Superconductivity with High Upper Critical Field in the Cubic Centrosymmetric $\eta$-Carbide Nb$_4$Rh$_2$C$_{1-\delta}$

The upper critical field is a fundamental measure of the strength of superconductivity in a material. It is also a cornerstone for the realization of superconducting magnet applications. The critical field arises because of the Copper pair breaking at a limiting field, which is due to the Pauli paramagnetism of the electrons. The maximal possible magnetic field strength for this effect is commonly known as the Pauli paramagnetic limit given as $\mu_0 H_{\rm Pauli} \approx 1.86{\rm [T/K]} \cdot T_{\rm c}$ for a weak-coupling BCS superconductor. The violation of this limit is only rarely observed. Exceptions include some low-temperature heavy fermion and some strongly anisotropic superconductors. Here, we report on the superconductivity at 9.75 K in the centrosymmetric, cubic $\eta$-carbide-type compound Nb$_4$Rh$_2$C$_{1-\delta}$, with a normalized specific heat jump of $\Delta C/\gamma T_{\rm c} =$ 1.64. We find that this material has a remarkably high upper critical field of $\mu_0 H_{\rm c2}{\rm (0)}$ =~28.5~T, which is exceeding by far its weak-coupling BCS Pauli paramagnetic limit of $\mu_0 H_{\rm Pauli}$~=~18.1 T. Determination of the origin and consequences of this effect will represent a significant new direction in the study of critical fields in superconductors.