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Basic concepts on magnetization reversal (1) Static properties : coherent reversal and beyond Stanislas ROHART Laboratoire de Physique des Solides Université Paris Sud and CNRS Orsay, France

Introduction: Hysteresis loop Manipulation of a magnetization : Application of a magnetic field Zeeman energy : E z =-µ 0 H . M S M Spontaneous Magnetization M S Remanent Magnetization M R Coercive field H C H S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 2 European School on Magnetism - Targosite 2011

Introduction: Soft and Hard materials Hard Materials SoftMaterials M M H H Applications : Transformer, flux Applications : Permanent magnets, guide (for electromagnets…), motors, magnetic recording magnetic shielding Ex: Cobalt, NdFeB, CoSm, Garnets Ex : Iron, FeCo, Permalloy (Fe 20 Ni 80 ) S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 3 European School on Magnetism - Targosite 2011

Introduction: Energies in magnetic systems Exchange energy Magnetocrystalline anisotropy energy ≠ r r r r ( ) 2 = = − E J S S E MC K m . e ex i j ( ) 2 = ∇ θ (simplest form, may be more complicated) A reflects the cristal symmetry Zeeman energy Dipolar energy r r r r   µ 3 ( m . r ) r r m   i ij ij 0 i = − − E . m   D j 5 3 π 4 r r   ij ij For practical use : shape anisotropy r r 1 = − µ E M . H D 0 d 2 r r r r 1 = − µ E Z M . H = − µ M .[ N ] M 0 0 2 S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 4 European School on Magnetism - Targosite 2011

Introduction: Micromagnetism: Typical Length Scales Bloch wall Exchange length -> Anisotropy vs. Exchange -> Dipolar coupling vs. Exchange Ex : Magnetic vortex ( ) 2 θ d 2 = + θ sin E A K dx 2 A Λ = Λ A ~ 2 . 6 δ B = 2 µ Bloch wall parameter M K 0 S Typical value : 5-10 nm A = π d B Bloch wall width K Quality factor σ = 4 AK Bloch wall energy B ( ) 2 hard > Q 1 Λ 2 K Typical value : 2-3 nm (hard) = = Q δ 2 µ M soft << Q 1 -> 100-1000 nm (soft) 0 S S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 5 European School on Magnetism - Targosite 2011

Introduction: Magnetic Domains Bulk materials Mesoscopic scale Nanometric scale Complex magnetic paterns Small number of possible Magnetic single domain configuration. but non collinearities are Well defined states still possible True collinear state at very reduced dimensions Self organization of domains (< few Λ ) 120 µm 130 µm (square dots – 500 nm) • Except at very small scales, dipolar energy plays an essential role [competition between dipolar energy (long Cowburn et al. PRL 81, 5414 (1998) range) and domain wall energy (local)]. Cowburn J.Phys.D: Appl. Phys. • Single domain state is observed well below 1 µm or for 33, R1 (2000) hard material . S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 6 European School on Magnetism - Targosite 2011

Contents I. Coherent reversal II. Magnetization reversal in nanostructures III. Domain nucleation and domain wall propagation IV. Conclusion S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 7 European School on Magnetism - Targosite 2011

Coherent reversal: Macrospin hypothesis Hypothesis : m ( r )=cte=M (strong approximation) Easy axis M Exchange energy is constant θ θ θ θ Dipolar energy equivalent to anisotropy energy r r r = − µ E G ( M ) M . H 0 Simplest model : Stoner and Wohlfarth 2 = θ − µ θ + θ E K sin M H cos( ) θ Η θ θ θ eff 0 S H Η Η Η H = + K K k eff mc d = µ Anisotropy field : H 2 K / M K eff 0 S sin 2 Dimensionless equation : = θ − θ + θ 2 cos( ) e h H Different names : Uniform rotation, coherent rotation, macrospin, Stoner and Wohlfarth model… S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 8 European School on Magnetism - Targosite 2011

Coherent reversal: Equilibrium states and switching (Field aligned with the anisotropy axis) θ = 0 H sin 2 = θ − θ e 2 h cos θ = − θ = θ cos h : ∂ e ⇒ m = 0 ∂ e ∂ θ θ = θ = π = θ θ + sin 0 : 0 or 2 sin (cos ) h ∂ θ h = -2 h = 1 h = 0 h = 2 h = -0.5 h = -1 Stability 4 4 4 4 4 4 2 2 2 2 2 2 2 ∂ e 2 2 = − θ + θ + θ 2 sin 2 cos 2 h cos 2 ∂ θ 0 0 0 0 0 0 e e e e e e 2 = θ − + θ 4 cos 2 2 h cos -2 -2 -2 -2 -2 -2 -4 -4 -4 -4 -4 -4 2 ∂ e <0 >0 = + −π/2 −π/2 −π/2 −π/2 −π/2 −π/2 0 0 0 0 0 0 π/2 π/2 π/2 π/2 π/2 π/2 π π π π π π 3π/2 3π/2 3π/2 3π/2 3π/2 3π/2 ( 0 ) 2 ( 1 h ) ϕ θ θ θ θ θ 2 ∂ θ 1.0 2 ∂ e 0.5 2 θ = − > ( ) 2 ( h 1 ) 0 m=M/M S Square hysteresis m 2 ∂ θ 0.0 loop 2 ∂ e -0.5 H switch = H K π = − ( ) 2 ( 1 ) h >0 >0 2 ∂ θ -1.0 -2 -1 0 1 2 h S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 9 European School on Magnetism - Targosite 2011

Coherent reversal: Equilibrium states sin 2 = θ − θ e 2 h cos 1.0 • Square hysteresis loop ->H switch = H K 0.5 m=M/M S 0.0 -0.5 Energy barrier ∆ = ϕ − e e ( ) e ( 0 ) -1.0 m ( ) ( -2 -1 0 1 2 ) 2 2 = − + − − 1 h 2 h 2 h h ( ) 2 = + 1 h For arbitrary angle : Important for thermally • no analytical solution activated switching • H K /2<H switch <H K • ∆ e=(1-h) α with α= 1.5 S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 10 European School on Magnetism - Targosite 2011

Coherent reversal: Hysteresis Loops sin 2 = θ − θ + θ e 2 h cos( ) H θ = 0 θ Η = π /3 1.0 1.0 0.5 0.5 0.0 m 0.0 m -0.5 -0.5 -1.0 4 -1.0 4 -2 -1 0 1 2 2 H switch =H C 2 h -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 0 e 0 e h -2 H switch -2 -4 -4 −π/2 0 π/2 π 3π/2 ϕ −π/2 0 π/2 π 3π/2 ϕ Switching field (or reversal field) 4 4 -> abrupt jump of magnetization angle 2 2 0 e 0 e Coercive field : M . H = 0 H C -2 -2 -> may not be equal to the switching field -4 -4 −π/2 0 π/2 π 3π/2 ϕ −π/2 0 π/2 π 3π/2 ϕ S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 11 European School on Magnetism - Targosite 2011

Coherent reversal: Switching field plot : astroids Astroid curve : Polar plot of H switch H K = H ( ) switch 3 / 2 2 / 3 2 / 3 θ + θ sin cos H H 0 0 1.0 1.0 330 330 30 30 300 300 60 60 0.5 0.5 0.0 0.0 0.0 0.0 270 270 90 90 0.5 0.5 240 240 120 120 210 210 150 150 1.0 1.0 180 180  −  π π θ ∈ π H if ; @ [ ]   switch H   4 4 = H J.C. Slonczewski (1956)   C π π 1 3 θ θ ∈ π sin 2 if ; @ [ ]   H H   2 4 4 S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 12 European School on Magnetism - Targosite 2011

Coherent reversal: Switching field plot : astroids r r r = − e G ( m ) 2 h . m Equilibrim condition r r ( ) r r v de = θ sin θ m cos , ( ) = − = G ' ( m ) 2 h . e 0 = − θ cos θ e sin , with θ d 2 stable states -> For given m : Straight line in the field space, tangente 1.0 Hard axis to the critial astroid curve, directed along m Stability condition 0.5 r r r d ² e = + > G " ( m ) 2 h . m 0 θ d ² Easy axis 0.0 h y ->For given m : Only one part of the line is stable -0.5 -1.0 -1.0 -0.5 0.0 0.5 1.0 h x 1 stable state J.C. Slonczewski Research Memo RM 003.111.224, IBM Research Center (1956) A. Thiaville JMMM 182, 5, (1998) S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 13 European School on Magnetism - Targosite 2011

Coherent reversal: Switching field plot : astroids -> To go further : • Complex type of anisotropy • Three dimensionnal extension = θ θ G sin ² cos ² (Cubic anisotropy) = θ θ + θ + π G sin ² cos ² sin ²( / 6 ) (Cubic anisotropy + uniaxial) Thiaville PRB 61, 12221 (2000) Thiaville JMMM 182, 5, (1998) S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 14 European School on Magnetism - Targosite 2011

Coherent reversal: Experimental relevance First observation (2D) : Co (D = 25 nm) cluster Wernsdorder et al. PRL 78, 1791 (1997) In 3 D: (same Co cluster) E. Bonnet et al PRL 83, 4188 (1999) 3 nm Co cluster : M. Jamet et al PRL 86, 4676 (2001) S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties 15 European School on Magnetism - Targosite 2011