Thermodynamic stability and properties of boron subnitrides from first principles

We use the first-principles approach to clarify the thermodynamic stability as a function of pressure and temperature of three different $\alpha$-rhombohedral-boron-like boron subnitrides, with the compositions of ${\mathrm{B}}_6\mathrm{N}, {\mathrm{B}}_{13}{\mathrm{N}}_2$, and ${\mathrm{B}}_{38}{\mathrm{N}}_6$, proposed in the literature. We find that, out of these subnitrides with the structural units of ${\mathrm{B}}_{12}$(N-N), ${\mathrm{B}}_{12}$(NBN), and ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$, respectively, only ${\mathrm{B}}_{38}{\mathrm{N}}_6$, represented by ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$, is thermodynamically stable. Beyond a pressure of about 7.5 GPa depending on the temperature, also ${\mathrm{B}}_{38}{\mathrm{N}}_6$ becomes unstable, and decomposes into cubic boron nitride and $\alpha$-tetragonal-boron-like boron subnitride ${\mathrm{B}}_{50}{\mathrm{N}}_2$. The thermodynamic stability of boron subnitrides and relevant competing phases is determined by the Gibbs free energy, in which the contributions from the lattice vibrations and the configurational disorder are obtained within the quasiharmonic and the mean-field approximations, respectively. We calculate lattice parameters, elastic constants, phonon and electronic density of states, and demonstrate that ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$ is both mechanically and dynamically stable, and is an electrical semiconductor. The simulated x-ray powder-diffraction pattern as well as the calculated lattice parameters of ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$ are found to be in good agreement with those of the experimentally synthesized boron subnitrides reported in the literature, verifying that ${\mathrm{B}}_{38}{\mathrm{N}}_6$ is the stable composition of $\alpha$-rhombohedral-boron-like boron subnitride.

Figure 1

Structural units of two different $\alpha \mathrm{R}$-boron-like boron subnitrides, proposed in the literature: (a) ${\mathrm{B}}_6\mathrm{N}$ or ${\mathrm{B}}_{12}$(N-N) with 14.28 at. % N, and (b) ${\mathrm{B}}_{13}{\mathrm{N}}_2$ or ${\mathrm{B}}_{12}$(NBN) with 13.33 at. % N. ${\mathrm{B}}_{38}{\mathrm{N}}_6$ with 13.63 at. % N can be viewed as an alloy of the two in the form of ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$. White and grey spheres represent boron and nitrogen atoms, respectively.

Figure 2

Formation enthalpy $\mathrm{\Delta }{H}^{\mathrm{form}}$ of $t-{\mathrm{B}}_{50}{\mathrm{N}}_2, {\mathrm{B}}_{12}$(NBN), ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$, and ${\mathrm{B}}_{12}$(N-N), corresponding to the compositions of ${\mathrm{B}}_{50}{\mathrm{N}}_2, {\mathrm{B}}_{13}{\mathrm{N}}_2, {\mathrm{B}}_{38}{\mathrm{N}}_6$, and ${\mathrm{B}}_6\mathrm{N}$ respectively, with respect to pure boron ($\alpha \mathrm{R}$ and $\gamma$ phases) and $c$-BN under pressures, ranging from 0 to 20 GPa. Filled and open symbols indicate compositions, which are lying on and above the convex hulls at different fixed pressures, respectively. The inset shows a full picture of the convex hull at $p=0$ GPa.

Figure 3

Electronic density of states of (a) ${\mathrm{B}}_6\mathrm{N}$, (b) ${\mathrm{B}}_{13}{\mathrm{N}}_2$, and (c) ${\mathrm{B}}_{38}{\mathrm{N}}_6$, represented by ${\mathrm{B}}_{12}$(N-N), (b) ${\mathrm{B}}_{12}$(NBN), and (c) ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$, respectively. The dashed lines at 0 eV indicate the highest energy states, occupied by the electrons.

Figure 4

Specific heat ($c_V$) of ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$, evaluated from Monte Carlo simulations, using simulation boxes of $30\times 30\times 30$ rhombohedral unit cells.

Figure 5

Superatomic configuration, representing the ordered structure of ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$, in which white and grey spheres stand for ${\mathrm{B}}_{12}$(N-N) superatoms and ${\mathrm{B}}_{12}$(NBN) superatoms, respectively.

Figure 6

Phonon density of states (PDOS) of (a) ${\mathrm{B}}_{12}$(N-N), (b) ${\mathrm{B}}_{12}$(NBN), and (c) ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$, corresponding to the compositions of ${\mathrm{B}}_6\mathrm{N}, {\mathrm{B}}_{13}{\mathrm{N}}_2$, and ${\mathrm{B}}_{38}{\mathrm{N}}_6$, respectively, at $p=0$ GPa. Negative numbers correspond to imaginary frequencies.

Figure 7

$p-T$ phase diagram at a composition of ${\mathrm{B}}_{38}{\mathrm{N}}_6$.

Figure 8

Simulated x-ray powder-diffraction pattern ($\text{Cu}K\alpha$) of ${\left[{\mathrm{B}}_{12}\left(\mathrm{N}-\mathrm{N}\right)\right]}_{0.33}{\left[{\mathrm{B}}_{12}\left(\mathrm{NBN}\right)\right]}_{0.67}$ at the composition of ${\mathrm{B}}_{38}{\mathrm{N}}_6$ (blue), compared with the experimental powder-diffraction data of ${\mathrm{B}}_6{\mathrm{N}}_{0.92}$ [30] (red) and ${\mathrm{B}}_{13}{\mathrm{N}}_2$ (black), taken from The International Centre for Diffraction Data (ICDD) cards: 00-050-1504 and 00-060-0209, respectively.