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Porous-B$_{18}$: An Ideal Topological Semimetal with Symmetry-Enforced Orthogonal Nodal-Line and Nodal-Surface States

Xiao-jing Gao, Yanfeng Ge, Yan Gao

Abstract

Topological semimetals (TSMs) featuring symmetry-protected band degeneracies have attracted considerable attention due to their exotic quantum properties and potential applications. While nodal line (NL) and nodal surface (NS) semimetals have been extensively studied, the realization of a material where both NL and NS coexist and are intertwined, particularly with an ideal electronic band structure, remains a significant challenge. Here, we predict via first-principles calculations and symmetry analysis a metastable boron allotrope, Porous-B$_{18}$ (space group $P6_3/m$, No.~176), as a pristine TSM hosting a NS and two straight NLs near the Fermi level. The structure, a honeycomb-like porous 3D framework, exhibits excellent dynamical, thermal (stable up to 1000~K), and mechanical stability. Its electronic band structure is remarkably clean: only the highest valence band (HVB) and the lowest conduction band (LCB) cross linearly within a large energy window of 1.84~eV, free from trivial-band interference. The nodal surface lies on the $k_z = \pm π$ planes, protected by combined time-reversal symmetry ($T$) and twofold screw-rotational symmetry ($S_{2z}$), yielding a full-plane Kramers-like degeneracy. The two nodal lines along $K$--$H$ and $K'$--$H'$ are protected by inversion and time-reversal symmetries, carry a quantized Berry phase of $\pm π$, and connect orthogonally to the nodal surface, forming an intertwined nodal network. Drumhead surface states on the $(1\bar{1}0)$ surface further confirm the nontrivial topology. Porous-B$_{18}$ thus provides an ideal platform for investigating the interplay between nodal-line and nodal-surface fermions and exploring novel quantum transport phenomena.

Porous-B$_{18}$: An Ideal Topological Semimetal with Symmetry-Enforced Orthogonal Nodal-Line and Nodal-Surface States

Abstract

Topological semimetals (TSMs) featuring symmetry-protected band degeneracies have attracted considerable attention due to their exotic quantum properties and potential applications. While nodal line (NL) and nodal surface (NS) semimetals have been extensively studied, the realization of a material where both NL and NS coexist and are intertwined, particularly with an ideal electronic band structure, remains a significant challenge. Here, we predict via first-principles calculations and symmetry analysis a metastable boron allotrope, Porous-B (space group , No.~176), as a pristine TSM hosting a NS and two straight NLs near the Fermi level. The structure, a honeycomb-like porous 3D framework, exhibits excellent dynamical, thermal (stable up to 1000~K), and mechanical stability. Its electronic band structure is remarkably clean: only the highest valence band (HVB) and the lowest conduction band (LCB) cross linearly within a large energy window of 1.84~eV, free from trivial-band interference. The nodal surface lies on the planes, protected by combined time-reversal symmetry () and twofold screw-rotational symmetry (), yielding a full-plane Kramers-like degeneracy. The two nodal lines along -- and -- are protected by inversion and time-reversal symmetries, carry a quantized Berry phase of , and connect orthogonally to the nodal surface, forming an intertwined nodal network. Drumhead surface states on the surface further confirm the nontrivial topology. Porous-B thus provides an ideal platform for investigating the interplay between nodal-line and nodal-surface fermions and exploring novel quantum transport phenomena.

Paper Structure

This paper contains 9 sections, 4 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. METHOD
  3. RESULTS
  4. Crystal Structure and Stability
  5. The Ideal Two-Band Crossing
  6. Symmetry-Enforced Nodal Surface
  7. Straight Nodal Lines
  8. Topological Surface States
  9. DISCUSSION AND CONCLUSION

Figures (4)

  • Figure 1: (Color online) Crystal structure and Brillouin zone (BZ) of Porous-B$_{18}$. (a) Top view of a $2\times2\times2$ supercell. (b) Perspective view of the primitive cell. (c) Side view of a $1\times1\times2$ supercell. (d) The B$_9$ cage as the basic structural unit. (e) The first BZ and the projected $(1\bar{1}0)$ surface BZ.
  • Figure 2: (Color online) Stability analysis of Porous-B$_{18}$. (a) Phonon dispersion spectrum, showing no imaginary frequencies. (b) Ab initio molecular dynamics (AIMD) simulation results at 600 K (green) and 1000 K (blue) for 8 ps. The inset shows the snapshot of the $2\times2\times3$ supercell after 8 ps at 1000 K, which maintains its structural integrity.
  • Figure 3: (Color online) Electronic properties of Porous-B$_{18}$. (a) Electronic band structure along high-symmetry paths. The irreducible representations (irreps) of the crossing bands are labeled. (b) Projected density of states (PDOS). (c) Charge density distribution of the crossing states at point P. (d) Brillouin zone showing the nodal surface on the $k_z = \pm \pi$ planes and nodal lines along the K--H and K$'$--H$'$ directions. (e) Band dispersion along the direction normal to the nodal surface at a generic point X on the nodal surface. (f) 3D band crossing visualization in the $k_z = \pi$ (left) and $k_z = 0.9\pi$ (right) planes.
  • Figure 4: (Color online) Surface states on the $(1\bar{1}0)$ surface. (a) Surface band structure along the high-symmetry paths. The drumhead surface states (TSS) are highlighted. Constant energy contours at (b) $E = 0.08$ eV and (c) $E = 0.18$ eV, respectively, showing the surface states confined between the projections of the bulk nodal lines.