Honeycomb Schroedinger operators in the strong binding regime

Understand

In this article, we study the Schr\"odinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane.

• The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues.
• We consider this Schr\"odinger operator in the regime of strong binding, where the depth of the potential wells is large.
• Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model (Wallace, 1947).