  Linear strings of trapped atomic ions held in radio-frequency (rf) traps
constitute one of the leading platforms for quantum simulation experiments,
allowing for the investigation of interacting quantum matter. However, linear
ion strings have drawbacks, such as the difficulty to scale beyond $\sim 50$
particles as well as the inability to naturally implement spin models with more
than one spatial dimension. Here, we present experiments with planar Coulomb
crystals of about 100 $^{40}$Ca$^+$ ions in a novel monolithic rf trap, laying
the groundwork for quantum simulations of two-dimensional spin models with
single-particle control. We characterize the trapping potential by analysis of
crystal images and compare the observed crystal configurations with numerical
simulations. We further demonstrate stable confinement of large crystals, free
of structural configuration changes, and find that rf heating of the crystal is
not an obstacle for future quantum simulation experiments. Finally, we prepare
the out-of-plane motional modes of planar crystals consisting of up to 105 ions
close to their ground state by electromagnetically-induced transparency
cooling, an important prerequisite for implementing long-range entangling
interactions.
