Scattering Matrix Approach to Electron Transmission Probability Through a Flat Semiconductor / Vacuum Interface

A scattering matrix technique is used to study the transmission probability through a flat semiconductor / vacuum interface considering the effective mass difference between the two materials. A comparison is made between the longitudinal and transverse energy dependence of the transmission probability calculated using the BenDaniel-Duke (BD) and Zhu-Kroemer (ZK) boundary conditions at the semiconductor / vacuum interface. For longitudinal energy between the minimum of the conduction band and a few ${{\mathrm{k}}_{\mathrm{B}}}\mathrm{T}$ above the Fermi level in the semiconductor, the transmission probability can be an order of magnitude smaller when using the Zhu-Kroemer boundary conditions compared to the BenDaniel-Duke boundary conditions. The Zhu-Kroemer boundary conditions should be used to calculate the field emission current through a semiconductor/vacuum interface.