G}_{21} & -\check {G}_{12} \\ -\check {G}_{21} & \check {G}_{12} \end {pmatrix} \begin{pmatrix} \mathfrak {v} _1 \\ \mathfrak {v} _2 \end {pmatrix}, \end{equation}

where

\begin{eqnarray} \check {G}_{ij} = \check {G}_i \check {G}_j/\sqrt {G^+_i G^+_j}. \end{eqnarray}

Here \(\check {G}_i\) is the interface conductance matrix from the MCT for interface \(i\). The potential vector in the FM layers was assumed to only have a charge component, i.e., \(\mathfrak {v} = (0,0,0,V_c)\), as the leads connected to the outer terminals do not have any spin accumulation. In SOT devices, assuming the thickness of the free layer is below the spin dephasing length, the SHE could break this assumption. Then, the spin-charge voltage in the FM layers has to be expressed as \(\mathfrak {v} = (V_{x},V_{y},V_{z},V_c)\), where \(V_{i}\) is the \(i\)’th polarization component of the spin potential.

Simone et al. proposed a model based on both the circuit theory approach and a NEGF approximation for a thick/strong barrier [73, 64], which simplifies the results of both while including the most prominent transport characteristics. This approach was shown to reproduce the angular dependence described by Slonczewski, as well as experimental results for the voltage dependence of the resistance and torques [73]. The tunneling charge current is given by

\begin{equation} j_c\cdot \bm {n} = \mathcal {G}_0(1+P_1 P_2 \bm {m_1}\cdot \bm {m_2})\Delta V_c, \end{equation}

where \(\mathcal {G}_0 = (\mathcal {G}_p + \mathcal {G}_{AP})/2 \) is the average conductance of the TB, \(R_P = 1/\mathcal {G}_P\) and \(R_{AP} = 1/\mathcal {G}_{AP}\) are the resistances measured when the MTJ is in a parallel or antiparallel state, respectively, and \(P_i\) are the in-plane Slonczewski polarization parameter of interface \(i\), respectively. The tunneling spin current can then be expressed in terms of the tunneling charge current as

\begin{equation} \tilde {J}_{s}\bm {n} = -\frac {\mu _B}{e}\frac {\bm {j_c}\cdot \bm {n}}{1+P_{RL}P_{FL}(\bm {m_{1}}\cdot \bm {m_{2}})}\left [ a_{1}P_{1}\bm {m_{1} } + a_{2}P_{2}\bm {m_{2}} +\frac {1}{2}(P_{1}P^\eta _{1} - P_{2}P^\eta _{2})(\bm {m_{1}}\times \bm {m_{2}}) \right ], \end{equation}

where \(a_i\) describe the influence of the spin-mixing conductance and \(P_i^\eta \) are out-of-plane polarization parameters for interface \(i\).