Calorimetric evidence of nodal gaps in the nematic superconductor FeSe

Superconductivity in FeSe has recently attracted a great deal of attention because it emerges out of an electronic nematic state of elusive character. Here we study both the electronic normal state and the superconducting gap structure using heat-capacity measurements on high-quality single crystals. The specific-heat curve, from 0.4 K to $T_c=9.1\mathrm{K}$, is found to be consistent with a recent gap determination using Bogoliubov quasiparticle interference [P. O. Sprau et al., Science 357, 75 (2017)]; however, only if nodes are introduced on either the electron or the hole Fermi-surface sheets. Our analysis, which is consistent with quantum-oscillation measurements, relies on the presence of one hole and one electron band only, and thus the fate of the theoretically predicted second electron pocket remains mysterious.

Figure 1

(a) Resistivity of samples 1 and 2. The inset shows a magnification of the low-temperature region. The dashed line is a linear extrapolation of the normal-state resistivity to $T=0.$ (b) Heat capacity of samples 1 and 2. (c) Comparison of the magnetic susceptibility of sample 1 to that of ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ reproduced from Ref. [27]. In (b) and (c), the insets show data around $T_S=91$ K on an enlarged scale.

Figure 2

(a) Low-temperature $C_e\left(T\right)$ of samples 1, 2, 3, and 4 in 0 and 14 T. The dashed line represents an entropy–conserving construction that yields $T_c\approx$ 9.1 K. The inset shows the low-temperature data on an enlarged scale, revealing a linear $C_e/T$ for $T\le 2$ K indicative of line nodes (solid line). (b) Mixed-state heat capacity $\gamma (T,H)$ of samples 1 and 5 at $T=0.6$ K. The inset shows $\gamma (T,H)$ as a function of $H^{1/2}$ in the low-field region.

Figure 3

(a) Angle-resolved nodeless hole and electron superconducting gaps, ${\mathrm{\Delta }}_{h,e}\left(\varphi \right)$, reported by BQPI/STM [8] (green and blue symbols, respectively). $\varphi$ is measured with respect to the $a$ orthorhombic axis. (d), (g), (j) Nodal gap structures used for heat-capacity calculations. Solid lines are fits within the leading angular harmonic approximation. (b), (e), (h), (k) Comparison of the heat capacity (red line) calculated using the gap structures shown in panels (a), (d), (g), (j), respectively, to the measured $C_e\left(T\right)$ of sample 2. (c), (f), (i), (l) show the respective low-temperature part on an enlarged scale. The green and blue lines are the individual hole and electron heat-capacity contributions, respectively.

Figure 4

(a) $C_e\left(T\right)$ of samples 1, 2, 5, and 6 with different amounts of disorder. The inset shows the low-temperature part in normalized units on an enlarged scale.

Figure 5

Temperature dependence of the specific heat of the seven single crystals discussed in this paper in 0 T (closed symbols) and 14 T (open symbols) for H $\parallel$ c. For clarity, curves are vertically shifted from each other by $20\mathrm{mJ}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-2}$. Red curves represent fits to the 14 T data using Eq. (B1). The inset shows the resulting electronic heat capacity for $H=14\mathrm{T}$.

Figure 6

Comparison of $C_e\left(T\right)$ of FeSe to that of multiband superconductors ${\mathrm{MgB}}_2$ and ${\mathrm{KFe}}_2{\mathrm{As}}_2$ taken from Refs. [31, 32]. The arrow indicates the hump related to the small gaps ${\mathrm{\Delta }}_S\left(T\right)$, i.e., where ${\mathrm{\Delta }}_S\left(T\right)\approx k_{B}T$.

Figure 7

Comparison of published ${\mathrm{C}}_{\mathrm{e}}\left(\mathrm{T}\right)$ [samples grown by solid-state reaction (gray) [33], ${\mathrm{KCl}:\mathrm{AlCl}}_3$ flux (dark yellow) [21], and chemical vapor transport (red) 9, 22, 23, 24, 66], to our samples with similar $T_c$ values. The inset shows data below 3 K on an enlarged scale. The dotted line is an entropy-conserving construction to determine $T_c$ in our samples. The continuous red lines shown in (b) and (c) are Schottky contributions $C_{\text{Sch}}(T,n)$ calculated with Eq. (D1) for the samples of Sun et al. [24] and Jiao et al. [22], respectively. The difference $C_e\left(T\right)-C_{\mathrm{Sch}}(T,n)$ for both samples is compared in (f) to $C_e\left(T\right)$ of sample 5 and Lin et al. [21].

Figure 8

(a) Comparison of the low temperature $C_e/T$ of samples 2 and 7 to that of Chen et al. [9]. The 1 K anomaly is only observed in our samples for measurements performed upon slow cooling. (b) Comparison of the mixed-state heat capacity reported by Sun et al. [24] (red curve) to that of sample 1 (light green curve). The dark green curve shows the erroneous heat capacity of sample 1 obtained when the field dependence of the addenda is not taken into account. (c) Field dependence of the addenda (dark yellow curve) and sample (light green curve) contributions to the total heat capacity (dark blue curve). We find that the addenda in our setup have a strong field dependence in low fields.