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Numerical Analysis and Innovative Simulation
Techniques for Designing Advanced MRAM

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6.5 Validation Results

6.5.1 Spin-Accumulation and Spin Torque Solutions

The spin-accumulation and torque obtained from the parameters of Table 6.1 are presented in Figure 6.2 (a) and (b). The structure employed consists of semi-infinite ferromagnetic leads and a TB of 1 \(\si {\nano \meter }\) thickness. The magnetization of the RL points in the \(x\)-direction, while that of the FL points in the \(z\)-direction. A comparison between the FE solution and the analytical solutions demonstrates that the additional boundary terms in the FE implementation yield perfect agreement.

The inclusion of the tunneling spin-current terms creates a discontinuity jump between the values of the spin-accumulation components parallel to the magnetization at the left and right interfaces of the TB. This is a manifestation of the MTJ polarization effects on the spin-current.

Figure 6.2 (c) shows the angular dependence of the damping-like torque with the inclusion of the spin-current boundary conditions, computed in the same structure. The typical sinusoidal dependence of the torque acting on the FL in an MTJ is reproduced exactly, for various values of the RL|TB interface spin polarization. The structure is biased by a fixed voltage of \(V = 1.3\) \(\si {\volt }\), so that the torque is independent of the TB|FL polarization, and linearly depends only on the RL|TB polarization.

6.5.2 Effect of Dephasing Length on Torque Localization

While employing Equation (6.28) fixes the spin current in the TB to the value expected in MTJs, the length parameters entering the spin-accumulation equation still determine the region over which the transverse spin-accumulation components are absorbed. They also determine the behavior of the torque in the bulk of the FM layers.

(image)

Figure 6.2: Validation of the spin drift-diffusion model with tunneling spin-current boundary conditions for semi-infinite ferromagnetic layers. (a) spin-accumulation and (b) spin-transfer torque components computed along the stack direction, comparing the analytical solution with the FE implementation. The magnetization is oriented along \(x\) in the RL and along \(z\) in the FL. (c) Angular dependence of the damping-like torque for several values of the RL spin polarization (\(P_\mathrm {RL} = 0.4\), \(0.7\), \(0.9\)), reproducing the expected sinusoidal behavior characteristic of MTJs. Figure (c) adapted from  [182].
Table 6.1: Parameters used in the drift-diffusion simulations with the TB boundary terms.
.
Parameter Value
Conductivity polarization, \(\beta _\sigma \) 0.7
Diffusion polarization, \(\beta _D\) 0.8
NM diffusion coefficient, \(D_{\mathrm {e},\mathrm {NM}}\) \(10^{-2}\,\si {\meter \squared \per \second }\)
FM diffusion coefficient, \(D_{\mathrm {e},\mathrm {FM}}\) \(2.0\times 10^{-3}\,\si {\meter \squared \per \second }\)
TB diffusion coefficient, \(D_\mathrm {S}\) \(2.0\times 10^{-8}\,\si {\meter \squared \per \second }\)
NM conductivity, \(\sigma _\mathrm {NM}\) \(5.0\times 10^{6}\,\si {\siemens \per \meter }\)
FM conductivity, \(\sigma _\mathrm {FM}\) \(1.0\times 10^{6}\,\si {\siemens \per \meter }\)
TB conductance, \(G_0\) \(4.76\times 10^{-5}\,\si {\siemens }\)
Polarization factors, \(P_\mathrm {RL} = P_\mathrm {FL}\) 0.707
In-plane torque reduction, \(a_\mathrm {RL} = a_\mathrm {FL}\) 1.0
Out-of-plane polarization, \(P_\mathrm {RL}^\eta = P_\mathrm {FL}^\eta \) 0.2
Spin-flip length, \(\lambda _\mathrm {sf}\) 10 \(\si {\nano \meter }\)
Spin exchange length, \(\lambda _J\) 1 \(\si {\nano \meter }\)
Spin dephasing length, \(\lambda _\varphi \) 0.4 \(\si {\nano \meter }\)

Figure 6.3 shows the spin-accumulation and torque obtained in a symmetrical MTJ structure including 50 \(\si {\nano \meter }\) thick non-magnetic (NM) layers, where the FM layers are 20 \(\si {\nano \meter }\) thick and the TB is 1 \(\si {\nano \meter }\) thick. For \(\lambda _\varphi = 5\) \(\si {\nano \meter }\), the torque components are not completely absorbed in the FL, contrary to what is usually expected in strong ferromagnets, and the spin-accumulation components transverse to the local magnetization penetrate inside the NM contacts. Setting an effective dephasing length of \(\lambda _\varphi = 0.4\) \(\si {\nano \meter }\), it guarantees a faster decay of the transverse spin-accumulation components, ensuring that the torque acts only in the proximity of the TB interface.