Quantum coherence in an exchange-coupled dimer of single-molecule magnets

1. Quantum coherence in an exchange-coupled dimer of single-molecule magnets 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 PART I • Introduction to single-molecule magnets • Emphasis on quantum magnetization dynamics PART II • Monomeric Mn4 single-molecule magnets • Electron Paramagnetic Resonance technique, with examples • Focus on [Mn4]2 dimer • Quantum mechanical coupling within a dimer • Evidence for quantum coherence from EPR Supported by: NSF, Research Corporation, & University of Florida

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

3. MESOSCOPIC MAGNETISM Classical Quantum macroscale nanoscale nanoparticles permanent micron clusters molecular Individual superparamagnetism magnets particles clusters spins 100 nm 10 nm 1 nm 23 10 8 6 5 4 3 2 S = 10 10 10 10 10 10 10 10 10 1 Mn12-ac Ferritin multi - domain single - domain Single molecule nucleation, propagation and uniform rotation quantum tunneling, annihilation of domain walls quantum interference 1 1 1 Fe 8 0.7K S S S 0 0 0 M/M M/M M/M 0.1K 1K -1 -1 -1 -40 -20 0 20 40 -100 0 100 -1 0 1 m m m H(mT) H(T) H(mT) 0 0 0 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. Easy-axis anisotropy due to Jahn-Teller distortion on Mn(III) • Crystallizes into a tetragonal structure with S4 site symmetry • Organic ligands ("chicken fat") isolate the molecules The first single molecule magnet: Mn12-acetate Lis, 1980 Mn(III) S = 2 S = 3/2 Mn(IV) Oxygen Carbon [Mn12O12(CH3COO)16(H2O)4]·2CH3COOH·4H20 R. Sessoli et al. JACS 115, 1804 (1993) • Ferrimagnetically coupled magnetic ions (Jintra 100 K) Well defined giant spin (S = 10) at low temperatures (T < 35 K)

6. Spin projection - ms Energy E-4 E4 E-5 E5 Eigenvalues given by: E-6 E6 E-7 E7 • Small barrier - DS2 • Superparamagnet at ordinary temperatures E-8 E8 DE  DS2 10-100 K E-9 E9 "up" "down" |D | 0.1 - 1 K for a typical single molecule magnet E-10 E10 Quantum effects at the nanoscale (S = 10) Simplest case: axial (cylindrical) crystal field 21 discrete ms levels Thermal activation

7. Spin projection - ms Energy E-4 E4 E-5 E5 E-6 E6 • msnot good quantum # • Mixing of msstates • resonant tunneling (of ms) through barrier • Lower effective barrier E-7 E7 E-8 E8 E-9 E9 "up" "down" DEeff < DE E-10 E10 Quantum effects at the nanoscale (S = 10) Break axial symmetry: HT interactions which do not commute with Ŝz Thermally assisted quantum tunneling

8. Spin projection - ms Energy E-4 E4 E-5 E5 E-6 E6 • Ground state degeneracy lifted by transverse interaction: • splitting  (HT)n "down" "up" E-7 E7 E-8 E8 • Ground states a mixture of pure "up" and pure "down". E-9 E9 "up" "down" E-10 E10 Do { } Tunnel splitting ± Pure quantum tunneling Quantum effects at the nanoscale (S = 10) Strong distortion of the axial crystal field: • Temperature-independent quantum relaxation as T0

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

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

11. Tunneling "on" Tunneling "off" Hysteresis and magnetization steps Mn12-ac Low temperature H=0 step is an artifact • Friedman, Sarachik, Tejada, Ziolo, PRL (1996) • Thomas, Lionti, Ballou, Gatteschi, Sessoli, Barbara, Nature (1996) 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. MnIII: 3 × S = 2 - MnIV: S = 3/2  S = 9/2 Mn4 single molecule magnets (cubane family) [Mn4O3Cl4(O2CEt)3(py)3] C3v symmetry 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 from which the dimers are made.

14. Fairly typical SMM: exhibits resonant MQT B//z 1W. Wernsdorfer et al., PRB 65, 180403 (2002). 2W. Wernsdorfer et al., PRL 89, 197201 (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.

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

16. q Cavity perturbation Cylindrical TE01n (Q~104 -105) 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)

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

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

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

20. Routes to incredible # of SMMs • Core ligands (X): Cl-, Br-, F- NO3-, N3-, NCO- OH-, MeO-, Me3SiO- Jahn-Teller points towards core ligand • Peripheral ligands: (i) carboxylate ligands: -O2CMe, -O2CEt (ii) Cl-, py, HIm, dbm-, Me2dbm-, Et2dbm-

21. Antiferromagnetic exchange in a dimer of Mn4 SMMs No H = 0 tunneling (-9/2,7/2) (-9/2,-9/2) (9/2,-5/2) Energy (K) (9/2,-7/2) (-9/2,9/2) -10 (1) (2)(3) (9/2,9/2) (4)(5) µ H (T) 0 z -20 To zeroth order, the exchange generates a bias field Jm'/gmB which each spin experiences due to the other spin within the dimer -30 -40 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 [Mn4O3Cl4(O2CEt)3(py)3] m1 m2 D1 = -0.72 K J = 0.1 K Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409

22. 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). • 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 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. Heisenberg: JŜ1.Ŝ2

23. S1 = S2 = 9/2; multiplicity of levels = (2S1 + 1) (2S2 + 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 Exchange bias [Mn4O3Cl4(O2CEt)3(py)3] Monomer -1/2 to 1/2 Monomer

25. 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

26. Full exchange calculation for the dimer 1st order correction lifts degeneracies between states where m1 and m2 differ by ±1 • 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...

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

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.

30. Clear evidence for coherent transitions involving both molecules 9 MHz  tf > 1 ns f = 145 GHz Experiment Simulation S. Hill et al., Science302, 1015 (2003)

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

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

33. 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" Light "on" + - Control of exchange

34. Decoherence – role of nuclear spins The role of dipolar and hyperfine fields was first demonstrated via studies of isotopically substituted versions of Fe8. [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. Chicken Fat Mn4 Mn4 º ,

36. Chicken Fat Mn4 Mn4 º ,

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

38. 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? What do we currently understand? • Quantum tunneling is extremely sensitive to SMM symmetry • Transverse anisotropies provide tunneling matrix elements • Magnetization dynamics controlled by nuclear and electron spin-spin interactions • Fluctuations drive SMMs into and out of resonance • Such interactions represent unwanted source of decoherence  Pulsed/time-domain EPR

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