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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 usingthe random $S$ matrix model. The roughness effects on the self-consistent orderparameter, the surface mass current and the surface density of states arestudied using the quasi-classical theory. We find that the surface mass currentis suppressed by the surface roughness. The surface density of states shows aquite similar behavior to that of superfluid ${}^3$He B phase. When the surfaceis specular, the surface Andreev bound states form a band which fills the bulkenergy gap $\Delta_{\rm bulk}$. When the surface becomes diffusive, thereoccurs a sharp upper edge of the surface bound states band and there opens asub-gap between the edge and the bulk energy gap. We show that this sub-gap isinduced by the repulsion between the surface bound states and the propagatingBogoliubov quasi-particles through the second order process of roughness.

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