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Introduction to Micromagnetic Simulation

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.

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Introduction to Micromagnetic Simulation

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

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

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

  4. H Magnetic Materials Diamagnetic: M 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.

  5. CGS and SI Units

  6. Hysteresis Loop

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

  8. 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.

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

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

  11. 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:

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

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

  14. DC Field Simulations

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

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

  17. 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.

  18. Fast Multipole Method Source Near Field Middle Field Far Field

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

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

  21. Exchange Field • 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

  22. 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

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

  24. 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.

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

  26. Default Simulation Parameters

  27. Various c-axis distributions

  28. Various exchange constants

  29. Various anisotropy constants

  30. Various thickness

  31. Mixture of two anisotropy constants

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

  33. 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.

  34. Thermal effects

  35. Domain Domain wall Domain Wall Simulation (Side View)

  36. Bloch wall Domain Wall Simulation (Top View)

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

  38. Question?

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