Transcript Document

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
systems are synthesized already)
100
Mn, Fe, Ni, Co, … based macromolecules;
S = 0, 1/2, 1, … , 33/2
Crystals of magnetic molecules
 15 Å
The clusters are assembled in a
crystalline structure, with relatively
small (dipolar) inter-cluster interactions
Magnetic Anisotropy
2
ˆ
Hanisotr  DSz
Sz
-10
-5
0
5
0
-10
Energy (K)
z
-20
-30
-40
-50
-60
Th-A T
QT
-70
The magnetic moment of the molecule is preferentially
aligned along the z – axis.
10
Magnetic Anisotropy
Sz
-10
-5
0
5
10
0
Energy (K)
-10
-20
-30
-40
-50
-60
Th-A T
 10-11 K
QT
-70
The actual eigenstates of the
molecular spin are quantum
superpositions of macroscopically
different states
2
ˆ
H  100Dsz  sx ,
0
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
Energy
Single nuclear spin (H)
 10-2 K
At low temperature, the field produced by the electrons on the
nuclei is quasi-static
Zeeman energy distribution (55Mn)
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)
Hˆ i  Ei Siz
Interaction of nuclear spins
Magnetic dipole moments

Vij 
mi m j   3n n 
3
ij
R
~ 10 K  Ei , j
6
1
  
ˆ
H tot   Ei si  Vij si s j
2 i , j , , 
i
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)
Sz=-1/2
B  40T
 ~ 10-6 K
 ~ 0.01K
Sz=1/2
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
Transition can take place
if Zeeman energies are in
“resonance” |1-2| < 
m1m2 u0
V~ 3 ~ 3
R12
R12
Transition probability is
given by (Landau Zener)
1
2
 2V 2T 1 

1  exp 
 

T1 is the spectral
diffusion period
Transition rate induced by spectral diffusion
W (t ) 



t
ndR
f ( E )dE f ( E )u0 n
T1


 2u 2T1 

1  exp  6
 R u n 

0 




u
t
u0 n
T1
E
E
dR
t 8  21/ 4 3 2 1/ 4
W (t ) 
T1
9
u0 nT1
2
f
( E )dEu0 n



 2
3
Self-diffusion rate
Overall relaxation rate is determined by the external rate plus the
stimulated rate
1 1 1 8 2
~  ~
T1 T1 T1
1/ 4
3 
9
2
1/ 4
~
u0 nT1
2
f
( E )dEu0 n


Solution:
1
1
64  21/ 2  4 1/ 2 u0 n
, k
~  k  1
T1k
27

T1
1
T1   yields ~  k
T1
 f
2
( E )dEu0 n

2
Nuclear spin relaxation and decoherence rates
d=3, agrees with Morello, et al, Phys. Rev. Lett. 93, 197202 (2004)
1
T1diff
64  21/ 2  4 1/ 2 u0 n

27

 f
2

2
( E )dEu0 n  100s 1
1
dE
 2 u0 n 8 2 u0 n 2
1



f
(
E
)
dE
u
n

1000
s
0
T2
dt
3 T1
9
 
3
d=2
u0 n 3 / 2
1
 1000
T1

u0 n 3 / 2
1
 100
T2

 f
 f
2
2
( E )dEu0 n
( E )dEu0 n
3/ 2
3/ 2
  10
6
8
  10
3
2
s 1
s 1
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


2

u
3T1
 2 u0 n
f ( E )dEf ( E )
t
ndR 1  exp  6
 R u n 2
3
T1

0





u






u0 n
 2
1

f ( E )dEf ( E )
t
n  dR 1 

4u02 3T1
3
T1
1 6

R u0 n 2










u
n
1
2
0
f ( E )dEt
n  dx1 
1
T1 0 
1 2

x





4 2 2
9
4u02 3T1
u0 n 2

4 2 2
9
4u0 n3T1
 2 




u0 n
1 
2
f ( E )dEt
dx1 


1 
T1 0 
1 2

x 
t 8  21/ 4 3 2 1/ 4
W (t ) 
T1
9
u0 nT1
2
f
( E )dEu0 n


1 64 21/ 2  4 1/ 2 u0 n

T1
27

 f
2
( E )dEu0 n

2
Non-adiabatic “Floquet” Regime
Level crossing when E1-E2-n =0, n=0,1,-1,2,-2, …a/
Transition amplitudes: V12,n= (a/)1/2U0/R3

E
E1
Level crossing neighbors

TPU0
E
Spectral
diffusion
covers level splitting:
<TPU0  inevitable
E1
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)21/(TPU0)
Self-consistent transition rate 1/1=a(PU0)21/(TPU0)1Ncr ~
a(PU0)2 coincides with the non-adiabatic Landau-Zener
expression
Current Status
Frequency
1/1
1/2
Mechanism
<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)
a(PU0)2<<T(PU0)
TPU0<
Adiabatic field
control
a(PU0)2
(aT)1/2(PU0)3/2
Non-adiabatic
“Landau-Zener”
or “Floquet”
regimes
?
?
?
Non-Linear Self-Consistent Regime TPU0 < 

TPU0
E
E1
Level
crossing
is
permitted with the only
one of n=a/ Floquet
states,
transition
amplitude goes down by
n-1/2:
RENORMALIZATION:
PPa/, U0U0(/a)1/2
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:

aT ( PU0 )  1
Wtr 

.
v
TPU0
2
0p
1
Transition rate:
 1,sd

(Remember many-body theory)
1
1
2
Wtr  (a /  )T ( PU0 )
4
3
Summary
Frequency
1/1
1/2
Mechanism
<T2(PU0)4/a
T(PU0)3
T(PU0)2
Quasistatic field
linear regime
T2(PU0)4/a<<a(PU0)2
a(PU0)2<<T(PU0)
(a)1/2(PU0)
(T2a)1/4(PU0)
Adiabatic field
control
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 selfconsistent 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==104K. 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