A time-dependent embedding calculation of surface electron emission

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