Superconducting Properties of the Non-Fermi-Liquid System $\beta $-YbAlB$_{4}$.

{\$}$\backslash $beta{\$}-YbAlB{\$}{\_}4{\$} is the first Yb-based heavy fermion superconductor with the transition temperature of {\$}T{\_}{\{}$\backslash $rm c{\}} = 80{\$} mK [1]. Our study using high-purity single crystals indicates that the clean limit superconductivity directly emerges from the pronounced non-Fermi-liquid state [2]. The critical fields are anisotropic and probably have a paramagnetic effect under field along the {\$}c{\$}-axis because of the divergent behavior of the Ising like {\$}c{\$}-axis susceptibility. The strong pair-breaking effect, most likely due to nonmagnetic impurities, suggests that {\$}$\backslash $beta{\$}-YbAlB{\$}{\_}4{\$} is an unconventional superconductor. We discuss the detailed properties of superconductivity and quantum criticality. [1] S. Nakatsuji, K.Kuga, Y. Machida, T. Tayama, T. Sakakibara, Y. Karaki, H. Ishimoto, S. Yonezawa, Y. Maeno, E. Pearson, G. G., Lonzarich, L.Balicas, H. Lee and Z. Fisk, \textit{Nature Physics}, \textbf{4}, 603 (2008). [2] K. Kuga, Y. Karaki, Y. Matsumoto, Y. Machida, and S. Nakatsuji,\textit{ Physical Review Letters},\textbf{ 101}, 137004 (2008).