Deeper insight into phase relations in ultrathin Pb films

Films of $\mathrm{Pb}(111)$ with thickness ranging from 1 to 16 monolayers (ML) were grown on $\mathrm{Ge}(111)$ and examined by angle-resolved photoemission spectroscopy. The measured thickness dependences of the work function and surface energy from the same set of samples were both well described by damped sinusoidal oscillatory functions with the same period, but the oscillation phases differ by $1/4$ period between the two quantities, as judged by the node positions. This difference agrees with the prediction of a generic quantum-well model. The varied node positions of damped oscillations of the work function and surface energy were examined in terms of the details of Pb subband dispersions, indicating the different roles of the density of states at the Fermi level to both thin-film properties. The surface dipoles of Pb films were found to have no relevant effects on the phase shift of thickness-dependent work functions except for the amplitudes of oscillations. The total boundary phase shifts of the quantum well states, extracted from the energy positions of the quantum-well states for both $\mathrm{Pb}/\mathrm{Ge}(111)$ and freestanding Pb films, are compared and related to the thickness-dependent work function and surface energy.