  We have investigated the trends in the formation of carbon vacancies in
binary transition metal (TM) carbides using density functional calculations for
two common TM carbide crystal structures, namely the B1 (rock-salt) and WC
structure types. The TM are taken from group IV (Ti, Zr, Hf), V (V, Nb, Ta) and
VI (Cr, Mo, W) of the periodic table, as well as Sc from group III. For
B1-structured TM carbides, the general trend is that it is easier for C
vacancies to be formed as the number of valence electrons in the system
increases. The exception is ScC where C vacancies are rather easy to form. It
is also clear that the formation of C vacancies depends on the growth
conditions: For TM-rich conditions, B1-structured carbides will always favour C
vacancy formation. For C-rich conditions it is energetically favourable to form
C vacancies in ScC, VC, NbC, CrC, MoC and WC. Experimentally large C vacancy
concentrations are found in B1-structured TM carbides and our calculations are
in line with these observations. In fact, TiC, ZrC, HfC and TaC are the only
B1-structured TM carbides that do not spontaneously favour C vacancies and then
only for C-rich conditions. C vacancies may still be present in these systems,
however, due to the high temperatures used when growing TM carbides. In the
case of WC-structured TM carbides, C vacancy formation is not energetically
favourable irrespective of the growth conditions, which is the reason why this
structure is only found close to a one-to-one metal-to-carbon ratio. ...
