Density functional study of carbon vacancies in titanium carbide

Abstract

It is well established that TiC contains carbon vacancies not only in carbon-deficient environments but also in carbon-rich environments. We have performed density functional calculations of the vacancy formation energy in TiC for C- as well as Ti-rich conditions using several different approximations to the exchange-correlation functional, and also carefully considering the nature and thermodynamics of the carbon reference state, as well as the effect of varying growth conditions. We find that the formation of carbon vacancies is clearly favorable under Ti-rich conditions, whereas it is slightly energetically unfavorable under C-rich conditions. Furthermore, we find that the relaxations of the atoms close to the vacancy site are rather long-ranged, and that these relaxations contribute significantly to the stabilization of the vacancy. Since carbon vacancies in TiC are also experimentally observed in carbon-rich environments, we conclude that kinetics may play an important role. This conclusion is consistent with the experimentally observed high activation energies and sluggish diffusion of vacancies in TiC, effectively causing a freezing in of the vacancies.

Carbon Vacancy Formation in Binary Transition Metal Carbides from Density Functional Theory

“…Both the B1-and WC-structured supercells contain 250 atomic positions and the B1-structured supercells have been used previously to model defects in TiC. 18 We note that this size of supercell provides an effective vacancy concentration of 0.8%. The use of a finite size supercell introduces an error in the defect formation energies since defects in neighbouring cells will have a non-physical interaction with each other.…”

“…In a previous study we used a finite size scaling procedure to calculate the C vacancy formation energy in TiC. 18 The C vacancy formation energy was evaluated for a set of supercells of increasing size and thereafter interpolated to evaluate the vacancy formation energy of a single C vacancy in an infinite supercell. 18 The difference in the C vacancy formation energy between the extrapolated result and the result obtained for the 5×5×5 supercell was found to be 0.12 to 0.17 eV depending on which density functional approximation was used, where the extrapolated value is the lower value.…”

“…18 The C vacancy formation energy was evaluated for a set of supercells of increasing size and thereafter interpolated to evaluate the vacancy formation energy of a single C vacancy in an infinite supercell. 18 The difference in the C vacancy formation energy between the extrapolated result and the result obtained for the 5×5×5 supercell was found to be 0.12 to 0.17 eV depending on which density functional approximation was used, where the extrapolated value is the lower value. We consider it to be likely that this difference the same or similar for all TM carbides in our study and since we are interested in the trends in the vacancy formation energies among the TM carbides we will use the 5×5×5 supercell to evaluate the C vacancy formation energy in the TM carbides.…”

“…50 In a recent study it was shown that it is possible to simulate graphite using standard density functionals by evaluating the energy of single sheets of graphene separately using density functional calculations and to add the interatomic van der Waals interaction energy evaluated using the random phase approximation (RPA). 18 As in Ref. 18 we use the RPA interlayer interaction energy obtained by Lebègue et al 51 of E C vdW−RPA = 48 meV.…”

“…18 As in Ref. 18 we use the RPA interlayer interaction energy obtained by Lebègue et al 51 of E C vdW−RPA = 48 meV. By using this convention to calculate the total energy of graphite it is possible to avoid the evaluation of the van der Waals bonding in this system by DFT.…”