Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM
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6.3 Tunneling Spin-Current Boundary Conditions
The drift-diffusion approach accounts only for semiclassical transport properties and must be adapted to describe tunneling across an MTJ. The Nonequilibrium Green Functions (NEGF) formalism is frequently employed to
compute expressions for the charge and spin-current flowing through the TB of an MTJ. However, performing such calculations at every iteration of the FE solver would be computationally prohibitive. Using a circuit-theory
approach and analytical solutions of the NEGFs equations, the expressions for the tunneling spin and charge-currents can be simplified for practically thick TBs to capture the most prominent transport characteristics by using only
a few key parameters.
6.3.1 Tunneling Current Expressions
With the magnetization in the RL pointing in the \(x\)-direction and the one in the FL lying in the \(xz\)-plane at an angle \(\theta \) from the RL, the tunneling charge- and spin-currents can be expressed as [118]:
\(\seteqnumber{0}{6.}{22}\)
\begin{align}
J_\mathrm {C}^\mathrm {TB} &= J_0(V)\left (1 + P_\mathrm {RL}P_\mathrm {FL}\cos \theta \right ), \label {eq:tunnel-charge}\\ J_{\mathrm {S},x}^\mathrm {TB} &= -\frac {a_\mathrm
{RL}P_\mathrm {RL} + a_\mathrm {FL}P_\mathrm {FL}\cos \theta }{1 + P_\mathrm {RL}P_\mathrm {FL}\cos \theta }\frac {\hbar }{2e}J_\mathrm {C}^\mathrm {TB}, \label {eq:tunnel-spin-x}\\ J_{\mathrm
{S},y}^\mathrm {TB} &= -\frac {\frac {1}{2}\left (P_\mathrm {RL}P_\mathrm {RL}^\eta - P_\mathrm {FL}P_\mathrm {FL}^\eta \right )\sin \theta }{1 + P_\mathrm {RL}P_\mathrm {FL}\cos \theta
}\frac {\hbar }{2e}J_\mathrm {C}^\mathrm {TB}, \label {eq:tunnel-spin-y}\\ J_{\mathrm {S},z}^\mathrm {TB} &= -\frac {a_\mathrm {FL}P_\mathrm {FL}\sin \theta }{1 + P_\mathrm {RL}P_\mathrm
{FL}\cos \theta }\frac {\hbar }{2e}J_\mathrm {C}^\mathrm {TB}, \label {eq:tunnel-spin-z}
\end{align}
where \(J_0(V)\) contains the voltage-dependent portion of the current density, \(P_\mathrm {RL}\) and \(P_\mathrm {FL}\) are the in-plane Slonczewski polarization parameters, \(P_\mathrm {RL}^\eta \) and \(P_\mathrm
{FL}^\eta \) are out-of-plane polarization parameters, and \(a_\mathrm {RL}\) and \(a_\mathrm {FL}\) describe the influence of the interface spin-mixing conductance on the transmitted in-plane spin-current. The spin-current
is expressed in units of \(\si {\joule \per \meter \squared }\).
Equation (6.23), describing the dependence of the charge-current on the RL and FL polarization parameters and on the relative angle
between their magnetization vectors, is already included in the model through Equation (6.22). The spin-current component must also
be accounted for. The FE approach to the drift-diffusion equations, based on the Galerkin method, enforces the continuity of the spin-accumulation and current across all interfaces. The tunneling expressions must therefore be
included while preserving the continuous nature of both quantities.
6.3.2 Finite Element Boundary Terms
To obtain the solution through the FE solver, the following boundary terms must be added on the RHS of the weak formulation (5.25):
\(\seteqnumber{0}{6.}{26}\)
\begin{align}
&-\int _{\mathrm {RL|TB}} \mathbf {J}_\mathrm {S}^\mathrm {TB} \cdot \mathbf {w} \, d\mathbf {x} + \int _{\mathrm {TB|FL}} \mathbf {J}_\mathrm {S}^\mathrm {TB} \cdot \mathbf {w} \,
d\mathbf {x}, \label {eq:fe-boundary-integral}\\ \mathbf {J}_\mathrm {S}^\mathrm {TB} &= -\frac {\mu _B}{e}\frac {\mathbf {J}_\mathrm {C}\cdot \mathbf {n}}{1 + P_\mathrm {RL}P_\mathrm
{FL}\mathbf {m}_\mathrm {RL}\cdot \mathbf {m}_\mathrm {FL}}\Big [a_\mathrm {RL}P_\mathrm {RL}\mathbf {m}_\mathrm {RL} + a_\mathrm {FL}P_\mathrm {FL}\mathbf {m}_\mathrm {FL} \nonumber \\
&\quad + \frac {1}{2}\left (P_\mathrm {RL}P_\mathrm {RL}^\eta - P_\mathrm {FL}P_\mathrm {FL}^\eta \right )\mathbf {m}_\mathrm {RL}\times \mathbf {m}_\mathrm {FL}\Big ], \label
{eq:fe-spin-current-bc}
\end{align}
where RL|TB and TB|FL are the interfaces between the TB and the RL and FL, respectively, \(\mathbf {n}\) is the interface normal (which for the employed meshes points in the positive \(x\)-direction for both interfaces), and
\(\mathbf {m}_\mathrm {RL(FL)}\) is the unit magnetization vector of the RL (FL) at the interface. This formulation is consistent with physics-based factorization approaches for MTJ modeling [183]. The diffusion
coefficient of the TB \(D_\mathrm {S}\) is also taken to be low, proportional to the conductivity from Equation (6.22), as no diffusive
transport occurs across the barrier.
By solving the weak formulation with the inclusion of Equation (6.27) and the low value of \(D_\mathrm {S}\), the spin-current
in the TB is fixed to the value prescribed by Equation (6.28) when \(\mathbf {J}_\mathrm {C}\) flows through the MTJ. This is
crucial for accurately describing the spin-currents and the spin-accumulations in the RL and FL of an MTJ.
6.3.3 Implementation of Interface Mapping
The computation of the boundary term associated with Equation (6.27) requires knowledge of the magnetization vector on the
opposite interface. To access these values, the coefficient describing the boundary integral is initialized as follows:
-
(i) For each integration point on the RL|TB interface requiring the computation of the tunneling spin-current, the solver iterates through the integration points of the TB|FL interface.
-
(ii) The TB|FL point with coordinates closest to the current RL|TB one is selected.
-
(iii) The integration point index and element index associated with the identified TB|FL point are mapped to the coordinates of the RL|TB point.
-
(iv) The mapping procedure is repeated for the TB|FL interface.
In a transient simulation, the search is performed only during solver initialization. At every time step, the data necessary for the computation of Equation (6.28) can be accessed through the generated maps, without the need to repeat the search procedure.