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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
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