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