Exciton-Phonon Coupling in the Ultraviolet Absorption and Emission Spectra of Bulk Hexagonal Boron Nitride

We present an ab initio method to calculate phonon-assisted absorption and emission spectra in the presence of strong excitonic effects. We apply the method to bulk hexagonal BN, which has an indirect band gap and is known for its strong luminescence in the UV range. We first analyze the excitons at the wave vector $\overline{q}$ of the indirect gap. The coupling of these excitons with the various phonon modes at $\overline{q}$ is expressed in terms of a product of the mean square displacement of the atoms and the second derivative of the optical response function with respect to atomic displacement along the phonon eigenvectors. The derivatives are calculated numerically with a finite difference scheme in a supercell commensurate with $\overline{q}$. We use detailed balance arguments to obtain the intensity ratio between emission and absorption processes. Our results explain recent luminescence experiments and reveal the exciton-phonon coupling channels responsible for the emission lines.

Figure 1

Finite-$q$ calculations in hexagonal boron nitride. (a) $G{W}_0$ quasiparticle band structure of bulk hBN, displayed in the relevant part of the BZ. The dashed horizontal black line represents the position of the lowest-bound direct exciton, while the dashed red lines correspond to the indirect excitons of momentum $\overline{q}=|K-M|$. The exact valence band maxima are labeled $T_1$ and $T_2$. (b) The phonon dispersion in bulk hBN. The teal vertical line highlights the frequencies of the phonon modes with momentum $\overline{q}$. The notation is $Z$ for out-of-plane modes, $L$ for in-plane longitudinal modes, and $T$ for in-plane transverse modes. $O$ ($A$) stands for optical (acoustic) modes.

Figure 2

Phonon-assisted absorption in hexagonal boron nitride. Imaginary part of the dielectric function including contributions of transitions at $q=\overline{q}$ mediated by a single phonon [dashed red line, ${ϵ}_{\overline{q}2}^{(2)}(\omega )$ in the text]. The orange (green) peaks originate from the phonon couplings to $i1$ ($i2$), and the phonon modes responsible for them are labeled. The peak broadenings are set to 1.5 meV for the temperature of 0 K. The dashed red vertical lines indicate the positions of the two pairs of $A_1+B_1$ excitons at $\overline{q}$ (labeled $i1$ and $i2$). The thick gray vertical line is at the position of the main optically active $q=0$ exciton.

Figure 3

Phonon-assisted emission in bulk hexagonal boron nitride. (a) Spectral function $R_{\overline{q}}^{\mathrm{sp}}=\sum _{\lambda }{R}_{\lambda \overline{q}}^{\mathrm{sp}}$ (green solid line) at 5 K. The $\lambda$ components of the spectrum, belonging to the different phonon modes, are also plotted in various colors. The exciton-phonon couplings are labeled. (b) Comparison with the normalized experimental spectrum (black dots) at 10 K from Ref. [14]. The multiphonon overtones are denoted as “o.” Notice that here our spectrum is blueshifted by 0.322 eV to match the experimental one. The temperature-dependent peak widths are described according to a linear model [23] (parameters taken from the experimental fit; see Supplemental Material [32]). The experimental excitonic temperature $T_{\mathrm{exc}}=55\mathrm{K}$ (see the text) is used.