Quantum-well and modified image-potential states in thin Pb(111) films: an estimate for the local work function

Quantum-confined electronic states such as quantum-well states (QWS) inside thin Pb(111) films and modified image-potential states (IPS) above the Pb(111) films grown on Si(111)7$\times$7 substrate were studied by means of low-temperature scanning tunnelling microscopy (STM) and spectroscopy (STS) in the regime of constant current $I$. By plotting the position of the $n-$th emission resonances $U^{\,}_n$ versus $n^{2/3}$ and extrapolating the linear fit for the dependence $U^{\,}_n(n^{2/3})$ in the high-$n$ limit towards $n=0$, we estimate the local work function for the Pb(111) film: $W\simeq 3.8\pm 0.1\,$eV. We experimentally demonstrate that modifications of the shape of the STM tip can change the number of the emission peaks associated with the resonant tunnelling via quantized IPS levels for the same Pb terrace; however it does not affect the estimate of the local work function for the flat Pb terraces. We observe that the maxima in the spectra of the differential tunnelling conductance $dI/dU$ related to both the QWS and the modified IPS resonances are less pronounced if the STM tip becomes more blunt.