Understanding magnetic relaxation
Despite the breakthroughs that have led to record breakingU eff values, T B, and magnetic hysteresis temperature, an unequivocal understanding of the relationship between structure and relaxation profile (\(\tau^{-1}\) vsT ) is still missing. The combined efforts of the Mills and Chilton Groups, together with Manchester-based and international colleagues, have taken aim at these phenomena by developing a systematic, and synergic synthetic/computational strategy to address specific issues affecting relaxation dynamics.
Spin-phonon coupling. Highly axial Dy(III)-based SMMs that present \(U_{\text{eff}}\) values larger than 1000 K and an energy gap between the ground- and first excited mstates of around 500 K are now routinely reported.[14] With such energy separations between the electronic states involved in the relaxation of magnetization, the question is why these systems cannot retain magnetization at room-temperature and where the energy that bridges those states is coming from? The answer to that is the coupling between the spin and motion degrees of freedom (phonons) of the molecule and crystal,i.e. spin-phonon coupling.
The mathematics describing spin-phonon coupling and magnetic relaxation was pioneered by the likes of van Vleck and Orbach in the in the post-war years,[26] and until recently only parametric models where used to interpret magnetic relaxation.[27] Modern computational methodologies have advanced to the point where an ab-initio description of spin-phonon coupling is now possible.[28] Theab-initio spin dynamics formalism to calculate spin-phonon couplings was initially proposed to explain the extraordinary magnetic properties of dysprosocenium (2 ).[14a]The theoretical framework was deeply influenced by the experimental observation that the relaxation dynamics of pure crystalline, magnetically dilute crystalline, and also amorphous samples were superimposable in the Orbach regime, exemplifying the tight relationship between experiment and theory. This observation implies that in this temperature range, the processes that govern relaxation are of molecular origin (i.e. nothing intrinsic to the lattice), greatly simplifying the theoretical treatment of the problem. In a significant simplification, the method completely ignores acoustic phonons and phonon dispersion, focusing only on the localized molecular vibrational modes. Using Fermi’s golden rule, derived from time-dependent perturbation theory, one can use these normal modes to monitor the response of the \(m_{J}\) states as the molecule is distorted and calculate spin-phonon couplings. Then, these couplings can be turned into a relaxation profile (\(\tau\) vs T ) by constructing and solving the master matrix containing the coupling elements between all states.[1] Currently, this approach describes only the Orbach regime (first term in eq.1), as it uses expressions that couple states via single-phonons.
During recent years, efforts have been dedicated to the control and design of static properties, i.e. how to optimize the coordination environments to stabilize the magnetic states of interest. The new generation of highly-axial Dy(III) SMMs have brought enormous\(U_{\text{eff}}\) values and thus, design criteria must now shift towards considering the dynamic properties of molecules and crystals. To that end, predictive tools such as the one developed by the Chilton Group, as well as in other groups,[15,25,30,31]are necessary to drive the field forward.
Dipolar interactions. Although the vast majority of Dy(III) SMMs are monometallic complexes, through space dipolar interactions between magnetic moments in proximity to each other can cause state-mixing and consequently faster loss of magnetic information. A typical strategy to reduce the effect of dipolar magnetic interactions is to dilute the spin in a diamagnetic matrix, so the magnetic ions do not “see” each other. For instance, assuming a perfectly axial Dy(III) (g z = 20, g x =g y = 0) in a crystal lattice whose nearest neighbor is a co-parallel Dy(III) center sitting at x = 10.4 Å, the coupling constant is ca. -0.31 cm-1, which reduces to ca. -0.18 cm-1 when the distance increases to 12.5 Å. From an experimental point of view this poses some challenges in terms of material characterization and analysis. One approach for heavy Ln SMMs (Tb-Yb) is to use a Y analogue as diamagnetic host.[9,14a,14b,15] This is due to the many chemical similarities between Dy and Y, such as their dominant oxidation state (+3) and similar ionic size, which translates to an essentially superimposable coordination chemistry. Y complexes are doped with a small amount of Dy (usually < 5%) and recrystallized in order to ensure homogeneity To validate the effectiveness of doping protocols, a combination of experimental techniques are employed to determine the purity of crystalline phase (single-crystal and powder XRD) and metal composition (elemental analysis, ICP-MS, ICP-OES, X-ray fluorescence). This can be a challenging task in the case of extremely air- and moisture sensitive species like the low-coordinate, or organometallic SMMs above.
To address the relative effect of dipolar interactions on the relaxation of magnetization, we, along with an international collaborative team, produced samples of known SMMs at pure Dy(III), and 5%Dy@Y doping levels.[29] The systems were chosen as the magnetic properties (temperature at onset of magnetic hysteresis,T H most notably) of each differed significantly, though all possess high U eff values, and all had been studied as pure Dy(III) samples (Figure 3a).[29] Comparison of data from both pure, and doped samples would hint towards the origin of the different behaviori.e. if dipolar interactions were the sole driver of QTM, doping 5%Dy@Y should engender different improvements in each sample as they each possess different U eff values. Hysteresis loops measured at 2 K showed that the effect of magnetic dilution was clearly visible as a reduction in the QTM step at zero field in2 .[29] The same behavior was also observed in the other systems, [K(18-crown-6)(THF)2][Dy(BIPM)2] (3 , BIPM = C{PPh2NSiMe3}2) and [Dy(tBuO)Cl(THF)5][BPh4] (4 ). However, in these examples a significant drop in remnant magnetization is still present in the absence of an external magnetic field, which hints at the fact that dipolar interactions are not the main cause of the zero-field step in these high-barrier SMMs.[29]