Micromagnetics at Finite Temperature

Micromagnetics include two main parts: “magnetization curve theory” and “domain theory” [1]. In 1935, the Landau-Lifshitz equations were brought up to analyze the domain wall motion [2]. Since the early works by Brown Jr. in 1960s, micromagnetics had considerable progress in 1980s, including the utilization of the Landau-Lifshitz-Gilbert (LLG) equations to calculate the thin film's magnetization curve [3], and the establishment of the Finite-Difference-Method Fast-Fourier-Transform (FDM-FFT) micromagnetics to highly speed up the computation of the most time consuming magnetostatic interactions among total number $N$micromagnetic cells [4]. Recent developments of computational magnetism require the ability to calculate the magnetics at finite temperature [5], to calculate domains at large scales, i.e., three-dimensional micromagnetics at finite temperature are the target for the development of computational magnetics. In 2016, the author and collaborators developed a new micromagnetic method based on the Hybrid Monte Carlo (HMC) algorithm, which can calculate M-H loops and domains at finite temperature below Curie point [6], after two years' testing in explaining experiments, and in the characteristics of computational costs, it is found that the simulations using the HMC micromagnetics method can explain M-H loops and domains from 0K to the Curie temperature, even with superparamagnetic properties calculated. It costs roughly $\mathrm {O}( N)$computational time with a faster convergence time than the conventional Landau-Lifshitz equations. The Hybrid Monte Carlo method is a numerical simulation method used extensively in lattice field theory community which is capable of generating a Boltzmann-like distribution of the form $\mathrm {e}^{{- {S}} {[\Phi ]}}$, where $\Phi $denotes the collection of all degrees of freedom of the system and $S [ \Phi ]$is known as the Euclidean action (energy) of the system. The way to perform this simulation is to mimic what Nature does for the molecules in the air. For our purpose, the field variables $\Phi $are just all magnetization vectors $\{ M_{i}\}$for various micromagnetic cells and the action $S [ \Phi ]$is replaced by $F[ \{ M_{i}\}] {(!{-}!)}$the Landau magnetic free energy divided by $\mathrm {k}_{B}T$. In the micromagnetic simulation, the ferromagnetic material is discretized into a regular mesh, similar as the FDM-FFT method. The Hamiltonian for the HMC algorithm is given by Eq. (2) in Ref. [6], where $V_{c}$is the volume of a micromagnetic cell, $M_{i}$is the magnetization vector in the $i$th cell and $\Pi _{i}$is the corresponding conjugate momentum, two of which forming a conjugate pair similar to that of coordinate and momentum in mechanics. The equation of motion for the system is Hamilton equations Eq. (3) in Ref. [6]for the magnetization vector $M_{i}$and the corresponding conjugate momentum $\Pi _{i}$. The whole simulation time is divided into trajectories. In each trajectory, this set of Hamilton equations will be utilized to generate the configurations of $\{ M_{i}\}$versus Monte Carlo time $\tau $according to Eq. (2); at the end of the trajectory, a Monte-Carlo judgment is performed (if the total energy is lower than that at the beginning of the trajectory, the new configuration is accepted, if the total energy is higher than that at the beginning of the trajectory, the new configuration is accepted with a probability $\mathrm {e}^{\Delta{E}/kT})$, finally at a certain temperature $T$the equilibrated distribution $\mathrm {e}^{{-F[\{ {Mi}} {\}]/k {T}}}$can be achieved. The expression for the effective field $\mathrm {H}_{eff}^{i}$, as we will see below, is quite similar to the effective field in traditional micromagnetic models using LL equations, except for terms related to the newly added constraint potential to confine the magnitude of $M_{i}$near a sphere $\vert M_{i} \vert =$Ms(T). The magnetic free energy is Eq. (4)-(9) in Ref. [6], where the anisotropy energy and the constraint potential form a double-well potential that usually appears in the Landau's second-order phase transition theory, with $\{ M_{i}\}$as the order parameter. For a (40nm $) ^{3}$magnetic cube with 8*8*8 micromagnetic cells, the simulation result is unchanged at a chosen K for $\lambda =$bK in the shaded area, as seen in Fig.1. HMC micromagnetic method can also be utilized to study non-equilibrium problems such as time-dependent coercivity. If we compare the simulated $M_{i}(t)$versus the Sharrock's law, we can determine that a trajectory $( \tau = 10 ^{-3})$in the HMC algorithm roughly equals to $t=1$ns in the real time scale [7]. We cooperate with NIMS Japan [8]to study the M-H loops of FePt-C media at different temperature. It is found that the power $\eta$for $K(T)/ \mathrm {K}(0) =[Ms(T)/Ms(0)]^\eta$has correlation to the alloys, in FePt-C media $\eta$is around 2.5, the calculated magnetization loops at temperatures 300K, 400K, 500K, 600K agree well with the experimental results; at 700K, no experimental M-H loops are measured because the magnetic signal is so weak comparing with the noise, but using HMC micromagnetics, the superparamagnetic properties can be simulated. This confirms the validity of the HMC micromagnetic method in the polycrystalline thin film.