Superconducting specific-heat jump $\Delta C_{\mathrm{el}}\propto T_c^{\beta }(\beta \approx 2)$ for ${\text{K}}_{1-x}$${\text{Na}}_x$${\text{Fe}}_2$${\text{As}}_2$

We present a systematic study of the electronic specific-heat jump ($\Delta C_{\mathrm{el}}$) at the superconducting transition temperature $T_c$ of ${\text{K}}_{1-x}$${\text{Na}}_x$${\text{Fe}}_2$${\text{As}}_2$. Both $T_c$ and $\Delta C_{\mathrm{el}}$ monotonously decrease with increasing $x$. The specific heat jump scales approximately with a power law, $\Delta C_{\mathrm{el}}\propto T_c^{\beta }$, with $\beta \approx 2$ determined by the impurity scattering rate, in contrast to most iron-pnictide superconductors, where the remarkable Bud'ko-Ni-Canfield (BNC) scaling $\Delta C_{\mathrm{el}}\propto T^3$ has been found. Both the $T$ dependence of $C_{\mathrm{el}}\left(T\right)$ in the superconducting state and the nearly quadratic scaling of $\Delta C_{\mathrm{el}}$ at $T_c$ are well described by the Eliashberg theory for a two-band $d$-wave superconductor with weak pair breaking due to nonmagnetic impurities. The disorder induced by the Na substitution significantly suppresses the small gaps, leading to gapless states in the slightly disordered superconductor, which results in a large observed residual Sommerfeld coefficient in the superconducting state for $x>0$.

Figure 1

(a) Temperature dependence of the volume susceptibility ${\chi }_v$ measured in a magnetic field of $H\parallel ab$ = 5 Oe. (b) Temperature dependence of the molar susceptibility ${\chi }_m$ measured in a magnetic field of $H\parallel ab$ = 10 kOe. Inset: Magnetization curves measured in an external field $H\parallel ab$. For explanation of the fitting curves see [20].

Figure 2

(a) Total specific heat $C/T$ of ${\text{K}}_{1-x}$${\text{Na}}_x$${\text{Fe}}_2$${\text{As}}_2$ vs $T$. Open symbols: zero field data; closed symbols: data in an applied magnetic field $H\parallel c$ = 15 kOe; solid lines: the fit of the normal-state specific heat (see text). The experimental data for $x=0.29$ at $T<$ 0.4 K are taken from Ref. [12]. Inset: the specific heat $\Delta C_{\mathrm{el}}/T=[C\left(0\right)-C\left(H\right)]/T$ vs $T$ for a sample with $x=0$ and 0.29. Dots: experimental data; filled rhombus: data from Ref. [16] shown for comparison; solid line: calculation within two-band Eliashberg theory for $d$-wave superconductors with nonmagnetic impurities and $T_{c0}$ = 3.44 K for the adopted clean limit. (b) $\Delta C_{\mathrm{el}}/T$ near $T_c$ for different samples of ${\text{K}}_{1-x}$${\text{Na}}_x$${\text{Fe}}_2$${\text{As}}_2$. Solid lines are plotted according to the entropy equal-area construction method for determining $T_c$ and $\Delta C_{\mathrm{el}}/T_c$.

Figure 3

(a) The dependence of $\Delta C_{\mathrm{el}}$ vs $T_c$ for different iron-pnictide superconductors. Solid curve: obtained within Eliashberg theory; dotted curve: the best fit of our experimental data by $\Delta C_{\mathrm{el}}\propto T_c^{\beta }$; and dashed curve: BNC scaling $\Delta C_{\mathrm{el}}\propto T_c^3$ [1, 10]. The data are taken from Refs. [21, 22, 23, 24, 26] for ${\text{KFe}}_2$${\text{As}}_2$, ${\text{CsFe}}_2$${\text{As}}_2$ [38], ${\text{RbFe}}_2$${\text{As}}_2$ [37], (Ba,K)${\text{Fe}}_2$${\text{As}}_2$ and Ba(Fe,TM)${}_2$${\text{As}}_2$ [10] (TM – transition metal), LiFeP [28], ${\text{Ca}}_{0.32}$${\text{Na}}_{0.68}$${\text{Fe}}_2$${\text{As}}_2$ [44], (Ba,Na)${\text{Fe}}_2$${\text{As}}_2$ [45], ${\text{SrFe}}_2$(${\text{As}}_{0.65}$${\text{P}}_{0.35}$)${}_2$ [46], Li(Fe,Cu)As [31, 32], ${\text{Ba}}_{0.55}$${\text{K}}_{0.45}$${\text{Fe}}_{1.95}$${\text{Co}}_{0.05}$${\text{As}}_2$ [47], ${\text{LaFeAsO}}_{0.9}$${\text{Fe}}_{0.1}$ [20, 48], ${\text{BaFe}}_2$(${\text{As}}_{0.7}$${\text{P}}_{0.3}$)${}_2$ [29]. (b) $\Delta C_{\mathrm{el}}/T_c$ vs $T_c$ for (K,Na)${\text{Fe}}_2$${\text{As}}_2$. Inset: $T_{\mathrm{c}}$ vs residual resistivity ${\rho }_0$ and the Na substitution level $x$. Solid line: fit using the Abrikosov-Gor'kov formula [20], and the dashed curve: guide to the eye.