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)