Understanding first-order Raman spectra of boron carbides across the homogeneity range

Boron carbide, a lightweight, high temperature material, has various applications as a structural material and as a neutron absorber. The large solubility range of carbon in boron, between $\approx 9\%$ and 20%, has been theoretically explained by some of us by the thermodynamical stability of three icosahedral phases at low temperature, with respective carbon atomic concentrations: 8.7% (${\mathrm{B}}_{10.5}\mathrm{C}$, named ${\mathrm{OPO}}_1$), 13.0% (${\mathrm{B}}_{6.7}\mathrm{C}$, named ${\mathrm{OPO}}_2$), whose theoretical Raman spectra are still unknown, and 20% (${\mathrm{B}}_4\mathrm{C}$), from which the nature of some of the Raman peaks are still debated. We report theoretical and experimental results of the first-order, nonresonant, Raman spectrum of boron carbide. Density functional perturbation theory enables us to obtain the Raman spectra of the ${\mathrm{OPO}}_1$ and ${\mathrm{OPO}}_2$ phases, which are perfectly ordered structures with however a complex crystalline motif of 414 atoms, due to charge compensation effects. Moreover, for the carbon-rich ${\mathrm{B}}_4\mathrm{C}$, with a simpler 15-atom unit cell, we study the influence of the low energy point defects and of their concentrations on the Raman spectrum, in connection with experiments, thus providing insights into the sensitivity of experimental spectra to sample preparation, experimental conditions, and setup. In particular, this enables us to propose a new structure at 19.2% atomic carbon concentration, ${\mathrm{B}}_{4.2}\mathrm{C}$, that, within the local density approximation of density functional theory (DFT-LDA), lies very close to the convex hull of boron carbide, on the carbon-rich side. This new phase, derived from what we name the “3+1” defect complex, helps in reconciling the experimentally observed Raman spectrum with the theory around 1000 ${\mathrm{cm}}^{-1}$. Finally, we predict the intensity variations induced by the experimental geometry and quantitatively assess the localization of bulk and defect vibrational modes and their character, with an analysis of “chain” and “icosahedral” modes.

Figure 1

Typical experimental Raman spectrum of carbon-rich boron carbide compared to the theoretical first-order Raman spectrum of the (${\mathrm{B}}_{11}\mathrm{C})\mathrm{C}\text{−}\mathrm{B}\text{−}\mathrm{C}$ structure. The experimental one presented here has been obtained on a sample with small grains and is thus randomized over light polarization directions, as the theoretical one, with an incident laser with a wavelength of 633 nm. The frequencies of the main experimental peaks are reported.

Figure 2

Maps of the computed Raman intensity vs polar angle $\varphi$ for various choices of the azimuthal angle $\theta$. $\theta$ and $\varphi$ identify the geometrical relationship between the incident light polarization and the crystal lattice orientation. $\theta$ is given in degrees.

Figure 3

Comparison of simulated spectra for selected values of the azimuthal ($\theta$) and polar ($\varphi$) angles. The spectrum corresponding to a polycrystalline sample with randomly oriented grains, shown here for comparison (yellow curves), is reduced by a factor of 10.

Figure 4

Experimental Raman spectra of boron carbide across a grain boundary. (a) Micrograph showing the Raman collection line and the grain boundary. (b) Spatial map of the spectrum along the line shown in (a). (c) Comparison of the spectra of grains 1 and 2.

Figure 5

Theoretical bulk spectrum of carbon-rich boron carbide ${\mathrm{B}}_4\mathrm{C}$ in the 15-atom ground state structure $\left({\mathrm{B}}_{11}{\mathrm{C}}^p\right)\mathrm{C}\text{−}\mathrm{B}\text{−}\mathrm{C}$. (a) The calculated first-order Raman spectrum (broadening: $10{\mathrm{cm}}^{-1}$) with chain/icosahedron character enhanced by a factor $\alpha =100$ following Eq. (3). (b) The localization of each mode (irrespective of its Raman activity) given by Eq. (2) as a function of frequency (all $\mathrm{\Gamma }$-point modes corresponding to a $2\times 2\times 2$ supercell are included). (c) The normalized phonon density of states, obtained with three different q-point grids in the BZ, for comparison. The infrared active mode at $1641{\mathrm{cm}}^{-1}$ is the antisymmetric chain stretching. Its Raman intensity is approximately 1/100 of the intensity of the largest peak and for this reason it is not shown in (a).

Figure 6

The labeling of the atomic sites of the 15-atom $\left({\mathrm{B}}_{11}\mathrm{C}\right)\mathrm{C}\text{−}\mathrm{B}\text{−}\mathrm{C}$ unit cell (in blue). The two arrows indicate the approximate position of the $2c$ and $6g$ EIWS (extra interstitial Wyckoff sites) of the $R\overline{3}m$ rhombohedral space group. Among icosahedral atoms, equatorial borons B5–B8 are bound to neighboring C-B-C chains, while polar atoms B2–B4 and C3 bind to neighboring icosahedra.

Figure 7

(a) Theoretical Raman spectrum of the negatively charged boron chain vacancy, at various defect concentrations. (b) Mode localization compared to undefected bulk ${\mathrm{B}}_4\mathrm{C}$.

Figure 8

Theoretical Raman spectrum of two negatively charged carbon antisites (a) and (c) and one positively charged boron antisite (e), at various defect concentrations, and their mode localization (b), (d), and (f) compared to undefected bulk ${\mathrm{B}}_4\mathrm{C}$.

Figure 9

Theoretical Raman spectrum of two types of boron interstitials in charge states +1 and +3, at various defect concentrations (a), (c), (e), and (g), and mode localization (b), (d), (f), and (h) compared to undefected bulk ${\mathrm{B}}_4\mathrm{C}$. The huge Raman activity of the boron interstitial in the Wyckoff position $2c$ with charge +1 (a) is due to the small Fermi level range of stability of this defect, which questions the reliability of the underlying approximations of the present calculation (first-order nonresonant Raman spectrum in the Placzek approximation).

Figure 10

(a) Theoretical Raman spectrum of the neutral bipolar complex, at various defect concentrations, and (b) mode localization of this defect compared to undefected bulk ${\mathrm{B}}_4\mathrm{C}$.

Figure 11

Theoretical Raman spectrum of the $\mathrm{B}{\langle }_{\text{B}}^{\text{B}}\rangle \mathrm{B}$ blocks in charge states +1 (a) and +3 (c), at various defect concentrations, and mode localization of this defect complex compared to undefected bulk ${\mathrm{B}}_4\mathrm{C}$ (b) and (d)). The charge transition level between these two charge states is at approximately 0.6 eV above the valence band top.

Figure 12

The Raman spectrum ${\mathrm{B}}_4\mathrm{C}$ containing or not a 3+1 defect complex. The three spectra were obtained with a direct DFPT calculation. Those labeled ${\mathrm{B}}_{4.8}\mathrm{C}$ and ${\mathrm{B}}_{4.2}\mathrm{C}$ refer, respectively, to cells containing 406 and 121 atoms, built by inserting in ${\mathrm{B}}_4\mathrm{C}$ a 3+1 defect complex. The former structure, with 19.2% atomic carbon concentration, lies on the convex hull in DFT-LDA and thus defines a stable phase at that stoichiometry. The corresponding spectrum is the one in the middle. The one at the top, corresponding to a higher concentration of the 3+1 defect complex, is shown for comparison.

Figure 13

Raman spectrum calculated for the two boron-rich phases called ${\mathrm{OPO}}_1$ (a) and ${\mathrm{OPO}}_2$ (b) and for the ${\mathrm{OPO}}_2$ variant named ${\mathrm{OPO}}_2^{\text{panto}}$ (c) featuring the $\mathrm{C}{\langle }_{\text{B}}^{\text{B}}\rangle \mathrm{C}$ pantograph structure instead of $2c$ boron interstitials. The unit cells of these phases all contain 414 atoms.