Excitons in periodic potentials
The energy band structure of excitons is studied in periodic potentials produced by the short-range interaction between the exciton and electrons of Wigner lattices. Treating the exciton as a pointlike dipole that interacts with the periodic potential, we can solve a simple one-body problem that provides valuable information on excitons in many-body problem settings. By employing group theory, we identify the excitonic energy bands that can couple to light and then quantify their energy shifts in response to a change in the period of the potential. This approach allows us to emulate the response of optically active exciton and trion states to a change in electron density. In agreement with experimental results, the exciton (trion) state shows a relatively strong (weak) energy shift when the electron density increases. This behavior stems from the difficulty of exciton states to remain extended, whereas the trion state remains largely the same as long as its radius is smaller than the average distance between resident electrons. Here, we identify an important relation between the energy blueshift of bright excitons and the presence of a Wigner crystal, where the blueshift magnitude is proportionate to the symmetry of the electron lattice.