A delta barrier in a well and its generalization for emission studies

The transmission coefficient for a delta-function barrier is a convenient model for many technologically important applications relying on photoemission, simulations of wave packets, or modeling the narrow barrier of a normal-superconducting point contact. We examine an extension of the model to treat instead a delta-function sequence (a rectangular barrier that approaches the behavior of a delta-function in the limit of vanishing width). It is shown how the eigenstates of the sequence converge on the delta-function barrier eigenstates, but more importantly, how the even and odd parity states depart from the delta-function limiting case. The exact eigenstates enable the time evolution of exponentially attenuated tunneling to be exactly evaluated, in contrast to numerical methods. The application is the inclusion of tunneling time effects in simulations of time-varying electron emission.