Phonon dispersion curves of stable and metastable B C3 honeycomb epitaxial sheets and their chemical bonding: Experiment and theory

We have investigated experimentally and theoretically the phonon-dispersion curves along the $\overline{\ensuremath{\Gamma}}\text{\ensuremath{-}}\overline{M}$ axis in two-dimensional surface Brillouin zones of the stable and metastable $\mathrm{B}{\mathrm{C}}_{3}$ honeycomb sheets with different lattice constants. Most of the observed curves agreed with the theoretical ones calculated on the basis of the ab initio theory. There exists a common relationship between the nearest-neighbor stretching force constants and bond lengths of two-dimensional honeycomb structures in various compounds of boron, carbon, and nitrogen; the two-dimensional bonds are discriminated clearly from the one- and three-dimensional bonds. We detected a librational mode of carbon hexagons in $\mathrm{B}{\mathrm{C}}_{3}$; a peculiar mode for a large unit cell of honeycomb sheets such as the $2\ifmmode\times\else\texttimes\fi{}2$ structure of monolayer graphite and $\mathrm{B}{\mathrm{C}}_{3}$. The restoring force related with the bond bending is insensitive to the change in the B-C bond length.