An interactive dashboard for lanthanide SIMs
In this work, we built a dataset of the most relevant chemical and physical properties of 1405 lanthanide SIM samples collected from 451 scientific articles (Supplementary Section 7) published between 2003 and 2019 and developed a user-friendly dashboard-style web application named SIMDAVIS (Single Ion Magnet DAta VISualization) to host it. The dataset contains over 10000 independent pieces of chemical information, as well as over 5000 independent pieces of physical (magnetic) information. Furthermore, the dataset is hierarchically clustered into magnetostructural “taxonomies” (see Supplementary Sections 4 and 6) in order to pave the way for further analysis, including Machine Learning studies.
SIMDAVIS allows the chemical community to visualize the key relationships between chemical structures and physical properties in our catalogue of SIMs. Our interactive dashboard can be directly invoked by accessing the internet site where it is located.33 It is organized in 6 main tabs: Home, ScatterPlots, BoxPlots, BarCharts, Data (View Data and Download Data) and About SIMDAVIS (Variables, Authors, Feedback&Bugs, Changelog and License) as we can observe in Fig. 3.
In the SIMDAVIS dashboard, the most versatile source of graphical information is the “ScatterPlots” tab, where an example plot is explained in Fig. 3. The next two tabs display the data in complementary ways. The “BoxPlots” tab allows to examine the distribution of each SIMs quantitative property vs a categorization criterion, e.g. we can see the distribution of Ueff values as a function of the coordination elements. The boxplot for each category is shown, including the median and the interquartile range. The “BarCharts” tab allows the exploration of the frequency of different qualitative variables in our dataset. Stacked bar graphs allow the simultaneous analysis of two qualitative variables, e.g. we can display, for each chemical family, the number of samples which present magnetic hysteresis. The “Data” tab is a powerful interface to browse the dataset, featuring the possibility to choose the data columns to show, ordering in ascending or descending order, and filtering by arbitrary keywords; it also permits downloading all data, including links to the CIF files, when available. Finally, the “About SIMDAVIS” tab contains information about the variables contained in the dataset.
Statistically-driven chemical design of SIMs
SIMDAVIS allows the visualization of the relationships between chemical and physical variables in SIMs, and thereby enables determining the main variables that the synthetic chemist needs to consider to obtain the desired physical properties. We will first analyze this qualitatively employing a series of boxplots, violin plots and bar charts (see Fig. 4 and Supplementary Figs. 11.1-11.6, 12). The full statistical analysis is presented in Supplementary Sections 4, 5 and 6.
First, let us focus on the effective energy barrier Ueff and the blocking temperature TB3 (the temperature for maximum out-of-phase ac susceptibility χ’’ at 103 Hz, see Fig.1). From Fig. 4 and Supplementary Figs. 11.1-11.4, we can see that the only chemical family with a clearly distinct behaviour is the LnPc2 family, with median values of Ueff > 300 K and TB3 > 30 K. Equivalently, one can see that Dy3+ and Tb3+ are somewhat better than the rest, and that in general oblate ions perform better than prolate ions, for both properties. In addition, non-Kramers ions present higher median TB3 but similar Ueff values compared with Kramers ions.
Now, let us analyse the maximum hysteresis temperature Thyst which has been much less studied despite being the main justification for this whole field. The only chemical family with a distinct positive behaviour is the metallocene family. More surprisingly, Er3+ complexes have distinctly high hysteresis temperatures, markedly with a higher median than Dy3+ or Tb3+ complexes. This is in sharp contrast with their relative TB3 values which are consistently much lower in the case of Er3+ complexes. This not only indicates that searching for equatorial environments, precisely the ones that favour good magnetic properties in Er3+ complexes,30 often results in more rigid ligands, but also indicates an underexplored territory. It is certainly possible that chemical modifications of [Er(COT)2]− (or other Er3+ record-bearing complexes) designed to optimise the detrimental effect of molecular vibrations may achieve records that are competitive with DyCp2. Prolate ions are consistently -and surprisingly- better than the oblate ones, having a higher median value for Thyst. This is again in contrast with the opposite behaviour which is observed for TB3 and Ueff, and possibly again due to the influence of Er3+ complexes with their more rigid equatorial environments. Finally, the coordination number and the number of ligands do have an influence on the statistically expected hysteresis temperature, with the best ones being 2 and 7 in the case of the coordination number and just 7 for the number of ligands. As we will discuss below, there are chemical insights to be gained from this.
To put all these trends into perspective, it is important to numerically analyse the connexion between the different variables and the clustering of our data. A lognormal analysis (see Supplementary Section 4.3) shows that the three main chemical variables, namely the chemical family, the lanthanide ion and the coordination elements, are sufficient to reasonably explain the variation of values of the others, meaning there is a limit on the information one can independently extract from the rest of the chemical variables. Multiple correspondence analysis (see Supplementary Sections 4.1, 4.2) suggests a chemical clustering that consists in three small groups, namely Gd3+ complexes, metallocenes and LnPc2 double deckers, and two much larger groups with a large overlap with oblate and prolate ions respectively. A factorial analysis of mixed data considering also all magnetic information available (see Supplementary Section 6) simplifies the clustering to three groups. Again, the two distinct families present a large overlap with metallocenes and LnPc2 double decker chemical families, both of them presenting significantly better properties than the other kinds of samples.
Further insight is provided by bar charts representing the reported presence of magnetic hysteresis, whether full or pinched, as a function of different chemical variables (Fig. 4). Note that we are limited by the minority of the samples where hysteresis or its absence is reported; in the vast majority of the cases this information is lacking. Nevertheless, here it is apparent that certain families such as LnPc2 (and metallocenes) tend to display (pinched) hysteresis.
The effective energy barrier: oversimplified yet meaningful
A key question is how much the analyses in this field have been affected by the simplified assumption that SIMs relax via an Orbach mechanism, which is characterised by τ0 and Ueff. It has been pointed out that frequently, as Ueff increases, τ0 decreases, leaving relaxation times essentially constant.5 The classical text of Abragam and Bleaney offered the following relation between the two parameters for the two-phonon Orbach process:34
where n = 3, reasonable parameters for rare earth elements resulted in an Orbach rate ROr ≈ 104 K-3·s-1, and early experimental results were in the range 103 K-3·s-1 < ROr < 105 K-3·s-1. Fitting τ0 vs Ueff in our dataset to equation (1) results in n ≈ 2.4, ROr ≈ 103 K-3s-1 (see Supplementary Table 5). This minor discrepancy with the expected exponent serves as an independent evaluation of the limitations of a simple Orbach model. We also find ROr(prolate) ≈ 5·ROr(oblate), meaning that, for comparable Ueff, relaxation for oblate ions is on average substantially slower than that for prolate ions. This is consistent with the observation that complexes of oblate ions present values of TB3 higher than expected considering their Ueff (see Supplementary Fig. 11.3). The limited (<100) data points of Ueff,ff, τ0,ff pairs, where all relaxation processes were considered, present a better agreement on the exponent, with n ≈ 3 and lower Orbach rates ROr ≈ 150 K-3s-1.
The remaining crucial issue is to quantify up to what level the value of Ueff and τ0 are well correlated with the slow relaxation of the magnetisation, or to determine whether one would need to employ Ueff,ff instead. Let us proceed with increasing the order of complexity. A visual inspection in SIMDAVIS shows that, in a few cases where there is simultaneous information on Ueff and Ueff,ff, their values are very similar (Fig. 5a). Furthermore, this partial information is corroborated by the very similar dependencies of TB3 or Thyst vs either Ueff or Ueff,ff, as well as in the numerical correlations (see Supplementary Sections 3.2 and 5.3). A categorical analysis (Figs. 5b, c) shows that the data dispersion is large, meaning that it is impossible to predict the experimental behaviour for an individual sample merely from its Ueff value. However, it demonstrates that, statistically, samples which present a maximum in the out-of-phase susceptibility χ’’, or hysteresis, also present higher Ueff values. A more thorough numerical analysis (see Supplementary Section 6) confirms these trends.
An in-depth statistical analysis of all physical parameters based on the Akaike Information Criterion (see Supplementary Section 5.3) concludes that Ueff derived from a simple Arrhenius plot is the best single predictor for the magnetic behaviour in our dataset. This means that, whether we are discussing in terms of the out-of-phase component of the ac susceptibility or magnetic hysteresis, Ueff is a better predictor than τ0, τ0,ff, Ueff,2 and, in practice, than Ueff,ff. Factorial analysis of mixed data (see Supplementary Section 6) also reveals the predictive power of Ueff compared with τ0. Note that this does not contradict previous studies which demonstrated that a variation in the Orbach barrier does not fully explain the differences in retention of magnetisation,14 since we have not explicitly considered other relaxation mechanisms in the present work.