Electronic, dynamical, and thermal properties of ultra-incompressible superhard rhenium diboride: A combined first-principles and neutron scattering study

Rhenium diboride is a recently recognized ultra-incompressible superhard material. Here we report the electronic $\left(e\right)$, phonon $\left(p\right)$, $e\text{−}p$ coupling, and thermal properties of $\mathrm{Re}{\mathrm{B}}_2$ from first-principles density-functional theory calculations and neutron scattering measurements. Our calculated elastic constants ($c_{11}=641\mathrm{GPa}$, $c_{12}=159\mathrm{GPa}$, $c_{13}=128\mathrm{GPa}$, $c_{33}=1037\mathrm{GPa}$, and $c_{44}=271\mathrm{GPa}$), bulk modulus $(B\approx 350\mathrm{GPa})$ and hardness $(H\approx 46\mathrm{GPa})$ are in good agreement with the reported experimental data. The calculated phonon density of states agrees very well with our neutron vibrational spectroscopy result. Electronic and phonon analysis indicates that the strong covalent B-B and Re-B bonding is the main reason for the super incompressibility and hardness of $\mathrm{Re}{\mathrm{B}}_2$. The thermal expansion coefficients, calculated within the quasiharmonic approximation and measured by neutron powder diffraction, are found to be nearly isotropic in $a$ and $c$ directions and only slightly larger than that of diamond in terms of magnitude. The excellent agreement found between calculations and experimental measurements indicate that first-principles calculations capture the main interactions in this class of superhard materials, and thus can be used to search, predict, and design new materials with desired properties.

Figure 1

(Color online) (a) A unit cell of $\mathrm{Re}{\mathrm{B}}_2$. (b) The electron localization function of the $(11-20)$ crystal plane. The four types of chemical bonds are denoted using dashed lines. The strong directional, covalent B-B (labeled “1”) and Re-B (labeled “2” and “3”) bonding are apparent.

Figure 2

(Color online) (a) The calculated electronic band structure of $\mathrm{Re}{\mathrm{B}}_2$, along high-symmetry directions in the Brillouin zone. $\Gamma$, $A$, $H$, $K$, $M$, and $\Lambda$ represent $k$ points (0 0 0), (0 0 0.5), $(-1∕3,2∕3,0.5)$, $(-1∕3,2∕3,0)$, (0, 0.5, 0), and (0, 0.5, 0.5), respectively. (b) The total and partial density of states (DOS) of $\mathrm{Re}{\mathrm{B}}_2$. The main feature of the electronic structure of $\mathrm{Re}{\mathrm{B}}_2$ is a strong hybridization of $\mathrm{Re}\text{−}d$ and $\mathrm{B}\text{−}p$ orbitals, which results in the strong covalent Re-B bond.

Figure 3

Least-square fit of the total energy data vs strain parameters. The dots are from the first-principles calculations and the solid lines are the least-square fit. The deformation matrices for each distortion are also shown. The five hexagonal elastic constants of $\mathrm{Re}{\mathrm{B}}_2$, along with the bulk modulus and shear modulus are summarized in the right-bottom panel.

Figure 4

(a) Calculated phonon dispersion curves along high-symmetry directions in the Brillouin zone for $\mathrm{Re}{\mathrm{B}}_2$ and the phonon density of states. (b) Inelastic neutron scattering spectrum from $\mathrm{Re}{\mathrm{B}}_2$ (at $4\mathrm{K}$) along with the calculated spectrum. The agreement is excellent.

Figure 5

(Color online) (a) The calculated (lines) and experimental (dots) temperature dependence of the structural parameters $a$ and $c$ of $\mathrm{Re}{\mathrm{B}}_2$. (b) The calculated (lines) and experimental (dots) linear thermal expansion coefficient of $\mathrm{Re}{\mathrm{B}}_2$. For comparison purpose, the calculated linear thermal expansion coefficient of diamond is also shown. The thermal expansion of $\mathrm{Re}{\mathrm{B}}_2$ is nearly isotropic and the magnitude is in the same order as that of diamond.