nuclear spin irreversible dynamics in crystals of magnetic molecules alexander burin department of...
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Nuclear spin irreversible dynamics in crystals of magnetic molecules
Alexander Burin
Department of Chemistry, Tulane University
Motivation
1. Nuclear spins serve as a thermal bath for electronic spin
relaxation
2. Nuclear spins form fundamentally interesting modeling
system to study Anderson localization affected by weak
long-range interaction
3. Nuclear spins can be used to control electronic spin
dynamics (slow down or accelerate). It is sensitive to
electronic polarization and dimension
Outline
1. Crystals of magnetic molecules; frozen electronic
spins
2. Nuclear spins in distributed static field
3. Spectral diffusion and self-diffusion
4. What next?
5. Acknowledgement
Magnetic molecules
Molecular magnets (more than 100 systems are synthesized already)
Mn, Fe, Ni, Co, … based macromolecules; S = 0, 1/2, 1, … , 33/2
The clusters are assembled in a crystalline structure, with relatively small (dipolar) inter-cluster interactions
15 Å
Crystals of magnetic molecules
The magnetic moment of the molecule is preferentially aligned along the z – axis.
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
z
Magnetic Anisotropy
2ˆzanisotr DSH
-70
-60
-50
-40
-30
-20
-10
0
-10 -5 0 5 10
Th-A T
QT
Sz
En
erg
y (
K)
The actual eigenstates of the molecular spin are quantum superpositions of macroscopically different states
10-11 K
Magnetic Anisotropy
0
,100ˆ 2
xz sDsH
Outline
1. Crystals of magnetic molecules; frozen electronic
spins
2. Nuclear spins in distributed static field
3. Spectral diffusion and self-diffusion
4. What next?
5. Acknowledgement
Single nuclear spin (H)
En
ergy
At low temperature, the field produced by the electrons on the nuclei is quasi-static
10-2 K
Zeeman energy distribution (55Mn)
izii SEH ˆ
Inuclear = 5/2
Three NMR lines corresponding to the three non-equivalent Mn sites
Finite width of lines due to interaction with all electronic spins f(E)
Interaction of nuclear spins
Magnetic dipole moments
ji
ij
jiij EK
R
nnmmV ,
63
10~3
,,,2
1ˆji
jiiji
iitot ssVsEH
Outline
1. Crystals of magnetic molecules; frozen electronic
spins
2. Nuclear spins in distributed static field
3. Spectral diffusion and self-diffusion
4. What next?
5. Acknowledgement
Scenario for spin self-diffusion
Assume the presence of irreversible dynamics in ensemble of nuclear spin.
Transitions of spins stimulates transitions of other spins due to spin-spin interactions
Can this process result in self-consistent irreversible dynamics?
Mechanism of spin diffusion
Single spin evolution (H)
B 40T
~ 0.01K
Sz=1/2
Sz=-1/2
~ 10-6 K
Single spin flip is not possible because the energy fluctuation ~10-6K due to “dynamic” nuclear spin interaction is much smaller than the static hyperfine energy splitting ~10-1K
Two spin “flip-flop” transition
12
312
0312
21 ~~R
u
R
mmV
Transition can take place if Zeeman energies are in “resonance” |1-2| <
Transition probability is given by (Landau Zener)
122
exp1TV
T1 is the spectral diffusion period
Transition rate induced by spectral diffusion
3
2
)( tW
dEEf )(1
0)(T
tnuEf
Rnd
unuR
Tu
06
122
exp1
E
E1
0 T
tnu
dR
nudEEfnTu
T
ttW 0
2104/124/1
1
)(9
328)(
Self-diffusion rate
nudEEfTnu
TTT0
2104/124/1
111
)(~
9
328~11
~1
Overall relaxation rate is determined by the external rate plus the stimulated rate
Solution:
kT
T
nudEEfnu
kkT
kT
1
1
2
020
2/142/1
11
~1
yields
)(27
264 ,
11~
1
Nuclear spin relaxation and decoherence rates
10
203
1
0
2
12
020
2/142/1
1
1000)(9
28
3
21
100)(27
2641
snudEEfnu
T
nu
dt
dE
T
snudEEfnu
T diff
d=3, agrees with Morello, et al, Phys. Rev. Lett. 93, 197202 (2004)
d=2
1232/30
22/3
0
2
1862/30
22/3
0
1
10)(1001
10)(10001
snudEEfnu
T
snudEEfnu
T
Outline
1. Crystals of magnetic molecules; frozen electronic
spins
2. Nuclear spins in distributed static field
3. Spectral diffusion and self-diffusion
4. What next?
5. Acknowledgement
What next?
Spin tunneling is suppressed in 2-d: subject for experimental verification?
Isotope effect in T2 can be predicted, subject to test
Effect of polarization on the nuclear spin relaxation: 1/T2~1/<M2>, 1/T1diff~1/<M2>2 to be tested
Outline
1. Crystals of magnetic molecules; frozen electronic
spins
2. Nuclear spins in distributed static field
3. Spectral diffusion and self-diffusion
4. What next?
5. Acknowledgement
Acknowledgement
To coworkers: Igor Tupitsyn & Philip Stamp
To Tulane Chemistry Department Secretary Ginette Toth for help in organizing this meeting
Funding by Louisiana Board of Regents, Tulane Research and Enhancement Fund and PITP
2
341
11
3
2)()(
06
1201
0
nuR
Tudn
T
nutEdEfEf
R
201
02
0
120
2
11
11)(
2
34
9
24
x
dxnT
nutdEEf
nu
Tu
unuR
Tund
T
nutEdEfEf
2
32exp1
3
2)()(
06
12
1
0
R
nudEEfnTu
T
ttW
x
dxT
nutdEEf
Tnu
0210
4/124/1
1
201
02102
)(9
328)(
11
11)(
2
34
9
24
2020
2/142/1
1
)(27
2641nudEEf
nu
T
Non-adiabatic “Floquet” Regime
E1
E
Level crossing when E1-E2-n =0, n=0,1,-1,2,-2, …a/
Transition amplitudes: V12,n= (a/)1/2U0/R3
Level crossing neighbors
E1E
TPU0
Spectral diffusion covers level splitting:
<TPU0 inevitable level crossing when |E1-E2|<a, otherwise (>TPU0) a special consideration is needed
Non-adiabatic Transitions between Floquet States
Number of level crossings during the spectral diffusion cycle (1): Ncr~TPU0/
Transition probability per single crossing Ptr ~ Vtr2/(TPU0/1) ~
~ (U0Pa)2(/a)1/(TPU0)=a(PU0)21/(TPU0)
Self-consistent transition rate 1/1=a(PU0)21/(TPU0)1Ncr ~ a(PU0)2 coincides with the non-adiabatic Landau-Zener expression
Current Status
Frequency 1/1 1/2Mechanism
<T2(PU0)4/a T(PU0)3 T(PU0)2 Quasistatic field, Linear Regime
T2(PU0)4/a<<a(PU0)2 (a)1/2(PU0) (T2a)1/4(PU0) Adiabatic field control
a(PU0)2<<T(PU0) a(PU0)2 (aT)1/2(PU0)3/2 Non-adiabatic “Landau-Zener”
or “Floquet” regimes
TPU0< ? ? ?
Non-Linear Self-Consistent Regime TPU0 <
Level crossing is permitted with the only one of n=a/ Floquet states, transition amplitude goes down by n-1/2: RENORMALIZATION:
PPa/, U0U0(/a)1/2E1E
TPU0
Relaxation rate
Rate of the energy change
v = Amplitude/Quasi-period = TPU0/1,
Transition amplitude 0p ~ U0(/a)1/2 PTPU0a/
Non-adiabatic case: a/(T(PU0)2)2 < v = TPU0/1
Transition probability
per one crossing:
Transition rate:
(Remember many-body theory)
.)(
0
14
022
0
TPU
PUaT
vW p
tr
30
1,1
)()/(11
PUTaWtrsd
Summary
Frequency 1/1 1/2Mechanism
<T2(PU0)4/a T(PU0)3 T(PU0)2 Quasistatic field linear regime
T2(PU0)4/a<<a(PU0)2 (a)1/2(PU0) (T2a)1/4(PU0) Adiabatic field control
a(PU0)2<<T(PU0) a(PU0)2 (aT)1/2(PU0)3/2 Non-adiabatic regime
TPU0<<a (a/)T(PU0)3 (a/)1/2T(PU0)2 Non-linear self-consistent regime
a<<T T(PU0)3 T(PU0)2 Fast field linear regime
Conclusion
(1) Interaction induced relaxation is very complicated under the realistic conditions, non-linearity takes place at a>TPU0
~10-5K (10mK). For an elastic field a==104K. One needs ~10-9 – 10-8 for the true linear regime. For an electric field a=el, assuming ~1D wanted el ~ 40V/m. Looks almost impossible (see, however, Pohl and coworkers, 2000).
(2) Theory predicts both linear temperature dependence and/or the absence of any temperature dependence. A careful treatment of existing measurements is needed (backgrounds, etc.)
(3) It is not clear whether the thermal equilibrium of phonons and TLS is fully established. This can change the way of the treatment of experimental data
Acknowledgement
(1) Yuri Moiseevich Kagan
(2) Leonid Aleksandrovich Maksimov
(3) Il’ya Polishchuk
(4) Fund TAMS GL 211043 through the Tulane University
Dedication
To Professor Siegfried
Hunklinger with the
best wishes of Happy
65th birthday and
the further great successes
in all his activities