Low temperature magnetization of the quantum critical heavy fermion superconductor $\beta$-YbAlB$_4$

$\beta$-YbAlB$_4$ is the first Yb-based heavy fermion superconductor with $T_{\rm c} = 80$ mK. We measured low temperature magnetization of high-purity single crystals down to $T=$ 25 mK. The measurements have revealed a considerable amount of volume fractions of the superconductivity and the upper critical field $B_{c2}$ curve under field along the c axis, consistent with the previous results. In the normal state, the previously observed divergent behavior in the temperature dependence of the magnetization has been confirmed using higher quality samples and under a low field of 22 mT. In addition, the measurements have revealed a power law behavior, namely, $dM/dT\propto T^{3/2}$, which has a slightly higher exponent than the previous results.