We present a dynamical mean-field theory (DMFT) study of the charge and orbital correlations in finite-size La$_{0.5}$Ca$_{0.5}$MnO$_3$ (LCMO) nanoclusters. Upon nanostructuring LCMO to clusters of 3 nm diameter, the size reduction induces an insulator-to-metal transition in the high-temperature paramagnetic phase. This is ascribed to the reduction in charge disproportionation between Mn sites with different nominal valence [Das et al., Phys. Rev. Lett. 107, 197202 (2011)]. Here we show that upon further reducing the system size to a few-atom nanoclusters, quantum confinement effects come into play. These lead to the opposite effect: the nanocluster turns insulating again and the charge disproportionation between Mn sites, as well as the orbital polarization, are enhanced. Electron doping by means of external gate voltage on few-atom nanoclusters is found to trigger a site- and orbital-selective Mott transition. Our results suggest that LCMO nanoclusters could be employed for the realization of technological devices, exploiting the proximity to the Mott transition and its control by size and gate voltage.
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