Machine Learned Interatomic Potential for AlN

• The paper demonstrates that MLIP models for AlN achieve near-DFT accuracy at orders of magnitude lower computational cost.
• It utilizes diverse methodologies—NNP, ACE, UF³, and charge-aware models—to accurately capture atomic energetics, defect dynamics, and thermal behaviors.
• These MLIPs enable efficient simulation of epitaxial growth, strain engineering, and high-temperature phase transitions in device-relevant environments.

A machine learned interatomic potential (MLIP) for aluminum nitride (AlN) is a parameterized model, trained on first-principles data, that allows fast and accurate prediction of the atomic-scale energetics and forces governing AlN crystal behavior. MLIPs for AlN have become essential tools for simulating defect kinetics, structural properties, phonon dynamics, and growth processes in device-relevant regimes—enabling predictive molecular dynamics well beyond the reach of direct density functional theory (DFT) calculations. Several MLIP families have been developed for AlN, including neural-network potentials (NNP, DeepMD-type), atomic cluster expansions (ACE), ultra-fast force field (UF³), and symmetry-adapted models augmented for charged defects. These models differ in their mathematical formalism, database construction, training protocol, and validation metrics, but all deliver near-DFT accuracy at several orders of magnitude less computational cost.

1. Mathematical Foundations of AlN MLIPs

MLIPs for AlN typically decompose the total potential energy $E_{\rm tot}$ into atom-centered terms reliant on local structural descriptors:

Neural Network Potentials (DeepMD/Behler–Parrinello)

The total energy is written as a sum over atomic neural-network predictions:

$E_{\rm tot} = \sum_{i} E_i(G_i)$

where $G_i$ encodes the local environment (neighbor matrix, radial/angular info within cutoff $r_c$). DeepMD architecture uses two subnetworks: "embedding" (neighbor descriptors to latent vector) and "fitting" (vector to atomic energy), with layer sizes typically (25, 50, 100) → (240, 240, 240), tanh activation, and Adam optimization (learning rate $1\times10^{-3}$ decaying to $1\times10^{-8}$) (Reza et al., 29 Oct 2025).

Atomic Cluster Expansion (ACE)

In ACE, the atomic environment is projected onto body-ordered, permutation- and rotation-invariant basis functions via radial and angular terms:

$$E_{\rm tot} = \sum_{i} E_i = \sum_{i} \sum_{\alpha} c_{\alpha} B_{i,\alpha}$$

with expansion coefficients $c_{\alpha}$ fit by regularized linear regression to DFT data. ACE produces fast, strictly linear-scaling potentials (Yang et al., 2023).

Ultra-Fast Force Field (UF³)

UF³ represents $E$ as a sum of learned two-body and three-body B-spline interactions:

$$E[\{r_i\}] = \sum_{i<j} V^{(2)}_{\alpha_i\alpha_j}(r_{ij}) + \sum_{i<j<k} V^{(3)}_{\alpha_i\alpha_j\alpha_k}(r_{ij}, r_{ik}, \theta_{jik})$$

where $V^{(2)}, V^{(3)}$ are constructed from spline coefficients fitted linearly to DFT energies/forces with additional ridge and curvature regularization (Taormina et al., 11 Nov 2025).

NNPs for Charged Defects

Modified NNPs for charged AlN include an explicit "system charge" input node alongside standard radial and angular symmetry functions, allowing the network to adjust predictions for defect-laden, non-neutral configurations (Dou et al., 2024).

2. Training Data Generation and Fitting Protocols

The fidelity of an MLIP hinges on an expansive and diversified DFT-based training set:

• Bulk and Strained Structures: All approaches sample wurtzite and cubic AlN under various volume strains, random atomic displacements, and anisotropic deformations. DeepMD-based NNPs for AlN use $10^5-10^6$ supercell configurations spanning endpoints and alloys (e.g., Al$_x$Ga$_{1-x}$N, $x=0.25$–0.75) (Reza et al., 29 Oct 2025).
• Defects and Dislocations: Point defects (vacancies, interstitials, Frenkel pairs) and, in the UF³ case, dislocation dipoles, are modeled by explicit insertion and DFT relaxation (Taormina et al., 11 Nov 2025).
• High-Temperature Coverage: DPA-Semi and ACE sample up to $7200$ K and beyond melting conditions to ensure robust liquid-state transferability (Liu et al., 2023, Yang et al., 2023).
• Phonon-Relevant Sampling: ACE and NNPs targeting phonon transport focus on structures spaced across MD trajectories (100–1000 K) and systematically scaled lattices (Yang et al., 2023, Dou et al., 2024).
• Defect Charge States: NNPs for phonon scattering with charged defects generate configurations for vacancy/interstitial variants (V$_\text{N}^{3+}$, V$_\text{Al}^{3-}$, etc.) with jellium background (Dou et al., 2024).
• Fitting Approach: DeepMD and DPA-Semi employ direct stochastic gradient descent with dynamic weighting of energy and force errors; UF³ uses an Optuna-driven search for regularization and error balancing; ACE applies linear regression with Tikhonov regularization (Reza et al., 29 Oct 2025, Taormina et al., 11 Nov 2025, Yang et al., 2023).

3. Validation Metrics and Benchmarking Against DFT

Performance of AlN MLIPs is quantitatively assessed by comparison to DFT or experiment:

Model/Framework	Energy RMSE (meV/atom)	Force RMSE (meV/Å)	Structural Properties	Thermal Conductivity (W/mK)
DeepMD NNP (Reza et al., 29 Oct 2025)	≤2	≤50	$a=3.112$ Å, $c=4.982$ Å, $B=213$ GPa	–
ACE (Yang et al., 2023)	0.13	5.01	$a=3.1156$ Å, $c=4.9815$ Å, $B=205$ GPa	310 (in-plane, 300 K)
DPA-Semi (Liu et al., 2023)	10.18	283	$a=4.410$ Å, $B=200$ GPa	–
UF³ (Taormina et al., 11 Nov 2025)	30 (wz set)	7 (wz set)	$a=3.02$, $c=4.90$ Å, $B=379$ GPa	–
NNP (charge states) (Dou et al., 2024)	3.55	69	$a=3.119$ Å, $c=4.985$ Å	256.9 (at 293 K)

Structural metrics—lattice constants, elastic constants $C_{ij}$, cohesive energies, and surface energies—are captured within 1–5% of DFT or experiment, with phonon band structures and thermal conductivities additionally reproduced to high fidelity.

Defect formation energies and migration barriers from DeepMD NNPs give:

• Al Frenkel pair: $11.05$ eV; N Frenkel: $11.25$ eV; Schottky: $6.06$ eV
• Vacancy migration: V$_\text{Al}$ in-plane $2.23$ eV, out-of-plane $2.76$ eV; V$_\text{N}$ $2.69/3.12$ eV (Reza et al., 29 Oct 2025)

Point-defect energies, migration barriers, and defect-induced phonon properties remain within 0.2–0.6 eV or 10% error versus DFT. UF³ uniquely reproduces the experimentally validated 8-atom edge-dislocation core, a key benchmark for atomistic realism (Taormina et al., 11 Nov 2025).

4. Physical Insights and Predictive Capabilities

MLIPs enable simulation and exploration of physical processes inaccessible to direct DFT:

• Elastic and Structural Response: All models reliably predict pressure-dependent volumes, equilibrium equations of state, and non-linear elastic behavior in alloys (Al$_x$Ga$_{1-x}$N) (Reza et al., 29 Oct 2025).
• Defect Dynamics: Migration barriers elucidate the compositional insensitivity of Ga/Al vacancy motion vs. strong local-chemistry dependence for N defects (Reza et al., 29 Oct 2025). Charge-dependent NNPs reveal unexpected trends in phonon-defect scattering, highlighting that structural distortions can dominate over simple mass-difference effects (Dou et al., 2024).
• Phonon and Thermal Phenomena: ACE and NNPs yield DFT-level predictions for specific heat, thermal expansion, and lattice conductivity (in-plane 310 W/mK; cross-plane 230 W/mK), allowing predictive strain engineering (tensile strain can suppress conductivity by 40%) (Yang et al., 2023).
• Epitaxial Growth: UF³ predicts homoepitaxial layer-by-layer AlN growth with correct wurtzite structure and morphology, surpassing previous SW/GaN potentials (Taormina et al., 11 Nov 2025).
• High-Temperature and Phase Coverage: DPA-Semi models retain GGA-quality results up to $7200$ K, with accurate liquid-state phase behavior (Liu et al., 2023).

5. Specialty Features: Charged Defects and Strain Engineering

Recent advances involve explicit encoding of charge states to accurately treat defect-laden AlN. NNPs with system-charge nodes model vacancy and interstitial charge effects on phonon transport and thermal properties, capturing subtle defect-induced changes in phonon frequencies and scattering rates (Dou et al., 2024).

ACE-based and DeepMD models have enabled analysis of thermal conductivity tuning via biaxial strain, providing quantitative guidance for heat-management in AlN-based power electronics (e.g., $+4$% strain reduces $K$ by 40%, highlighting strain as a major design lever) (Yang et al., 2023).

6. Limitations, Transferability, and Perspectives

Most AlN MLIPs exhibit transferability across bulk polymorphs and typical device temperatures, but certain limitations persist. ACE lacks built-in long-range Coulombic interactions, approached via non-analytical corrections. Charge encoding in NNPs is global, not atom-specific, limiting accurate treatments of long-range electrostatics (Yang et al., 2023, Dou et al., 2024). UF³, while efficient, underpredicts specific elastic constants (C$_{13}$, C$_{33}$) and has systematic errors of up to 0.05 eV/atom in certain defect energetics (Taormina et al., 11 Nov 2025). Expansion to amorphous phases and complex ternary alloys requires strategic retraining.

Significantly, advances in universal models (DPA-Semi) and physically interpretable frameworks (UF³, ACE) afford the ability to generalize with minimal new data, supporting defect, surface, and high-temperature regime studies. This points toward the imminent feasibility of sub-DFT-cost, device-scale, and composition-spanning simulations essential for materials design.

7. Comparative Synopsis and Application Domains

AlN MLIPs now deliver near-ab initio precision for structural, elastic, defect, and thermal property predictions, with demonstration across multiple phase regimes and device-relevant phenomena. DeepMD-type NNPs and ACE offer comprehensive bulk property and phonon modeling; UF³ uniquely excels in epitaxial and dislocation core simulation; dedicated charge-state NNPs resolve subtle phonon-defect interactions. All approaches dramatically reduce the computational demands compared to plane-wave DFT, supporting atomistic simulations at scale.

A plausible implication is that these frameworks collectively enable predictive materials engineering for nitride-based electronics, optoelectronics, and thermal management, and their continued refinement will expand simulation reach to amorphous states, heterostructures, and high-defect-concentration environments.