Large and homogeneous mass enhancement in the rattling-induced superconductor KOs${}_2$O${}_6$

We have determined the Fermi surface in KOs${}_2$O${}_6$ ($T_c=9.6$ K and $B_{c2}\sim 32$ T) via de Haas–van Alphen (dHvA) oscillation measurements and a band structure calculation. We find effective masses up to 26(1)$m_e$ ($m_e$ is the free electron mass), which are unusually heavy for compounds where the mass enhancement is mostly due to electron-phonon interactions. Orbit-resolved mass enhancement parameters ${\lambda }_{\text{dHvA}}$ are large but fairly homogeneous, concentrated in the range 5–8. We discuss origins of the large homogeneous mass enhancement in terms of rattling motion of the K ions.

Figure 1

(a) Magnetic torque in KOs${}_2$O${}_6$ at $T=0.05$ K for the field direction $\theta ={79}^{\circ }$. Both field-up and field-down sweeps are shown. The slight difference in the oscillation phase between the up and down sweeps for $B\lesssim 32$ T is not intrinsic but due to some experimental problem. (b) Corresponding Fourier transforms in 1/$B$ for a field range 30.9–35.1 T, which gives a frequency resolution of $\Delta F=258$ T. Fundamental dHvA frequencies labeled with greek letters and their harmonics and combinations are resolved. Inset: $T$ dependence of the amplitude $A$ of the frequency $\rho$. The associated effective mass $m^{*}$ is estimated to be 26(1)$m_e$ from a fit to the standard Lifshitz-Kosevich formula (solid curve).

Figure 2

(a) Angular dependences of the experimental and calculated dHvA frequencies. For the experimental ones, those assigned to fundamentals are shown by solid circles, while those assigned to harmonics or combinations are shown by open circles. The mark sizes are based on the oscillation amplitudes logarithmically. Error bars attached to some data points are based on half of the frequency resolution, i.e., $\pm$129 T. (b) Calculated Fermi surface of KOs${}_2$O${}_6$. The hole surface (left) is shown in the repeated zone scheme. Orthogonal pillars constituting the hole surface cross at X points. The observed dHvA orbits are shown in the figures except for the $\zeta$ and $\lambda$ orbits, both of which are orbits on the hole surface. $\zeta$ is a $\mathsf{V}$-shaped orbit extending from an X point toward adjacent X points along the $k_x$ and $k_y$ directions. $\lambda$ is centered approximately at a middle of two vertically adjacent X points and extends between the two X points.

Figure 3

Orbit-resolved dHvA mass enhancement parameters ${\lambda }_{\text{dHvA}}$ in KOs${}_2$O${}_6$ normalized to the Fermi-surface averaged specific-heat mass enhancement parameter $\langle \lambda \rangle$ as a function of the dHvA frequency. Data for CsOs${}_2$O${}_6$ (Refs. 11 and 36), LuNi${}_2$B${}_2$C (Refs. 46 and 47), and MgB${}_2$ (Refs. 48 and 49) are shown for comparison. Different symbols are used for different bands, as indicated by the legend in the figure.