Determining the Thickness of Pb Film Similar to Bulk with Energy Dispersion Derived from Quantum Well States

Abstract

It is known that the energy spacing between adjacent empty quantum well (QW) states in Pb islands on Cu(111) would reveal the shrinking characteristic originating from the effect of the image potential. Using the phase accumulation model, including a phase factor contributed from the image potential, the shrinking energy spacing can be quantitatively explained with the assumption of the parabolic energy versus wave vector (E–k) dispersion. However, an experimental dispersion acquired from analyzing the energies of the QW state reveals a linear E–k relationship corresponding to the Pb bulk band structure, implying the assumed parabolic dispersion is not appropriate. By combining the linear dispersion with the image potential effect in the calculation, it is found that the calculated values of energy spacing of island thickness below eight atomic layers are not in agreement with the experimental measurements. This implies that the electronic structure of Pb islands would be similar to that of the bulk when their thicknesses reach eight-atomic layers.