Effects of high-order Van Hove singularities on exciton and trion energy dispersions
We investigate the effects of Van Hove singularities in the electronic band structure of two-dimensional semiconductors on the energy dispersion of excitons and positive trions. In particular, we study valence band energy dispersions which possess (i) a typical logarithmic Van Hove singularity, (ii) a line high-order Van Hove singularity (HOVHS) from a Mexican-hat dispersion or (iii) a point HOVHS such as a monkey saddle. We find that the density of states (DOS) of excitons and trions containing such singularities is dramatically enhanced and shows, in general, how the HOVHS in the valence band can strongly affect and be mirrored in the DOS of excitons and trions. This leads to new states that govern the optical properties of the system. In addition, we study a set of materials, InSe, GaSe and $α$-SnAs, from a class of materials in which the topmost valence band has an inverted Mexican-hat shape. The most favourable exciton occurs when the singularity is at the $Γ$-point, as in the example of monolayer $α$-SnAs, which hosts a HOVHS. Our work thus provides a pathway to engineer specific bound states in two-dimensional materials that host such singularities, thereby opening new avenues for potential applications.