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Computation of Torques
in Magnetic Tunnel Junctions

Chapter 9 Summary and Outlook

MRAM devices have gained growing interest due to their nonvolatile nature, high speed, excellent endurance and compatibility with CMOS technology. Spin-transfer torque MRAM shows promise for IoT and automotive applications, and as a replacement for flash memory in embedded DRAM and last level caches.

Development of accurate simulation tools is a valuable help for improving design and performance of emerging STT-MRAM devices. This thesis was thus devoted to the development and study of different approaches to the torque acting in STT-MRAM cells and entering the LLG equation describing the magnetization dynamics.

First, a finite difference implementation of the LLG equation, with the STT torque computed by employing the simplified Slonczewski expression under the assumption of a uniform and constant current density, was described. By deriving an analytical solution for the current density flowing in an MTJ, it was shown that it can be highly nonuniform during switching. The FD solver was thus extended to address this behavior, by including two more realistic approaches that compute the torque term with a fixed voltage and a fixed total current.

The FD solver was employed to compare switching results obtained with the fixed current density, fixed total current and fixed voltage approaches. It was shown that a correction to the value of the fixed current applied in the first two approaches is required to match the switching time distribution of the fixed voltage one. The dependence of the current correction on the TMR, temperature, and structure diameter was investigated, showing how the correction increases with all three parameters. By performing macrospin simulations, it was observed that all the results can be explained by a dependence of the correction on the switching time, with a shorter switching time requiring a larger correction.

Understanding the behavior of the current correction allows for the simple constant current density approach, most commonly employed, to correctly reproduce the switching time distribution. This can lead to the inclusion of the current correction in a compact model, and will allow the development of fast simulation tools for switching realizations in a single free layer.

Recently proposed STT-MRAM devices are often composed of multiple ferromagnetic, nonmagnetic and tunneling layers. Computing the spin accumulation in the whole MRAM stack gives the possibility of deriving all torques contributions from a unified expression. This can be achieved by employing the spin and charge drift-diffusion formalism. The drift-diffusion equations were implemented in an open-source finite element software, and coupled to the LLG equation. The FE method was chosen as it can easily address the more complex structure of modern MRAM devices.

The FE implementation of the drift-diffusion formalism was extended to include the transport properties of MTJs. The TMR effect, and the current density redistribution in the presence of nonuniform magnetization, were successfully reproduced by modeling the tunnel barrier as a poor conductor, with conductivity locally depending on the relative magnetization orientation in the ferromagnetic layers.

The FE solution obtained for the spin accumulation in spin-valves was tested against literature results, showing a very good agreement. The dependence of the torque on several system parameters was investigated, and it was shown that a proper set of effective parameters can be employed to match the spin torque magnitude expected in MTJs. A unique set of parameter, however, does not allow to properly reproduce all the torque properties. The FE implementation was thus further improved with the introduction of appropriate boundary conditions at the tunnel barrier interfaces, to account for the tunneling spin current polarization. It was shown how the solver was then able to reproduce both the angular and voltage dependence observed in MTJs.

The solver was further updated with the possibility of computing an iterative solution of the charge and spin accumulation equations, and it was shown how this procedure allows to account for the GMR effect in a spin-valve, while it is not necessary in structures containing only MTJs.

Furthermore, the presented approach was employed to compute the torque acting in recently proposed ultra-scaled devices, with elongated ferromagnetic layers. It was shown how the torque generated by the MTJ and the one coming from magnetization textures are not independent, so that the computation of the spin accumulation is necessary in order to account for the interplay.

The solver was applied to compute switching simulations of recently proposed ultra-scaled MRAM cells with elongated layers. It was shown how the switching performance is improved by using a free layer composed of two segments, instead of a single one, as the different parts are able to switch one at a time. Investigating the switching of a structure with a free layer composed of three segments revealed an even faster switching process, and suggested the possibility of employing these structures as multi-bit memory cells. Overall, the obtained results validate the use of the proposed approach to help in the design of advanced MRAM devices.

Finally, by deriving a solution for the spin accumulation in the presence of ballistic corrections to the spin current, based on analytical expressions, a more complex oscillatory behavior of the torque, predicted theoretically, was reproduced.

The possibility to apply the developed approach to structures with an arbitrary number of ferromagnetic layers, tunnel barriers or nonmagnetic spacers of different shapes will enable its application in predicting the switching behavior of realistic MRAM stacks consisting of several layers of different materials. Moreover, the presented solver can be extended with additional terms in the spin current expression to account for spin-orbit coupling effects, opening the possibility of applying it to MRAM cells based on spin-orbit torques.