Odd-frequency Cooper pairs and zero-energy surface bound states in superfluid 3He

We study the odd-frequency Cooper pairs formed near the surface of superfluid 3He. The odd-frequency pair amplitude is closely related to the local density of states in the low energy limit. We derive a formula relating explicitly the two quantities. This formula holds for arbitrary boundary condition at the surface. We also present some numerical results on the surface odd-frequency pair amplitude in superfluid 3He-B. Those analytical and numerical results allow one to interpret the midgap surface density of states, observed recently by transverse acoustic impedance measurements on superfluid 3He-B, as the manifestation of the surface odd-frequency state.

Sub-gap in the Edge States of 2-D Chiral Superconductor with Rough Surface

We discuss the rough surface effects on a two-dimensional chiral $k_x+ik_y$ superconductor. The atomic scale roughness at the surface is considered using the random $S$ matrix model. The roughness effects on the self-consistent order parameter, the surface mass current and the surface density of states are studied using the quasi-classical theory. We find that the surface mass current is suppressed by the surface roughness. The surface density of states shows a quite similar behavior to that of superfluid ${}^3$He B phase. When the surface is specular, the surface Andreev bound states form a band which fills the bulk energy gap $Delta_{rm bulk}$. When the surface becomes diffusive, there occurs a sharp upper edge of the surface bound states band and there opens a sub-gap between the edge and the bulk energy gap. We show that this sub-gap is induced by the repulsion between the surface bound states and the propagating Bogoliubov quasi-particles through the second order process of roughness.

Proximity effect between a dirty Fermi liquid and superfluid 3He

The proximity effect in superfluid $^3$He partly filled with high porosity aerogel is discussed. This system can be regarded as a dirty Fermi liquid/spin-triplet p-wave superfluid junction. Our attention is mainly paid to the case when the dirty layer is in the normal state owing to the impurity pair-breaking effect by the aerogel. We use the quasiclassical Greens function to determine self-consistently the spatial variations of the p-wave order parameter and the impurity self-energy. On the basis of the fully self-consistent calculation, we analyze the spatial dependence of the pair function (anomalous Greens function). The spin-triplet pair function has in general even-frequency odd-parity and odd-frequency even-parity components. We show that the admixture of the even- and odd-frequency pairs occurs near the aerogel/superfluid $^3$He-B interface. Among those Cooper pairs, only the odd-frequency s-wave pair can penetrate deep into the aerogel layer. As a result, the proximity-induced superfluidity in a thick aerogel layer is dominated by the Cooper pair with the odd-frequency s-wave symmetry. We also analyze the local density of states and show that it has a characteristic zero-energy peak reflecting the existence of the odd-frequency s-wave pair, in agreement with previous works using the Usadel equation.