Field dependence of thermal conductivity in the iron-based superconductor KFe$_{2}$As$_{2}$: Evidence of a $d$-wave state

Pairing symmetry in the iron-arsenide superconductor KFe$_{2}$As$_{2}$ is the subject of active debate. Thermal conductivity at milliKelvin temperatures is a sensitive and directional probe of the superconducting gap structure. Thermal conductivity measured along both [100] and [001] directions reveal that all Fermi surfaces in KFe$_{2}$As$_{2}$ must have vertical line nodes. This stringent constraint is automatically satisfied by $d$-wave symmetry, but not likely to be satisfied by an $s$-wave state, where nodes are accidental. Here, we report a detailed study of the magnetic field dependence of thermal conductivity in KFe$_{2}$As$_{2}$, measured in the $T =$ 0 limit. The data are found to be in excellent agreement with $d$-wave calculations. Our data are also compatible with low-temperature specific heat data as a function of field, within a multi-band $d$-wave scenario. Using Fermi surface parameters from quantum oscillations, we estimate the thermal conductivity expected of the gap structure extracted from ARPES measurements on KFe$_{2}$As$_{2}$ for the different Fermi surface sheets. We find a result that is incompatible with our thermal conductivity data, and conclude that the superconducting state at the surface, accessed by ARPES, must be different from the state in the bulk, accessed by transport and thermodynamic measurements.