Flat Bands and Giant Light-Matter Interaction in Hexagonal Boron Nitride

Dispersionless energy bands in $k$ space are a peculiar property gathering increasing attention for the emergence of novel electronic, magnetic, and photonic properties. Here, we explore the impact of electronic flat bands on the light-matter interaction. The van der Waals interaction between the atomic layers of hexagonal boron nitride induces flat bands along specific lines of the Brillouin zone. The macroscopic degeneracy along these lines leads to van Hove singularities with divergent joint density of states, resulting in outstanding optical properties of the excitonic states. For the direct exciton, we report a giant oscillator strength with a longitudinal-transverse splitting of 420 meV, a record value, confirmed by our ab initio calculations. For the fundamental indirect exciton, flat bands result in phonon-assisted processes of exceptional efficiency, that compete with direct absorption in reflectivity, and that make the internal quantum efficiency close to values typical of direct band gap semiconductors.

Figure 1

(a) Electronic band structure evaluated in the framework of many-body GW calculations. The shaded areas indicate the paths that are parallel to the c-axis in the Brillouin zone of bulk hBN. Arrows in dark gray (dashed black) indicate the parallel direct (indirect) transitions contributing to the formation of the direct (dX) and indirect (iX) excitons, respectively. (b) Color map in the Brillouin zone of bulk hBN of the weighted oscillator strengths $A_{\lambda }^t{\rho }^t(\mathbf{q})$ [as defined in Eq. (1)] building the dX exciton. For the sake of clarity, the map is limited to the points of the prevailing direct independent-particle transitions $t$, that contribute in total to 80% of the dX absorption. The irreducible part of the Brillouin zone is delimited by red lines, which are thicker for the KH and ML paths.

Figure 2

Reflectivity (black) and transmission (green) of a high-quality hBN crystal, recorded close to normal incidence at 8 K. The crystal thickness is $4.5\mu \mathrm{m}$. The shaded area indicates the excitonic reststrahlen band in hBN. Inset: microscope image of high-quality hBN synthesized by the metal flux method.

Figure 3

Comparison between reflectivity (black) and photoluminescence (red) of bulk hBN at 8 K. The vertical dashed line at 5.955 eV indicates the energy of the indirect exciton (iX). The dotted blue spectrum corresponds to the mirror image of the PL spectrum with respect to the iX energy. Inset: schematic representation of absorption (blue) and emission (red) assisted by phonon emission (black arrow). The blue and red dashed lines indicate the virtual states involved in the phonon-assisted optical processes. $TA$, $LA$, $TO$, and $LO$ indicate the modes of the emitted phonons in reflectivity.

Figure 4

(a) Experimental reflectivity spectrum at 8 K. (b) Calculated reflectivity spectrum (red solid line) taking into account the direct exciton ($dX$) only, with $\gamma /{\mathrm{\Delta }}_{\mathrm{LT}}=0.41$, where $\gamma$ is the $dX$ linewidth and ${\mathrm{\Delta }}_{\mathrm{LT}}$ is the longitudinal-transverse splitting. ${\mathrm{\Delta }}_{\mathrm{LT}}=420\mathrm{meV}$ in hBN. The red dashed line is the reflectivity spectrum calculated for $\gamma /{\mathrm{\Delta }}_{\mathrm{LT}}={10}^{-4}$ in order to better visualize ${\mathrm{\Delta }}_{\mathrm{LT}}$, which corresponds to the width of the reflectivity stop band when $\gamma /{\mathrm{\Delta }}_{\mathrm{LT}}\ll 1$. (c) Calculated reflectivity spectrum (green solid line) taking into account the indirect ($iX$) and direct ($dX$) excitons ($\gamma /{\mathrm{\Delta }}_{\mathrm{LT}}=0.41$). $E_{iX}=5.955$ and $E_{dX}=6.125\mathrm{eV}$.