Ex. 3: Current-induced dynamics

When an electric current flows through a ferromagnetic material, it acquires a spin polarization through which the conduction electrons interact with the magnetization. This effect is described by the so-called spin-transfer torque (STT). In this example, we simulate the STT-induced magnetization dynamics in a square Permalloy platelet. The micromagnetic problem simulated here is based on a proposal by Najafi et al., published in Ref. [1].

The sample is a Permalloy platelet of 100 nm \(\times\) 100 nm \(\times\) 10 nm size. We use FreeCAD to define the geometry and to generate an irregular FEM mesh with the netgen plugin. In our example, the mesh size is set to 2 nm, resulting in 39,246 irregularly shaped terahedral finite elements.

Calculating the initial configuration

First we simulate the initial configuration – a vortex structure at zero external field and without spin-polarized current. The keyword intial state = vortex_xy can be used to generate a vortex-type initial structure, circulating in the \(xy\) plane, from which the magnetization can be relaxed. The simulation.cfg for this static part of the simulation is

name = sp_stt
scale = 1.e-9
mesh type = vtk
alpha = 1
initial state = vortex_xy
time step = 2 # demag refresh interval in ps
duration = 5000  # simulation time in ps
solver type = gpu
torque limit = 1.e-4
remove precession = yes

By performing the static simulation, we quickly obtain the converged state, stored in the file sp_stt.vtu, which we rename to sp_stt_vortex.vtu. This will be the initial state of the subsequent dynamic simulation wit STT effects.

../_images/static_vortex.png

Spin-transfer-torque–driven magnetization dynamics

According to the problem specification, we apply a spatially homogenous dc current density \(j = 10^{12}\) A/m\(^2\) flowing along the positive \(x\) direction. The damping constant is set to \(\alpha=0.1\) and the non-adiabaticity parameter to \(\xi=0.05\). The degree of spin polarization \(P\) is not specified in the article by Nafaji et al. For the lack of a better value, we set \(P=100\)%.

The above-mentioned parameters are specified in the simulation.cfg file:

name = sp_stt
scale = 1.e-9
mesh type = vtk
alpha = 0.1
initial state = fromfile_sp_stt_vortex.vtu
time step = 0.1
duration = 14000
solver type = gpu

current type = dc
     spin polarization = 1.0 # degree of spin polarization in stt dynamics
     current density =  1.0 # current density for STT dynamics [in 10^12 A/m^2]
     beta = 0.05 # non-adiabaticity parameter in STT term (also known as xi)
     current theta = 90 # polar angle of electric current flow [in degrees]
     current phi = 0 # azimuthal angle of electric current flow [in degrees]

The dynamic simulation with STT yields a displacement of the vortex along a spiralling orbit, with a radius that decreases in time. The oscillatory convergence towards a new equilibrium state can be recognized in the plots of the spatially averaged \(x\) and \(y\) components of the magnetization as a function of time:

../_images/mxmy_vs_t.png

These results are in good agreement with the data reported in Fig. 6 of the article by Najafi et al. [1]. Minor deviations can be attributed to the relatively coarse mesh used in this simulation, and to the fact that tetmag uses the electron gyromagnetic ratio \(\gamma=2.21276148\times 10^5\) rad m/ As, whereas the problem specification indicates a slightly different value of \(2.211\times 10⁵\) rad m / As. Other data that the authors of Ref. [1] suggest for comparison are the average \(x\) and \(y\) components of the magnetization of the equilibrium state reached after 14 ns. Our simulation yields \(\langle M_x\rangle = -1.7009\times 10^5\) A/m and \(\langle M_y\rangle = 1.566 \times 10^4\) A/m, which compares rather well with the data reported in Table 1 of Ref. [1 ]. Here, too, we expect to obtain better agreement when using smaller cell sizes.