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Materials Cartography: Representing and Mining Material Space Using Structural and Electronic Fingerprints
Olexandr Isayev, Denis Fourches, Eugene N. Muratov, Corey Oses, Kevin Rasch, Alexander Tropscha, Stefano Curtarolo
TL;DR
Materials databases are growing faster than the knowledge extracted from them, motivating computational methods for scientific discovery. The paper combines structural and electronic fingerprints with similarity searches, graph-based materials cartography, and QMSPR modeling, demonstrating the framework on superconducting critical temperatures. Fingerprints support qualitative clustering, while adapted SiRMS descriptors provide quantitative models.
Problem
The paper addresses the widening gap between accumulated materials information and derived knowledge in rapidly growing databases.
Method
The framework combines structural and electronic materials fingerprints with similarity searches, graph theory, materials cartography, and machine-learning QMSPR models.
Results
The continuous SiRMS model achieved Q2 = 0.66, while the classification model achieved balanced accuracy of 0.97 under five-fold external cross-validation.
Takeaways & Limitations
Materials cartograms identify regions with distinct physical and chemical properties, while fingerprints and QMSPR models support searching and modeling materials with desired properties.
Takeaways & Limitations
B- and D-fingerprint QMSPR models were unsatisfactory for quantitative prediction, so more sophisticated SiRMS descriptors were required.
Abstract
from arXiv · showhide
As the proliferation of high-throughput approaches in materials science is increasing the wealth of data in the field, the gap between accumulated-information and derived-knowledge widens. We address the issue of scientific discovery in materials databases by introducing novel analytical approaches based on structural and electronic materials fingerprints. The framework is employed to (i) query large databases of materials using similarity concepts, (ii) map the connectivity of the materials space (i.e., as a materials cartogram) for rapidly identifying regions with unique organizations/properties, and (iii) develop predictive Quantitative Materials Structure-Property Relation- ships (QMSPR) models for guiding materials design. In this study, we test these fingerprints by seeking target material properties. As a quantitative example, we model the critical temperatures of known superconductors. Our novel materials fingerprinting and materials cartography approaches contribute to the emerging field of materials informatics by enabling effective computational tools to analyze, visualize, model, and design new materials.
Introduction
Materials research faces a widening gap between accumulated data and derived knowledge, motivating materials-informatics methods for navigating, comparing, and designing materials. The paper introduces fingerprints combined with similarity searches, graph theory, and machine learning to characterize and explore materials space.
- Motivation: Materials research is accumulating vast datasets, including more than 160,000 ICSD entries, while desired-property design remains challenging.Properties depend on constitutive, crystallographic, geometrical, and electronic variables.
- Motivation: Materials-informatics methods adapt data-mining and visualization approaches to define, visualize, and navigate materials space.The paper situates these approaches alongside successful cheminformatics applications.
- Approach: The proposed framework combines materials fingerprints with graph theory, similarity searches, and machine learning.Fingerprints encode band structures, density of states, crystallographic information, and constitutional information.
- Approach: Materials cartography visualizes materials space and identifies clusters of materials with similar properties.The framework is intended to support characterization, comparison, visualization, and design.
- Application: Fingerprint-based QMSPR models are used to discover materials with desired properties in databases.The paper applies this modeling framework to target material properties.
Methods
AFLOWLIB provides a large repository of density-functional-theory calculations for crystal structures, generated with AFLOW and VASP-based computational settings.
- AFLOWLIB: AFLOWLIB contains calculations characterizing over 20,000 crystals, approximately one quarter of the ICSD.Roughly half of the characterized systems are metallic and half insulating.
- Computational setup: AFLOWLIB is managed by AFLOW and uses VASP with PAW pseudopotentials and the PBE exchange-correlation functional.The calculations determine crystal total energies.
Data set of superconducting materials
The superconductivity dataset was curated from multiple experimental sources and filtered to standard-pressure, zero-field records with more reliable critical-temperature values. Separate continuous, classification, and structural datasets support complementary predictions of superconducting critical temperature.
- Data curation: More than 700 superconductivity records were compiled from three experimental sources before curation.Sources included the Handbook of Superconductivity, CRC Handbook of Chemistry and Physics, and SuperCon Database.
- Data curation: Only records measured at standard pressure without external magnetic fields were retained.Records with unreliable high variability were discarded, while variations below 3 K were averaged.
- Curated dataset: The curated dataset contains 465 materials spanning Tc values from 0.1-133 K.Most records vary by ±1 K between sources.
- Model datasets: The continuous dataset contains 295 unique materials with log(Tc) ranging from 0.30-2.12.It excludes materials with Tc values below 2 K and predicts Tc continuously.
- Model datasets: The classification dataset contains 464 materials divided by the 20 K threshold, including 29 above and 435 at or below it.Lanthanum cuprate was included despite previously discarded variability because it satisfies the classification criteria.
- Model datasets: The structural model uses the continuous-model dataset to identify structural components influencing Tc.The classification and continuous models address threshold position and quantitative Tc, respectively.
Materials fingerprints
The paper represents materials through structural and electronic fingerprints that encode composition, topology, geometry, band structure, and density of states. These numerical descriptors support similarity-based analysis and machine-learning models across materials.
- Descriptor basis: Materials are assumed to have properties determined by structure, with structurally similar materials likely to share physical-chemical properties.Similarity considers constitutional, topological, spatial, and electronic characteristics.
- Electronic fingerprints: Electronic structures are encoded as symmetry-dependent B-fingerprints from band structures and symmetry-independent D-fingerprints from density of states.These arrays enable cheminformatics and machine-learning approaches for mining, visualization, and modeling.
- Electronic fingerprints: B-fingerprints discretize energies at high-symmetry Brillouin-zone points into 32 bins.For a cubic lattice, four encoded points produce an array of length 128.
- Electronic fingerprints: D-fingerprints sample density-of-states diagrams in 256 bins with 32-bit magnitudes, totaling 1024 bytes.Only symmetry-independent D-fingerprints were generated because B-fingerprints are more complex and limited by symmetry dependence.
- SiRMS descriptors: SiRMS descriptors characterize materials through compositional, topological, and spatial characteristics based on simplex fragments.The approach was modified from small-molecule descriptors to handle materials and mixtures.
- SiRMS descriptors: Material SiRMS descriptors count simplex-fragment occurrences across crystal-unit-cell constituents while incorporating stoichiometric ratios and lattice structure.Mixture descriptors are weighted by the smallest stoichiometric ratio, and unbounded simplexes can span up to four components.
Network Representation (Material Cartograms)
The materials library is represented as a similarity network in which fingerprint-encoded materials form nodes and sufficiently similar pairs form edges. Degree distributions are analyzed to assess whether these networks are scale-free.
- Each material is encoded as a fingerprint-based node, with edges linking pairs whose Tanimoto similarity reaches the threshold S = 0.7.The network is defined as G(V, E), where edges connect materials with similarity sim(ν1, ν2) ≥ T.
- Degree distributions are examined to determine whether material networks follow a power-law connectivity pattern characteristic of scale-free networks.The tested form is p(x) = kx−α, with k as the normalization constant and α as the exponent.
Similarity search in the materials space
Fingerprint representations support similarity searches across large materials databases, retrieving compositionally different materials with similar properties and identifying duplicate records. Demonstrative searches cover electronic, ferroelectric, and topological-insulator cases, while systematic DFT errors limit direct property accuracy but may preserve similarity relationships.
- Fingerprint construction: Electronic structures from AFLOWLIB are converted into symmetry-dependent B-fingerprints and symmetry-independent D-fingerprints for materials comparison.B-fingerprints sample band energies at high-symmetry reciprocal points, whereas D-fingerprints encode DOS strengths across 256 bins spanning [-10, 10] eV.
- Similarity search: Fingerprint similarity searches retrieve materials with similar properties but different compositions and rapidly identify duplicate records.Identical fingerprints were found for several BaTiO3 records, including ICSD #15453, #27970, #6102, and #27965.
- Scope and caveat: Standard DFT has severe limitations for excited states, but comparable errors among similar systems may remain irrelevant when the goal is only material similarity.This caveat particularly concerns the characterization of semiconductors and insulators.
- Similarity search: The five materials retrieved as most similar to GaAs using D-fingerprints are GaP, Si, SnP, GeAs, and InTe.GaAs served as the reference against a virtual screening library of more than 20,000 AFLOWLIB materials.
- Similarity search: Among six BaTiO3 materials with S > 0.8, five are known for optical properties and cubic YbSe is largely unexplored.The B-fingerprint search used BaTiO3 with the perovskite structure as the reference material.
- Similarity search: Topological insulators showed exceptionally high band-structure similarities despite most AFLOWLIB DFT calculations omitting spin-orbit coupling.The example is presented as evidence that B-fingerprint searches can identify related materials in this challenging class.
- Implications: The examples support fingerprint-based similarity searches as rapid tools for identifying materials with similar properties in large databases.The paper connects fingerprint similarity with shared properties such as ferroelectricity or insulating behavior.
Visualizing and exploring the materials space
Materials cartograms and fingerprint networks organize the materials space by complexity, connectivity, electronic character, and superconducting critical temperature. These maps expose compositional regions, communities, highly connected materials, and a compact high-Tc region dominated by layered cuprates.
- Materials cartograms: The D-fingerprint cartogram maps materials by unit-cell complexity, with unary systems confined to a small region and quaternaries concentrated farthest from unary materials.Binary materials occupy a compact region, ternaries mainly populate the center, and higher-complexity compounds lie toward the top of the network.
- Network connectivity: The connectivity distribution exhibits a power-law form, while the most connected D-fingerprint nodes include many bimetallic and polymetallic materials.Al3FeSi2 has the highest reported D-fingerprint connectivity, with connectivity 946.
- Electronic communities: The B-fingerprint network separates metals from insulators and identifies four large material communities.Group-A contains approximately 3000 materials, Group-B approximately 2500, Group-C approximately 500, and Group-D includes approximately 300 small-band-gap materials plus approximately 500 semimetals and semiconductors.
- Electronic communities: Group-A consists predominantly of insulating compounds and semiconductors, whereas Group-B is dominated by polymetallic materials.Group-A is 63% insulating compounds and 10% semiconductors; Group-B is 70% polymetallic among approximately 2500 materials.
- Network connectivity: Lithium Scandium Diphosphate has the highest B-fingerprint connectivity at 746, and the B-fingerprint connectivity distribution also follows a power law.Highly connected materials are nearly evenly distributed between Groups A and B, forming dense central clusters.
- Superconductivity: The superconductivity map places all high-Tc superconductors in a compact region centered on Ba2Cu3XO7 layered cuprates.The two highest-Tc materials in the set are Ba2Ca2Cu3HgO8 with Tc =133 K and Ba2CaCu2HgO6 with Tc =125 K.
- Superconductivity: All top 15 high-Tc superconductors are layered cuprates with conserved band features near -6, -1, and 4 eV relative to the Fermi energy at Γ.These compounds are categorized as Charge-Transfer Mott Insulators within the described DFT+U picture.
- Superconductivity: The 15 materials with the lowest Tc have a random fingerprint distribution, contrasting with the localized high-Tc region.The paper uses this contrast to visualize the relationship between band-structure features and superconductivity.
Predictive QMSPR Modeling
The study develops continuous, classification, and structural QMSPR models for superconducting critical temperatures using SiRMS descriptors and fingerprint-based materials representations. Classification was highly accurate, while quantitative modeling required more detailed structural descriptors and remained subject to data and material-specific limitations.
- Descriptor choice: B- and D-fingerprints were effective for qualitative clustering but insufficient for quantitative prediction, motivating the use of adapted SiRMS chemical-fragment descriptors.The fingerprint-based QMSPR attempts for both datasets were not satisfactory.
- Continuous model: Q2 = 0.66 for the continuous model using a consensus RF- and PLS-SiRMS approach with five-fold external cross-validation.Materials with log(Tc)≤1.3 were scattered but within the correct range, whereas higher values generally received greater accuracy.
- Classification model: BA = 0.97 for the classification model predicting whether Tc exceeded the 20 K threshold under five-fold external cross-validation.The model used the RF-SiRMS technique and classified Tc as above or below Tthr.
- Classification model: 98% and 90% accuracies were obtained for Tc ≤Tthr and Tc > Tthr, respectively, yielding 94% cumulative accuracy.Ten high-Tc materials were predicted below the threshold, while two low-Tc materials were predicted above it.
- Structural model: Q2 = 0.61 for the structural model using SiRMS descriptors, PLS, and five-fold external cross-validation.The model converts descriptor contributions into atomic contributions related to material structures.
- Structural interpretation: Atomic contributions to Tc were nonlocal because they depended strongly on the surrounding atomic environment and could change substantially after substitution.Mo6PbS8 and Mo6NdS8 differed by a substitution while showing substantially different Tc values.
Conclusion
The paper presents fingerprinting and materials cartography as data-analysis tools for navigating materials databases and relating structure to physical properties. Applied to AFLOWLIB data and superconductivity, the framework supports clustering, property modeling, and exploration of candidate materials, while remaining an initial demonstration.
- Conclusion: The work addresses the widening gap between accumulated materials information and derived knowledge as high-throughput research expands.It motivates adapting data-analysis approaches from cheminformatics and bioinformatics.
- Materials cartography: Materials cartograms represent compounds as nodes and similarities as connections, revealing regions with distinct physical and chemical properties.These regions are intended to help search for interesting, previously unknown compounds.
- Framework: The framework combines atomic composition, bond topology, structure geometry, and electronic properties from AFLOWLIB with materials fingerprints and cheminformatics models.Band-structure and DOS fingerprints can locate metals, semiconductors, topological insulators, piezoelectrics, and superconductors.
- QMSPR modeling: More complex QMSPR models address qualitative and quantitative superconducting critical temperatures and geometrical features that help or hinder criticality.The approach includes SiRMS descriptors adapted for materials.
- Conclusion: The fingerprinting cartography demonstrates utility on an initial set of problems and supports insight into relationships between material structure and physical properties.The authors identify further database analysis and exploration as a possible foundation for rational materials design.