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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
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
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
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
• Size distribution approximately independent of D
•Clusters of 30 to 4000 atoms
Co55
Co147
Co561
Co2057
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
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)
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
+
-
+
-
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
K/L at the surface ~ 10(K/L) in bulk
L becomes much more anisotropic at the surface
(mL – mL||)
LK
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
but larger blocking temperature
and larger orbital magnetic moment
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))
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
Series of samples: N = 1, 2, 3, ..., 20 prepared under identical conditions
The size distribution is almost independent of N
Experimental results
The average U increases
one layer 30 layers
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
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
• 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)
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
ZFS7 – 14 K
Anisotropy
Hsingle = -DSz2 – E(Sx
2 – Sy2)
Giant spin model: anisotropy and quantum tunnelling
Tunnelling
U
z
S
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
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
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
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)
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)
• 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
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
Results Calculated
•Weak dependence on 0
•Opposite signs!!!
Experimental
?
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
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
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))
Experiments: time-dependent specific heat (Leiden)(spin-lattice relaxation
time )
(“relaxation” or “experimental” time e
Adjustable: 0.1 – 1000 seconds)
C = e/R
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!!
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!
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
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
Arigato
Thank you!
¡Gracias!
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
TB = U/kBln(t/0) decreases
No blocking at all when B > 1.7 T ! tunnelling via the ground state
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)
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
+
-
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
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
Experimental study: giant isotope effect in Fe8
Spin of Fe nuclei:
• Natural I = 0
• 57Fe I = 1/2
-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
TB (20 s) < TB(1 s)