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 Magnetization M S p l N Magnetic dipole moment p m pl Magnetization magnetic dipole moment per unit volume m M V Magnetic Materials Diamagnetic M Most elements in the periodic table including copper silver and gold Paramagnetic M Include magnesium molybdenum lithium and tantalum H M Ferromagnetic Iron nickel and cobalt Ferrimagnetic Ferrites CGS and SI Units 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 H M M s The Landau Lifshitz Gilbert LLG equation M 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 dM L M H M M H 2 dt 4 M s The Landau Lifshitz Gilbert LLG equation dM dM G M H M dt Ms dt where 4 L M s G L 1 2 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 L dM L M H M M H dt Ms dM s2 0 dt Field Analysis Applied field DC AC Demagnetizing field time consuming Effective field E H eff M H eff x E E E H eff y H eff z M x M y M z Anisotropy field uniaxial and cubic Exchange field quantum mechanic effect Other fields DC Field Solution DC field only single grain The solution is H0 0 M y 0 x sin cos L H 0 t 0 M M s sin sin L H 0 t 0 cos where 0 tan tan exp L H 0 t 2 2 DC Field Simulations Small Applied AC Field Small ac field single grain resonance Hx h cos t Hy h sin t h H0 H0 M y x Demagnetizing Field long distance n S j M r r r 2 3 N i 1 H dem d r d r Dij M j 3 Vi vi j 1 S j r r j 1 r r n 2 1 T r d r 3 Dij Tdxdydz where r r S v vi N j 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 Fast Fourier Transform N i H dem Dij M j Convolution j 1 Symmetry in geometries v b row v a row 0 3 0 2 0 1 0 0 2 1 2 0 0 3 3 3 2 3 3 2 2 2 1 2 1 1 1 0 1 3 0 2 0 0 u a column 3 2 2 1 1 1 1 0 3 3 2 3 2 2 1 2 0 1 3 1 3 0 1 3 2 0 3 1 3 0 u b column 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 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 H x i 1 M s i j NN 2 Aij m j mi 2ij 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 m 1 m z m m z z y y z y x m z x y x z x z 30 x x x 30 ODE Solver Runge Kutta embedded 4th 5th method 2 i 1 g ni f t n ci hn y n hn aij g nj j 1 i 1 s t n 1 t n hn s y q n 1 y n hn bq i g ni i 1 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 Top View Side View d Layer 2 a c axis dz y Layer 1 x Default Simulation Parameters a d dz 0 5 1 0 1 1 0 Ms K1 emu cc ergs cc 233 5 105 heh hev 0 05 0 05 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 L dM L M H M M H dt Ms L L M dh M M dh Ms Where h is a stochastic field with the property E h t 0 2 k BT E hi t h j t ij t s 2 ij t s M s v 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 Domain Domain Wall Simulation Top View Bloch wall Summary Basic ideas in micromagnetic simulation Algorithms in micromagnetic modeling Micromagnetic simulation results