Sign-changing nodal $s$-wave gap in heavily over doped (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ evidenced from thermal-transport measurement

The superconducting state of hole-doped Fe-based superconductors, (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$, changes from a fully-gapped state near the optimally doping ($x\sim$0.5) to a nodal one at the end material ($x$= 1)[1,2]. Here we report the results of thermal-transport measurements for heavily overdoped $x$=1, 0.93, 0.88, 0.76 crystals and discuss the doping evolution of the superconducting gap. For $x$=0.88, 0.93 and 1, the $T$-dependence of thermal conductivity in zero field shows a finite $\kappa _0/T$ in the zero-temperature limit. In low magnetic fields, $\kappa/T$ ($T\to 0$ K) increases as $\propto \sqrt{H}$. These results indicate the presence of gap nodes in the gap function. We find that the residual $\kappa _0/T$ exhibits a non-monotonic $x$-dependence, which is inconsistent with $d$-wave symmetry. We show that the observed $x$-dependence can be explained by nodal $s$-wave pairing with sign change between zone centered hole pockets. [1] K. Hashimoto {\it et al.}, Phys. Rev. B {\bf 82}, 014526 (2010). [2] K. Okazaki {\it et al.}, Science {\bf 337}, 1314 (2012).