We ask the question, what are the ideal characteristics of a near-field thermophotovoltaic cell? Our search leads us to a reformulation of near-field radiative heat transfer in terms of the joint density of electronic states of the emitter-absorber pair in the thermophotovoltaic system. This form reveals that semiconducting materials with narrowband absorption spectra are critical to the energy-conversion efficiency. This essential feature is unavailable in conventional bulk semiconductor cells but can be obtained using low-dimensional materials. Our results show that the presence of matched van Hove singularities resulting from quantum confinement in the emitter and absorber of a thermophotovoltaic cell boosts both the magnitude and spectral selectivity of radiative heat transfer, dramatically improving energy-conversion efficiency. We provide a model near-field thermophotovoltaic system design making use of this idea by employing the van Hove singularities present in carbon nanotubes. Shockley-Queisser analysis shows that the predicted heat transfer characteristics of this model device are fundamentally better than existing thermophotovoltaic designs. Our work paves the way for the use of quantum dots, quantum wells, two-dimensional semiconductors, semiconductor nanowires, and carbon nanotubes as future materials for thermophotovoltaic cells.