2002 2002 Agilent Agilent Technologies Technologies Europhysics Prize Lecture Europhysics Prize Lecture on Bernard Barbara, L. Néel Lab, Grenoble, France Jonathan R. Friedman, Amherst College, Amherst, MA, USA Dante Gatteschi, University of Florence, Italy Roberta Sessoli, University of Florence, Italy Wolfgang Wernsdorfer, L. Néel Lab, Grenoble, France Budapest 26/08/2002 Quantum Dynamics of Nanomagnets
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Quantum Dynamics of Nanomagnets · 2002-10-30 · Roberta Sessoli, University of Florence, Italy Wolfgang Wernsdorfer, L. Néel Lab, Grenoble, France Budapest 26/08/2002 Quantum Dynamics
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Bernard Barbara, L. Néel Lab, Grenoble, FranceJonathan R. Friedman, Amherst College, Amherst, MA, USADante Gatteschi, University of Florence, ItalyRoberta Sessoli, University of Florence, ItalyWolfgang Wernsdorfer, L. Néel Lab, Grenoble, France
Budapest 26/08/2002
Quantum Dynamics of Nanomagnets
the miniaturization process
Single Domain Particles
coherent rotation of all the spins
θ
Ener
gy
θ
∆E
Quantum effects in thedynamics of the magnetizationFirst evidences of Quantum Tunneling innanosized magnetic particles
(difficulties due to size distribution)
0 3 nmQuantum Coherence in ferrihydriteconfined in the ferritin mammalianprotein
(inconclusive due to distribution of ironload)
= metal ions = oxygen = carbon
The molecules are regularly arranged in the crystal
Mn(IV)S=3/2
Mn(III)S=2
Total Spin=10
Mn12acetateMn12acetate
T. Lis Acta Cryst. 1980, B36, 2042.
high spin molecules
and low spin molecules
Uniaxial magnetic anisotropy H=-DSz
2
H=0
M
M=+S
M=-S+1M=S-1
0
E
-S+S
If S is large
H=-DSz2+gµBHzSz
H≠0
M=-S
M=S
return to the equilibriumthermal activated mechanismthermal activated mechanism
H=0
∆∆∆∆E=DS2
M=-SM=S
τ=τ0exp(∆E/kBT) τ0≈10-7
∆E=63 K
time=0
J. Villain et al. Europhys. Lett.1994, 27, 159
return to the equilibriumthermal activated mechanismthermal activated mechanism
H=0
∆∆∆∆E=DS2
M=-SM=S
τ=τ0exp(∆E/kBT) τ0≈10-7
∆E=63 K
time=∞
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1 τ
Sessoli et al. Nature 1993, 365, 141
τ0=2x10-7 s
∆E/kB=61 K
Temperature dependence of the relaxation time of Mn12acetate
Temperature dependence of the relaxation time of Mn12acetate
0
0.1
0.2
0.3
2 6
1 year1 day
1 s
1 ms
TEMPERATURE (K)
[log(τ/τ 0
)]-1
deviations fromthe Arrhenius law
Barbara et al. J. Magn. Magn. Mat. 1995, 140-144, 1825
return to the equilibriumtunnel tunnel mechanismmechanism
H=0
M=S M=-S
terms in Sx and Sy of the spin Hamiltonian
return to the equilibriumtunnel tunnel mechanismmechanism
H=0
M=S M=-S
terms in Sx and Sy of the spin Hamiltonian
What is the difference ?
Four fold axisTetragonal (E=0)
Mn12 Fe8
HH = = µµBB S.g.BS.g.B - D - D SSzz22 + + E (E (SSxx
22--SSyy22)) + B + BSSzz
44 + + C (C (SS++44++SS--
44))
Stot=10
What is the difference ?
Four fold axisTetragonal (E=0)
Two fold axisRhombic (E≠0)
Mn12 Fe8
HH = = µµBB S.g.BS.g.B - D - D SSzz22 + + E (E (SSxx
22--SSyy22)) + B + BSSzz
44 + + C (C (SS++44++SS--
44))
Hysteresis loops for Mn12
Friedman et al.,PRL, 1996;Hernandez et al,EPL, 1996;Thomas et al.,Nature, 1996
-0.4-0.3
-0.2
-0.1
00.1
0.2
0.3
0.4
-30 -20 -10 0 10 20 30
2.0 K2.2 K2.4 K2.6 K2.8 K3.0 Κ
M (e
mu)
H (kOe)
Hysteresis loops for Mn12
−5
0
5
10
15
20
25
30
0 5 10 15 20
2.0 K2.2 K2.4 K2.6 K2.8 K3.0 K
dM/d
H (1
0-5 e
mu/
Oe)
H (kOe)
Friedman et al.,PRL, 1996;Hernandez et al,EPL, 1996;Thomas et al.,Nature, 1996
Uniform spacing betweensteps
0
5
10
15
20
25
30
0 1 2 3 4 5 6
Hef
f (kO
e)
step number n
Step spacing: ~4.5 kOe
-0.4-0.3
-0.2
-0.1
00.1
0.2
0.3
0.4
-30 -20 -10 0 10 20 30
2.0 K2.2 K2.4 K2.6 K2.8 K3.0 Κ
M (e
mu)
H (kOe)
Hysteresis loops for Mn12
Enhanced Relaxation at Step Fields
10-3
10-2
0 2000 4000 6000
9.5 kOe9.0 kOe
(Msa
t - M
) (e
mu)
t (s)
Higher energy barrier
Yet faster relaxation!
Enhanced Relaxation at Step Fields
10−5
10−4
10−3
10−2
0 5 10 15 20
2.0 Κ2.6 Κ
Γ (s-1)
H (kOe)
Thermally Assisted ResonantTunneling
m = -10
m = -9
m = 10
m = 9
Thermalactivation
Fast tunneling
Tunneling occurs when levels in opposite wells align.
Hamiltonian for Mn122z BDS gµ= − − ⋅S H
The field at which (in the left well) crosses (in the right well):
m nm+−
Bnmm g
DnHµ−=+−,
Steps occur at regular intervals of field, as observed.
Step occurs every 4.5 kOe ⇒ D/g = 0.31 K
Compare with ESR data:D = 0.56 K, g = 1.93 D/g = 0.29 K(Barra et al., PRB, 1997)
Hamiltonian for Mn122z BDS gµ= − − ⋅S H 4
zBS−Spectroscopic studies revealed a 4th-order longitudinal anisotropy term B ~1.1 mK. (ESR: Barra et al., PRB, 1997 and Hill et al., PRL, 1998; INS:Mirebeau et al., PRL, 1999, Zhong et al., JAP, 2000 and Bao et al., cond-mat, 2000)
⇒Different pairs of levels cross at slightly different fields.
⇒Allows for the Examination of the Crossover from Thermally Assisted toPure Quantum Tunneling.
Crossover to Ground-stateTunneling
Abrupt “first-order” transition betweenthermally assisted and ground statetunneling.
Theory: Chudnovsky and Garanin, PRL,1997; Exp’t: Kent, et al., EPL, 2000, Merteset al., JAP, 2001.
2 2, 1 ( )m m
B
Dn BH m mg Dµ′
′= + +
Level crossing fields:
Fe8 Hamiltonian in Zero Field
2 2 2 4 4( ) ( )z x yDS E S S C S S+ −= − + − + +
Easy Axis Hard Axis
Spin wants to rotate in the y-z plane
Two Paths for MagnetizationReversal
Easy axis
Hard axis
Z
Y
XH
ϕϕϕϕ
A
B
ClockwiseCounterclockwise
Destructive Topological InterferenceEasy axis
Hard axis
Z
Y
XH
ϕϕϕϕ
A
B
Equivalence between paths ismaintained when H is appliedalong the Hard Axis.
Topological (Berry’s) phasedepends on solid angle Ωenscribed by the two paths.
Complete destructiveinterference occurs for certaindiscrete values of Ω.
Theoretical Prediction: A. Garg., 1993.
Solid AngleΩΩΩΩ
Destructive Topological Interference
A. Garg., 1993.
Modulation of Tunnel Splitting:
where Ω depends on the field along the Hard Axis.
When SΩΩΩΩ = ππππ/2, 3ππππ/2, 5ππππ/2…, tunneling is completely suppressed!
Interval between such destructive interference points:
cos( ),S∆ = Ω
2 2 ( )B
H E E Dgµ
∆ = +
Measured Tunnel Splitting
0 0.2 0.4 0.6 0.8 1 1.2
0.1
1
10
Tunn
el s
pitti
ng ²(
10-7
K)
Magnetic tranverse field (T)
0°
ϕϕϕϕ = 90°50°
30°20°
10°
5°
0 0.2 0.4 0.6 0.8 1 1.2 1.40.1
1
10
Tunn
el s
plitt
ing
²(10
-7 K
)
Magnetic transverse field (T
M = -10 -> 10
ϕϕϕϕ - 0°
ϕϕϕϕ - 7°
ϕϕϕϕ - 20°ϕϕϕϕ - 50°ϕϕϕϕ - 90°
experimentalcalculated with
D = -0.29, E = 0.046, C = -2.9x10-5 K
W. Wernsdorfer and R. Sessoli, Science, 1999.
Parity Effect: Odd vs. EvenResonances
-1 -0.5 0 0.5 10.1
1
10
²tu
nn
el(1
0-8
K)
µ 0Htrans (T)
n = 0
n = 1
n = 2
ϕϕϕϕ - 0°
W. Wernsdorfer and R. Sessoli, Science, 1999.
What Causes Tunneling andWhy the Parity Effect in Fe8
• Tunneling is produced by terms in theHamiltonian that do not commute withSz.
• For Fe8, these terms are
• Selection rule:• Every other tunneling resonance is
forbidden!
2 2 2 2( ) ( )2x yEE S S S S+ −− = +
,...)3,2,1(2 =±=∆ ppm
What Causes Tunneling andWhy the Parity Effect in Fe8