Magnetization reversal in nanostructures:Quantum dynamics

in nanomagnets andsingle-molecule magnets

Wolfgang Wernsdorfer,Laboratoire de

Magnétisme Louis NéelC.N.R.S. - Grenoble

S = 102 to 106 S = 1/2 to 30

Mesoscopic physics in magnetism

S = 1020 1010 108 106 105 104 103 102 10 1

clustersindividual

spinsmolecularclusters

nanoparticlesmicron

particlespermanentmagnets

macroscopic nanoscopic

-1

0

1

-40 -20 0 20 40

M/M

S

H(mT)

multi - domainnucleation, propagation andannihilation of domain walls

-1

0

1

-100 0 100

M/M

S

H(mT)

single - domainuniform rotation

curling

-1

0

1

-1 0 1

M/M

S

H(T)

Fe8

1K 0.1K

0.7K

giant spinquantum tunneling,

quantizationquantum interference

Photons Phonons

Spin bath- nuclear spins- paramagnetic spins- intermolecular interactions

Interactions in magnetic mesoscopic systems

Conductionelectrons

etc.

Magnetic quantum system- spin- molecule- nanoparticle- chain

OutlinePart I: classical magnetism

1. Magnetization reversal by uniform rotation (Stoner-Wohlfarth model) • theory • experiment (3 nm Co clusters)2. Influence of the temperature on the magnetization reversal (Néel-Brown model)3. Magnetization reversal dynamics (Landau-Lifshitz-Gilbert)

• magnetization reversal via precession • dynamical astroid

OutlinePart II: quantum magnetism

1. A simple tunnel picture• Giant spin model• Landau Zener tunneling

• Spin parity • Berry phase

2. Interactions with the environment• Intermolecular interactions

• Interaction with photons

Conclusion

z

MH

θ

ψϕ

ba -1

0

1

2

-45° 0° 45° 90° 135° 180°

E( )

h = 0

h > 0

h = h0

B B

T = 0 K

E = K sin2θ− µ0MSH cos(θ − ϕ)

K = K1 +12

µ0MS2 (N b − Na )

Uniform rotation of magnetization:Stoner - Wohlfarth model (1949)

• single domain magnetic particle• one degree of freedom: orientation θ of magnetization M

• potential:

Stoner - Wohlfarth switching field

-1

0

1

-1.5 -1 -0.5 0 0.5 1 1.5

M

h

0°10°30°45°

70°

90° 0

0.2

0.4

0.6

0.8

1

0°

30°

60°90°

120°

210°

240°270°

300°

330°

hsw

hsw = sin2/3θ + cos2/3θ( )−3/2

Stoner - Wohlfarth astroid

easyaxis

hardaxis

Generalisation of the Stoner - Wohlfarth model

• from 2D to 3D • for arbitrary anisotropy functions

André Thiaville, JMMM 182, 5 (1998)PRB 61, 12221 (2000)

• potential St-W:

• potential 3D:

E = K sin2θ − µ0MSH cos(θ − ϕ)

E = E0 θ,( ) − µ0r M

r H

E = E0 r m ( ) − µ0MS

r m

r H

E0 (r m ) = Eshape(

r m ) + Ecristal(

r m ) + Esurface (

r m ) + Emag.elastic (

r m )

Switching fields of uniaxial anisotropy

E0 = − mz2

easy axis

Switching fields of biaxial anisotropy

E0 = − mz2 + 0.5mx

2

Switching fields of cubic anisotropies

E0 = + mx2my

2 + mx2mz

2 + my2mz

2[ ]E0 = − mx2my

2 + mx2mz

2 + my2mz

2[ ]

Micro-SQUID magnetometry

• sensitivity : 10-4 o

102 - 103 µB i.e. (2 nm)3 of Co

10-18 - 10-17 emu

• fabricated by electron beam lithography(D. Mailly, LPM, Paris)

particle

Josephson junctions

stray field

≈ 1 µmB

Roadmap of the micro-SQUID technique

1

100

104

106

108

1010

1993 1995 1997 1999 2001 2003

S( B)

Quantum limit of a SQUID

informationstorage

0 3 nm ?

Years

3 nm cobalt clusterDPM - Villeurbanne: LASER vaporization and inert gas condensation source

Low Energy Cluster Beam Deposition regime

HRTEL along a [110] directionfcc - structure, faceting

Ideal case: truncated octagedron with 1289 or 2406 atoms for diameters of 3.1 or 3.8 nm

blue: 1289-atoms truncated octahedrongrey: added atomes, total of 1388 atomes

Low energy cluster beam deposition

Nb

Si

Nb

Si

embeddedclusters

clusters

Micro-SQUID magnetometry

SQUID is fabricated by electron beam lithographyD. Mailly, LPN-CNRS

sensitivity : ≈ 102 - 103 µB i.e. (2 nm)3 of Coi.e. ≈ 10-18 - 10-17 emu

clusters in Nb - matrixM. Jamet, V. Dupuis, A. Perez, DPM-CNRS, Lyon

Acknowledgment: B. Pannetier, F. Balestro, J.-P. Nozières

Nb

Si

embeddedclusters

Josephson junctions

1 µm

embedded clusters≈ 3 nm

3D switching fields of a 3nm Co cluster

-0.4

µ0Hx(T) 0.4

0

0.3 0.3

0

µ0Hz(T)

µ0Hy(T)

0Hz

Hx

Hy

Finding the anisotropy function from 3D switching field measurements

H = −mz2 + 0.4mx

2

Finding the anisotropy from 3D switching field measurements

E0 = −mz2 + 0.4mx

2 − 0.1 mx2my

2 + mx2mz

2 + my2mz

2[ ]

Magnetic anisotropy energy

E0 (r m ) = Eshape(

r m ) + Ecristal(

r m ) + Esurface (

r m ) + Emag.elastic (

r m )

Eexperiment v = −K1 mz2 + K2 mx

2 K1 ≈ 2.5 ×105 J / m32.5×106 erg/cm3( )

K2 ≈ 0.7 ×105J / m30.7 ×106 erg /cm3( )

Eshape v = −K|| mz2 + K⊥ mx

2 K|| ≈ K⊥ ≈ 0.1− 0.3 × 105 J / m3

Ecristal v = −Kcristal mx2my

2 + mx2mz

2 + my2mz

2( ) KcristalCo−bulk ≈ 1.2 ×105 J / m3

however the measured 4th order is much smaller

Kmag.elastic < 0.1 ×105 J / m3

⇒ Esurfaceseems to be the main contribution to the magnetic anisotropy

Model systems for studying the magnetisation reversal

Quasi-static measurements• hysteresis loops

• switching fields (angles)

Magnetic anisotropy• forme• crystal• surface

• • •

Magnetisation reversal mode• nucleation, propagation, annihilation of domain walls

• uniform rotation, curling, nucleation in a small volume• thermal activation, tunneling

Dynamical measurements• relaxation (T,t)

• switching fields (T,dH/dt)• switching probabilities (T,t)

Activation volumeDamping factor

• material• defects

Temperature dependence of the switching fields of a 3nm Co cluster

=> in agreement with the Néel Brown theory

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

0H

z (T)

0Hy (T)

0.04 K

1 K2 K

4 K8 K

12 K

TB 14 K∆t ≈ 1 s

Temperature dependence of the magnetisation reversal-- Escape from a metastable potential well --

-1

0

1

2

-45° 0° 45° 90° 135° 180°

E( )

h = 0

h > 0

h = h0

B B

T = 0 K

Stoner - Wohlfarth model

0

0.4

0.8

1.2

-45° 0° 45° 90° 135°

E( )h > 0

T 0 K

E k BT

thermal activation

MQT

Néel Brown model

P(t)=e−t /

= 0e∆E /kT

∆E = E0 1− H sw

H sw0

(angle) =1.5 ↔ 2

Experimental methods for the study of the Néel–Brown model

Waiting time Switching field Telegraph noise

P(t)

t0

0

1

τ(H,T)2σ(dH/dt,T)

<Hsw(dH/dt,T)>counts

H t

M

tw

τ(Hy,T)

H = const.T = const.

dH/dt = const.T = const.

Hy = const.T = const.

Synopsis of the switching field measurement at a given temperatureH

M

T

t

t

+1

-1

0.04 K

0.04 K < Tmeasure < 30 K

Hsat

-Hsat(i)testH

t

t

Temperature dependence of the switching fields of a 3 nm Co cluster

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

TB 14 K0.04 K

1 K2 K

4 K8 K

12 K

Hz

µ0 Hy (T)

Hz

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

E0

0.9 E0

0.8 E0

0.7 E00.6 E0

0.5 E0

0.4 E0

0.3 E0

0.2 E0

0.1 E0

µ0 Hy (T)

τ = τ0e∆E/kBT ∆E = ln(τ/τ0) kBT

Néel Brown theory:P(t)=e−t /

= 0e∆E /kT

∆E = E0 1− H sw

H sw0

(angle) =1.5 ↔ 2

Temperature dependence of the switching fields of a 3 nm Co cluster

0

0.04

0.08

0.12

0.16

0 5 10 15

∆t = 0.01s∆t = 0.1s∆t = 1s

0H

(T)

T(K)

θ = 0°

θ = 45°

T1/2

T2/3

0

0.04

0.08

0.12

0.16

0 0.2 0.4 0.6 0.8 1

∆t = 0.01s∆t = 0.1s∆t = 1s

0H

(T)

(T/TB)1/a

θ = 0°

θ = 45°

T1/2

T2/3

Néel Brown theory:P(t) = e−t / τ

τ = τ0eE / kT

E = E0 1− Hsw

Hsw0

α

α(angle) = 1.5 ↔ 2

Application of the Néel-Brown-Coffey modelto find the damping parameter

0.08

0.12

0.16

0.2

0 15 30 45 60 75 90

f( )IHD

angle

= 0.5

0

0.05

0.1

0.15

0 15 30 45 60 75 90

f( )LD

angle

= 0.035

W.T. Coffey, W. Wernsdorfer, et al, PRL, 80, 5655 (1998)

20 nm Co nanoparticle 10 nm BaFeO nanoparticle

Magnetization reversal dynamics

Landau-Lifshitz-Gilbert equation

St.-W. particle (macrospin) : ||M|| = Msγ : gyromagnetic ratioMs : saturation magnetizationα : the damping parameterHeff : effective field (derivative of the free energy density with respect to M)

d

r M dt

= −r

M ×r H

eff( ) +M

S

r M × d

r M dt

Numerical integration using Runge-Kutta

Switching of magnetization by non-linear resonance studied in single nanoparticles

C. Thirion,W. Wernsdorfer,

D. MaillyNature Mat. (1.Aug. 2003)

Energ

y

Magnetizationangle

H

HDynamical astroid

Synopsis of the switching field measurement with microwavesH

M

H

t

t

+1

-1

10 kHz < f < 40 Ghz

Hsat

-Hsat(i)testH

t

t

Switching field measurement with microwavesMagnetization reversal via precession

tE Static field+

RF-pulse

0

0.05

0.1

0.15

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

∆H = 0

0H

x(T

)

0Hy(T)

4.4 GHz≈ 10 dBm

∆t > 20 ns3.2 GHz2.4 GHz

0

0.05

0.1

0.15

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

∆H = 010 dBm8 dBm6 dBm5 dBm

0H

x(T

)

0Hy(T)

4.4 GHz∆t > 20 ns

Magnetization reversal dynamics

Landau-Lifshitz-Gilbert equation

St.-W. particle (macrospin) : ||M|| = Msγ : gyromagnetic ratioMs : saturation magnetizationα : the damping parameterHeff : effective field (derivative of the free energy density with respect to M)

d

r M dt

= −r

M ×r H

eff( ) +M

S

r M × d

r M dt

Numerical integration using Runge-Kutta

Simulation of magnetization reversal via precession

0

0.5

1

0 1 2 3 4 5 6 7 8

H/Ha

t (ns)

= 1 decreases to 0.05 = 0.05

H H + H sin( t)

-1

-0.5

0

0.5

1

0 1 2 3 4 5 6 7 8

M

t (ns)

Mx

My

Mz-1

-0.5

0

0.5

1

0 1 2 3 4 5 6 7 8

M

t (ns)

Mx

My

Mz

t

= 10.8°

Magnetization reversal via precession

H = 0.615Ha, H = 0.02Ha, = 10.8° = 0.05, f = 3 GHz

H = 0.64Ha, H = 0.02Ha, = 1.55° = 0.01, f = 3 GHz

Magnetization reversal via precession

H = 0.625Ha, H = 0.02Ha, = 11.4° = 0.05, f = 3 GHz

0

1

10

easy axis

hard axis

0 1 2 3 4 5 6 ns

unstable

stable

H = 0.020 Ha

= 0.05f = 3 GHz

Dynamical astroid

0

0.05

0.1

0.15

-0.4 -0.2 0 0.2 0.4

static4.4 GHz3.2 GHz2.4 GHz

0H

z(T)

0Hx(T)

∆t = 10 ns

Pulse length dependance

0

0.05

0.1

0.15

-0.4 -0.2 0 0.2 0.4

static4.4 GHz3.2 GHz2.4 GHz

0H

z(T)

0Hx(T)

∆t = 10 ns

Switching field measurement with microwavesMagnetization reversal via precession

0

0.05

0.1

0.15

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

∆H = 0

0H

x(T

)

0Hy(T)

4.4 GHz≈ 10 dBm

∆t > 20 ns3.2 GHz2.4 GHz

Conclusion

Coming soon (or later)

• Studying smaller particles (molecules)

Measurements of magnetization reversal of nanoparticles containing about a thousand atoms

• The magnetic anisotropy is governed by surface anisotropy

• Switching field as a function of temperature are in agreement with the Néel Brown model

• Measurements probing nanosecond timescales (pulse fields, microwaves)

OutlinePart I: classical magnetism

1. Magnetization reversal by uniform rotation (Stoner-Wohlfarth model) • theory • experiment (3 nm Co clusters)2. Influence of the temperature on the magnetization reversal (Néel-Brown model)3. Magnetization reversal dynamics (Landau-Lifshitz-Gilbert)

• magnetization reversal via precession • dynamical astroid