1. Stephen Hill, Rachel Edwards Nuria Aliaga-Alcalde and George Christou
Electron Paramagnetic Resonance Studies of Quantum Coherence in Dimers of Mn4 Single-Molecule Magnets
Stephen Hill, Rachel Edwards
Department of Physics, University of Florida, Gainesville, FL32611
Nuria Aliaga-Alcalde and George Christou
Department of Chemistry, University of Florida, Gainesville, FL32611
Monomeric Mn4 single-molecule magnets and quantum tunneling
Electron Paramagnetic Resonance technique, with examples
Focus on [Mn4]2 dimer
Evidence for exchange bias from EPR
Quantum mechanical coupling within a dimer
EPR spectrum for the dimer
Concluding remarks
Support from: US National Science Foundation, Research Corporation

2.  Mn4 single molecule magnets (cubane family) S = 9/2
[Mn4O3Cl4(O2CEt)3(py)3]
C3v symmetry
MnIII: 3 × S = 2 -
MnIV: S = 3/2 
S = 9/2
Distorted cubane
MnIII (S = 2) and MnIV (S = 3/2) ions couple ferrimagnetically to give an extremely well defined ground state spin of S = 9/2.
This is the monomer (C3v symmetry) from which the dimers are made.

3. Fairly typical SMM: exhibits resonant MQT
B//z
1W. Wernsdorfer et al., PRB 65, (2002).
2W. Wernsdorfer et al., PRL 89, (2002).
Barrier  20D  18 K
Spin parity effect1
Importance of transverse internal fields1
Co-tunneling due to inter-SMM exchange2
Note: resonant MQT strong at B  0, even for half integer spin.

4. Zeeman diagram for D < 0 system, B//z
B // z-axis of molecule
Note frequency range

5. Cavity perturbation Cylindrical TE01n (Q ~ 104 - 105) f = 16  300 GHz
S. Takahashi
Rev. Sci. Inst.
Submitted (2004)
Cavity perturbation
Cylindrical TE01n (Q ~ )
f = 16  300 GHz
Single crystal 1 × 0.2 × 0.2 mm3
T = 0.5 to 300 K, moH up to 45 tesla
(now 715 GHz!)
We use a Millimeter-wave Vector Network Analyzer (MVNA, ABmm) as a spectrometer
M. Mola et al., Rev. Sci. Inst. 71, 186 (2000)

6. Zeeman diagram for D < 0 system, B//z
B // z-axis of molecule
Note frequency range

7. HFEPR for high symmetry (C3v) Mn4 cubane
[Mn4O3(OSiMe3)(O2CEt)3(dbm)3]
Field // z-axis of the molecule (±0.5o)

8. Fit to easy axis data - yields diagonal crystal field terms
Note (0,0) intercept
Kramers theorem
See posters: PB-122 and PB-110

9. Antiferromagnetic exchange in a dimer of Mn4 SMMs
[Mn4O3Cl4(O2CEt)3(py)3]
Monomer Zeeman diagram m1 m2
D = -0.75(1) K
B04 = 5 × 10-5 K
J  0.12(1) K

10. Antiferromagnetic exchange in a dimer of Mn4 SMMs
[Mn4O3Cl4(O2CEt)3(py)3]
Monomer Zeeman diagram
B//z m1
 
  m2
D = -0.75(1) K
B04 = 5 × 10-5 K
J  0.12(1) K
Multiplicity increases from
(2S +1) to (2S +1)2

11. Antiferromagnetic exchange in a dimer of Mn4 SMMs
[Mn4O3Cl4(O2CEt)3(py)3]
Dimer Zeeman diagram m1
EPR
Bias should shift the single spin (monomer) EPR transitions.
 
  m2
D = -0.75(1) K
B04 = 5 × 10-5 K
J  0.12(1) K
To zeroth order, the exchange generates a bias field BJ = Jm'/gmB which each spin experiences due to the other spin within the dimer
Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002,

12. Effect of exchange bias on the quantum tunneling
[Mn4O3Cl4(O2CEt)3(py)3]
-40
-30
-20
-10
-1.2
-0.8
-0.4
0.4
0.8
1.2
Energy (K) H z
(T)
(1)
(4)(5)
(9/2,9/2)
(-9/2,9/2)
(-9/2,-9/2)
(-9/2,7/2)
(9/2,-7/2)
(9/2,-5/2)
(2)(3)
No H = 0 tunneling m1 m2
Just like the EPR, the exchange bias shifts the positions of the magnetic quantum tunneling resonances
Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002,
D1 = K
J  0.1 K

13. Systematic control of coupling between SMMs - Entanglement
This scheme in the same spirit as proposals for multi-qubit devices based on quantum dots
D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998).
Tuesday
plenary
lecture
by Seigo
Tarucha
Heisenberg:
JŜ1.Ŝ2
Quantum mechanical coupling caused by the transverse (off-diagonal) parts of the exchange interaction Jxy(Ŝx1Ŝx2 + Ŝy1Ŝy2).
This term causes the entanglement, i.e. it truly mixes mz1,mz2 basis states, resulting in co-tunneling (WW2) and EPR transitions involving two-spin rotations.
CAN WE OBSERVE THIS?
Zeroth order bias term, JŜz1Ŝz2, is diagonal in the mz1,mz2 basis.
Therefore, it does not couple the molecules quantum mechanically, i.e. tunneling and EPR involve single-spin rotations.

14. S1 = S2 = 9/2; multiplicity of levels = (2S1 + 1) (2S2 + 1) = 100
Look for additional splitting (multiplicity) and symmetry effects (selection rules) in EPR.

15. S1 = S2 = 9/2; multiplicity of levels = (2S1 + 1) (2S2 + 1) = 100
Look for additional splitting (multiplicity) and symmetry effects (selection rules) in EPR.

16. First clues: comparison between monomer and dimer EPR data
[Mn4O3Cl4(O2CEt)3(py)3]
Exchange bias
Monomer
-1/2 to 1/2
Monomer

17. Full exchange calculation for the dimer}
Monomer Hamiltonians
Isotropic exchange
Apply the field along z, and neglect the transverse terms in ĤS1 and ĤS2.
Then, only off-diagonal terms in ĤD come from the transverse (x and y) part of the exchange interaction, i.e.
The zeroth order Hamiltonian (Ĥ0D) includes the exchange bias.
The zeroth order wavefunctions may be labeled according to the spin projections (m1 and m2) of the two monomers within a dimer, i.e.
The zeroth order eigenvalues are given by

18. Full exchange calculation for the dimer
One can consider the off-diagonal part of the exchange (Ĥ') as a perturbation, or perform a full Hamiltonian matrix diagonalization
Ĥ' preserves the total angular momentum of the dimer, M = m1 + m2.
Thus, it only causes interactions between levels belonging to a particular value of M. These may be grouped into multiplets, as follows...
1st order correction
lifts degeneracies between
states where m1 and m2
differ by ±1

19. 1st order corrections to the wavefunctions
The symmetries of the states are important, both in terms of the energy corrections due to exchange, and in terms of the EPR selection rules.
Ĥ' is symmetric with respect to exchange and, therefore, will only cause 2nd order interactions between states having the same symmetry, within a multiplet.

20. 1st and 2nd order corrections to the energies
Exchange
bias
Full Exchange
1st order
-18.5
-22.5
-24.5
-26.5
1st order
-32.5
-40.5
Magnetic-dipole interaction is symmetric

21. 1st and 2nd order corrections to the energies
Exchange
bias
Full Exchange
-18.5
2nd order
-22.5
-24.5
2nd order
-26.5
-32.5
-40.5
Magnetic-dipole interaction is symmetric

22. 1st and 2nd order corrections to the energies
Exchange
bias
Full Exchange
(4) - S
(5) - A
(6) - S
(7) - A & S
(8) - A
(9) - S
(d)
(1) - S
(2) - A
(3) - S
(a)
(b)
(c)
(e)
-18.5
-22.5
-24.5
-26.5
Matrix element
very small
-32.5
-40.5
Magnetic-dipole interaction is symmetric

23. Variation of J, considering only the exchange bias
Variation of J, considering the full exchange term
Simulations clearly demonstrate that it is the off diagonal part of the exchange that gives rise to the EPR fine-structure.
Thus, EPR reveals the quantum coupling.
Above simulations consider only lowest 20 states.

24. Clear evidence for coherent transitions involving both molecules
f = 145 GHz
Experiment
Simulation
9 GHz
 tf > 1 ns
Jz = Jxy = 0.12(1) K
S. Hill et al., Science 302, 1015 (2003)

25. Clear evidence for coherent transitions involving both molecules
D = -0.75(1) K
B04 = 5 × 10-5 K
J  0.12(1) K
S. Hill et al., Science 302, 1015 (2003)

26. A comparison between two dimers - NA11 (hexane) and NA3 (MeCN)
J = 0.12 K
J = 0.10 K
NA11, 145 GHz
NA3, f = 185 GHz

27. Confirmed by hole-digging (minor loop) experiments
R. Tiron et al., Phys. Rev. Lett. 91, (2003)

28. Summary/conclusions
Experiments on the dimer of [Mn4]2 show that, on the time scale of EPR (nanoseconds), the dimer really behaves as a coupled quantum system
Exciting prospects for quantum information processing if one could find a way to modulate the exchange
Science 302, 1015 (2003)
Future work:
Control of exchange coupling
Time-domain EPR  measure tf

30. Spectra
Simulation takes 57 ms per increment in the magnetic field. The simulations above took 34 s to calculate.
Real data from: Hill, et all. Science, Vol 302, Issue 5647, , 7 November 2003

31. Confirmation of the exchange bias for NA3
Single spin transitions should be shifted in field by an amount JkBm'/gmB (the "exchange bias"), where m' denotes the state of the other half of the dimer.
Splitting of ground state EPR transition due to extra multiplicity of transitions due to the exchange bias.
Shift in spectral weight between peaks in agreement with the model of Wernsdorfer
Notice the
frequency range

32. Magnetic field dependence
This figure does not show all levels
The effect of Ĥ' is field-independent, because the field does not change the relative spacings of levels within a given M multiplet.