Coupling between the transverse and longitudinal componentsof an electron in resonant tunneling

In this paper we present formulas for the transmission coefficient in multibarrier heterostructures when taking into account the space-dependent electron effective mass and the coupling between components of the motion of an electron in directions parallel and perpendicular to the interfaces. We find that the coupling leads to a decrease of the effective barrier height seen by the electron. The numerical calculations are carried out for the double- and triple-barrier heterostructures consisting of GaAs/${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$ ${\mathrm{Al}}_{\mathrm{x}}$ As. Our results show that the coupling effect leads not only significantly to a shift of resonant peaks toward the low-energy region, but also causes the broadening of resonant peaks and the reduction of the peak-to-valley ratio in the transmission spectrum. The coupling becomes more pronounced for higher-lying resonant states. In addition, the variation of the coupling strength with the concentration x of Al is also investigated.