Debye temperature and stiffness of carbon and boron nitride polymorphs from first principles calculations

A theoretical investigation has been made on the phonon spectrum and heat capacity of polymorphs of carbon and boron nitride with special interests on the variation of Debye temperature and stiffness with temperature. A part of optical phonon branches of graphite exhibits higher frequencies than those of diamond. As a consequence, graphite shows smaller heat capacity and higher Debye temperature than diamond above a crossover temperature of $1000\mathrm{K}$. This supports experimental reports of heat capacity although available experimental data are widely scattered. The higher Debye stiffness of graphite at above $1000\mathrm{K}$ is not contradictory to the fact that conventional stiffness of diamond is much larger than that of graphite, since the Debye stiffness is determined by both acoustic and optical phonons, whereas only acoustic phonons contribute to the conventional stiffness. The same trend was found between hexagonal and cubic boron nitrides with a crossover temperature of $600\mathrm{K}$.

Figure 1

Phonon dispersion relations and phonon density of states of four crystals. Solid curves show the calculated dispersions. Experimentally, dispersion relations over the Brillouin zone have been obtained by measurements such as neutron scattering, electron energy-loss spectroscopy, and x-ray scattering techniques. Experimental values from different data sources are discriminated by different symbols (Refs. 9, 38, 39, 40, 41, 42, 43, 44, 45). Frequencies at $\Gamma$ point have been determined by Raman and IR spectrum experiments, which are denoted by crosses (Refs. 46, 47, 48, 49, 50).

Figure 2

(Color online) Temperature variation of weighted phonon DOS. (a) Weighted phonon DOS of diamond. (b) Weighted phonon DOS of graphite. The weighted phonon DOS, $g\left(\nu \right)W(h\nu ∕k_{B}T)$, is the integrand in the formula of heat capacity and represents the portion of phonon DOS that contributes to the heat capacity. Lower panels show cross sections of the weighted phonon DOS at several temperatures for (c) diamond, and (d) graphite. Curves for 300, 1000, $2000\mathrm{K}$ are offset for $x$ and $y$ directions by 4 THz and $0.041∕\mathrm{THz}$, respectively. Heat capacity is proportional to the cross section at a given temperature.

Figure 3

(Color online) Heat capacity and difference in heat capacity of related polymorphs. (a) Heat capacity of carbon allotropes. (b) Heat capacity of boron nitride polymorphs. Solid and dashed curves represent the theoretical values of $C_p$ and $C_V$, respectively. Experimental values from different data sources are denoted by different symbols (Refs. 51, 52, 53, 54). (c) Difference in heat capacity of two carbon allotropes calculated as, $\Delta C=C\left(\text{graphite}\right)-C\left(\text{diamond}\right)$. (d) Heat capacity difference of boron nitride polymorphs given as $\Delta C=C\left(h\text{−}\mathrm{B}\mathrm{N}\right)-C\left(c\text{−}\mathrm{B}\mathrm{N}\right)$. Experimental values of heat capacity differences were obtained from smoothed curves available in literature.

Figure 4

(Color online) Temperature dependence of Debye temperature ${\Theta }_D$ for four crystals. (a) diamond and graphite; (b) cubic and hexagonal boron nitride. Theoretical and experimental values are shown by curves and symbols, respectively. Theoretical values of ${\Theta }_D$ are determined in such a way that the heat capacity by the Debye model [Eq. (3)] corresponds to the calculated values of $C_V$. Experimental values of ${\Theta }_D$ presented here were obtained from heat capacity measurements (Ref. 2).