Hybrid $s$-wave superconductivity in ${\mathrm{CrB}}_2$

In a metal with multiple Fermi pockets, the formation of $s$-wave superconductivity can be conventional due to electron-phonon coupling or unconventional due to spin fluctuations. We analyze the hexagonal diboride ${\text{CrB}}_2$, which is an itinerant antiferromagnet at ambient conditions and turns superconducting upon increasing pressure. While the high-pressure behavior of $T_c$ suggests conventional $s$-wave pairing, we find that spin fluctuations promoting unconventional $s$-wave pairing become important in the vicinity of the antiferromagnetic dome. As the symmetry class of the $s$-wave state is independent of its underlying mechanism, we argue that ${\text{CrB}}_2$ is a realization of a hybrid $s$-wave superconductor where unconventional and conventional $s$-wave mechanisms team up to form a joint superconducting dome.

Figure 1

Proposed schematic temperature-pressure phase diagram for a hybrid $s$-wave superconductor. Spin fluctuation limit: Antiferromagnetic region with maximum $T_{\text{AFM}}$ at ambient pressure (blue region), dropping rapidly with pressure and followed by (possibly overlapping with) a smaller unconventional superconducting dome near the quantum critical point (red region). Electron-phonon limit: $T_c$ is only weakly dependent on pressure and dominates the high-pressure part of the phase diagram (light-orange region). The hybrid $s$ wave (green) appears in the crossover region between the two limits where the pair-breaking impact of spin fluctuations is vital for explaining the drop of $T_c$ towards ambient pressure.

Figure 2

Electronic band structure and density of states (DOS) at 100 GPa. Fat bands and DOS of Cr $3d$ states (cyan) and B $2p$ states (red) show that both states contribute to the formation of the Fermi surface.

Figure 3

Spin-fluctuation pairing. Eigenvalues ${\lambda }_i$ of the leading and subleading instabilities as a function of pressure as calculated with $U=0.124\text{eV}$ where the critical pressure is $p_c=16\text{GPa}$. All calculations are at fixed temperature $T=0.02\mathrm{eV}$. Insets: Gap function $g_i\left(\mathbf{k}\right)$ on the Fermi surface of the leading $s$-wave (black circles) state and $d/g$-wave (red circles) state for two representative pressures of 26 and 60 GPa as indicated by the arrows. Red and blue regions represent two opposite signs of the gap function.

Figure 4

Phonon dispersion, electron-phonon (EP) linewidth, spectral function $\left[{\alpha }^{2}F\left(\omega \right)\right]$, and phonon density of states (ph DOS) of the relaxed structure at 100 GPa. Normalized percentage of EP linewidth (orange) is equal to $\gamma /{\gamma }_{\max }\times 100\%$ and is strongly peaked at zone center, $\mathrm{\Gamma }$.