  Using low-energy muons, we map the charge carrier concentration as a function
of depth and electric field across the \SiOSi interface up to a depth of
\SI{100}{\nano\meter} in Si-based MOS capacitors. The results show that the
formation of the anisotropic bond-centered muonium \MuBCz state in Si serves as
a direct measure of the local changes in electronic structures. Different
band-bending conditions could be distinguished, and the extension of the
depletion width was directly extracted using the localized stopping and probing
depth of the muons. Furthermore, electron build-up on the Si side of the
\SiOO/Si interface, caused by the mirror charge induced by the fixed positive
charge in the oxide and the image force effect, was observed. Our work
represents a significant extension of the application of the muon spin rotation
technique ($\mu$SR) and lays the foundation for further research on direct
observation of charge carrier density manipulation at technologically important
semiconductor device interfaces.
