Specific Heat Measurements of ${\mathrm{Ba}}_{0.68}{\mathbf{K}}_{0.32}{\mathrm{Fe}}_2{\mathrm{As}}_2$ Single Crystals: Evidence for a Multiband Strong-Coupling Superconducting State

The specific heat of high-purity ${\mathrm{Ba}}_{0.68}{\mathrm{K}}_{0.32}{\mathrm{Fe}}_2{\mathrm{As}}_2$ single crystals with the highest reported superconducting $T_c=38.5\mathrm{K}$ was studied. The electronic specific heat $C_p$ below $T_c$ shows two gap features, with ${\Delta }_1\approx 11\mathrm{meV}$ and ${\Delta }_2\approx 3.5\mathrm{meV}$ obtained from an $\alpha$-model analysis. The reduced gap value, $2{\Delta }^{\max }/k_B{T}_c\approx 6.6$, the magnitude of the specific-heat jump, $\Delta C_p(T_c)/T_c$, and its slope below $T_c$ exhibit a strong-coupling character. We also show that an Eliashberg model with two hole and two electron bands gives the correct values of $T_c$, the superconducting gaps, and the free-energy difference.

Figure 1

(a) Temperature dependence of the specific heat $C_p/T$ of ${\mathrm{Ba}}_{0.68}{\mathrm{K}}_{0.32}{\mathrm{Fe}}_2{\mathrm{As}}_2$ (squares) and $\mathrm{Ba}({\mathrm{Fe}}_{0.88}{\mathrm{Mn}}_{0.12}{)}_2{\mathrm{As}}_2$ (circles). The inset shows a plot of $C_p/T$ vs $T^2$ at low $T$ for both samples. The lines represent the best fit to $C_p(T)/T=\gamma (0)+\beta T^2$. (b,c) Temperature dependence of $C_p$ of ${\mathrm{Ba}}_{0.68}{\mathrm{K}}_{0.32}{\mathrm{Fe}}_2{\mathrm{As}}_2$ near $T_c$ measured at different external magnetic fields applied perpendicular to the $ab$ plane.

Figure 2

Superconductivity-induced specific-heat difference of ${\mathrm{Ba}}_{0.68}{\mathrm{K}}_{0.32}{\mathrm{Fe}}_2{\mathrm{As}}_2$. The black solid curve is the result of a fit according to the two-gap $\alpha$ model. The dashed and dash-dotted curves represent the partial contributions of the two bands. The light gray line is the electronic specific heat calculated from the four-band Eliashberg model. Inset: Temperature dependence of the thermodynamic critical field (see text).

Figure 3

Comparison of the dimensionless superconducting jump, energy gap at $T=0$, and slope of the specific heat near $T_c$ calculated by the semiphenomenological expressions in Ref. 19 (lines) with our results for ${\mathrm{Ba}}_{0.68}{\mathrm{K}}_{0.32}{\mathrm{Fe}}_2{\mathrm{As}}_2$ (open circles), and for ${\mathrm{Pb}}_{0.8}{\mathrm{Bi}}_{0.2}$ (triangles) and In (squares), after Ref. 19. The open and solid circles represent the maximum gap values obtained by the two-band $\alpha$ and four-band Eliashberg model, respectively. Shaded areas represent the uncertainty in ${\omega }_{\ln }/T_c$ according to the error bars in our data points.

Figure 4

Free-energy difference between normal and superconducting states, experimentally obtained (open circles) and calculated from the four-band Eliashberg model (light gray solid line). The dashed, dash-dotted, and dotted lines represent the partial contributions of the individual bands. Inset: Spin-fluctuation coupling function $B(\Omega )$.