Two-photon absorption in two-dimensional materials: The case of hexagonal boron nitride

We calculate the two-photon absorption in bulk and single-layer hexagonal boron nitride ($h$-BN) both by an ab initio real-time Bethe-Salpeter approach and by a real-space solution of the excitonic problem in tight-binding formalism. The two-photon absorption obeys different selection rules from those governing linear optics and therefore provides complementary information on the electronic excitations of $h$-BN. Combining the results from the simulations with an analysis of the crystal symmetries, we show that two-photon absorption is able to probe the lowest-energy $1s$ state in the single-layer $h$-BN and the lowest degenerate exciton of bulk $h$-BN. This result indicates that in $h$-BN multilayer stackings with inversion symmetry one can measure the Davydov splitting by means of a combination of one and two-photons excitations. The same analysis can be applied to other two-dimensional materials with the same point-group symmetry—such as the transition metal chalcogenides.

Figure 1

Two-photon absorption and imaginary part of the dielectric constant in single layer (a) and bulk $h$-BN (b), obtained from real-time ab initio dynamics. The two curves have been rescaled in such a way to have the same intensity at the maximum position. Vertical lines in (b) indicate the position of the maximum.

Figure 2

(a) Excitonic joint density of states of the 2D BN monolayer; (b) zoom in the exciton region; top: excitonic joint density; middle: one-photon optical spectrum; bottom: two-photon optical spectrum. For convenience, a broadening of ${10}^{-2}$ eV has been applied, via the imaginary part of $z$ in the Green functions. The $A_1$ exciton is only dark in the one-photon spectrum. The origin of the energy scale is taken at the position of the lowest exciton, which can also be labeled as a $1s$ exciton whereas the other ones derive from $2s$ and $2p$ states. Full ab initio calculations including exchange effects predict the $A_1$ level to be at a higher energy [14].

Figure 3

One- (green) and two-photon (blue) absorption spectrum of $A{A}^{\prime } h$-BN. The results are in agreement with the ab initio results in Fig. 1.

Figure 4

Allowed transitions towards the lowest excitonic states with circularly polarized light. (Left) Monolayer; the ${\mathrm{\Phi }}^{+}\left({\mathrm{\Phi }}^{-}\right)$ is excited with a one-photon (two-photon) process, where ${\mathrm{\Phi }}^{\pm }$ denote the two circularly components of the degenerate $1s$ exciton. (Right) Bulk $A{A}^{\prime }$ stacking; there are now two antisymmetric (AS, odd, one-photon allowed) and symmetric (S, even, two-photon allowed) degenerate states, separated by a Davydov splitting.