Deep Potential Models for TaN

• Deep potential models for TaN are advanced machine learning frameworks that map local atomic environments to per-atom energy contributions with near-DFT accuracy.
• These models predict key properties such as thermal conductivity, elastic moduli, and superconducting behavior by simulating crystalline, amorphous, and nanostructured phases.
• Incorporating symmetry and physically-informed neural network architectures ensures scalability and transferability, driving innovations in electronics and quantum device design.

Deep potential models for tantalum nitride (TaN) represent the convergence of high-accuracy ab initio electronic structure calculations, ML frameworks, and atomistic molecular dynamics (MD) simulation methodologies. These models enable precise predictions of physical properties—including thermal conductivity, elastic behavior, and T,P-violating phenomena—across crystalline and amorphous TaN phases, thereby informing the design and optimization of TaN-containing devices for advanced electronic, spintronic, and quantum information technologies.

1. Foundations of Deep Potential Models for TaN

Deep potential (DP) models constitute a data-driven approach to constructing many-body potential energy surfaces (PES) for materials systems. Such models utilize deep neural networks to map local atomic environments, encoded via symmetry-invariant descriptors, onto per-atom energy contributions, with the total energy obtained as a sum over all atoms: $$E_{\text{total}} = \sum_{i=1}^{N} E_i\left(\left\{\frac{1}{r_{ij}}, \cos \theta_{ijk}, \sin \phi_{ijk}, \cos \phi_{ijk} \right\}\right)$$ where the inputs are constructed in local reference frames centered on each atom and with respect to its neighbors within a cutoff radius, incorporating {1/r, angular features} to capture spatial and chemical correlations (Han et al., 2017).

Crucially, these models are trained on data calculated from density functional theory (DFT) or high-level correlated wavefunction methods (e.g., coupled-cluster techniques), ensuring "first-principle" accuracy. This strategy avoids empirical guesswork, permitting high-fidelity interpolation and, with suitable augmentation, enhanced transferability.

2. Model Construction and Training Procedures

The latest DP models for TaN—including θ-phase (hexagonal P63/mmc), cubic, and amorphous phases—employ an active learning paradigm, as realized in the DP-GEN software stack (Zong et al., 5 Aug 2025). Model development encompasses:

• Initial Dataset Generation: Configurational diversity is introduced by sampling different phase structures, introducing atomic displacements to span the relevant configurational space.
• Ab Initio Labeling: Energies and forces for these structures are computed via DFT (with plane-wave cutoffs typically set at 600 eV for TaN) to serve as training labels.
• Active Learning Loop: A model ensemble is trained; input structures with high variance in predicted forces identify regions of poor coverage, and corresponding atomic geometries are recalculated at the DFT level. This iterative loop continues until model accuracy saturates (target ~99% for energy/force matching).
• Validation: Rigorous comparisons of DP-predicted energy, force, and phonon spectra (for crystalline TaN) to DFT benchmarks confirm the reliability of the resulting PES, especially in the acoustic phonon branches crucial for thermal transport (Zong et al., 5 Aug 2025).

This ensures that both short-range ordering and many-body effects—including those pertinent to phonon transport and defect energetics—are accurately captured, for both ordered and disordered (amorphous) TaN.

3. Symmetry Considerations and Neural Network Architecture

Deep potential models for TaN explicitly incorporate:

• Translational and Rotational Invariance: Achieved by encoding neighbor positions in local frames that respect the underlying lattice or local disorder.
• Permutational Symmetry: Inputs for neighboring atoms are grouped and sorted by chemical species and distance; weights within the neural network are tied for atoms of the same species. This architecture ensures that the total energy is invariant under permutation of indistinguishable atoms (Han et al., 2017).
• Scalability: The local (atom-centered) construction ensures computational cost scales linearly with system size, making simulations of large, realistic systems tractable.

Network architectures vary by system complexity. For molecular and condensed-phase benchmarks, deep fully connected feedforward networks with tens to hundreds of nodes per layer are standard; batch normalization and ensemble averaging augment robustness.

4. Application to TaN: Structure, Properties, and Phases

Thermal Transport

Recent applications (Guo et al., 2017, Zong et al., 5 Aug 2025) establish DP models as the premier tool for simulating TaN thermal conductivity with near-DFT accuracy at a fraction of the computational expense:

• Crystalline θ‑TaN: Equilibrium MD simulations using the DP models yield bulk thermal conductivities of ~530 ± 90 W/(m·K) (x), 430 ± 90 W/(m·K) (y), 470 ± 90 W/(m·K) (z) at room temperature (Zong et al., 5 Aug 2025). These values are significantly higher than experimental results (47.5–90 W/(m·K)), likely due to extrinsic defect and grain boundary scattering in polycrystalline samples, and match well with theoretical BTE predictions (~995 W/(m·K)) that account for four-phonon scattering (Guo et al., 2017).
• Anisotropy: Pronounced anisotropy is observed, consistent with the crystal structure. The high thermal conductivity is attributed to a large acoustic-optical phonon gap (6.05 THz), originating from the substantial mass difference between Ta and N (mass factor δ = 11.92) and leading to long phonon lifetimes due to inhibited acoustic-optical scattering.
• Size Effects in Nanofilms: Non-equilibrium MD simulations reveal strong thickness dependence; in θ‑TaN nanofilms, thermal conductivity rises rapidly with film thickness owing to the reduced scattering of long mean free path phonons. For amorphous TaN, conductivity remains low (~1.6 ± 0.2 W/(m·K)) and nearly independent of thickness, indicating the localization of vibrational modes.

Phase	Bulk κ (W/m·K)	Nanofilm Size Effect
θ‑TaN	430–530	Strong, uptrend
a‑TaN	~1.6	Negligible

Mechanical and Superconducting Properties

DP frameworks support the prediction of elastic moduli and phase stability by modeling the energy landscape across diverse local environments:

• Elastic Constants: First-principles and DP models both yield high bulk modulus (260 GPa), shear modulus (233 GPa), and C44 (215 GPa), confirming TaN as hard and incompressible (Guo et al., 2017).
• Superconducting Phases: Growth-optimized films exhibit critical temperatures up to 5 K (SiO₂ substrate; up to 7.7 K on sapphire) and maximum critical fields μ₀Hc2 = 13.8 T (Müller et al., 2021). Deep potentials can be trained on configurations spanning cubic, hexagonal, and other polymorphs to guide optimization strategies for superconducting device applications.

T,P-Violating Effects and Fundamental Physics

The TaN molecule, owing to the high nuclear charge of Ta and enhanced MQM, is uniquely sensitive to T,P-violating interactions (Skripnikov et al., 2015). Deep potential models, when trained on high-level coupled-cluster data, can encode effective Hamiltonians that incorporate:

• The electron EDM (de) interaction with internal fields,
• Scalar–pseudoscalar nucleus–electron interactions (kSP),
• MQM-electron couplings (WM × M).

Incorporating these effects into DP models enables simulation of responses to external fields and extraction of observables relevant for searches for CP violation and new physics. The computed parameters (e.g., Wa, WSP, WM) relate directly to experimental energy shifts and can be cross-validated with spectroscopic data.

5. Advances: Physically-Informed Neural Networks and Transferability

Physically-informed neural network (PINN) approaches integrate ML regression directly with a bond-order potential (BOP) backbone (Lin et al., 2021). Instead of mapping directly from coordinates to energy, the network predicts environment-specific corrections to global BOP parameters: $E_i = \Phi(\mathbf{R}_i, \mathbf{p}_i)$ where $\mathbf{p}_i = \mathbf{p}^0 + \delta \mathbf{p}_i$, and $\Phi$ is the energy as a function of atomic positions and bond-order parameters. PINN approaches offer:

• Improved Extrapolation: By retaining a physics-motivated baseline (global BOP), the model extrapolates more robustly to novel environments than pure-ML potentials.
• Systematic Validation: For elemental Ta, RMSE values of 2.8 meV/atom and accurate reproduction of defect, deformation, and phase behavior (including melting at 3000 K) are reported. By extension, multi-component PINN models can be developed for TaN by extending the training set to Ta–N configurations and adapting the BOP parameterization (Lin et al., 2021).
• Computational Efficiency: PINN and DP models enable large-scale simulation (orders of magnitude faster than DFT) with only modest computational overhead compared to analytic force fields.

6. Implications and Future Research Directions

Deep potential models for TaN have enabled:

• High-Throughput Simulation and Materials Optimization: They allow for accurate, atomistic-scale simulations of large systems over experimental timescales, critical for device design.
• Resolution of Experimental/Theoretical Discrepancies: By simulating ideal crystalline, amorphous, and nanostructured samples, DP models clarify the sources of variation in measured thermal conductivities and can predict the influence of defects, grain boundaries, and heterointerfaces (Zong et al., 5 Aug 2025).
• Multi-Scale Spintronic and Superconducting Modeling: DP models, particularly when parameterized to capture spin–orbit coupling and nontrivial band structures, inform spintronic device engineering and elucidate temperature-dependent torque generation and conversion mechanisms in TaN heterostructures (Müller et al., 2021).

Continued progress is anticipated via:

• Expansion of multi-component and multi-phase DP and PINN models to describe TaN alloys and complex heterostructures.
• Incorporation of higher-order anharmonicity and electron–phonon coupling effects, particularly near device-relevant operating conditions.
• Integration with experimental characterization (e.g., time-domain thermoreflectance, high-resolution electron microscopy) for robust benchmarking and inverse design.

7. Summary Table: Deep Potential Model Applications in TaN

Application	Property Modeled	Reference
Bulk θ‑TaN	Thermal conductivity (κ)	(Zong et al., 5 Aug 2025)
Amorphous TaN (a‑TaN)	κ, size dependence	(Zong et al., 5 Aug 2025)
Mechanically hard phases	B, G, C44 (elastic moduli)	(Guo et al., 2017)
Superconducting behavior	Tc, μ₀Hc2, spin-orbit torques	(Müller et al., 2021)
Fundamental physics (e.g., CPV)	T,P-odd Hamiltonian, EDM, MQM	(Skripnikov et al., 2015)
PINN for multi-component	Transferability, Ta–N energetics	(Lin et al., 2021)

Deep potential models for TaN, whether applied to bulk, nanostructures, or molecules, establish a rigorous and efficient framework for property prediction and atomic-scale insight, essential for both fundamental research and technological innovation in next-generation electronic and quantum devices.