Introduction to Micromagnetic Simulation Feng Xie Ph.D. student Major advisor: Dr. Richard B. Wells

Contents Introduction to magnetic materials. Ideas in micromagnetics. Physical equations. Field analysis. ODE solver and coordinate selection. Simulations for ideal cases. Thermal effects Summary

(magnetic dipole moment Magnetization Magnetic dipole moment Magnetization (magnetic dipole moment per unit volume) S N  +p -p l  

Magnetic Materials Diamagnetic: M H M Most elements in the periodic table, including copper, silver, and gold. Ferromagnetic: Iron, nickel, and cobalt. Ferrimagnetic: Ferrites. Paramagnetic: M Include magnesium, molybdenum, lithium, and tantalum.

CGS and SI Units M H K m 0 Magnetization Magnetic field Quantity Symbol CGS unit SI unit CGS value / SI value Magnetization M emu/cc A•m-1 10-3 Magnetic field H Oe A-turn•m-1 410-3 Anisotropy constant K ergs/cc J•m-3 10 Magnetic charge density m unit pole/cm3 Wb•m-3 100/(4) Permeability of vacuum 0 =1 H•m-1 107/(4)

Hysteresis Loop

Scale Comparison A magnetic force microscopy (MFM) image showing Domain structure Micromagnetic explanation of Domain structure (Phenomenology) Electron Spins (Quantum theory)

Why Micromagnetics? To provide magnetization pattern inside the material. To explain some experimental results. To simulate new materials. To realize new properties of materials. To provide material parameters to designers.

Micromagnetic Assumptions Magnitude: The Landau-Lifshitz-Gilbert (LLG) equation Magnetic fields: Externally applied field Exchange field Demagnetizing field Anisotropy field Stochastic field or the stochastic LLG equation

Physical Equations The Landau-Lifshitz (LL) equation: The Landau-Lifshitz-Gilbert (LLG) equation: where

Comments on Equations When <<1, LG=22.8 M (radHz/Oe). From either the LL or the LLG equation: In analysis, we prefer the form:

Field Analysis Applied field: DC + AC. Demagnetizing field: time consuming. Effective field: Anisotropy field: uniaxial and cubic. Exchange field: quantum mechanic effect. Other fields.

DC Field Solution DC field only + single grain The solution is where x y 0 0 H0 where

DC Field Simulations

Small Applied AC Field Small ac field + single grain H0 resonance Hx = h cos(t) Hy = h sin(t) h << H0 x y   H0

Demagnetizing Field Consuming most of computation time. where long distance where Consuming most of computation time.

Fast Algorithm Two computational methods are in discussion: Fast Multipole Method (FMM) and Fast Fourier Transform (FFT). FMM is good for very big sample size. It can be applied on either asymmetric or symmetric geometries. FFT is good for small sample size. It can only applied on symmetric geometries.

Fast Multipole Method Source Near Field Middle Field Far Field

Fast Fourier Transform Convolution Symmetry in geometries 0, 0 1, 0 2, 0 3, 0 0, 1 1, 1 2, 1 0, 2 3, 1 1, 2 3, 3 2, 2 3, 2 2, 3 1, 3 0, 3 u_b(column) v_b(row) u_a(column) 0, 0 1, 0 2, 0 3, 0 0, 1 1, 1 2, 1 0, 2 3, 1 1, 2 3, 3 2, 2 3, 2 2, 3 1, 3 0, 3 v_a(row)

Anisotropy Field Uniaxial Anisotropy: E=K0u+K1usin2+K2usin4+ Magnetocrystalline Anisotropy H  M Uniaxial Anisotropy: E=K0u+K1usin2+K2usin4+ Cubic anisotropy: E=K0c+K1c(cos21cos22 +cos22cos23+ cos23cos21)+

Exchange Field The effective exchange field is It is mainly from electron spin coupling. It is short-range so that we take into consideration only exchange energy between nearest-neighbor grains. The effective exchange field is

Adjustable Parameters Crystalline anisotropy HCP or FCC K1, K2, … distribution of c_axis (how good is good) Exchange constant A: Different materials have different As. Different parts may have different As (poly). Sample size and shape. Anisotropy . Nonuniform Ms

Coordinates + y x z m x x z < 30 |m|=1  y x z m   y x z m  = ?  y x z m z z + x < 30

ODE Solver Runge Kutta embedded 4th-5th method [2]. Adaptive time step. No need value of previous steps. [2] J. R. Cash and A. H. Karp, “A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides,” ACM Transactions on Mathemathical Software, vol. 16, no. 3, pp.201-222, September 1990.

Geometry x y a d Top View Layer 1 Side View c-axis  dz Layer 2 

Default Simulation Parameters  a d  dz heh hev 0.5 1 0.1 -0.05 Ms (emu/cc) K1 (ergs/cc)   233 5105 [0,20 ] [0 ,360 ]

Various c-axis distributions

Various exchange constants

Various anisotropy constants

Various thickness

Mixture of two anisotropy constants

Stochastic LLG Equation Due to thermal fluctuation. Stochastic Landau-Lifshitz-Gilbert equation Where is a stochastic field with the property

SDE Solver Stochastic LLG equation is a stochastic ODE with multidimensional Wiener process. The strong order of Runge-Kutta methods cannot exceeds 1.5 [3]. Heun scheme is applied. [3] K. Burrage and P.M. Burrage, “High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations,” Applied Numerical Mathematics 22 (1996) 81-101.

Thermal effects

Domain Wall Simulation (Side View)

Domain Wall Simulation (Top View) Bloch wall

Summary Basic ideas in micromagnetic simulation. Algorithms in micromagnetic modeling. Micromagnetic simulation results.

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