We present a theoretical study of the constant current scanning tunneling spectroscopy of quantum well states localized in Pb(111) overlayers on Cu(111) surfaces. The distance-voltage characteristic of the tunneling junction is obtained with a mixed approach. The wave packet propagation technique is applied to describe electron tunneling from the tip into the sample, and the density functional calculations provide the necessary inputs for the one-dimensional model potential representation of the system. The excited-state population decay mechanisms via inelastic electron-electron and electron-phonon interactions are taken into account with a bias-dependent absorbing potential introduced in the metal. Our results are in good agreement with recent experimental studies [Phys. Rev. Lett. 102, 196102 (2009), Phys. Rev. B 81, 205438 (2010)] over the energy range where the free-electron description of the Pb overlayer used here applies. We find that at high bias the quantum well states experience a Stark energy shift and partially acquire a character of field emission resonances. The present model study also sheds light at the experimentally observed departure of the energies of the quantum well states from the particle-in-a-box prediction for the bias above 4 eV. The measured trend can be consistently explained as due to the departure of the realistic Pb band structure in the $\Gamma -\text{L}$ direction from free-electron parabola when the electron momentum approaches the $\Gamma$ point.

• Received 17 June 2011

DOI:https://doi.org/10.1103/PhysRevB.84.115422