Symmetry imposed line nodes in the superconducting gap of KFe$_2$As$_2$

The variation with temperature of the London penetration depth, $\Delta\lambda(T)$, was measured in single crystals of K$_{1-x}$Na$_x$Fe$_2$As$_2$ ($x=0,~0.07$) by using a tunnel diode resonator down to 50 mK. Electrical resistivity measurements on the same samples show that isoelectronic Na-substitution significantly increases residual resistivity $\rho_0$ (from 0.2 to 1 $\mu\Omega$), without changing the shape of $\rho(T)$, and changes $T_c$ from 3.6 K to 2.9 K for $x=0$ and 0.07 samples. In both pure and doped compounds, the penetration depth follows the power-law function, $\Delta\lambda(T)=AT^n$ below $0.3\,T_c$ with $n\approx 1.5$ and 2.0 for $x=0$ and 0.07 samples, respectively. This behavior is consistent with presence of line nodes in the superconducting gap with moderate scattering for $x=0$ evolving into dirty limit for $x=0.07$. The normalized superfluid density, $\rho_s=\lambda^2(0)/\lambda^2(T)$ was calculated with $\lambda(0)=200$ nm and 500 nm for $x=0$ and $x=0.07$, respectively. Detailed investigation of the calculated $\rho_s$ strongly supports the existence of symmetry imposed line nodes in KFe$_2$As$_2$ superconductor and is consistent with thermal conductivity data that concluded $d-$wave pairing in this compound.