Landau-Lifshitz-Gilbert-Brown - UPMCleticia/SEMINARS/cergy-15.pdf · Landau-Lifshitz-Gilbert-Brown...

Landau-Lifshitz-Gilbert-Brown

Leticia F. Cugliandolo

Université Pierre et Marie CurieSorbonne Universités

[email protected]/ leticia

C. Aron, D. Barci, LFC, Z. González-Arenas, G. S. Lozano,

J. Stat. Mech. (2014) P08020 & in preparation

F. Romá, LFC, G. S. Lozano,

Phys. Rev. E 90, 023203 (2014) & arXiv : 1412.7564

LFC, P-M. Déjardin, G. S. Lozano & F. van Wijland

arXiv : 1412.6497

January 2015, Cergy-Pontoise, France

1

Plan

•Macromagnetism: the stochastic Landau-Lifshitz-Gilbert-Brown (sLLGB)

equation (precession, dissipation, noise and torque).

• Discretisation schemes for multiplicative white noise (Markov) stochas-

tic differential equations. Asymptotic measure and drift term.

• Back to the sLLGB equation, analytic approach:

drift term & functional formalism in Cartesian and spherical coordi-

nates ; exact results.

• Back to the sLLGB equation, numerical simulations:

analysis of a benchmark.

• Future work.

2

Plan










• Future work.

3

Magnetisation precessionBloch equation

Evolution of the time-dependent 3d magnetisation density per unit vo-

lume, M = (Mx,My,Mz), with constant modulus Ms = |M|

dtM = −γ0 M ∧Heff

γ0 ≡ γµ0 is the product of γ = µBg/ℏ, the gyromagnetic ratio, and µ0, the

vacuum permeability constant (µB Bohr’s magneton and g Lande’s g-factor)

For the initial condition M(ti) = Mi

the magnetisation precesses around Heff

with 2M · dtM = dt|M|2 = 0

and dt(M ·Heff) = 0 (if Heff = ct)

Bloch 32

4

Dissipative effectsLandau-Lifshitz & Gilbert equations

dtM = − γ01 + η2γ20

M ∧[Heff +

ηγ0Ms

(M ∧Heff)

]Landau &

Lifshitz 35

dtM = −γ0 M ∧(Heff − η

MsdtM

)Gilbert 55

2nd terms in RHS : dissipative mechanisms slow

down the precession and push M towards Heff


and dt(M ·Heff) > 0

5

Magnetic fieldConservative & non-conservative contributions

Local magnetic field Heff = Hceff +Hnc

eff

Conservative contribution Hceff = −µ−1

0 ∇MU

with U the energy per unit volume, can originate from an external (possibly time-

dependent) magnetic field Hext, and a crystal field Hani associated to the crys-

talline anisotropy,

U = −µ0M ·Hext + Vani(M)

Vani(M) is the anisotropy potential (per unit volume), e.g.

Vani(M) = K∑

i=j M2i M

2j (cubic crystalline structure)

Vani(M) = K(M2s −M2

z ) (uniaxial symmetry)

Bertotti, Mayergoyz & Serpico 09 ; Coffey & Kamykov 12

6

Magnetic fieldNon-conservative component

Local magnetic field Heff = Hceff +Hnc

eff

Non-conservative contribution Hnceff = −µ−1

0 ∇MU

Spin-torque term in Gilbert equation

Hnceff = −gµBℏJ(t)P

2M2s de

(M ∧ p)

J(t) the current per unit area,

P the (dimensionless) polarisation function of the fixed layer,

p is a unit vector in the direction of the current,

d the interlayer separation

e is the electric charge of the carriers.Berger 96 ; Slonczewski 96

7

Thermal fluctuationsÀ la Langevin in Gilbert’s formulation

dtM = −γ0M ∧(Heff +H− η

Ms

dtM

)

H is a white random noise, with zero mean ⟨Hi(t)⟩ = 0 and correlations

⟨Hi(t)Hj(t′)⟩ = 2Dδijδ(t− t′)

For the moment, the (diffusion) parameter D is free.

The noise H multiplies the magnetic moment M.

This is the Markov stochastic Landau-Lifshitz-Gilbert-Brown (sLLGB) multi-

plicative white noise stochastic differential equation.

Subtleties of Markov multiplicative noise processes are now posed.Brown 63

8

sLLGBAim

• Set up a path-integral formalism.

• Set up a numerical integrator.

Use them.

9

Plan










• Future work.

10

Stochastic calculusDiscretization prescriptions

dtx(t) = f [x(t)] + g[x(t)]H(t)

with ⟨H(t)⟩ = 0 and ⟨H(t)H(t′)⟩ = 2D δ(t− t′) means

x(t+ dt) = x(t) + f [x(t)] dt+ g[x(t)]H(t)dt

withx(t) = αx(t+ dt) + (1− α)x(t)

and 0 ≤ α ≤ 1. Particular cases are α = 0 Ito ; α = 1/2 Stratonovich.

The chain rule for the time-derivative is

dtY (x) = dtx dxY (x) +D(1− 2α) g2(x) d2xY (x)

For α = 1/2 (Stratonovich) one recovers the usual expression.

Stratonovich 67 ; Gardiner 96 ; Øksendal 00 ; van Kampen 07

11

Stochastic calculusFokker-Planck & stationary measure

The Fokker-Planck equation

∂tP (x, t) = −∂x[(f(x) + 2Dαg(x)dxg(x))P (x, t)]

+D∂2x[g

2(x)P (x, t)]

has the stationary measure

Pst(x) = Z−1 [g(x)]2(α−1) e1D

∫ x f(x′)g2(x′) = Z−1e−

1DUeff(x)

with Ueff(x) = −∫ x f(x′)

g2(x′) + 2D(1− α) ln g(x)

Remark : the potential Ueff(x) depends upon α and g(x).

Noise induced phase transitions

Stratonovich 67 ; Sagués, Sancho & García-Ojalvo 07

12

Stochastic calculusFokker-Planck & stationary measure

e.g. Ueff(x) = V (x) + 2D(1− α) ln g(x)Ueff

xUeff

x

x2 + 2D(1− α) ln x x2 + 2D(1− α) ln(1− x2)

g(x) = x g(x) = (1− x2)

13

Stochastic calculusDrift

The Gibbs-Boltzmann equilibrium

PGB(x) = Z−1 e−βU(x)

is approached if

f(x) 7→ −g2(x)dxU(x)︸︷︷︸−2Dαg(x)dxg(x)︸︷︷︸Potential drift

Remark: the drift is also needed for the Stratonovich mid-point scheme.

Important choice : if one wants the dynamics to approach thermal equi-

librium independently of α and g the drift term has to be added.

14

Stochastic calculusPath-integral representation

The initial state at time−T is drawn from a probability distributionPi(x−T ).

The noise generates random trajectories with probability densityP (x;α).

P (x;α) = ⟨∏

t δ(xt − xsolt )⟩ where xsol

t is a solution to the Langevin

equation and depends on the noise H .

The constraint can be written J δ(∏

t Eqt[x,H;α]) with the Jacobian

J = dettt′

[δEqt[x,H;α]

δxt′

]Using the exponential representation of the delta δ(y) ∝

∫dy eiyy

We write the probability density of a trajectory xt from t = −T to t = T

P (x;α) = Pi(x−τ ) ⟨ J∫

D[x] e∫ T−T dt ixt Eqt[x,H;α] ⟩

15

Stochastic calculusPath-integral representation

P (x;λ, α) ∝∫

D[x] P (x, x;λ, α) =

∫D[x] eS(x,ix;λ,α)

and the Martin-Siggia-Rose-Janssen 79 action

S(x, ix;λ, α) ≡ lnPi(x−T , λ−T )

+

∫ [ixt(dtxt − ft + 2Dαgtdxgt) +D(ixt)

2g2t − αdxft︸︷︷︸]

From the Jacobian and the integration over the noise

cfr. Langouche, Roekaerts & Tirapegui 79 ; Lau & Lubensky 07

Remark: The action depends on α and g.

Observable averages can now be calculated as

⟨A(xt)⟩ =∫

D[x] P (x;λ, α)A(xt) (and do not depend on α)

16

Path probabilitiesPath-integral representation

P (T x, T x;λ, α)

P (x, x;λ, α)= e∆S(x,x;λ,α)

where T x and T x are transformed trajectories,

λ the transformed parameter in the potential,

α a different discretization parameter ;

and from here obtain relations between observables by averaging this re-

lation : equilibrium fluctuation dissipation (∆S = 0), or out of equilibrium

theorems (∆S = 0).

Jarzinsky 97, Crooks 00, & many others

17

Stochastic calculusTransformations in the path-integral representation

Let us define

d(α)t xt ≡ dtxt − 2β−1(1− 2α)gtdxgt

and group two terms in the action due to the coupling to the bath

Sdiss[x, ix] =

∫ixt [d

(α)t xt + β−1ixtg

2t ]

This expression suggests to use the generalized transformation on the

time-dependent variables {xt, ixt}

Tc =

xt 7→ x−t ,

ixt 7→ ix−t − βg−2−t d

(α)t x−t ,

18

Stochastic calculusConsequences of the transformation

For initial conditions drawn from Pi(x) = Z−1e−βU(x) and

f(x) = −g2(x)dxU(x) + 2Dαg(x)dxg(x) one proves

Sdet+jac[Tcix, Tcx;α] = Sdet+jac[ix, x;α]

that implies

P [Tcix, Tcx;α] = P [ix, x;α]

From this result we can prove exact equilibrium relations such as the

fluctuation-dissipation theorem

R(t, t′) = δ⟨xt⟩δht′

∣∣∣h=0

= β ∂t′⟨xtxt′⟩ θ(t− t′)

19

Stochastic calculusConsequences of the transformation

For initial conditions drawn from Pi(x) = Z−1e−βU(x) and

f(x, λt) = −g2(x)∂xU(x, λt) + 2Dαg(x)dxg(x) one proves

P [Tcix, Tcx;α, λt]

P [ix, x;α, λt]= eβW−β∆F

with

W =

∫dt dtλt ∂λU(x, λ)

∆F = lnZ(λT )− lnZ(λ−T )

the work, and free-energy difference between initial and fictitious final states.

Exact out of equilibrium relations such as the Jarzinsky relation follow

⟨e−βW ⟩ = e−β∆F

20

Plan










• Future work.

21

Magnetisation dynamicsConservation of magnetisation modulus

dtM = −γ0M ∧ (. . . ) → M · dtM = 0

butdt|M|2 = dtM

2s = 4D(1− 2α)

γ20

1 + η2γ20

M2s = 0

because of theα-dependent chain rule associated to the Markov Landau-

Lifshitz-Gilbert-Brown stochastic equation.

The sLLGB equation has to be modified to ensure the modulus conser-

vation. This is achieved by replacing the derivative by the operator

dt 7→ D(α)t = dt + 2D(1− 2α)

γ20

1 + η2γ20

22

Magnetisation dynamicsFokker-Planck equation and Gibbs-Boltzmann equilibrium

With this change, the probability density P (M, t) is determined by an

α-independent Fokker-Planck equation.

If only conservative fields are applied Heff = µ−10 ∇MU one proves

that its asymptotic solution is of the Gibbs-Boltzmann form

PGB(M) = Z−1 e−βV U(M)

with V the sample volume and U the potential energy per unit volume,

provided the diffusion coefficient in the noise-noise correlation be chosen

to satisfy

D =ηkBT

MsV µ0

23

FormalismChoices

Path-integral construction:

Landau-Lifshitz & Gilbert formalism ; Cartesian & spherical coordinates.

Aron, Barci, LFC, González-Arenas & Lozano 14

Generic results on equilibrium and out of equilibrium theorems:

We preferred to use the Gilbert formalism in Cartesian coordinates.


Numerical simulations.

We preferred to use the Landau-Lifshitz formalism in spherical coordinates.

Romá, LFC & Lozano 14

24

The actionin the Gilbert formalism in Cartesian coordinates

SG = SG,det + SG,diss + SG,jac

with

SG,det = lnPi[M(ti),Heff(ti)] +

∫iM∥ ·D

(α)t M

+

∫iM⊥ ·

(M−2

s D(α)t M ∧M+ γ0Heff

)SG,diss =

∫γ0iM⊥ ·

(Dγ0iM⊥ − η

MsD

(α)t M

)SG,jac =

αγ01 + η2γ20

1

Ms

∫ [2ηγ0M ·Heff +MsϵijkMk∂jH

nceffi

−ηγ0(M2s δij −MiMj)∂jHeffi

]

25

The transformationin the Gilbert formalism in Cartesian coordinates

Tc =

Mt 7→ −M−t

γ0 iM⊥t 7→ −γ0 iM

⊥−t − βV µ0 dtM−t

iM∥t 7→ iM

∥−t

andα = 1− α

if one simultaneously changes the sign of the effective field

Hceff t 7→ −Hc

eff−t .

This change is a consequence of the transformation Mt 7→ −M−t when the

effective field derives from a potential Heff = −µ−10 ∇MU with U an even

function of M.

26

The transformationin the Gilbert formalism in Cartesian coordinates

With this formalism one proves

the equilibrium fluctuation-dissipation theorem.

out of equilibrium fluctuation relations.

One could also apply these ideas in the Lifshitz-Landau formulation and

in spherical coordinates.

Aron, Barci, LFC, González-Arenas & Lozano, arXiv:1412.7564

27

Plan










• Future work.

28

FormalismChoices

Path-integral construction:

Landau-Lifshitz & Gilbert formalism ; Cartesian & spherical coordinates.


Generic results on equilibrium and out of equilibrium theorems:

We preferred to use the Gilbert formalism in Cartesian coordinates.


Numerical simulations.

We preferred to use the Landau-Lifshitz formalism in spherical coordinates.

Romá, LFC & Lozano 14

29

Magnetisation dynamicsThe equations in spherical coordinates

The vector M defines the local basis (er, eθ, eϕ) with M ≡ Ms er(θ, ϕ)

and the Cartesian components Mx(t) = Ms sin θ(t) sinϕ(t), My(t) =

Ms sin θ(t) cosϕ(t) and Mz(t) = Ms cos θ(t) .

The sLLGB equation in this system of coordinates becomes dtMs = 0,

dtθ =D(1− 2α)γ201 + η2γ20

cot θ +γ0

1 + η2γ20[Heff,ϕ +Hϕ

+ηγ0(Heff,θ +Hθ)] ,

sin θ dtϕ =γ0

1 + η2γ20[ηγ0(Heff,ϕ +Hϕ)− (Heff,θ +Hθ)] ,

withHθ = Hx cos θ cosϕ+Hy cos θ sinϕ−Hz sin θ andHϕ = −Hx sinϕ+

Hy cosϕ. Similarly for Heff .

30

Magnetisation dynamicsAdimensionalisation and α-points

We introduce m = M/Ms, heff = Heff/Ms, h = H/Ms

τ = γ0Mst, η0 = ηγ0, and D0 = Dγ0/Ms

We define the α-prescription angular variables

θατ ≡ αθ(τ +∆τ) + (1− α)θ(τ) ,

ϕατ ≡ αϕ(τ +∆τ) + (1− α)ϕ(τ) ,

with 0 ≤ α ≤ 1. The effective fields at the α-point are

hαeff,θ ≡ heff,θ(θατ , ϕ

ατ ) hαeff,ϕ ≡ heff,ϕ(θ

ατ , ϕ

ατ )

We first draw the Cartesian components of the fields as ∆Wi = hi∆τ =

ωi

√2D0∆τ where the ωi are Gaussian random numbers with mean zero and

variance one, and we then calculate ∆Wϕ = hϕ∆τ and ∆Wθ = hθ∆τ .

31

Magnetisation dynamicsThe equations in spherical coordinates

The discretized dynamic equations now read Fθ = 0 and Fϕ = 0 with

Fθ ≡ − (θτ+∆τ − θτ ) +D0∆τ(1− 2α)

(1 + η20)cot θατ

+∆τ

1 + η20

[hαeff,ϕ + η0h

αeff,θ

]+

1

1 + η20[∆Wϕ + η0 ∆Wθ]

Fϕ ≡ − (ϕτ+∆τ − ϕτ )

+∆τ

1 + η20

[η0 h

αeff,ϕ − hαeff,θsin θατ

]+

1

1 + η20

[η0 ∆Wϕ −∆Wθ

sin θατ

]We used a Newton-Raphson routine and we imposed F 2

θ + F 2ϕ < 10−10 to

find ϕτ+∆τ and θτ+∆τ .

To avoid singular behavior when the magnetization gets too close to the z axis

we apply a π/2 rotation of the coordinate system around the y axis.

32

Magnetisation dynamicsThe benchmark

Uniformly magnetised ellipsoid with volume V = 6.702× 10−26 m3.

The potential energy per unit volume is

U(M) = −µ0 M ·Hext +µ0

2(d2xM

2x + d2yM

2y + d2zM

2z )

We set Hext = 0 and dx = dy = 0.4132 and dz = 0.0946

After adimensionalisation

the friction coefficient becomes η0 = ηγ0 ≪ 1

the time-step ∆τ = 1 corresponds to ∆t = 3.2 ps

We set the temperature to T = 300 K ; thenkBT

V∆U≃ 0.153

d’Aquino, Serpico, Coppola, Mayergoyz & Bertotti 06

33

Magnetisation dynamicsTypical trajectories of the three magnetisation components

0 6-1

0

1

(c)

time [µs]

mz

0 6-1

0

1

(b)

time [µs]

my

0 6-1

0

1

time [µs]

mx

(a)

34

Magnetisation dynamicsDecay of the averaged mz component

Stratonovich scheme

0 1x106

2x106

3x106

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Prescription: α = 0.5

Time step: ∆τ = 0.5

τ

<m

z>

0 3x106

-0.003

0.003

τ

<m

y>

0 3x106

-0.003

0.003

τ

<m

x>

35

Magnetisation dynamicsDistribution of the mz component

Stratonovich scheme, comparison to the analytic form (solid line)

-1.0 -0.5 0.0 0.5 1.0

10-2

10-1

100

101

102

103

τmax

=3.2x106

τmax

=2x105

τmax

=1x105

m

z

P(m

z )

(a)

0 1x10-5

0.00

0.25

Sτ

-1

max

36

Magnetisation dynamicsDecay of the mz component

Dependence on ∆τ ; α = 1/2 (main) and α = 0 (inset)

cfr. Kloeden & Platen Numerical solutions of stochastic differential equations

0.0 5.0x104

1.0x105

1.5x105

0.5

0.6

0.7

0.8

0.9

1.0

<m

z

>

τ

∆τ=0.5

∆τ=0.05

∆τ=0.005

α = 0.5

0 1x105

0

1

α = 0.0

<m

z

>

τ

37

Magnetisation dynamicsDistribution of the mz component, Ito scheme

-1.0 -0.5 0.0 0.5 1.010

-3

10-2

10-1

100

101

102

103

α = 0.0

∆τ=0.5

∆τ=0.05

∆τ=0.005

P(m

z )

m

z

-1 0 10.0

1.5

P

(mx )

m

x

For α = 0, ∆τ = 0.05 is needed to get close to the equilibrium pdf.

38

ConclusionsNumerics

• All discretisation schemes converge to Gibbs-Boltzmann equilibrium

P (M) = Z−1e−βV U(M)

once the correct drift term has been added.

• The magnetisation modulus is conserved by the numerical integration

with no need for artificial rescaling.

• The Stratonovich convention is the most efficient one in the sense that

one can use larger values of ∆τ and stay close to the continuous

time limit.

We will study the dependence of the algorithms’ precision, for different

α, in the future

39

Plan










• Future work.

40

Related & future workSome projects en route

Analytics : Dean-Kawasaki stochastic equation for the spatial magneti-

sation density of magnetic moments in pair interaction and controlled by

the sLLGB microscopic equation.

LFC, Déjardin, Lozano & van Wijland, arXiv:1412.6497

Numerics : spin-torque effect, experimentally relevant situations.

Better understanding of the algorithms’ precision.

Romá, LFC & Lozano

41

Dissipative effectsLandau-Lifshitz & Gilbert equations

dtM = − γ01 + η2γ20

M ∧[Heff +

ηγ0Ms

(M ∧Heff)

]Landau &

Lifshitz 35

2nd terms in RHS : dissipative mechanisms slow

down the precession and push M towards Heff


and dt(M ·Heff) > 0

Proof : take the scalar product with Heff and use (M ·Heff)2 < M2

sH2eff

42

Magnetisation dynamicsDistribution of the mx component

Stratonovich scheme, Gaussian pdf

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

τmax

=3.2x106

τmax

=2x105

τmax

=1x105

m

x

P(m

x )

(b)

43

Magnetisation dynamicsDecay of the average mz component

0.0 5.0x104

1.0x105

1.5x105

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

α = 0.5

∆τ = 0.5

η0 = 0.005

η0 = 0.01

η0 = 0.02

η0 = 0.04

η0 = 0.08

<m

z

>

τ

∆τ = 0.005

η0 = 0.005

η0 = 0.01

η0 = 0.02

η0 = 0.04

η0 = 0.08

(a)

Stratonovich scheme

dependence on the friction coefficient η0 and the time-step ∆τ

44

Magnetisation dynamics⟨mz⟩ under an applied field Hext = 0

For α = 0.5, little dependence on ∆τ = 0.05 vs ∆τ = 0.5

Note that a very large value ∆τ = 0.5 can be used.

45

Magnetisation dynamics⟨mz⟩ under an applied field Hext = 0

Dependence on α cured by using smaller ∆τ :

poorer precision for α = 0.5 confirmed.

46

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