Domains(and(domain(walls( Andrs'Cano' - PowerPoint PPT Presentation

Domains(and(domain(walls( Andrés'Cano' ICMCB.CNRS,'University'of'Bordeaux,'France' Department'of'Materials,'ETH'Zürich,'Switzerland'

References( On#the#theory#of#the#dispersion#of#magne2c#permeability#in#ferromagne2c#bodies## L.D.'Landau'and'E.'Lifshitz'' Phys.'Z.'Sowjet.'8,'153'(1935);'Collected'papers'of'L.D.'Landau,'pp.101.114 '' ' Physical#theory#of#ferromagne2c#domains## C.'KiYel'' Rev.'Mod.'Phys.'21,'541'(1949)' ' Magne2c#domains:#the#analysis#of#magne2c#microstrctures## A.'Hubert'and'R.'Schafer'(Springer,'Berlin,'1998)' # Microscopic#approach#to#current:driven#domain#wall#dynamics## G.'Tatara.'H.'Kohno,'J.'Shibata' Physics'Reports'468,'213'(2008)'

Outline( 1.(History(&(mo9va9on( ( 2.(Observa9on(techniques( ( 3.(The(origin(of(domains( ( 4.(Domain(walls( ( 5.(Domain(wall(mo9on(

domain'wall' domain'

The'concept'of' domain:( ( ' Postulated' by' Pierre' Weiss' in' 1907' to' explain' why' ferromagnebc'bodies'can'appear'non.magnebc.''' 110(aniversary!

F = a ( T − T c ) M 2 + bM 4 M + M 0 ferromagnebc paramagnebc T T c - M 0 Two'possible'states'below' T c#

M Barkhausen(effect( H

The'disbnct'response'of'ferromagnets'is'' inherently'related'to'domains'(and'domain'walls) ' M H

Observa9on(techniques( Magneto.opbcal'Kerr'effect'(MOKE) ' D = ε ( E + iQ M × E ) weak'(but'detectable)'dependence'on'the'magnebzabon'of'the'opbcal'constants '

Observa9on(techniques( Transmission'Electron'Microscopy'(TEM) ' F Lorentz = q ( v × B ) electrons'are'deflected'by'the'Lorentz'force' '

Observa9on(techniques( Transmission'Electron'Microscopy'(TEM) '

Observa9on(techniques( r · B =0 z }| { r · H = �r · M longitudinal'variabons'of' M 'are'a'source'of'magnebc'field'(stray'field)'' Magnebc'force'microscopy'(MFM)' F = µ 0 ( m tip · r ) H stray

Observa9on(techniques( Spin.polarized'scanning.tunneling'microscopy'(SP.STM)' 90ML'Dy/W(110)

Observa9on(techniques( Spin.polarized'scanning.tunneling'microscopy'(SP.STM)' 1ML'Fe/W(110)

Observa9on(techniques(

The(origin(of(domains( On#the#theory#of#the#dispersion#of#magne2c#permeability#in#ferromagne2c#bodies## Landau'&'Lifshitz,'Phys.'Z.'Sowjet.'8,'153'(1935)'

The(origin(of(domains( E magnetostatic = µ 0 Z H 2 r · H = �r · M d ( M ) dV � ! 2 domains'form'to'minimize'the'magnebc'energy'' r · M = 0 n · M | surface = 0 ˆ

d Size(of(domains(( d )( � ⇒ ∂ E ( d, L ) � E ( d, L ) = = 0 � L ∂ d � d 0 strategy :'compute'the'total'magnebc'energy'of'the'system'and' determine' d 'from'the'principle'of'minimum'energy' E 1 . 7 M 2 d E ( d, L ) = + ε dw ( L/d ) | {z } | {z } magnetostatic domain wall KiYel's'law d 0 ∼ L 1 / 2 d d 0

d Size(of(domains(( d )( L 1 . 7 M 2 d E stripe ( d, L ) = + ε dw ( L/d ) | {z } | {z } magnetostatic domain wall E flux clousure ( d, L ) = K 2 M 2 d + ε dw ( L/d ) | {z } | {z } domain wall anisotropy KiYel's'law d 0 ∼ L 1 / 2 To'know'the'actual'size'of'the'domains'we'need'' to'determine'the' energy(of(the(domain(walls(

Domain(walls( Micromagnebc'formalism' discrete continuous z }| { z }| { S i = S ( r i ) S ( r ) = M ( r ) /M s � ! 1 � 1 2( ∆ r i · r S | r = r i ) 2 S ( r i ) · S ( r i + ∆ r i ) � ! Z h ⇣ M ( r ) ⌘i 2 X H ex = � J ij S i · S j A dv ( A ⇠ J/a ) � ! r M s ij

Domain(walls( Uniaxial'ferromagnet'( m #=# M /M s )' Z n o � µ 0 A ( r m ) 2 + K ( m 2 x + m 2 E = y ) 2 M · H d ( M ) dv | {z } | {z } | {z } exchange anisotropy stray field � � m = 0 , sin θ ( x ) , cos θ ( x ) Bloch(wall ( r · M = 0) Z 1 ( A θ 0 2 + K sin 2 θ ) dx ε dw = �1 A θ 00 − K sin θ cos θ = 0 dx ) 2 + K cos 2 θ d A ( d θ ⇥ ⇤ = 0 dx dx ) 2 + cos 2 θ = 1 K ( d θ A θ − g ¨ l sin θ cos θ = 0 q p 1 − cos 2 θ = sin θ A d θ ± dx = K q q d θ K ln tan θ K sin θ = ± 2 = A ( x − X ) A dx →

Domain(walls( Uniaxial'ferromagnet'( m #=# M /M s )' Z n o � µ 0 A ( r m ) 2 + K ( m 2 x + m 2 E = y ) 2 M · H d ( M ) dv | {z } | {z } | {z } exchange anisotropy stray field � � m = 0 , sin θ ( x ) , cos θ ( x ) Bloch(wall ( r · M = 0) Z 1 ( A θ 0 2 + K sin 2 θ ) dx ε dw = �1 A θ 00 − K sin θ cos θ = 0 cos θ sin θ θ ( x ) = 2 arctan[exp( x/w )] � p w = A/K � / � - 4 - 2 2 4 √ ε dw = 4 AK

d Domains(&(domain(walls( L Fe,'Ni w ∼ 10 nm d 0 ( L = 1 cm) ∼ 10 µ m E d = d 0 ⇠ E d = d 0 magnetostatic ⌧ E d →∞ magnetostatic dw singleNdomain(state( if d 0 ∼ ( wL ) 1 / 2 L . w − →

Domain(walls(in(thin(films( Bloch(wall Neel(wall r · M 6 = 0 r · M = 0 top'view L ! L ! demagnebzing'factors � W L N Bloch = N N´ eel = W + L W + L 5 � / � 1 2 3 4

Domain(walls(in(thin(films( L ! ''''''''''''.'Mulb.dimensional'descripbon'due'to'the'stray'fields' ''''''''''''.'Addibonal'length'scales # ''''''''''''.'Analybcal'.>'numerical'calculabons'&'ansatzs'+'variabonal'procedures (

Domain(walls(in(thin(films( crossN9e(wall Neel(wall r · M 6 = 0

Bloch(walls Bloch(lines(&(Bloch(points two'(equivalent)'' rotabon'senses'

rotabon'senses' two'(equivalent)'' Bloch(lines(&(Bloch(points Bloch(walls

Domain(wall(mo9on( H eff M γ = µ 0 ge ˙ M = − γ M × H e ff (gyromagnetic ratio) 2 m e | {z } torque Z [ A ( r M ) 2 � KM 2 E = z � M · H tot ] dv H tot = H ext + H d Z [( A r 2 M + KM z ˆ δ E = e z + H tot ) · δ M ] dv | {z } H eff Landau.Lifshitz.Gilbert'equabon:' ˙ − α M × ˙ M = − γ M × H e ff M | {z } | {z } torque damping

Domain(wall(mo9on( m = − γ m × H e ff − α m × ˙ ˙ m y x z Z [ A ( r m ) 2 � Km 2 K d m 2 � M · H ext ] d 3 x E = z + y | {z } K d = µ 0 m y = 0 → stray-field-free wall 2 M 2 s m = − γ (sin θ cos φ , sin θ sin φ , cos θ ) sin θ r · (sin 2 θ r φ ) + K d θ � α ˙ ˙ φ sin θ = 2 γ A M s [ � 2 sin θ sin 2 φ ] � K + K d sin 2 φ φ sin θ + α ˙ ˙ θ = 2 γ � ⇥ r 2 θ � 1 2 sin 2 θ ( r φ ) 2 ⇤ sin 2 θ + γ H sin θ A M s 2

Domain(wall(mo9on( m = − γ m × H e ff − α m × ˙ ˙ m y x z Z [ A ( r m ) 2 � Km 2 K d m 2 � M · H ext ] d 3 x E = z + y | {z } K d = µ 0 m y = 0 → stray-field-free wall 2 M 2 s m = − γ (sin θ cos φ , sin θ sin φ , cos θ ) sin θ r · (sin 2 θ r φ ) + K d θ � α ˙ ˙ φ sin θ = 2 γ A M s [ � 2 sin θ sin 2 φ ] � K + K d sin 2 φ φ sin θ + α ˙ ˙ θ = 2 γ � ⇥ r 2 θ � 1 2 sin 2 θ ( r φ ) 2 ⇤ sin 2 θ + γ H sin θ A M s 2 θ = θ ( x, t ) and φ = conts.

Domain(wall(mo9on( m = − γ m × H e ff − α m × ˙ ˙ m y x z Z [ A ( r m ) 2 � Km 2 K d m 2 � M · H ext ] d 3 x E = z + y | {z } K d = µ 0 m y = 0 → stray-field-free wall 2 M 2 s m = − γ (sin θ cos φ , sin θ sin φ , cos θ ) sin θ r · (sin 2 θ r φ ) + K d θ � α ˙ ˙ φ sin θ = 2 γ A M s [ � 2 sin θ sin 2 φ ] � K + K d sin 2 φ φ sin θ + α ˙ ˙ θ = 2 γ � ⇥ r 2 θ � 1 2 sin 2 θ ( r φ ) 2 ⇤ sin 2 θ + γ H sin θ A M s 2 θ = θ ( x, t ) and φ = conts. � α K d x θ − K + K d sin 2 φ � � � = 2 γ A ∂ 2 γ M s sin 2 φ − H sin θ sin 2 θ M s 2 | {z } | {z } =0 =0 Walker's(solu9on(( θ ( x, t ) = 2 arctan { exp[ ± ( x ± vt ) /w ∗ ] } , sin 2 φ = H/H c q A/ ( K + K d sin 2 φ ) , w ∗ = H c = α 2 M s , v = γ α w ∗ H cos φ sin φ ⇣ ⌘ m = cosh[( x − vt ) /w ∗ ] , cosh[( x − vt ) /w ∗ ] , ± tanh[( x − vt ) /w ∗ ]

Domain(wall(mo9on( The'wall'moves'at'a'constant'speed'(~' H# for'low'fields).' ' If'the'speed'increases'the'angle'increases'.>'stray'field'&'wall'narrowing.' ' There'is'a'maximum'velocity.' ' There'is'a'cribcal'field'above'which'this'solubon'is'not'valid.' ( (( Walker's(solu9on(( θ ( x, t ) = 2 arctan { exp[ ± ( x ± vt ) /w ∗ ] } , sin 2 φ = H/H c q A/ ( K + K d sin 2 φ ) , w ∗ = H c = α 2 M s , v = γ α w ∗ H cos φ sin φ ⇣ ⌘ m = cosh[( x − vt ) /w ∗ ] , cosh[( x − vt ) /w ∗ ] , ± tanh[( x − vt ) /w ∗ ]

Domain(wall(mo9on( Longitudinal(suscep9bility(( Z t γ γ α w ∗ α w ∗ i ω H ω e i ω t L z α w ∗ H ω e i ω t → ∆ x = i ω H ω e i ω t → ∆ M ω = v = γ vdt = d × Surface 0 χ l ( ω ) ≡ 1 H ω e i ω t = γ w ∗ ∆ M ω (relaxation behavior with no resonance) V i ω α d Walker's(solu9on(( θ ( x, t ) = 2 arctan { exp[ ± ( x ± vt ) /w ∗ ] } , sin 2 φ = H/H c q A/ ( K + K d sin 2 φ ) , w ∗ = H c = α 2 M s , v = γ α w ∗ H cos φ sin φ ⇣ ⌘ m = cosh[( x − vt ) /w ∗ ] , cosh[( x − vt ) /w ∗ ] , ± tanh[( x − vt ) /w ∗ ]

Domains(and(domain(walls( Andrs'Cano' - PowerPoint PPT Presentation

Domains(and(domain(walls( Andrs'Cano' ICMCB.CNRS,'University'of'Bordeaux,'France' Department'of'Materials,'ETH'Zrich,'Switzerland' References( On#the#theory#of#the#dispersion#of#magne2c#permeability#in#ferromagne2c#bodies##

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