Superconducting pairing symmetries in the 3-K and bulk phases of Sr$_2$RuO$_4$

We report recent progress on our single-particle tunneling and phase-sensitive measurements on 3-K and bulk phases of Sr$_2$RuO$_4$. The latter refers to an eutectic phase of Sr$_2$RuO$_4$ that features Ru microdomains embedded in a single crystal of Sr$_2$RuO$_4$. These Ru microdomains are of a mesoscopic size and varying shape. Therefore their pairing symmetries are not subject to the same set of constraints as those in the bulk. We have performed tunneling measurements on the 3-K phase to identify $all$ possible pairing states in this unique superconducting system. The junctions used earlier were prepared by pressing In wire onto a cleaved $ab$ face of a Ru-containing Sr$_2$RuO$_4$ single crystal containing multiple Ru microdomains. More recently we focused on tunnel junctions prepared on pre-selected single Ru microdomains. Possible existence of an intrinsic mixed pairing state in the interior of a Ru microdomain featuring simultaneous presence of both the $s$- and $p$-wave superconducting condensates. We will also discuss briefly our current effort in the phase-sensitive measurements on bulk Sr$_2$RuO$_4$, focusing on detecting possible $/k_z$ dependence of the order parameter and the existence of domains.