Lévy distributions and disorder in excitonic spectra

We study analytically the spectrum of excitons in disordered semiconductors like transition metal dichalcogenides, which are important for photovoltaic and spintronic applications. We show that ambient disorder exerts a strong influence on the exciton spectra. For example, in such a case, the well-known degeneracy of the hydrogenic problem (related to Runge-Lenz vector conservation) is lifted so that the exciton energy starts to depend on both the principal quantum number n and orbital l. We model the disorder phenomenologically substituting the ordinary Laplacian in the corresponding Schrödinger equation by the fractional one with Lévy index μ, characterizing the degree of disorder. Our variational treatment (corroborated by numerical results) shows that an exciton exists for 1 < μ≤ 2. The case μ = 2 corresponds to the "ordered" hydrogenic problem, while in the opposite case μ = 1 the exciton collapses. The exciton spectrum is dominated by the sample areas with moderate disorder. Our theory permits controlled predictions to be made of the excitonic properties in semiconductor samples with different degrees of disorder.