Oscillatory crossover from two-dimensional to three-dimensional topological insulators

We investigate the crossover regime from three-dimensional topological insulators Bi₂Te₃ and Bi₂Se₃ to two-dimensional topological insulators with quantum spin Hall effect when the layer thickness is reduced. Using both analytical models and first-principles calculations, we find that the crossover occurs in an oscillatory fashion as a function of the layer thickness, alternating between topologically trivial and nontrivial two-dimensional behavior.