Comparison of Computational Speed between Landau-Lif-shitz- Gilbert and Hybrid-Monte-Carlo Micromagnetics.

The Landau-Lifshitz-Gilbert (LLG) equation is widely utilized to analyze the magnetic properties in micromagnetics [1]. However, the calculation of micromagnetics with Landau-Lifshitz-Gilbert equations is time-consuming, so the FDM-FFT method was brought up to speed up the simulation in large scale [2]. Recently, we have 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 [3]. Some simulation results have been put forward to confirm the validity of the HMC algorithm [4]–[6]. Furthermore, we also wish to focus on the computational efficiency to compare HMC micromagnetics with traditional LLG micromagnetics. Comparison of computational speed has been made of these two methods in various micromagnetic scales in this work. In LLG or HMC micromagnetics, a regular mesh with cell size $5 \times 5 \times 5$ nm 3 is utilized. Other important parameters are: anisotropy energy constant $K(0 \mathrm {K}) = 1 \times 10 ^{5}\mathrm {J}/ \mathrm {m}^{3},$ saturation magnetization at zero temperatur $e M_{s}(0 \mathrm {K}) = 7.98 \times 10 ^{5}\mathrm {A} /\mathrm {m}$, exchange constant $A^{\ast }= 1.0 \times 10 ^{-11}\mathrm {J} /\text{m}$. In LLG micromagnetics, the iterations can go endless unless the spin error $\vert \Delta\vert ($ the spin maximum deviation between two simulation steps) is set to be a limited number. To ensure the stability and accuracy, $\vert \Delta\vert $ is set as $10 ^{-6}$ in LLG algorithm. In HMC micromagnetics, the Monte Carlo time t is not real time. Based on the feature of the leap-frog algorithm used for iteration of the Hamilton equations, in HMC micromagnetics, the spin error $\vert \Delta\vert $ stands for the spin deviation of two adjacent trajectories. To compare the computational speed of the LLG and the HMC micromagnetic methods, the spin error $\vert \Delta\vert $ in HMC MuMag is also set as $10 ^{-6}$. The M-H loops are same under these two micromagnetic algorithms. Fig. 1 shows the computational time to reach $\vert \Delta \vert \sim 10 ^{-6}$ at each external magnetic field $(\mathrm {H}_{ext})$, with 16*8*16 micromagnetic cells. The computational time for Hybrid Monte Carlo MuMag keeps around 59 seconds for different $\mathrm {H}_{ext}$; while for Laudau-Lifshitz-Gilbert MuMag, the simulation time is typically several hundred of seconds and shows a sharp rising near coercivity to about 1904 seconds, which is different from HMC MuMag. In Fig. 2, the computational speed of the two micromagnetic methods is presented. The horizontal axis is ${log}_{2}(N_{cell})$, and the vertical axis is the total computation time for a loop in Fig. 1 (in seconds). The blue line is the speed of the LLG MuMag. In the Ref. 7, the LLG algorithm has a running time proportional to the number of particles N, which yields ${O(Nlog}_{2}N)$ computational time. We have compared the computational time of LLG algorithm with ${O(Nlog}_{2}N)$. The triangle-dot line is the computation time vs. ${log}_{2}(N_{cell})$, which shows great agreement with ${O(Nlog}_{2}N)($ black line in Fig. 2). The red dotted line shows the computation time consumed by the HMC algorithm, which show great coincidence with $O(N)$ in the insert figure. From the comparison, the HMC algorithm has a significant advantage for the computational speed. When the number of micromagnetic cells rise to 32*32*32, LLG algorithm spends 16 times more than HMC algorithm.