Introduction to Numerical Micromagnetism. Application to Mesoscopic - PowerPoint PPT Presentation

Introduction to Numerical Micromagnetism. Application to Mesoscopic Magnetic Systems Liliana Buda-Prejbeanu CEA / DRFMC / SPINTEC Grenoble, France Jean-Christophe Toussaint CNRS – Laboratoire Louis Néel Grenoble, France Summer School Magnetism of Nanostructured Systems and Hybrid Structures Bra ş ov 2003

Outline Micromagnetics – theoretical background - hypothesis & limits - total free energy minimization (variational principle) - static and dynamic equations Micromagnetics – overview of the numerical implementation - current state of the art - finite difference approximation (fields & energies) - errors & accuracy & validation Application for mesoscopic ferromagnetic elements - circular Co dots - self-assembled epitaxial submicron Fe dots References

Length Scale Co (hcp) L, l, e <1 µm magnetic device ~ 4 Å Quantum Mechanics Micromagnetism Bulk Atomic nanoscopic mesoscopic macroscopic scale scale scale scale 10 µm 1 Å 1 Å 1 nm 1 nm 10 nm 10 nm 100 nm 100 nm 1 µm 1 µm 10 µm bulk individual nanoparticules thin films micrometric spins clusters objects sub-microns objects ultra-thin films

Experimental Scale Macroscopic studies Microscopic studies Hysteresis curves Local imaging MFM, Lorentz microscopy,… r - spatial resolution limited (>20 nm) Mean values of the magnetization M - several possible configurations MOKE, SQUID,... i Vellekoop et al., JMMM. 190, 148 (1998).

Hypothesis 1907 P. Weiss / magnetic domains 1963 - W. F. Brown Jr. 1935 Landau-Lifshiz / domain walls Classical theory of continuous ferromagnetic material - smooth spatial variation of the magnetization vector Continuous functions (space & time) Individual spins r M r ( r , t ) - magnetization r H r ( r , t ) - fields r r [ ] ( ) E m r , t - energies ≡ Magnetization constant amplitude vector r r r r = M ( r , t ) M m ( r , t ) s r r = m ( r , t ) 1 Thermal fluctuations neglected Continuous material A ( T ), M ( T ), K ( T ) ex s u i J. F. Brown , Jr. : Micromagnetics, J. Wiley and Sons, New York (1963)

Total Free Energy (Gibb’s free energy) ( ) r Exchange interaction ∫ r ∇ 2 A m dV - magnetic order (T<T c ) ( QM) ex - parallels spins V next neighbors [ ] Magneto-crystalline anisotropy ∫ r r ( ) − ⋅ 2 K 1 1 u m dV - the crystal symmetry axis K - easy direction V local interaction [ ] r ∫ r Zeeman coupling − ⋅ µ M m H dV 0 s app - external applied field - magnetization rotation V local interaction Magnetostatic interaction [ ] r 1 µ ∫ r r ( ) − ⋅ - Maxwell’s equations M m H m dV 0 s dem 2 - magnetic charges distribution V - magnetic domains formation long range interaction Others contributions : surface coupling, ….

Micromagnetic equations m ( r ) magnetization distribution minima r { r r r r } ( ) = ∈ = m m r r V , m 1 energy total free energy functional Space of configurations r r [ ] ( ) dV ∫ = ε E m m V magnetic stable state = minimum of the total free energy functional r r r → + δ m m m r [ ] δ = E m 0 r ⇐ variational principle = m 1 2 r [ ] δ > r r E m 0 2 ⋅ δ = m m 0

Micromagnetic equations – static equilibrium equations r ( ) r r r ∂   m r r ∫ ∫ δ = − × ⋅ δ θ + × ⋅ δ θ   E µ M m H dV 2 A m dS 0 s eff ex ∂  n  V S r r r δ = θ δ × m m r effective field n r r δ = − δ ⋅ ∫ E µ M m H dV r 0 s eff m V S r r r 2 A 2 K r r r r r ( ) V = ∆ + ⋅ + + + H ex m 1 u m u H H C m eff K K app D µ M µ M 0 s 0 s Brown’s equations r ∂ m r = ∈ [ ] ( ) 0 , r S r r r r r × = ∀ ∈ ∂ m H r 0 r V n eff r r ∂ ∂ m m r = ∈ A A , r S 1 2 ∂ ∂ ex , 1 n ex , 2 n A. Hubert, R. Schäfer : Magnetic Domains (p. 149) i J. Miltat in Applied Magnetism (p. 221)

Micromagnetic equations m ( r, t ) { } r r r r r ( ) = ∈ ≥ = space & time dependence → m m r , t r V , t 0 , m 1 energy 1 2 Space of configurations magnetization trajectory between two magnetic states

Micromagnetic equations - dynamics Landau-Lifshitz-Gilbert Equation (LLG) precession relaxation r [ ] ∂ r r ( ) ( ) m r r r ( ) + α = − γ × − αγ × × 1 m µ H m m µ H 2 ∂ 0 eff 0 eff t H H m m g e γ = γ = × × γ = > µ g 1 . 105 10 m /( As ) 0 5 γ = gyromagnetic ratio 0 0 2 m e ≅ g 2 g = Landé factor α ≅ ÷ 0 . 001 1 . 0 α = damping parameter

Magnetostatic Equations r r = − ∇ φ Scalar potential formalism: H D r r ( ) ( ) ∆ φ = − ρ r r m r r r ( ) ( ) r ∈ S φ = φ r r 3D int ext r r r r [ ] ( ) ( ) r ⋅ ∇ φ − φ = σ n r r r ∈ S int ext m r r φ → → ∞ ( r ) 0 , r r r r 1 r r & ∇ = − = G ( r ) Green’s function formalism : G ( r ) r r π 3 π 4 r 4 r r r r r r r r r ( ) ( ) ( ) ( ) φ = ρ − + σ − ∈ ℜ ∫ ∫∫ ( r ) r ' G r r ' dV ' r ' G r r ' dS ' r 3 m m V S r r r r r r r r r r ( ) ( ) = − ∇ − ρ − ∇ − σ ∫ ∫∫ H ( r ) G ( r r ' ) r ' dV ' G ( r r ' ) r ' dS ' D m m V S i J. D. Jackson : Classical electrodynamics , New York (1962)

General Algorithm r r r { ( ) } Defining the problem m r , t , K , A , M , H -geometry description 0 ex s appl -material parameters -initial conditions r (time & space) r r r ⋅ r ∂ ∂ = ρ = − ∇ ⋅ σ = n M , m n 0 M , r r r r r State characterization = + + + H H H H H -magnetic charges eff app D ex k -magnetostatic field ε = ε + ε + ε + ε t → t+ δ t -fields & energy terms H D ex k r ∂ r [ r ] ( ) ( ) m r r r ( ) + α = − γ × − αγ × × 1 m µ H m m µ H 2 LLG time integration ∂ 0 eff 0 eff t - amplitude conservation r false r × ≤ ε max Check equilibrium criteria m H M rr eff s true Stable state- numerical solution r { } m , E eq eq

Solution static & dynamic equations r ← = � non-linear m 1 Second order r integro-differential equations � non-local ← H D ← ∂ , ∂ � coupled partial differential 2 analytical treatment - macrospin approximation - Bloch domain wall in bulk - by linearisation - near the saturation limit - nucleation and switching of domains - ferromagnetic resonance numerical treatment - powerful and efficient tools (if some rules are respected! )

Current State of the Art Finite difference Finite element method approximation (FDA) (FEM / BEM) -regular mesh -irregular meshes -restrictive geometry -adaptive mesh refinement W.F. Brown Jr (1965) Schabes et al., (1988) Fredkin & Koehler Berkov et al. Fidler & Schrefl Bertram et al. Hertel & Kronmuller Donahue et al. Ramstöck et al. Miltat et al. ……. Nakatani et al. Toussaint et al. Scheinfein et al. J. -G. Zhu et al………….

Finite Difference Approximation (FDA) Numerical discretisation discretisation Numerical { } r r r { } r r r r r ( ) = = = m m i 1 .. N , m 1 = ∈ = m m r r V , m 1 i i { } r r { } r r r r ( ) = = H H i 1 .. N = ∈ H H r r V eff eff , i eff eff N – total number of mesh nodes 3D 2D -orthorhombic cells -infinite prisms : dots, wires, platelets, … : e.g. thin films

Finite Difference Aproximation (FDA) [ ] r r r r r r [ ] ( ) ( ( ) )   ∇ + − ⋅ − A m r K 1 u m r 2 2   r ex 1 K [ ] = ∫ E m dV   [ ] [ ] r r 1 r r r r r r r ( ) ( ) ( ) ( ( ) ) − ⋅ − µ ⋅ µ M m r H r M m r H m r   V   0 s app 2 0 s dem magnetic r r r 2 A 2 K r r r r ( ) charges = ∆ + ⋅ + + H m u m u H H ex 1 ( ) r r eff µ M µ M K K H D ρ = − ∇ ⋅ M s m 0 s 0 s m r ⋅ r ( ) σ = M s m n m Taylor expansion ∂ ∂ 2 m 1 m ( ) ( ) ( ) ( ) + = + + 2 + 3 m i 1 , j m i , j i , j h i , j h O ( h ) x x x ∂ ∂ 2 x 2 x 2D ∂ ∂ 2 m 1 m ( ) ( ) ( ) ( ) − = − + + 2 3 m i 1 , j m i , j i , j h i , j h O ( h ) ∂ x ∂ x x 2 x 2 x ( ) ( ) ∂ + − − m m i 1 , j m i 1 , j ( ) ≅ i , j ∂ x 2 h x ( ) ( ) ( ) ∂ + − + − 2 m m i 1 , j 2 m i , j m i 1 , j ( ) ≅ i , j ∂ 2 2 x h x The accuracy is dependent on the Taylor expansion order ! i M. Labrune, J. Miltat, JMMM 151 , 231 (1995).