Transmission coefficient of electrons through a single graded barrier

We present a model to describe the effective mass, generalized kinetic-energy operator, and barrier potential to study the motion of an electron across a nonabrupt barrier of GaAs/${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As/GaAs. With this model, we calculate the transmission coefficient using three different methods. The results obtained with these methods show a significant change of the transmission coefficient when compared with that of an abrupt barrier. Numerical results are obtained for different values of the interfacial width, and compositional variation of the aluminum.