Dirac Fermions and Flat Bands in Phosphorus Carbide Nanotubes: Structural and Quantum Phase Transitions in a Quasi-One-Dimensional Material

Chemically realistic quasi-one-dimensional (1D) materials in which Dirac fermions and highly degenerate flat bands coexist intrinsically at the Fermi level are exceedingly rare, while representing a highly desirable platform for correlated and topological quantum phenomena. Here, using specialized symmetry-adapted first-principles calculations we predict a new class of nanomaterials – phosphorus carbide nanotubes ($\text{P}_2\text{C}_3$NTs) – obtained by rolling monolayer $\text{P}_2\text{C}_3$, a two-dimensional material shown in a previous letter to host “double Kagome bands”. Both armchair and zigzag $\text{P}_2\text{C}_3$NTs are stable at room temperature and feature the rare coexistence of Dirac crossings and multiple flat bands at the Fermi level inherited from the underlying honeycomb-Kagome lattice, with the flat bands resilient to elastic deformations. Under large strain, the structure transforms from honeycomb-Kagome to “brick-wall,” accompanied by multiple coupled structural and quantum phase transitions. We also uncover localized edge states, spin splitting from vacancies and dopants, and strain-tunable magnetism. Together, these results establish $\text{P}_2\text{C}_3$NTs as a chemically specific and mechanically tunable 1D material platform with potential applications in quantum hardware and spintronics.

💡 Research Summary

In this work the authors predict and thoroughly characterize a new class of quasi‑one‑dimensional nanomaterials—phosphorus carbide nanotubes (P₂C₃NTs)—obtained by rolling a previously reported two‑dimensional P₂C₃ sheet that hosts “double Kagome” bands. Using a symmetry‑adapted real‑space density‑functional framework (Helical DFT) together with conventional plane‑wave codes, they explore both armchair (n,n) and zigzag (n,0) chiralities. Cohesive‑energy calculations show that the tubes become more stable with increasing radius (‑5.35 eV to ‑5.46 eV per atom), placing them energetically between phosphorene nanotubes and carbon nanotubes, and suggesting feasible synthesis. Ab‑initio molecular‑dynamics simulations at 315 K for up to 10 ps confirm thermal stability of several tube sizes.

Mechanical analysis reveals linear elastic response for axial strains up to ±3.3 % and torsional twists up to 4.5° nm⁻¹. The axial stiffness of a 1.85 nm‑radius armchair tube is ≈2.7 × 10³ eV nm⁻¹, while a 0.80 nm‑radius zigzag tube shows ≈1.3 × 10³ eV nm⁻¹. Torsional stiffness is markedly lower than that of comparable carbon nanotubes (≈2.1 × 10² eV nm⁻¹ for zigzag, ≈9.6 × 10² eV nm⁻¹ for armchair), reflecting the much smaller bending modulus of the P₂C₃ sheet (≈0.14 eV) relative to graphene (≈1.5 eV).

Electronic structure calculations demonstrate that the rolled geometry preserves the honeycomb‑Kagome lattice symmetry, leading to the simultaneous presence of Dirac crossings and multiple flat bands exactly at the Fermi level. The flat bands are remarkably robust against both axial strain and torsional deformation, indicating “elastic‑resilient flat bands.” When a large compressive strain (~10 %) is applied, the lattice undergoes a structural transformation from the honeycomb‑Kagome motif to a brick‑wall configuration. This transition is accompanied by a coupled quantum phase transition: Dirac points disappear, flat bands reorganize, and the band topology changes, providing a controllable platform for studying correlation‑driven phenomena and topological phase changes in a 1D setting.

Defect and doping studies reveal that carbon vacancies generate localized magnetic moments, while phosphorus‑to‑carbon substitution introduces spin splitting, leading to spontaneous magnetism. Importantly, the magnitude and orientation of this magnetism can be tuned continuously by applying strain, offering a route to strain‑controlled spintronic functionality.

Overall, the paper establishes P₂C₃ nanotubes as chemically realistic, mechanically flexible, and electronically multifunctional one‑dimensional materials. Their unique combination of Dirac fermions, highly degenerate flat bands, strain‑induced structural/quantum phase transitions, and tunable magnetism makes them promising candidates for quantum hardware (e.g., platforms for strongly correlated electron phases, unconventional superconductivity, or fractional quantum Hall analogues) and spintronic devices where mechanical actuation can modulate spin states. The work also showcases the power of helical symmetry‑adapted DFT for efficiently treating large‑periodicity nanostructures, paving the way for systematic exploration of other rolled‑up 2D lattices with exotic electronic features.