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Coherent Spin Dynamics in Molecular Cr8Zn Wheels Alberto Ghirri, Alessandro Chiesa, Stefano Carretta, Filippo Troiani, Johan Van Tol, Stephen Hill, Inigo J. Vitorica-Yrezabal, Grigore A. Timco, Richard E. P. Winpenny, and Marco Affronte J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b02527 • Publication Date (Web): 03 Dec 2015 Downloaded from http://pubs.acs.org on December 5, 2015

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The Journal of Physical Chemistry Letters

Coherent Spin Dynamics in Molecular Cr8Zn Wheels Alberto Ghirri, †, * Alessandro Chiesa,§ Stefano Carretta, § Filippo Troiani,† Johan van Tol,⊥ Stephen Hill,⊥, ‡ Inigo Vitorica-Yrezabal,∥ Grigore A. Timco,∥ Richard E. P. Winpenny∥ and Marco Affronte. †, ‡ †

Istituto Nanoscienze-CNR, via G. Campi 213A, 41125 Modena, Italy

§

Dipartimento di Fisica e Scienze della Terra, Università di Parma, Parco Area delle Scienze 7/a,

43123 Parma, Italy ⊥National

‡

High Magnetic Field Laboratory, Tallahassee, Florida 32310, United States

Department of Physics, Florida State University, Tallahassee, Florida 32306, United States

∥School

of Chemistry, University of Manchester, Oxford Road, Manchester M13 9PL, United

Kingdom ‡

Università di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e

Matematiche, via Campi 213A, 41125 Modena, Italy Corresponding Author

ACS Paragon Plus Environment

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*Phone: +39 059 2058123. Email: [email protected]

ABSTRACT: Controlling and understanding transitions between molecular spin states allows selection of the most suitable ones for qubit encoding. Here we present a detailed investigation of single crystals of a polynuclear Cr8Zn molecular wheel using 241 GHz Electron Paramagnetic Resonance (EPR) spectroscopy in high magnetic field. Continuous wave spectra are well reproduced by spin Hamiltonian calculations, which evidence that transitions in correspondence to a well-defined anti-crossing involve mixed states with different total spin. We studied, by means of spin echo experiments, the temperature dependence of the dephasing time (T2) down to 1.35 K. These results are reproduced by considering both hyperfine and intermolecular dipolar interactions, evidencing that the dipolar contribution is completely suppressed at the lowest temperature. Overall, these results shed light on the effects of the decoherence mechanisms, whose understanding is crucial to exploit chemically engineered molecular states as a resource for quantum information processing.

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Molecular clusters show discrete patterns of energy levels and quantum features of spin states that can be exploited for the implementation of solid-state quantum bits.1, 2 In these systems the magnetic interactions can be introduced in a controlled way at the synthetic level, giving rise to thermal entanglement at low temperature, and enabling the implementation of two-qubit quantum gates.3-11 With respect to isolated spins in crystals,12, 13 the functionalization of external ligands offers more flexibility and precision in linking molecular spins to each other or for controlling their deposition on surfaces.14 A critical point is the control of decoherence. Spin dephasing, which is induced by dipolar interactions with neighboring electronic and nuclear spins, is the dominant decoherence mechanism at low temperature, where the spin-lattice relaxation rate becomes vanishingly small.2, 15, 16, 17 Careful optimization of the molecular species allowed the improvement of the dephasing time (T2) in mononuclear18-21 and polynuclear complexes,22-24 by choosing suitable organic ligands, minimizing nuclear magnetic moments and motional degrees of freedom. Intermolecular dipolar couplings are usually suppressed by diluting molecular spins in nonmagnetic matrices. A characteristic of molecular spin clusters is the presence of anisotropic terms in the Hamiltonian that may mix pure spin states and thus make accessible otherwise forbidden EPR transitions. The contribution of such terms to decoherence is not yet studied, because this information is not accessible for randomly oriented clusters in frozen solutions. In some cases the orientation of the spin clusters can be preserved after the dilution in isostructural diamagnetic matrices.24, 25 Another approach is provided by high frequency – high field pulsed Electron Paramagnetic Resonance (EPR) spectroscopy, which allows the direct investigation of


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non-diluted single crystals. At low temperature, the magnetic field can polarize the molecular spins and suppress the dephasing induced by the spin-bath fluctuations.15

Figure 1. (a) Structure of the Cr8Zn molecular wheel. Color scheme: Cr, green; Zn, cyan; F, yellow; O, red; C, black; N, blue. H atoms omitted for clarity. (b) Cw-EPR spectra measured at 241 GHz and different angles. Black and red lines respectively display experimental and simulated spectra.

Here we report the results of continuous wave (cw) and pulsed EPR experiments on a single crystal of [i(C3H7)2NH2][Cr8ZnF9(O2CCtBu)18], hereafter Cr8Zn. Cr8Zn belongs to a family of heterometallic wheels26 and is characterized by a cyclic structure with eight Cr3+ ions (sCr = 3/2) and one non-magnetic Zn2+ (Figure 1a). The pattern of the energy levels has been previously investigated by Inelastic Neutron Scattering (INS), thermodynamic and NMR measurements.27-30 In zero field, the ground state is a singlet (total spin S = 0), with first (S = 1) and second (S = 2) excited multiplets respectively separated by energy gaps of 3.5 K and 12 K. Due to the Zeeman effect, the ground state changes from |S, M> =|0, 0> to |1, -1>, and from |1, -1> to |2, -2>, in applied perpendicular magnetic fields of ~2.5 and ~6.5 T, respectively.30 Thanks to these characteristics of the energy diagram of the Cr8Zn wheel, the polarization can be maximized by


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exploiting the |2, -2> state in high magnetic field and the coherent spin dynamics of mixed states can be successfully explored by means of 241 GHz pulsed EPR spectroscopy.

Continuous wave and pulsed EPR experiments were performed on hexagonal crystals, which contain two molecules per unit cell. For the measurements presented in Figure 1b, the hexagonal face of the crystal was aligned approximately perpendicular or parallel to the applied magnetic field (H). These directions correspond to an effective angle θ = 15° and 75° between the normal to the wheels and the applied field (Supplementary Information). Cw-EPR spectra measured at 2.5 K on a single-crystal of Cr8Zn show three EPR transitions between 8 and 10 T, whose resonance fields and intensities differ for θ = 15˚and 75˚ (Figure 2). A further (weak) excitation is visible at 4.3 T.

The results of INS and thermodynamic measurements27-30 have been modeled with the spin Hamiltonian:

7

8

8

i=1

i=1

i=1

8

8

8

H = J ∑ si ⋅ si+1 + d ∑ szi2 + e∑( sxi2 − syi2 ) + ∑ ∑ si ⋅ Dij ⋅ s j + gµ B H ⋅ ∑ si i=1 j=i+1

(1)

i=1

where the first term describes the isotropic exchange interactions between nearest neighboring Cr3+ ions, the second and third terms account for single-ion zero-field splitting,31 and the last term is a Zeeman interaction with the external magnetic field. Also included are intramolecular magnetic dipole-dipole interactions (fourth term in Eq. 1), with couplings Dij calculated in the point dipole approximation (no free parameters). A good fit to the experimental data is found with J = 9.92 cm-1, d = -0.23 cm-1, e = 0.02 cm-1 and g = 1.98 (Figure 2 and Supplementary


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Information).32 These parameters are close to those reported for other Cr8Zn and Cr8Cd derivatives.27-30, 33

Figure 2. (a) Calculated energy levels as a function of H (θ = 15˚). The vertical black lines are the observed EPR transitions. The horizontal dashed line indicates the frequency of the microwave excitation (241 GHz). (b) Population of the lowest lying energy levels as a function of H, calculated for T = 1.5 K (solid lines) and 2.5 K (dashed lines).

Figure 2a shows the pattern of the lowest-lying energy levels calculated for θ = 15°. As in Cr8Cd crystals,34 there is a level anti-crossing at about 8.5 T, which is due to the mixing of the total spin eigenstates |S, M'> =|0, 0> and |2, -1>, being M' the component of the total spin along H. This feature emerges as a combined effect of zero-field splitting (Eq. 1) and a finite transverse field component (θ ≠ 0°). The comparison between experimental and calculated spectra allows the


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labeling of observed EPR transitions (Figures 2 and 3). Resonance A is a transition with ∆M = 2 between |1, -1> and |1, 1>, which is possible for θ ≠ 0°. At higher field, two resonances, labeled B and C, are observed near the anti-crossing at 8.5 T. The transitions calculated from Eq. 1 are (apart from small corrections) from the ground state |2, -2> to the mixed state α|0, 0>+β|2, -1>. At different H, the relative composition of the mixed state varies and the simulations indicate 0.77|0, 0>+0.64|2, -1> for B, and 0.37|0, 0>+0.93|2, -1> for C. The compositions in terms of different total-spin states directly reflect the significantly different observed intensities. Indeed, the larger intensity of the B transition results from the larger component of the magnetic |2,-1> state in the excited level. As evidenced by the cw-EPR study at different temperature (Supplementary Information), the fourth resonance (D) belongs to excited multiplets, and therefore is weak at low temperature.

Efficient suppression of electron spin bath fluctuations without dilution requires the polarization of the Cr8Zn crystal.15 In Figure 2b we report the Boltzmann population Pj = e

− E j / k BT

/Z

at temperature T of the lowest-lying energy levels as a function of H. In the

range 8 - 12 T the population of |E0> =|2, -2> is maximized. In correspondence to the resonance field of B and C, it reaches respectively 0.84 and 0.94 at 1.5 K. At 2.5 K, these values decrease to 0.69 and 0.73 respectively, showing a steep reduction within a temperature range of 1 K. The nuclear spin bath, conversely, is not polarized and can be assumed to be in an infinitetemperature state.


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Figure 3. (a) Echo-detected EPR spectrum measured at 1.35 K and 2.17 K (θ=15˚). (b) Decay of the integrated echo area measured at 8.436 T (transition B, red squares) and 9.116 T (transition C, black circles). Solid lines show the fit to a single-exponential function. (c) Temperature dependence of the decay rate 1/T2 extracted for transitions B and C.

We thus performed spin-echo experiments with the two-pulse Hahn echo sequence (π/2-τ-π-τecho) to investigate the coherent dynamics of transitions B and C. The transitions in the echodetected field-swept EPR spectrum (Figure 3a) match those of the cw-EPR spectrum in Figure 1b, and are labeled accordingly. At 1.35 K we observed a significant enhancement of the echo signal with respect to the 2.17 K data. The transition C is still the strongest, while the intensity of B is about one third (Figure 3a). The decay of the echo intensity, measured by varying the delay 2τ, can be fit with a monoexponential function I=I0 e-2τ/ T2 (Figure 3b), from which we derived T2(B) = 657 ns and T2(C) = 475 ns from least-square fitting. The two transitions thus have similar decay rates with a ratio T2(B)/T2(C) ≃ 1.4 at the lowest temperature, which is reasonable for the not too different


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composition of the involved states. Spin echo measurements were repeated for 1.35 < T < 2.17 K, and the rate 1/T2 extracted from the mono-exponential fits was plotted as a function of T (Figure 3c). We note that 1/T2 has a stronger temperature dependence for C than for B. In the following we discuss this behavior on the basis of a model for decoherence in molecular spin clusters. In the case of non-diluted single-crystals of Fe8 in high magnetic field, the dominant relaxation mechanism has been ascribed to collective excitations mediated by intermolecular dipolar interactions and by phonons, while the contribution from nuclei was estimated to be low. For 6.5 ≲ H ≲ 13.6 T, Cr8Zn has a S = 2 ground state. Thus the intermolecular dipolar interaction is significantly reduced in comparison to Fe8 (S = 10). However, its effects are still present in a non-diluted crystal, as suggested by the strong temperature dependence of the echo decay in the range 1.35 - 2.17 K (Figure 3c). The phonon contribution to the spin-echo decay was investigated within the theoretical framework of Ref. 35 and based on the irreversible evolution of the density matrix produced by spin-phonon interactions. Since we deal with times much longer than those characterizing coherent dynamics, we can apply the secular approximation and decouple the evolution of the diagonal terms of the density matrix from the off-diagonal ones. Within this picture, we consider as the main source of relaxation the modulation of local crystal fields by phonons, which can be modeled in terms of the rate matrix W, whose matrix elements Wst represent the probability per unit time of a transition between the eigenstates |t> and |s> of the molecular Hamiltonian. By assuming a spherically symmetric magnetoelastic coupling of each ion, we obtain:

Wst = γ 2π 2 ∆2st n(∆ st )

∑

i , j =1, N q1 ,q2 = x , y , z

s Oq1 ,q2 (si ) t t Oq1 ,q2 (s j ) s ,


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where

(

)

n ( x ) = e hx / k BT − 1

−1

,

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(

)

∆ st = (Es − Et ) / h and O q ,q (s i ) = s q i s q ,i + s q ,i s q ,i / 2 are 1

2

1,

2

2

1

quadrupolar operators.36 γ is proportional to the spin-phonon coupling strength, and in AF rings is usually determined by fitting NMR data. By using the same value of γ obtained for Cr7Ni,37 we find that phonons give only a marginal contribution to level lifetimes at low temperature (2.2 K or lower). Each Cr8Zn molecule experiences the combined effects of the external magnetic field and a random time-dependent field resulting from the surrounding electron and nuclear spins. The decay of the echo intensity is due primarily to this time-dependent field that, in contrast to static effects, cannot be refocused by spin-echo techniques. As the measurements were carried out in high magnetic fields, single spin-flip processes are suppressed by the large Zeeman energy gaps, and spin bath fluctuations are dominated by energy-conserving flip-flops transitions between single-molecule eigenstates |j> and |k>. These occur with a probability proportional to the product of their populations 1 = C ∑ M j ,k Pj Pk + Γ T2 j >k .

Pj = e

− E j / k BT

/Z

at temperature T:15

(2)

To account for inter-molecular interactions while keeping the model as simple as possible, we follow Ref. 15 and represent each molecule as a magnetic dipole of spin S. Within this approximation, the matrix elements Mj,k are given by:

M j,k = jk S1x ' Sx2' + S1y' Sy2' kj

2

where x’ and y’ are

orthogonal to H and jk = j ⊗ k is the tensor product of single-molecule eigenstates |j> and |k>, belonging to a pair of dipolar-interacting molecules. Other terms bilinear in the spin operators are not included, because they do not induce energy-conserving flip-flop transitions. Γ is a temperature-independent term, which originates from the interaction with the magnetic nuclei.


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The temperature dependence of 1/T2 predicted by this model is shown in Figure 3c. Here C and Γ are transition dependent fitting parameters. To reduce the number of free parameters we assumed the same C/Γ ratio for both transitionsGiven the simplicity of the model, the agreement is remarkably good and reproduces the correct temperature dependence. At low temperature, decoherence is mainly due to the hyperfine interaction [Γ term in Eq. (2)] and we obtain ΓB = 1.30 MHz, ΓC = 1.97 MHz, while the dipolar contribution almost vanishes at 1.35 K. The measured T2 is comparable to the one measured for diluted and non-perdeuterated Cr7Ni and Cr7Mn rings,22 confirming that, at 1.35 K, the spin bath fluctuation is effectively suppressed. Conversely, the decay of T2 upon increasing the temperature is caused by inter-molecular electron dipole-dipole interactions. In conclusion, we have investigated two different 241 GHz EPR transitions in a single-crystal of Cr8Zn. The 1/T2 dephasing rate, measured by spin echo experiments for two transitions involving superposition of different total-spin states shows similar values for two examined EPR transitions. On the basis of a theoretical model that includes intermolecular dipolar and hyperfine interactions, we correctly reproduce its temperature dependence. By applying a high magnetic field we show that the dipolar interactions are completely suppressed at the lowest temperature, even in a non-diluted crystal. These results clarify the origin of decoherence mechanisms in molecular clusters and show how these can be coherently manipulated by high frequency-high field EPR.

ASSOCIATED CONTENT


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Supporting Information. Experimental methods, synthesis, crystallographic data, cw-EPR temperature study, saturation recovery measurements. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Notes The authors declare no competing financial interests. ACKNOWLEDGMENT This work was funded by FIRB project RBFR12RPD1 of the Italian Ministry of Research, EPSRC(UK) and by the US AFOSR/AOARD program, contract 134031 FA2386-13-1-4029. Work at the National High Magnetic Field Laboratory is supported by the NSF (grant numbers DMR-1157490 and DMR-1309463) and the State of Florida.

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(29) Bianchi, A.; Carretta, S.; Santini, P.; Amoretti, G.; Guidi, T.; Qiu, Y.; Copley, J. R. D.; Timco, G.; Muryn, C.; Winpenny, R. E. P. Rotational Bands in Open Antiferromagnetic Rings: A Neutron Spectroscopy Study of Cr8Zn. Phys. Rev. B 2009, 79, 144422. (30) Adelnia, F.; Chiesa, A.; Bordignon, S.; Carretta, S.; Ghirri, A.; Candini, A.; Cervetti, C.; Evangelisti, M.; Affronte, M.; Sheikin, I.; et al. A. Low Temperature Magnetic Properties and Spin Dynamics in Single Crystals of Cr8Zn Antiferromagnetic Molecular Rings. submitted. (31) The minimal model assumed for the non-axial single-ion anisotropy is a simple effective way to account for the presence of non-axial terms arising from the non-regular ring structure and/or from the non-collinearity of local easy axes. We have checked that a more complex form of the local zero-field splitting tensors (with easy axes tilted with respect to the ring axis and rotated from site to site) yields very similar EPR spectra. (32) To account for the relative intensity of the measured peaks we have assumed a crystal misalignment of 3° with respect to the directions perpendicular (15°) and parallel (75°) to the hexagonal face. (33) The zero field splitting parameters are the same reported in Ref. 29 for the deuterated Cr8Zn variant, while a slight reduction of J (less than 7%) has been applied to fit EPR spectra. (34) Guidi, T.; Gillon, B.; Mason, S. A.; Garlatti, E.; Carretta, S.; Santini, P.; Stunault, A.; Caciuffo, R.; van Slageren, J.; Klemke, B.; et al. Direct Observation of Finite Size Effects in Chains of Antiferromagnetically Coupled Spins. Nat. Commun. 2015, 6, 7061. (35) Bianchi, A.; Carretta, S.; Santini, P.; Amoretti, G.; Lago, J.; Corti, M.; Lascialfari, A.; Arosio, P.; Timco, G.; Winpenny, R. E. P. Phonon-Induced Relaxation in the Cr7Ni Magnetic Molecule Probed by NMR. Phys. Rev. B 2010, 82, 134403. (36) Santini, P.; Carretta, S.; Liviotti, E.; Amoretti, G.; Carretta, P.; Filibian, M.; Lascialfari, A.; Micotti, E. NMR as a Probe of the Relaxation of the Magnetization in Magnetic Molecules. Phys. Rev. Lett. 2005, 94, 077203.


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(37) Carretta, S.; Santini, P.; Amoretti, G.; Affronte, M.; Candini A.; Ghirri, A.; Tidmarsh, I. S.; Laye, R. H.; Shaw, R.; McInnes, E. J. L. High-Temperature Slow Relaxation of the Magnetization in Ni10 Magnetic Molecules. Phys. Rev. Lett. 2006, 97, 207201.


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Coherent Spin Dynamics in Molecular Cr8Zn Wheels - The Journal of

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