A scattering matrix approach to the effective mass dependence of tunneling current through heterojunctions

A scattering matrix technique is used to calculate the longitudinal and transverse energy dependence of the transmission probability through various heterostructures using both the BenDaniel–Duke (BD) and the lesser known Zhu–Kroemer (ZK) boundary conditions to take into account the spatial dependence of the effective mass. We first illustrate the large difference in the transmission probabilities calculated using both boundary conditions for the simple problems of tunneling through a potential step, a single rectangular barrier, and a resonant tunneling device. Then, we present numerical calculations of the external electric field dependence of the field emission (FE) current from a n-doped GaAs semiconductor/vacuum interface using both boundary conditions, showing that the BD boundary conditions underestimate the FE current for large values of the applied external electrostatic field. A comparison of calculated FE characteristics with FE data may be a way to determine the appropriate boundary conditions to solve tunneling problems through heterostructures with spatially varying effective mass.