From a single molecule to an ensemble of molecules at T ~0 :

From a single molecule to an ensemble of molecules at T ~0 : Both tunneling rate and decoherence increase

ener

gy

magnetic field

²

| S, -m >

| S, m-n >

1 P

1 - P

| S, -m >

| S, m-n >

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

Spin-bath (Prokofiev and Stamp):

PSB ~ (2/0)e-││/0.n(ED) >> PLZ

0= hyperfine energy = tunnel window

Large spins Mesoscopic tunneling (slow)

Nuclear spins Observation possible Strong decoherence.

H= - DSz2 - BSz

4 - E(S+2 + S-

2) - C(S+4 + S-

4) - gBSzHz

Barrier in zero field (symmetrical)H= - DSz

2 - BSz4 - 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

Ener

gy

ener

gy

magnetic field

²

| S, -m >

| S, m-n >

1 P

1 - P

| S, -m >

| S, m-n >

H // -M

New resonances at gBHn = nD (B=0)

Thermally activated tunneling

Landau-Zener transition at avoided level crossing

(single molecule)

Tunneling probability:

P=1 – exp[-(/ħ)2/c]

c = dH/dt

Coexistence of tunneling and hysteresis

Proposal of Morello, Stamp, Tupitsyn

-1

-0.5

0

0.5

1

-0.5 0 0.5 1 1.5 2 2.5

0°10°19°32°44°56°64°81°

M/M

S

B0L (T)

T=1.75K

Effect of a tilted field (Mn12-ac)

J. Appl. Phys. (1997)

Easy axis

өBBL

BT

Transverse field with constant transverse field (Fe8)

-1

-0.5

0

0.5

1

-0.2 0 0.2 0.4 0.6 0.8

M/M

S

µ0Hz(T)

Htrans =

0.000 TdHz/dt = 14 mT/s0.056 T0.112 T0.196 T

H= - DSz2 - BSz

4 - E(S+2 + S-

2) - C(S+4 + S-

4) - gBSzHx - gBSzHz

~ DS2(┴ / Il)2S/p with ┴ << Il

2 (E/D)S

4 (CS2/D)S/2

1 (Hx/DS)2S

(Parity)

36

40

44

48

52

56

60

64

68

-2,5 -2 -1,5 -1 -0,5 0

/kB=67K

/kB=60K

/kB=59K

ef

f (K)

B (T)

n=4n=3

n=0n=1

n=2

n=5n=6

Mn12-ac

No effect of S = 9

A (small) parity effect on thermally activated tunneling (S=10)

-(S-1)

- S

S-1

S

-(S-1)

-S

S-2

S-1

S

n= 0, 2…

n=1, 3…

JMMM (1999)

4 (E/D)S/2

0

-1 0

Large parity effect and quantum phase interference at low temperature (Fe8)

[Mn12]-2e

S = 10

W. Wernsdorfer et al, PRL (2005), Science (1999)

-1 -0.5 0 0.5 10.1

1

10² t

unne

l(10

-8 K

)

µ0Htrans(T)

n = 0

n = 1

n = 2 0°

= ° cosor = ° sin

gBHx/[2E(E+D)]1/2

(e.g. review Tupitsyn, BB)

Z

Y

XH

A

B

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°

Dephasing

How the system escapes from the quantum regime (Mn12-ac)

Chiorescu et al, PRL, 83, 947 (1999)

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

B n (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)

Bn/n = D –B[(m-n)2+n2] . Sharp or continuous transition

Crossover From Quantum to Classical 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=0n=1

n=2n=3n=4

n=6n=7

n=8n=9n=10

B 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)

Classical Thermal Activation

Tblocking

Ground-state Tunneling

Tc-o

(Mn12-ac)

Shorter timescales (ac susceptibility): Tunneling moves to higher temperatures

spin down spin up

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

Ground state tunneling

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

|S,S> |S,-S>

SZ

Ener

gy

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5

(M+

Ms)

/(2M

s)

B0 (T)

2

3

5

1

67 4

T=2.1 K1. B

0=-0.691 T

2. B0=-0.794 T

3. B0=-0.824 T

4. B0=-0.841 T

5. B0=-0.856 T

6. B0=-0.868 T

7. B0=-0.909 T

First relaxation curves (Mn12-ac)

Scaling of the Quantum Dynamics of Mn12-acM/Ms= f(t/(H,T))

Exponential to Square Root Relaxation N. Prokofiev and P. Stamp, PRL 80, 5794 (1998)

0

0.2

0.4

0.6

0.8

1

10 2 10 4 10 6 10 8

2.0 K2.1 K2.2 K2.3 K2.4 K2.5 K2.6 K2.7 K2.8 KM

/Ms

t (s)

t1/2 (s1/2)

0.96

0.98

0 100 200 300

M/Ms

2.0 K

1.7 K

1.5 K

1.8 K

1.9 K

L. Thomas et al, J. Low Temp. Phys. (1998); PRL (1999).Paulsen et al J. Low Temp (1998).

t/(T)

Sqrt(t) at in H// and H┴

1 10 100 1000-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

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

0,4

0,8

1,2

1,6

2,0 M / MS

M|| / MS

T = 0.5 Kn = 0B

T = 4.42 T

T = 0.9 Kn = 8B

L = 4.02 T

norm

aliz

ed m

agne

tizat

ion

t (s)

exponential regime

square root regime

0

2

3L

T

(1/s)

10

10 T (K)

0 1 2 3 4 5 6 7 8 9 10-140

-120

-100

-80

-60

-40

-20

0

20

E (K

)

transverse field (T)

Emin

Emax

Calculated Energy Spectrum Measured relaxation

Chiorescu et al, PRL (2000)

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 4,150

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

B n (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

sq

rt(s

-1)

µ0H(T)

M in = -0.2 M s

-0.005 0 0.0054 10-6

6 10-68 10-6

10-5

2 10-5 t0=0s

t0=10st0=5s

t0=20st0=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 (x105) Strong decoherence

Environmental effects

Central molecule spinMn12, Fe8

Spin-bathEnvironmental spins

Enhance tunnelingMesoscopic spins

Decoherence

Phonon-bath

Spin-phonons transitionBottleneck (TB>>T1)

V15

From Large to Low Spin Molecules

Large spins Low spins Mn12 , Fe8 V15

Order Parameter Ferro. Antiferro. (S = 10) (N =15/2, S=1/2) Barrier DS2 Large Small Tunnel Splitting Small Large Dipolar interactions 50mT 1mT

Spins bath Essential Important

Phonons bath Depends on T Important

Time Reversal Symmetry

=0 (Kramers Theorem)Experimentally: ~80 mK.D ~Jg /g ~ 50mK (Also hyperfine interactions ~20 mK)

V15 : a large molecule with collective spin ½ 15 spins ½ with AF coupled (DH=215)

-1 0 1 2 3 4 5

-1

0

1

2

3 0.1 K 0.3 K 0.9 K 4.2 K fit, diff. T

M (µ

B)

applied field (T)

-4 -2 0 2 4B

0(T)

S=3/2

S=1/2

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

Diagonalization of the 15-Spin ½ Hamiltoninan H = JijSiSj (I. Tupitsyn)

200 calculated levels.

The 8 levels lowest levels frustrated 3-spins ½ triangle

Effective hamiltonian:

H = |J | (S1S2 + S2S3 + S3S4) – gBB(S1 + S2 + S3)

Measurements of M(H) and (T) confirm this picture

Dissipative spin reversal in a two-level system ( T<0.1K)Effects of the phonon bath at low temperatureLow sweeping rates / Strong coupling to the cryostat

LZS transition at Finite Temperature (dissipative)

botl1 > meas

Hysteresis (≠Orbach process).

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=Tph (K)

(c)M

(µ

B)

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)Abragam and Bleaney (Oxford, 1970)

M(H): Irreversible

Equilibrium (Reversible)

M(H)=Msth{H/2kT}

Spin temperature: n1/n2=exp(H/kTs)

nT= number of phonons with ћ =

Ts = T

Ts << T

Ts T (n1/n2= constant)

nTph = nT

nTph increases rapidly

hole in the phonons density nTph ~ 0

0

Time-scales: B >> 1 (v = dB/dt) B=(/H

2)tanh2(H/2kT)

< 0

In the presence of a barrier (large spins)Similar phonons emission:Recovery to the ground-state by Inelastic tunneling ?inev2

3(1+n(H))

Now: fast sweeping rates / weak coupling to the cryostat

Adiabatic LZS Spin Rotation is recovered (Ts~0, reversible but out of equilibrium)

Fit to M = (1/2)(gB)2H/2+(gBH)2 80 mK

Chiorescu et al PRB, 2003

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

0,2

0,4

0,6

0,8

1,0

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

0,2

0,4

0,6

0,8

1,0

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

0,2

0,4

0,6

0,8

1,0

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

0,2

0,4

0,6

0,8

1,0M

/MS

B0(T)

= 130

0.014 T/s 0.1 K 0.2 K

M/M

S

B0(T)

= 130

60 mK 0.14 T/s 0.14 mT/s

M/M

S

B0(T)

= 0.09

60 mK 0.28 T/s 0.14 mT/s

M/M

S

B0(T)

= 0.09

0.14 T/s 0.1 K 0.2 K

Relaxation Experiments

0 2000 4000 6000 8000 100000,00

0,05

0,10

0,15

0,20

0,25

0,30

0 2000 4000 6000 8000 100000,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

M/M

S

t (s)

B0=0.014 T

0.15 K

H: fit=551s / th=1323s

0.05 K

H: fit=1507s / th=8716s

M/M

St (s)

B0=0.07 T

0.15 K

H: fit=970s / th=997s

0.05 K

H: fit=3883s / th=3675s

Inside Outside

B << calculated value B (B,T) ~ calculated value Nuclear spin-bath affects bottleneck Bottleneck only

Fit of M(t) to the Bottleneck model B (B,T)

Environmental effects

Central molecule spinMn12, Fe8

Spin-bathEnvironmental spins

Enhance tunnelingMesoscopic spins

Decoherence

Phonon-bath

Spin-phonons transitionBottleneck (TB>>T1)

Electromagnetic radiation bath

Spin-photons transitions(incoherent)

V15