Coherence Studies of Mn4 Single-Molecule Magnets

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

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 (C3v symmetry) from which the dimers are made.

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.

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

S. Takahashi Rev. Sci. Inst. Submitted (2004) 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)

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

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

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

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

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

Antiferromagnetic exchange in a dimer of Mn4 SMMs EPR     To zeroth order, the exchange generates a bias field BJ = Jm'/gmB which each spin experiences due to the other spin within the dimer • Bias should shift the single spin (monomer) EPR transitions. Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409 [Mn4O3Cl4(O2CEt)3(py)3] Dimer Zeeman diagram m1 m2 D = -0.75(1) K B04 = 5 × 10-5 K J 0.12(1) K

Effect of exchange bias on the quantum tunneling 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 Just like the EPR, the exchange bias shifts the positions of the magnetic quantum tunneling resonances -30 -40 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 Wolfgang Wernsdorfer, George Christou, et al., Nature, 2002, 406-409 [Mn4O3Cl4(O2CEt)3(py)3] m1 m2 D1 = -0.72 K J 0.1 K

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 (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. Tuesday plenary lecture by Seigo Tarucha Heisenberg: JŜ1.Ŝ2

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

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

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

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...

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.

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

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

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

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.

Clear evidence for coherent transitions involving both molecules 9 GHz  tf > 1 ns f = 145 GHz Experiment Simulation Jz = Jxy = 0.12(1) K S. Hill et al., Science302, 1015 (2003)

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., Science302, 1015 (2003)

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

Confirmed by hole-digging (minor loop) experiments R. Tiron et al., Phys. Rev. Lett. 91, 227203 (2003)

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 Science302, 1015 (2003) • Future work: • Control of exchange coupling • Time-domain EPR  measure tf

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, 1015-1018 , 7 November 2003

Confirmation of the exchange bias for NA3 Notice the frequency range 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

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.

Coherence Studies of Mn4 Single-Molecule Magnets