Influence of high-energy local orbitals and electron-phonon interactions on the band gaps and optical absorption spectra of hexagonal boron nitride

We report ab initio band diagram and optical absorption spectra of hexagonal boron nitride ($h$-BN), focusing on unravelling how the completeness of the basis set for $GW$ calculations and electron-phonon interactions (EPIs) impact on them. The completeness of the basis set, an issue which was seldom discussed in previous optical spectra calculations of $h$-BN, is found crucial in providing converged quasiparticle band gaps. In the comparison among three different codes, we demonstrate that by including high-energy local orbitals in the all-electron linearized augmented plane waves based $GW$ calculations, the quasiparticle direct and fundamental indirect band gaps are widened by $\sim 0.2$ eV, giving values of 6.81 eV and 6.25 eV, respectively at the $G{W}_0$ level. EPIs, on the other hand, reduce them to 6.62 eV and 6.03 eV respectively at 0 K, and 6.60 eV and 5.98 eV respectively at 300 K. With clamped crystal structure, the first peak of the absorption spectrum is at 6.07 eV, originating from the direct exciton contributed by electron transitions around $K$ in the Brillouin zone. After including the EPIs-renormalized quasiparticles in the Bethe-Salpeter equation, the exciton-phonon coupling shifts the first peak to 5.83 eV at 300 K, lower than the experimental value of $\sim 6.00$ eV. This accuracy is acceptable to an ab initio description of excited states with no fitting parameter.

Figure 1

Electronic band structure calculation by means of LAPW method shows LDA (black solid lines) and $G{W}_0$ without/with HLOs (red/blue solid lines) results of $h$-BN. The orange arrow line represents direct gap, and the green one represents fundamental gap.

Figure 2

Comparison of KS band energies of $h$-BN at the $\mathrm{\Gamma }$ point obtained from using the default LAPW basis and those with additional high-energy local orbitals with $n_{\mathrm{LO}}=1$ and $l_{\max }^{\mathrm{LO}}=4$. The inset shows the convergence of the $G_0{W}_0$ band gap of $h$-BN (calculated with $N_k=2\times 2\times 1$) as a function of the number of bands considered with $n_{\mathrm{LO}}=0$ and $n_{\mathrm{LO}}=1$, respectively.

Figure 3

$T$-dependent direct band gap of $h$-BN calculated using dynamical HAC theory. The band gap renormalization is added to the LAPW $+$ HLOs corrected $G{W}_0$ gap, obtained in Sec. 3a, which is 6.82 eV in yambo code.

Figure 4

$T$-dependent fundamental band gap of $h$-BN calculated using dynamical HAC theory. The band gap renormalization is added to the LAPW $+$ HLOs $G{W}_0$ gap, obtained in Sec. 3a, which is 6.27 eV in yambo code.

Figure 5

(a) Generalized electron-phonon Eliashberg function for v (black dashed line), c (red dotted line), and band edge (blue solid line) of direct band gap. (b) Phonon dispersion along selected symmetry points. (c) Eliashberg function of fundamental indirect band gap. (d) The direct band-edge Eliashberg function projected on each phonon mode. Only the most representative phonon modes are considered (e) below 900 ${\mathrm{cm}}^{-1}$ and (f) above 1200 ${\mathrm{cm}}^{-1}$.

Figure 6

Comparison between measured and calculated optical absorption of $h$-BN: The experimental spectra [61] (red circles), calculation without EPIs at RPA (black dotted line) and BSE (blue dashed line) level, and calculation with EPIs at BSE level at 0 K (green solid line) and 300 K (orange solid line).