Ferrotoroidic ground state in a heterometallic {CrIIIDyIII6} complex displaying slow magnetic relaxation

Toroidal quantum states are most promising for building quantum computing and information storage devices, as they are insensitive to homogeneous magnetic fields, but interact with charge and spin currents, allowing this moment to be manipulated purely by electrical means. Coupling molecular toroids into larger toroidal moments via ferrotoroidic interactions can be pivotal not only to enhance ground state toroidicity, but also to develop materials displaying ferrotoroidic ordered phases, which sustain linear magneto–electric coupling and multiferroic behavior. However, engineering ferrotoroidic coupling is known to be a challenging task. Here we have isolated a {CrIIIDyIII 6} complex that exhibits the much sought-after ferrotoroidic ground state with an enhanced toroidal moment, solely arising from intramolecular dipolar interactions. Moreover, a theoretical analysis of the observed sub-Kelvin zero-field hysteretic spin dynamics of {CrIIIDyIII 6} reveals the pivotal role played by ferrotoroidic states in slowing down the magnetic relaxation, in spite of large calculated single-ion quantum tunneling rates.


Supplementary Note 2. Single-ion relaxation mechanism
A qualitative mechanism for the magnetic relaxation originating from the Dy1 site, obtained from the ab initio calculations, is shown in Supplementary

Supplementary note 3. How does our analysis of the experimental results exclude a nontoroidal arrangement?
To probe the robustness of our conclusion i.e. that the ground state in our system is ferrotoroidically coupled, we varied one of the key results of our CASSCF-RASSI-SO calculations, which plays a crucial role in determining the ferrotoroidic ground state, namely the direction of the local anisotropy axes of the Dy ions, and used the resulting modified model to simulate the experimental magnetization. From our calculations the local anisotropy axes turn out to be almost exactly contained in the two triangles' planes, and directed along the local tangent to the wheel's circumference. To set up models that depart from this ab initio result, we generalized our exchange + dipolar coupling Hamiltonian introducing two angles: an angle  measuring the departure of the anisotropy axis from an in-plane configuration, and an angle measuring the departure of the in-plane projection of the anisotropy axis from a locally tangential direction. To comply with the D 3d pseudo-symmetry of the metal core of the complex, we demanded that the angle  be the same for all Dy ions, while the angle  should have opposite signs for the two wheels, due to inversion symmetry.
We explored two significant scenarios departing from our parameter-free ab initio model, and reported the resulting powder magnetization curves obtained at 2 K in the figure below, together with the results of our parameter-free ab initio model (orange curve in the picture) and the experimental data points (blue data points in the picture): (i)  = 30°,  = 0°, i.e. a significant departure from in-plane tangential configuration of the magnetic axes, which will determine a significant out-of-plane magnetic moment for the Antiferrotoroidic (AFT) configuration only, but a zero out-of-plane magnetic moment for the Ferrotoroidic (FT) configuration. Such out-of-plane magnetic component of the AFT state will also be coupled antiferromagnetically to the Cr magnetic moment, thus stabilizing the AFT with respect to the non-magnetic FT state. For  = 30°, the appearance of a significant anisotropic out-of-plane magnetic moment in the AFT state, determines a strong Cr-Dy 6 antiferromagnetic stabilisation energy contribution which makes the AFT configuration the ground state, and the FT state the first excited state. However, the powder magnetization we calculate in this scenario is reported in the picture below (green curve), and evidently it does not match the experimental data, which instead support our finding that at low field the only source of magnetic response comes from the Cr ion. Any additional (anisotropic) magnetism from the Dy-triangles would make the low-field magnetization steeper than what is observed experimentally, which supports our finding of the FT configuration (implying a zero magnetic moment on the wheels) as the ground state.
(ii)  = 0°,  = 90°, i.e. the axes are still perfectly in-plane (contained in the planes defined by the two triangles), but they are now directed radially instead of tangentially to the triangle's circumference. In such configuration it is still possible to achieve a non-magnetic noncollinear ground state on the Dy wheels, for which the magnetism solely arises from isotropic to make definitive statements about the ferrotoroidic nature of its ground state. We note here that the model Hamiltonian used here to simulate the magnetic data for CrDy 6 includes dipolar coupling (which in fact is shown to dominate the resulting energy spectrum), and contains no fitting parameter.
2) Lin et al., 3 have presented an interesting study of a Dy 6 -2 cluster that can be viewed as two very closely spaced Dy 3 triangular units with two edges, one from each triangle, directly facing each other. The system is not exactly co-planar, but displays a 29º dihedral angle between the two triangles' planes. In that work the magnetic coupling is modeled including We note however that there are a few features of the Hamiltonian used in that paper that would seem to need further testing before the conclusions drawn about the nature of the ground state be unambiguously confirmed. First of all, we note that the choice of a single exchange coupling parameter, especially given that this is a fitting parameter not derived from a theoretical model or an ab initio calculation, can be in principle criticized. In particular, we note that while antiferromagnetic coupling between ions belonging to the same triangle is known to stabilize a toroidal moment, given the geometry of Dy 6 -2, antiferromagnetic coupling between nearest neighbor ions on different triangles (e.g. Dy 1 and Dy 2 in Fig. S1 of that paper) will in fact stabilize counter-rotating toroidal states on different triangles, hence an antiferrotoroidic ground state. Given that the distance between Dy 1 and Dy 2 in Fig. S1 is shorter (3.34 Å) than any intra-triangular Dy-Dy distance (3.39Å, 3.51Å, 3.54Å), it could be argued that a stronger inter-molecular antiferromagnetic exchange between such ions could flip the energetic order of the con-rotating and counter-rotating coupled-toroidal states. On the other hand, an antiferromagnetic diagonal interaction (e.g. between Dy 1 -Dy 3 in Fig. S1 of that paper) will indeed stabilize a con-rotating toroidal configuration, but the distance between the Dy ions is in CrDy6 (1) much longer, thus weakening such interaction (Dy 1 -Dy 3 distance is 4.7Å). Such important competing effects are clearly not captured by a single fitting parameter, and these issues are not discussed by Lin et al. 3 (We present alternative calculations on Dy 6 -2 and these are discussed below).
Aside from the details of the Hamiltonian parameterisation, assuming that the arrangement of local Dy magnetic moments in the ground state of Dy 6 -2 leads to a maximization of the overall toroidal moment of the molecule, to characterize such ground state as a ferrotoroidically-coupled state still seems somewhat problematic on the grounds of two issues, which are in fact related to each other: (i) the triangle-triangle distance is shorter than two of the intra-triangle's Dy-Dy distances, so that the detailed connectivity of the central Nevertheless, to the best of our knowledge, the CrDy 6 (1) system presented here, according to our parameters-free model, provides the first example of a well-defined ferrotoroidic ground state resulting from the coupling of two separate toroidal subunits, maximizing the total toroidal moment, and characterized by low-lying pure toroidal excitations to antiferrotoroidic states (counter-rotating toroidal moments resulting in zero toroidal moment), well separated from higher-energy magnetic excitations.
Alternative calculations on Dy 6 -2 complex: To test alternative scenarios for Dy 6 -2 we have set up an approximate model Hamiltonian in which the two triangles are considered equilateral using average experimental bond distances, the 29º dihedral angle between the two triangles' planes is explicitly taken into consideration, and the deviation from coplanarity of the Dy anisotropy axes is also included in the model using the data reported in that paper. Dipolar coupling is explicitly included in the model as is exchange coupling (see the green curve is associated to the parameters discussed above). It can be seen that the simulation of the experimental data is hardly changed in the two settings, but for a single parameter the ground state consists of con-rotating toroidal states separated by a large energy gap from the counter-rotating toroidal state (transition indicated with a blue arrow in Supplementary Fig. 9), almost at the same energy as the first magnetic excitation (transition indicated with a red arrow in Supplementary Fig. 9), while with the new parameters tried here the con-rotating and counter-rotating energy ordering is inverted, and the toroidal excitation is much smaller than the magnetic excitation.

Supplementary Note 5. Further analysis of the theoretical dynamical magnetization
To further analyse our simulation of the dynamical magnetization, we report the plots of the contributions to Tr arising from the ferrotoroidic and antiferrotoroidic states  Fig. 11(c)). If we now solve Equation (8)/ Equation (11) for a slightly different set of parameters, still preserving the proposed hierarchy, but using faster 2-flip transitions, so that  Cr = 3  10 4 Hz/(cm -1 ) 3 >> 1 = 3.33 10 -7  Cr > 2 = 10 -2  1 , and  Cr = 410 15 Hz 2 >> 1 = 2  10 12 Hz 2 > 2 = 10 -1  1 , we obtain a hysteretic magnetization reported in Supplementary Figure 12(a), which still reproduces the zero-field hysteresis loop, with an almost closed hysteresis at high fields, but now dominated by the onion states at high fields (see Supplementary Figure 12(c)), and displaying a sizeable contribution from AFT states at low-fields, almost as large as that of the FT states (see Supplementary Figure 12(b)), but otherwise not changing the main conclusions drawn above.
Supplementary Figure 11. a) Single-crystal experimental magnetization (blue curve) measured at T = 0.03 K and a sweep rate of 0.1T/s for a magnetic field oriented parallel to the triangles' planes and along the easy-axis (y axis in Figure 6), superimposed to the simulated dynamical magnetization at the same temperature, sweep rate and field orientation, by solving Equation (