A Model of Photoemission Delay with Heterostructure and Coating Barriers*

High performance Free Electron Lasers (FEL) and x-ray FELs (xFEL) seek high quantum efficiency (QE) photocathodes (typically semiconductors), but simultaneously demand low emittance. Photocathodes incorporating heterostructures and/or coatings such as graphene or boron nitride have been under consideration as a means to improve QE (as well as robustness and lifetime) while allowing for methods beyond simple bulk transport, pulse shaping, and surface modifications to tailor emittance behavior. Modifications induced by coatings and heterostructures have lead to an analysis of the transmission probability of the more complex barrier profiles. The potentials over which emission occurs are provided using a ``macro-averaging'' technique applied to the output of density functional theory simulations. This presentation will analyze the behavior of delayed emission effects for short pulses where field conditions at the surface change over the duration of the pulse. The models are in development for the Particle-in-Cell code MICHELLE used to model beam formation and transport in rf injectors and electron sources.