Scaling up Electronic Spin Qubits into a Three-Dimensional Metal-Organic Framework

Transcription

1 SUPPORTING INFORMATION Scaling up Electronic Spin Qubits into a Three-Dimensional Metal-Organic Framework Tsutomu Yamabayashi, Matteo Atzori, Lorenzo Tesi, Goulven Cosquer, Fabio Santanni, Marie-Emmanuelle Boulon, Elena Morra, Stefano Benci, Renato Torre,, Mario Chiesa, Lorenzo Sorace, Roberta Sessoli,#, * and Masahiro Yamashita,,, * Department of Chemistry, Graduate School of Science, Tohoku University, 6-3 Aramaki Aza-Aoba, Aoba-ku, Sendai, Japan. Dipartimento di Chimica Ugo Schiff & INSTM RU, Università degli Studi di Firenze, Via della Lastruccia 3, I50019 Sesto Fiorentino (Firenze), Italy. Dipartimento di Chimica e NIS Centre, Università di Torino, Via P. Giuria 7, I10125 Torino, Italy. ǁ European Laboratory for Non-Linear Spectroscopy, Università degli Studi di Firenze, Via Nello Carrara 1, I50019 Sesto Fiorentino (Firenze), Italy. Dipartimento di Fisica ed Astronomia, Università degli Studi di Firenze, Via G. Sansone 1, I50019 Sesto Fiorentino (Firenze), Italy. # ICCOM-CNR, Research Area Firenze, Via Madonna del Piano 10, I50019 Sesto Fiorentino (Firenze), Italy. WPI, Advanced Institute for Materials Research, Tohoku University, Katahira, Aoba-Ku, Sendai , Japan. School of Materials Science and Engineering, Nankai University, Tianjin , China. Corresponding authors: roberta.sessoli@unifi.it yamasita.m@gmail.com S1

2 Crystal Structures Figure S1. View of a portion of the crystal structure of 1 along the c axis (a) and along the ab plane (b). Figure S2. Portion of the crystal structure of 3 along the ab plane (a) and along the c axis (b). S2

3 Powder X-ray Diffraction Crystallography Figure S3. Comparison between experimental and simulated PXRD patterns (5 40 ) for 1 and 1. Figure S4. Comparison between experimental and simulated PXRD patterns (5 40 ) for 2, 3, 2, 2. S3

4 Alternate Current Susceptometry 1 + (ωτ) 1 α sin ( πα χ (ω) = χ S + (χ T χ S ) 2 ) 1 + 2(ωτ) 1 α sin ( πα (S1) 2 ) + (ωτ)2 2α Equation S1. Equation of the Debye model used to extrapolate the relaxation time from the AC susceptibility measurements. is the in-phase susceptibility, is the angular frequency, s is the adiabatic susceptibility, T is the isothermal susceptibility, and is the distribution width of the relaxation time. (ωτ) 1 α cos ( πα χ (ω) = (χ T χ S ) 2 ) 1 + 2(ωτ) 1 α sin ( πα (S2) 2 ) + (ωτ)2 2α Equation S2. Equation of the Debye model used to extrapolate the relaxation time from the AC susceptibility measurements. is the in-phase susceptibility, is the angular frequency, s is the adiabatic susceptibility, T is the isothermal susceptibility, and is the distribution width of the relaxation time. Figure S5. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 1 as a function of the temperature ( K) under an applied static magnetic field of 1.0 T. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). S4

5 Figure S6. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the temperature ( K) under an applied static magnetic field of 1.0 T. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). Figure S7. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the temperature ( K) under an applied static magnetic field of 2.0 T. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). S5

6 Figure S8. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the temperature ( K) under an applied static magnetic field of 1.0 T. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). Figure S9. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 1 as a function of the static magnetic field ( T) at T = 5.0 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). S6

7 Figure S10. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 1 as a function of the static magnetic field ( T) at T = 7.5 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). Figure S11. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 1 as a function of the static magnetic field ( T) at T = 10.0 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). S7

8 Figure S12. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the static magnetic field ( T) at T = 5.0 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). Figure S13. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the static magnetic field ( T) at T = 7.5 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). S8

9 Figure S14. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the static magnetic field ( T) at T = 10.0 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). Figure S15. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the static magnetic field ( T) at T = 5.0 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). S9

10 Figure S16. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the static magnetic field ( T) at T = 7.5 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). Figure S17. Frequency dependence of the in-phase component χ' (left) and the out-of-phase component χ'' (right) of the magnetic susceptibility of 2 as a function of the static magnetic field ( T) at T = 10.0 K. The continuous lines represent the best-fit obtained by simultaneously fitting χ' and χ'' through the generalize Debye model equations (Eq. S1 and Eq. S2). S10

11 Figure S18. Magnetic field dependence of for 1 at different temperatures (see legend). Figure S19. Magnetic field dependence of for 2 at different temperatures (see legend). S11

12 Figure S20. Magnetic field dependence of for 2 at different temperatures (see legend). Figure S21. Magnetic field dependence of for 2 at T = 5.0 K. Red points refer to a second measurement performed to exclude that this feature was originated by instrumental issues. S12

13 Figure S22. Comparison between the temperature dependence of the spin-lattice relaxation time of 2 under applied magnetic fields of 1.0 T and 2.0 T. Figure S23. Magnetic field dependence of τ for compounds 1, 2 and 2 at 7.5 K. The continuous lines represent the best-fit of the Brons-van Vleck model with a power law related to the direct mechanism having m = 3 (see main text). S13

14 Figure S24. Magnetic field dependence of τ for compounds 1, 2 and 2 at 10.0 K. The continuous lines represent the best-fit of the Brons-van Vleck model with a power law related to the direct mechanism having m = 3 (see main text). Figure S25. Magnetic field dependence of τ for compounds 1 and 2 at 5.0 K. The continuous lines represent the best-fit of the Brons-van Vleck model assuming a power law related to the direct mechanism of relaxation with m = 3 and 4 (see main text). S14

15 Figure S26. Magnetic field dependence of τ for compounds 1 and 2 at 5.0 K. The continuous lines represent the best-fit of the standard and extended Brons-van Vleck models assuming a power law related to the direct mechanism of relaxation m = 3 (see main text). Table S1. Best-fit parameters of the models used to reproduce the magnetic field dependence of the magnetization relaxation rate for 1 at T = 5.0, 7.5, 10.0 K (Equation 1), and for 2 at T = 5.0, 7.5, 10.0 K (Equation 2). Compound T (K) c (T -3 μs -1 ) d (ms -1 ) e (T -2 ) f (T -2 ) g (ms -1 ) h (T -2 ) (2) 9(1) 19(1) 187(34) (3) 17(3) 22(2) 220(51) (4) 24(5) 25(2) 266(76) (4) 13(1) 8(1) 580(189) 2.3(6) 33(11) (4) 16(2) 11(1) 660(206) 3.2(7) 36(11) (4) 18(4) 13(2) 819(527) 4(1) 42(17) S15

16 CW EPR spectroscopy Figure S27. Experimental CW-EPR spectrum of 2 and simulations of the three different species observed using the parameters reported in Table S2. Table S2. Best simulation parameters for EPR spectrum of 2 and corresponding weights of the three contributions. Attribution g x g y g z A x (MHz) A y (MHz) A z (MHz) % [VO(TPP)] (2) 169(2) 480(2) 96.7 radical ligand Ti(III) Pulsed EPR spectroscopy HYSCORE experiments HYSCORE is a two-dimensional experiment where correlation of nuclear frequencies in one electron spin (ms) manifold to nuclear frequencies in the other manifold is created by means of a mixing pulse. X-band HYSCORE experiments were performed at two different magnetic field settings for both samples 1 and 2. Owing to the large anisotropic character of the EPR spectrum of VO 2+ and the limited bandwidths of the microwave pulses, HYSCORE experiments carried out at different fields result in the selection of vanadyls with specific orientations. The two selected observer positions correspond to quasi-single-crystal positions S16

17 (A ( 51 V) for the MI = 7/2), where vanadyl units with a narrow sets of orientations ( 0 12, where is the angle between the g axes of the VO 2+ g tensor assumed to be oriented along the V=O double bond of the vanadyl unit and the external magnetic field) are selected (Figure S28) and the so-called powder position (MI = 1/2) where all orientations of the vanadyls with respect to the external magnetic field are excited (Figure S29). The spectra show a rich set of cross peaks arising from the interaction of the unpaired electron with 14 N (I = 1) and 1 H (I = 1/2) nuclei in the local environment. A ridge in the (+,+) quadrant, centred at the 1 H nuclear Larmor frequency, with a width of about 3.5 MHz is due to the hyperfine interaction with protons of the porphyrin ring. The observed maximum coupling of about 3.5 MHz is in line with observations on similar molecular systems. S1 Beside the proton ridge, the spectra show clear correlation peaks in both (,+) and (+,+) quadrants arising from hyperfine and nuclear quadrupole interactions of the coordinating nitrogen nuclei. For a nuclear spin I = 1 with a substantial nuclear quadrupole interaction, such as 14 N, six nuclear frequencies, three in each MS manifold, are present. Two such frequencies, υ α,β sq1 and υ α,β sq2 correspond to single-quantum transitions ( MI ±1), whereas the third, υ α,β dq, is related to the double-quantum transition ( MI ±2). Cross peaks can then arise between any nuclear frequency in the electron spin manifold and any nuclear frequency in the manifold, including frequencies of transitions with MI ±2. Moreover, when several nuclei contribute to the modulation pattern, combination frequencies of the types υ i α (k) ± υ j α (l) and υ i β (k) ± υ j β (l) where the indices k and l represent different nuclei coupled to the same electron spin, may appear. Given the remarkable resolution of the HYSCORE spectra recorded at the single-crystal-like position for both 1 and 2 such combination peaks are very well resolved and fully assigned in Figure S30, where the simulation of different nuclear frequencies based on the spin Hamiltonian parameters reported in Table S3 is also shown. The values extracted from the simulation fit very well with typical values reported for vanadyl phthalocyanine and porphyrin molecular complexes, indicating that the local structure of the monomer is preserved in the solid system. Moreover, the negligible orientation distribution at the single-crystal-like position, resulting in narrow HYSCORE cross peaks, seems to indicate that the 14 N quadrupole and hyperfine tensors are nearly collinear with the 51 V hyperfine tensor. S17

18 Figure S28. HYSCORE spectra of 1 (blue lines) and 2 (green lines) recorded at 10 K and B 0 = mt corresponding to the M I = 7/2 transition with = 100 ns. Figure S29. HYSCORE spectra of 1 (blue lines) and 2 (green lines) recorded at 10 K and B 0 = mt corresponding to the M I = 1/2 transition (powder like position) with = 100 ns. S18

19 Figure S30. HYSCORE spectrum of 1 recorded at 10 K and B 0 = mt corresponding to the M I = 7/2 transition with = 100 ns and computer simulation of the different cross peaks. Cross peaks relative to single quantum transitions are in red. Green crosses simulate the double quantum transitions (υ α,β dq ), while in blue, cross peaks related to single-double quantum transitions are shown. In cyan combination peaks of the form (υ α dq (k) + υ α dq (l), υ β dq (k)) where k and l refer to two different nitrogen nuclei are simulated. The corresponding (υ β dq (k) + υ β dq (l), υ α dq (k)) cross peaks at [( 12.3,+9.7),( 9.7,+12.3)] MHz are depicted in magenta, while purple crosses refer to combination peaks of the form (υ α dq (k) + υ α dq (l), υ β dq (k) + υ β dq (l) ). The arrows indicate sum and difference combination peaks between the 1 H Larmor frequency and 14 α,β α,β N double quantum frequencies of the form (υ dq ± υh ). The spin-hamiltonian parameters used for the simulation of the 14 N HYSCORE spectra are reported in Table S3. Table S3. 14 N principal hyperfine values (A) and nuclear quadrupole values (P) obtained from the simulation of the HYSCORE spectra of 2 and comparison with those of related vanadyl phthalocyanine and porphyrin molecular complexes a Compound Ax Ay Az Px Py Pz Ref. 2 + [VO(TPP)] (toluene) this work + S1 [VO(OEP)] + S2 VO(F 64-Pc) + S1 (ethanol) a TTP = tetraphenylporphyrin, OEP = octaphenylporphyrin, F 64-Pc = perfluorophthalocyanine. S19

20 Inversion Recovery and Echo decay experiments Table S4. Extracted relaxation times (T 1 and T m) for 1 at X-band frequency (OP1, m I = 1/2). T (K) Inversion recovery Echo decay T 1 ( s) stretch T m ( s) Table S5. Extracted relaxation times (T 1 and T m) for 2 at X-band frequency (OP1, m I = 1/2). T (K) Inversion recovery Echo decay T 1 ( s) stretch T m ( s) S20

21 Simulation of the temperature dependence of T1 Besides the phenomenological equation (5) of the main text, another common way to reproduce the temperature dependence of the spin-lattice relaxation time induced by the Raman process is to introduce the Debye Transport Integral S3 according to T 1 1 = a dir T x + a ram ( T 9 ) J θ 8 ( θ D D T ) (S4) where θ D /T J 8 ( θ D T ) = e x x8 (e x 1) 2 dx 0 (S5) The first term of eq. (S4) accounts for the direct relaxation mechanism (see main text), while the second term describes the Raman process through the parameter aram, that quantifies the efficiency of the process, with ϑd, the Debye temperature. The J8 function of eq. (S5) is known as the Transport Integral. To fit the experimental data with MATLAB R2015b, the Transport Integral has been calculated as a series of polylogarithm functions using the code generated through θ DT e x x 8 (e x 1) 2 dx = 0 = 8 (7 z 6 Li 2 (e z ) 42 z 5 Li 3 (e z ) z 4 Li 4 (e z ) 840 z 3 Li 5 (e z ) z 2 Li 6 (e z ) 5040 z Li 7 (e z ) z8 8 + z7 log(1 e z ) Li 8 (e z )) z8 e z Li 8(1) where Li n (x) is the polylogarithm function and z = θ D T. S21

22 Figure S31. Experimental T 1 data of the MOF (1) and VO(TPP) (2) from EPR inversion recovery experiments at X-bands (same data as Figure 5) reproduced (solid lines) with the power law reported in eq. (5) of the main text (a). Same data reproduced by using the Debye Transport Integral as reported in eq. (S4) (b). Best-fit parameters are reported in Table S6. Table S6. Best parameters in the simulation of the temperature dependence of T 1, extracted from EPR inversion recovery experiments, performed using the power law of eq. (5), the Debye Transport Integral of eq. (S4), and the local modes of eq. (7) in addition to the direct process of relaxation. adir (ms -1 K -x ) x aram (ms -1 K -n ) n Eq. (5) (1) 0.54(6) 3(2) (1) 2 0.4(1) 0.3(3) 1.5(8) (1) adir (ms -1 K -x ) x aram (ms -1 ) ϑd (K) Eq. (S4) 1 1.0(1) 0.64(4) 2.7(8) (37) 2 0.4(1) 0.4(3) 1.1(1) (10) adir (ms -1 K -x ) x aloc (ms -1 ) (cm -1 ) Eq. (7) (1) 0.65(1) 2.95(9) (2) 2 0.4(2) 0.3(2) 34(7) 6(2) (fixed) 303(35) S22

23 Nutation experiments Figure S32. (a) Rabi oscillations recorded for 2 at 50 K for different microwave attenuations (X-band). (b) Fourier Transform of the Rabi oscillations. (c) Linear dependence of the Rabi frequency ( R) as a function of the relative intensity of the oscillating field B 1. TD-THz spectroscopy Figure S33. Vibrational spectra in the THz range (50 80 cm ) of compounds 2 and 3 recorded at selected temperatures (see legend). S23

24 SUPPORTING INFORMATION REFERENCES S1. H. Moons, H. H. Patel, S. M. Gorun, S. Van Doorslaer Z. Phys. Chem. 2017, 231, 887. S2. K. Fukui, H. Ohya-Nishiguchi, H. Kamada, J. Phys. Chem. 1993, 97, S3. Eaton, S. S.; Eaton, G. R. Distance Measurements in Biological Systems by EPR; Berliner, L. J., Eaton, G. R., Eaton, S. S., Eds.; Biological Magnetic Resonance; Springer US: Boston, MA, 2002; Vol. 19. S24