A time-dependent embedding calculation of surface electron emission

Abstract

The Dirac–Frenkel variational principle is used to derive the embedding method for solvingthe time-dependent Schrödinger equation. Embedding allows the time evolution of thewavefunction to be calculated explicitly in a limited region of space, the region of physicalinterest, the embedding potential ensuring that the wavefunction satisfies the correctboundary conditions for matching on to the rest of the system. This is applied to a study ofthe excitation of electrons at a metal surface, represented by a one-dimensional modelpotential for Cu(111). Time-dependent embedding potentials are derived for replacing thebulk substrate, and the image potential and vacuum region outside the surface, so that thecalculation of electron excitation by a surface perturbation can be restricted tothe surface itself. The excitation of the Shockley surface state and a continuumbulk state is studied, and the time structure of the resulting currents analysed.There is a distinction between emission from the localized surface state, wherethe charge is steadily depleted, and the extended continuum state, where thecurrent emitted into the vacuum is compensated by current approaching the surfacefrom the bulk. The time taken for the current to arrive outside the surface isstudied.