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## Simulation of thermal noise in micromagnetics

Dear all,

I have been trying to simulate thermal noise in micromagnetic systems with COMSOL 5.3a. Using Fluctuation-dissipation theorem, I derived the amplitude of thermal field:

H_rms=sqrt(2 alpha kB T/(gama mu0 Ms dV dt))

with the Gilbert damping constant alpha in LLG equation, temperature T, saturated magnetization Ms, time step dt and the volume of each lattice dV. The program was developed by my senior using weak form of LLG equation:

-test(m1) m1t+alpha test(m1) (m2 m3t-m3 m2t)-gama (m2 h3-m3 h2) test(m1)

-test(m2) m2t+alpha test(m2) (m3 m1t-m1 m3t)-gama (m3 h1-m1 h3) test(m2)

-test(m3) m3t+alpha test(m3) (m1 m2t-m2 m1t)-gama (m1 h2-m2 h1) test(m3)

Here I have omitted the interaction between lattices since I plan to simulate macrospin in finite temperature. The thermal noise is incorporated by effective field (h1,h2,h3), whose components have the RMS value H_rms and are uncorrelated. I also exerted a constant magnetic field h0=0.388e5 A/m along m3 direction. After a sufficient number of time steps, I make a histogram of energy E=-mu0 Ms h0 dV m3, plotting the lnP(E)-E curve and fitting it linearly to check whether it matches Boltzmann distribution (good linearity and accurate fitted value of T). Here I have already considered the density of states g(E).

To make the results more presuasive, I have simulated for more than 10 million time steps during one simulation, which reveals very good linearity (with adjusted R^2 up to 0.9999), but the fitted value of temperature T is always more than 10% higher the given value. The following factors might affect the result, and I have been rather confused.

1. Does the configuration of meshes affect the result? It seems that the magnet moments cannot be directly influenced by the size of meshes, and dV is merely a coefficient that determines the strength of stochastic fields (which is mathematically equivalent to Ms). However, free tetrahedral meshes perform much better than swept cubic meshes, the latter outputting an even higher temperature; furthermore, if I change the value of dV, the output values of T will also be quite different.

2. The quality of random numbers. Here I use only one random function of 4 variables:

h1=H_rms*rn1(x/lx,y/ly,z/lz,t/tt)

h2=H_rms*rn1(t/tt,x/lx,y/ly,z/lz)

h3=h0+H_rms*rn1(z/lz,t/tt,x/lx,y/ly)

which seems not bad since I can hardly see any spatial or temporal correlation between h1, h2 and h3. But will it be better if I use 3 functions rn1, rn2 and rn3 with different random seeds for h1, h2 and h3?

1. Is the result strongly dependent on transient solvers, such as the choice of damping factors and tolerance?

I am struggling with these problems since I can hardly find any reference on this topic. I would appreciate that if you may express your opinion!

0 Replies Last Post Nov 17, 2020, 11:57 AM EST

Hello 允文 刘