Ideal Nodal-Sphere Semimetal in the Three-Dimensional Boron Allotrope CT-B$_{24}$

TL;DR

This study addresses the challenge of realizing an ideal nodal-sphere semimetal (NSSM) in real materials by combining first-principles calculations with symmetry analysis to identify a boron allotrope, $CT-B_{24}$, as a nearly ideal NSSM. It reveals two bands crossing near the Fermi level to form a quasi-nodal sphere around the $Γ$ point with a record-low HOTTs gap of about $0.008$ meV and a sixfold mirror-symmetry-derived nodal-ring skeleton, along with drumhead surface states on the (001) surface. The nodal sphere is tunable via strain and pressure, undergoing transitions to a Dirac semimetal and then to a trivial insulator, highlighting potential for strain-engineered topological devices. Overall, $CT-B_{24}$ provides the closest material realization to an ideal NSSM in a light-element system, with clear experimental signatures and opportunities for ARPES verification and device applications.

Abstract

Nodal-sphere semimetals (NSSMs), featuring spherical band degeneracies in momentum space, constitute a fascinating class of topological materials. However, their realization in real materials is severely hampered by discrete crystallographic symmetry constraints, often resulting in gapped ``pseudo'' nodal spheres. Here, combining first-principles calculations and symmetry analysis, we predict a new three-dimensional boron allotrope, CT-B$_{24}$, as a nearly ideal NSSM. Its structural stability is systematically confirmed by phonon calculations, \textit{ab initio} molecular dynamics simulations at 600~K, and elastic constant analysis. Notably, the electronic structure of CT-B$_{24}$ exhibits two bands crossing linearly near the Fermi level, forming a quasi-nodal sphere around the $Γ$ point. The maximum energy gap is merely 0.008~meV, which is two orders of magnitude smaller than the gaps reported in previous pseudo-NSSMs. Furthermore, the (001) surface hosts pronounced drumhead-like surface states located outside the projected nodal sphere, providing distinct signatures detectable by angle-resolved photoemission spectroscopy (ARPES). The nodal sphere also demonstrates remarkable robustness and tunability under external strain, driving a topological phase transition from an NSSM to a Dirac semimetal and finally to a trivial insulator. Our work not only presents a superior material platform for exploring nodal-sphere physics but also suggests potential for strain-tunable topological devices.

Figures (5)

• Figure 1: (Color online) Crystal structure and Brillouin zone of CT-B$_{24}$. (a) Top view and (c) side view of a $2\times2\times1$ supercell. (b) Perspective view of the primitive cell. (d) B$_{3}$ triangle and (e) B$_{9}$ cage building blocks. (f) The bulk Brillouin zone (BZ) and the projected (001) surface BZ.
• Figure 2: (Color online) Stability analysis of CT-B$_{24}$. (a) Phonon dispersion curves. (b) Ab initio molecular dynamics (AIMD) simulation at 600 K: potential energy fluctuation over time (main panel) and structural snapshots before (left inset) and after (right inset) the 8000 fs simulation. (c) Energy-volume curves comparing CT-B$_{24}$ with other representative boron allotropes.
• Figure 3: (Color online) Electronic and topological properties of CT-B$_{24}$. (a) Band structure along high-symmetry paths. Inset: 3D visualization of the band crossing in the $k_1$--$k_2$ plane. (b) Projected density of states (PDOS). (c) Top and side views of the charge density distribution at the P point labeled in Fig. \ref{['fig_structure']}(a). (d) Density of states (DOS) near the Fermi level ($E_{\text{F}}$), with the dashed line indicating a constant value. (e) Band crossing at a gapless D point on the spherical surface. (f) Band structure around the $\Gamma$ point along the K-$\Gamma$-M path, showing the mirror eigenvalues of the crossing bands. (g) Band crossing at the $X_1$ point with a maximum gap of 0.008 meV, where $X_1$ is located on the spherical surface. Band gap maps on the (h) $k_x$--$k_z$ and (i) $k_y$--$k_z$ planes, with white curves highlighting the mirror-symmetry-protected nodal rings in the $C_{6v}$ point group. Top (j) and perspective (k) views of the quasi-nodal sphere in the BZ. Yellow dashed lines depict the three $\sigma_v$ and three $\sigma_d$ mirror planes of the $C_{6v}$ point group; blue and red solid curves represent the mirror-symmetry-protected nodal rings residing on these planes.
• Figure 4: (Color online) Surface states on the (001) surface of CT-B$_{24}$. (a) Surface spectral function along the $\bar{\text{K}}$-$\bar{\Gamma}$-$\bar{\text{M}}$ high-symmetry path. The bright-red intensity denotes topological surface states. (b) Constant-energy contour at the Fermi level ($E_{\text{F}}$), revealing drumhead surface states.
• Figure 5: (Color online) Diagram of the band structure evolution of CT-B$_{24}$ under external strain. (a) Uniaxial strain along the $c$-axis. (b) Hydrostatic pressure. The plots trace the topological phase transition from a nodal-sphere semimetal (0% strain, 0 GPa) to a Dirac semimetal ($-1.2\%$ strain, $-2.6$ GPa) and then to a trivial insulator.