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.
Categories: cond-mat.supr-con cond-mat.str-el