Exciton Band Structure in Two-Dimensional Materials

Low-dimensional materials differ from their bulk counterparts in many respects. In particular, the screening of the Coulomb interaction is strongly reduced, which can have important consequences such as the significant increase of exciton binding energies. In bulk materials the binding energy is used as an indicator in optical spectra to distinguish different kinds of excitons, but this is not possible in low-dimensional materials, where the binding energy is large and comparable in size for excitons of very different localization. Here we demonstrate that the exciton band structure, which can be accessed experimentally, instead provides a powerful way to identify the exciton character. By comparing the ab initio solution of the many-body Bethe-Salpeter equation for graphane and single-layer hexagonal boron nitride, we draw a general picture of the exciton dispersion in two-dimensional materials, highlighting the different role played by the exchange electron-hole interaction and by the electronic band structure. Our interpretation is substantiated by a prediction for phosphorene.

Figure 1

$\mathrm{Im}{\epsilon }_M(\mathbf{q},\omega )$ of (a) graphane and (b) $h$-BN calculated for several values of $\mathbf{q}$ (in ${\AA }^{-1}$) along $\mathrm{\Gamma }M$. For the sake of clarity, each spectrum has been multiplied by $q^2$. The red arrows mark the QP band gap.

Figure 2

Electronic distribution of the wave function ${\mathrm{\Psi }}_{\mathbf{q}}^{\lambda }$ of the lowest-energy exciton of graphane and $h$-BN for the fixed position of the hole (black sphere) calculated for $\mathbf{q}\to 0$ (left) and $0.4{\AA }^{-1}$ (right). The black arrow is the direction of $\mathbf{q}\ne 0$. White and green spheres represent N and B atoms in $h$-BN and H and C atoms in graphane.

Figure 3

(a) Calculated dispersion of the lowest-energy bright excitons in graphane and $h$-BN along $\mathrm{\Gamma }M$ and phosphorene along $\mathrm{\Gamma }X$. (b) Singlet and triplet exciton dispersion in $h$-BN in a two-band model with flat bands and (c) with the real electronic band structure.