Exciton energy-momentum map of hexagonal boron nitride

Understanding and controlling the way excitons propagate in solids is a key for tailoring materials with improved optoelectronic properties. A fundamental step in this direction is the determination of the exciton energy-momentum dispersion. Here, thanks to the solution of the parameter-free Bethe-Salpeter equation (BSE), we draw and explain the exciton energy-momentum map of hexagonal boron nitride (h-BN) in the first three Brillouin zones. We show that h-BN displays strong excitonic effects not only in the optical spectra at vanishing momentum $\mathbf{q}$, as previously reported, but also at large $\mathbf{q}$. We validate our theoretical predictions by assessing the calculated exciton map by means of an inelastic x-ray scattering (IXS) experiment. Moreover, we solve the discrepancies between previous experimental data and calculations, proving then that the BSE is highly accurate through the whole momentum range. Therefore, these results put forward the combination BSE and IXS as the tool of choice for addressing the exciton dynamics in complex materials.

Figure 1

Dynamic structure factor at three Brillouin zone boundaries $\mathbf{q}=A, K$, and $M$ calculated in the GW-RPA and from the solution of the BSE compared to the NRIXS data from Ref. [19].

Figure 2

(a) GW band structure of h-BN in the (b) first Brillouin zone with the irreducible part delimited by yellow lines and shaded.

Figure 3

Dynamic structure factor for $\mathbf{q}$ along the $\mathrm{\Gamma }K$ direction calculated in the GW-RPA and from the solution of the BSE compared to the NRIXS data from Ref. [19]. The $\mathbf{q}=2.5$ Å${}^{-1}$ is the $M^{\prime }$ point (see text).

Figure 4

(a) Color map of $S(\mathbf{q},\omega )$ calculated from the BSE for fixed $\omega =7$ eV and for all $\mathbf{q}$ spanning the first three Brillouin zones with ${\mathbf{q}}_z=0$. To enhance its visibility, $S(\mathbf{q},\omega )$ in the first BZ has been magnified by a factor 3. The high-symmetry points relevant for the discussion in the main text are explicitly indicated (in parentheses those belonging to the ${\mathbf{q}}_z=0.5$ plane). Adding $\mathbf{q}=\mathrm{\Gamma }M$ (violet arrow) to $\mathbf{k}=M\left(L\right)$ one obtains another ${\mathbf{k}}^{\prime }=\mathbf{k}+\mathbf{q}=M\left(L\right)$ (see dashed violet line). Equivalently adding $\mathbf{q}=\mathrm{\Gamma }K$ (red arrow) to $\mathbf{k}=K\left(H\right)$ one obtains again ${\mathbf{k}}^{\prime }=K\left(H\right)$ (see dashed red line). (b) Comparison between calculated and measured $S(\mathbf{q},\omega )$ for $\mathbf{q}$ along the $\mathrm{\Gamma }M$ direction.

Figure 5

Dynamic structure factor along three high-symmetry lines in the first Brillouin zone $\mathrm{\Gamma }A, \mathrm{\Gamma }K$, and $\mathrm{\Gamma }M$ calculated in the GW-RPA (dashed lines) and from the solution of the BSE (solid lines), compared to the NRIXS data (dotted lines) from Ref. [19]. The distance from $\mathrm{\Gamma }$ to the borders of the first BZ along each direction is indicated in the labels.