1. Quantum coherence in an exchange-coupled dimer of single-molecule magnets Supported by: NSF, Research Corporation, & University of Florida PART I  Introduction to single-molecule magnets Emphasis on quantum magnetization dynamicsEmphasis on quantum magnetization dynamics PART II  Monomeric Mn 4 single-molecule magnets  Electron Paramagnetic Resonance technique, with examples  Focus on [Mn 4 ] 2 dimer Quantum mechanical coupling within a dimerQuantum mechanical coupling within a dimer Evidence for quantum coherence from EPREvidence for quantum coherence from EPR Stephen Hill and Rachel Edwards Department of Physics, University of Florida, Gainesville, FL32611 Nuria Aliaga-Alcalde, Nicole Chakov and George Christou Department of Chemistry, University of Florida, Gainesville, FL32611

2. PART I Introduction to single-molecule magnets Emphasis on their quantum dynamics

3. MESOSCOPIC MAGNETISM macroscale nanoscale permanent magnets micron particles nanoparticles clustersmolecular clusters Individual spins S = 10 23 10 8 6 5 4 3 2 10 1 multi - domainsingle - domain Single molecule nucleation, propagation and annihilation of domain walls uniform rotationquantum tunneling, quantum interference 0 1 -40-2002040 M/M S   H(mT) 0 1 -1000100 M/M S   H(mT) 0 1 01 M/M S   H(T) Fe 8 1K 0.1K 0.7K Mn 12 -ac Ferritin 1 nm10 nm100 nm superparamagnetism Classical Quantum size W. Wernsdorfer, Adv. Chem. Phys. 118, 99-190 (2001); also arXiv/cond-mat/0101104.

4. A. J. Tasiopoulos et al., Angew. Chem., in-press.

5. The first single molecule magnet: Mn 12 -acetate [Mn 12 O 12 (CH 3 COO) 16 (H 2 O) 4 ] · 2CH 3 COOH · 4H 2 0 Mn(IV) Mn(III) Oxygen Carbon S = 2 S = 3/2 Lis, 1980 Easy-axis anisotropy due to Jahn-Teller distortion on Mn(III)Easy-axis anisotropy due to Jahn-Teller distortion on Mn(III) Crystallizes into a tetragonal structure with S 4 site symmetryCrystallizes into a tetragonal structure with S 4 site symmetry Organic ligands ("chicken fat") isolate the moleculesOrganic ligands ("chicken fat") isolate the molecules R. Sessoli et al. JACS 115, 1804 (1993) Ferrimagnetically coupled magnetic ions (J intra  100 K)Ferrimagnetically coupled magnetic ions (J intra  100 K) Well defined giant spin (S = 10) at low temperatures (T < 35 K)

6. "up" "down" E  10 E9E9E9E9 E8E8E8E8 E7E7E7E7 E 10 E9E9E9E9 E8E8E8E8 E7E7E7E7 E6E6E6E6 E5E5E5E5 Spin projection - m s E6E6E6E6 E5E5E5E5 Energy E4E4E4E4 E4E4E4E4 21 discrete m s levels Quantum effects at the nanoscale (S = 10) Simplest case: axial (cylindrical) crystal field Eigenvalues given by: Small barrier - DS 2Small barrier - DS 2 Superparamagnet at ordinary temperaturesSuperparamagnet at ordinary temperatures  E  E  DS 2 10-100 K |D |  0.1  1 K for a typical single molecule magnet Thermal activation

7. "up" "down" E  10 E9E9E9E9 E8E8E8E8 E7E7E7E7 E 10 E9E9E9E9 E8E8E8E8 E7E7E7E7 E6E6E6E6 E5E5E5E5 Spin projection - m s E6E6E6E6 E5E5E5E5 Energy E4E4E4E4 E4E4E4E4 Thermallyassistedquantumtunneling Break axial symmetry: H T  interactions which do not commute with Ŝ z Quantum effects at the nanoscale (S = 10)  E eff <  E m s not good quantum # m s not good quantum # Mixing of m s statesMixing of m s states  resonant tunneling (of m s ) through barrier Lower effective barrierLower effective barrier

8. "up" "down" E  10 E9E9E9E9 E8E8E8E8 E7E7E7E7 E 10 E9E9E9E9 E8E8E8E8 E7E7E7E7 E6E6E6E6 E5E5E5E5 Spin projection - m s E6E6E6E6 E5E5E5E5 Energy E4E4E4E4 E4E4E4E4 Pure quantum tunneling Strong distortion of the axial crystal field: Temperature-independent quantum relaxation as T  0Temperature-independent quantum relaxation as T  0 Ground states a mixture of pure "up" and pure "down".Ground states a mixture of pure "up" and pure "down". "down""up" ± { } Ground state degeneracy lifted by transverse interaction:Ground state degeneracy lifted by transverse interaction: splitting  ( splitting  ( H T ) n  Tunnel splitting Quantum effects at the nanoscale (S = 10)

9. Spin projection - m s Applied field represents another source of transverse anisotropy Zeeman interaction contains odd powers of Ŝ x and Ŝ y Several important points to note: For now, consider only B//z : (also neglect transverse interactions) "down" "up" Magnetic quantum tunneling is suppressedMagnetic quantum tunneling is suppressed Metastable magnetization is blocked ("down" spins)Metastable magnetization is blocked ("down" spins) System off resonance X X Application of a magnetic field

10. Spin projection - m s System on resonance Applied field represents another source of transverse anisotropy. Zeeman interaction contains odd powers of Ŝ x and Ŝ y. Several important points to note: For now, consider only B//z : (also neglect transverse interactions) "down" "up" Resonant magnetic quantum tunneling resumesResonant magnetic quantum tunneling resumes Metastable magnetization can relax from "down" to "up"Metastable magnetization can relax from "down" to "up" Increasing field

11. Hysteresis and magnetization steps Friedman, Sarachik, Tejada, Ziolo, PRL (1996)Friedman, Sarachik, Tejada, Ziolo, PRL (1996) Thomas, Lionti, Ballou, Gatteschi, Sessoli, Barbara, Nature (1996)Thomas, Lionti, Ballou, Gatteschi, Sessoli, Barbara, Nature (1996) Mn 12 -ac Tunneling "off" Tunneling"on" Low temperature H=0 step is an artifact This loop represents an ensemble average of the response of many molecules

12. PART II Quantum Coherence in Exchange-Coupled Dimers of Single-Molecule Magnets

13. Mn 4 single molecule magnets (cubane family) [Mn 4 O 3 Cl 4 (O 2 CEt) 3 (py) 3 ] Mn III (S = 2) and Mn IV (S = 3/2) ions couple ferrimagnetically to give an extremely well defined ground state spin of S = 9/2.Mn III (S = 2) and Mn IV (S = 3/2) ions couple ferrimagnetically to give an extremely well defined ground state spin of S = 9/2. This is the monomer from which the dimers are made.This is the monomer from which the dimers are made. Distorted cubane Mn III : 3 × S = 2  Mn IV : S = 3 / 2  S = 9 / 2 C 3v symmetry

14. Fairly typical SMM: exhibits resonant MQT B//z Note: resonant MQT strong at B=0, even for half integer spin. 1 W. Wernsdorfer et al., PRB 65, 180403 (2002). 2 W. Wernsdorfer et al., PRL 89, 197201 (2002). Barrier  20D  18 KBarrier  20D  18 K Spin parity effect 1Spin parity effect 1 Importance of transverse internal fields 1Importance of transverse internal fields 1 Co-tunneling due to inter-SMM exchange 2Co-tunneling due to inter-SMM exchange 2

15. Energy level diagram for D < 0 system, B//z B // z-axis of molecule Note frequency range

16. Cavity perturbation Cylindrical TE01n (Q ~ 10 4 - 10 5 ) f = 16  300 GHz Single crystal 1 × 0.2 × 0.2 mm 3 T = 0.5 to 300 K,  o H up to 45 tesla M. Mola et al., Rev. Sci. Inst. 71, 186 (2000)  We use a Millimeter-wave Vector Network Analyzer (MVNA, ABmm) as a spectrometerWe use a Millimeter-wave Vector Network Analyzer (MVNA, ABmm) as a spectrometer (now 715 GHz!)

17. Energy level diagram for D < 0 system, B//z B // z-axis of molecule Note frequency range

18. HFEPR for high symmetry (C 3v ) Mn 4 cubane Field // z-axis of the molecule (±0.5 o ) [Mn 4 O 3 (OSiMe 3 )(O 2 CEt) 3 (dbm) 3 ]

19. Fit to easy axis data - yields diagonal crystal field terms

20.  Core ligands (X): Cl -, Br -, F - NO 3 -, N 3 -, NCO - OH -, MeO -, Me 3 SiO -  Peripheral ligands: (i) carboxylate ligands: - O 2 CMe, - O 2 CEt (ii) Cl -, py, HIm, dbm -, Me 2 dbm -, Et 2 dbm - Routes to incredible # of SMMs Jahn-Teller points towards core ligand

21. Antiferromagnetic exchange in a dimer of Mn 4 SMMs m1m1m1m1 m2m2m2m2 D 1 =  0.72 K J = 0.1 K To zeroth order, the exchange generates a bias field Jm'/g   which each spin experiences due to the other spin within the dimer [Mn 4 O 3 Cl 4 (O 2 CEt) 3 (py) 3 ] No H = 0 tunneling Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409

22. 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). Systematic control of coupling between SMMs - Entanglement Zeroth order bias term, JŜ z1 Ŝ z2, is diagonal in the  m z1,m z2  basis.Zeroth order bias term, JŜ z1 Ŝ z2, is diagonal in the  m z1,m z2  basis. Therefore, it does not couple the molecules quantum mechanically, i.e. tunneling and EPR involve single-spin rotations.Therefore, it does not couple the molecules quantum mechanically, i.e. tunneling and EPR involve single-spin rotations. Quantum mechanical coupling caused by the transverse (off- diagonal) parts of the exchange interaction J xy (Ŝ x1 Ŝ x2 + Ŝ y1 Ŝ y2 ).Quantum mechanical coupling caused by the transverse (off- diagonal) parts of the exchange interaction J xy (Ŝ x1 Ŝ x2 + Ŝ y1 Ŝ y2 ). This term causes the entanglement, i.e. it truly mixes  m z1,m z2  basis states, resulting in co-tunneling and EPR transitions involving two-spin rotations.This term causes the entanglement, i.e. it truly mixes  m z1,m z2  basis states, resulting in co-tunneling and EPR transitions involving two-spin rotations. CAN WE OBSERVE THIS?CAN WE OBSERVE THIS?Heisenberg: JŜ 1.Ŝ 2

23. S 1 = S 2 = 9 / 2 ; multiplicity of levels = (2S 1 + 1) (2S 2 + 1) = 100 Look for additional splitting (multiplicity) and symmetry effects (selection rules) in EPR or tunneling experiments.

24. First clues: comparison between monomer and dimer EPR data Monomer Monomer - 1 / 2 to 1 / 2 Exchange bias [Mn 4 O 3 Cl 4 (O 2 CEt) 3 (py) 3 ]

25. Full exchange calculation for the dimer } Isotropic exchange Monomer Hamiltonians Apply the field along z, and neglect the transverse terms in Ĥ S1 and Ĥ S2.Apply the field along z, and neglect the transverse terms in Ĥ S1 and Ĥ S2. Then, only off-diagonal terms in Ĥ D come from the transverse ( x and y ) part of the exchange interaction, i.e.Then, only off-diagonal terms in Ĥ D come from the transverse ( x and y ) part of the exchange interaction, i.e. The zeroth order Hamiltonian ( Ĥ 0D ) includes the exchange bias.The zeroth order Hamiltonian ( Ĥ 0D ) includes the exchange bias. The zeroth order wavefunctions may be labeled according to the spin projections ( m 1 and m 2 ) of the two monomers within a dimer, i.e.The zeroth order wavefunctions may be labeled according to the spin projections ( m 1 and m 2 ) of the two monomers within a dimer, i.e. The zeroth order eigenvalues are given byThe zeroth order eigenvalues are given by

26. Full exchange calculation for the dimer 1st order correction lifts degeneracies between states where m 1 and m 2 differ by ±1 One can consider the off-diagonal part of the exchange ( Ĥ' ) as a perturbation, or perform a full Hamiltonian matrix diagonalizationOne 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 = m 1 + m 2.Ĥ' preserves the total angular momentum of the dimer, M = m 1 + m 2. Thus, it only causes interactions between levels belonging to a particular value of M. These may be grouped into multiplets, as follows...Thus, it only causes interactions between levels belonging to a particular value of M. These may be grouped into multiplets, as follows...

27. 1st order corrections to the wavefunctions Ĥ' is symmetric with respect to exchange and, therefore, will only cause 2nd order interactions between states having the same symmetry, within a multiplet.Ĥ' is symmetric with respect to exchange and, therefore, will only cause 2nd order interactions between states having the same symmetry, within a multiplet. 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.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.

28.  40.5  32.5  26.5  24.5  22.5  8.5 Exchange bias Full Exchange (1) - S (2) - A (3) - S (4) - S (5) - A (6) - S (7) - A & S (8) - A (9) - S (a) (b) (c) (d) (e) 1st and 2nd order corrections to the energies Magnetic-dipole interaction is symmetric Matrix element very small

29. 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.The effect of Ĥ' is field-independent, because the field does not change the relative spacings of levels within a given M multiplet.

30. Clear evidence for coherent transitions involving both molecules Experiment Simulation S. Hill et al., Science 302, 1015 (2003) f = 145 GHz 9 MHz    > 1 ns

31. 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.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.Thus, EPR reveals the quantum coupling. Coupled states remain coherent on EPR time scales.Coupled states remain coherent on EPR time scales.

32. R. Tiron et al., Phys. Rev. Lett. 91, 227203 (2003) Next session: B25.011 Confirmed by hole-digging (minor loop) experiments

33. Control of exchange This scheme is 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). "off" "on"  +  Light

34. Decoherence – role of nuclear spins The role of dipolar and hyperfine fields was first demonstrated via studies of isotopically substituted versions of Fe 8. [Wernsdorfer et al., Phys. Rev. Lett. 82, 3903 (1999)] Reduce via nuclear labeling – may require something other than Mn Also have to worry about intermolecular interactions

35. Mn 4 Chicken Fat ,

36. Mn 4 , Chicken Fat

37. 1.Control over magnitude and symmetry of the anisotropy through the choice of molecule: Choice of magnetic ion, modifications to molecular core, etc.Choice of magnetic ion, modifications to molecular core, etc. 2.Reduce electronic spin-spin interactions by adding organic bulk to the periphery of the SMM, or by diluting with non-magnetic molecules. 3.Reduce electron-nuclear cross-relaxation by isotopic labeling. 4.Move the tunneling into frequency window where decoherence may be less severe: Achieved with lower spin and lower symmetry molecules,Achieved with lower spin and lower symmetry molecules, or with a transverse externally applied field,or with a transverse externally applied field, or by deliberately engineering-in exchange interactions.or by deliberately engineering-in exchange interactions. 5.Move over to antiferromagnetic systems, e.g. the dimer: Quantum dynamics of the Néel vector - harder to observe!Quantum dynamics of the Néel vector - harder to observe! The molecular approach is the key Immense control over the magnetic unit and its coupling to the environmentImmense control over the magnetic unit and its coupling to the environment

38. What do we currently understand?  Quantum tunneling is extremely sensitive to SMM symmetry Transverse anisotropies provide tunneling matrix elementsTransverse anisotropies provide tunneling matrix elements  Magnetization dynamics controlled by nuclear and electron spin-spin interactions Fluctuations drive SMMs into and out of resonanceFluctuations drive SMMs into and out of resonance  Such interactions represent unwanted source of decoherence What do we not understand?  What are the dominant sources of quantum decoherence?  What are typical decoherence times for various quantum states based on SMMs which could be useful?  How can we reduce decoherence?  Can we control the spin dynamics coherently?  Pulsed/time-domain EPR

39. UF Physics Rachel Edwards Alexey Kovalev Konstantin Petukhov Susumu Takahashi Jon Lawrence Norman Anderson Tony Wilson Cem Kirman Shaela Jones Sara Maccagnano UF Chemistry George Christou Nuria Aliaga-Alcalde Monica Soler Nicole Chakov Sumit Bhaduri Muralee Murugesu Alina Vinslava Dolos Foguet-Albiol UCSD Chemistry David Hendrickson En-Che Yang Evan Rumberger FSU Chemistry Naresh Dalal Micah North David Zipse Randy Achey Chris Ramsey NYU Physics Andy Kent Enrique del Barco Many collaborators/students involved...illustrates the interdisciplinary nature of this work...illustrates the interdisciplinary nature of this work Also: Kyungwha Park (NRL) Marco Evangelisti (Leiden) Hans Gudel (Bern) Wolfgang Wernsdorfer (Grenoble) Mark Novotny (MS State U) Per Arne Rikvold (CSIT - FSU)