Magnetisation dynamics at different timescales: dissipation and - - PowerPoint PPT Presentation

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

O.Chubykalo-Fesenko

Instituto de Ciencia de Materiales de Madrid, Spain

Different timescales:

10-3s ms 10-6s s 10-9s ns 10-11s ps 10-14s fs 10-0s s 103s hs 106s month 109s years

All-optical laser-pulsed experiments Electron-spin relaxation processes. Magnetisation precession. Fast-Kerr measurements

Hysteresis measurements.

Conventional magnetometers (VSM, SQUID)

Magnetic viscosity experiments. Long-time thermal stability for magnetic recording. Langevin dynamics

• n atomistic level

Langevin dynamics

• n micromagnetic

level kinetic Monte Carlo with energy barriers calculations dynamics acceleration techniques

Outline of the talk

 Long-time (>s) magnetisation dynamics

• The concept of energy barriers and switching for an individual

nanoparticle

• Calculation of energy barriers for more complicated systems
• Energy barriers for systems of interacting nanoparticles
• Kinetic Monte Carlo for thermal decay evaluation in a completely

interacting system.

• Ultra-short timescale magnetisation dynamics

(fs-ps)

• The ultra-fast pump-probe experiment
• Modelling
• Introduction of the correlated noise approach.

Long timescale (up to years)

The Stoner-Wolfarth particle

  cos cos MHV KV E   

2

2 2 1 1 1

m f m f dt dm    /

) / exp( ) (  t t m f f   

1 2

Easy axis H M 

 

 

 

1

f E kT exp( / ) 

=f1,2

• Master equation

s K K

M K H H H KV E / , 2 1

2

         

The Arrhenius-Neél law

2 4 6 8 10 10 10

1

10

2

10

3

10

4

10

5

10

6

Asymptote integration of the Landau-Lifshitz equation LLL-MC

Trelax 10

• 13 s

Ebarr/kBT

Energy barrier:

  cos cos MH KV E   

2

Brown’s asymptote:

 

T k h KV h h T k KV M K

B B s

/ ) ( exp ) )( ( ) (

2 2 2

1 1 1 1 2 1          

• In general, f0=1/0=F(H, but in most of cases the approximation

f0=109-1011 s-1 is sufficient For an independent small particle with applied field parallel to anisotropy The reversal probability is obtained from the solution of the Fokker-Planck equation The general problem does not have solution.

Relaxation in complex systems

 



      E d E E t t m ) ( ) ( / exp ) (  

 

E E E      ,

 

const E   

If in some interval

) log( ) ( t S M E t m    

 

 

 

1

f E kT exp( / ) 

• widely observed behavior
• Based on the assumption of

Homogeneous energy barrier distribution which is constant in time

M(t) log (t/t0)

H=const T=const

) (log / t d dM S 

• magnetic viscosity

S H

) /(

s B act

HM T k V  

Activation volume Hc

Vact

Superparamagnetism

 The relaxation time of a grain is given by the Arrhenius-Neel

law

 where f0 = 109s-1. and E is the energy barrier  This leads to a critical energy barrier for superparamagnetic

(SPM) behaviour

 where tm is the ‘measurement time’  Nanoparticles with E < Ec exhibit thermal equilibrium (SPM)

behaviour - no hysteresis

 

 

 

1

f E kT exp( / ) 

E KV k Tln t f

c c B m

  ( )

KV>25kBT – for stability at room temperature, KV>60kBT – for magnetic recording

Slow processes: k

B

T <<E (Energy barrier)

• Energy barrier calculation is essential part for determination of

long-time thermal stability and slow thermal relaxation

• This is important from the point of view of magnetic recording

applications.

• Evalulation of energy barriers should be done in a

multidimensional space and is a difficult problem

• in an interacting system energy barriers are dynamical and

should be constantly recalculated.

) / exp( T k E t t

B i

 

Slow processes: k

B

T <<E (Energy barrier)

Energy barriers should satisfy conditions: grad E =0 Only one (lowest) eigenvalue of the Hessian matrix 

j i m

m E   2

Ridge optimisation method

Ridge optimization method The obtained point is checked:

• To have a unique negative eigenvalue
• To separate the basins of attractions of

the two minima from which one is initial Similar method –elastic band

Constrained (Lagrangian multiplier) method:

), (       N H F

  

i i i i

m m 

(for simple cases only)

Altura D=2r

Fe dot

0.0 0.2 0.4 0.6 0.8 1.0

• 34.840
• 34.835
• 34.830
• 34.825
• 34.820
• 34.815

0.2 0.4 0.6 0.8 1.0



/

 

/



4 6 8 10 12 14 16 18 20 22 24 26

80 120 160 200 240 280 320 360 400 440 Lagrangian method KV EB / (kB TRoom) L(nm)

Varying the length-> different saddle point configurations corresponding to different reversal mechanism One atomistic FePt grain

Energy barriers in a single FePt grain

0.0 0.2 0.4 0.6 0.8 20 40 60 80 100 120 140 160 180

4S(A (K+M

2 Hard)) 1/2

4S(A K)

1/2

KhardVhard / 300 kB

MHard=1100 emu/cm

3

MSoft=1270 emu/cm

3

E(kBT) Js/J

Individual energy barriers soft layer goes first hard layer follows Collective energy barrier is defined by the energy of the domain wall in the hard layer For Js>0.1J energy barriers are collective and larger than individual energy barrier of the hard material saddle point configuration saddle point configuration Energy barriers as a function

• f Js: soft/hard grain

Energy barriers for systems of nanoparticles

The Pfeiffer approximation

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.00 0.05 0.10 0.15 frequency EB/2KV exact Pfeiffer

 

                   



) , ( ) , ( ) ( 14 . 1 86 . ) ( ) ( sin ) ( [cos ) ( ) ( / 1 [

int int 1 min 2 min 2 / 3 3 / 2 3 / 2 ) ( int

h M h M E E E E E g k g g h KV E

Pf Pf B k Pf

         



 is the angle between anisotropy and interaction field

The Pfeiffer approximation is slightly displaced to smaller values

Co particles

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

E/KV

hex/hK

Angle between easy axes solid symbols- approxim (Pfeiffer). empty symbols - exact Angles between anistoropy axes /4 /2

H.Pfeiffer Phys Status Solidi A 118, 295 (1990):

Multidimensional energy barrier distribution for Co and FePt particles

(only magnetostatic interactions)

0,48 0,50 0,52 0,54 0,0 0,1 0,2 0,3

frequency

E/(2KV)

annealed regular

0.2 0.4 0.6 0.8 1.0 1.2 0.00 0.05 0.10 0.15

frequency EB/(2KV)

annealed structure regular array

Co FePt

Multidimensional energy barrier distribution for annealed FePt particles array in the presence of exchange

0.4 0.6 0.8 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 frequency E/(2KV)

all barriers FePt structure1 Hex=0 Hex/HK=0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

probability EB/(2K<V>)

Volume c=0.05 c=0.2 c=0.36 c=0.56

Multidimensional energy barrier distributions evaluated at the remanence

Magnetostatic interactions:

• Broaden distributions
• Displace the center to

larger values This is consistent with experimental observations that with the increase of the strength of interactions:

• Magnetisation decay starts

earlier

• The blocking temperature

increases

c=0.2 c=0.36 c=0.56

Kinetic Monte Carlo

• Evaluate all energy barriers in multidimensional space
• Evaluate all transition rates, according to the Arrhenius law
• Choose a particle (cluster) with the probability proportional to its transition

rate and invert it

• Approximate the waiting time from the exponential distribution
• Recalculate all the energy barriers

in practice is possible for only small interaction

 Only plausible, if a good initial guess for all the clusters is known  Energy barrier distribution is a dynamical property and requires a large computational effort.

) / exp( T k E f f

B i

  

dt ft f dt t D ) exp( ) (  





i

f f

For initial guess, we use the Metropolis MC with simulated annealing

Thermal decay for an emsemble of 2D Co particles:

( starting from the remanent state, in-plane field 2D easy random easy axes distribution)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

probability EB/(2K<V>)

Volume c=0.05 c=0.2 c=0.36 c=0.56

Initial energy barrier distributions

1E-11 1E-10 1E-9 1E-8 1E-7 1E-6 0.3 0.4 0.5 0.6 0.7 0.8 0.9 magnetisation

time c=0.05 c=0.2 c=0.56

Energy barrier and magnetic relaxation calculated for conventional Co longitudinal recording medium

0.0 1.0x10

• 7 2.0x10
• 7 3.0x10
• 7 4.0x10
• 7 5.0x10
• 7 6.0x10
• 7

200 400 600 800 1000 1200 1400 granos sin barrera Numero de granos Energia de barreras (unidades internas)

J=0.015 erg/cm

J=0.075 erg/cm

10

• 1010
• 8 10
• 6 10
• 4 10
• 2 10

0 10 2 10 4 10 6 10 8 10 10

0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 J=0.015 erg/cm

J=0.75 erg/cm

M/M(inicial)

tiempo (s) 0.0 1.0x10

• 7 2.0x10
• 7 3.0x10
• 7 4.0x10
• 7 5.0x10
• 7 6.0x10
• 7

200 400 600 800 1000 1200 1400 granos sin barrera Numero de granos Energia de barreras (unidades internas)

J=0.015 erg/cm

J=0.075 erg/cm

10

• 1010
• 8 10
• 6 10
• 4 10
• 2 10

0 10 2 10 4 10 6 10 8 10 10

0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 J=0.015 erg/cm

J=0.75 erg/cm

M/M(inicial)

tiempo (s)

Important conclusions: Untextured medium with small amount of

exchange is initially less stable but more stable at large time scale Textured medium is more stable in the presence of small amount of exchange

textured untextured

Perpendicular CoCrPt recording medium

(100 grains, full micromagnetic contributions)

1E-13 1E-8 1E-3 100 1E7 1E12 1E17 0.5 0.6 0.7 0.8 0.9 100 s, 10% decay 20 years, 9% decay H=-0.3HK H=0

magnetisation time (s) CoCrPt recording medium

Textured, random easy axes, <V>= 6.5nm, K=2.4 10

6

erg/cm

3

, M

s

= 442 emu/cm

3

Ultrafast timescale: femto-pico seconds

“Spin-flip” as a fundamental problem

Via field pulses: Via field pulses:

2000

Motivation

s) (10 ds femtosecon

• 15

 

s picosecond  

• Model the 3 regions of ultrafast spin dynamics experiments:

E.Beaurepaire et al.

• Phys. Rev. Lett. 76, 4250 (1996)
• M. Van Kampen et al, Phys. Rev. Lett. 95, 267207 (2005)
• I. Fast demagnetisation
• II. Magnetisation recovery

s nanosecond  

• III. Damped precession

Delay between Pump and probe pulses

Can light reverse M?

Magnetization reversal induced by a single 40 fs laser pulse. Magnetization reversal induced by a single 40 fs laser pulse.

.Each domain is written with a single 40 fs laser pulse. .Magnetization reversal must occur within 1 ps. .Femtosecond (THz) opto-magnetic switch is faisible.

C.D. Stanciu et al.:

First demostration of all-optical Magnetic recorsing – 5microns bits

Femtosecond dynamics chalenges:

 The physics of the dynamics even in simple 3d metal such as Ni

is not understood.

• Direct spin-momentum transfer to electron system is discarded
• Excitation of non-magnetic states mediated by enhanced spin-orbit

coupling

• No inverse Faray effect
• Thermal mechanism ?!

Femtosecond pump-probe experiment allows to investigate the dynamics

• f electron, phonon and spin

systems in non-equilibrium conditions

Experimental measurements: Kerr signal and Reflectivity dynamics in Ni thin films

• 10

10 20 30 40 50 60 0.0 2.5 5.0 7.5

R [a.u.]  [ps]

50 mJ/cm

40 mJ/cm

30 mJ/cm

20 mJ/cm

15 mJ/cm

10 mJ/cm

• 10

10 20 30 40 50 60 0.6 0.7 0.8 0.9 1.0

k/k,0  [ps]

• 10

10 20 30 40 50 0.5 0.6 0.7 0.8 0.9 1.0

k/k,0  [ps]

• 10

10 20 30 40 50 2 4 6 8 10

R [a.u.]  [ps]

50 mJ/cm

40 mJ/cm

30 mJ/cm

20 mJ/cm

15 mJ/cm

10 mJ/cm

10nm 15nm

• Fs demagnetisation

+ ps recovery

• Slowing down of the

magnetisation rates as a function

• f pump fluency
• Excitation of incoherent stress

Waves at high pump fluency

Schematics of the model

Schematics: Laser excitation … in a thermal macrospin model

Theoretical model: 2T model

) ( d d ) ( ) ( ) ( d d

l e el l l th l l e el e e

T T G t T C K T t P T T G t T C          300

• Assumptions (simplification):
• Fitting results:

K m J 10 3 ,

3 2

    

e e

T C

ps K m J K m W 50 10 1 3 10 10

3 6 3 7

    

th l el

C G  , . ,

5 10 15 20 300 400 500 600

Tl

20mJ/cm

2

F= 50mJ/cm

2

T [K]

 [ps]

TC=631 K

Te

Three main approaches: (all based on the Langevin dynamics)

 Atomistic model based on the LLG (Langevin)

equation

 Atomistic model based on the Landau-Lifshitz-

Miyasaki-Seki equation

• > to take into account

electron-electron corelations

 Micromagnetic approach based on the Landau-

Lifshitz-Bloch (Langevin) equation

• > to extend modelling size

Atomistic model

 Uses the Heisenberg form of exchange  Dynamics governed by the Landau-Lifshitz-Gilbert

(LLG) equation.

 Random field term introduces the temperature

(Langevin Dynamics).

 Variance of the random field determined by the

electron temperature Tel.

j i i j ij exch i

S S J E   .







The Landau-Lifshitz-Bloch equation

[D.Garanin,PRB 55 (1997) 3050] :

         

eff eff eff

m m H m m H m H m m       

  2 2

   m ) (

||

Oe cm emu K 1500 500 631

2

  

app S C

M T H

• Using two relaxations parallel and

perpendicular

• Magnetisation magnitude is no

conserved

• Entropy correction

||  2T / 3TC   [1T / 3TC] (300K)  0.04    0.045

• LLB is coupled to 2T model
• Temperature-dependent

parameters from MFA

300 400 500 600 300 600 900

|| [fs]

T [K]

20 40 60 1.0 1.1 1.2

Te

max/Tl max

F [mJ/cm

2]

Critical slowing down for strong demagnetization Slowing down of the electron temperature increase

• Nonlinear electron specific heat

dependence on temperature?

• Phonon contribution to reflectivity?
• Excitation of coherent stress waves?

) ( ) , (

|| || ||

T T H     

Simulation and experimental results for Ni 15nm thin film

2 15 30 45 0.6 0.7 0.8 0.9 1.0

20 40 60 100 200 300 20 40 60 2 4

50mJ/cm

2

F=10mJ/cm

2

k/k,0 (mx/mx,0)

 [ps]

m [fs] E [ps]

Fluence [mJ/cm

2]

slowing down slowing down

Thermal character of demagnetization mechanism revealed

Modelling of optically-induced precession.

0.30 0.40 0.50 0.60 0.70 0.80 0.04 0.06 0.08 0.10 0.12 0.14

• 0.06
• 0.04
• 0.02

0.00 0.02 0.04 0.06 0.08

m z m y m x

easy axis

M.van Kampen et al, PRL 88 (2002)227201 (experiment)

Precessional frequency decreases with laser pulse fluency

Langevin dynamics based on the LLG: the white noise approximation is not always valid

 The electron-electron correlation time in metals

is of the of 10 fs

 The electron-phonon correlation time is of the order of 1ps  Strong fields (including exchange field) have characteristic

frequencies of the order of inverse correlation time

 The spin and phonon systems are not at equilibrium, the

fluctuation-dissipation theorem should be avoided. It is necessary to introduce correlated noise!

The Ornstein-Uhlenbeck process

i i i

ε H H    

Equilibrium angle distributions:

fs T k H

c B

10      , /

• For small temperatures (additive noise), the diffusion coefficient is re-normalised
• For large temperatures (multiplicative noise), the distribution is not Bolzman

   T k D

B



Diffusion coefficient Correlation time

ij c c j i i

t t D t t t       ) / | ' | exp( ) ' ( ) ( , ) (      

Boltzman distr.

)) ( ( 1 ) ( 1

2 2

t H S S t H S S

i i i i i i

            



   

Landau-Lifshitz-Miyazaki-Seki (LLMS) equations

• The coupling term describes

the adjustment of the noise to the spin direction

• When c ->0, the LLG equation is recovered
• We generalize the LLMS equations to many

spin case.

Longitudinal relaxation as a function of correlation time

64x64x64 spins (Ni)

10

• 16

10

• 15

10

• 14

1 10 100

5x10

• 13

1x10

• 12

2x10

• 12

0.6 0.8 1.0

mz time (s)

c = 10

• 15 s

c = 5 10

• 15s

c = 10

• 14 s

T=5K T=300K T=600K

||/ ||,LLG c [s]

Modelling of the ultra-fast pump-probe experiment for different correlation times

Main features of this new approach and conclusions

 The fluctuation-dissipation theorem is applied to the electron

system only

 The spin and electron systems do not need to be

in the equilibrium with each other

 It is thermodynamically consistent  The exact values of corr and are subject of ab-initio

calculations

 If corr ~10 fs the longitudinal relaxation is affected

by correlations but the perpendicular relaxation (LLG) is not.

• The slowing down of demagnetisation rates as a function of

correlation time (different materials have different demagnetisation rates, Should be possible to control with dopping, observed in half metals)

Reference: U.Atxitia et al Phys. Rev. Lett. 102 (2009) 055013.

CONCLUSIONS



For the long-timescale:

• Energy barrier determination is essential for the long-time magnetisation

decay

• Energy barriers of nanoparticles which are not circular or elliptical are not

KV

• Energy barriers are changing in time due to magnetic interactions
• The kinetic Monte Carlo combination with simulated annealing is capable

to determine magnetisation decay for arbitrary timescale.

CONCLUSIONS

 For ultra-short timescale:



We have shown that the Langevin dynamics approach adequately describes all stages of laser-induced dynamics:

• Femtosecond linear demagnetisation.
• Picosecond magnetisation recovery.
• Laser-induced precession.
• The main contribution to the slowing down of ultrafast demagnetisation rate

comes from the slowing down of the longitudinal relaxation approaching Tc



In some extreme conditions with characteristic timescale of the order of electron-electron correlation time a coorelated noise approach is necessary



We have introduced an approach based on the Landau-Lifshitz-Miyazaki- Seki equation.