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Single Molecule Magnets: Exploring Quantum Magnetization Dynamics at the Nanoscale

Single Molecule Magnets: Exploring Quantum Magnetization Dynamics at the Nanoscale. Quantum versus classical magnetic clusters Single molecule magnets (SMMs) In particular, Mn 12 and Fe 8 Quantum magnetization dynamics Quantum tunneling, hysteresis, and the role of the environment

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Single Molecule Magnets: Exploring Quantum Magnetization Dynamics at the Nanoscale

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  1. Single Molecule Magnets: Exploring Quantum Magnetization Dynamics at the Nanoscale • Quantum versus classical magnetic clusters • Single molecule magnets (SMMs) • In particular, Mn12 and Fe8 • Quantum magnetization dynamics • Quantum tunneling, hysteresis, and the role of the environment • High frequency Electron Paramagnetic Resonance • Solution to the tunneling mechanism in Mn12-acetate • Quantum coherence in a supramolecular dimer of SMMs

  2. 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

  3. down Up • Energy barrier prevents switching - hence bistability.... • ....unless DE < kBT superparamagnetism [t = toexp(DE/kBT)] Magnetic anisotropy  bistability, hysteresis The case of a classical, single domain, ferromagnetic particle: Stoner-Wohlfarth DE = K× Volume  anisotropy × # of spins K = energy density associated with the anisotropy Energy density DE • Anisotropy due to crystal field - in this case, uniaxial.

  4. down Up • Energy barrier prevents switching - hence bistability.... • ....unless DE < kBT superparamagnetism [t = toexp(DE/kBT)] Magnetic anisotropy  bistability, hysteresis The case of a classical, single domain, ferromagnetic particle: Stoner-Wohlfarth DE = K× Volume  anisotropy × # of spins K = energy density associated with the anisotropy Energy density DE • Anisotropy due to crystal field - in this case, uniaxial. • Switching involves rotation of all spins together. • Classical continuum of states between "up" and "down".

  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. How is this evidenced? Mn30 - the largest single molecule magnet displaying a temperature independent relaxation rate Prokof'ev and Stamp Phys. Rev. Lett. 80, 5794 (1998) W. Wernsdorfer, G. Christou, et al., cond-mat/0306303 The physics behind this short-time relaxation requires some detailed explanation (to follow). • Temperature-independent magnetization relaxation as T  0 K. • Highly axially symmetric systems show the slowest relaxation, e.g. Mn12-ac (G-1 ).

  10. Another well characterized single molecule magnet [Fe8O2(OH)12(tacn)6]Br8.9H20 Fe3+ S=5/2 S=10 Wieghardt, 1984

  11. How do these two S = 10 systems differ? z-axis is out of the screen Mn12-ac, S = 10 Fe8, S = 10 Two-fold axis Rhombic Four-fold axis Tetragonal

  12. Spin projection - ms Spin projection - ms Thermally assisted quantum tunneling Below TB 3 K, t  (H = 0) • This is why tunneling in high spin (classical) systems is not observed. Example:  for Fe8, HT mixes states for which Dms= ±2 in first-order, Dms= ±4 in second-order, and so on. Thus, the ms= ±10 states interact only in 10th order • Many potential sources of transverse anisotropy for Mn12-ac: • Internal dipolar, exchange and hyperfine fields; • Higher order crystal field interactions; • Disorder, lattice defects, etc.. • However, after 10 years, the mechanism remains poorly understood. Pure quantum tunneling = 0.1 mK Mn12-ac, S = 10 Fe8, S = 10 D  0.6 K D  0.2 K D/E  5 G0 10-3 Hz 115 GHz or 5.5 kB 300 GHz or 14.4 kB • To first-order, HT mixes states with Dms= ±k, where k is the power to which Sx or Sy appear in HT.

  13. 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

  14. 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

  15. Tunneling "on" Tunneling "off" Hysteresis and magnetization steps Mn12-ac Low temperature H=0 step is an artifact This loop represents an ensemble average of the response of many molecules Friedman et al., PRL (1996) Thomas et al., Nature (1996)

  16. Both even and oddDmssteps are observed +10 -9 +10 -8 +10 -7 +10 -6 +10 -5 +10 -4 +10 -3 Periodicity of the magnetization steps • Therefore,HT must contain odd powers of Ŝx and Ŝy. • This, in turn, suggest that transverse fields play a crucial role in the tunneling. [Chudnovsky and Garanin, PRL 87, 187203 (2001)] -500 Tunneling steps are periodic in Hz

  17. (1) (3) (2) (4) (5) M<Msat x Msat 3) Some molecules will relax, thus burning a hole in the dipolar field distribution; this relaxation results in a change in the magnetization of the sample. 1) Prepare the sample in a fully magnetized state. 5) Dynamic nuclear fluctuations and electron/nuclear spin cross-relaxation broaden the magnetic energy levels (even at mK temperatures). 2) Put the system in resonance so that the magnetization can begin to relax. Note: due to internal dipolar fields, not all molecules will be in resonance - depends on sample shape. 4) The change in magnetization drives those molecules that were initially in resonance, out of resonance. Meanwhile, other molecules are brought into resonance, and the hole becomes broader. So, what about internal fields? • These are too weak to explain tunneling at odd resonances in Mn12. • Nevertheless, they play a crucial role in the tunneling mechanism. field dipolar exchange hyperfine Mn12 56Fe8 20 mT  0 12 mT 50 mT 10 mT 1.2 mT • Relaxation is a complex many-body problem. This combination of "dipolar shuffling" and the coupling to the nuclear bath governs the time-evolution of the magnetization, leading to the short time t-1/2 relaxation. [Prokof'ev and Stamp, Phys. Rev. Lett. 80, 5794 (1998)]

  18. Here, we see the importance of the environment 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)]

  19. For Fe8: rhombic term, E(Ŝx2 - Ŝy2), is the dominant source of transverse anisotropy. This has been verified via measurements of tunnel splittings D, using an elegant Landau-Zener method. [Wernsdorfer, Sessoli, Science 284, 133 (1999)] For Mn12: dominant source of transverse anisotropy not known; 4th order term, B44(Ŝ+4 + Ŝ-4), believed to play a role. Landau-Zener studies fail to observe clear tunnel splittings; instead, broad distributions are found. [del Barco et al., Europhys. Lett. 60, 768 (2002)] For both: odd tunneling resonances suggest additional source of transverse anisotropy, possibly related to disorder. [Chudnovsky and Garanin, PRL 87, 187203 (2001)] • Nuclear and dipolar couplings control magnetization dynamics. • These couplings also represent unwanted source of decoherence. What have we learned thus far? HTprovides the tunneling mechanism (tunnel matrix elements)

  20. Single-crystal, high-field/frequency EPR Reminder: field//z z, S4-axis Hz ms represents spin- projection along the molecular 4-fold axis • Magnetic dipole transitions (Dms = ±1) - note frequency scale! • EPR measures level spacings directly, unlike Landau-Zener

  21. s z, S4-axis Hxy Single-crystal, high-field/frequency EPR Rotate field in xy-plane and look for symmetry effects In high-field limit (gmBB > DS), ms represents spin- projection along the applied field-axis

  22. Subsequent outcomes: Determination of significant D-strain effect in Mn12 and Fe8. [Polyhedron 20, 1441 (2001); PRB 65, 014426 (2002)] Measurement of dipolar couplings in Mn12 and Fe8, and first evidence for significant intermolecular interactions in Fe8. [PRB 65, 224410 (2002); PRB 66, 144409 (2002)] Location of low-lying S = 9 state in Fe8. [cond-mat/030915; PRB in-press (2003)] The first single-crystal study PRL 80, 2453 (1998) • The advantages are two-fold: • One can perform angle dependent studies to look for symmetry of HT. • One can also carry out detailed line-shape analyses; this is harder to do for powder averages. Sample: 1 × 0.2 × 0.2 mm3

  23. q Susumu Takahashi (2003) two-axis rotation capability 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)

  24. Four-fold line shifts due to a quartic transverse interaction in HT • Previously inferred from neutron studies • Mirebeau et al., PRL 83, 628 (1999) • B44 is the only free parameter in our fit • PRL 90, 217204 (2003) four-fold line shape/width modulation due to a quadratic transverse interaction caused by solvent disorder PRL 90, 217204 (2003) f Determination of transverse crystal-field interactions in Mn12-Ac Hard-plane (xy-plane) rotations f = 50 GHz 0 30 60 Angle (degrees) 90 Easy 120 150

  25. Organic surrounding (not directly bound to core): 4 water molecules 2 acetic acid molecules Solvent disorder (rogueness)

  26. +E(Sx2 - Sy2) -E(Sx2 - Sy2) +E -E Equal population of molecules with opposite signs of E A. Cornia et al., Phys. Rev. Lett. 89, 257201 (2002) Disorder lowers the symmetry of the molecules

  27. b9 does not appear until 2.5 degrees of rotation a10 vanishes in the first degree of rotation Rotation away from the hard plane q

  28. a10 2.15o spread Hard plane b9 1.25o spread Rotation away from the hard plane 0.18o increments Implies about 1.5o distribution in the orientations of the easy axes of the Mn12 molecules This is the solution to the odd tunneling resonance problem

  29. The molecular approach is the key • Immense control over the magnetic unit and its coupling to the environment Returning to issue of decoherence and the environment • Mn12 (Fe8?) probably not ideal quantum system for future work • 1st single molecule magnets - novel, yet contrasting behaviors • Physicists have focused almost exclusively on these two systems • Timescale assoc. with nuclear and dipolar fluctuations is slow • Strong coupling to slow spin dynamics  quantum decoherence • Move tunneling into a "coherence window" - different frequency range • [Stamp and Tupitsyn, cond-mat/0302015] • Larger tunnel splittings via transverse applied field, or smaller spin S • Other ways to beat decoherence • Isotopically label with I = 0 nuclei • Focus on antiferromagnetic S = 0 systems (immune to dipolar effects) • Isolate molecules, either by dilution, or with "chicken fat"

  30. Chicken Fat S = 4 Ni4 system with strong quartic (Ŝ+4 + Ŝ-4) anisotropy (Ni also predominantly I = 0) Ni4 Ni4 º , E-C. Yang, JACS (submitted, 2003)

  31. Site-selective reactivity o N c N N N Mn Mn Mn Mn Mn Mn O O O O O O Mn R Mn O O N Br O Mn Mn Mn Mn Mn Mn C O O O O O O O O Mn Mn Mn Mn Mn O O O H H Mn H C Cl Mn Mn O Mn O O F Mn O Mn Mn Mn Mn Mn O O Mn O O O O Mn Mn Mn4S = 9/2 An untapped gold mine

  32. No H = 0 tunneling To lowest order, the exchange generates a bias which each spin experiences due to the other spin within the dimer Exchange coupling in a dimer of S = 9/2 Mn4 clusters Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409

  33. Including the full exchange into the calculation

  34. Clear evidence for coherent transitions involving both molecules Experiment Simulation (submitted, 2003)

  35. UF Chemistry George Christou Nuria Aliaga-Alcalde Monica Soler Nicole Chakov Sumit Bhaduri UCSD Chemistry David Hendrickson En-Che Yang Evan Rumberger Many collaborators/students to acknowledge... ...illustrates the interdisciplinary nature of this work UF Physics Rachel Edwards Alexey Kovalev Susumu Takahashi Sabina Kahn Jon Lawrence Andrew Browne Shaela Jones Neil Bushong Sara Maccagnano FSU Chemistry Naresh Dalal Micah North David Zipse Randy Achey NYU Physics Andy Kent Enrique del Barco Also: Kyungwha Park (NRL) Wolfgang Wernsdorfer (Grenoble) Mark Novotny (MS State U) Per Arne Rikvold (CSIT - FSU)

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