Magnetisation dynamics at Magnetisation dynamics at different - - PowerPoint PPT Presentation

Magnetisation dynamics at Magnetisation dynamics at different timescales: different timescales: dissipation and thermal dissipation and thermal processes. processes.

Numerical modelling methodology. Numerical modelling methodology.

O.Chubykalo-Fesenko O.Chubykalo-Fesenko

Instituto de Ciencia de Instituto de Ciencia de Materiales de Madrid, Spain Materiales de Madrid, Spain

Fe elongated nanoparticles prepared by extrusion Lithographed Fe antidots Self-organized Co nanoparticles FePt nanoparticles Prepared by laser ablation CoCrPt magnetic recording media FePt nanoparticles

Objective: large-scale modelling of complex ferromagnetic materials

SmCo for hard magnets Very soft magnetic material: Finemet

Patterned FePt magnetic media

Objective: modelling of Objective: modelling of technological processes technological processes

Conventional magnetic recording Ultra-fast (fs) Kerr dynamics Heat-assisted magnetic recording All-optical magnetic recording

Introduction Introduction

• Magnetic system is not isolated, the magnetisation change

can occur at any timescale.

• Magnetism is a quantum phenomena.
• Ab-initio calculations, although rapidly developing, at the

present state of art are not capable to calculate magnetisation dynamics in complex materials at arbitrary timescale and temperature.

• At larger spatial scale,

relatively large magnetisation volumes (10nm) can be considered as classical particles.

The exchange term: micromagnetics The exchange term: micromagnetics versus spin models versus spin models

• Micromagnetics calculates the magnetostatic fields exactly but

which is forced to introduce an approximation to the exchange valid

• nly for long-wavelength magnetisation fluctuations.
• The exchange energy is essentially short ranged and involves a

summation of the nearest neighbours. Assuming a slowly spatially varying magnetisation the exchange energy can be written Eexch = Wedv, with We = A(m)2 with (m)2 = (mx)2 + (my)2 + (mz)2 The material constant A = JS2/a for a simple cubic lattice with lattice constant a. A includes all the atomic level interactions within the micromagnetic formalism.

• Atomistic models are discrete and use the Heisenberg

form of exchange

j i i j ij exch i

S S J E   .







Micromagnetic models of Micromagnetic models of nanostructured materials nanostructured materials

Models need nanostructure and micromagnetic parameters from experiment

Natural Natural magnetisation magnetisation dynamics: dynamics: 100 pico- 100 100 pico- 100 nano-second nano-second timescale timescale

Outline for today: 100ps- Outline for today: 100ps- 100ns (natural) dynamics 100ns (natural) dynamics

• Non-thermal dynamics:
• Ferromagnetic resonance
• Basic dynamical equation: the Landau-Lifshitz-

Gilbert

• The problem of magnetic damping (): main

processes

• Thermal dynamics:
• Principles of the Langevin dynamics.
• Modelling of thermal spinwaves
• The Landau-Lifshitz-Bloch micromagnetics for

dynamics close to Tc

Ferromagnetic resonance(FMR): Ferromagnetic resonance(FMR): (Arkadiev, 1911; Kittel, 1947) (Arkadiev, 1911; Kittel, 1947)

 



M N N H M N N H

z y z x

) ( ) (        Torque on magnetisation

The absorption line width contains Information on damping processes

A ferromagnetic body under applied field has a maximum absorption in frequencies:

The absorption peak contains information about anisotropy field.

Ferromagnetic resonance Ferromagnetic resonance

• The experiment is normally

performed in almost saturated conditions.

• The absorption peak contains

information about anisotropy field.

• The linewidth contains information

about dissipation processes.

FMR tecniques as a probe of FMR tecniques as a probe of magnetisation dynamics magnetisation dynamics

Courtesy of G.Kakazei et al

The Landau-Lifshitz (LL) and the The Landau-Lifshitz (LL) and the Landau-Lifshitz-Gilbert (LLG) equations Landau-Lifshitz-Gilbert (LLG) equations

• f motion
• f motion

     

H M M M H M dt M d

s LL

           

'

'   

Landau-Lifshitz damping, 1935

 

           dt M d M M H M dt M d

s G

       0

Gilbert damping, 1955 LL equation Gilbert equation (physically more reasonable for large damping) How the Gilbert equation could be transformed into the LL equation

2 2

1 1

G G LLG G

          , '

LLG equation

   

2 s s G

M M M dt M d dt M d M M dt M d M M M H M M dt M d M                                        

The LL eq. is equivalent to G equation with substitutions (for magnetization vector):

Convenient form of the LLG Convenient form of the LLG equation equation: :

 

s K K s

M K H m E H H h KV E E M M m 2 2          , / ,

int

    

• Anisotropy field

 

2

1

G Kt

H      /

   

h m m h m d m d              

Contains all contributions: anisotropy, Exchange, magnetostatic, Zeeman, depends on M

The Bloch-Bloembinger The Bloch-Bloembinger damping: damping:

 

Y X Y X Y X

M T H M dt M d

, , , 2

1              

 

) (

Z s Z Z

M M T H M dt M d           

1

1    

Transverse relaxation Longitudinal relaxation

The problem of damping The problem of damping: :

• Different relaxation processes:
• Magnon-magnon scattering
• Magnon-electron interactions (especially in

metals)

• Phonon-magnon interactions

(magnetostriction)

• Impurities
• Extrinsic factors (grain boundary, surface

roughness, etc.)

• Temperature disorder

Theory of magnetic Theory of magnetic damping constant damping constant  ) ) 

Uniform motion Spin waves Electron system Dissipation in lattice Impurities Surrounding body

Ferromagnets and their spin excitations

• 1. Uniform precession (ellipsoid)

 

    

i i B j i j i

g J H H S S S 

,

H0

); (

eff

t H M M      

.

eff

H   

. / M H  

tot eff

E  

More generally: 

• 2. Spin Waves

2

) ( ) ( Ak    k

E k 

M M k=2 k=2 / /  Heisenberg Heisenberg Hamiltonian Hamiltonian





i i B

V g S M 

Courtesy of K.Guslienko

Kittel formula for spinwaves Kittel formula for spinwaves dispersion relation: dispersion relation:

  

k s A A

M Ak H H Ak H H    

2 2 2 2

2 sin              

Applied field Anisotropy field Exchange interaction Magnetostatic interaction

Anisotropic single crystal ferromagnet:

Angle between M and k

 k 

Magnons and their Magnons and their interactions: interactions:

• Classical spinwaves correspond to quasiparticles

called magnons.

• Homogeneous magnetisation (FMR mode)

corresponds to magnon with k=0.

• Linear normal modes (magnons) do not interact.

Nonlinear processes correspond to magnon- magnon interactions.

1 2 3 4 magnon scattering Magnon decay 1 2 3 3 1 2 Two magnon merging These interactions define kinetic effects (e.x. heat conductivity) and width and shape of the FMR line and magnon lifetime

Nonlinear phenomena: Suhl Nonlinear phenomena: Suhl instabilities. instabilities.

• For large excitation power - FMR

saturation occurs

• If the density of magnons gets higher

than critical value – the homogeneous

• scillations become unstable

) ( ) ( ) ( k k k     2 2    

k=0 k

• k

The occurrence of the instability depends on the system geometry and is governed by the applied field.

) ( ) ( ) ( k k k     2    

Second condition, more easy to meet

Inherent relaxation processes Inherent relaxation processes (via spin-wave instabilities) (via spin-wave instabilities)

• Even without external dissipation it is possible to

reach magnetisation reversal via spin-wave instabilities.

Main non-inherent relaxation Main non-inherent relaxation processes: processes:

• Direct spin-lattice relaxation due to

nonuniformities

• Heterogeneity of composition
• Polycrystalline structure (grain coundaries, etc.)
• Nonuniform stresses, dislocations
• Geometrical roughness: pores, surfaces etc.
• Indirect spin-lattice relaxation
• Via ions with strong spin-orbital coupling
• Via charge carriers

The problem of damping ( The problem of damping ( ) )

• Although there exist theories trying to

evaluate the damping parameter basing

• n a particular mechanism, the

comparison with experiment remains poor.

• Normally the Gilbert damping  is a

phenomenological parameter, taken from the experiment.

• The values from FMR and direct

measurement of magnetisation switching (fast Kerr measurements) not always coincide.

W.K.Hiebert etal, PRB, PRL, Nature (2002) Scanning optical microscope

Observation of the precessional dynamics:

Simulation with LLG

Mx

Dinamical effects: Precesional switching:

Faster and less field.

Experiment with ps field pulses perpendicular to the magnetisatrion (C.Back et al, Science, 1999)

Simulación LLG Landau-Lifshitz-Gilbert (LLG):

     

eff eff

H M M H M M             

• H || M –non-precessional switching
• Prcessional switching is faster, however, the ringing

phenonema occur. Fe/GaAs

Comparison of Comparison of Patterns Patterns

Observed (SEMPA) Calculated (fit using LLG)

Anisitropies same as FMR Damping  = 0.017 4x larger than FMR WHY? Additional angular momentum dissipation? - spin current

pumped across interface into paramagnet causes additional damping (SPIN ACCUMULATION)

100 m

From Ch.Stamm- SLAC overview

Thermal effects Thermal effects

Thermal fluctuations play very important Thermal fluctuations play very important role in magnetisation dynamics: role in magnetisation dynamics: At the microscopic level:

• At the equilibrium they are responsible for thermally

excited spinwaves.

• Spinwaves are responsible for thermal magnetisation

reversal via the spinwave instabilities and energy transfer to main reversal mode.

At more macroscopic level

• Thermal fluctuations are responsible for random walk in

a complex energy landscape

• Eventually energy barriers could be overcome with the

help of thermal fluctuations leading to magnetisation decay.

The theory of thermal magnetization fluctuations of single domain, non-interacting particles was introduced by W.F.Brown (W.F.Brown Phys Rev 130 (1963) 1677)

“We now suppose that in the presence of thermal agitation, “the effective field” describes only statistical (ensemble) average of rapidly fluctuating random forces, and that for individual particle this expression must be augmented by a term h(t) whose statistical average is zero” <h

i

(t)>=0, <h

i

(t)h

j

(t+)>=

ij

i,j =x,y,z “The random-field components are formal concepts, introduced for convenience, to produce the fluctuations M” W.F.Brown outlined two methods:

• Based on the fluctuation-dissipation theorem
• Imposing the condition that the equilibrium solution of the Fokker-Planck

equation is the Boltzman distribution As a result of both in a non-interacting system:

) 1 ( 2

2

    

i s B

V M T k

Thermal micromagnetics Thermal micromagnetics

W.F.Brown, Phys Rev 130 (1963) 1677.

• Initially introduced

for nanoparticles

• This was brought to

micromagnetics. No correlations Between time and different particles!!

Note on the damping and Note on the damping and thermal processes. thermal processes.

• In principle, the Gilbert (or other) form of

damping is as a result of spin coupling with the oscillator thermal bath, in this sense, the thermal fluctuations are already included into the damping term.

• In some approximations, the undamped LL

equation is coupled to a system of

• scillators (phenomenological phonon

bath) and the resulting LLG damping is derived.

Fokker-Plank equation for Fokker-Plank equation for isolated nanoparticle : isolated nanoparticle :

 

P m m m D h m m h m m P                                       

 

T k m E m P

B eq

/ ) ( exp ) (    

Diffusion coefficient (strength of fluctuations)

V M T k D

s B

 

Boltzmann distribution in the equilibrium The noise can be introduced either to precessional term or to both damping and precessional terms

Problem of numerical scheme Problem of numerical scheme

• The noise is multiplicative although for small deviations – additive.
• Ito & Stratonovich interpretation of stochastical differential

equations- two different interpretations of stochastical integrals:

• The Ito evaluates the integral on the lower point of the integration

interval while the Stratonovich – in the middle one.

• The Ito intepretation produces a stochastical drift.
• Stratonovich interpretation should be used, for example the Heun

numerical scheme*.

• However, if after each integration step the magnetisation is

renormalized – normal scheme could be used**.

2 2 2 z i y i x i i i

m m m m m

, , ,

  

 

n n n n n t t t

W m t B m t B dW m t B

n n

  

 





) ( ) ( ) , (

, , 1 1

1

2 1 

*J. Garcia-Palacios et al, Phys Rev B 58 (1998) 14937 **D.Berkov et al J.Phys:Cond Mat 14(2002) 281

thermal

H H H     

int

Generalisation of the Langevin Generalisation of the Langevin dynamics to many spin problem: dynamics to many spin problem:

• The main assumption is that the noise

is uncorrelated in time (no memory effects, separation of timescales.

• Around the equilibrium the formalism of the

Onsager coefficients can be done for many spin system which shows that for particular damping (LLG) for many spin system no correlation between particles exist.

• In a general case – Fokker-Plank equation – no

solution exists.

Although the thermal fluctuations properties were derived for

• nly non-interacting particles, the same form of the

Langevin-LLG equation is used to calculate the switching properties even in an interacting system. O.Chubykalo et al J. Magn.Magn.Mat 226, (2003) 28

Langevin dynamics based on the Langevin dynamics based on the Landau-Lifshitz-Gilbert equation. Landau-Lifshitz-Gilbert equation.

could be formulated for both

• Atomistic spins (localized classical magnetic moments 

in the Heisenberg description with J and on-site anisotr. d), defines coupling to thermal bath Characteristic timescale is determined by exchange; fs-ps)

• Micromagnetic units (averaged magnetisation, Ms(T)), A(T),

K(T) The temperature in this case is included twice:

• The damping  contains already thermal averaging: 
• Langevin dynamics defines different trajectories

Characteristic timescale is determined by anisotropy; (ps-ns)

Modelling of thermal spinwaves Modelling of thermal spinwaves

• Langevin dynamics calculations have been carried out

for approximately 10 precessional periods

• Fourier transform in both space and time has been

performed

26 nm

0.00E+000 5.00E+012 1.00E+013 1.50E+013 2.00E+013 2.50E+013 0.00E+000 1.00E+011 2.00E+011 3.00E+011 4.00E+011 5.00E+011 6.00E+011

k=(0,0,5) |f(,k)|

2

 (s

• 1)

 

 



 



 

,

exp ) , ( ) , (

k k k x

t r k i k f t r m



   

5 10 15 20 25 30 35

• 2.00E+013

0.00E+000 2.00E+013 4.00E+013 6.00E+013 8.00E+013 1.00E+014 1.20E+014 1.40E+014

=(H+HK(T)+2A(T)/ (Ms(T)*d

2) sin 2(kd/2))

(1/s) k=(0,0,k) theory (T=0K)

=(H+HK+2J/ssin

2(ka/2))

T=10 K T=200 K T=350 K

Thermal Langevin dynamics: Thermal Langevin dynamics: micromagnetics versus atomistic spin (Heisenberg) micromagnetics versus atomistic spin (Heisenberg) model model

Atomistic Heisenberg model gives correct Tc Atomistic (classical) Heisenberg model for FePt (parametrised through ab-initio)

• N. Kazantseva et al.,
• Phys. Rev. B 77, 184428 (2008)

Langevin dynamics based on the Langevin dynamics based on the micromagnetic Landau-Lifshitz- micromagnetic Landau-Lifshitz- Gilbert equation. Gilbert equation.

 

) / )( / 1 ( ), 1 /(

2 i i s i i i i i i i

M E V M H t H M M H M d M d                         

ij i s B j i i therm

V M T k t h t h h h H H       ) ( ) 1 ( 2 ) ( ) ( ,

2 int

          

G.Grinstein and R.H.Koch, PRL 90 (2003) 207201.

Langevin dynamics for the micromagnetics does not correctly describe spinwaves:

• The spectrum is cut

and Tc is not correct

• density of states is

not correct.

Scaling approaches – correctly scale M(T), K(T), A(T) with discretization size within micromagnetics.

Atomistic modelling of magnetisation reversal

643 magnetic moments

• n cubic lattice

Field applied at 30o Field applied at 135o

• Magnetisation magninute is not

conserved

• Damping is enhanced at high T

Temperature-dependent magnetisation dynamics cannot be described within standard LLG approach.

O.Chubykalo-Fesenko et al, Phys Rev B 74 (2006) 094436

Longitudinal and transverse relaxation at high T

LLB equation LLB equation

Transverse (LLG) term Longitudinal term introduces fluctuations of M

D.Garanin Phys Rev B, 55 (1997) 3050.

LLB versus LLG equation: LLB versus LLG equation:

• Magnetisation length is not

conserved

• Temperature dependent

micromagnetic parameters

• Two relaxations: transverse and

longitudinal

• Damping parameters dependence on

temperature

• Valid both below and above Tc

Langevin dynamics based on the Landau-Lifshitz-Bloch equation.

 

 

 

 

 }

{

2 || 2 ||  

             

eff eff eff

H m m m m H m m H m m          

) ' ( ) ( 2 ) ' ( ) (

|| || ||

t t V T M T k t t

ij i s B j i

       

) ' ( ) ( 2 ) ' ( ) ( t t V T M T k t t

ij i s B j i

  

  

    

D.Garanin, O.Chubykalo-Fesenko, Phys.Rev B 70 (2004) 212409.

100 200 300 400 500 600 700 800 0.0 0.2 0.4 0.6 0.8 1.0

8

3 x 1.5nm discretization

Langevin + LLB

• Eq. magnetization

Temperature (K) MFA A(T) ~ me A(T) ~ me

1.4

A(T) ~ me

2

100 200 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

|M| time (*J/0)

T=1.3 *J/(1.44 J)*Tc

T=1.4 *J/(1.44 J)*Tc

T=1.35 *J/(1.44 J)*Tc

T=1.43 *J/(1.44 J)*Tc

T=1.438 *J/(1.44 J)*Tc

100 200 300 400

• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

M/M(t=0) time (*J/0) T=1.0*J /1.44 J*Tc

MFA

T=1.3*J /1.44 J*Tc

MFA

T=1.4*J /1.44 J*Tc

MFA

COMPARISON BETWEEN ATOMISTIC AND ONE-SPIN LLB SIMULATIONS atomistic

• ne-spin LLB

Multscale approach Multscale approach

Multiscale modelling: Multiscale modelling:

all the parameters were evaluated from atomistic all the parameters were evaluated from atomistic modelling for FePt with ab-initio input parameters modelling for FePt with ab-initio input parameters (Tc= 650K) (Tc= 650K) solid line – one spin LLB

Longitudinal relaxation Transverse relaxation

• The usual formalism for large-scale calculations of

magnetic properties is Micromagnetics.

• Although different theories of magnetic damping

parameters exist, due to a complexity of the problem, the damping parameter remains phenomenological.

• Thermal effects can be introduced, but the limitation of

long-wavelength fluctuations means that the standard micromagnetics cannot reproduce phase transitions.

• The Landau-Lifshitz-Bloch equation is a valid

micromagnetic formalism for high temperatures.

CONCLUSIONS