Electronic and optical properties of crystalline nitrogen versus black phosphorus: A comparative first-principles study

Crystalline black nitrogen (BN) is an allotrope of nitrogen with the black phosphorus (BP) structure recently synthesized at high pressure by two independent research groups [Ji et al., Sci. Adv. 6, eaba9206 (2020); Laniel et al., Phys. Rev. Lett. 124, 216001 (2020)]. Here, we present a systematic study of the electronic and optical properties of BN focusing on its comparison with BP. To this end, we use the state-of-the-art quasiparticle self-consistent $GW$ approach with vertex corrections in both the electronic and optical channels. Despite many similarities, the properties of BN are found to be considerably different. Unlike BP, BN exhibits a larger optical gap (2.5 vs 0.26 eV), making BN transparent in the visible spectral region with a highly anisotropic optical response. This difference can be primarily attributed to a considerably reduced dielectric screening in BN, leading to enhancement of the effective Coulomb interaction. Despite relatively strong Coulomb interaction, exciton formation is largely suppressed in both materials. Our analysis of the elastic properties shows exceptionally high stiffness of BN, comparable to that of diamond.

Figure 1

Top: Schematic crystal structure of BN (left) and BP (right). The numbers correspond to the bond lengths (in Å). Bottom: Brillouin zone with high-symmetry points for primitive (left) and conventional (right) unit cells of BN/BP containing four and eight atoms, respectively.

Figure 2

Pressure and total energy as a function of the primitive cell volume calculated for BN with the GGA and optB86b functionals. Experimental pressures from Ref. [7] are shown for comparison. The vertical line corresponds to the experimental volume ($V_{\exp }\approx 20$ Å${}^3$) used in this work. Zero total energy corresponds to the energy calculated at $V_{\exp }$.

Figure 3

Single-particle band structure of BN (left) and BP (right) calculated using the DFT (LDA), $\mathrm{QS}GW$, and $\mathrm{QS}G\widehat{W}$ approaches (from top to bottom) along high-symmetry directions of the conventional Brillouin zone (Fig. 1). The colors correspond to the contributions of $p_x$ (blue), $p_y$ (red), and $p_z$ (green) orbitals.

Figure 4

Density of states projected onto different orbital states calculated for BN (top) and BP (bottom) using the $\mathrm{QS}G\widehat{W}$ approach.

Figure 5

Imaginary part of the diagonal components of the dielectric tensor calculated as a function of frequency using RPA@$\mathrm{QS}G\widehat{W}$ (top) and BSE@$\mathrm{QS}G\widehat{W}$ (bottom) for BN (left) and BP (right).

Figure 6

Effective Coulomb interaction as a function of the interatomic distance in BN and BP. Violet and green symbols correspond to the bare ($V$) and screened ($W$) interactions, respectively. Lines correspond to the classical $V\left(r\right)=k_{e}e/r$ and screened classical $W\left(r\right)=V\left(r\right)/\varepsilon$ Coulomb interaction, where $\varepsilon$ is the effective screening constant.

Figure 7

Phonon dispersion and phonon density of states calculated along the high-symmetry direction of the primitive Brillouin zone (Fig. 1) for BN and BP. The labels at the $\mathrm{\Gamma }$ points indicate the symmetries of the corresponding optical modes according to irreducible representations of the $D_{2h}$ point group. Red arrows denote frequencies of the Raman-active modes.

Figure 8

Orientation dependence of Young's modulus $E\left(\mathbf{n}\right)$ in BN and BP.

Figure 9

(a) and (b) The convergence of the RPA@$\mathrm{QS}G\widehat{W}$ and BSE@$\mathrm{QS}G\widehat{W}$ dielectric functions ${\varepsilon }_2^{yy}\left(\omega \right)$ in BN with respect to the k-point mesh. (c) The convergence of the deepest-lying dark $e$-h eigenvalue.