On the transfer matrix method and WKB approximation for systems with spatial-dependent effective mass

A set of coupled differential equations is derived by considering the continuous limit of the transfer matrix method, which is a numerical approach for the one-dimensional structures such as the semiconductor heterostructures. By decoupling such a set of equations, an extension to the Wentzel-Kramers-Brillouin (WKB) method is obtained to incorporate effects due to the spatial-dependent effective mass. For a traveling wave, the decoupling is to ignore the reflection resulting from the variations of both the potential and effective mass. By considering a solvable fully-quantized system, it is shown that the extended WKB method provides good approximation for the states with the high eigenenergies.