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