Nonsymmorphic symmetry protected hourglass Dirac chain topology and conventional superconductivity in ZrIrGe

TL;DR

ZrIrGe is a stoichiometric, nonsymmorphic crystal that hosts symmetry-enforced hourglass band dispersions and a Dirac nodal ring due to glide mirror symmetry, and simultaneously becomes a bulk type-II superconductor with a full gap. First-principles calculations show a nontrivial $Z_2$ topology and robust surface states on the (100) surface, including drumhead states linked to the bulk Dirac ring. Experimental measurements confirm bulk superconductivity with $T_c\approx2.84$ K, weak-coupling $s$-wave behavior, and parameters consistent with conventional BCS physics, while SOC drives sizable band splitting and topological features near $E_F$. The coexistence of conventional superconductivity and symmetry-protected topological band structure makes ZrIrGe a compelling platform to study intrinsic topological superconductivity and the interplay between superconductivity and topological surface states in a clean, stoichiometric material.

Abstract

Ternary transition-metal germanide superconductors with nonsymmorphic symmetries offer promising platforms for symmetry-protected topological phases. In this work, we investigate ZrIrGe, which crystallizes in the nonsymmorphic TiNiSi-type structure. Electrical, magnetic, and specific heat measurements confirm bulk type-II superconductivity with a full gap and a transition temperature of 2.84(7) K, consistent with weak-coupling BCS behavior. First-principles calculations reveal hourglass-shaped bulk band dispersions and a Dirac chain composed of symmetry-protected fourfold-degenerate Dirac points, leading to drumhead-like surface states near the Fermi level. Additionally, ZrIrGe exhibits a nontrivial $\mathbb{Z}_2$ topological character, resulting in helical surface states that cross the Fermi level, making it a strong candidate for proximity-induced topological superconductivity. The coexistence of conventional superconductivity and topological band features establishes ZrIrGe as a rare stoichiometric system for exploring intrinsic topological superconductivity.

Figures (4)

• Figure 1: Structural and bulk superconductivity characterization of ZrIrGe: (a) Powder XRD patterns of ZrIrGe with Rietveld refinement, where the red markers represent experimental data, the black line indicates the theoretical refinement, and the vertical bars correspond to the Bragg positions. The line at the bottom depicts the difference between the observed and calculated data. The inset represents the TiNiSi-type crystal structure of the ZrIrGe compound. (b) Temperature-dependent electrical resistivity [$\rho(T)$] measured at H = 0 mT for ZrIrGe, with inset showing an enlarged view of $\rho(T)$, confirming the superconductivity at $T_{\rm c}$ = 2.83(5) K. (c) Magnetization as a function of temperature at an applied magnetic field H = 1.0 mT, measured in ZFCW and FCC mode, confirming superconductivity. The inset depicts the magnetic field-dependent magnetization at 1.8 K.
• Figure 2: Lower and upper critical fields and specific heat analysis of ZrIrGe: (a) Temperature dependence of the lower critical field $H_{c1}$, fitted using the Ginzburg-Landau (GL) relation, with the inset illustrating the low-field magnetization behavior at various isotherms. (b) The upper critical field $H_{c2}$, was determined from resistivity and magnetization measurements, fitted to the GL relation. The inset presents resistivity curves at different magnetic fields for ZrIrGe. (c) The temperature-dependent electronic specific heat C$_{el}$, obtained after subtracting the phonon contribution, fitted with isotropic s-wave model. The inset showing specific heat C/T vs $T^{2}$ plot confirms superconductivity, with the data in the normal state well-fitted using the Debye model, allowing extraction of the phonon contribution.
• Figure 3: Electronic band structure, hourglass dispersions, Dirac rings and surface states of ZrIrGe: (a) Bulk electronic band structure of ZrIrGe calculated in the absence of spin-orbit coupling (SOC). (b) Corresponding nodal ring centered at the $\Gamma$ point in the $k_x = 0$ plane. (c) The three-dimensional bulk Brillouin zone (BZ) and its projection onto the (100) surface, with high-symmetry points and paths marked by red dots and blue lines, respectively. (d) Orbital-resolved density of states of ZrIrGe without including spin-orbit coupling (SOC). (e) Fermi surface including SOC, showing multiple sheets across the Brillouin zone. (f) Electronic band structure with SOC included. (g,h) Hourglass-like band dispersions along the high-symmetry directions $S-R$, and $S-K$, where $K$ denotes the midpoint between $T$ and $R$. The inset in (h) schematically depicts a fourfold-degenerate Dirac ring, formed by neck points (red dots) of the hybrid hourglass dispersion on the $k_x = \pi$ plane. (i) Distribution of the hourglass Dirac ring (shown in white) encircling the $S$ point, with the color scale representing the local band gap. (j) Surface state spectrum along high-symmetry paths of the projected (100) two-dimensional surface Brillouin zone. (k) Constant-energy slice of the surface spectrum at $-0.05$ eV, revealing topological surface Fermi arcs.
• Figure 4: The Uemura plot between T$_{c}$ and T$_{F}$ for ZrIrGe is indicated by a red diamond marker positioned near the conventional superconductivity range.