In the presence of elongated FLs, simultaneous switching of the entire layer is not guaranteed: domain walls or magnetization textures may form, and their propagation through the FL can affect the switching behavior. In such
cases, the additional spin torques arising from magnetization gradients in the bulk of the ferromagnetic layers must be taken into account. These torques are modeled by the Zhang and Li (ZL) equation [184]:
where \(\epsilon = (\lambda _J/\lambda _\mathrm {sf})^2\). This expression is derived from Equation (6.10) by
setting \(\nabla \mathbf {S} = \mathbf {0}\) and taking the cross product with \(\mathbf {m}\) twice. This approximation is strictly valid only when the changes in \(\mathbf {S}\) are relatively slow, namely, when they
occur over length scales longer than \(\lambda _\mathrm {sf}\).
Figure 6.5: (a) Non-uniform magnetization texture with the magnetization orientation changing from \(x\) to \(-z\). (b) Comparison of the spin torque \(\mathbf {T}_\mathrm {S}\) to the ZL torque \(\mathbf {T}_\mathrm
{ZL}\) for an approximately 100 nm long magnetization texture, with the parameters of Table 6.1. (c) Comparison for an
approximately 3 nm long magnetization texture with \(\lambda _\varphi = 0.4\) nm. The shorter dephasing length quickly dissipates
the transverse spin-accumulation, ensuring agreement between the ZL approximation and the drift-diffusion solution. The two approaches show good agreement in both cases.
By applying the same procedure to Equation (6.19), the ZL expression can be generalized to include \(\lambda _\varphi \):
where \(\epsilon ' = (\lambda _J/\lambda _\varphi )^2\) accounts for the additional contributions of the term involving \(\lambda _\varphi \).
6.7.1 Comparison with Drift-Diffusion Solution
To test the ZL approximation, a magnetization profile with the magnetization orientation changing from \(+z\) to \(-x\) (shown in Figure 6.5 (a)) serves
to compute both \(\mathbf {T}_\mathrm {S}\) and \(\mathbf {T}_\mathrm {ZL}\), with the parameters of Table 6.1. The structure used in the simulations consists
of a single ferromagnetic layer 450 \(\si {\nano \meter }\) thick. A constant current density value of \(J_{\mathrm {C},x} = -1.33\times 10^{12}\) \(\si {\ampere \per \meter \squared }\), flowing in
the \(x\)-direction, is employed for both approaches.
Figure 6.5 (b) demonstrates that, for the given magnetization profile with a width of approximately 100 \(\si {\nano \meter }\), \(\mathbf
{T}_\mathrm {S}\) is well reproduced with Equation (6.35). Figure 6.5 (c) reports the torques obtained using the same parameters and a shorter magnetization texture of approximately 3 \(\si {\nano \meter }\), in an
FM layer of 70 \(\si {\nano \meter }\) thickness. With the standard dephasing length, the spin-accumulation gradients neglected in Equation (6.35) become significant, leading to a large deviation of \(\mathbf {T}_\mathrm {S}\) from \(\mathbf {T}_\mathrm {ZL}\), particularly for the field-like torque
(\(y\)-component). Employing a shorter spin dephasing length of \(\lambda _\varphi = 0.4\) \(\si {\nano \meter }\) ensures the rapid absorption of transverse spin-accumulation components, restoring good
agreement between the \(\mathbf {T}_\mathrm {ZL}\) approximation and the drift-diffusion solution.
6.7.2 Interaction Between Tunneling and Bulk Torques
In MRAM cells with elongated FLs, magnetization textures and domain walls can form along the layer during switching. In this case, the polarized tunneling spin-current can interact with such magnetization textures, generating
torques that further deviate from the ZL approximation.
To verify this, the torque is computed in an experimental MTJ structure with a 5 \(\si {\nano \meter }\) RL, 0.9 \(\si {\nano \meter }\) TB, and an elongated FL of 15 \(\si {\nano \meter }\)
with a magnetization profile similar to that shown in Figure 6.5 (a), with the magnetization vector transitioning from the \(z\)-direction to the
\(-x\)-direction over the length of the layer. The magnetization in the RL is pointing toward the \(x\)-direction and 50 \(\si {\nano \meter }\) NM contacts are included.
The torque acting in the FL for this magnetization profile is shown in Figure 6.6 (a). Near the TB the tunneling spin-current generates an observable
Slonczewski torque contribution, while the magnetization texture causes the torque contribution in the bulk. Figure 6.6 (b) shows a close-up of the bulk
portion of \(\mathbf {T}_\mathrm {S}\), compared with the result obtained by applying the ZL expression (6.36) to the
magnetization configuration of the FL, for the exact value of the current density. The comparison reveals a substantial difference between the torques obtained using the presented model and those obtained with the ZL
approximation. This implies that the Slonczewski and ZL torque contributions cannot simply be added together, but influence and interact with each other.
Figure 6.6: (a) Torque computed for an MRAM cell with elongated RL and FL and a magnetization profile in the FL similar to that of Figure 6.5 (a),
with a width of approximately 3 \(\si {\nano \meter }\). The brown vectors indicate the magnetization direction in the RL and in two parts of the FL. (b) Close-up of the spin torque \(\mathbf {T}_\mathrm {S}\)
compared to the ZL torque \(\mathbf {T}_\mathrm {ZL}\). The presence of the MTJ also influences the bulk torque component, making the unified approach the most suitable for ultra-scaled MTJs with elongated ferromagnetic
layers. Figure (a) originally published in [182].
