Fully gapped $s$-wave superconductivity in KOs$_2$O$_6$

The discovery of superconductivity in the $\beta$-pyrochlore oxides AOs$_2$O$_6$ (A=Cs, Rb, and K) has attracted so much attention, because the geometric spin frustration inherent to their pyrochlore crystal structures is supposed to give rise to unconventional superconductivity via magnetic spin fluctuations. Until now experimental results suggest, however, that CsOs$_2$O$_6$ ($T_c=3.3$ K) and RbOs$_2$O$_6$ ($T_c=6.3$ K) are fully gapped $s$-wave superconductors. On the other hand, the experimental data of KOs$_2$O$_6$ ($T_c=9.6$ K) show somewhat unusual behaviors, pointing out in some cases to unconventional superconductivity. In this talk we will discuss magnetic penetration depth data of single crystals of KOs$_2$O$_6$ down to 30 mK. The data clearly indicate that KOs$_2$O$_6$ is a fully single-gapped $s$-wave superconductor. This implies that all of the geometrically spin-frustrated compounds known until now respond as conventional superconductors, which would suggest that spin frustration does not lead to unconventional pairing as expected.