Specific heat of ${\mathrm{Ca}}_{0.32}{\mathrm{Na}}_{0.68}{\mathrm{Fe}}_2{\mathrm{As}}_2$ single crystals: Unconventional $s_{\pm }$ multiband superconductivity with intermediate repulsive interband coupling and sizable attractive intraband couplings

We report a low-temperature specific heat study of high-quality single crystals of the heavily hole-doped superconductor ${\mathrm{Ca}}_{0.32}{\mathrm{Na}}_{0.68}{\mathrm{Fe}}_2{\mathrm{As}}_2$. This compound exhibits bulk superconductivity with a transition temperature $T_{\mathrm{c}}\approx 34$ K, which is evident from the magnetization, transport, and specific heat measurements. The zero-field data manifest a significant electronic specific heat in the normal state with a Sommerfeld coefficient $\gamma \approx 53$ mJ/mol K${}^2$. Using a multiband Eliashberg analysis, we demonstrate that the dependence of the zero-field specific heat in the superconducting state is well described by a three-band model with an unconventional s${}_{\pm }$ pairing symmetry and gap magnitudes ${\Delta }_i$ of approximately 2.35, 7.48, and $-$7.50 meV. Our analysis indicates a non-negligible attractive intraband coupling, which contributes significantly to the relatively high value of $T_c$. The Fermi surface averaged repulsive and attractive coupling strengths are of comparable size and outside the strong coupling limit frequently adopted for describing high-$T_c$ iron pnictide superconductors. We further infer a total mass renormalization of the order of five, including the effects of correlations and electron-boson interactions.

Figure 1

(a) The $T$ dependence of the magnetization of CaFe${}_2$As${}_2$ and Ca${}_{0.32}$Na${}_{0.68}$Fe${}_2$As${}_2$ single crystals, measured under an applied magnetic field of 1 T parallel to the crystallographic basal plane in zero-field cooled conditions. (b) $T$ dependence of the in-plane electrical resistivity in zero field up to 300 K. The inset presents a zoom of the resistivity data around $T_c$ for the Ca${}_{0.32}$Na${}_{0.68}$Fe${}_2$As${}_2$ sample.

Figure 2

The temperature dependence of the specific heat $C_p$ measured in zero-field conditions for CaFe${}_2$As${}_2$ and Ca${}_{0.32}$Na${}_{0.68}$Fe${}_2$As${}_2$. The inset shows the low-temperature behavior of $C_p/T$ as a function of $T^2$. The straight lines are the linear fits to $C_p/T=\gamma +\beta T^2$ (see text).

Figure 3

The $T$ dependence of the specific heat $C_p/T$ of Ca${}_{0.32}$Na${}_{0.68}$Fe${}_2$As${}_2$ single crystal measured in magnetic fields applied along the crystallographic $c$ axis. For reference, the specific heat of CaFe${}_2$As${}_2$ measured in zero-field conditions is also shown. The inset shows $C_p/T$ of Ca${}_{0.32}$Na${}_{0.68}$Fe${}_2$As${}_2$ near $T_c$.

Figure 4

The electronic specific heat of Ca${}_{0.32}$Na${}_{0.68}$Fe${}_2$As${}_2$ after subtracting the phonon contribution as a function of reduced temperature $t=T/T_{\mathrm{c}}$. In the inset, the normal and superconducting state entropies are shown.

Figure 5

(a) The assumed spectrum of phonons (black/dark) and spin fluctuations (red/light). (b) The temperature dependence of the gap functions ${\Delta }_i({\omega }_n=\pi /\beta )$. The dashed red line in (b) is a rescaled version of the solid red line, to show the non-BCS-like temperature dependence of the superconducting gap for this band (see main text). (c) A comparison between the calculated (thick black line) and measured (open $◯$) change in electronic specific heat $\Delta C_{\mathrm{el}}\left(T\right)$ as a function of the reduced temperature $T/T_c$. The individual band contributions are also shown for $T<T_c$, following the color scheme of (b).

Figure 6

(a) The electronic density of state (DOS) from LDA-FPLO calculation for Ca${}_{0.32}$Na${}_{0.68}$Fe${}_2$As${}_2$ with the various elemental contributions from Fe 3$d$, As 4$p$, and Ca 4$s$ states. (b) The Fermi surface sheets (FSSs) band-resolved contributions. Bands $h_{1\text{−}3}$ include the three hole-type FSSs centered around the $\Gamma$ point (see Fig. 7) and $e_{1,2}$ are the two electron-type FSSs (see Fig. 8), respectively. (c) Orbital resolved partial DOS for the outer hole FSSs $h_3$ (see Fig. 7), lower panel). The 3$d_{xz}$ and 3$d_{yz}$ are degenerate in the present tetragonal symmetry and have the same weight. The notation of orbitals is the same as that used in the ARPES and dHvA literature for Fe pnictides, i.e., there is a 45${}^{\circ }$ rotation of the $x$ and $y$ axes with respect of the original tetragonal axis of the the Bravais cell.

Figure 7

The three hole Fermi surface sheets (FSSs) around the $\Gamma$ point formed by the bands $h_1$ (upper), $h_2$ (middle), and $h_3$ (bottom) according to our LDA calculation. The color denotes the magnitude of the Fermi velocities. Within scenario I, we tentatively assign the upper ($h_1$) and the middle ($h_2$) FSSs to band 1 in our three-band Eliashberg analysis (see subsection D) and the lower FSS ($h_3$) to band 2. Within scenario II, the band $h_2$, only, forms band 1, whereas the two remaining hole bands $h_1$ and $h_3$ form the effective band 2. The color scale denotes the magnitude of the calculated Fermi velocities.

Figure 8

The two electron FSSs resulting from bands $e_1$ and $e_2$ according to both scenarios I and II are tentatively assigned to to the effective band 3 in our Eliashberg analysis of the specific heat (see Sec. 3 D).