Strong-coupling superconductivity in NaFe${}_{1-x}$Co${}_x$As: Validity of Eliashberg theory

We study the normal-state and superconducting properties of the NaFe${}_{1-x}$Co${}_x$As system by specific heat measurements. Both the normal-state Sommerfeld coefficient and superconducting condensation energy are strongly suppressed in the underdoped and heavily overdoped samples. The low-temperature electronic specific heat can be well fitted by either a one-gap or a two-gap BCS-type function for all the superconducting samples. The ratio ${\gamma }_N{T}_c^2/H_c^2\left(0\right)$ can nicely associate the neutron spin resonance as the bosons in the standard Eliashberg model. However, the value of $\Delta C/T_c{\gamma }_N$ near optimal doping is larger than the maximum value the model can obtain. Our results suggest that the high-$T_c$ superconductivity in the Fe-based superconductors may be understood within the framework of boson-exchange mechanism but significant modification may be needed to account for the finite-temperature properties.

Figure 1

(a) Phase diagram of NaFe${}_{1-x}$Co${}_x$As that shows $T_c$ (black solid square), $T_N$ (red open circle), and $T_s$ (blue open triangle). The dash lines are guided to the eye. (b) Specific heats of some samples plotted as $C$/$T$ vs $T$. (c) Fitted results of several models on the specific heat of the $x=0.2$ sample. The differences between the data and each model are given in (d).

Figure 2

The electronic specific heat (black open circle) of serial samples obtained as described in the text. The red lines blow $T_c$ are the fitted results of one-gap BCS function except for the $x=0$ sample, which shows no superconducting jump. Above $T_c$, the horizontal red lines are fitted ${\gamma }_N$.

Figure 3

Doping dependence of (a) ${\gamma }_N$ (black solid squares) and ${\gamma }_0$ (red open circles), (b) the condensation energy, (c) 2$\Delta \left(0\right)/k_B{T}_c$ and (d) $A_1/(A_1+A_2)$. All the dashed lines are guided to the eye.

Figure 4

(a) Theoretical result of $({\gamma }_N-{\gamma }_0)T_c^2/H_c^2\left(0\right)$ calculated from Eq. (2) as shown by the red line. The values of conventional superconductors fall into the shaded area. The open circles represent the values obtained in this paper, which give the corresponding $T_c/{\omega }_{ln}$ (b) The $T_c$ dependence of the resonance energy $E_R$, ${\omega }_{ln}$ and 2${\Delta }_2$. Both the solid and dashed lines are guides to the eye. (c) The doping dependence of $\Delta C/T_c({\gamma }_N-{\gamma }_0)$. (d) The corresponding $({\gamma }_N-{\gamma }_0)T_c^2/H_c^2\left(0\right)$ and $\Delta C/T_c({\gamma }_N-{\gamma }_0)$ for each sample. The solid line is the linear relationship between these two values as calculated from Eqs. (1) and (2).