

BiographyMohamed Mohamedou was born on October 1991 in Akjoujt, Mauritania. He graduated from ParisSaclay University in 2016 where he obtained a Master’s Degree in Numerical Analysis/Applied Mathematics. Afterwards, he worked on numerical modeling for different industrial and academic projects spanning electromagnetism and mechanics engineering. In September 2019 Mohamed joined the Institute for Microelectronics where he started working towards his doctoral degree. His research is focused on developing high performance numerical simulationsbased micromagnetics approaches to investigating nonvolatile magnetic memory devices. 
Numerical Integration of the LandauLifshitzGilbert Equation with the Inclusion of SpinTransferTorque
The evolution of magnetization in a ferromagnetic material is usually described by the LandauLifshitzGilbert (LLG) equation. The timedependent magnetization dynamics under an effective field H_{eff} are modeled using the LLG equation. The effective field (H_{eff}) encapsulates the contributions of the external field (H_{Ext}), the anisotropy field (H_{ani}), the exchange field (H_{ex}), the demagnetizing field (H_{dem}) and the temperature field (H_{trm}). In order to have switching of the magnetization in the free layer, a torque must be exerted on the magnetization by a spinpolarized current. To model this switching phenomenon, an additional term, T_{s}(m), must be added to the LLG equation:
When doing simulations of spintransfertorque switching, the currentdependent torque can be approximated by a Slonczewskilike term that does not require the computation of spin accumulation. This does not allow for a precise torque computation, however, or for the computation of the torque in the reference layer, only in the free layer. To know the torque in all the magnetic layers in the structure, one needs to know the spin accumulation, and the equations that describe its behavior are the spindriftdiffusion equations. When dealing with a magnetic tunnel junction, we know that the macroscopic dependence of the conductivity is described by eq.(5). The idea is to have a charge current that redistributes according to the local relative orientation of the magnetization in the two ferromagnetic layers.
To numerically solve this firstorderintime LLG equation without torque, a common schema based on a finite difference method is often used in the literature. The time integration of this equation, eq.(1), presents two main challenges: the strong nonlinearity and the nonconvex constraint m^{2} =1. To treat the nonlinearity with a finite difference method is computationally expensive as adaptation of such an approach to complex structures is not always a straightforward task.
To tackle these difficulties, we use a semiimplicit method based on an equivalent reformulation of the model in the tangent space, Ͳ_{m}, discretized by finite elements and require only the solution of one linear system per timestep. The nonconvex constraint is satisfied at the discrete level by applying a mapping projection to the computed solution at any timestep.
Fig. 1: Structure of the mesh simulation.
Fig. 2: Simulation results with: α = 0.02, γ = 2.213× 10^{5}
m/A.s, M_{s} = 8× 10^{5}
A/m, A_{ex}= 1.3× 10^{11}
J/m, H_{Ext} = (0,0,10^{4}) A/m, S = (0,0,0) A/m.