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. z Magnetic Anisotropy

 The actual eigenstates of the molecular spin are quantum superpositions of macroscopically different states  K Magnetic Anisotropy

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) Energy At low temperature, the field produced by the electrons on the nuclei is quasi-static  K

Zeeman energy distribution ( 55 Mn) I nuclear = 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

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 S z =1/2 S z =-1/2  ~ K Single spin flip is not possible because the energy fluctuation  ~10 -6 K due to “dynamic” nuclear spin interaction is much smaller than the static hyperfine energy splitting  ~10 -1 K

Two spin “flip-flop” transition 11 22 Transition can take place if Zeeman energies are in “resonance” |  1 -  2 | <  Transition probability is given by (Landau Zener) T 1 is the spectral diffusion period

Transition rate induced by spectral diffusion E E dR

Self-diffusion rate Overall relaxation rate is determined by the external rate plus the stimulated rate Solution:

Nuclear spin relaxation and decoherence rates d=3, agrees with Morello, et al, Phys. Rev. Lett. 93, (2004) d=2

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 T 2 can be predicted, subject to test Effect of polarization on the nuclear spin relaxation: 1/T 2 ~1/, 1/T 1diff ~1/ 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

Non-adiabatic “Floquet” Regime E1E1 E Level crossing when E 1 -E 2 -n  =0, n=0,1,-1,2,-2, …a/  Transition amplitudes: V 12,n = (a/  ) 1/2 U 0 /R 3 

Level crossing neighbors E1E1 E  TPU 0 Spectral diffusion covers level splitting:  TPU 0 ) a special consideration is needed

Non-adiabatic Transitions between Floquet States Number of level crossings during the spectral diffusion cycle (  1 ): N cr ~TPU 0 /  Transition probability per single crossing P tr ~ V tr 2 /(TPU 0 /  1 ) ~ ~ (U 0 Pa) 2 (  /a)  1 /(TPU 0 )=a(PU 0 ) 2  1 /(TPU 0 ) Self-consistent transition rate 1/  1 =a(PU 0 ) 2  1 /(TPU 0 )  1 N cr ~ a(PU 0 ) 2 coincides with the non-adiabatic Landau-Zener expression

Current Status Frequency 1/  1 1/  2 Mechanism  <T 2 (PU 0 ) 4 /a T(PU 0 ) 3 T(PU 0 ) 2 Quasistatic field, Linear Regime T 2 (PU 0 ) 4 /a<  <a(PU 0 ) 2 (a  ) 1/2 (PU 0 )(T 2 a  ) 1/4 (PU 0 ) Adiabatic field control a(PU 0 ) 2 <  <T(PU 0 ) a(PU 0 ) 2 (aT) 1/2 (PU 0 ) 3/2 Non-adiabatic “Landau-Zener” or “Floquet” regimes TPU 0 <  ???

Non-Linear Self-Consistent Regime TPU 0 <  Level crossing is permitted with the only one of n=a/  Floquet states, transition amplitude goes down by n -1/2 : RENORMALIZATION: P  Pa/ , U 0  U 0 (  /a) 1/2 E1E1 E  TPU 0

Relaxation rate Rate of the energy change v = Amplitude/Quasi-period = TPU 0 /  1, Transition amplitude  0p ~ U 0 (  /a) 1/2 PTPU 0 a/  Non-adiabatic case: a/  (T(PU 0 ) 2 ) 2 < v = TPU 0 /  1 Transition probability per one crossing: Transition rate: (Remember many-body theory)

Summary Frequency 1/  1 1/  2 Mechanism  <T 2 (PU 0 ) 4 /a T(PU 0 ) 3 T(PU 0 ) 2 Quasistatic field linear regime T 2 (PU 0 ) 4 /a<  <a(PU 0 ) 2 (a  ) 1/2 (PU 0 )(T 2 a  ) 1/4 (PU 0 ) Adiabatic field control a(PU 0 ) 2 <  <T(PU 0 ) a(PU 0 ) 2 (aT) 1/2 (PU 0 ) 3/2 Non-adiabatic regime TPU 0 <  <a(a/  )T(PU 0 ) 3 (a/  ) 1/2 T(PU 0 ) 2 Non-linear self- consistent regime a<  <T T(PU 0 ) 3 T(PU 0 ) 2 Fast field linear regime

Conclusion (1)Interaction induced relaxation is very complicated under the realistic conditions, non-linearity takes place at a>TPU 0 ~10 -5 K (10mK). For an elastic field a=  =10 4  K. One needs  ~10 -9 – 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 through the Tulane University

Dedication To Professor Siegfried Hunklinger with the best wishes of Happy 65 th birthday and the further great successes in all his activities