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

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

Jan 21, 2023 135 likes •627 views

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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Domains(and(domain(walls(

Andrés'Cano'

ICMCB.CNRS,'University'of'Bordeaux,'France' Department'of'Materials,'ETH'Zürich,'Switzerland'

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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)'

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Outline(

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

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domain'wall'

domain'

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The'concept'of'domain:( ( '

Postulated' by' Pierre' Weiss' in' 1907' to' explain' why' ferromagnebc'bodies'can'appear'non.magnebc.'''

110(aniversary!

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Tc T +M0

  • M0

M

F = a(T − Tc)M 2 + bM 4

Two'possible'states'below'Tc#

ferromagnebc paramagnebc

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H M

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H M

Barkhausen(effect(

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H M

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

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Outline(

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

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Observa9on(techniques(

Magneto.opbcal'Kerr'effect'(MOKE)'

D = ε(E + iQM × E)

weak'(but'detectable)'dependence'on'the'magnebzabon'of'the'opbcal'constants'

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Observa9on(techniques(

Transmission'Electron'Microscopy'(TEM)' electrons'are'deflected'by'the'Lorentz'force''

FLorentz = q(v × B)

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Observa9on(techniques(

Transmission'Electron'Microscopy'(TEM)'

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Observa9on(techniques(

longitudinal'variabons'of'M'are'a'source'of'magnebc'field'(stray'field)''

Magnebc'force'microscopy'(MFM)'

F = µ0(mtip · r)Hstray

r·B=0

z }| { r · H = r · M

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90ML'Dy/W(110)

Observa9on(techniques(

Spin.polarized'scanning.tunneling'microscopy'(SP.STM)'

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1ML'Fe/W(110)

Observa9on(techniques(

Spin.polarized'scanning.tunneling'microscopy'(SP.STM)'

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Observa9on(techniques(

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Outline(

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

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The(origin(of(domains(

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

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domains'form'to'minimize'the'magnebc'energy''

r · M = 0 ˆ n · M|surface = 0

The(origin(of(domains(

r · H = r · M

  • !

Emagnetostatic = µ0 2 Z H2

d(M)dV

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Size(of(domains((d)(

E(d, L) = ⇒ ∂E(d, L) ∂d

  • d0

= 0

strategy:'compute'the'total'magnebc'energy'of'the'system'and' determine'd'from'the'principle'of'minimum'energy' d L

E(d, L) = 1.7M 2d | {z }

magnetostatic

+ εdw(L/d) | {z }

domain wall

d0 ∼ L1/2 KiYel's'law

d0 d E

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Size(of(domains((d)(

d L

To'know'the'actual'size'of'the'domains'we'need'' to'determine'the'energy(of(the(domain(walls( Estripe(d, L) = 1.7M 2d | {z }

magnetostatic

+ εdw(L/d) | {z }

domain wall

Eflux clousure(d, L) = K 2 M 2d | {z }

anisotropy

+ εdw(L/d) | {z }

domain wall

d0 ∼ L1/2 KiYel's'law

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Outline(

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

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Domain(walls(

Micromagnebc'formalism'

discrete

z }| { Si = S(ri)

  • !

continuous

z }| { S(r) = M(r)/Ms S(ri) · S(ri + ∆ri)

  • !

1 1 2(∆ri · rS|r=ri)2 Hex = X

ij

JijSi · Sj

  • !

Z A h r ⇣M(r) Ms ⌘i2 dv (A ⇠ J/a)

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Domain(walls(

Uniaxial'ferromagnet'(m#=#M/Ms)'

Bloch(wall

m =

  • 0, sin θ(x), cos θ(x)
  • εdw =

Z 1

1

(Aθ02 + K sin2 θ)dx Aθ00 − K sin θ cos θ = 0 E = Z n A(rm)2 | {z }

exchange

+ K(m2

x + m2 y)

| {z }

anisotropy

µ0 2 M · Hd(M) | {z }

stray field

  • dv

± q

A K dθ dx =

p 1 − cos2 θ = sin θ

A K ( dθ dx)2 + cos2 θ = 1 d dx

⇥ A( dθ

dx)2 + K cos2 θ

⇤ = 0

dθ sin θ = ±

q

K A dx

→ ln tan θ

2 =

q

K A (x − X)

(r · M = 0)

¨ θ − g

l sin θ cos θ = 0

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Domain(walls(

Uniaxial'ferromagnet'(m#=#M/Ms)'

Bloch(wall

m =

  • 0, sin θ(x), cos θ(x)
  • εdw =

Z 1

1

(Aθ02 + K sin2 θ)dx Aθ00 − K sin θ cos θ = 0 E = Z n A(rm)2 | {z }

exchange

+ K(m2

x + m2 y)

| {z }

anisotropy

µ0 2 M · Hd(M) | {z }

stray field

  • dv

θ(x) = 2 arctan[exp(x/w)] w = p A/K εdw = 4 √ AK

  • 4
  • 2

2 4

  • /

sin θ cos θ

(r · M = 0)

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Domains(&(domain(walls(

d L

Fe,'Ni

w ∼ 10 nm d0(L = 1 cm) ∼ 10 µm Ed=d0

dw

⇠ Ed=d0

magnetostatic ⌧ Ed→∞ magnetostatic

d0 ∼ (wL)1/2 − → L . w singleNdomain(state(if

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Domain(walls(in(thin(films(

Bloch(wall

top'view

r · M = 0 Neel(wall r · M 6= 0 L! L!

demagnebzing'factors

NBloch = W W + L NN´

eel =

L W + L

1 2 3 4 5 /

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Domain(walls(in(thin(films(

''''''''''''.'Mulb.dimensional'descripbon'due'to'the'stray'fields' ''''''''''''.'Addibonal'length'scales# ''''''''''''.'Analybcal'.>'numerical'calculabons'&'ansatzs'+'variabonal'procedures(

L!

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Domain(walls(in(thin(films(

Neel(wall r · M 6= 0 crossN9e(wall

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

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

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Outline(

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

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Domain(wall(mo9on(

˙ M = −γ M × Heff | {z }

torque

γ = µ0ge 2me

(gyromagnetic ratio)

δE = Z [(Ar2M + KMzˆ ez + Htot) | {z }

Heff

· δM]dv E = Z [A(rM)2 KM 2

z M · Htot]dv

Htot = Hext + Hd

Heff M

˙ M = −γ M × Heff | {z }

torque

− αM × ˙ M | {z }

damping

Landau.Lifshitz.Gilbert'equabon:'

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˙ θ α ˙ φ sin θ = 2γ

Ms [ A sin θr · (sin2 θrφ) + Kd 2 sin θ sin 2φ]

˙ φ sin θ + α ˙ θ = 2γ

Ms

  • A

⇥ r2θ 1

2 sin 2θ(rφ)2⇤

K+Kd sin2 φ

2

sin 2θ + γH sin θ

Domain(wall(mo9on(

E = Z [A(rm)2 Km2

z +

Kdm2

y

| {z }

Kd= µ0

2 M 2 s

M · Hext]d3x

˙ m = −γ m × Heff − αm × ˙ m

m = −γ(sin θ cos φ, sin θ sin φ, cos θ)

my = 0 → stray-field-free wall x y

z

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˙ θ α ˙ φ sin θ = 2γ

Ms [ A sin θr · (sin2 θrφ) + Kd 2 sin θ sin 2φ]

˙ φ sin θ + α ˙ θ = 2γ

Ms

  • A

⇥ r2θ 1

2 sin 2θ(rφ)2⇤

K+Kd sin2 φ

2

sin 2θ + γH sin θ

Domain(wall(mo9on(

E = Z [A(rm)2 Km2

z +

Kdm2

y

| {z }

Kd= µ0

2 M 2 s

M · Hext]d3x θ = θ(x, t) and φ = conts.

˙ m = −γ m × Heff − αm × ˙ m

m = −γ(sin θ cos φ, sin θ sin φ, cos θ)

my = 0 → stray-field-free wall x y

z

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˙ θ α ˙ φ sin θ = 2γ

Ms [ A sin θr · (sin2 θrφ) + Kd 2 sin θ sin 2φ]

˙ φ sin θ + α ˙ θ = 2γ

Ms

  • A

⇥ r2θ 1

2 sin 2θ(rφ)2⇤

K+Kd sin2 φ

2

sin 2θ + γH sin θ

Domain(wall(mo9on(

E = Z [A(rm)2 Km2

z +

Kdm2

y

| {z }

Kd= µ0

2 M 2 s

M · Hext]d3x θ = θ(x, t) and φ = conts.

˙ m = −γ m × Heff − αm × ˙ m

m = −γ(sin θ cos φ, sin θ sin φ, cos θ) γ αKd

Ms sin 2φ − H

  • |

{z } sin θ

=0

= 2γ

Ms

  • A∂2

xθ − K+Kd sin2 φ 2

sin 2θ | {z }

=0

  • m =

⇣ cos φ cosh[(x − vt)/w∗], sin φ cosh[(x − vt)/w∗], ± tanh[(x − vt)/w∗] ⌘ Walker's(solu9on(( θ(x, t) = 2 arctan{exp[±(x ± vt)/w∗]}, sin 2φ = H/Hc

w∗ = q A/(K + Kd sin2 φ), Hc = α

2 Ms,

v = γ

αw∗H

my = 0 → stray-field-free wall x y

z

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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.'

( (( m = ⇣ cos φ cosh[(x − vt)/w∗], sin φ cosh[(x − vt)/w∗], ± tanh[(x − vt)/w∗] ⌘ Walker's(solu9on(( θ(x, t) = 2 arctan{exp[±(x ± vt)/w∗]}, sin 2φ = H/Hc

w∗ = q A/(K + Kd sin2 φ), Hc = α

2 Ms,

v = γ

αw∗H

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Domain(wall(mo9on(

m = ⇣ cos φ cosh[(x − vt)/w∗], sin φ cosh[(x − vt)/w∗], ± tanh[(x − vt)/w∗] ⌘ Walker's(solu9on(( θ(x, t) = 2 arctan{exp[±(x ± vt)/w∗]}, sin 2φ = H/Hc

w∗ = q A/(K + Kd sin2 φ), Hc = α

2 Ms,

v = γ

αw∗H

Longitudinal(suscep9bility((

v = γ

αw∗Hωeiωt → ∆x =

Z t vdt =

γ αw∗

iω Hωeiωt → ∆Mω =

γ αw∗

iω Hωeiωt Lz d × Surface χl(ω) ≡ 1 V ∆Mω Hωeiωt = γw∗ iω αd (relaxation behavior with no resonance)

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Domain(wall(mo9on(

FieldN(vs.(currentNinduced(mo9on((

M = −gµB a3 S = −~γ a3 S Hint = −Js-d Z d3xS · (c†σc) | {z }

s

Exchange'interacbon'between'localized'(3d)'and'ibnerant'spins((

Spin-transfer torque Field-like torque Incident electron Outgoing electron

˙ S = γ S ⇥ (Heff + H0) α S S ⇥ ˙ S a3 2eS (js · r)S | {z }

spin-transfer torque

a3β 2eS2 [S ⇥ (js · r)S] | {z }

field-like torque

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Domain(wall(mo9on(

FieldN(vs.(currentNinduced(mo9on((

, r e

Current Current

Domain'wall

y e

the'angular'moment'lost'by'the'electrons'is'transferred'to'the'domain'wall((

spin.transfer'torque

˙ S = γ S ⇥ (Heff + H0) α S S ⇥ ˙ S a3 2eS (js · r)S | {z }

spin-transfer torque

a3β 2eS2 [S ⇥ (js · r)S] | {z }

field-like torque

n s

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Domain(wall(mo9on(

FieldN(vs.(currentNinduced(mo9on((

˙ S = γ S ⇥ (Heff + H0) α S S ⇥ ˙ S a3 2eS (js · r)S | {z }

spin-transfer torque

a3β 2eS2 [S ⇥ (js · r)S] | {z }

field-like torque

ll e n

  • e

n g s

  • 1

1 1 1 1 1 Writing Reading Shifting Write pulse Shift pulse racetrackNmemory(concept(

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Domain(wall(mo9on(

X(t)'can'be'understood'as'the'posibon'of'the'wall' ' What(is(the(conjugate(momentum?(

Dissipa9on(func9on( Spin(Lagrangian( LandauNLifshitzNGilbert(equa9on(⟷(EulerNLagrange(equa9ons( d dt ∂LS ∂ ˙ q + r · ∂LS ∂rq ∂LS ∂q = ∂WS ∂ ˙ q q = (θ, φ) LB is'a'spin'Berry'phase LS = LB − HS

  • M = −γ ~S

a3 n

  • θ = 2 arctan

h exp ⇣ ± x ± X(t) w∗ ⌘i φ0 = constant LB = Z d3x a3 ~S ˙ φ(cos θ 1) HS = S2 2 Z d3x a3 [J(rn)2 Kn2

z + K⊥n2 y + 2γ~ S n · H]

WS = α~S 2 Z d3x a3 ˙ n = α~S 2 Z d3x a3 ( ˙ θ2 + ˙ φ2 sin θ) Walker's(solu9on(

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Domain(wall(mo9on(

Spin(Lagrangian(&(dissipa9on(func9on( Walker's(solu9on( θ = 2 arctan h exp ⇣ ± x ± X(t) w∗ ⌘i φ0 = constant LS = −~NS w∗ ˙ φ0X + K⊥Sw

2~

sin2 φ0 − γXH

  • WS = ~NS

w∗ αw∗ 2 h⇣ ˙ X w∗ ⌘2 + ˙ φ2 i

X and ϕ0'are'conjugate'variables'

'

non.linear'relabon'due'to'internal'ϕ0 degree'of'freedom'(even'if'the'wall'is'rigid)'

X(t)'can'be'understood'as'the'posibon'of'the'wall' ' What(is(the(conjugate(momentum?(

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Domain(wall(mo9on(

Spin(Lagrangian(&(dissipa9on(func9on( LS = −~NS w∗ ˙ φ0X + K⊥Sw

2~

sin2 φ0 − γXH

  • WS = ~NS

w∗ αw∗ 2 h⇣ ˙ X w∗ ⌘2 + ˙ φ2 i Equa9ons(of(mo9on(for(the(rigid(wall(

1 2 3 4 /

  • (

)

  • φ0(t) = arctan

⇣ D tanh(t/τ) τ −1 + C tanh(t/τ) ⌘

τ −1 =

1 1+α2

p [αK⊥S/(2~)]2 − (γH)2 D = γH 1 + α2 , C = αK⊥S/(2~) 1 + α2

Transient(behavior(

1 w∗ ˙

X − α ˙ φ0 = κ⊥ sin 2φ0 ˙ φ0 +

α w∗ ˙

X = γH

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Domain(wall(mo9on(

Pinning( Equa9ons(of(mo9on(for(the(rigid(wall( Vpinning = − Z d3x a3 ∆K (Sa)3 2 δ(r) sin2 θ

local'change'of'the'' easy.axis'anisotropy'

  • 4
  • 2

2 4 /

  • Vpinning = ∆KS2

2 1 cosh2(X/w) → M 2 Ω2(X2 − w2)Θ(w − |X|)

1 w∗ ˙

X − α ˙ φ0 = κ⊥ sin 2φ0 ˙ φ0 +

α w∗ ˙

X = γH − νpin X w∗ Θ(w − |X|) | {z }

Fpinning

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Domain(wall(mo9on(

Equa9ons(of(mo9on(

¨ X + 2ακ⊥ ˙ X + 2νpinκ⊥X = 2γκ⊥w∗H domain'wall'mobon''⟶ mobon'of'an'effecbve'point'parbcle''

1 w∗ ˙

X − α ˙ φ0 = κ⊥ sin 2φ0 ˙ φ0 +

α w∗ ˙

X = γH − νpin X w∗ Θ(w − |X|) | {z }

Fpinning

Linear(regime(