Interface induced states at the boundary between a $3D$ topological insulator and a normal insulator

We show that, when a three-dimensional ($3D$) narrow-gap semiconductor with inverted band gap (“topological insulator”, TI) is attached to a $3D$ wide-gap semiconductor with non-inverted band gap (“normal insulator”, NI), two types of bound electron states having different spatial distributions and spin textures arise at the TI/NI interface. Namely, the gapless (“topological”) bound state can be accompanied by the emergence of the gapped (“ordinary”) bound state. We describe these states in the framework of the envelope function method using a variational approach for the energy functional; their existence hinges on the ambivalent character of the constraint for the envelope functions that correspond to the “open” or “natural” boundary conditions at the interface. The properties of the ordinary state strongly depend on the effective interface potential, while the topological state is insensitive to the interface potential variation.