Executive Summary
The organization of polymorph energy landscapes — not merely the count of structures — determines phase accessibility and synthesis windows in materials design. This framework quantifies landscape topology using high-throughput analysis of 700,000+ polymorphs from the AFLOW repository across seven anion families: oxides, sulfides, carbides, nitrides, phosphides, borides, and halides.
Two topology descriptors are introduced: (1) the tailness index T quantifying the extension of high-energy states, and (2) the compression exponent β from power-law scaling ΔEspan ∝ (1 + N50)^−β. Analysis reveals three distinct landscape organizations with clear predictive implications for synthesis accessibility.
Novel Descriptors
Quantifies how energetic dispersion evolves with increasing near-ground-state crowding. High β (oxides, nitrides ≈ 3) indicates strong compression — low-energy polymorphs suppress the global energy span. Low β (borides < 1) indicates rigid, incompressible landscapes.
Captures the relative extension of high-energy polymorphs beyond the near-ground-state core. T > 1 indicates an extended metastable tail. Minimal sensitivity to compositional filtering — reflects intrinsic topological property of polymorph landscapes.
Together, β and T encode complementary and non-redundant aspects of landscape structure. β describes compression efficiency of the core; T describes the extension of the metastable periphery. Families with comparable β may differ substantially in T, as observed in the oxide–nitride comparison.
Compression Exponent β — Interactive
Hover each bar to see the compression exponent and fit quality (R²) for each anion family. The hierarchy is monotonic: Oxides ≈ Nitrides > Carbides > Halides > Sulfides > Phosphides > Borides.
| Anion Family | β (compression exponent) | R² | Landscape Type | Interpretation |
|---|---|---|---|---|
| Oxides | ≈ 3.0 | ≈ 0.40 | Continuous | Strong crowding–flattening behavior |
| Nitrides | ≈ 3.0 | ≈ 0.38 | Continuous | Comparable to oxides, ionic coordination |
| Carbides | ≈ 2.4 | ≈ 0.28 | Intermediate | High β but low T — rigid core |
| Halides | ≈ 2.0 | ≈ 0.22 | Fragmented | Sensitive to mixed-anion filtering |
| Sulfides | ≈ 1.8 | ≈ 0.18 | Fragmented | Weak, incoherent scaling |
| Phosphides | ≈ 1.6 | ≈ 0.15 | Fragmented | Low compression, decoupled topology |
| Borides | < 1.0 | < 0.10 | Rigid | No power-law scaling — static landscape |
Tailness Index T
The tailness index T is robust to compositional filtering — unlike β, which decreases 17–28% after strict single-anion filtering, T changes less than 2% for phosphides, borides, and halides. This indicates that tail organization is an intrinsic topological property, not a compositional artifact.
Landscape Classification
Combining β and T reveals three characteristic landscape types. This classification is robust with respect to dataset size and filtering strategy.
High β, moderate-to-high T, and strong scaling coherence. Smooth energetic gradients connect polymorphs. Proliferating low-energy states drive coordinated compression of the global energy span. Core and tail are mechanically coupled.
Intermediate β, moderate T, and reduced scaling coherence. Clustered low-energy states separated from sparse high-energy configurations. Phase purity intrinsically difficult to maintain — "leaky" core landscape.
Low β, low T, and absence of clear scaling. Structurally constrained, strongly bonded frameworks. Minimal crowding–flattening response. Directional covalent bonding prevents tail formation.
Effect of Strict Single-Anion Filtering
Strict compositional filtering (removing mixed-anion compounds) reveals that mixed-anion chemistry contributes disproportionately to apparent compression efficiency. Despite representing only 10–20% of polymorph entries, mixed-anion compounds substantially inflate β in phosphides, borides, and halides.
| Family | β (Unfiltered) | β (Strict) | Change | T Change | Interpretation |
|---|---|---|---|---|---|
| Oxides | ≈ 3.0 | ≈ 2.95 | −2% | <2% | Dominated by intrinsic anion properties |
| Nitrides | ≈ 3.0 | ≈ 2.92 | −3% | <2% | Robust — ionic landscape character |
| Carbides | ≈ 2.4 | ≈ 2.1 | −13% | <2% | Moderate mixed-anion contribution |
| Sulfides | ≈ 1.8 | ≈ 1.4 | −22% | <2% | Mixed-anion enhances flexibility |
| Phosphides | ≈ 1.6 | ≈ 1.2 | −25% | <2% | Compositional mixing critical |
| Halides | ≈ 2.0 | ≈ 1.5 | −25% | <2% | Mixed-anion drives 25–28% of β |
| Borides | < 1.0 | < 0.85 | −17% | <2% | Rigid regardless of filtering |
Implications for Materials Discovery
The β–T framework provides a practical synthesis heuristic. Families with high β exhibit efficient energy compression — controllable phase selection via temperature or pressure tuning is feasible. Low-β families show limited tunability through polymorph crowding alone.
| Landscape Type | β–T Signature | Synthesis Implication | Key Families |
|---|---|---|---|
| Coherent | High β, moderate-high T | Metastability accessible but competitive — precise energy control required | Oxides, Nitrides |
| Decoupled | Low β, moderate T | Phase purity intrinsically difficult — high-energy metastable phases unexpectedly synthesizable | Sulfides, Phosphides |
| Rigid | Low β, low T | Limited structural diversity — phase selection constrained by covalent framework | Borides, Carbides |
All analysis codes and processed datasets are released open-source to enable community validation and extension to additional material families.
Framework code and processed datasets available on GitHub. Cross-database validation using Materials Project (MD-6K sulfide validation set) confirms robustness of topological classification.