We study the effect of electron spill-out and of nonlocality on the propagation of light inside a gap between two semi-infinite metallic regions. We first present a simplified physical model for the spill-out phenomenon, an approach sufficient to show that the propagation of the gap-plasmon becomes impossible in the tunneling regime. However, in the limit of very small gaps, only a Quantum Hydrodynamic Theory (QHT) approach, taking into account both the electron spill-out and nonlocality, is able to accurately model the gap-plasmon characteristics and to correctly retrieve the refractive index of the bulk metal as the limit of the effective index of the gap-plasmon for vanishing gaps. Finally, we analyze the relation between different models and show that up to a certain size it is possible to predict the correct gap-plasmon effective index by considering a properly resized effective gap.
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