2002 Agilent Technologies 2002 Agilent Technologies

Europhysics Prize Lecture Europhysics Prize Lecture on

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

Ene

rgy

E

Quantum effects in the dynamics of the magnetization

First evidences of Quantum Tunneling in nanosized magnetic particles

(difficulties due to size distribution)

0 3 nmQuantum Coherence in ferrihydrite confined in the ferritin mammalian protein

(inconclusive due to distribution of iron load)

= 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+gBHzSz

H0

M=-S

M=S

return to the equilibrium thermal activated thermal activated

mechanismmechanism

H=0

E=DS2

M=-SM=S

=0exp(E/kBT) 010-7 E=63 K

time=0

J. Villain et al. Europhys. Lett.1994, 27, 159

return to the equilibrium thermal activated thermal activated

mechanismmechanism

H=0

E=DS2

M=-SM=S

=0exp(E/kBT) 010-7 E=63 K

time=

0

0.1

0.2

0.3

2 6

1 year1 day

1 s

1 ms

TEMPERATURE (K)

[log(/

0)]-1

0

0.1

0.2

0.3

2 6

1 year1 day

1 s

1 ms

TEMPERATURE (K)

[log(/

0)]-1

Sessoli et al. Nature 1993, 365, 141

0=2x10-7 s

E/kB=61 K

Temperature dependence of the relaxation time of

Mn12acetate

Mn12acetate: Hysteresis loopMn12acetate: Hysteresis loop

-3 -2 -1 0 1 2 3

-20

-10

0

10

20

T=2.1K

M

AG

NE

TIZ

AT

ION

( B

)

MAGNETIC FIELD (T)

magnetic hysteresis without cooperativity

High Spin Clusters

Single Molecule Magnets

applications?

0

0.1

0.2

0.3

2 6

1 year1 day

1 s

1 ms

TEMPERATURE (K)

[log(/

0)]-1

0

0.1

0.2

0.3

2 6

1 year1 day

1 s

1 ms

TEMPERATURE (K)

[log(/

0)]-1

0=2x10-7 s

E/kB=61 K

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(/

0)]-1

deviations from the Arrhenius law

Barbara et al. J. Magn. Magn. Mat. 1995, 140-144, 1825

return to the equilibriumtunnel mechanismtunnel mechanism

H=0

M=S M=-S

terms in Sx and Sy of the spin Hamiltonian

return to the equilibriumtunnel mechanismtunnel mechanism

H=0

M=S M=-S

terms in Sx and Sy of the spin Hamiltonian

What is the difference ?

Four fold axis Tetragonal (E=0)

Mn12 Fe8

HH = = BB 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 axis Tetragonal (E=0)

Two fold axis Rhombic (E0)

Mn12 Fe8

HH = = BB 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

0

0.1

0.2

0.3

0.4

-30 -20 -10 0 10 20 30

2.0 K2.2 K2.4 K2.6 K2.8 K

M (

emu)

H (kOe)

Hysteresis loops for Mn12

0 5 10 15 20

2.0 K2.2 K2.4 K2.6 K2.8 K3.0 K

dM/d

H (

10-5

em

u/O

e)

H (kOe)

Friedman et al., PRL, 1996; Hernandez et al, EPL, 1996; Thomas et al., Nature, 1996

Uniform spacing between steps

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

0

0.1

0.2

0.3

0.4

-30 -20 -10 0 10 20 30

2.0 K2.2 K2.4 K2.6 K2.8 K

M (

emu)

H (kOe)

Hysteresis loops for Mn12

Enhanced Relaxation at Step Fields

10-3

10-2

9.5 kOe9.0 kOe

(Msa

t - M

) (

emu)

t (s)

Higher energy barrier

Yet faster relaxation!

Enhanced Relaxation at Step Fields

0 5 10 15 20

(s-1)

H (kOe)

Thermally Assisted Resonant Tunneling

m = -10

m = -9

m = 10

m = 9

Thermal activation

Fast tunneling

Tunneling occurs when levels in opposite wells align.

Hamiltonian for Mn12

2z BDS g S HH

The field at which (in the left well) crosses (in the right well): 

m nm

Bnmm g

DnH

,

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 Mn12

2z BDS g S HH 4

zBSSpectroscopic 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 to Pure Quantum Tunneling.

Crossover to Ground-state Tunneling

Abrupt “first-order” transition between thermally assisted and ground state tunneling.

Theory: Chudnovsky and Garanin, PRL, 1997; Exp’t: Kent, et al., EPL, 2000, Mertes et al., JAP, 2001.

2 2, 1 ( )m m

B

Dn BH m m

g D

Level crossing fields:

Fe8 Hamiltonian in Zero Field

2 2 2 4 4( ) ( )z x yDS E S S C S S H

Easy Axis Hard Axis

Spin wants to rotate in the y-z plane

Two Paths for Magnetization Reversal

Easy axis

Hard axis

Z

Y

XH

A

B

ClockwiseCounterclockwise

Destructive Topological InterferenceEasy axis

Hard axis

Z

Y

XH

A

B

Equivalence between paths is maintained when H is applied along the Hard Axis.

Topological (Berry’s) phase depends on solid angle enscribed by the two paths.

Complete destructive interference occurs for certain discrete values of .

Theoretical Prediction: A. Garg., 1993.

Solid Angle

Destructive Topological Interference

A. Garg., 1993.

Modulation of Tunnel Splitting:

where depends on the field along the Hard Axis.

When S = /2, 3/2, 5/2…, tunneling is completely suppressed!

Interval between such destructive interference points:

cos( ),S

22 ( )

B

H E E Dg

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°

= 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

0°

7°

20° 50° 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. Even Resonances

-1 -0.5 0 0.5 1

0.1

1

10

²tu

nne

l(1

0-8

K)

µ0Htrans(T)

n = 0

n = 1

n = 2

0°

W. Wernsdorfer and R. Sessoli, Science, 1999.

What Causes Tunneling and Why the Parity Effect in Fe8

• Tunneling is produced by terms in the Hamiltonian that do not commute with Sz.

• For Fe8, these terms are

• Selection rule:• Every other tunneling resonance is

forbidden!

2 2 2 2( ) ( )2x yE

E S S S S

,...)3,2,1(2 ppm

What Causes Tunneling and Why the Parity Effect in Fe8

n = 1

-9 8

9

10-10

n = 0

-10

9-9

10

2m p 2m p Tunneling Allowed Tunneling Forbidden

Parity Effect: Odd vs. Even Resonances

-1 -0.5 0 0.5 1

0.1

1

10

²tu

nne

l(1

0-8

K)

µ0Htrans(T)

n = 0

n = 1

n = 2

0°

W. Wernsdorfer and R. Sessoli, Science, 1999.

Crossover From Classical to Quantum Regime

0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,00,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

4,5

5,0

n=5

n=0

n=1

n=2

n=3

n=4

n=6

n=7

n=8

n=9

n=10B

n

T (K)

Activated Tunneling

Measured ( ) and Calculated ( ) Resonance Fields

Barbara et al, JMMM 140-144, 1891 (1995) and J. Phys. Jpn. 69, 383 (2000) Paulsen, et al, JMMM 140-144, 379 (1995); NATO, Appl. Sci. 301, Kluwer (1995)

Classical Thermal Activation

Tblocking

Ground-state Tunneling

Tc-o

(Mn12-ac)

The Tunnel Window: An effect of weak Hyperfine Interactions

• Chiorescu et al, PRL, 83, 947 (1999)• Barbara et al, J. Phys. Jpn. 69, 383

(2000)• Kent et al, EPL, 49, 521 (2000)

3,75 3,80 3,85 3,90 3,95 4,00 4,05 4,10 4,15

0

1

2

3

4

n=8T=0.95 K

dm /

dB0

B0 (T)

8-1 8-0

Inhomogeneous broadening of Two resonances: Dipolar fields

Data points and calculated lines

Level Scheme

0,4 0,6 0,8 1,0 1,2 1,4

3,0

3,5

4,0

4,5

5,0 10-010-1

9-09-1 9-2

8-08-1 8-2

7-07-1 7-2

6-06-1 6-2

Bn (

T)

T(K)3,0 3,5 4,0 4,5 5,0

-30

-20

-10

0

10

20

(n-p) : -S+p S-n-p

9-2 10-1

9-1 10-0

9-0

8-2

8-1

8-0

7-2

7-1

7-0

6-0

6-1

6-2

E (K)

B0 (T)

-0.04 -0.02 0 0.02 0.04 0.06 0.0810-7

10-6

10-5

sqrt(s

-1)

µ0H(T)

M in = -0.2 M s

-0.005 0 0.0054 10-6

6 10-6

8 10-6

10-5

2 10-5 t0=0s

t0=10s

t0=5s

t0=20s

t0=40s

Homogeneous broadening of nuclear spins: Tunnel window

• Wernsdorfer et al, PRL (1999)

Effects of Strong Hyperfine Interactions:

Tetragonal symmetry (Ho in S4)

J = L+S = 8; gJ=5/4

Dipolar interactions between Ho3+ << mT

Case of Rare-earth ions: Ho3+ in Y0.998Ho0.002LiF4

HCF-Z = -B20 O2

0 - B40 O4

0 - B44 O4

4 - B60O6

0 - B64O6

4 - gJBJH Bl

m : acurately determined by high resolution optical spectroscopy

Sh. Gifeisman et al, Opt. Spect. (USSR) 44, 68 (1978); N.I. Agladze et al, PRL, 66, 477 (1991)

Hysteresis loop of Ho3+ ions in YLiF4

Mn12-ac

Thomas et al, Nature (1996) Giraud et al, PRL, 87, 057203-1 (2001)

Friedman et al, PRL (1996), Hernandez et al, EPL (1996)

Steps at Bn = 450.n (mT) Steps at Bn = 23.n (mT) Tunneling of Mn12-ac Molecules Tunneling of Ho3+ ion

-80 -40 0 40 80 120

-1,0

-0,5

0,0

0,5

1,0

200 mK 150 mK 50 mK

M/M

S

0H

z (mT)

-20 0 20 40 60 800

100

200

300

n=0n=3

n=1

n=-1

n=2

dH/dt > 0

1/ 0

dm

/dH

z (1/

T)

Ho3+

-1

-0,5

0

0,5

1

-3 -2 -1 0 1 2 3

1.5K

1.6K

1.9K

2.4K

M/M

S

BL (T)

Comparison with Mn12-ac

Role of Strong Hyperfine Interactions H = HCF-Z + A.I.J

-80 -40 0 40 80 120

-1,0

-0,5

0,0

0,5

1,0

200 mK 150 mK 50 mK

M/M

S

0H

z (mT)

-20 0 20 40 60 800

100

200

300

n=0n=3

n=1

n=-1

n=2

dH/dt > 0

1/ 0

dm/d

Hz (

1/T)

-200 -150 -100 -50 0 50 100 150 200

-180,0

-179,5

-179,0

-178,5

E (

K)

0H

z (mT)

-7/2

7/2

5/2

-7/2

7/2

3/2

5/2

3/2

-5/2 -3/2-1/2

1/2

-5/2 -3/2

Induce Tunneling of Electronic Moments

-1/2 1/2

Avoided Level Crossings between |, Iz and |+, Iz’ if DI= (Iz -Iz’ )/2 integer

Co-Tunneling of Electronic and Nuclear Spins:

Electro-nuclear entanglement

Exchange-biased quantum tunnelling in a dimer of Mn4 molecule

W. Wernsdorfer et al, Nature 416, 406 (2002)

V15 : The Archetype of Low spin Molecules

A Mesoscopic Spin S=1/2

Anisotropy of g-factor:

~ 0.6%Ajiro et al, J..Low. Temp.

Phys. to appear (2003)

Barra et al,J. Am. Chem. Soc. 114, 8509 (1992)

Exchange interactions: Antiferromagnetic ~

several 102K

Müller, Döring, Angew. Chem. Intl. Engl., 27, 171 (1988)

Barbara et al, cond-mat / 0205141 v1; submited to PRL.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

M/M

S

B0(T)

= 130

60 mK 0.14 T/s 0.14 mT/s

V15

M(H) : Reversible and out of equilibrium

80 mK

en

erg

y

magnetic field

²

| S, -m >

| S, m-n >

1 P

1 - P

| S, -m >

| S, m-n >

Adiabatic Landau-Zener Spin Rotation

« Isolated V15 » : A two-level system « without dissipation »

M(H) = dE(H)/dH

Fast sweeping rate / Weak coupling to the cryostat

Nuclear Spin-Bath :Weak Level Broadening

Low sweeping rate / Strong coupling to the cryostat

« Non-Isolated V15 » : A two-level system « with dissipation »

0,0

0,2

0,4

0,6

0,8

1,0

-0,6 -0,3 0,0 0,3 0,60,00

0,05

0,10

0,15

T=0.1 K

B0 (T)

TS=T

ph (K)

(c)

M (

µB)

M (

µB) T = 100 mK

0.14 T/s 0.07T/s 4.4 mT/s

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,70,0

0,2

0,4

0,6

0,8

1,0(d)

B0 (T)

Measured

Calculated

Chiorescu et al, PRL 84, 3454 (2000)

M(H): IrreversibleLZS transition at Finite Temperature

(dissipative)

Phonon-bath bottleneck modelAbragam, Bleaney, 1970; Chiorescu et al, 1999.

Nuclear spin-bath level broadeningStamp, Prokofiev, 1998.

V15: a Gapped Spin ½ Molecule

Dzyaloshinsky-Moriya interactions: HDM= - DijSixSj

The Multi-Spin Character of the Molecule (15 spins)

+

Time Reversal Symmetry D = 0 (Kramers Theorem)

Magnetism: From Macroscopic to Single atoms

S = 1020 1010 108 106 105 104 103 102 10 1

clusters spinsmoleculesNano-particlesNano-wiressubmicron

0 3 nm

Ho

Macroscopic Quantum Tunneling of Magnetization

of Single Nanoparticles easy axis

Barium ferrite Nanoparticle (10 nm)

Wernsdorfer et al. et al, PRL, 79, 4014, (1997)

Tc=0.31 K

Stoner-Wohlfarth astroid

0.2

0.4

0.6

0.8

1

1.2

0° 15° 30° 45° 60° 75° 90°

Tc(

)/T

c(4

5°)

angle

Tc() 0Ha 1/4 cot 1/ 6

1 cot 2/3 1

Miguel and Chudnovsky, PRB (1995)

Dissipation control of LZSMolecule spins 1/2 : Gapped

V15

Ho3+ , Mn4 pairs:Cross-spin transitions, Co-tunneling

Mn4 Fe8

Quantum DysnamicsBerry Phases

Quantum Classical crossoverQuantum Dynamics, Spin Bath

Mn12-ac

Conclusion and Perspectives

Quantum Tunneling at the Mesoscopic Scale(Environmental Effects on Quantum Mechanics)

Evidence for Quantum Coherence (, Rabbi oscillations, … )

Manipulations of Quantum Spins, Spins Qbits(Quantum Informations and Computers)

Tunneling of single Ho3+ ions Entangled I-J states

Ho3+

AcknowledgementsUniv. Florence & Modena:Andrea CaneschiClaudio SangregorioLorenzo SoraceAngelo RettoriAnna FortAndrea Cornia

L. Neél Lab. CNRS GrenobleE. BonetI. Chiorescu R. Giraud L. Thomas C. Thirion R. Tiron

CCNY:Myriam SarachikYicheng Zhong

U. Barcelona:Javier TejadaJoan Manel HernandezXixiang Zhang (now Hong Kong)Elias Molins

XeroxRon Ziolo

Lehman College CUNYEugene Chudnovsky

Univ. Rio de Janeiro M. Novak

Italian INFMA. LascialfariF. BorsaR. CaciuffoG. AmorettiM. Affronte

CNRS GrenobleJ. VillainA. L. BarraC. PaulsenV. Villar A. BenoitCNRS Bagneux D. Mailly

Lehman College CUNYEugene Chudnovsky

Univ. Calif. San Diego:David HendricksonSheila AubinEvan RumbergerE. Yang

Los Alamos:Rob Robinson(now ANSTO, Australia)Tim KelleyHeinz NakotteFrans Trouw (Argonne) Wei Bao

Univ. FloridaN. Aliaga, S. Bhaduri, C. Boskovic, C. Canada, C._Sanudo, M. Soler, G. Christou

Univ. P. et M. Curie, ParisV. Marvaud, M. Verdaguer,

Univ. BielefeldH. Bögge, A. Müller

U. St. PetersburgA.M. Tkachuk

Univ. Manchester R.E.P. Winpenny

Univ. Kyoto H. Ajiro, T. Goto, S. Maegawa

Univ. OkkaidoY. Furukawa

AcknowledgementsJ. F. Fernandez

A. Garg

M. Leuenberger D. Loss,

A. Mukhin

S. Miyashita

N.V. Prokof'ev

A. Zvezdin

L. J. de Jongh

J. Schweizer

P.C.E. Stamp

I. Tupitsyn