QUANTUM SPIN DYNAMICS OF RARE-EARTHS IONS

B. Barbara,

W. Wernsdorfer, E. Bonet, L. Thomas (IBM),

I. Chiorescu (FSU), R. Giraud (LPN)

Laboratory Louis Néel, CNRS, Grenoble

Collaborations with other groups

B. Malkin (Kazan)

A.M. Tkachuk (St Petersburg)

H. Suzuki (Tsukuba)

D. Gatteschi (Florence)

A. Müller (Bielefeld)

D. Mailly (LPN, Marcoussis)

Nanometer scale

NanoparticleCluster

20 nm3 nm1 nm 2 nm

Magnetic ProteinSingle Molecule

50S = 10 103 106

The molecules are regularly arranged in the crystal

Mn(IV)S=3/2

Mn(III)S=2

Total Spin =10

Mn12acetateMn12acetate

SINGLE MOLECULE « MAGNET »Energy barrier in zero field (symmetrical)

H= - DSz2 - BSz

4 - E(S+2 + S-

2) - C(S+4 + S-

4)

spin down spin up

|S,S-2> |S,-S+2>

Ground state tunneling

|S,S-1> |S,-S+1>

|S,S> |S,-S>

SZ

En

erg

y

en

erg

y

magnetic field

²

| S, -m >

| S, m-n >

1 P

1 - P

| S, -m >

| S, m-n >

Thermally activated tunneling

If applied field // -M

non-symmetrical barrierNew resonances at gBHn = nD

Simplified picture of an isolated spin: Landau-Zener model

Molecular magnets (S. Miyashita)large spins give extremely small

splittings

Tunneling probability:PLZ=1 – exp[-(/ħ)2/c]

c = dH/dt

Tunneling of magnetization in Mn12-ac:

-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)

ICM’94 Barbara et al, JMMM (1995); NATO-ASI, QTM’94 ed. Gunther and Barbara; Thomas et al Nature (1996); Friedman et al, PRL (1996);

…. Slow quantum spin dynamics of molecule magnets….

« Technical » hysteresis loop + resonant tunneling

Steps at Hn =450.n mT

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

(Phonon Bath)

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)

Ground-state Tunneling

(Spin-Bath)

(Mn12-ac)

Resonance width and tunnel window Effects of magnetic couplings and 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 dipolar broadening and the electronic spin-bathData 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 the tunnel window by nuclear spins:

• Wernsdorfer et al, PRL (1999) Prokofiev and Stamp (1998) Weak HF coupling: Broadens the tunnel window (105) Decoherence mechanisms

Landau-Zener model

For an isolated spinen

ergy

magnetic field

²

| S, -m >

| S, m-n >

1 P

1 - P

| S, -m >

| S, m-n >

Tunneling probability:PLZ = 1 – exp[-(/ħ)2/c]

c = dH/dt

Single Molecule Magnets: large spins

very small tunnel splittings: ~ (E/D)

-2S

very small tunnel probabilities:

PLZ ~ 2/c ~ (E/D)4S/-c

PPS ~ (2/0)e-││/0

Larger tunneling rateStrong decoherence

H= - DSz2 - BSz

4 - E(S+2 + S-

2) - C(S+4 + S-

4) - gBSzHz

For an ensemble of spins

V15 , a molecule with S=1/2Dipolar interactions 103 times smaller but I=7/2

1

1

)(21

)(2

)(

)(

B

B

BM

BM

eq

-1

-0.5

0

0.5

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0 s0.1 ms0.5 ms1 ms2 ms3 ms

M/M

s

µ0H (T)

0.04 K11 GHz

0.001 T/s

period: 10 ms

Absorption of sub-centimetric waves

Max ~ 5 s-1

I. Chiorescu, W. Wernsdorfer, A. Müller, H. Boggë, and B. Barbara et al, PRL (2000)W. Wernsdorfer, D.Mailly, A. Müller, and B. Barbara, EPL, 2004

Gaussian absorption lines

W. Wernsdorfer, D.Mailly, A. Müller, and B. Barbara, EPL, 2004

Important broadening by nuclear spins and other molecule spins Loss of coherence

R ~ b ~ 30 kHz2~ ~ 0.2 GHz

Rabi oscillations, require much larger b.N = BMax/2 = B2/ ~20

Precession ~ 20 turns

tbBbB

bP 2

1222

22

2

)(2

1sin

)()(

)(

)()(4

2LL Bfb

0.2 % Ho3+ in substitution of Y3+

In YLiF4Tetragonal symmetry (Ho in S4); (J = L+S = 8; gJ=5/4)

Dipolar interactions

~ 20 mK << 200 mK (levels separation)

Tunneling of the angular momentum of Isolated Rare-earths ions

(ensemble measurements of « paramagnetic » ions)

An extention of the slow quantum dynamics studies of SMM to the cases of strong spin-orbit and hyperfine coupling

-6 -4 -2 0 2 4 6-200

-150

-100

-50

0

50

100

150

-9 -6 -3 0 3 6 9-240

-200

-160

-120

-80

-40b)a) E (K)

<Jz>

E (K)

0H

z (T)

R. Giraud, W. Wernsdorfer, D. Mailly, A. Tkachuk, and B. Barbara, PRL, 87, 057203-1 (2001)

B20 = 0.606 K, B40 = -3.253 mK, B44 =- 42.92 mK, B60 =-8.41mK, B64 =- 817.3mK Sh. Gifeisman et al, Opt. Spect. (USSR) 44, 68 (1978);

N.I. Agladze et al, PRL, 66, 477 (1991)

Barrier short-cuts

Energy barrier ( ~ 10 K)

Strong mixing

Singlet excited state+

Doublet ground-state+

Large 1 (Orbach

process)

CF levels and energy barrier of Ho3+ in Y0.998Ho0.002LiF4

46

46

44

44

06

06

04

04

02

02 OBOBOBOBOBHCF

Hysteresis loop of weakly interacting Ho3+ ions in YLiF4

-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

dH/dt=0.55 mT/s

-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)

Many steps !

L.Thomas, F. Lionti, R. Ballou, R. Sessoli, R. Giraud, W. Wernsdorfer, D. Mailly, A.Tkachuk,

D. Gatteschi,and B. Barbara, Nature, 1996. and B. Barbara, PRL, 2001

Steps at Bn = 450.n (mT) Steps at Bn = 23.n (mT)

Tunneling of Mn12-ac Molecules Tunneling of Ho3+ ion

… Nuclear spins…

Quasi-Ising CF Ground-state + Hyperfine Interactions

H = HCF-Z + A{JzIz + (J+ I- + J- I+ )/2}

-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

I = 7/2

E (

K)

0H

z (mT)

-7/2

7/2

7/2

-7/2

3/2

-7/2

Co-Tunneling of electronic and nuclear momenta: Electro-nuclear entanglement (2-bodies)

The ground-state doublet 2(2 x 7/2 + 1) = 16 states

-5/2

5/2

gJBHn = n.A/2 A = 38.6 mK, Linewidth ~ 10 mK ~ Dip. Int.

Avoided Level Crossings between |, Iz and |+, Iz’ if I= (Iz -Iz

’ )/2= odd

-5/2

-75 -50 -25 0 25 50 75-1.0

-0.5

0.0

0.5

1.0

T = 30 mKv = 0.6 mT/s

HT=190 mT

HT=170 mT

HT=150 mT

HT=130 mT

HT=110 mT

HT=90 mT

HT=70 mT

HT=50 mT

HT=30 mT

HT=10 mT

M/M

S

0H

z (mT)

Application of a transverse magnetic field:

(slow sweeping field: sample at the cryostat temperature)

Acceleration of quantum dynamics the remanent magnetization vanishes Quantum fluctuations destroy the local moment Transition from « Classical » to Quantum Paramagnet (QPT) Nature of the mixing: entangled electro-nuclear states

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

-180,0

-179,5

-179,0

-178,5

I = 7/2

E (

K)

0H

z (mT)

50 mK0.3 T/s

120 160 200 240

0

4

8

-150 -75 0 75 150 225

0

20

40

60

-300 -200 -100 0 100 200 300-1,0

-0,5

0,0

0,5

1,0

-8 -6 -4 -2 0 2 4 6 8 10-180

-120

-60

0

60

120

180

240

n = 6

n = 7n = 8

n = 9

b)

dH/dt<0

n=1

n=0

1/ 0

dm

/dH

z (1/

T)

0H

z (mT)

a)

M/M

S

0H

z (mT)

integer n half integer n

linear fit

0H

n = n x 23 mT

0H

n (

mT

)

n

Giraud et al, PRL 87, 057203 1 (2001)

50 mK0.3 T/s

Simultaneous tunneling of Ho3+ pairs due to dipolar interactions (4-bodies entanglement)Two Ho3+ Hamiltonian avoided level crossings at Hn = (23/2).n

Additional steps at fields: Hn = (23/2).n (mT)(single Ho3+ tunneling being at avoided level crossings at

Hn = 23.n mT)

Ac susceptibility (SQUID measurements)

Tunneling rates andac measurement frequency

Co-tunneling

Single-ion and dipolar-biasTunneling

R. Giraud, A. Tkachuk, B. Barbara, PRL, 2003.

R. Giraud, A. Tkachuk, and B. Barbara, PRL (2003).

Single-ion level structure En = E gBHn

Tunneling: gBHn = (n’-n)/2

Co-tunneling: gBHn=(n’-n+1/2)/2

A= Ho hyperfine constant)

Two-ions Level structure

Electronic Spin-bath:Co-tunnelingBiais tunneling Diffusive tunneling

Nuclear spin-bath (Li, F, Y): Linewidths

-2000 -1000 0 1000 2000

-180.0

-179.5

-179.0

-178.5

-2000 -1000 0 1000 2000

-360

-359

-358

-357

0 100 200 300 400 500

-360.0

-359.6

n=-9b)

a)

n=-8 n=3/2

. . .

. . .

mI=+5/2

mI=+7/2

mI=+5/2

mI=+7/2

I = 7/2E

ner

gy

(K)

Hz (Oe)

87654

32

1

0

En

erg

y (K

)

Hz (Oe)

n = 0

Hbias

n = 2n = 3/2n = 1/2

n = 1

En

erg

y (K

)

Hz (Oe)

.

G. Shakurov, B. Malkin, B. Barbara, Appl. Magn. Res. 2005

0,2 0,4

0

50

Magnetic field (T)

175 GHz

1

2

3 4

2 1

2+3

Ho-dimer satellites in the EPR signal in 7LiYF4 (1% Ho):Bias-tunneling transitions only

Boris Malkin group, Kazan

0,4 0,6

-300

0

300

Magnetic field (T)

250 GHz

3 42

21

In the 7Li 0.1% sample the width of single ions ~3.5 mT and of dimers ~ 2mT

Toy model of two coupled effective spins, with gz /gx >> 1

H/J = ijSi

zSjz +

ij(Si

+Sj- + Sj

+Si-)/2 + ij (Si

+Sj+ + Sj

-Si-)

with

= (Jx + Jy)/4J = (Jx - Jy)/4J

This is why dipolar interactions induce co-tunneling

Co-tunnelingDiffusive tunneling

0 1 2 3 4 5 6 70

20

40

60

80

100

, G

Hz

B, KGs

7LiYF4:Ho 0.1% B||B1||c

2,8 3,2 3,6 4,0 4,4 4,8 5,2 5,6 6,0 6,4 6,8 7,2 7,640

60

80

100

120

140

160

180

200

220

240

Magnetic field (kOe)

J

frequency 100 GHz

0,00 0,05 0,10 0,15 0,20 0,25 0,30-250

-225

-200

-175

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

m=0

Ener

gy (G

Hz)

Magnetic field B (T)

m=2

Hyperfine sublevels of Ho3+ ion in LiYF4

Direct check of hyperfine sublevels from EPR In Ho:YLiF4 (B. Malkin group)

G. Shakurov, B. Malkin, B. Barbara, Appl. Magn. Res. 2005

0,00 0,05 0,10 0,15

210.80

210.55

210.35

210.90

Magnetic field B (T)

210.25

(GHz)

Direct observation of levels repulsionsHyperfine sublevels (m=2) in the EPR spectra

7

G. Shakurov, B. Malkin, B.Barbara, Appl. Magn. Res. 2005

80 100 120 140 160 1800.00

0.01

0.02(b)

1/T

1 (m

se

c-1)

Hz(mT)

19F_NMRM. J. Graf, A. Lascialfari, F. Borsa, A. M. Tkachuk, and B. Barbara (cond-mat 2005)

Phenomenological fit:1/T1 = B2 W/ [W2 + (N - )2] , = [1.3x1018 (H-23n)2 + n

2 ]1/2 with n ~20 mK,

Levels broadening at crossing is extremely small (~ 2 mT):

Decoherence strongly suppressed : possible to measure directly level repulsion

-1

-0.5

0

0.5

1

-0.08 -0.04 0 0.04 0.08

0.136 mT/s0.068 mT/s0.034 mT/s0.017 mT/s

M/M

s

µ0H (T)

0.04 K

n=1n=2

Case of a metallic matrix: Ho3+ ions in Y0.999Ho0.001Ru2Si2

n=0

These steps come from tunneling transitions of J+I of single Ho3+ ions,in a sea of free electrons.

B. Barbara, R. Giraud, W. Wernsdorfer, D. Mailly, A. Tkachuk, H. Suzuki, ICM-Rome, JMMM (2004)

CONCLUSION

Molecular magnetsCoexistence of classical hysteresis loop and resonant quantum tunneling

non-adiabatic Landau-Zener (single-ion picture)Observation of tunneling made possible by environmental spins (nuclear spins)Spin tunneling asssited by photons (photons bath)Strong decohrence by environmental spins (nuclear spins)

Highly diluted Ho3+ in LiYF4

Tunneling of the total angular momentum J = L+S of Ho3+ single ions two-bodies entanglement Quasi-isolated Ho3+ ions: J and I tunnel simultaneously (in a metal also: Ho in YSi2Ru2).

Relevant quantum number of Ho3+ is not J but I+J (Kramers, QPT…).

Co-tunneling, bias-tunneling, spin-diffusion in Ho3+ dimmers four-bodies entanglements, Co-tunneling of dimmers is observed.

Crucial role of the anisotropic character of dipolar interactions. Microscopic basis for the study of QPT (concentrated systems) and coherent quantum dynamics.

…. Molecular magnets with Rare-Earths R-E Double-Deckers also show single-ion tunneling on electro-nuclear states (M. Ruben)

Some perspectives

Higher order many-body tunneling and decoherence by the environment (quantum phase transitions)

Spin-echo experiment and Rabi oscillations on electronic states of

- Molecular magnets(intra-molecules hyperfine interactions ~10 mK)

- Entangled E-N pairs of Ho3+ (dipolar interactions, hyperfine interactions ~1 mK)

Metallic systems : Decoherence by free carriers on spin tunneling in metals, Injection of polarized spins..

(Tunneling, Kondo, Heavy fermions, Spintronics)

Spin qubits manipulated by photons

…..

Collaborations:J. Bonvoisin e t C. Joachim (CEMES, Toulouse)

F. Ciontu et Ph. Jorrand (IMAG, Grenoble)

Remerciements: J.P. Sutter, M. Kahn.

e-Photon h1

Photon h2

Photon h3

Qubit de spins coupled by the injection of an electron and manipulated by transfert of photo-electrons

Far infra-red : variations of charges (S)Sub-centimeter: variations of spin projections(mS)

Manipulating the exchange interactions between two spins

MANY THANKS

FOR

YOUR ATTENTION !