Abstract
We propose a phase prediction method for the pattern formation in the uniaxial two-dimensional kinetic Ising model with the dipole-dipole interactions under the time-dependent Ginzburg-Landau dynamics. Taking the effects of the material thickness into account by assuming the uniformness along the magnetization axis, the model corresponds to thin magnetic materials with long-range repulsive interactions. We propose a new theoretical basis to understand the effects of the material parameters on the formation of the magnetic domain patterns in terms of the equation of balance governing the balance between the linear- and nonlinear forces in the equilibrium state. Based on this theoretical basis, we propose a new method to predict the phase in the equilibrium state reached after the time-evolution under the dynamics with a given set of parameters, by approximating the third-order term using the restricted phase-space approximation [R. Anzaki, K. Fukushima, Y. Hidaka, and T. Oka, Ann. Phys. 353, 107 (2015)] for the $\phi^4$-models. Although the proposed method does not have the perfect concordance with the actual numerical results, it has no arbitrary parameters and functions to tune the prediction. In other words, it is a method with no a priori assumptions on domain patterns.
Highlights
Magnetic materials were of great interest even before the beginning of the application of quantum physics in solid-state physics [1]
Jagla [6] and Kudo and Nakamura [7] performed numerical simulations using similar models to reproduce the magnetic domain patterns on two-dimensional magnetic materials. The latter proposed a relation between the sweep rate of the external magnetic field and the final magnetic domain patterns in the equilibrium state. They utilized a two-dimensional kinetic Ising spin system with spins on the square lattice lying on the xy plane, while magnetization was restricted in the z direction, which is normal to the xy plane
One of the most convenient choices is to eliminate the dimensionful saturation magnetization ρ. In this case, using the time variable τ, spatial coordinate ζ, Laplacian ∂2, and magnetization φ, we find that the normalized time-dependent GinzburgLandau (TDGL) equation for the dynamics under the external magnetic field swept from B0 > 0 to zero at a (a) 50 y
Summary
Magnetic materials were of great interest even before the beginning of the application of quantum physics in solid-state physics [1]. Jagla [6] and Kudo and Nakamura [7] performed numerical simulations using similar models to reproduce the magnetic domain patterns on two-dimensional magnetic materials The latter proposed a relation between the sweep rate of the external magnetic field and the final magnetic domain patterns in the equilibrium state. Garel and Doniach [23] analyzed the behavior of a similar system under a finite temperature T and external magnetic field H thermodynamically and plotted the T -H phase diagram with three phases labeled uniform, bubble, and striped These phases were defined using simple analytic functions with few parameters. We use a numerical method to construct an effective two-dimensional Green’s function by analytically averaging the dipole-dipole interactions along the z direction for each grid point on the xy plane, as proposed by [8], enabling us to consider the effects of the thickness more precisely
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