Introduction to Magnetism (2) : Magnetism today or How performs - PowerPoint PPT Presentation

Introduction to Magnetism (2) : Magnetism today or How performs the micromagnetic theory now? André Thiaville Laboratoire de Physique des Solides Université Paris-Sud & CNRS, Orsay European School of Magnetism, 1 Constanta, 2005: André THIAVILLE

1- Introduction 2 - Micromagnetic theory and applications statics domain walls dynamics 3 - Micromagnetics of nano-elements macrospin limit quasi-uniform structures dynamics 4 - Beyond micromagnetics molecular magnetism clusters of a few atoms spin-polarized transport European School of Magnetism, 2 Constanta, 2005: André THIAVILLE

Nanometric size structures are already used European School of Magnetism, 3 Constanta, 2005: André THIAVILLE

for 10 3 kbit/in 1 bit = 25 nm European School of Magnetism, 4 Constanta, 2005: André THIAVILLE

European School of Magnetism, 5 Constanta, 2005: André THIAVILLE

G. Herzer, Amorphous and nanocrystalline Soft magnets, in Magnetic hysteresis in novel materials, G.C. Hadjipanayis Ed., Nato ASI E338 (Kluwer, Dordrecht, 1997) European School of Magnetism, 6 Constanta, 2005: André THIAVILLE

"Standard" Micromagnetics European School of Magnetism, 7 Constanta, 2005: André THIAVILLE

The “micromagnetic” description of magnetism y atomic spins x 1 m y 0.8 continuous distribution m x 0.6 m 0.4 0.2 0 position x Assumes that structures to describe are large compared to atomic sizes European School of Magnetism, 8 Constanta, 2005: André THIAVILLE

Magnetic Interactions H = − ⋅ E J S S i j i j European School of Magnetism, 9 Constanta, 2005: André THIAVILLE

Micromagnetic equations = = M M ( T ) m no thermal fluctuations m 1 s 1 = ∇ + − µ ⋅ − µ ⋅ E A ( m ) KG ( m ) M m H M m H 2 s s 0 0 D 2 exchange anisotropy applied field demagnetizing field v v = H eff x m 0 Statics : minimise ∫ E Brown equations r r ∂ m = + boundary conditions 0 r ∂ n effective = + + + H H H H H eff applied demag aniso exchange field δ 1 E = − H eff δ µ 0 m 2 A M ∆ m s µ M European School of Magnetism, 10 0 s Constanta, 2005: André THIAVILLE

Magnetostatics of matter r r ( ) r r r r = = div B 0 = µ + with rot H j and B H M 0 r r = = − div H 0 div H div M ext D r = = rot H j rot H 0 D ext ext + boundary conditions r r r r r ext int − ⋅ = ⋅ ( H H ) n M n D D r r r ext int − ⋅ = ( H H ) t 0 D D demagnetizing field applied field European School of Magnetism, 11 Constanta, 2005: André THIAVILLE

Magnetostatic energy ( ) r r r 2 1 1 ∫ ∫ = − µ ⋅ = µ ≥ E M H H 0 3 2 2 D 0 D 0 D V R r r = − ∇ φ H proof : introduce the scalar potential D r r r r ⋅ div M M n 1 1 ∫ ∫ ∆ φ = φ = − + div M r r r r π π 4 − 4 − r r ' r r ' ∂ V V and transform by integration by parts both expressions into r r r 1 1 ∫ ∫ µ ⋅ φ − µ φ ( M n ) div M 2 2 0 0 ∂ V V European School of Magnetism, 12 Constanta, 2005: André THIAVILLE

Characteristic lengths A ∆ = Bloch wall width parameter K ∆ = 1 - 100 nm A=10 -11 J/m, K=10 2 – 10 5 J/m 3 2 A Λ = exchange length µ M 2 s 0 M s = 10 6 A/m Λ = some nm 2 Quality factor Λ   2 K = =   Q  ∆  2 µ M Q > 1 hard material 0 s Q << 1 soft material European School of Magnetism, 13 Constanta, 2005: André THIAVILLE

The Bloch wall (1932) Easy axis z y θ m x x y = div m 0 No demag energy   0   r = m sin θ ( x )     cos θ ( x )   2 θ   ( ) ( ) d 2 = + θ θ − ∞ = θ + ∞ = π   E A K sin 0 ,   dx European School of Magnetism, 14 Constanta, 2005: André THIAVILLE

2 θ d Energy minimization − + θ θ = 2 A 2 K sin cos 0 2 equation dx θ d ∫ × and dx dx 2 θ   d 2 st − + θ = =   A K sin C 0 First integral   dx A θ θ d sin ∆ = = ± Bloch wall width parameter K ∆ dx 1 1 0.75   −   x x ( )   θ = + π 0 2 Atan  exp  0.5   ∆       0.25 π ∆ 0 Linear width : -5 -4 -3 -2 -1 0 1 2 3 4 5 European School of Magnetism, 15 Constanta, 2005: André THIAVILLE

Properties of the Bloch wall 2 A = 2 AK Integrated exchange energy = ∆ ∆ = Integrated anisotropy energy = 2 K 2 AK Integrated hard axis ∫ ∫ ∫ = θ = ∆ θ = π ∆ m y dx sin dx d component 2 ∫ ∫ ∫ 2 = θ = ∆ θ θ = ∆ m y dx sin dx sin d 2 etc. European School of Magnetism, 16 Constanta, 2005: André THIAVILLE

The vortex 2D magnetization =   y / r div m 0   = − m x 0/ r   ⋅ n = m 0   ( ) 2 2 = ∇ = E ech A m A / r Divergence of the exchange energy 3D magnetization approximation = θ   div m 0 sin ( r ) y / r   = − θ m sin ( r ) x / r     ⋅ n ≠ m 0 θ cos ( r )     2  θ  d 2 2 = θ +   E ech A sin / r     dr   European School of Magnetism, 17 Constanta, 2005: André THIAVILLE

−  sin θ ( r ) y / r   r = m sin θ ( r ) x / r     cos θ ( r )     2 2   d θ sin θ = +   E ech A     dr 2   r 2 µ M 2 = 0 s E cos θ 2 dem E. Feldtkeller, H. Thomas, Phys. kondens. Materie 4 , 8 (1965) 2   d ( 2 θ ) d ( 2 θ ) 1 1 1 2 A   Λ = + + − = sin 2 θ 0   µ M r dr 2 2 2 2 Λ   s dr r 0 European School of Magnetism, 18 Constanta, 2005: André THIAVILLE

Variational calculation Normalized widths W / Λ Normalized thickness D / Λ A.Hubert et R. Schäfer Magnetic Domains (Springer, 1998) European School of Magnetism, 19 Constanta, 2005: André THIAVILLE

Walls in films with perpendicular anisotropy Λ = 4 nm Bloch Néel Néel European School of Magnetism, 20 Constanta, 2005: André THIAVILLE

The Néel wall (1955) Thin film without anisotropy, or small in-plane anisotropy Bloch wall Néel wall European School of Magnetism, 21 Constanta, 2005: André THIAVILLE

Approximate analytical model m x The wall has logarithmic tails x/D Q = 0.04, 1.88 D = 2.5 Λ Q = 2-2.5 10 -4 D = 20, 2.5 et 1.45 Λ H. Riedel, A. Seeger, phys. stat. sol. 46 377 (1971) European School of Magnetism, 22 Constanta, 2005: André THIAVILLE

Walls in soft thin films European School of Magnetism, 23 Constanta, 2005: André THIAVILLE

2D instability of the Néel wall : cross-tie map of the magnetic charges - + - + - + - + - + - + - + electron holography image A. Tonomura et al., Phys. Rev. B25 6799 (1982) European School of Magnetism, 24 Constanta, 2005: André THIAVILLE

Magnetization dynamics µ g e = − γ γ gyromagnetic ratio (>0) γ = = B L M / g 2 m h H Angular momentum d L = Γ m dynamics dt Γ = µ × M s m H 0 d m = γ × H m dt 0 5 γ = µ γ ≈ 2 . 2 10 S . I . 0 0 Can be found directly from 0.28 GHz/ mT quantum mechanics European School of Magnetism, 25 Constanta, 2005: André THIAVILLE

Dynamics of a magnetization continuum Effective = + + + H H H H H eff applied demag anisotropy exchange field δ 2 A 1 E ∆ = − m H eff δ µ µ 0 m M M 0 s s d m d m = γ 0 × + α × H m m Landau-Lifshitz-Gilbert eff dt dt α : Gilbert γ     damping = × + α × × 0 H m m  H m    eff eff   2   + α parameter 1 European School of Magnetism, 26 Constanta, 2005: André THIAVILLE

another magnetization dynamics equation d m d m = γ 0 × + α × H m m Landau-Lifshitz-Gilbert (1955) eff dt dt   d m = γ × + λ × × Landau-Lifshitz (1935) H m m  H m  eff eff   dt L are mathematically equivalent dm/dt dm/dt γ H x m γ H x m m α m x dm/dt λ m x (H x m) H LL LLG European School of Magnetism, 27 Constanta, 2005: André THIAVILLE

Properties of the magnetization dynamics r 2 r d ( m ) r d m 1) = = 2 m . 0 Conservation of the magnetization modulus dt dt   r dE d m d m = − µ = − αµ   M H . M H . m x 2) eff eff dt dt dt 0 s 0 s   2     r d m d m = − αµ = − αµ γ   M .  H x m  ( M / ) eff   dt dt 0 s 0 s   Decrease of the energy with time : the magnetic system is not isolated European School of Magnetism, 28 Constanta, 2005: André THIAVILLE

Micromagnetics & Nano-objects European School of Magnetism, 29 Constanta, 2005: André THIAVILLE