  We investigate the possible occurrence of field-effect induced
superconductivity in the hydrogenated $(111)$ diamond surface by
first-principles calculations. By computing the band alignment between bulk
diamond and the hydrogenated surface we show that the electric field exfoliates
the sample, separating the electronic states at the valence band top from the
bulk projected ones. At the hole doping values considered here, ranging from
$n=2.84\times10^{13}$cm$^{-2}$ to $n=6\times 10^{14} $ cm$^{-2}$, the valence
band top is composed of up to three electronic bands hosting holes with
different effective masses. These bands resemble those of the undoped surface,
but they are heavily modified by the electric field and differ substantially
from a rigid doping picture. We calculate superconducting properties by
including the effects of charging of the slab and of the electric field on the
structural properties, electronic structure, phonon dispersion and
electron-phonon coupling. We find that at doping as large as $n=6\times 10^{14}
$ cm$^{-2}$, the electron-phonon interaction is $\lambda=0.81$ and
superconductivity emerges with $T_{\text{C}}\approx 29-36$K. Superconductivity
is mostly supported by in-plane diamond phonon vibrations and to a lesser
extent by some out-of-plane vibrations. The relevant electron-phonon scattering
processes involve both intra and interband scattering so that superconductivity
is multiband in nature.
