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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 PART I Introduction to single-molecule magnets Emphasis on their quantum dynamics MESOSCOPIC MAGNETISM Classical macroscale permanent magnets nanoparticles micron particles size Quantum nanoscale clusters molecular superparamagnetism clusters 100 nm 23 S = 10 1010 10 8 10 6 10 nm 10 5 10 4 Individual spins 1 nm 10 3 10 2 10 1 Mn12-ac multi - domain single - domain nucleation, propagation and annihilation of domain walls uniform rotation 1 Ferritin Single molecule quantum tunneling, quantum interference 1 1 S S S M/M M/M M/M Fe8 0 0 0.7K 0 1K -1 -40 -1 -20 0 20 m 0H(mT) 40 0.1K -1 -100 0 m 0 H(mT) 100 -1 W. Wernsdorfer, Adv. Chem. Phys. 118, 99-190 (2001); also arXiv/cond-mat/0101104. 0 m 0H(T) 1 ~1.2 nm A. J. Tasiopoulos et al., Angew. Chem., in-press. The first single molecule magnet: Mn12-acetate Lis, 1980 Mn(III) S=2 Mn(IV) S = 3/2 Oxygen Carbon [Mn12O12(CH3COO)16(H2O)4]·2CH3COOH·4H20 R. Sessoli et al. JACS 115, 1804 (1993) •Ferrimagnetically coupled (Jintra 100 Well defined giant spin (S = magnetic 10) at low ions temperatures (T <K)35 K) •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 Quantum effects at the nanoscale (S = 10) Energy E ms Hˆ DSˆz2 ( D 0) E-4 E-5 E4 E5 E-6 E6 E(ms ) - D m E-7 E7 •Small barrier - DS2 E-8 "up" Simplest case: axial (cylindrical) crystal field Spin projection - ms DE DS2 10-100 K E-9 E-10 2 s •Superparamagnet at ordinary temperatures E9 "down" |D | 0.1 - 1 K E8 E10 21 discrete ms levels Thermal activation Eigenvalues given by: for a typical single molecule magnet Quantum effects at the nanoscale (S = 10) Energy E ms Hˆ DSˆz2 HˆT HT interactions which do not commute with Ŝz E-4 E-5 E4 E5 E-6 E6 • ms not good quantum # E-7 E-8 "up" Break axial symmetry: Spin projection - ms E-9 E-10 Thermally assisted quantum tunneling DEeff < DE E7 E8 •Mixing of ms states resonant tunneling (of ms) through barrier •Lower effective barrier E9 "down" E10 Quantum effects at the nanoscale (S = 10) Energy E ms E-4 E-5 E4 E5 E-6 E6 E-7 E7 E-8 "up" Strong distortion of the axial crystal field: Spin projection - ms E-9 "up" 1 2 { "down" ± } Do Tunnel splitting •Ground state degeneracy lifted by transverse interaction: splitting (HT)n E8 •Ground states a mixture of pure "up" and pure "down". E9 "down" Pure quantum tunneling E-10 •Temperature-independent quantum relaxation as T0 E10 Application of a magnetic field Spin projection - ms "up" "down" Hˆ DSˆz2 HˆT g m B B Sˆ B Sˆ Bx Sˆx By Sˆ y Bz Sˆz Several important points to note: •Applied field represents another source of transverse anisotropy X System off resonance X •Zeeman interaction contains odd powers of Ŝx and Ŝy For now, consider only B//z : (also neglect transverse interactions) 2 s s B s E(m ) - D m g m Bm •Magnetic quantum tunneling is suppressed •Metastable magnetization is blocked ("down" spins) Application of a magnetic field Spin projection - ms "up" "down" Hˆ DSˆz2 HˆT g m B B Sˆ B Sˆ Bx Sˆx By Sˆ y Bz Sˆz Several important points to note: Increasing field System on resonance •Applied field represents another source of transverse anisotropy. •Zeeman interaction contains odd powers of Ŝx and Ŝy. For now, consider only B//z : (also neglect transverse interactions) 2 s s B s E(m ) - D m g m Bm •Resonant magnetic quantum tunneling resumes •Metastable magnetization can relax from "down" to "up" Hysteresis and magnetization steps Mn12-ac Tunneling "on" Low temperature H=0 Tunneling "off" •Friedman, Sarachik, step is an artifact This loop represents an ensemble average of the response of many molecules Tejada, Ziolo, PRL (1996) •Thomas, Lionti, Ballou, Gatteschi, Sessoli, Barbara, Nature (1996) PART II Quantum Coherence in Exchange-Coupled Dimers of Single-Molecule Magnets Mn4 single molecule magnets (cubane family) C3v symmetry [Mn4O3Cl4(O2CEt)3(py)3] MnIII: 3 × S = 2 MnIV: S = 3/2 Distorted cubane S = 9/ 2 •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. Fairly typical SMM: exhibits resonant MQT B//z “down” Hˆ o DSˆz2 m B B.g.Sˆ Hˆ T Hˆ ' “up” -S (S-1) S B<0 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. Energy level diagram for D < 0 system, B//z Hˆ o DSˆz2 m B B.g.Sˆ 400 S = 9/2 300 B // z-axis of molecule Frequency (GHz) 200 100 0 -100 M = 5 - / to 3 S 2 -/ M = 7 2 M = 9 S - / to 5 S 2 -/ -/ 7 t 2 2 o/2 -200 -300 -400 -500 0 1 2 3 Note frequency range Magnetic field (tesla) 4 5 q Cavity perturbation Cylindrical TE01n (Q ~ 104 - 105) f = 16 300 GHz (now 715 GHz!) Single crystal 1 × 0.2 × 0.2 mm3 T = 0.5 to 300 K, moH up to 45 tesla •We use a Millimeter-wave Vector Network Analyzer (MVNA, ABmm) as a spectrometer Au SS B M. Mola et al., Rev. Sci. Inst. 71, 186 (2000) Energy level diagram for D < 0 system, B//z Hˆ o DSˆz2 m B B.g.Sˆ 400 S = 9/2 300 B // z-axis of molecule Frequency (GHz) 200 100 0 -100 M = 5 - / to 3 S 2 -/ M = 7 2 M = 9 S - / to 5 S 2 -/ -/ 7 t 2 2 o/2 -200 -300 -400 -500 0 1 2 3 Note frequency range Magnetic field (tesla) 4 5 HFEPR for high symmetry (C3v) Mn4 cubane Cavity transmission (arb. units - offset) [Mn4O3(OSiMe3)(O2CEt)3(dbm)3] Field // z-axis of the molecule (±0.5o) 1 3 1 - /2 to - /2 1 - /2 to /2 5 24 K 18 K 14 K 8K 6K 4K 3 - /2 to - /2 7 5 - /2 to - /2 9 f = 138 GHz 7 - /2 to - /2 0.0 1.0 2.0 3.0 Magnetic field (tesla) 4.0 5.0 Fit to easy axis data - yields diagonal crystal field terms ˆ , where Oˆ 0 Sˆ 2 Sˆ 2 Sˆ 4 Hˆ o DSˆz2 B40Oˆ 40 m B g zz BS 4 z z z 100 Frequency (GHz) 150 S = 9/2 -1 D = -0.484 cm 0 -1 B4 = -0.000062 cm gz = 2.00(1) 50 0 0 1 2 3 Magnetic field (tesla) 4 5 Routes to incredible # of SMMs Core ligands (X): Cl-, Br-, F- NO3-, N3-, NCOOH-, MeO-, Me3SiO- Jahn-Teller points towards core ligand Peripheral ligands: (i) carboxylate ligands: -O2CMe, -O2CEt (ii) Cl-, py, HIm, dbm-, Me2dbm-, Et2dbm- Antiferromagnetic exchange in a dimer of Mn4 SMMs [Mn4O3Cl4(O2CEt)3(py)3] -10 a 1 0.14 T/s (-9/2,-9/2) m1 s M/M Energy (K) D1 = -0.72 K J = 0.1 K No H = 0 tunneling 0.5 -20 (9/2,-5/2) 0 -30 -0.5 m2 (-9/2,7/2) (9/2,-7/2) (-9/2,9/2) (1) -40 -1 -1.2 -1.2 (9/2,9/2) -0.8 -0.8 -0.4 -0.4 (2)(3) 0.4 00 0.4 z (T) µµ 00 HH (T) 0.04 K 0.3 K 0.4 K 0.5 K 0.6 K 0.7 K 0.8 K (4)(5) 1.0 K 0.80.8 1.21.2 Hˆ Hˆ 1 Hˆ 2 JSˆ1 Sˆ2 Hˆ 1 Hˆ 2 JSˆz1Sˆz 2 E D m12 m22 g m B B m1 m2 Jm1m2 To zeroth order, the exchange generates a bias field Jm'/gmB which each spin experiences due to the other spin within the dimer Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409 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). Heisenberg: JŜ1.Ŝ2 •Quantum mechanical coupling by the transverse (off•Zeroth order bias term, JŜz1Ŝcaused z2, is diagonal in the mz1,mz2 basis. diagonal) parts of the exchange interaction Jxy(Ŝx1Ŝx2 + Ŝy1Ŝy2). •Therefore, it does not couple the molecules quantum mechanically, •This term causes the involve entanglement, i.e. itrotations. truly mixes mz1,mz2 i.e. tunneling and EPR single-spin basis states, resulting in co-tunneling and EPR transitions involving two-spin rotations. •CAN WE OBSERVE THIS? S1 = S2 = 9/2; multiplicity of levels = (2S1 + 1) (2S2 + 1) = 100 Energy (K) -20 -30 9 9 9 7 9 5 9 9 (- /2,- /2) (- /2,- /2)S,A -40 0.0 (- /2,- /2)S,A (- /2, /2) 0.2 0.4 0.6 0.8 1.0 1.2 Magnetic field (tesla) Look for additional splitting (multiplicity) and symmetry effects (selection rules) in EPR or tunneling experiments. First clues: comparison between monomer and dimer EPR data [Mn4O3Cl4(O2CEt)3(py)3] NA11 frequency dependence NA11 145 GHz easy axis T-dep 9 /) ,2- 2 / o( t ) 9 / 9 /) 2 2 3 / ,1 / ,2 2 ((+ to 9 /) 9 /) 2 2 3 / ,, 1 / 2 (+ (- 2 to 9 /) 2 1 / ,2 (+ 180 Frequency (GHz) 18 K 15 K 10 K 8K 6K 4K 2K 100 80 60 40 0 0 1 2 3 4 5 6 Cavity transmission (arb. units - offset) 150 1 3 1 Monomer 1 - /2 to /2 - /2 to - /2 3 - /2 to - /2 7 5 - /2 to - /2 9 Monomer f = 138 GHz 7 - /2 to - /2 1.0 2.0 3.0 24 K 18 K 14 K 8K 6K 4K 4.0 5.0 2 3 4 Magnetic field (tesla) 5 100 5 Magnetic field (tesla) 1 120 Magnetic field (tesla) 0.0 0 140 20 Frequency (GHz) Cavity transmission (arb. units) 160 1 S = 9/2 -1 D = -0.484 cm 0 -1 B4 = -0.000062 cm gz = 2.00(1) 50 6 0 0 1 2 3 Magnetic field (tesla) 4 5 Full exchange calculation for the dimer } Hˆ D Hˆ S1 Hˆ S 2 JSˆ1.Sˆ2 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. ˆ Hˆ D Hˆ S1 Hˆ S 2 JSˆz1Sˆz 2 12 J Sˆ1 Sˆ2- Sˆ1- Sˆ2 Hˆ 0 D H' •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. m1 , m2 •The zeroth order eigenvalues are given by E0 D D2 m12 m22 D4 m14 m24 m B g z B m1 m2 Jm1m2 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' 12 J Sˆ1 Sˆ2- Sˆ1- Sˆ2 •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... M m1 m2 m1 , m2 M -9 - 92 , - 92 M -8 - 92 , - 72 , - 72 , - 92 M -7 - 92 , - 52 , - 52 , - 92 , - 72 , - 72 M -6 - 92 , - 23 , - 23 , - 92 , - 72 , - 52 , - 52 , - 72 M -5 - 92 , - 12 , - 12 , - 92 , - 72 , - 23 , - 23 , - 72 , - 25 , - 25 1st order correction lifts degeneracies between states where m1 and m2 differ by ±1 1st order corrections to the wavefunctions 1S M -9 - 92 , - 92 2 A M -8 1 - 92 , - 72 - - 72 , - 92 2 3 S M -8 1 - 92 , - 72 - 72 , - 92 2 4 S M -7 1 1 2 1 2 2 5 A M -7 1 - 92 , - 52 - - 52 , - 92 2 6 S M -7 1 1 2 2 - - 7 2 9 2 •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. , - 52 - 52 , - 92 - 2 - 72 , - 72 , - 72 - 92 , - 52 - 52 , - 92 •Ĥ' is symmetric with respect to exchange and, therefore, will only cause 2nd order interactions between states having the same symmetry, within a multiplet. 7A M -6 1 1 2 1 '2 - 9 2 , - 32 - - 32 , - 92 - ' - 72 , - 52 - - 52 , - 72 7 S M -6 1 1 2 1 '2 - 9 2 , - 32 - 32 , - 92 - ' - 72 , - 52 - 52 , - 72 8 A M -6 1 1 2 1 '2 - 7 2 , - 52 - - 52 , - 72 ' - 92 , - 32 - - 32 , - 92 9 S M -6 1 1 2 1 '2 - 7 2 , - 52 - 52 , - 72 ' - 92 , - 32 - 32 , - 92 ' 3J 0.472 D 12 J 21 0.578 0.273 8 1st and 2nd order corrections to the energies - 72 , - 52 -18.5 - 92 , - 32 -22.5 - 72 , - 72 -24.5 - -26.5 9 2 ,- 5 2 Exchange bias Full Exchange (9) - S (8) - A (e) (d) (7) - A & S (6) - S (5) - A (4) - S (b) - 92 , - 72 (c) Matrix element very small (3) - S (2) - A -32.5 (a) - 92 , - 92 -40.5 (1) - S Magnetic-dipole interaction is symmetric Magnetic field dependence 7 -0 Energy (K) 9 (- /2, /2) 9 9 (- /2, /2) 1 9 (+ /2,- /2) 5 5 ( - / 2, - / 2) 3 7 ( - / 2, - / 2) -0 1 9 ( - / 2, - / 2) 5 7 ( - / 2, - / 2) 3 9 ( - / 2, - / 2) -0 7 7 5 9 ( - / 2 , - / 2) -0 ( - / 2 , - / 2) 7 9 9 9 (- /2,- /2) -100 This figure does not show all levels 0 1 2 3 4 Magnetic field (tesla) 5 (- /2,- /2) 6 •The effect of Ĥ' is field-independent, because the field does not change the relative spacings of levels within a given M multiplet. Clear evidence for coherent transitions involving both molecules Hˆ D Hˆ S1 Hˆ S 2 JSˆz1Sˆz 2 12 J Sˆ1 Sˆ2- Sˆ1- Sˆ2 Experiment f = 145 GHz Simulation Cavity transmission (arb. units) NA11 145 GHz easy axis T-dep 24 K 18 K 4.2 4.8 5.4 9 MHz Magnetic field (tesla) tf > 1 ns S. Hill et al., Science 302, 1015 (2003) Variation of J, considering only the exchange bias Variation of J, considering the full exchange term J=0K J = 0.03 K J = 0.06 K J = 0.1 K J = 0.13 K Absorption (arb. units) Absorption (arb. units) J=0 J = 0.03 K J = 0.06 K J = 0.1 K J = 0.13 K T = 6 K, f = 160 GHz T = 6 K, f = 160 GHz 0 1 2 3 4 Magnetic field (tesla) 5 6 0 1 2 3 4 5 Magnetic field (tesla) • 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. 6 Confirmed by hole-digging (minor loop) experiments R. Tiron et al., Phys. Rev. Lett. 91, 227203 (2003) Next session: B25.011 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" Light "on" + - 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 Chicken Fat Mn4 Mn4 OH , OH Chicken Fat Mn4 Mn4 OH , OH The molecular approach is the key • Immense control over the magnetic unit and its coupling to the environment 1. Control over magnitude and symmetry of the anisotropy through the choice of molecule: • 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, • or with a transverse externally applied field, • 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! 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 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 Many collaborators/students involved ...illustrates the interdisciplinary nature of this work UF Physics FSU Chemistry UF Chemistry Rachel Edwards Alexey Kovalev Konstantin Petukhov Susumu Takahashi Jon Lawrence Norman Anderson Tony Wilson Cem Kirman Naresh Dalal George Christou Micah North David Zipse Randy Achey Chris Ramsey Nuria Aliaga-Alcalde Monica Soler Nicole Chakov Sumit Bhaduri Muralee Murugesu Alina Vinslava Dolos Foguet-Albiol Shaela Jones Sara Maccagnano Enrique del Barco NYU Physics Andy Kent Also: Kyungwha Park (NRL) Marco Evangelisti (Leiden) Hans Gudel (Bern) Wolfgang Wernsdorfer (Grenoble) Mark Novotny (MS State U) Per Arne Rikvold (CSIT - FSU) UCSD Chemistry David Hendrickson En-Che Yang Evan Rumberger