Numerical Coupling of Landau-Lifshitz-Gilbert Equation with Maxwell's Equations to Simulate Electromagnetic Wave Interaction with Ferromagnetic Materials

The Landau-Lifshitz-Gilbert (LLG) theory, based on the microscopic quantum theoretical description of spin dynamics, describes magnetic properties on an atomic level as \begin{equation*}\frac{\partial M}{\partial t}-\frac{Cy}{\vert M\vert }M\times\frac{\partial M}{\partial t}+\mu_{0}\gamma M\times H=0\tag{1a}\end{equation*} where
$\alpha$
is the damping term,
$\gamma$
is the gyromagnetic ratio,
$H$
is the effective magnetic field and
$M$
is the magnetization field. The fundamental property of ferromagnetic materials is their constitutive nonlinear behaviour, where the permeability/susceptibility of the material is a function of the magnetic field intensity. This nonlinear magnetic constitutive relation yields complicated and nonlinear electromagnetic problems to solve in practical applications. To provide a deeper insight into these physical and engineering problems that includes ferromagnetic materials, the numerical solver would become an important analysis tool. In this work, we study a numerical model (Maxwell-LLG model) to bridge two theories for simulating the ferromagnetic materials in time domain. Our main objective is to treat the cooperative phenomena such as magnetization processes of the ferromagnetic material and then, provide a solver for interaction dynamic electromagnetic waves with magnetic spin behaviours. To best of our knowledge, no existing method effec-tively addresses the dynamic magnetic behaviour within numerical schemes. The majority of the methods in the literature use Polder's permeability tensor, which is equivalent to the linearising (1) by saturating the material in one direction. In this study, we firstly focus on a first-order numerical scheme for the solution of Maxwell-LLG model in the finite-difference time-domain (FDTD) method. In the conference, we will present the simulation results and discuss the advantages and disadvantages of this approach. The central focus will be on predicting the dynamic magnetic behaviour of the material and investigating the stability of the proposed scheme.