Exploring Charge Density Waves in two-dimensional NbSe2 with Machine Learning

TL;DR

This work develops a physically informed workflow for training Allegro-based machine-learning interatomic potentials to capture charge density wave phenomena in NbSe2, including monolayer and bilayer forms. By combining Bayesian active learning, large-cell training, and enhanced cutoffs, the authors achieve accurate structural reconstructions and vibrational properties, and they use SSCHA to estimate CDW transition temperatures around 50–60 K with weak dimensionality dependence. The study reveals that CDW lattice distortions are relatively easy to learn while phonon properties—and in particular long-range force constants—pose greater challenges, necessitating diverse, large datasets and refined training strategies. The results enable reliable simulations of commensurate and incommensurate CDW phases, shedding light on electron–phonon coupling, dimensionality effects, and stacking in 2D materials, and offering a framework applicable to other layered CDW systems.

Abstract

Niobium diselenide (NbSe$_2$) has garnered significant attention due to the coexistence of superconductivity and charge density waves (CDWs) down to the monolayer limit. However, realistic modeling of CDWs-capturing effects such as layer number, twist angle, and strain-remains challenging due to the high computational cost of first-principles methods. Here, we develop a physically informed workflow for training machine-learning interatomic potentials (MLIPs) based on the E(3)-equivariant Allegro architecture, tailored to capture the subtle structural and dynamical signatures of CDWs in mono- and bilayer NbSe$_2$.We find that while CDW lattice distortions are relatively easy to learn, modeling vibrational properties remains more challenging. It requires targeted dataset design and careful hyperparameter tuning, pushing the boundaries and testing the extensibility of current MLIP frameworks. Our MLIPs enable reliable simulations of commensurate and incommensurate CDW phases, including their sensitivity to dimensionality and stacking, as well as CDW dynamics, phonons, and transition temperatures estimated via the stochastic self-consistent harmonic approximation. This work opens new possibilities for studying and tuning CDWs in NbSe$_2$ and other two-dimensional systems, with implications for electron-phonon coupling, superconductivity, and advanced materials design.

Figures (6)

• Figure 1: Development of a machine-learned interatomic potential (MLIP) for NbSe$_2$ in the normal state (large smearing, no CDW).a - Structure of monolayer $\mathrm{NbSe_2}$ in the normal phase, visualized using Ovito stukowski2009visualization, showing a $3\times3$ supercell. Orange sticks represent Se atoms, and green sticks represent Nb atoms. The range of interactions ($r_\mathrm{max} = 5$ Å) is indicated schematically. b - Computational workflow employed to develop the MLIP. c - Phonon dispersion relations from DFT (solid lines) and Allegro MLIP (dashed lines), demonstrating close agreement. d - Energy mean absolute error (MAE) per simulation frame from molecular dynamics (MD) simulations at 200 K in the NVT ensemble, comparing Allegro MLIP predictions with DFT. e - Corresponding force MAEs; values are reported per element as shown in the legend. Further details are given in Methods and SI.
• Figure 2: Development of a MLIP for NbSe$_{2}$ in the normal and CDW phases.a - Structure of the hollow CDW phase of $\mathrm{NbSe_2}$, with the $3\times3$ supercell shown. CDW distortions in the Nb atom positions are highlighted with additional sticks between Nb atoms, using a cutoff of 3.45 Å. b - Energy MAE per simulation frame from MD simulations at 200 K in the NVT ensemble, comparing Allegro MLIP predictions with DFT. c - Phonon dispersion curves for the normal state, computed using DFT (solid lines) and Allegro MLIP (dashed lines), demonstrating close agreement. d - Structure of the filled CDW phase, shown for a $3\times3$ supercell with distortions highlighted as in panel a. e - Force parity plot for Nb and Se atoms, comparing MLIP and DFT forces. f - Phonon density of state, computed at $\Gamma$, for the hollow CDW phase, comparing DFT and MLIP. Further details are provided in the SI.
• Figure 3: Relaxations of commensurate and incommensurate supercells of NbSe$_2$ monolayers.a – MD NVT simulation at 10 K, starting from a $9\times9$ supercell initialized with a mixture of hollow, normal, and filled motifs. The system evolves toward the hollow CDW structure. b – Relaxations of incommensurate $7\times7$ and $11\times11$ supercells. The $7\times7$ case shows mainly filled-type distortions with some hollow features, while the $11\times11$ case displays alternating hollow and filled motifs. c – Incommensurate $3\times7$ and $3\times8$ supercells also show coexistence of hollow and filled patterns, with the $3\times8$ case exhibiting less well-developed distortions.
• Figure 4: Development of a MLIP for bilayer NbSe$_{2}$.a - Binding energy curves for high-symmetry stackings of the 180° $\mathrm{NbSe_2}$ bilayer. Filled symbols represent Allegro MLIP results, and empty symbols represent DFT results. b Energy MAE per simulation frame from MD simulations at 200 K in the NVT ensemble, comparing Allegro MLIP predictions with DFT. c - Phonon dispersion curves for the normal phase of the bilayer in the MM stacking. A comparison is shown between DFT (solid lines) and Allegro MLIP (dashed lines). d - Binding energy curves for high-symmetry stackings of the 0° $\mathrm{NbSe_2}$ bilayer. Filled symbols represent Allegro MLIP results, and empty symbols represent DFT results. e - Force parity plot for Nb and Se atoms, comparing MLIP and DFT forces. f -Phonon dispersions of the monolayer evaluated with the bilayer MLIP, compared with DFT. Further details are provided in the SI.
• Figure 5: Results for the MLIP applied to NbSe$_{2}$ bilayers. MD NVT simulation at 10 K, starting from a 9$\times$9 supercell prepared with a mixture of hollow, normal, and filled phases in both layers. Intermediate frames illustrate the evolution of the system during the relaxation process, with the final structure corresponding to the stable configuration.