Symmetry-unprotected nodes or gap minima in the $s_{++}$ state of monocrystalline FeSe

We report a study on the superconducting pairing mechanism of FeSe via the pair-breaking effects induced by ${\mathrm{H}}^{+}$ irradiation combined with low-temperature specific heat measurements. A multigap structure with nodes or gap minima is suggested in a clean FeSe by the specific heat results. The suppression of critical temperature $T_c$ with increasing the defect density manifests a two-step behavior. When the increase in the residual resistivity is small, $\mathrm{\Delta }{\rho }_0\lesssim 4.5\mu \mathrm{\Omega }\mathrm{cm}$, $T_c$ is gradually suppressed with increasing the density of the scattering centers, suggesting the presence of symmetry-unprotected nodes or gap minima. However, for $\mathrm{\Delta }{\rho }_0\gtrsim 4.5\mu \mathrm{\Omega }\mathrm{cm}$, $T_c$ is almost independent of the scattering, which indicates that the nodes or gap minima are lifted and the order parameter becomes almost isotropic without a sign change. Thus, the superconductivity in FeSe is found to be realized in a symmetry-unprotected nodal or highly anisotropic $s_{++}$ state.

Figure 1

(a) Specific heat divided by temperature, $C/T$ vs $T^2$, measured under 0 and 9 T. The solid line represents the fit to the normal-state specific heat. The inset is the temperature dependence of magnetic susceptibility $\chi$. (b) Normalized zero-field electronic specific heat $C_e/{\gamma }_{n}T$ vs $T$ together with the fit lines of $s+s$ wave ($2{\mathrm{\Delta }}_L/k_B{T}_c=4.35$, $2{\mathrm{\Delta }}_S/k_B{T}_c=0.74$, ${\gamma }_1=0.74$, ${\gamma }_2=0.26$), $s+\text{nodal}s$ wave ($2{\mathrm{\Delta }}_s/k_B{T}_c=4.35$, $2{\mathrm{\Delta }}_{ns}/k_B{T}_c=1.06$, ${\gamma }_1=0.74$, ${\gamma }_2=0.26$), and $s+\text{extended}s$ wave ($2{\mathrm{\Delta }}_s/k_B{T}_c=4.35$, $2{\mathrm{\Delta }}_e/k_B{T}_c=0.85$, ${\gamma }_1=0.71$, ${\gamma }_2=0.29$, $\alpha =0.72$). The inset is the enlarged low-temperature part. (c) XRD patterns for FeSe before and after irradiating ${\mathrm{H}}^{+}$ up to $5\times {10}^{16}{\mathrm{ions}/\mathrm{cm}}^2$. The inset is the enlarged part of the (004) peaks.

Figure 2

(a) Temperature dependence of the resistivity for FeSe irradiated by ${\mathrm{H}}^{+}$ with doses of 0, 0.2, 0.5, 1, 2, 5, and $10\times {10}^{16}{\mathrm{ions}/\mathrm{cm}}^2$. Dashed lines are linear extrapolations to zero temperature for estimating the residual resistivity ${\rho }_0$. (b) Dose dependence of $T_c^{\text{onset}}$ and $T_c^{\text{mid}}$, and the difference in the residual resistivity before and after irradiation ($\mathrm{\Delta }{\rho }_0$).

Figure 3

(a) $T_c^{\text{onset}}$ and $T_c^{\text{mid}}$ as a function of $\mathrm{\Delta }{\rho }_0$ for FeSe. The dashed lines show the definition of the crossover point of $\mathrm{\Delta }{\rho }_0^{*}$ by linearly extrapolating the $T_c$ in two steps. The inset is the normalized $T_c/T_{c0}$ as a function of $\mathrm{\Delta }{\rho }_0$. (b)–(d) Schematic evolution of order parameters as the density of the scattering centers is increased.