Uncommonly high upper critical field of the pyrochlore superconductor $\mathrm{K}{\mathrm{Os}}_2{\mathrm{O}}_6$ below the enhanced paramagnetic limit

The entire temperature dependence of the upper critical field $H_{\mathrm{c}2}$ in the $\beta$-pyrochlore $\mathrm{K}{\mathrm{Os}}_2{\mathrm{O}}_6$ is obtained from high-field resistivity and magnetic measurements. Both techniques identically give $H_{\mathrm{c}2}(T\simeq 0\mathrm{K})$ not only surprisingly high $(\sim 33\mathrm{T})$, but also the approach to it is unusually temperature linear all the way below $T_{\mathrm{c}}(=9.6\mathrm{K})$. We show that, while $H_{\mathrm{c}2}\left(0\right)$ exceeds a simple spin-singlet paramagnetic limit $H_{\mathrm{P}}$, it is well below an $H_{\mathrm{P}}$ enhanced due to the missing spatial inversion symmetry reported recently in $\mathrm{K}{\mathrm{Os}}_2{\mathrm{O}}_6$, ensuring that the pair breaking here is executed by orbital degrees. Ab initio calculations of orbital $H_{\mathrm{c}2}$ show that an unusual temperature dependence is reproduced if dominant $s$-wave superconductivity resides on the smaller closed Fermi surfaces.

Figure 1

(Color online) Field dependence of the resistivity in $\mathrm{K}{\mathrm{Os}}_2{\mathrm{O}}_6$. In the experiments, a block of small single crystals was cut into two pieces: one was used for resistivity and the other for magnetic measurements, as those in Fig. 2. Inset: typical field-pulse profiles used. Magnetic fields up to $\sim 50\mathrm{T}$ were applied in the current direction.

Figure 2

(Color online) Time derivative $\mathrm{d}f∕\mathrm{d}t$ of the frequency shift of the tunnel diode oscillator containing the $\mathrm{K}{\mathrm{Os}}_2{\mathrm{O}}_6$ sample. The full up-field (dashed line) and down-field cycle is shown for $T=0.54\mathrm{K}$, for which the frequency shift $\Delta f$ vs field is also displayed in the inset. At other temperatures (color coded as in Fig. 1), only down-field sweeps are shown and each curve is vertically shifted for clarity.

Figure 3

(Color online) Temperature dependence of the upper critical field determined by the resistivity (solid squares) and magnetic (open circles) measurements. The solid line is the expected orbital $h^{*}\left(t\right)$ from the conventional Werthamer-Helfand-Hohenberg (WHH) formulation. The dotted and dashed lines are ab initio calculations for the maximum $h^{*}\left(t\right)$ ([111] direction) by fitting the initial slope at $T_c$ and taking into account all the Fermi surfaces (FS) and the closed FS only, respectively. Inset: schematic Os network in the noncentrosymmetric $\left(F\overline{4}3m\right)$ structure.