Basic concepts on magnetization reversal (2)magnetism.eu/esm/2011/slides/rohart-slides2.pdf ·...

Basic concepts on magnetization reversal (2)Slow dynamics and thermal related processes

Stanislas ROHARTLaboratoire de Physique des SolidesUniversité Paris Sud and CNRSOrsay, France

Introduction: Application considerations

All applications need to work at room temperatureThermal activation may become problematic for long time scales

Energy barrier criteria :

TkKVE B)6025(

Thermal activation is crutial for nanoparticles (<25 nm)It is also relevant for thin films (domain nucleation and domain wall propagation)

Thermal activation is crutial for nanoparticles (<25 nm)It is also relevant for thin films (domain nucleation and domain wall propagation)

S. ROHART : Basic Concepts on Magnetization Reversal : Slow Dynamics European School on Magnetism ‐ Targosite 2011

2

Contents

I. SuperparamagnetismII. NucleationIII. Domain wall propagation in disordered

magnetic filmsIV. Conclusion


3

Contents




4

Superparamagnetism: Energy barrier for macrospin

Stoner and Wohlfarth hamiltonian

)cos(sin 02

HSeff HMKE Easy axisM

H

Volumic quantities

-0.5

0.0

0.5

1.0

1.5

E/K

E = KV

2

2

22

1

1

221

)0()(

K

m

HHKVE

h

hhh

eee Energy barrier (field aligned with easy axis)

2/3

1

SWHHKVE

General case

Victoria PRL 63, 457 (1989)


5

Superparamagnetism: Thermodynamics of a macrospin

cossin 02 HVKEV eff

Macrospin moment

Partition function :

HZ

ZkT

ddkTEVZ

0

sinexp

Average moment

projected on easy axis:

Isotropic case : K = 0

kTHx

xx

dkTHZ

0

0

0

1tanh

1

sincosexp2

(Langevin function)

Infinite anisotropy: =0 or

kTHxx

kTH

kTHZ

0

00

tanh

expexp

(Brillouin ½ function)

-4 -2 0 2 4

-1.0

-0.5

0.0

0.5

1.0

<>/

x=µ0µH/kT

Langevin Brillouin

Initial susceptibility

Scaling with 1/T (like Curie law)

kTor

kT

20

20

3


6


Case with finite anistropy

t

K dxxtErfiKVkTtHHh

thErfi

thErfi

th

th

th

0

2

2

)exp()(;/;/

11

1exp2sinh2

Brillouin like

Langevin like

Usefull : at low temperatureUsefull : at low temperature

KVkT

kT1

20

Chantrell et al. JMMM 53, 1999 (1985)Fruchart et al. JMMM 239,224 (2002)


7

10-3 10-2 10-1 100 101 102

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.10.025 0.25 2.5 25 250 2500

T/TB [if ln(t/0)=25]

kT/µ 0µ2

kT/KV


0 30 60 90 120 150 1800.00

0.01

0.02

Pro

babi

lity

dens

ity

kT/KV = 0.1 0.2 0.3 0.5 1.0 5.0 isotropic (sin)

Occupancy probability for a Stoner‐Wohlfarth model (H = 0)


8

Superparamagnetism: Temperature induced switchingPhenomenological: Arrhenius law

Switching time )/exp(0 kTE

0E calculated with the macrospin approch

attempt time (~1 ns) : linked with magnetization dynamics

Néel’s model : (Phonon and magnetostriction approach)

GkTVDMG

mHe

Se

K

23 2

01

0

Young’s modulus magnetostrition Fluctuation of the demag. field

Néel Ann. Geophys.5, 99 (1949)

Brown’s model : (Langevin fluctuation of a macrospin)

kTVHMH KS

K

212 0

021

0

well

Brown Phys. Rev. 130, 1677 (1963)See also Coffey et al. PRL 80, 5655 (1998)


9

Superparamagnetism: Temperature induced switching)/exp(0 kTE

Stability criterion : tmeasuremen

If ns1.00 1s 1min 1h 1day 1year 10 year

E/kT 23 27 31 34 40 43

Hard drive applications

This problem drives an intense effort of reseach for high anisotropy materials. Best candidates are Pt‐bases ordered alloys like FePt and CoPt (K~10 MJ/m3 in bulk phase <‐> Dlimit ~ 3 nm)See e.g. Sun et al. Science 287, 1989 (2000)

This problem drives an intense effort of reseach for high anisotropy materials. Best candidates are Pt‐bases ordered alloys like FePt and CoPt (K~10 MJ/m3 in bulk phase <‐> Dlimit ~ 3 nm)See e.g. Sun et al. Science 287, 1989 (2000)

Blocking temperature

0ln

tkKVT

BB


10

Warning: it is difficult to predict TB from volumic anisotropy measurements as in nanoparticles K is generally dominated by surface effects/low coordinated atomsSee Jamet et al. PRL 86, 4676 (2001)

Gambardella etal. Science 300, 1130 (2003)

Warning: it is difficult to predict TB from volumic anisotropy measurements as in nanoparticles K is generally dominated by surface effects/low coordinated atomsSee Jamet et al. PRL 86, 4676 (2001)

Gambardella etal. Science 300, 1130 (2003)

Superparamagnetism: Characteristic time in experiments

Stability criterion : tmeasuremen

Blocking temperature

0ln

tkKVT

BB


11

• Loop measurment in SQUID ‐> ~ 10min ‐ 1h• ZFC‐FC in squid ‐> ~ 1s• magneto optical kerr effect (resistive coils) ‐> ~1s• SP‐STM / microSQUID : 100 ms• ac susceptibility : 1ms ‐ 10s• Moessbauer : 0.1 µs

In sweeping (field or temperature) experiments, charateristic time is ambiguous‐> rather speak of sweeping rate

see e.g. Kurkijarvi PRB 6 832 (1972) (field sweep)Rohart et al PRB 74, 104408 (2006) (temperature sweep)

Superparamagnetism: Rate equation model

-0.5

0.0

0.5

1.0

1.5

E/K

EEH 0

Brown’s model is difficult to use.Simpler model: rate equations

1111

ndt

dnnn

dtdn

nndt

dn

nn

111

)/exp(

with

tn

First order decay law with relaxation time ‐ If tmeas

kT

Hn

0tanh12

For constant field and temperature (switching rates are constant)

This equation can be used for any field orientation (if switching rates are known…)


12

Superparamagnetism: Hysteresis loopSimulation of mean hysteresis loops with rate equations

Vsweep = 0.375HK/s

Vsweep = 0.375HK/s

Vsweep = 0.375HK/s

Vsweep = 3.75HK/s

Coercive field is not intrinsic‐> depend on temperature AND sweeping rate

BK

measKC

TTH

tKVkTHH

1

ln10

Scharrok JAP 76, 6413 (1994)IEEE Trans Mag 26, 193 (1999)


13

Superparamagnetism: experimental evidence

Anisotropy precisely determined from astroid at 35 mKVolume determined by SEM imagingFirst perfect agreement with Néel‐Brown model


14

First observation (2D) : Co (D = 25 nm) clusterWernsdorder et al. PRL 78, 1791 (1997)

In 3 D: (same Co cluster) E. Bonnet er al PRL 83, 4188 (1999)

3 nm Co cluster : M. Jamet PRL 86, 4676 (2001)


15

Superparamagnetism: experimental evidence

Jamet et al. PRL 86, 4676 (2001)

Vouille et al. JMMM 272‐276, e1237 (2004)(with exp. Data from Jamet et al.)

Simulation of macrospin dynamics (Brown’s model)


16

Superparamagnetism: Experimental observation in practical


17

‐> Single particle measurments are difficult‐> Hysteresis loops are generally long to measure‐> Single particle measurments are difficult‐> Hysteresis loops are generally long to measure

Temperature variation of initial susceptibility Field coolded‐Zero field coolded measurements

)/exp(111

1)(

0 kTEiTieq

: Pulsation of the magnetic field

dtKdTkTtwithTt

TTM

/)(/exp11)(

2

Model for zfc, measuring from low to high T

Co nanodots on Au(111) with increasing sizeRohart et al. PRL 104, 137202 (2010)

Pt clusters in different matricesRohart et al. PRB 74, 104408 (2006)See also Tamion et al; APL 95, 062503 (2009)

Superparamagnetism: beyond macrospin?Superparamagnetism and nucleation‐propagation magnetization reversalSuperparamagnetism and nucleation‐propagation magnetization reversal

0

0 )/exp()(

kTAHBraun PRL 1993(same approach as Brown 1963)

If L>> : it is possible to nucleate and propagate a reversed domain

‐Strongly depends on H (<‐> size of the critial nucleus to reverse magnetization)‐Proportionnal to L (nucleation occurs in the particle – edges are neglected)


18

Superparamagnetism: beyond macrospin?

Superparamagnetism and nucleation‐propagation magnetization reversalExperimental check :Superparamagnetism and nucleation‐propagation magnetization reversalExperimental check :

Single particle measurement by spin polarized STM

Monte Carlo simulation0 decrease with lenght, increase with the widthedge nucleation

Krause et al. PRL 103, 127202 (2009)


19

Superparamagnetism: beyond macrospin?Superparamagnetism and spin wavesSuperparamagnetism and spin waves


20

0 200 400 600 8000

20

40

60

80

100

1 2 3 4 5 6 7 8D (nm)

Nat

0 (GHz)

E a (meV

)

Nat

0

0 200 400 600 8000

200

400

600

0 200 400 600 800

1

10

0 200 400 600 800

1

10

(b1)

(b2)

(a1)

PSDmax

PSD

(a.u

.)

Frequency (GHz)

(a)

(b)

min

PSD

(a.u

.)

Frequency (GHz)

5.180E-11

6.090E-11

Atomic scale micromagnetic calculation on Co flat nanodotsAtomic scale micromagnetic calculation on Co flat nanodots

•Homogeneous magnetization ground state•Coherent rotation at T = 0 But coherent reversal over estimates switching rates at T≠0

At finite T, high excitation modes are excitedRohart et al. PRL 104, 137202 (2010)

Explains experiments from Rohart et al. 2010 and Rusponi et al. Nature Mat. 2003

Contents




21

Nucleation processes: Droplet model


22

Energy :

HMtrtrrE S02 22)(

rDomain wall energy cost

Zeeman energy gain

H

r

E

rc

S

Scc

HMtE

HMrr

drdE

0

20

2

20)(

E

Important process for thin filmsEnergy barrier scales with 1/H

)/exp()( 0 kTERHR Nucleation rate

Barbara JMMM 129, 79 (1994)Vogel et al. CR Physique 7, 977 (2006)

‐> 2D problem : apply to thin films with weak disorder

Nucleation processes: Magnetization reversal in constant magnetic field


23

‐> Nucleation rate vs. Domain expansion :

)exp(1)( 00 RtNNRNNdtdN

20)( tvrtS cN

Domain nucleation

Surface of the domain nucleated at t=0

Pertienent parameter : defines if the magnetization reversal is propagation or nucleation limitedcRr

vK 0

Variation of the non reversed area for different K

Labrune et al. JMMM 80, 211 (1989)Fatuzzo Phys. Rev. 127, 1999 (1962)

RNN0

0v

Nucleation site densityNucleated domainesNucleation rate

Domain wall velocity

Nucleation processes: Magnetization reversal in constant magnetic field


24

Labrune et al. JMMM 80, 211 (1989)

Nucleation processes: Coercive field and droplet model


25

Raquet et al. PRB 54, 4128 (1996)

Moritz et al. PRB 71, 100402(R) (2005)Pt/Co/Pt films and nanostructures

Au/Co/Au films and nanostructures

Nucleation processes: Droplet model in reduced dimensions


26

Nucleation is generally easier at the edge

Critical droplet size :

c

extextc

Sc

RwithRL

HM

2

0

11

2(same as thin films)

Magnetic configuration at the maximum energy

(Same as Braun 1993)µ0H (mT)

Energy barrie

r (eV

)

coherent

droplet

Numerical application: Parameters for Pt/Co(0.5 nm)/Pt film

JP. Adam PhD thesis, Orsay 2008 (unpublished)See also 3D theory : Hinzke and Nowak PRB 58 265 (1998)

Confined droplet model (circular dot)

Nucleation processes: Coalescence between nucleated domains


27

Pt/Co(0.5 nm)/PtTime between frame : 400 sNucleation field : 5 OePropagation field : 3Oe (to reduce DW velocity)

The experiment validates the picture of nucleation/propagation

Coalescence between domains is difficultNucleation of 360° domain walls

Dipolar repulsion between domains

Saturation may be difficult to reachPoison for magnetization reversal

Bauer et al. PRL 94, 207211 (2005)See also : Schrefl, et al. J. Appl. Phys. 87, 5517 (2000) (in plane magnetization)

Hehn et al. APL 92, 072501 (2008) (in MRAM nanoelements)

50 µm

Contents




28

Domain wall motion in disordered films


29

50 µm

Pt/Co(0.5 nm)/Pt with perpandicular magnetizationH = 42 OeTime between frame : 100s

Domain wall is not extended as a straight line (minimisation of length) but also feels some weak defects.

Competition between elasticity and pinning

Driving force :

Elasticity :

Pinning : pinfAKtt

H

4

(roughness, thickness variation…)

VIDEO

CreepElasticity/Pinning

competition

Depinning thermal activation

Viscous regimeDefects and disorder are

not pertinent‐> DW velocity given by magnetization dynamics

theory

f = H for magnetic DW

Domain wall motion in disordered filmsFrom Creep to viscous motion


30

Chauve et al PRB 62 6241 (2000)

Universal law‐> Apply to ferroelectric DW‐> Superconducting vortices‐> Charge density waves‐> Fluid propagation in porous media

Universal law‐> Apply to ferroelectric DW‐> Superconducting vortices‐> Charge density waves‐> Fluid propagation in porous media

Domain wall motion in disordered filmsPropagation and roughness


31

H << Hcrit: very small velocity DW adapts to defects Significant roughness

d

v = d/t

H >> Hcrit: large velocity Limited role of disorder Small roughness

v (m

/s)

H (Oe)

Hcrit

0 500 10001E‐101E‐91E‐81E‐71E‐61E‐51E‐41E‐31E‐21E‐11E+01E+1

H (Oe)0 500 10000

5

10

V(m/s)

S. Lemerle et al, PRL.80, 849 (1998)Velocity with universal behavior on 10 orders of magnitude.Velocity with universal behavior on 10 orders of magnitude.

Domain wall motion in disordered filmsElastic energy


32

L

Fext

x

(x,t)

For a DW length L:

~ (x,t)mean of transverse fluctuations

LL

elastic dxdxd

E2

0

2

2

Elastic energy: increase in DW length

L = 2L1‐L = 2 / L

L

L1

Roughness and fluctuation Elastic energy cost Roughness and fluctuation Elastic energy cost

Low curvature:

L1= L/2 (1 +42 / L2)1/2

Domain wall motion in disordered filmsPinning


33

Lfpin

x

(x,t)

: Correlation length of disorder

ni = pinning center density fpin : pinning force of one center

Random force of individual forces

Only density fluctuations can collectively pin the domain wall

= « collective force » coming from disorder

Pinning energy : LLnfE ipinpinning2

ipinnf 2

Domain wall motion in disordered filmsCompetition between pinning and elasticity


34

x

L

Small scales:Disorder caracteristic length

~ grain size~Transverse fluctuations

31

22

c

cL

Larkin length LC ~ 0.025 µm(for Pt/Co/Pt ultrathin films)

c

ccc L

L2

2

Two critical lengths: LC andC)()( celasticcpinningC LELEE

1E-3 0.01 0.1 1

10-13

10-12

10-11

Eelastic

Epinning

Eelastic

+Epinning

En

ergi

e (e

rg/

cm3 )

L (µm)

Domain wall motion in disordered filmsScaling law – Renormalization group theory


35

For L > Lc > nC

= 2/3 : universal exponent (for a 1D interface in motion in a 2D medium with weak disorder)

)(CLL

c

Pinning energy 12)( CLL

Cpinning EE

1

CCSSZeeman L

LHVMHLtME

Zeeman energy

22

cLelastic L

LE c Elastic energy

tLVvolumeCritical ccc :

Scaling law for fluctuations

)()( CelasticCpinningC LELEE

H

Hcrit

Domain wall motion in disordered filmsActivation energy and regime of motion


36L. Roters, S. Lübeck and K.D. Usabel, Phys. Rev. E, 63, 026113 (2001)

In the vicinity of the depinning field

)( critHHV

CSCcrit VMEH /

)()( CZeemanCpinning LELE

pinningZeemanactivation EEE

41

-21-2

critC H

H E

E

12)( CLL

CcS EHVME

With = 2/3

Thermally activated regime in creep regime

Activation energy as a function of external field

4/1

0 )(expH

HTk

Evv crit

B

C

Flow regime µHv

Domain wall motion in disordered filmsex: Pt/Co/Pt ultrathin films


37

0 500 10001E‐12

1E‐9

1E‐6

1E‐3

1

v (m

/s)

H (Oe)

0 500 10000

5

10non irrad.

irrad.v (m

/s)

1.5 2.0 2.5 3.0

10‐6

10‐1

non irradiatedirradiated

(H/Hcrit)1/4

10‐12

v (m

/s)

Ec non irr= 46 kT Ec irr= 26 kT

For H<Hcrit/5

4/1

0 )(expH

HTk

Evv crit

B

C

Hcrit non irr= 700 Oe Hcrit irr= 230 Oe

)( critHHv for H >Hcrit

Depining regime Depining regime

Unpublished resultsSee also S. Lemerle et al, PRL.80, 849 (1998)

Creep regime Creep regime

tCo = 0.8 nm

0 500 1000 1500 20000

20

40

60

80

v (m

/s)

H (Oe)

Domain wall motion in disordered filmsex: Pt/Co/Pt ultrathin films


38

0 500 1000 1500 20000

20

40

60

80

v (m

/s)

H (Oe)

tCo = 0.5 nm

0 500 1000 1500 20000

20

40

60

80

v (m

/s)

H (Oe)

tCo = 0.6 nm

0 500 1000 1500 20000

20

40

60

80

v (m

/s)

H (Oe)

tCo = 0.7 nm

P. J. Metaxas, Phys. Rev. Lett. 99 217208 (2007)

Hv

Mobilité/ champ critique/ champ de Walker


39

Mouvement visqueux observé = régime précessionnel

Paramètre d’amortissement de Gilbert ~0.25 compatible avec résultats FMR et valeurs utilisées pour des simulations numériques

Hv p 21

P. J. Metaxas, Phys. Rev. Lett. 99 217208 (2007)

changement de dynamique interne masqué par le piégeage

Hµv P

t(Co) (nm)0.5 100 470 0.240.6 210 590 0.300.7 230 705 0.320.8 220 650 0.31

H W (Oe) H dep (Oe)

Interprétation cohérente des observations expérimentales à fort champ

Evaluer Hw , confinement pris en compte

Domain wall motion in disordered filmsDivergence of energy barrier


40

Evidence of energy barrier for H /Hcrit <<1

3/4

1H

EL Ljump

Ljump is defined at the energy barrier

3)(5

31

CCCSL

LC L

LLHtMEE

C

41

critC )

H

H( EE

Jump length

Barkhausen jumps with 1/H4/3 jump lengthMotion through avalanche processes

V. Repain et al., EuroPhysicsLett 68 (2004) 10213

H= 2 Oe ( 0.01 Hcrit) Dt = 200 secondes H = 4 Oe (0.02 Hcrit) Dt =1 seconde

Domain wall motion in disordered filmsRoughness analysis


41

3/422 )()]()([)(CLL

LLxxLC

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

C2 (L

)

L/LC

)(Cc LL

LL = 2/3

Correlation on a given length L

101 102

100

101

102

103 2=4/3

<[

(x+

L)-(

x)]2 >

L (pixel)

H/Hc=

0.9 0.8 0.6 0.4 0.3 0.13 0.1 0.06

Critial exponent is independent of H (for H<0.9HC)Sligh increase of LC for H/HC>0.2

A. Mougin et al. unpublished results

Domain wall motion in disordered filmsCreep in confined geometry


42

Domain wall propagation in nanowires (Pt/CoFe0.3nm/Pt)Domain wall propagation in nanowires (Pt/CoFe0.3nm/Pt)Cross over between creep motion (2D like) and hoping (1D like)

Universal scaling :

c

col

Ccol

LLw

HLwLw 4/3/

Wire width

Segment length for jump

Larkin length

w

Kim et al. Nature 458, 740 (2009)

Domain wall motion in disordered filmsMagnetization reversal with strong pining


43

Epitaxial FePt(40nm)/Pt film : presence of micromacle than pins the domain walls.

Domain wall propagation is channeled between micromacles : fast fractal‐like domain growth.Saturation needs to overcome strong pinning barriers :Slow high field saturationThermal activation (slow dynamics)

Attane et al. PRL 93, 257203 (2004)

Some readings

•Skomski and Coey Permanent magnetism (Taylor & Francis Group 1999)•Aharoni Introduction to the theory of ferromagnetism (Oxford 1996)•Skomski Nanomagnetics J. Phys. Cond. Mat. 15, R841 (2003)•Fiorani Surface effects in Magnetic Nanoparticles (Springer 2005)•P. Metaxas, PhD thesis (Creep), Orsay, 2009 : http://trove.nla.gov.au/work/39007513

Some readings

•Skomski and Coey Permanent magnetism (Taylor & Francis Group 1999)•Aharoni Introduction to the theory of ferromagnetism (Oxford 1996)•Skomski Nanomagnetics J. Phys. Cond. Mat. 15, R841 (2003)•Fiorani Surface effects in Magnetic Nanoparticles (Springer 2005)•P. Metaxas, PhD thesis (Creep), Orsay, 2009 : http://trove.nla.gov.au/work/39007513

To go further

•Toward quantum fluctuations : already observed in molecular magnets‐> quantum fluctuation and tunneling of a domain wall is still a challenge…

Wernsdorfer and Sessoli Science 284, 133 (1999)Gunther and Barbara Quantum Tunneling of Magnetization‐QTM’94 (Kluwer, Netherlands,

1995)•Magnetization reversal under spin polarized current‐> Creep law with a different driving force (spin transfert torque)

Yamanouchi et al. Nature 428, 539 (2004)

To go further

•Toward quantum fluctuations : already observed in molecular magnets‐> quantum fluctuation and tunneling of a domain wall is still a challenge…

Wernsdorfer and Sessoli Science 284, 133 (1999)Gunther and Barbara Quantum Tunneling of Magnetization‐QTM’94 (Kluwer, Netherlands,

1995)•Magnetization reversal under spin polarized current‐> Creep law with a different driving force (spin transfert torque)

Yamanouchi et al. Nature 428, 539 (2004)

Take home message

•When dealing with long time scales, thermal activation plays an essential role.•Many properties (like coercive field) depend on the time scale (field sweeping time, waiting time)…

Take home message

•When dealing with long time scales, thermal activation plays an essential role.•Many properties (like coercive field) depend on the time scale (field sweeping time, waiting time)…

Conclusion


44

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