# Relaxation of electrons in quantum-confined states in Pb/Si(111) thin films from master equation with first-principles-derived rates

## TL;DR

This study models electron relaxation in Pb/Si(111) thin films, revealing the dominant role of electron-electron scattering and the significance of phonon-specific effects, including surface phonons, with results aligning with experimental data.

## Contribution

It introduces a first-principles-based master equation approach to analyze electron relaxation, incorporating detailed band structures and electron-phonon interactions in quantum-confined Pb/Si(111) films.

## Key findings

- Electron-electron scattering dominates relaxation dynamics.
- Electron-phonon scattering is highly phonon-mode-specific.
- A phonon bottleneck occurs at 0.3 eV above the Fermi level.

## Abstract

Atomically thin films of Pb on Si(111) provide an experimentally tunable system comprising a highly structured electronic density of states. The lifetime of excited electrons in these states is limited by both electron-electron (e-e) and electron-phonon (e-ph) scattering. We employ the description by a master equation for the electronic occupation numbers to analyze the relative importance of both scattering mechanisms. The electronic and phononic band structures, as well as the matrix elements for electron-phonon coupling within deformation potential theory were obtained from density functional calculations, thus taking into account quantum confinement effects. For the relaxation dynamics, the contribution of impact ionization processes to the lifetime is estimated from the imaginary part of the electronic self-energy calculated in the GW approximation. By numerically solving rate equations for the occupations of the Pb-derived electronic states coupled to a phononic heat bath, we are able to follow the distribution of the electronic excitation energy to the various modes of Pb lattice vibrations. While e-e scattering is the dominant relaxation mechanism, we demonstrate that the e-ph scattering is highly phonon-mode-specific, with a large contribution from surface phonons. At electron energies of about 0.3 eV above the Fermi surface, a 'phonon bottleneck' characteristic of relaxation in nanostructures with well-separated electronic states is observed. The time scales extracted from the simulations are compared to data from pump-probe experiments using time-resolved two-photon photoemission.