Self-Similar Magnetoresistance of a Semiconductor Sinai Billiard

We investigate the transition to a Sinai geometry by introducing a circular pattern at the center of a square mesoscopic billiard defined in a high quality AlGaAs/GaAs crystal. The transition induces a novel quantum interference structure in the magnetoresistance with a characteristic field scale over an order of magnitude smaller than previously reported in mesoscopic billiards. A systematic comparison of fine and coarse structures, which differ by an order of magnitude in field scale, demonstrates the first observation of geometry-induced ``self-similarity'' in the magnetoresistance of a semiconductor system.