1. F. Branzoli ¶, P. Carretta ¶, M. Filibian ¶, S. Klytaksaya ‡ and M. Ruben ‡ ¶ Department of Physics "A.Volta", University of Pavia-CNISM, Via Bassi 6, I-27100, Pavia (Italy) ‡ Institute of Nanotechnology, Forschungszentrum, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen (Germany) Low energy excitations in the neutral [LnPc 2 ] 0 single molecule magnets from μSR and NMR

2. Single Molecule Magnets New class of Single Molecule Magnets (SMM) [1]; magnetic anisotropy Highest magnetic anisotropy ever achieved in a molecular magnet [2]; Slow relaxationquantum tunneling Slow relaxation and quantum tunneling of magnetization; Energy separation ∆ Energy separation ∆ between the double degenerate ground state and the first excited levels of several hundreds of kelvin; Correlation time(τ c ) Correlation time (τ c ) for the spin fluctuations reaching several µs at liquid nitrogen temperatures ; Applications: Applications: - molecular spintronic devices [3] - logic units in quantum computers [4] - contrast agents [5]. Fig. 1: Schematic rapresentation of the TbPc 2 molecule. On the left, the energy level succession of Tb 3+ ion ground multiplet. [LnPc 2 ] 0 SMM:

3. muon polarization P µ (t) The time decay of the muon polarization P µ (t) shows a different behaviour for temperatures above and below T* ≈ 90 K for [TbPc 2 ] 0 and T* ≈ 60 K for [DyPc 2 ] 0 : − T > T* → high frequency regime: with β ≈ 0.5 and A ≈ 20 %. − T < T* → onset of very low-frequency fluctuations: − T < T* → onset of very low-frequency fluctuations: only the long tail of a static Kubo-Toyabe function can be detected due to the too fast initial decay of P µ (t): with α(H) the field dependent initial asymmetry. Fig 2: Time evolution of the muon polarization in [TbPc 2 ] 0 sample normalized to its value for t  0 at six selected temperatures. Muon polarizations P μ (t)

4. [TbPc 2 ] 0 and [DyPc 2 ] 0 NMR relaxation rates: [TbPc 2 ] 0 and [DyPc 2 ] 0 NMR relaxation rates: − The linewidth at half intensity increases from about 37 kHz at 202 K to a few MHz at 19 K  evidence of the slowing down of the spin dynamics; T > 130 K − T > 130 K → 1/T 1 progressively increases on cooling; T < 40 K − T < 40 K → 1/T 1 slowly decreases on cooling. − 140 K ≥ T ≥ 40 K − 140 K ≥ T ≥ 40 K → the observation of the signal is prevented by the too short relaxation time. The small peak in 1/T 1 for [YPc 2 ] - TBA + sample should be ascribed to the TBA + molecular motions. In fact, the Y 3+ ion is non magnetic and no high intensity peak is expected. Fig. 3: T dipendence of 1 H NMR 1/T 1 in [TbPc 2 ] 0 (black diamond), [DyPc 2 ] 0 (green squares) and [YPc 2 ] - TBA + (blue circles) for H = 1 T. Spin-lattice relaxation rates 1/T 1 from 1 H NMR [6]

5. [TbPc 2 ] 0 and [DyPc 2 ] 0 μSR relaxation rates: [TbPc 2 ] 0 and [DyPc 2 ] 0 μSR relaxation rates: − Peak at T m ≈ 90 K for [TbPc 2 ] 0 and T m ≈ 60 K for [DyPc 2 ] 0, in agreement with NMR findings. − The intensity of the peak is observed to scale with the inverse of the field intensity  the frequency of the fluctuations is close to Larmor frequency ω L at T = T m. Fig. 4: T dependence of the muon longitudinal relaxation rate in [TbPc 2 ] 0 and [DyPc 2 ] 0 for H = 1000 Gauss (black and green circles) and for H = 6000 Gauss (blue squares). Spin-lattice relaxation rates λ from μSR [6]

6. Owing to spin-phonon scattering processes each CF level is characterized by a finite life-time τ m which yields a lorentian broadening with the mean-square amplitude of the hyperfine field fluctuations and E m the eigenvalues of the CF levels. The life-time τ m can be expressed in terms of the transition probabilities between m, m ± 1 levels: with C the spin-phonon coupling constant. If ∆ >> T over all the the explored T range, Eq. (3) can be simplified in the form: with The µSR ( 1 H NMR ) spin-lattice relaxation rate λ (or 1/T 1 ) can be written as [7]: Estimating the correlation time of [LnPc 2 ] 0 SMM

7. [TyPc 2 ] 0 correlation time: [TyPc 2 ] 0 correlation time: − T > 50 K → high T activated regime: − T > 50 K → high T activated regime: it describes spin fluctuations among m = ± 6 and m = ± 5 levels. − T < 50 K → low T regime: − T < 50 K → low T regime: the constant correlation time signals tunneling processes among m = +6 and m = -6 levels. The sum of the fluctuation rates associated with each process gives: [DyPc 2 ] 0 correlation time: [DyPc 2 ] 0 correlation time: − T > 40 K  high T activated regime: ∆ ≈ 610 KC ≈ 1780 Hz/K 3 ∆ ≈ 610 K and C ≈ 1780 Hz/K 3. ∆ ≈ 880 K Energy barrier: ∆ ≈ 880 K C ≈ 3000 Hz/K 3 Spin-phonon coupling constant: C ≈ 3000 Hz/K 3 (1/τ c ) tunn ≈ 11 ms -1 Tunneling rate: (1/τ c ) tunn ≈ 11 ms -1 Fig. 5: T dependence of the correlation time for the spin fluctuations in [TbPc 2 ] 0 and [DyPc 2 ] 0 (for T > T*), derived from λ data reported in Fig. 4 on the basis of Eq. (5). The solid lines are the best fits according to Eq. (7). − T < 40 K  − T < 40 K  Eq. (3) should be resorted: the two lowest doublets are separated by an energy barrier in the tens of K range. The high energy barrier extimated corresponds to the one between the first and the second excited doublets. Estimating the correlation time of [LnPc 2 ] 0 SMM

8. Fig.6: T dependence of χT in [DyPc 2 ] 0 for H = 1000 gauss (open circles) and calculated curve (line). The analysis of the static uniform spin susceptibility allows to determine the low energy level structure of the Dy 3+ ion spin multiplet J = 15/2. The lowest energy splittings which allow to better reproduce the experimental data for χT are: - ∆ 1 = 115 K; - ∆ 2 = 547 K; - ∆ 3 = 57 K. These values are larger than the ones deduced for [DyPc2] - on the basis of crystal field calculations [8]. C 1  0 Hz/K 3 C 2 ≈ 2400 Hz/K 3 C 3 ≈ 3100 Hz/K 3 The muon relaxation rate behaviour can be correctly modelled with Eq. (3) by considering the first four CF levels and by using three different spin-phonon constants C 1  0 Hz/K 3, C 2 ≈ 2400 Hz/K 3 and C 3 ≈ 3100 Hz/K 3, associated with the four lowest transitions. Spin susceptibility

9. References: [1] Ishikawa N., Sugita M., Ishikawa T., Koshihara S. and Kaizu Y., J. Phys. Chem. B, 108, 11265 (2004). [2] Branzoli F., Carretta P., Filibian M., Zoppellaro G., Graf M. J., Galan-Mascaros J. R., Fhur O., Brink S. and Ruben M., J. Am. Chem. Soc., 131, 4387 (2009). [3] Bogani L. and Wernsdorfer W., Nat. Mater., 7, 179 (2008). [4] Leuenberg M. and Loss D., Nature, 410, 789 (2001). [5] Cage B., Russek S. E., Shoemaker R., Barker A. J., Stoldt C., Ramachandaran V. and Dalan N., Polyhedron, 26, 2413 (2007). [6] Branzoli F., Carretta P. and Filibian M., Phys. Rev. B, 79, 220404(R) (2009). [7] [7] Lascialfari A., Jang Z. H., Borsa F., Carretta P. and Gatteschi D., Phys. Rev. Lett., 81, 3773 (1998). [8] Ishikawa N., Tomochika I. and Kaizu Y., J. Phys. Chem. A, 106, 9543 (2002).