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Luminescent solar concentrators (LSCs) are large-area devices that absorb incident sunlight and emit luminescence photons with high quantum efficiency, which will finally be collected by a small photovoltaic (PV) system. The light-harvesting area of the PV system is much smaller than that of the LSC system, potentially reducing the cost of solar cells. Here, we present a theoretical description of the luminescent process in nanoscale LSCs where the conventional ray-optics model is no longer applicable. We demonstrate that a slot waveguide consisting of a nanometer-sized low-index slot region sandwiched by two high-index regions provides a broadband enhancement of light harvesting by the luminescent centers in the slot region. This is because the slot waveguide can 1) greatly enhance the spontaneous emission due to the Purcell effect, 2) dramatically increase the effective absorption length of luminescent centers, and 3) strongly improve the fluorescence quantum yield of luminescent centers. It is found that about 80% solar photons can be re-emitted even for a low fluorescent quantum yield of 0.5, and 80% re-emitted photons can be coupled to the slot waveguide. This LSC has the potential to be constructed in to a tandem structure which can absorb nearly full-spectrum solar photons, and also may be of special interest for building integrated nano-solar-cell applications.