Structure and energy of point defects in TiC: An ab initio study

We employ first-principles calculations to study the atomic and electronic structure of various point defects such as vacancies, interstitials, and antisites in the stoichiometric as well as slightly off-stoichiometric ${\mathrm{Ti}}_{1-c}{\mathrm{C}}_c$ (including both C-poor and C-rich compositions, $0.49\le c\le 0.51$). The atomic structure analysis has revealed that both interstitial and antisite defects can exist in split conformations involving dumbbells. To characterize the electronic structure changes caused by a defect, we introduce differential density of states (dDOS) defined as a local perturbation of the density of states (DOS) on the defect site and its surrounding relative to the perfect TiC. This definition allows us to identify the DOS peaks characteristic of the studied defects in several conformations. So far, characteristic defect states have been discussed only in connection with carbon vacancies. Here, in particular, we have identified dDOS peaks of carbon interstitials and dumbbells, which can be used for experimental detection of such defects in TiC. The formation energies of point defects in TiC are derived in the framework of a grand-canonical formalism. Among the considered defects, carbon vacancies and interstitials are shown to have, respectively, the lowest and the second-lowest formation energies. Their formation energetics are consistent with the thermodynamic data on the phase stability of nonstoichiometric TiC. A cluster type of point defect is found to be next in energy, a titanium [100] dumbbell terminated by two carbon vacancies.

Figure 1

Calculated total DOS (black line) and site-projected DOS of pristine TiC. The Ti states are shown in blue and those of C in red. The energy is given relative to the Fermi level $E_F$ (green dashed line). Vertical black dashed lines indicate the edges (top $E_T$ or bottom $E_B$) of the lower (region I) and the upper (regions II and III) valence bands.

Figure 2

Computed atomic and electronic structure of C-depleting point defects in TiC. The blue and brown balls represent the Ti and C atoms, respectively. Gray arrows show the direction and magnitude (exaggerated) of the atomic displacements around the defect. Interatomic distances are indicated by numbers (Å) and red arrows. Vertical and horizontal thin lines show the positions of lattice planes in the perfect TiC. In dDOS plots, the vertical dashed lines indicate the band edges and Fermi level (green) as in Fig. 1, and the horizontal line indicates the zero level of dDOS.

Figure 3

Computed atomic and electronic structure of C-enriching point defects in TiC. The blue and brown balls represent the Ti and C atoms, respectively. Gray arrows show the direction and magnitude (exaggerated) of the atomic displacements around the defect. Interatomic distances are indicated by numbers (Å) and red arrows. Vertical and horizontal thin lines show the positions of lattice planes in the perfect TiC. In dDOS plots, the vertical dashed lines indicate the band edges and Fermi level (green), as in Fig. 1, and the horizontal line indicates the zero level of dDOS.

Figure 4

Energy of formation of the ${\mathrm{Ti}}_{1-c}{\mathrm{C}}_c$ compound, $\Delta {\varepsilon }^{\mathrm{form}}$, calculated as a function of carbon atomic fraction $c$ in defect-free and defect-bearing supercells, relative to hcp titanium and diamond carbon (all energies are normalized per atom). The values of chemical potential $\mu$, corresponding to the slope of $\Delta {\varepsilon }^{\mathrm{form}}\left(c\right)$ in different regions of the Ti-C binary system, are indicated. Note that, for small concentration of carbon vacancies ${\mathrm{Va}}_{\mathrm{C}}$, the $\mu$ is positive. It turns to negative values at much higher ${\mathrm{Va}}_{\mathrm{C}}$ concentrations than those considered in the present work (see Ref. [44]).