Uncommonly High Upper Critical Field in the Superconducting KOs$_2$O$_6$ Pyrochlore

The entire temperature dependence of the upper critical field $H_{\rm c2}$ in the $\beta$-pyrochlore KOs$_2$O$_6$ is obtained from high-field resistivity and magnetic measurements. Both techniques identically give $H_{\rm c2}(T \simeq 0 {\rm K})$ not only surprisingly high ($\sim 33$ T), but also the approach to it unusually temperature-\emph{linear} all the way below $T_{\rm c}$ (= 9.6 K). We show that, while $H_{\rm c2}(0)$ exceeds a simple spin-singlet paramagnetic limit $H_{\rm P}$, it is well below an $H_{\rm P}$ enhanced due to the missing spatial inversion symmetry reported recently in KOs$_2$O$_6$, ensuring that the pair-breaking here is executed by orbital degrees. {\it Ab initio} calculations of orbital $H_{\rm c2}$ show that the unusual temperature dependence is reproduced if dominant s-wave superconductivity resides on the smaller closed Fermi surfaces.