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The approach of nanomagnets to thermal equilibrium
F. Luis, F. Bartolomé, J. Bartolomé, J. Stankiewicz, J. L. García-Palacios, V. González, and L. M. García
Instituto de Ciencia de Materiales de Aragón, Zaragoza, Spain
F. Petroff, V. Cross, and H. Jaffrès
Unité Mixte de Physique, CNRS-Thales, Orsay, France
F. L. Mettes, M. Evangelisti, and L. J. de Jongh
Kamerlingh Onnes Laboratory, Leiden University, The Netherlands
Kyoto 2003
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I Single nanomagnet: Anisotropy and its microscopic origin
U = KV
“up” “down”
III Material: Dipolar interactions
II Spin-bath interactions: phonons, electrons, decoherence
Thermal bathT
coherence
Spin-lattice relaxation
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Outline of the talk
• Magnetic relaxation in Co clusters
Size-dependent anisotropy and orbital magnetism
Influence of a Cu layer on K and L
Dipolar interactions and magnetic relaxation
• Spin-lattice relaxation of single-molecule magnets
Non-linear susceptibility in the thermally activated regime
Spin-lattice relaxation in the quantum regime
Long range dipolar order
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Surface anisotropy of Co clusters prepared by sequential deposition (Orsay)
• No trace of oxidation
• fcc crystal structure
• Good control of the average diameter between 0.7 and 6 nm
Co Al2O3
Si
tCo =0.1 - 1 nm
tAl2O3 = 3 nm
tCo =0.1 - 1 nm
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• Size distribution approximately independent of D
•Clusters of 30 to 4000 atoms
Co55
Co147
Co561
Co2057
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Activation energy: effective anisotropy
)(),(" 2 bbeqB UfUTTk
36/ D
DUK
M. I. Shliomis and V. I. Stepanov, Adv. Chem. Phys. 87, 1 (1994)
D
KKK s
bulk
6
bulk
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e-
K and L sensitive to the matrix (metallic or insulating) surrounding the cluster
Surface anisotropy and orbital magnetic moment
L S
(mL – mL||)L
• K S-O LS
Electron confinement
enhanced L
Surface
anisotropic L
P. Bruno, Phys. Rev. B 39, 865 (1989)
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D = 2.6 nm
XMCD study of the orbital moment
Sum rules mL/mS
• Circularly polarized X-rays• L2,3 edges of Co• Fluorescence and total electron yield
B < 5 T
+
-
+
-
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Bulk Co: mL/mS = 0.097
mL mL A mS mS D=
bulk
+
The orbital magnetic moment increases as D decreases
In bulk L = 0.15 B at the surface L 0.39 B
Lsurface
Lbulk
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K/L at the surface ~ 10(K/L) in bulk
L becomes much more anisotropic at the surface
(mL – mL||)
LK
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Effect of a metallic layer (Cu)
1.5 nm
Clusters covered by a thin layer of Cu
Samples with and without Cu show approximately the same equilibrium magnetic response
Same cluster size distribution
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but larger blocking temperature
and larger orbital magnetic moment
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CoCu
e-
L becomes larger at the Co/Cu interface
Agrees with experiments on Co/Cu layers (M.Tischer et al., Phys. Rev. Lett. (1995))
Modified DOS by hibridization with the Cu conduction band? (Wang et al. J.
Mag. Mag. Mater., 237 (1994))
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Interactions and magnetic relaxation
Self organized growth of the clusters in 3D
Babonneau et al., Appl. Phys. Lett. 76, 2892 (2000)
Control over dipolar interactions
• Number of layers N
• Interlayer separation
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Series of samples: N = 1, 2, 3, ..., 20 prepared under identical conditions
The size distribution is almost independent of N
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Experimental results
The average U increases
one layer 30 layers
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The blocking temperature increases almost linearly
with the number of nearest neighbours
F. Luis et al., Phys. Rev. Lett. 88, 217205 (2002)U = Ks S + A N
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z
1 2
3
1
32
Theoretical model Inspired in Dormann modelJ.L. Dormann et al., J. Phys. C 21, 2015 (1988)
• dominated by largest particles
• Nearest neighbors fluctuate rapidly
• Interaction energy is continuously minimized
= 0 expU + Edip
kBT
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• The anisotropy is two orders of magnitude larger than in bulk and it is mainly determined by the atoms located at the surface: U = KsS
• The enhanced K is related to an increase of the orbital moment L at the surface
• L at the surface is much more anisotropic than in bulk (K/L)surface 10 (K/L)bulk
• K and L can be enhanced by embedding the clusters in a metallic (Cu) matrix: potential for applications
• Dipolar interactions slow-down the relaxation process:
U = KsS + ANnn
Conclusions (Co clusters)
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Single-molecule magnetsD. Gatteschi et al., Science 265, 1054 (1994)
• Large intramolecular exchange interactions Net spin S
• Intermediate situation between paramagnetic atoms and magnetic nanoparticles
Mn12
Quantum world Classical world
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ZFS7 – 14 K
Anisotropy
Hsingle = -DSz2 – E(Sx
2 – Sy2)
Giant spin model: anisotropy and quantum tunnelling
Tunnelling
U
z
S
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Slow relaxation towards thermal equilibrium
Thermal bath(lattice)
Phonon-induced transitions between levels
• Fast intrawell transitions 10-7 s Cm0
Fe8
• Slow interwell transitions: >> Cmeq – Cm
0
No equilibrium when > eCm = Cm0 e- / + Cm
eq (1 – e-/e e
EquilibriumNo
Equilibrium
H = Hsingle + Hspin-lattice
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Dipole-dipole interactions
J. F. Fernández and J. Alonso, Phys. Rev. B 62, 53 (2000)
• Large molecular spins
• Super-exchange interactions can be neglected
• Fast spin-lattice relaxation (low anisotropy)
D 0.01 K
Mn6
S = 12
Tc ~ 0.1 – 0.5 K
long-range order
(Bdip)2,1
H = Hsingle + Hspin-lattice + Hdipolar
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Dipolar ferromagnet Tc = 0.17 K
A. Morello, et al. Phys. Rev. Lett. 90, 017206 (2002).
Equilibrium experiments down to very low T
T < Tc T > Tc
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UkBT
= 0 exp
Resonant tunneling via excited states (T > 1 K)
Multilevel Orbach process (Pauli Master equation)
<<
>
Tunnelling blocked by dipolar and hyperfine stray magnetic fields
U e
TB
UkBln(e/0)
TB =
F. Luis, J. Bartolomé, and J. F. Fernández, Phys. Rev. B 57, 505 (1998)
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Non-linear susceptibility of Mn12 clusters
M = 0H – 3 H3+...
Gives information on
Equilibrium: magnetic anisotropy
Non-equilibrium: spin-bath interaction
damping
2/L
0
2/L
0J. García-Palacios and P. Svedlindh, Phys. Rev. Lett. 85, 3724 (2000)
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• Third harmonic: (3)
• Second order coefficient in
() = () - 3 () H2 + ...
Experimental determination of 3
There are two possibilities
• hac sufficiently small not to induce any extra nonlinearity
• The same qualitative behavior in the classical limit
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A story of two Mn12 molecular crystals
U = 65 K for both compounds Same anisotropy
0 = 3×10-8 s Mn12 acetate Different spin-lattice interaction
0= 1.5×10-8 s Mn12 2-Cl benzoate (benzoate) 2 (acetate) < 10-3
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Results Calculated
•Weak dependence on 0
•Opposite signs!!!
Experimental
?
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Classical: 2/ H2 < 0
Quantum tunnelling: 2/H2 > 0
The classical 3 should be recovered at high fields
> 0.1 coherence < 0
Suitable method to ascertain if relaxation takes place via QT
Explanation: quantum non-linearity
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Application to more complex systems: natural ferritin
D = 7 nm
S 100
Tejada et al (1997): QT? Yes
Mamiya et al (2002): QT? No
Classical relaxation near TB
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Spin-lattice relaxation in the quantum regime (T < 1 K)
>> kBT?
×
Tunnelling induced by a fluctuating bias (Prokof’ev and Stamp, Phys. Rev. Lett. 80, 5794 (1998))
Two-level system
Spin reversal but ...No relaxation of energy
Thermal bath(lattice)
(Fernández and Alonso, Phys. Rev. Lett. 91, 047202 (2003))
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Experiments: time-dependent specific heat (Leiden)(spin-lattice relaxation
time )
(“relaxation” or “experimental” time e
Adjustable: 0.1 – 1000 seconds)
C = e/R
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Relaxation towards (ordered) equilibrium via quantum tunnelling: Mn4
S = 9/2
U = 14.5 K
H = – DSz2 – E(Sx
2 – Sy2)
R
Symmetry of the cluster
• R = Cl-(OAc)3(dbm)3
= 10-7 K
• R = (O2CC6H4-p-Me)4(dbm)3
= 10-4 K!!
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Relaxation rate: time-dependent specific heat: Mn4Cl
• becomes independent of T below 1 K: incoherent tunnelling
• Five order of magnitude faster than predicted for known processes!
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Conclusions (single-molecule magnets)
• Quantum tunnelling provides a mechanism for relaxation
to equilibrium for all T:
High T: resonant tunnelling via excited states
Low T: incoherent tunnelling mediated by phonons
and nuclear spins (challenge for theoreticians!)
• Long-range magnetic ordering induced purely by dipolar
interactions: Mn6 (isotropic) and Mn4 (Ising)
• Large quantum non-linear susceptibility
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Collaborations:Samples
• Fe8
J. Tejada, Departamento de Física Fonamental, Universitat de Barcelona, Spain
• Mn4, Mn6
G. Aromí, Universidad de Barcelona, SpainG. Christou, N. Aliaga, University of Florida, USA
• Mn12 D. Gatteschi, Department of Chemistry, University of Florence, Italy
• 57Fe8 R. Sessoli, Department of Chemistry, University of Florence, Italy
Theory
J. F. Fernández, Instituto de Ciencia de Materiales de Aragón, CSIC-Universidad de Zaragoza, Spain
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Arigato
Thank you!
¡Gracias!
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Tunnelling via lower lying states?
B
Leaves the symmetry intact
Increases for all ±m doublets
Tranverse magnetic field
m = ±10
m = ±9
m = ±8
m = ±7
m = ±6
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TB = U/kBln(t/0) decreases
No blocking at all when B > 1.7 T ! tunnelling via the ground state
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Direct measurement of the tunnel splitting: Fe8
kBT
F. Luis, F. L. Mettes, J. Tejada, D. Gatteschi, and L. J. de Jongh, Phys. Rev. Lett. 85, 4377 (2000)
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tun 0.5 ns
1 ms (phonons)
B = 0 () classical states
m = -10 or m = +10
A mesoscopic Schrödinger cat
B = 3 T () Quantum superpositions
+
-
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Common pehnomenon at the atomic scale
• Protons in hydrogen bonds• Ammonium molecule
But hard to conciliate with our macroscopic intuition
Possible technological applications: quantum computing
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The role of nuclear spin bath
• Quantum tunnelling of the electronic spin is forbidden by destructive interference when S = 9/2 (Loss et al., von Delft et al., 1992) at zero field
= 0
• Hyperfine interactions with nuclear spins can break the degeneracy
H = – DSz2 – E(Sx
2 – Sy2) + Ahf (IxSx + IySy + IzSz)
I S
For Mn nuclei I = 5/2
0Tunnelling
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Experimental study: giant isotope effect in Fe8
Spin of Fe nuclei:
• Natural I = 0
• 57Fe I = 1/2
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-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
t=0
II
I
l3
l2
l1
t <
t >
m
IV
III
l4
t 0
t > /t
m
F. Luis, J. Bartolomé, and J. F. Fernández, Phys. Rev. B 57, 505 (1998)
coherence << 0 > 0.1
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TB (20 s) < TB(1 s)