Domain wall dynamics in cubic magnetostrictive materials subject to Rashba effect and nonlinear dissipation (Q2677632)

The authors analyze the behavior of a bilayer piezoelectric-magnetostrictive heterostructure built with a thin magnetostrictive layer which is perfectly bonded on the top surface of a thick piezoelectric layer. The spatiotemporal evolution of magnetization dynamics in the magnetostrictive layer is described by the extended Landau-Lifshitz-Gilbert equation: \(\frac{\partial \mathbf{m}}{\partial t}=\gamma (\mathbf{h}_{\mathrm{eff}}\times \mathbf{m})+\mathbf{t} _{\mathrm{diss}}+\mathbf{t}_{\mathrm{stt}}+\mathbf{t}_{ra}\), where \(\mathbf{m}:\mathbb{R} \times \mathbb{\ R}^{+}\rightarrow \mathbb{S}^{2}\) is the normalized magnetization vector defined as \(\mathbf{m}(x,t)=\mathbf{M}(x,t)/M_{s}\), \( \mathbf{M}(x,t)\), \(M_{s}\), and \(\mathbb{S}^{2}\) representing the magnetization vector, saturation magnetization, and unit sphere in \(\mathbb{R }^{3}\), respectively, \(x\) and \(t\) correspond to the space and time variables, \(\mathbf{h}_{\mathrm{eff}}\) is the total effective magnetic field, \( \mathbf{t}_{\mathrm{diss}}\), \(\mathbf{t}_{\mathrm{stt}}\) and \(\mathbf{t}_{\mathrm{ra}}\) correspond to the three torque contributions arising from dissipation, spin-transfer-torque and Rashba effect, respectively. The authors give the expressions of the different terms and write \(m=(\cos \phi \sin \theta ,\sin \phi \sin \theta ,\cos \theta )\), where \(\phi (x,t)\) and \(\theta (x,t)\) are the azimuthal and polar angles. They end with a system of partial differential equations and the main part of the paper describes a numerical investigation of this system. The authors start computing the traveling wave profile using the Runge-Kutta method of order 4. They especially analyze the impact of the Rashba field value.