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Machine-learning Guided Search for Phonon-mediated Superconductivity in Boron and Carbon Compounds

Niraj K. Nepal, Lin-Lin Wang

TL;DR

This work develops a workflow that couples high-throughput DFPT calculations of electron-phonon coupling with machine-learning guided searches to identify boron/carbon–based superconductors at ambient pressure. Crucially, it includes compounds with imaginary phonon modes (dynamical instability) in the training data, using stabilization strategies to extract EPC properties and Tc estimates. The study compares CGCNN and ALIGNN, finding ALIGNN generally superior when unstable data are included, and reports promising Tc predictions for metastable materials such as Ca$_5$B$_3$N$_6$ and TaNbC$_2$, highlighting the role of soft, stabilized phonons in enhancing superconductivity. The methodology provides a robust screening tool for metastable phases and suggests that metastable boron/carbon compounds near the convex hull can host sizable Tc, with implications for discovering new phonon-mediated superconductors at ambient conditions.

Abstract

We present a workflow that iteratively combines \textit{ab-initio} calculations with a machine-learning (ML) guided search for superconducting compounds with both dynamical stability and instability from imaginary phonon modes, the latter of which have been largely overlooked in previous studies. Electron-phonon coupling (EPC) properties and critical temperature (T$_c$) of 417 boron, carbon, and borocarbide compounds have been calculated with density functional perturbation theory (DFPT) and isotropic Eliashberg approximation. Our study addresses T$_c$ convergence of Brillouin zone sampling with an ansatz test, stabilizing imaginary phonon modes for significant EPC contributions and comparing performance of two ML models especially when including compounds of dynamical instability. We predict a few promising superconducting compounds with formation energy just above the ground state convex hull, such as Ca$_5$B$_3$N$_6$ (35 K), TaNbC$_2$ (28.4 K), Nb$_3$B$_3$C (16.4 K), Y$_2$B$_3$C$_2$ (4.0 K), Pd$_3$CaB (7.0 K), MoRuB$_2$ (15.6 K), RuVB$_2$ (15.0 K), RuSc$_3$C$_4$ (6.6 K) among others.

Machine-learning Guided Search for Phonon-mediated Superconductivity in Boron and Carbon Compounds

TL;DR

This work develops a workflow that couples high-throughput DFPT calculations of electron-phonon coupling with machine-learning guided searches to identify boron/carbon–based superconductors at ambient pressure. Crucially, it includes compounds with imaginary phonon modes (dynamical instability) in the training data, using stabilization strategies to extract EPC properties and Tc estimates. The study compares CGCNN and ALIGNN, finding ALIGNN generally superior when unstable data are included, and reports promising Tc predictions for metastable materials such as CaBN and TaNbC, highlighting the role of soft, stabilized phonons in enhancing superconductivity. The methodology provides a robust screening tool for metastable phases and suggests that metastable boron/carbon compounds near the convex hull can host sizable Tc, with implications for discovering new phonon-mediated superconductors at ambient conditions.

Abstract

We present a workflow that iteratively combines \textit{ab-initio} calculations with a machine-learning (ML) guided search for superconducting compounds with both dynamical stability and instability from imaginary phonon modes, the latter of which have been largely overlooked in previous studies. Electron-phonon coupling (EPC) properties and critical temperature (T) of 417 boron, carbon, and borocarbide compounds have been calculated with density functional perturbation theory (DFPT) and isotropic Eliashberg approximation. Our study addresses T convergence of Brillouin zone sampling with an ansatz test, stabilizing imaginary phonon modes for significant EPC contributions and comparing performance of two ML models especially when including compounds of dynamical instability. We predict a few promising superconducting compounds with formation energy just above the ground state convex hull, such as CaBN (35 K), TaNbC (28.4 K), NbBC (16.4 K), YBC (4.0 K), PdCaB (7.0 K), MoRuB (15.6 K), RuVB (15.0 K), RuScC (6.6 K) among others.
Paper Structure (29 sections, 14 equations, 21 figures, 20 tables)

This paper contains 29 sections, 14 equations, 21 figures, 20 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Machine learning guided workflow
  4. Overview of DFPT calculations
  5. Training and testing ML models
  6. Comparison of ML models
  7. Dynamically stable systems
  8. Imaginary phonon modes and superconductivity
  9. Discussion
  10. METHODS
  11. Convergence with respect to Brillouin zone sampling (k-point mesh)
  12. Computational details
  13. Limitations
  14. Data availability
  15. Acknowledgements
  16. ...and 14 more sections

Figures (21)

  • Figure 1: Machine learning workflow. The numbers are the counts of compounds involved in each step. See main text for the detailed summary.
  • Figure 2: Statistical description of dynamically stable 113 compounds, whose (non-) superconductivity have been experimentally measured. (a) Number of compounds according to elements, each bar is partitioned into two color segments, with red representing the number of superconductors (SC), while blue denote non-superconductors (NSC).; (b) Proportion of boron and carbon compounds (c) Histogram for energy above the convex hull in (eV/atom), (d) Distribution according to space group, (e) Crystal structure of MgB$_2$. (f)-(m): Crystal structures of known superconductors. (f) NbC (10.03 K)WBM67, (g) Mo$_2$BC (7.5 K) ECCMHV16, (h) MgCNi$_3$ (8 K)HHSI01_MgNi3C, (i) Mo$_2$GaC (3.9 K)T67_Mo2GaC, (j) SrC$_6$ (1.65 K)KBORK07_SrC6, (k) YC$_2$ (3.89K)BJKSOS12, (l) YB$_6$ (7.2 K)LWTAMP06, and (m) LaPt$_2$B$_2$C (10 K)CBSKPCFTV94.
  • Figure 3: Graph neural network-based regression models for dynamically stable compounds. Predicting $\lambda$, $\omega_{log}$, T$_c$, and T$^{\prime}_c$ using CGCNN (red circles) and ALIGNN (blue upper triangle) models for the independent test set in comparison to the DFPT-calculated results in Run 1 (left panel, (a)-(d)) and Run 2 (right panel, (e)-(h)). Run 1 uses 250 training samples (45 space groups) and 58 test samples (27 space groups). Run 2 employs 323 training samples (54 space groups), with the test set unchanged. Similarly, Mean absolute error (MAE) are presented in inset. Red and blue arrows show the clustering pattern, discussed in the main text.
  • Figure 4: Graph neural network-based regression models similar to Fig. 3, but including both stable and stabilized unstable compounds. Predicting $\lambda$, $\omega_{log}$, T$_c$, and T$^{\prime}_c$ using CGCNN (red circles) and ALIGNN (blue upper triangle) models for the independent test set including both stable and dynamically unstable cases in comparison to the DFPT-calculated results in Run3. Run 3 employs 351 training samples (56 space groups) and 66 test samples (27 space groups). MAEs are presented in inset.
  • Figure 5: DFPT calculated EPC properties of a few dynamically stable compounds. (a) Crystal structure, (b) phonon dispersion, and (c) Eliashberg isotropic spectral function of TaNbC$_2$, respectively. (d)-(k) X-B-C compounds (X=Nb,Y): Comparison between DFPT computed T$_c$ values with experiments for (d) Nb-B-C and (e) Y-B-C systems. Experimentally known results are represented by blue circle, while red circles are not reported ones shown along the y$=$x dashed line as labeled. Crystal structures respectively for (f) Nb$_3$B$_3$C, and (g) Y$_2$B$_3$C$_2$. Atoms are highlighted by the colorized symbols. Phonon dispersion projected with mode-resolved $\lambda$ (green open circles) for (h) Nb$_3$B$_3$C, and (i) Y$_2$B$_3$C$_2$ respectively. Eliashberg spectral functions for (j) Nb$_3$B$_3$C, and (k) Y$_2$B$_3$C$_2$ respectively.
  • ...and 16 more figures