Database, Features, and Machine Learning Model to Identify Thermally Driven Metal-Insulator Transition Compounds

Fig.​ 3 illustrates the process by which we built the database for training the electronic classification models. We used a combination of domain knowledge and natural language processing (NLP) to generate both an initial materials database and a keyword list for the NLP pipeline. The materials database initially included all MIT compounds known by the authors along with related materials, including binary vanadates V_nO_m, $R$NiO₃ nickelates ($R$ a rare-earth metal), LaCoO₃, and some Ruddlesden-Popper oxides (e.g., Ca₂RuO₄), as well as lacunar spinels. Compounds with similar chemistries and stoichiometries that are exclusively metallic or insulating were then also added to the database.

To extend the database beyond the materials known to the authors, we have also used natural language procesing (NLP). The text corpus used for the NLP methods included 70,123 papers, which was down selected from an entire text corpus of over 4 million scientific papers and journals. Down selection to the highly specialized MIT text corpus was performed using keywords (Table S1), describing MITs and correlated electron systems from the authors’ domain knowledge, which matched words in the titles, abstracts, and introductory paragraphs of papers. New MIT compounds identified from two types of NLP searches (vide infra) were then verified manually by assessing experimental transport (and/or optical) data. The newly identified compounds were each assigned a metal, insulator, or MIT class label and added to the database. Based on these identifications, new keywords were also added to guide the NLP search or additional human searches to find more compounds relevant to the classification tasks. This process was repeated until the database acquired 343 unique entries.

Two NLP methods were used in our workflow. First, using the specialized MIT text corpus of approximately 70,000 papers and a state-of-the-art NLP pipeline,⁴⁴ we extracted chemical formulas of possible MIT compounds from each paper. This entity recognition process included tokenization of the relevant MIT text corpus, normalization to lemmatize each term, part-of-speech tagging using dependency parsers, and finally token classification. Using this NLP method, we identified the MIT compound PrRu₄P₁₂, which belongs to the skutterudite family frequently studied in thermoelectrics research. Its transition is attributed to the physics of Pr 4f electrons, rather than the Ru 4d electrons.⁴⁵ Although the authors were unaware of this compound prior to the NLP search, we were able to combine our MIT domain knowledge to perform a manual search of the literature and identified SmRu₄P₁₂ as another skutterudite MIT material, as well as a wide range of other skutterudite materials which exhibit solely metallic or insulating behavior.

Second, we employed a FastText model trained directly on the specialized MIT text corpus. The resulting word embeddings were then compared in terms of cosine similarity to previously identified MIT materials. We use a different cosine-similarity approach than that in Ref. 46 to identify compounds of interest. For each compound in the original dataset of identified materials, 100 words with highest cosine similarity in the trained FastText model were identified. Then using the metal, insulator, and MIT class labels present within the dataset, we grouped the closest 100 words for each compound into closest words for a given label. Because there were approximately 50 temperature-driven MITs in the original dataset, this grouping method resulted in $\approx$ 5,000 compounds with repeats, that were closest in cosine similarity of word embeddings to known MIT compounds. Then within the group of compounds for a given class label, the 20 most commonly occurring words were identified, resulting in 20 words for each classification label (60 words in total). Of the 60 words, the vast majority were chemical formulas with no noise.¹¹1Nine words were not exact chemical formulas of compounds and thus were considered to be noise. Although we assigned the words as noise, some words were still relevant; for example, the word “La$A$O₃” appeared in the 20 most common words associated with insulators, where $A$ represented an unidentified element on the periodic table. We further simplified the search by keeping only those words which were exact chemical formulas. After these compounds were identified, abstracts and titles were searched for these specific compounds. Among this subset of literature, we then searched the classifications for the compounds identified by the FastText model and added new materials to the database. The two NLP-assisted searches led to the addition of 116 compounds to the database (representing a 51 % increase in size from the initial dataset), which was then augmented further by additional human searches using the new domain knowledge. Our database was expanded from about 190 unique compounds to the current 343 with the addition of the mixed human + NLP search. This expansion led to identification, for example, of the skutterudite family in which the MIT mechanism is driven by the rare-earth cations rather than transition metal-driven physics characteristic of the other materials in our database.

As a result of this search, we have built a database that is as representative as possible. Fig.​ 4 shows the distribution of the complete dataset obtained from this workflow at the date of submission of this paper. As expected for a database of materials dominated by transition metal oxides, the dataset is imbalanced, with the number of insulators significantly greater than that of metals or MIT materials. All known MITs fall into a relatively restricted class of compounds and most frequently appear as complex ternary materials, often transition metal oxides and sulfides. In order for our database to be focused on thermal MITs, we have excluded compositionally-controlled MIT compounds. Our model then focuses on complex, inorganic materials involving d-shell or f-shell electrons as the valence states. Although an arbitrary increase in our database size to go beyond this limited scope may allow for an increase in our classifier scores, this increase would not be meaningful as we want the metals and insulators to have similar chemical composition and stoichiometry to the MIT compounds. The heatmap in Fig.​ 4 can be used as a quick guide to identify if a new compound to be tested contains elements that are already in the training data. This would better inform the decision as to how much trust one could place on the classification provided by our ML classifiers. Analogous heatmaps for metals, insulators or the MIT compounds can be found in the SI.

As most MITs are accompanied by a structural change coupled to the electronic transition, an important question in building our model was which structure to select for featurization: the structure corresponding to the low-temperature, often insulating phase, or the high-temperature, often, metallic phase, or both? Across the electronic transition, a symmetry lowering distortion typically occurs with the low-temperature phase comprising small atomic displacements absent in the high temperature structure (see local distortions illustrated in Fig.​ 1). For example, the rare-earth nickelates at low-temperature have a breathing distortion, i.e., small and large NiO₆ octahedra alternate in a 3D checkerboard pattern.^{47, 48} This type of microscopic model associated symmetry-breaking is usually known only in materials after the compound has been studied sufficiently to be labeled as an MIT material. Furthermore, most crystal structures are initially reported at room temperature, which can be far from $T_{\mathrm{MIT}}$. To build a model that is both mechanism-agnostic and can predict whether a material will have an MIT based on a simple theoretical or experimental structure before in-depth analysis could be performed, we opted to include only the high temperature structures in our dataset, if available. This allows our model to learn the susceptibility of a compound to undergo an MIT using more readily available high-temperature structures.

Constructing the machine learning model requires tabulating appropriate features to describe its properties for accurate class predictions. Our features include the Magpie⁴² composition feature set, oxidation state, Ewald energy, local and crystal structure parameters (i.e., variation in bond lengths and atomic volumes), and the global instability index (GII)⁴⁹, all of which are accessible from the Matminer⁵⁰ package. Certain descriptors known from the material physics community to be important in describing MIT compounds, however, are unavailable in these standard libraries designed for machine learning. To that end, we constructed additional structural features, intended to capture the displacive distortions shown in Fig.​ 1, as well as built featurizers that can provide an estimate of the electronic energy scales used in the Zaanen-Sawatzky-Allen framework⁵¹ to separate metals from insulators.⁵²

Now, we will describe some of the features determined to be important and their implementation. We begin by highlighting the range (or minimum) of the Mendeleev number and the average deviation of the covalent radius (see Fig.​ 5 for values used in this work). These two features both appear consistently as features with high importance in several training iterations, and have physically interpretable meanings. The Mendeleev number provides an alternative label beyond atomic number to distinguish elements with shared characteristics⁵³. The Mendeleev number generally (but now always) increases down the columns of the periodic table, then increases from left to right. This ordering is intended to bunch elements with similar chemical properties together by an expected oxidation state in most materials. To understand how this can be useful, consider the $AB$O₃ perovskite oxides with $A$ a rare-earth and $B$ a transition metal. The minimum of the Mendeleev number will characterize the $A$ cation and the maximum of the Mendeleev number will characterize the O anion. Thus, the range of the Mendeleev number will be the difference between the Mendeleev numbers of the O anion and the $A$ cation. Since most compounds in our dataset are oxides with just a few being sulfides, the maximum of the Mendeleev number is effectively fixed for a substantial portion of our dataset with only the minimum of the Mendeleev number varying between different compounds. This aspect leads to high correlation between the minimum and range Mendeleev number features: the range and minimum of the Mendeleev number have a linear correlation of $-0.995$ and can be considered equivalent features for most of our dataset. We chose to use the range of the Mendeleev number simply as it takes the minimum of the Mendeleev number into account, which may be important for other types of compounds (such as sulfides), and the effect of choosing one over the other is negligible on the performance of our models.

The average deviation of the covalent radius (ADCR) describes how different the covalent radii are for different elements in a compound:

where $R_{i}$ is the covalent radius of the element $i$, $\bar{R}$ is the average covalent radius of all elements in a specific compound, and $N$ is the total number of different elements. For example in an $A_{n}$X_m compound,

with the weighted average $\bar{R}=({nR_{A}+mR_{X}})/({n+m})$. We use covalent rather than ionic radii as features, because while ionic radii are known to underlie structural stability in ionic crystals according to Pauling’s rules,⁵⁴ they rely upon knowledge of oxidation states and coordination environment. Further, some of our compounds exhibit non-integer oxidation states, or have oxidation states that are the subject of ongoing research (including many of the skutterudites), which means that including them would have introduced additional ambiguity into our model.

Now, we turn to some of the features for which we have built our own featurizers. We begin with the structural global-instability-index (GII) descriptor, which is defined as

where $d_{i}=\mathrm{BVS}(i)-V_{i}$ is the difference between the bond valence sum (BVS) for the $i^{th}$ ion and its formal valence and $N$ is the number of ions in the unit cell.^{55, 56} The GII is the root-mean-square deviation of the bond valence sums from the formal valence averaged over all atoms in the cell. This can be understood as approximating an average structural stress inherent in a material, as it captures the average deviation in bond lengths from what would be experimentally expected. The stresses arise from a combination of over- and underbonded cation-ligand interactions and thus describe bond strains compatible with a given structure and crystallographic symmetry.⁵⁷

Next, we constructed additional structural features, which in turn allowed us to build the electronic features from the Zaanen-Sawatzky-Allen (ZSA) framework. The structural features include: the minimum, maximum, and mean distances between transition metal $M$ cations; the minimum, maximum, and mean distances between the transition metal and the ligand $L$; and the Madelung site potentials for the cations and anions. These features were then used to calculate approximations⁵² for the relevant ZSA energy scales: the difference between the highest occupied and lowest unoccupied metal orbitals (an estimated Hubbard $U$, hereafter $U_{0}^{\prime}$) and a charge-transfer energy gap ($\Delta_{0}$). Both electronic parameters originate from an ionic model involving local charge excitations with

where $I_{v}=A$ is the electron affinity of $M^{v+}$, $I_{v+1}$ is the ionization potential, and $e^{2}/d_{M-M}$ is the Coulomb attraction between the excited electron on one transition metal cation and the hole left behind on its nearest-neighbor cation. The estimated charge-transfer energy is

where $e\Delta{}V_{M}$ is the product of the electron charge and the difference in electrostatic Madelung site potentials between the cation and ligand sites, $I(L^{n-})$ is the ionization potential of the ligand in the $-n$ oxidation state, e.g., $I(\mathrm{O}^{2-})=-7.7$ eV for oxygen ligands. $I_{v}$ is as before and $e^{2}/d_{M-L}$ is the Coulomb attraction between the excited electron on the metal and the hole left behind on its ligand. Code for calculating $U^{\prime}_{0}$ and $\Delta_{0}$ was adapted from Ref. 58 and is available in Ref. 41. The ionization potentials and electron affinity values were web-scraped from the NIST Atomic Spectra Database⁵⁹.

We first present performance results for our three binary classifier models, as they perform significantly better than a single ternary classifier (see Figure S1 of SI). Classifier M distinguishes between metals and non-metals, which include compounds labeled as insulators (I) or MIT compounds (T). Classifier I is the analogous classifier for insulators with non-insulators comprised of M and T compounds. Classifier T distinguishes MIT compounds from non-MIT compounds, i.e., metals and insulators.

All model performance metrics presented are from models trained on the reduced feature set. The corresponding metrics from models trained on the full feature set are available at Ref. 41. Except for the I classifier, which has a slightly worse performance for the one trained on the reduced feature set than that trained on the full feature set, the M and T classifiers are able to retain the same level of performance when trained on the reduced feature set compared to those trained on the full feature set (see Figure S2).

To visualize model performances, ROC curves for our machine learning models are constructed across 5 random cross-validation runs (Fig.​ 6) under 10 random seeds. The ROC curve depicts the true positive rate, which is the ratio between the number of correctly identified positive class (true-positives) and all of the positive class (the sum of true-positives and false-negatives), against the false-positive rate, or the ratio of false-positives to the sum of false-positives and true-negatives. An ROC curve confined primarily to the upper left corner is an indication of a model correctly identifying instances of each class without many false positives. An area under the ROC curve (AUC) of 1 represents perfect separation whereas an AUC of 0.5 is equivalent to random guessing. Tighter bunching of the lines indicates less variance in model performance with varying random seeds. We obtain a median ROC area under curve (ROC-AUC) of 0.90 and 0.89 with interquartile ranges of 0.02 and 0.01 for the M and I classifiers, respectively. Remarkably, our novel T classifier exhibits a median ROC-AUC of 0.90 with an interquartile range of 0.03, indicating its overall accuracy is high.

Fig.​ 7 presents the precision (proportion of true positives to the sum of true positives and false positives) and recall (proportion of true positives to the sum of true positives and false negatives) curves of each binary classifier to better understand performance owing to imbalance among the classes. The median and interquartile range of the cross-validation weighted F₁ scores (harmonic mean of precision and recall that takes class imbalance into account) are 0.86 (0.03), 0.82 (0.02), and 0.88 (0.01) for the M, I, and T classifiers, respectively.

On one hand, we find that the area under the precision-recall curve for Classifier T is rather low. Indeed, MITs are the least represented class in the dataset owing to the small number of known thermally-driven MIT materials. The poor precision-recall performance could perhaps be overcome with additional data as seen in other works (Table S2). As more positive examples (MIT compounds) are added to the dataset, we expect the precision of Classifier T to improve because the under-representation of MITs in the training set may lead to under-prediction of MITs, which results in a smaller number of true positives and thus a lower precision. On the other hand, the performance of Classifier M and I is comparatively better than that of Classifier T since the dataset contains more metals and insulators. As a result, the models were able to better separate metals from non-metals and insulators from non-insulators. In other words, these two classifiers exhibit better performance, because for them the ratio of positive class to negative class is more balanced than that of Classifier T.

Compared with electronic state classifiers formulated in earlier works using various databases and different descriptors (Table S2), our models’ performance metrics for the M and I models are comparable. However, we note that the previous models were not intended to learn correlated electron materials, and their metrics if trained on our sparse dataset would likely be different. Therefore, the comparison is not strictly appropriate.

We also created a survey that let domain experts (e.g., materials scientists) classify the conductivity class of 18 compounds (6 metals, 7 insulators and 5 MIT compounds). The goal was to establish a human performance baseline for the 3 aforementioned classifiers and to evaluate whether identifying MIT compounds is a trivial task for human scientists. Unsurprisingly, the XGBoost classifiers outperform the average human scientist in every classification task (see Figure S3 of SI).

We now use a combination of domain knowledge, SHAP values, and Accumulated Local Effect (ALE) analysis to examine the role of different features in the T Classifier trained on the reduced feature set. We want to know how the model learns to differentiate an MIT compound from those that are exclusively metallic or insulating. SHAP values indicate which features are important and the effects of the features on the classification (i.e., how does changing the value of a feature change the classification). ALE plots play a similar role in elucidating the importance of each feature in classifying a material, as well as the role of the interactions between the features. Some of the most illustrative ALE plots can be found in the SI. Fig.​ 8 shows the rank-order SHAP importance of each feature as well each feature’s relative role in the MIT compound classification (e.g., whether a material having a small value of GII makes it more likely to be classified as an MIT compound or not).

The most important feature in our classification is the average transition metal - transition metal distance, which we refer to as metal-metal distance for brevity, as shown in Fig. 8. The ALE decomposition shows that the importance of this feature largely arises from its interaction with other features. This finding agrees well with our physical intuition. In transition metal compounds, the metal-metal distance may determine the electronic bandwidth $W$, which is in competition with other energy scales such as the Hubbard $U$, as described for example in the ZSA-classification scheme, that drive MITs. Indeed, we show in a 2D scatter plot the distribution of materials as a function of the metal-metal distance and Hubbard $U$ and the ALE plot for these two features in Fig. 9. We find that most of the most well-studied MIT materials (mostly perovskite compounds) fall within a narrow region identified as high in probability based on these two features. Interestingly, for some materials for which the mechanism is largely unknown, (SmRu₄P₁₂ and PrRu₄P₁₂), but widely assumed to be different than that of most other MIT materials, these two features do not contribute to the MIT classification in any significant amount. This agrees with previous theoretical work, which has suggested that the relevant MIT physics for these materials is, in fact, not driven by the transition metal ion electrons at all, and is driven by the Pr and Sm f-electrons instead.⁶⁵

The SHAP feature importances further find the relevant features for the classification are the GII, the charge transfer energy, and the transition metal-ligand distances (see Figure S10 with SHAP plots for these two compounds in the SI), providing a possible hint as to the relevant physics in these materials. The GII and ADCR are two features that we find to be consistently important, and novel, in our model. The GII has previously been related to MIT temperatures in certain materials families, for example in the the $Rn$Cu₃Fe₄O₁₂. ⁶⁶ Fig.​ 10 shows most MIT compounds exhibit ADCR values between $30\,\mathrm{pm}$ and $50\,\mathrm{pm}$ and GII values between 0.1 and 0.5. The moderate to high GII values for most MIT materials are consistent with our understanding that these thermally-driven MITs are assisted by a minor structural instability, which can alleviate bond stresses. Materials with high GII values may be too chemically unstable to support this type of mechanism. We find that the MIT compound with the lowest GII is V₂O₃. Overall, a low GII tends to favor an insulating state, and a higher GII favors a metallic or MIT state. For example, among binary oxides with rutile structure and composition $M$O₂, we find $\mathrm{GII\,(TiO}_{2})=0.11$, $\mathrm{GII\,(VO}_{2})=0.13$, and $\mathrm{GII\,(MoO}_{2})=0.32$. As most of our compounds are oxides, and most of those are insulators with a low GII, we deduce that materials with a low GII are highly stable from a bond-stress assessment and are unlikely MIT compounds, which is consistent with the GII SHAP data in Fig.​ 8.

We glean some understanding of the ADCR relevance by focusing on the perovskite $R$NiO₃ nickelates with $R$ a rare-earth cation.²²2A similar analysis can also be done on other perovskite families such as $R$CoO₃, or $Rn$Cu₃Fe₄O₁₂, with $R$ defined as before and $Rn$ further includes Ho, Tb, Tm. In cubic perovskites, ABO₃, including the RNiO₃, the ADCR is linearly correlated to the Goldschmidt tolerance factor ${t}=({r_{R}+r_{\mathrm{O}}})/(\sqrt{2}(r_{B}+r_{\mathrm{O}}))$. This tolerance factor is known to be associated with whether $B$O₆ octahedral rotations are likely to occur and distort the ideal cubic perovskite structure ($t=1$)⁶⁷. For $t<1$, the transition metal-oxygen octahedra rotate, making it more difficult for electrons to hop and favoring an MIT. Thus, a lower tolerance factor usually leads to a higher MIT temperature, while a higher $t$ can suppress MIT behavior altogether (e.g., LaNiO₃ is metallic). The ADCR plays a more important role in supporting the MIT classification for LuNiO₃ (lower ADCR) than it does for NdNiO₃ (higher ADCR), capturing the physical trend in the phase diagram in Fig.​ 11: the ADCR places NdNiO₃ close to metallic LaNiO₃, while it places LuNiO₃ significantly further away towards high $T_{\mathrm{MIT}}$. This physics is captured in the SHAP values in Fig.​ 12: NdNiO₃ is one of the nickelates with the highest ADCR and the lowest $T_{\mathrm{MIT}}$, making it close to the metallic class as identified by the model with a log-odds ratio of 5.08. In contrast, LuNiO₃ with an ADCR lower than NdNiO₃ has a 7.22 log-odds ratio. In other words, the classifier is more certain that LuNiO₃ is an MIT compound than it is for NdNiO₃. The ADCR is thus similar to a generalized tolerance factor, irrespective of the materials family studied. We note that the ADCR is also strongly correlated to the average deviation of electronegativity with a linear correlation of 0.919 (see Fig.​ 16), which we understand as a consequence of the electron affinity of an element being partially determined by its atomic radius.

Although we find the estimated Hubbard $U$ values $U_{0}^{\prime}$ and charge-transfer energies $\Delta_{0}$, which are important to the ZSA classification of correlated metals and insulators, are among the top 8 features, the MIT materials do not strongly cluster when they are plotted in a 2D space consisting of these features (Fig.​ 13). The GII, ADCR, and the range of the Mendeleev number (discussed later) lead to much stronger class clustering (or separation) than the ZSA-classification energies $U_{0}^{\prime}$ and $\Delta_{0}$, which have been used extensively over the last 30 years. The Hubbard $U$ is a strong counter-indicator of an MIT when large and somewhat less strongly predictive of an MIT when low, which gives some support to the findings in Ref. 52. The presence of materials with extremely high $U_{0}^{\prime}$ values arising from high ionization energies, e.g., titanates, distorts the SHAP color scale for the Estimated Hubbard $U$ row in Fig.​ 8. High values of $\Delta_{0}$ may also indicate that no MIT occurs, although sometimes the SHAP value of such a material is close to 0. The color scale for the charge transfer energy is also skewed by the presence of negative $\Delta_{0}$ values. These occur when the difference in metal and anion Madelung site potentials is small and/or the ionization energy of the metal is large. One reason that the ZSA classification energies may be difficult to interpret is that the energy estimates we use correspond to unscreened values. Dielectric screening and metal-ligand hybridization effects in solids can lead to significant renormalization of these values, but require electronic-structure based calculations such as cRPA ⁶⁸ to ascertain. Thus, although these numbers provide some information to our machine learning model, they are difficult to understand in isolation.

Two additional features of high importance include the average metal-metal distance and the average metal-nonmetal (metal-ligand) distance. We understand their role in relation to the energies $U_{0}^{\prime}$ and $\Delta_{0}$ within the ZSA framework, which are often scaled relative to the electron-hopping parameters describing correlated electron materials in the form of the d-orbital bandwidth in transition metal compounds. As the d-orbital bandwidth in the low-energy electronic structure is not directly available from the structure alone, a convenient proxy is the metal-metal distance and/or the metal-anion distance as the bandwidth is inversely proportional to the distance between the atomic pairs contributing states that hybridize by symmetry. This explains the strong role of inter-atomic distances in our model.

The Ewald energy reflects how stabilizing the ionic charge distribution in a crystal is due to the electrostatic potential imposed by oppositely charged ions in the atomic structure. The calculations as currently performed in the latest version of Matminer correspond to an electrostatic energy between ions modeled as point charges, with the charge approximated by the nominal ionic oxidation state. This feature, as implemented in Matminer has recently been updated to be normalized per atom. This Ewald energy per atom then, to first order, separates highly ionic materials from less ionic materials. Phosphates such as CoP₃ (which has an Ewald energy of $-124\,\mathrm{eV/atom}$) have strong ionic character in this picture, with P having a $3-$ charge, and Co a $9+$ charge, while sulfides tend to have lower absolute values of the Ewald energy (FeS₂ has an Ewald energy of $-10\,\mathrm{eV/atom}$). Fig.​ 15 shows oxides that exhibit intermediate values.

This separation based on the anion Ewald energy is similar to that based on the maximum Mendeleev number, which may explain why the maximum of the Mendeleev Number (which describes the anion in our compounds) does not have a role in our classification according to our SHAP scores whereas the range of the Mendeleev number is much more important. The range of the Mendeleev number essentially separates compounds that contain elements from the first three columns of the periodic table or from the lanthanide or actinide series (such as LaNiO₃ or EuO), from those that are binary transition metal compounds (such as FeS or NiO). The importance of this feature is clearer to discern through its interaction with other features, whereas the Ewald energy alone leads to a clear separation based on composition, particularly based on the anion type. From the combination of ADCR and the range of the Mendeleev number, we find strong clustering of MIT materials (Fig.​ 14). Particularly, a range of the Mendeleev number over 40 and an ADCR below 50 are likely an indicator of an MIT material. We also find that most known MIT compounds tend to be oxides containing yttrium or a lanthanide in their composition (as highlighted by the green dots enclosed in horizontal ellipses in Fig.​ 14).

Although the Ewald-energy-SHAP scores in Fig.​ 8 are difficult to interpret for the entire database as a whole, they are easier to understand if we focus on a particular family. Consider the V_nO_m binary vanadium oxide family. We find that V₂O₅ with an Ewald energy of $-57\,\mathrm{eV/atom}$ is insulating, while VO with an Ewald energy of $-24\,\mathrm{eV/atom}$ is metallic. The V₆O₁₃, VO₂, V₈O₁₅, V₆O₁₁, V₅O₉, V₄O₇, V₃O₅ and V₂O₃ vanadates exhibiting intermediate Ewald energies are all MIT compounds. This observation strongly suggests that the Ewald energy of the MIT compounds within a particular materials family is likely to lie in between that of the metallic and insulating members in that family. A similar analysis can be performed for the Ti_nO_m family (Fig.​ 15).

The Ewald energy is less useful, however, for differentiating materials with the same stoichiometry and structure type but comprised of different chemistry: LaNiO₃ has an Ewald energy of $-33\,\mathrm{eV/atom}$ and LuNiO₃ has an Ewald energy of $-34\,\mathrm{eV/atom}$ despite the very large differences between the two as illustrated in Fig.​ 11. As the materials classes have varying ranges of Ewald energy, it is then difficult to understand its role from the SHAP plot alone without focusing on a particular family.

Finally, we also included the average deviation of the electronegativity as a feature for our classifier. Although it is strongly linearly correlated with the ADCR, we surprisingly find that there is a strong clustering of the MIT compounds within this 2D feature space (Fig.​ 16).

Our pre-trained electronic classifiers are deployed and served to the larger materials science community through a cloud service called Binder⁶⁹. There are several Jupyter notebooks⁷⁰ hosted on the Binder server that may be used to easily reproduce the results presented herein. The Binder website offers interactive execution of these notebooks directly in a web browser without installing any dependency onto a local machine. All notebooks are also available at Ref. 41.

Here we present a brief demonstration of the Binder notebook at https://tinyurl.com/mit-classifiers, which enables a user to upload a structure in CIF format and immediately make a classification using our models, without the need to install any software. After uploading a lacunar spinel structure GaMo₄Se₈, which was identified computationally as a potential MIT material⁹, the notebook automatically featurizes the new structure and makes a prediction. Then after executing through the three binary classifiers trained on the reduced feature set, GaMo₄Se₈ is classified as an MIT material (Fig.​ 17). Note among the three possible conductivity classes, the assigned classifications may not be mutually exclusive since a single ternary classification is not made. The classifiers predict the following probabilities: 0.4761 for being a metal; 0.0479 for being an insulator; 0.5441 for being an MIT compound. The default threshold for making a positive classification is 0.5, which means a positive classification is made only if the predicted probability is greater than 0.5. In this case, GaMo₄Se₈ is predicted only to be an MIT compound. However it is also worth noting that although GaMo₄Se₈ does not obtain a positive metal classification, its probability for being a metal is 0.4761, which is close to 0.5.