Transferable Energy Model for Defect Formation

• The transferable energy model for defect formation is a comprehensive framework that integrates elastic, chemical, electronic, and topological aspects to predict defect energetics in various materials.
• It employs first-principles calculations, machine learning, and thermodynamic integration to quantify defect formation energies, activation barriers, and finite-temperature effects.
• The approach facilitates high-throughput screening and materials design by accurately accounting for local disorder, rehybridization, and electronic structure nuances.

A transferable energy model for defect formation is a set of theoretical and computational frameworks that link atomic-scale mechanisms of defect creation in solids to macroscopic material properties, enabling quantitative and predictive evaluation of defect energetics across materials families, structural motifs, and environmental conditions. Such models are designed to capture essential dependencies on bulk elasticity, chemistry, local disorder, electronic state, external stimuli, and finite-temperature effects, thus facilitating reliable simulation, high-throughput screening, and integration with large-scale thermodynamic or process models. Recent developments incorporate first-principles electronic structure calculations, machine learning, advanced thermodynamic methods, and explicit treatment of real-world complexities such as chemical disorder, compartmentalization of entropic contributions, and metastable defect configurations.

1. Bulk Property-Based Models

Historical and modern approaches posit a direct connection between bulk elastic parameters and defect thermodynamics. The cBΩ (“cBQ”) model expresses the defect formation free energy $g'_i$ in terms of the bulk modulus $B$, mean atomic volume $\Omega$, and a dimensionless process-dependent constant $c'_i$:

$g'_i = c'_i B \Omega.$

Thermodynamic derivatives yield entropy, enthalpy, and volume of defect formation, linking experimental observables (e.g., pressure and temperature derivatives of $B$, volume expansivity) to defect energetics:

$$s'_i = -c'_i \Omega \left(\beta B + \frac{dB}{dT}\right), \quad h'_i = c'_i \Omega \left[B - T\left(\beta B + \frac{dB}{dT}\right)\right],\ v'_i = c'_i \Omega \frac{dB}{dP}.$$

This framework has been validated across metals, ionic solids (including superionic conductors), high-$T_c$ superconductors, and glass-forming liquids, and is notably superior to earlier Zener-type models, which relate $g'_i$ to the shear modulus $u$ and fail to reproduce key empirical trends in defect concentrations, diffusion activation energies, and the anomalous pressure/temperature dependence of ionic conductivity (Varotsos, 2017, Varotsos, 2017).

2. Microscopic and Electronic Structure Corrections

For specific materials where local bonding rearrangement and curvature effects are dominant, such as carbon nanotubes, Stone–Wales (SW) defect energetics are governed by local curvature-induced rehybridization. Here, both the formation energy $E_f$ and kinetic activation barrier $E_a$ depend on the tube’s chirality, diameter, and defect orientation. Critical findings include:

• $E_f$ and $E_a$ both increase with diameter, converging to flat graphene limits (≈4.96 eV and ≈9.26 eV, respectively).
• Defect orientation strongly modulates energetics, with lattice symmetry-dependent variations.
• There exists an empirical Brønsted–Evans–Polanyi (BEP)-type linear correlation: $E_a = k_1 + k_2 E_f$ (reported fit: $E_a = (4.19\pm0.15\,\text{eV}) + (1.05\pm0.04)E_f$).

These relations grant predictive power based on easily calculated or measured quantities and reveal local physical origins (rehybridization quantified by a parameter $\tau$). Substitutional heteroatom doping (e.g., B, N, S) offers a mechanism to manipulate kinetic barriers by locally reducing bond strength and thus dramatically enhancing defect formation rates (Kabir et al., 2016).

3. Statistical and Computational Thermodynamics of Defect Formation

Recent models rigorously integrate configurational, vibrational, electronic, spin, and orientational entropic contributions into the finite-temperature free energy of defect formation,

$g_{f,P}(T) = h_{f,P}(T) - T s_{f,P}(T),$

where the entropy is constructed from harmonic/quasiharmonic phonons, statistical mechanics of configuration (including mixing and symmetry breaking), and Fermi–Dirac carrier statistics. Advanced methods employ thermodynamic integration (TI) to capture anharmonic free energy corrections:

$$\Delta A = \int_0^1 \langle \partial U(\lambda) / \partial \lambda \rangle_\lambda d\lambda,$$

connecting a harmonic reference to the anharmonic, defected system (Mosquera-Lois et al., 2023, Mosquera-Lois et al., 21 Dec 2024).

Machine learning force fields (MLFFs) are now routinely trained on first-principles energies, forces, and virials, enabling large-scale molecular dynamics sampling of vibrational, orientational, and configurational defect space with meV accuracy (Mosquera-Lois et al., 21 Dec 2024). These approaches capture the essential role of finite-T effects, which can lead to order-of-magnitude changes in defect solubilities and even shift dominant defect species.

4. Local Chemical Environment and Disorder Effects

For chemically disordered or multi-component systems, such as (U,Pu)O₂ mixed-oxide nuclear fuels, defect formation energies are not uniquely defined but depend on the local arrangement of, e.g., U and Pu cations surrounding the defect. Systematic sampling of all local configurations within a cutoff radius $R_D$ (here $\approx7$ Å, corresponding to three coordination shells) reveals a spread in defect formation energies up to nearly 1 eV. A concise interaction model based on the counts of U/Pu in defined shells enables analytical prediction of the full defect energy landscape:

$$EF(\text{BSD}) = A + A_0 N_1 + A_1 N_1' + A_{12} (N_1 N_1') + A_2 N_2 + A_{22} (N_2)^2,$$

where $N_j$ are the compositional counts and the $A$’s are fitted parameters (2206.13150). This methodology, validated by DFT+U benchmarking, demonstrates transferability across local compositional variations.

5. Electronic Structure Peculiarities and Defect Formation in Metals and Low-D Systems

In “gapped metals”—materials with a Fermi level at a band edge and a large internal band gap—defect formation can significantly alter band occupancy and the Fermi level. Accurate evaluation requires explicit band-filling corrections, potential alignment between supercells, and systematic size/k-point convergence studies:

$$E^{\text{corr}}_{\text{band filling}} = - \sum_{n,\mathbf{k}} w_{\mathbf{k}} [\theta(e_{n,\mathbf{k}}-E_F) Y_{n,\mathbf{k}} (e_{n,\mathbf{k}}-E_F) + \theta(E_F-e_{n,\mathbf{k}})(1-Y_{n,\mathbf{k}})(E_F-e_{n,\mathbf{k}})],$$

ensuring recovery of the dilute limit and robust quantitative predictions (Gopidi et al., 2023).

For low-dimensional semiconductors, conventional jellium background approaches to charged defect energetics lead to divergence as vacuum regions grow. A charge density correction replaces the unphysical uniform background with a band-edge-localized density:

$A_p = P_{\text{bg}} - P_{\text{edge}},$

yielding physically meaningful and converged formation and ionization energies (Zhu et al., 2020).

6. Integration with Thermodynamic Modeling (CALPHAD and DEF)

The Defect Energy Formalism (DEF) is a first-principles-based framework that reconciles microscopic defect energies with the CALPHAD Gibbs energy approach. DEF achieves explicit mapping between absolute defect formation energies (obtained from DFT or surrogate models) and the Gibbs energy parameters used for phase equilibria:

• Linear projection of “absolute” defect energies defines end-member Gibbs energies.
• A superposition principle enables multi-defect energies to be constructed as sums of individual defect contributions.
• The DEF constitutional space yields a unified platform for both chemically neutral and charged defects by including an auxiliary carrier sublattice.
• This approach abolishes the exponential parameter complexity of the Compound Energy Formalism (CEF) and eradicates the need for empirical model fits for each compositional/charge configuration (Movaffagh et al., 17 Jul 2024).

7. Machine Learning and Topological Descriptors

Recently, persistent homology features—topological invariants that encode the structural environment around defects—have been integrated with graph neural networks (GNNs) to create highly accurate, scalable, and supercell-converged defect energy prediction models. By constructing atom-specific persistent homology (ASPH) descriptors (summarizing Betti number statistics via min/max/mean/std/weight), and employing pooling operations (notably global max pooling), the resulting models achieve dramatic error reduction (e.g., mean absolute error reductions of 55% for O-based perovskites) and accurate resolution of defect–defect interactions (Fang et al., 6 Jul 2024).

This class of machine learning models, in conjunction with targeted DFT datasets or advanced MLFFs, enables rapid high-throughput screening of defect thermodynamics in technologically relevant classes of materials where complexity, disorder, and large databases present critical challenges.

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Summary Table: Key Paradigms in Transferable Defect Formation Energy Models

Model/Framework	Core Paradigm	Transferability Features
cBΩ/cBQ	Bulk moduli, volume control	Links to B, Ω; applies to many solids
Curvature-induced	Local orbital rehybridization	Predicts diameter, orientation trends
DFT/MLFF + TI	Thermodynamic integration	Captures finite-T, anharmonic, metastability
Defect Energy Formalism (DEF)	Explicit mapping to CALPHAD	General to multi-component, charged systems
Persistent homology + GNN	Local topology descriptors	High-throughput, scalable, supercell converged
First-principles corrections	Band-filling, jellium correction	Key for low-D, gapped metals, charged defects

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The transferable energy model for defect formation now consists of multi-tiered frameworks that incorporate elastic, electronic, chemical, thermodynamic, and topological features, enabling accurate, efficient, and extensible predictions of defect energetics as functions of structure, composition, thermodynamic state, and local environment. This progression supports both predictive scientific insight and materials-by-design initiatives across disciplinary boundaries.