Lattice relaxation as the origin of the insulating nature of the alkali/Si(111):B surface

\emph{Ab initio} density-functional theory calculations, photoemission spectroscopy (PES), scanning tunneling microscopy, and spectroscopy (STM, STS) have been used to solve the $2\sqrt{3} \times2\sqrt{3}R30$ surface reconstruction observed previously by LEED on 0.5 ML K/Si:B. It is found that the large K-induced vertical lattice relaxation obtain in the calculations and occurring only for $3/4$ of Si adatoms is shown to quantitatively explain both the chemical shift of $1.14$ eV and the ratio $1/3$ measured on the two distinct B 1s core levels. A gap is observed between valence and conduction surface bands by ARPES and STS which is shown to have mainly a Si-B character using the ab initio calculations. Finally, the calculated STM images agree with our experimental results. Therefore, the insulating character of alkali/Si:B interfaces has been captured to an excellent accuracy, from the low lying 1s state of boron, to the unoccupied states above the gap, within a one-electron approach. This work solves the controversy about the origin of the insulating ground state of alkali-metal/Si(111):B semiconducting interfaces which were believed previously to be related to many-body effects