Finite size effects on the quantum spin Hall state in HgTe quantum wells under two different types of boundary conditions

The finite size effect in a two-dimensional topological insulator can induce an energy gap Eg in the spectrum of helical edge states for a strip of finite width.In a recent work,it has been found that when the spin–orbit coupling due to bulk-inversion asymmetry is taken into account,the energy gap Eg of the edge states features an oscillating exponential decay as a function of the strip width of the inverted Hg Te quantum well.In this paper,we investigate the effects of the interface between a topological insulator and a normal insulator on the finite size effect in the Hg Te quantum well by means of the numerical diagonalization method.Two different types of boundary conditions,i.e.,the symmetric and asymmetric geometries,are considered.It is found that due to the existence of the interface between topological insulator and normal insulator this oscillatory pattern on the exponential decay induced by bulk-inversion asymmetry is modulated by the width of normal insulator regions.With the variation of the width of normal insulator regions,the shift of the Dirac point of the edge states in the spectrum and the energy gap Eg closing point in the oscillatory pattern can occur.Additionally,the effect of the spin–orbit coupling due to structure-inversion asymmetry on the finite size effects is also investigated.