Substrate-mediated umklapp scattering at the incommensurate interface of a monatomic alloy layer

Ultrathin Pb and Ge films deposited on Ag(111) surfaces have been investigated and compared. We found that at 1/3 ML, both films formed surface alloys, ${\mathrm{Ag}}_{2}\mathrm{Pb}$ and ${\mathrm{Ag}}_{2}\mathrm{Ge}$, with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}R{30}^{\ensuremath{\circ}}$ and $\frac{19}{20}\sqrt{3}\ifmmode\times\else\texttimes\fi{}\frac{19}{20}\sqrt{3}R{30}^{\ensuremath{\circ}}$ structures on Ag(111) but the surface electronic structures exhibit a most evident difference at the Ag(111) surface zone boundary ${\overline{M}}_{\mathrm{Ag}(111)}$, where the single band and the splitting ones were observed, respectively. Up to 1 ML, ${\mathrm{Ag}}_{2}\mathrm{Ge}$ subsequently develops into germanene with a striped phase and then a quasifreestanding phase, as previously reported [Lin et al., Phys. Rev. Mater. 2, 024003 (2018)], while ${\mathrm{Ag}}_{2}\mathrm{Pb}$ evolves to a dense Pb(111) phase that also reveals splitting bands at ${\overline{M}}_{\mathrm{Ag}(111)}$. We discover that the larger (smaller) atomic size of a Pb (Ge) atom with respect to an Ag atom causes the commensurate (incommensurate) interfaces and further demonstrate that the splitting bands of ${\mathrm{Ag}}_{2}\mathrm{Ge}$ surface alloy and 1-ML Pb film originated from the commonly incommensurate interface with Ag(111), which mediates umklapp scattering by inducing the mirror image of the pristine ${\mathrm{Ag}}_{2}\mathrm{Ge}$ and Pb(111) bands relative to ${\overline{M}}_{\mathrm{Ag}(111)}.$