Superconductivity and Magnetic Properties of high-quality single crystals of $A_{x}$Fe$_2$Se$_2$ ($A$ = K and Cs)

We successfully grew the high-quality single crystals of $A_{x}$Fe$_2$Se$_2$ ($A$ = K and Cs) by self-flux method. Sharp superconducting transition was observed for both types of crystals. The crystals show the onset superconducting transition temperatures ($T_{\rm c}$) of 31 K and 30 K for K- and Cs-compounds, respectively, with nearly 100% shielding fraction. The crystals show quite high resistivity in the normal state of more than 160 m$\Omega$ cm and 1300 m$\Omega$ cm maximum resistivity for $K_{0.86}Fe_2Se_{1.82}$ and $Cs_{0.86}Fe_{1.66}Se_{2}$ single crystals, respectively. Much larger upper critical field $H_{\rm c2}$ is inferred from low-temperature iso-magnetic-field magnetoresistance in these crystals than in FeSe. The anisotropy $H^{ab}_{\rm c2}$(0)/$H^{c}_{\rm c2}$(0) is around 3 for both of the two materials. Anisotropic peculiar magnetic behavior in normal state has been found for $Cs_{0.86}Fe_{1.66}Se_{2}$