Superconductivity in unconventional metals

Topological materials have drawn significant interest for their novel properties. Recently, a class of unconventional materials has been identified as topologically trivial with wannierizable valence Bloch states but featuring bands induced by an elementary band representation (EBR) on an empty site. Unconventional insulators of this kind are also termed obstructed atomic insulators. There also exist unconventional metals, appearing in contexts such as electrides, catalysis, hydrogen storage, and superconductivity. Transition metal dichalcogenides (TMDs) with 2H stacking show charge density wave (CDW) and superconductivity (SC). CDW transition temperatures decrease from ~120 K (2H-TaSe2) to 80 K (2H-TaS2) to 30 K (2H-NbSe2) and are absent in 2H-NbS2, while superconducting Tc increases from ~0.2 K (2H-TaSe2) to 7.2 K and 6 K in 2H-NbSe2 and 2H-NbS2. Despite extensive studies, the origin of SC remains unresolved. This work uses first-principles and band representation analysis to show monolayer 1H-MX2 is an unconventional metal with a half-filled EBR at an empty 1e site (RSI δ1@1e = 1). Phonons show a low-temperature CDW instability; BCS analysis and electron-phonon coupling (EPC) point to soft phonons as the origin of SC. Guided by this, SC is predicted in TaNS monolayer and 2H-TaN2 bulk.

Prior studies established CDW and SC in 2H TMDs, with trends in CDW transition temperatures and superconducting Tcs across TaSe2, TaS2, NbSe2, and NbS2. Soft-phonon behavior and CDW mechanisms in NbSe2 and related compounds have been investigated theoretically and experimentally, including dimensionality effects and quantum enhancement of CDW in monolayers. Recent theoretical frameworks on unconventional materials highlight empty-site EBRs and real-space invariants (RSIs), obstructed atomic insulators, and their roles in electrides, catalysis, and correlated phenomena. Tuning SC via intercalation (Li/Na/Cu) or substitution (Mo/W) in NbSe2 has been reported. These works motivate probing the link between empty-site EBR occupancy, EPC, CDW, and SC in TMDs.

First-principles density functional theory (DFT) calculations were performed with QUANTUM ESPRESSO (QE) using PAW pseudopotentials and the PBE generalized gradient approximation. Phonon properties and electron-phonon coupling (EPC) were computed within density functional perturbation theory (DFPT) in QE. Electronic smearing parameters were varied to emulate different temperatures in phonon dispersion calculations. Band representations (ABRs/EBRs) and irreducible representations were analyzed via topological quantum chemistry tools; irreps were computed using IR2PW, and ABRs were generated by POS2ABR. Electron doping effects were modeled by Na intercalation (bulk NaNbSe2) and Mo substitution (1H-MoSe2) to fill the empty-site EBR and assess phonon stability. Eliashberg spectral function α2F(ω) and integrated coupling λ(ω) were calculated, and mode-resolved EPC strengths λαv were mapped onto phonon dispersions. Superconducting transition temperatures Tc were estimated using the Allen-Dynes modified McMillan equation with an effective Coulomb parameter μ* (typically 0.10), using computed λ and ωlog from DFPT.

• 1H/2H-phase MX2 (M = Nb, Ta; X = S, Se, Te) are unconventional metals with a half-filled empty-site EBR A′@1e at EF; RSI δ1@1e = 1.
• In 1H-MX2, the Fermi-level band arises from hybridization of Nb/Ta d-orbital ABRs at 1c: A@1c (from d z2) and E′@1c (from dxy, dx2−y2), yielding A′@1e (half-filled) + E′@1e (empty).
• Phonon dispersions: at high electronic smearing (high T) NbSe2 and TaS2 show no imaginary modes (stable); at low smearing (low T) a soft phonon mode appears near M, indicating CDW instability not due to Fermi nesting but to EPC.
• Electron doping by Na intercalation (bulk NaNbSe2) or Mo substitution (1H-MoSe2) fully occupies the empty-site EBR (A1′@2c or A1′@1e) and removes soft modes, stabilizing the structures and suppressing CDW.
• In 1H-NbSe2, strong EPC (λ > 1) originates predominantly from soft phonons near ~70 cm−1 around M. The corresponding mode involves mainly in-plane vibrations of Nb atoms that strongly modulate the empty site, maximizing overlap with EF states centered at the empty site.
• Superconductivity mechanism: the partial filling of empty-site A1′@1e enhances EPC via spatial coincidence of EF charge centers and soft phonon distortions, driving SC.
• Predicted superconductors: • TaNS monolayer (SG 164): half-filled empty-site EBR at EF; phonons stable (no imaginary frequencies). EPC dominated by modes near ~100 cm−1; with μ* = 0.10 and λ = 0.72, estimated Tc ≈ 10 K (Allen-Dynes). • 2H-TaN2 bulk (SG 194): half-filled empty-site EBR (A@2c) with empty sites at 2c; phonons stable at σ = 0.02 Ry; with μ* = 0.10 and λ = 1.04, estimated Tc ≈ 26 K. λ(ω) mainly from ω < 400 cm−1, with notable contributions ~100 cm−1 along K–M. Phonon mode near M: Ta vibrates along z; N vibrates in-plane with larger amplitude; high ωlog assisted by light N atoms.
• Relation to known systems: 1H-TaSeS (Jenus) also an unconventional metal; its SC is experimentally confirmed. The σ-band in MgB2 exemplifies an empty-site EBR (A1g@3f) with partial filling, suggesting a broader link between unconventional electronic structure and strong EPC.

The study addresses the unresolved origin of superconductivity in 2H/1H TMDs by linking SC to the partial filling of an empty-site EBR at EF, which enhances EPC through soft phonon modes that spatially squeeze the empty sites. The presence of a soft phonon near M correlates with half-filled empty-site bands and CDW at low temperatures, while electron doping fills the empty-site EBR, removes soft modes, stabilizes the lattice, and suppresses CDW—clarifying the interplay between CDW and SC. Mode-resolved EPC and α2F(ω) show that low-frequency modes (~70–100 cm−1) dominate coupling in NbSe2, TaNS, and TaN2, consistent with strong coupling λ and observed/predicted superconductivity. The findings generalize a design principle: unconventional metals with partially filled empty-site EBRs provide fertile ground for strong EPC and phonon-mediated SC, supported by analogies to MgB2 and experimentally verified SC in related TMDs. The predictions for TaNS and 2H-TaN2 extend the framework and suggest that high ωlog from light elements (N) can elevate Tc even with moderate-to-strong λ.

This work establishes that 1H/2H TMD MX2 host half-filled empty-site EBRs at EF, producing soft phonons and strong EPC that underpin superconductivity. It elucidates the mechanism linking empty-site band occupancy to EPC and CDW, and demonstrates that electron doping stabilizes structures by fully occupying the empty-site EBR and suppressing soft modes. Guided by this principle, the study predicts superconductivity in TaNS monolayer (Tc ≈ 10 K) and 2H-TaN2 bulk (Tc ≈ 26 K). These results position unconventional metals with partially filled empty-site EBRs as promising platforms for discovering new superconductors. Future work should focus on experimental verification of TaNS and TaN2 superconductivity, systematic tuning via intercalation/substitution and dimensionality, and exploring quantum geometric contributions to EPC in empty-site band systems.

The superconducting transition temperatures are theoretical estimates based on DFPT-derived EPC parameters and the Allen-Dynes modified McMillan equation with an assumed μ* = 0.10. Low-temperature instabilities are inferred from electronic smearing as a proxy for temperature. The predictions for TaNS monolayer and 2H-TaN2 bulk have not yet been experimentally confirmed, and 2H-TaN2 has been proposed as a metastable high-pressure phase.