Nodeless superconductivity of single-crystalline ${\mathrm{K}}_x$Fe${}_{2-y}$Se${}_2$ revealed by the low-temperature specific heat

Low-temperature specific heat (SH) has been measured in ${\mathrm{K}}_x$Fe${}_{2-y}$Se${}_2$ single crystals with $T_c=32$ K. The SH anomaly associated with the superconducting transition is moderate and sharp, yielding a value of $\Delta C/T|{}_{T_c}=11.6\pm 1.0$ mJ/mol ${\mathrm{K}}^2$. The residual SH coefficient $\gamma (0)$ in the superconducting state at $T\to 0$ is very small with a value of about 0.39 mJ/mol ${\mathrm{K}}^2$. The magnetic-field-induced enhancement of the low-T SH exhibits a rough linear feature indicating a nodeless gap. This is further supported by the scaling based on the nodeless-gap approach of the low-T data at different magnetic fields. A rough estimate indicates that the normal-state SH coefficient ${\gamma }_n$ is about $6\pm 0.5$ mJ/mol ${\mathrm{K}}^2$, leading to $\Delta C/{\gamma }_{n}T|{}_{T_c}=1.93$ and placing this superconductor in the strong coupling region.

Figure 1

Raw data of the temperature dependence of specific heat for the ${\mathrm{K}}_x$Fe${}_{2-y}$Se${}_2$ at 0 and 9 T. The inset shows the temperature dependence of the dc magnetization for the same sample measured under a magnetic field of 20 Oe.

Figure 2

Main panel shows the SH data plotted as $C/T$ vs. $T^2$. The inset shows an enlarged view of the data together with the fit (see text) in the low-temperature region. No Schottky anomaly was detected.

Figure 3

Upper panel shows an enlarged view of the SH data near the transition temperature plotted as $C/T$ vs. $T$. The lower panel shows the difference in the SH data between 0 and 9 T. The blue solid lines here are just guides to the eye to show how we determine the height of the SH anomaly.

Figure 4

(a) Magnetic field background measured for the bare chip, $C_{\text{bkg}}(H)=C(H)-C(0)$, raw data are plotted as $C$ vs. $T$. (b) Enlarged view of the SH data of the sample in the low-T region for 9 T, before and after subtracting the background.

Figure 5

SH data in the low-temperature region under various magnetic fields. Inset (a): The difference in the SH data between a particular magnetic field and zero field. The dashed lines are the linear fit of this difference between 2.4 and 4 K. The slight bending down of the data at 5, 7, and 9 T was recognized as due to the unsuccessful removal of the magnetic field effect on the SH measuring chip. Inset (b): Field dependence of the field-induced term $\Delta \gamma (H)=[C(T,H)-C(T,0)]/T$ at $T=3$ K.

Figure 6

(a) Scaling of the data according to the nodeless-gap scenario (symbols) $C_{\text{cal}-s}=[C(H)-C(0)]/T^3$ vs. $T/\sqrt{H}$, the dashed line represents the theoretical expression. (b) Scaling of the data (symbols) based on $d$-wave prediction $C_{\text{cal}-d}=[C(H)-C(0)]/T\sqrt{H}$ vs. $T/\sqrt{H}$. No good scaling can be found for the $d$-wave case.