Doping evolution of the quasiparticle excitations in heavily hole-doped Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$: A possible superconducting gap with sign-reversal between hole pockets

To gain insight into the unconventional superconductivity of Fe pnictides with no electron pockets, we measure the thermal conductivity $\kappa$ and penetration depth $\lambda$ in the heavily hole-doped regime of Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$. The residual thermal conductivity ${(\kappa /T)}_{T\to 0\mathrm{K}}$ and $T$ dependence of $\lambda$ consistently indicate the fully gapped superconductivity at $x=0.76$ and the (line) nodal superconductivity at higher hole concentrations. The magnitude of $\frac{\kappa }{T}{T}_c{|}_{T\to 0\mathrm{K}}$ and the slope of $\lambda \left(T\right)$ at low temperatures, both of which are determined by the properties of the low-energy excitations, exhibit a highly unusual nonmonotonic $x$ dependence. These results indicate a dramatic change of the nodal characteristics in a narrow doping range. We suggest that the observed $x$ dependence is naturally explained by a doping crossover of the gap function between the $s$-wave states with and without sign reversal between $\Gamma$-centered hole pockets.

Figure 1

Low-temperature thermal conductivity in Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ single crystals. (a) $\kappa /T$ vs ${(T/T_c)}^2$ for $x=1.0$, 0.93, 0.88. Inset shows $\kappa /T$ plotted against ${(T/T_c)}^{1.25}$ for $x=0.76$. (b) Field dependence of $\kappa /T$ normalized to the normal-state values ${\kappa }_n/T$. Each point is determined from the extrapolation of the temperature dependence to $T\to 0$ K at each field applied along the $c$ axis. For comparison, the data for Nb is also plotted [24]. Inset is an expanded view at low fields. Dashed lines are the guide to the eyes.

Figure 2

Change in the penetration depth as a function of $T/T_c$ in Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$. Each curve is vertically shifted for clarity. The solid lines are fits to Eq. (1) with $T_{\mathrm{imp}}/T_c=0.17$, 0.08, 0.15, 0.18, 0.08 for $x=1.0$, 0.93, 0.91, 0.88, 0.86, respectively. Inset shows a comparison between the data for $x=0.86$ and 0.76, the latter of which is fitted to the temperature dependence of a fully gapped state (dashed line).

Figure 3

Doping evolution of low-energy QP excitations in Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$. (a) (Top) $x$ dependence of $\frac{\kappa \left(T\right)}{T}{T}_c{|}_{T\to 0\mathrm{K}}$(red squares, left axis) and the coefficient $\alpha$ in $\delta \lambda \left(T\right)$ [see Eq. (1)] (blue triangles, right axis). (Bottom) $x$ dependence of $T_c$. (b) We associate the above nonmonotonic behavior with the gap evolution with the parameter $r$ from predominantly $S_{+-}^{hh}$ ($r=0.6$) to predominantly $S_{++}^{hh}$ ($r=-0.6$) through intermediate regime with accidental nodes in ${\Delta }_2\left(\varphi \right)$ [see Eq. (3)]. (c) and (d) Theoretically determined residual $\kappa \left(T\right)/(T/T_c)$ (c) and $d\lambda \left(T\right)/d(T/T_c)$ (d) as functions of gap parameter $r$ with several values of impurity scattering ${\Gamma }_0$.