Numerical Analysis and Innovative Simulation
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6.6 GMR Effect in Spin-Valves
The approach detailed in the previous sections can compute both the TMR and the torques in an MTJ. Modern MRAM devices can, however, also include conductive spacer layers between the FM layers, either as part of the SAF
or to incorporate additional RLs and improve switching performance. In the presence of such a metallic spacer layer, the GMR effect becomes significant.
In the drift-diffusion formalism, the dependence of resistance on the relative angle between the magnetization vectors in the presence of a metallic spacer is accounted for by including the second RHS term of Equation (6.6 ) when computing the electrical potential \(V\). By imposing \(\nabla \cdot \mathbf {J}_\mathrm {C} = 0\) and expressing the electric
field as \(\mathbf {E} = -\nabla V\) in Equation (6.6 ) , the equation for \(V\) yields:
\(\seteqnumber{0}{6.}{31}\)
\begin{equation}
\nabla \cdot (\sigma \nabla V) = \nabla \cdot \left (\frac {e}{\mu _\mathrm {B}}\beta _D D_\mathrm {e}(\nabla \mathbf {S})^\mathrm {T}\mathbf {m}\right ). \label {eq:gmr-potential}
\end{equation}
Figure 6.3: Spin-accumulation and torque in a symmetric MTJ structure including NM contacts. (a) and (b) present results for \(\lambda _\varphi = 0.4\) \(\si {\nano \meter }\). The shorter dephasing length
guarantees faster decay of the transverse spin-accumulation components, so that the torque acts only in the proximity of the TB interface. Adapted from the figure [182].
In this formulation, the spin-accumulation and the electrical potential are coupled, yielding a coupled system of equations. To account for the interdependence of \(\mathbf {S}\) and \(V\), the FE solver is updated to iteratively
solve the system until a convergence threshold is reached:
\(\seteqnumber{0}{6.}{32}\)
\begin{equation}
\frac {\|\mathbf {S}_n\|_{L^2} - \|\mathbf {S}_{n-1}\|_{L^2}}{\|\mathbf {S}_n\|_{L^2}} < \epsilon , \label {eq:convergence}
\end{equation}
where \(\|\cdot \|_{L^2} = \sqrt {\int _\Omega |\cdot |^2 \, \mathrm {d}\mathbf {x}}\) denotes the \(L^2(\Omega )\) norm.
The iterative solution is computed by first obtaining an initial estimate for the spin-accumulation \(\mathbf {S}_0\) by solving the standard equations, including the RHS term of Equation (6.6 ) in the spin equation. This estimate is then used to compute a solution to Equation (6.32 ) , which in turn yields an updated spin-accumulation estimate \(\mathbf {S}_1\). These two steps are repeated until the solver converges. The dependence of the total
charge-current on the relative angle between the magnetization vectors in the FL and RL can be fitted using [117]:
\(\seteqnumber{0}{6.}{33}\)
\begin{equation}
I(\theta ) = \frac {V}{R_\mathrm {P}}\frac {1 + \chi \cos ^2(\theta /2)}{1 + \mathrm {GMR} + (\chi - \mathrm {GMR})\cos ^2(\theta /2)}, \label {eq:gmr-angular}
\end{equation}
where \(V\) is the applied bias voltage, \(R_\mathrm {P}\) is the resistance in the parallel state, and \(\chi \) and GMR are fitting parameters.
Figure 6.4 compares the angular dependence of the charge current for an MTJ with an MgO TB and a spin valve with a metallic spacer. Both structures employ
symmetric 2 nm FM layers separated by a 1 nm spacer,
with 50 nm NM contacts and a diameter of 40 nm . The
MTJ is biased at −1.3 V and simulated for two resistance configurations: \(R_\mathrm {P} = \SI {14}{\kilo \ohm }\), \(R_\mathrm {AP} =
\SI {42}{\kilo \ohm }\) (TMR \(= \SI {200}{\percent }\)) and \(R_\mathrm {P} = \SI {14}{\kilo \ohm }\), \(R_\mathrm {AP} = \SI {35}{\kilo \ohm }\) (TMR \(= \SI {150}{\percent }\)).
Both are well described by the Slonczewski conductance model (Equation (2.7 ) ). The spin valve is biased at −0.2 V , and the iterative coupling between the spin-accumulation and the electrical potential produces a GMR of approximately 11 % for a conductivity polarization \(\beta _\sigma = 0.7\) and 7.5 % for \(\beta _\sigma = 0.58\), both fitted by Equation (6.34 ) .
Since the voltage drop in an MTJ is localized at the TB, the iterative solution is not necessary for the correct computation of the current. In this case, the iterative solver always converges for \(n = 1\).
