Asymmetric Josephson effect in inversion symmetry breaking topological materials

+/- Topological materials which possess topologically protected surface states have attracted much attention in recent years. In this work, we study the critical current of superconductor/inversion symmetry breaking topological material/superconductor junctions. We found surprisingly that, in topological materials with broken inversion symmetry, the magnitude of the critical Josephson currents vertical bar I-c(+)(B)vertical bar at fixed magnetic field B is not the same for critical currents vertical bar I-c(-)(B)vertical bar flowing in the opposite direction. Moreover, the critical currents violate the vertical bar I-c(+/-)(B)vertical bar = vertical bar I-c(+/-)(-B)vertical bar relation and give rise to asymmetric Fraunhofer patterns. We call this phenomenon asymmetric Josephson effect (AJE). AJE can be used to detect inversion symmetry breaking in topological materials such as in quantum spin Hall systems and Weyl semimetals.