Probing a divergent van Hove singularity of graphene with a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Ca</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mi mathvariant="normal">N</mml:mi></mml:mrow></mml:math> support: A layered electride as a solid-state dopant

Layered electrides, as typified by Ca$_2$N, are a new class of quasi-two-dimensional materials with low work functions. Using first-principles calculations, we have shown that a graphene layer deposited on Ca$_2$N is doped to $n=5\times 10^{14}$ cm$^{-2}$ with its Fermi level aligned with the logarithmically divergent van Hove singularity in the graphene $\pi^*$ band. For bilayer graphene on Ca$_2$N, the inner graphene layer is doped to the same level, while the doping of the outer graphene layer is much more modest. This finding opens an interesting possibility of using layered electrides for the exploration of van Hove physics. The work function changes nonmonotonically with the number of graphene layers, which we explain in terms of the peculiar electronic structures of the constituent materials and their bonding.