Surface Andreev Bound States of $^3$He-B by Transverse Acoustic Impedance Measurements

Complex transverse acoustic impedance of the superfluid $^3$He was measured at the frequencies of 10 to 80 MHz from 6 up to 25 bar by a CW bridge method. The observed temperature dependence of it was well explained by the quasi-classical theory with random $S$-matrix model for a diffusive surface. The impedance was influenced by pair breaking and by quasi-particle density of states at the surface, which was drastically modified from the bulk one by the formation of surface Andreev bound states. In B phase, an additional gap in SDOS opened between the upper energy edge $\Delta^*$ of the surface Andreev bound states band and the bulk energy gap $\Delta$. Temperature dependence of $\Delta^*$ was measured and was about 30\% smaller than theoretical values. In A phase, flat and gapless SDOS was confirmed experimentally for the first time. It is demonstrated that the present spectroscopic method is a good tool to investigate the surface microscopic state, which has not been possible for the charge neutral {\it P}-wave superfluid.