Time-dependent embedding: surface electron emission

TL;DR

This paper introduces a time-dependent embedding method based on the Dirac-Frenkel variational principle to efficiently simulate electron emission at metal surfaces, capturing surface and bulk state dynamics.

Contribution

A novel embedding approach for solving the time-dependent Schrödinger equation that accurately models surface electron excitation with reduced computational domain.

Findings

Successfully modeled electron excitation at a Cu(111) surface

Differentiated emission from localized surface states and bulk continuum

Analyzed time-structure and non-linear effects of surface currents

Abstract

An embedding method for solving the time-dependent Schr\"odinger equation is developed using the Dirac-Frenkel variational principle. 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. Non-linear effects and the time taken for the current to arrive outside the surface are discussed. The method shows a clear 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.