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Designing advanced single-digit shape-anisotropy MRAM cells requires an accurate evaluation of spin currents and torques in magnetic tunnel junctions (MTJs) with elongated free and reference layers. For this purpose, we extended the analysis approach successfully used in nanoscale metallic spin valves to MTJs by introducing proper boundary conditions for the spin currents at the tunnel barrier interfaces, and by employing a conductivity locally dependent on the angle between the magnetization vectors for the charge current. The experimentally measured voltage and angle dependencies of the torques acting on the free layer are thereby accurately reproduced. The switching behavior of ultra-scaled MRAM cells is in agreement with recent experiments on shape-anisotropy MTJs. Using our extended approach is absolutely essential to accurately capture the interplay of the Slonczewski and Zhang-Li torque contributions acting on a textured magnetization in composite free layers with the inclusion of several MgO barriers.

The ever-improving semiconductor industry has relied, in recent years, on the down-scaling of its components. The presence of leakage currents has, however, caused an increase of the stand-by power consumption in traditional volatile memories like SRAM and DRAM1. Nonvolatile components would allow to avoid any stand-by power usage. Emerging nonvolatile spin-transfer torque (STT) magnetoresistive random access memory (MRAM) offers high speed and endurance and is attractive for stand-alone2, embedded automotive3, MCU, and IoT4 applications, as well as frame buffer memory5 and slow SRAM6. The core of an STT-MRAM cell consists of a magnetic tunnel junction (MTJ), cf. Fig. 1a, with two ferromagnetic layers separated by an oxide tunnel barrier (TB). The reference layer (RL) is fixed either by proper choice of materials or by antiferromagnetic pinning, while the magnetization of the free layer (FL) can be reversed. When the magnetization vectors in the two layers are parallel (P), the resistance is lower than in the anti-parallel state (AP), providing a way to store binary information. The percentage difference between the two resistance states is labeled tunneling magnetoresistance (TMR) ratio. In STT-MRAM, switching between the two stable configurations is achieved by running an electric current through the structure. The spin-polarization of the RL generates a spin current which, when entering the free layer, acts on the magnetization via the exchange interaction. When the magnetization vectors are not aligned, conservation of angular momentum causes the transverse spin current to be quickly absorbed, generating the spin-transfer torque7,8. Employing CoFeB for the ferromagnetic layers and MgO for the oxide layers allows to reach TMR values of up to 600%9. CoFeB and MgO also possess suitable properties for the fabrication of MTJs with perpendicular magnetic anisotropy (PMA), which present better thermal stability, better scalability, and a lower switching current10. In order to increase the interface PMA, provided by the MgO tunneling layer, the FL is often interfaced with a second MgO layer11. Recently, more advanced structures were proposed to boost the PMA even further, either by introducing more MgO layers in the FL or using the shape anisotropy of elongated FLs12, while also improving scalability thanks to a reduced diameter. Accurate simulation tools can provide valuable support in the design of these ultra-scaled MRAM cells, cf Fig. 1b. In order to model such devices, it is paramount to generalize the traditional Slonczewski13 approach for the torque computation, applicable only to thin FLs, to incorporate normal metal buffers or MgO barriers between multiple CoFeB free layers, as well as the barrier between the RL and FL, and the torques coming from magnetization textures or domain walls, which can be generated in elongated FLs. In this work, we present an extension of the drift-diffusion formalism for the computation of the torque in the presence of MTJs in the structure. The model is implemented in a finite element (FE) solver based on open-source software. We show how the proposed approach is able to reproduce the expected properties of the STT torque observed in MTJs. Moreover, we show how the STT contribution and the one coming from magnetization gradients in the bulk of the magnetic layers are non-additive, so that a unified treatment of the two contributions is necessary in order to describe the torque acting in the ultra-scaled MRAM devices. Finally, we present switching simulations carried out with the described approach. The parameters employed for all the simulations, unless specified differently in the text, are summarized in the supplementary material available online, together with the weak formulation employed by the FE solver.

(a) MTJ structure with non-uniform magnetization configuration. The structure is composed of a reference layer (dark red), a tunnel barrier (green), a free layer (yellow), and two non-magnetic contacts (light blue). The arrows represent the magnetization orientation. (b) Model examples of elongated ultra-scaled MRAM cells, with single (top) or composite (middle and bottom) free layer.

\(\textbf{m}\) is a unit vector pointing in the magnetization direction, \(\gamma \) is the gyromagnetic ratio, \(\mu _0\) is the magnetic permeability, \(\alpha \) is the Gilbert damping constant, \(M_S\) is the saturation magnetization, \(\mathbf {H_{eff}}\) is an effective field containing the contribution of external field, exchange interaction, and demagnetizing field, and \(\mathbf {T_S}\) is the STT term. We implemented the equation in a Finite Element (FE) solver based on the Open Source library MFEM14. The contribution of the demagnetizing field is evaluated only on the disconnected magnetic domain by using a hybrid approach combining the boundary element method and the FE method15. A complete description of the torque term, which allows to include all physical phenomena responsible for proper ultra-scaled MRAM operation, can be obtained by computing the non-equilibrium spin accumulation. For this purpose, the drift-diffusion (DD) formalism has already been successfully applied in a spin-valve structure with a non-magnetic spacer layer16,17,18. The drift-diffusion equations for charge and spin current density are19:

While the proposed approach is able to compute both the TMR and torque in an MTJ, in ultra scaled devices non-magnetic spacer layers can also be used to split the FL into two parts and avoid the formation of magnetization textures or domain walls. In a spin-valve with a metallic spacer layer, it is the Giant-Magnetoresistance (GMR) effect which causes the resistance of the structure to depend on the relative angle between the magnetization vectors. Such an effect can be accounted for by taking the magnetization-dependent contribution in (2a) into account when computing the current density. By taking \(\nabla \cdot \mathbf {J_C}=0\) (in the absence of current sources) and \(\textbf{E}=-\nabla V\) in (2a), one obtains the equation for the electrical potential.

(a) Torques computed for an MRAM cell with elongated RL and FL and a magnetization profile in the FL similar to the one of Fig. 8a. The brown vectors indicate the magnetization direction in the RL and in two parts of the FL. (b) Close-up of the spin torque \(\mathbf {T_S}\) compared to the Zhang-Li torque \(\mathbf {T_{ZL}}\). The presence of the MTJ influences also the bulk portion of the torque, making the unified approach the most suitable for dealing with ultra-scaled MTJs with elongated ferromagnetic layers.

The torque \(\mathbf {T_S}\) acting in the FL for this magnetization profile is shown in Fig. 10a. Both the interface contribution from the TB and the bulk ZL contribution are present. In Fig. 10b we show a close-up of the bulk portion of \(\mathbf {T_S}\), compared with the ZL torque \(\mathbf {T_{ZL}}\) computed in the FL for the same magnetization configuration. The comparison reveals a substantial difference between the torques obtained with our model and the traditional approach, where the ZL torque is simply added to the Slonczewski term, even in the presence of a short spin dephasing length. Our approach clearly demonstrates that, in an MTJ with elongated ferromagnetic layers, the Slonczewski and ZL torques are not independent: the presence of the TB also generates a spin accumulation component parallel to the magnetization, whose decay is dictated by \(\lambda _{sf}\), cf. Fig. 4a. This component interacts with the magnetization texture, modifying the ZL torque contribution. A unified treatment of the MTJ polarization process and FL magnetization texture is thus required to accurately describe the torque and switching in ultra-scaled MRAM.

Switching stages of an ultra-scaled STT-MRAM cell with composite free layer, showcasing how the different parts of the FL switch one at a time. The RL is the first section on the left of the structure, while the second and third sections are the two parts of the FL (from left to right, FL1 and FL2, respectively). AP to P switching is presented in (a), while P to AP switching in (b).

We presented a modeling approach to accurately describe the charge and spin currents, the torques, and the magnetization dynamics in ultra-scaled MRAM cells consisting of several elongated pieces of ferromagnets separated by multiple tunnel barriers. We showed how the fully 3D spin and charge drift-diffusion equations can be supplied with appropriate conditions at the tunneling layer to reproduce the TMR effect as well as the angular and voltage dependence of the torque expected in MTJs. We reported how an iterative solution of the charge and spin accumulation equations can be employed to account for the GMR effect. The advantage of the proposed approach is the possibility of computing all the torque contributions from a unified expression, so that the interactions between them can be evaluated, and the torque acting in the presence of multiple layers of varying thickness is automatically accounted for, even for non-uniform magnetization distributions. We demonstrated that the Slonczewski and Zhang and Li torques are not additive and must be derived from the spin accumulation to account for their interplay and correctly describe the torques on textured magnetization in elongated FLs with several MgO TBs. Finally, we applied the presented method to switching simulations of MRAM cells with elongated and composite FLs. The obtained results validate the use of the proposed simulation approach as support for the design of advanced ultra-scaled MRAM cells. 17dc91bb1f