# First-principles prediction of point defect energies and concentrations in
the tantalum and hafnium carbides

I. Khatri, R. K. Koju, and Y. Mishin

###### Abstract

First-principles calculations are combined with a statistical-mechanical model
to predict the equilibrium point-defect concentrations in the refractory
carbides TaC and HfC as a function of temperature and chemical composition.
Several different types of point defects (vacancies, interstitials, antisite
atoms) and their clusters are treated in a unified manner. The defect
concentrations either strictly follow or can be closely approximated by
Arrhenius functions with parameters predicted by the model. The model is
general and applicable to other carbides, nitrides, borides, or similar
chemical compounds. Implications of this work for understanding the diffusion
mechanisms in TaC and HfC are discussed.

Department of Physics and Astronomy, MSN 3F3, George Mason University,
Fairfax, Virginia 22030, USA

_Keywords:_ Refractory carbides; density-functional-theory; statistical
mechanics; point defects; atomic diffusion.

##  1 Introduction

The refractory carbides TaC and HfC belong to the class of ultra-high
temperature ceramics (UHTCs), which includes several other transition metal
carbides, nitrides, and borides. UHTCs are characterized by a high melting
temperature, large elastic moduli, large hardness, good thermal resistance,
and relatively low chemical reactivity. The TaC and HfC carbides have the
highest melting temperatures $T_{\mathrm{m}}$ among all UHTCs. In fact, HfC
has the highest melting temperature (about 3942∘C) of all materials known
today, with TaC’s $T_{\mathrm{m}}$ being only slightly lower (about 3900∘C)
[1].

Both carbides have the B1-ordered (NaCl prototype) crystal structure with
significant deviations from the 50-50 stoichiometry towards carbon-deficient
compositions. This off-stoichiometry is accommodated by constitutional
(structural) vacancies on the carbon sublattice. The chemical bonding in TaC
and HfC combines three contributions: metallic bonding due to the presence of
the metallic atoms, covalent metal-carbon bonds, and some degree of ionic
bonding caused by partial metal-carbon charge transfer [2]. The covalent bonds
are the strongest, making both carbides mechanically strong, hard and brittle,
leading to the high melting temperatures.

The TaC and HfC carbides are notoriously difficult to sinter due to the
extremely small diffusion coefficients. The rate of pore healing during the
sintering is kinetically controlled by diffusive mass transport. Thus, the
knowledge of diffusion coefficients in TaC and HfC is essential for optimizing
the synthesis and processing routes. Experimental information about the
diffusivity of either carbon or the metallic atoms in these carbides is scarce
and indirect. Diffusion measurements are highly challenging as they must be
conducted at temperatures exceeding 2000-2800∘C to generate reliable
concentration curves from which to extract the diffusion coefficients. Carbon
diffusion in TaC was estimated by back-calculation from the growth rate of an
oxide layer [3, 4] or carbide layer [5]. Interdiffusion coefficients in TaC
and HfC were extracted from evaporation data [6]. No experimental data is
available for metal self-diffusion in TaC or HfC.

Under the circumstances, calculations offer the only realistic option for
obtaining the diffusivities in these carbides (and perhaps all other carbides
of the UHTC family). Point-defect concentrations constitute an essential
ingredient for the diffusion calculations. Density-functional theory (DFT)
calculations have been performed for vacancies and vacancy clusters in TaC [7,
8, 9, 10, 11, 12, 13, 2, 14, 15] and HfC [14, 15]. Based on these and other
calculations [7, 8, 9, 10, 11, 14, 16], it is assumed that both carbon and
metallic atoms in TaC and HfC diffuse by vacancy-atom exchanges on the
respective sublattices. Among other findings, it was predicted that the
binding energy between the carbon and metal vacancies in HfC is much stronger
than in TaC, leading to the formation of vacancy clusters in which a Hf
vacancy is surrounded by several carbon vacancies [15]. However,
thermodynamically consistent calculations of the point-defect (and point-
defect cluster) formation energies and equilibrium concentrations have been
challenging. Such calculations must consider that point defects can only
appear and disappear by pairs or clusters that preserve the chemical
composition. Furthermore, different defect clusters must be in equilibrium
with each other with respect to composition-conserving dissociation-
recombination reactions. The only consistent treatment known to us was for
vacancies and antisite defects in the Ti, Zr and Hf carbides and nitrides
[16].

This paper aims to predict point-defect concentrations in TaC and HfC as a
function of temperature and deviation from stoichiometry. To this end, we
develop a methodology that combines DFT calculations with statistical
mechanics accounting for the point-defect energies and entropies, including
both configurational and orientational entropy contributions. This allows us
to predict the equilibrium concentrations of single point defects and point-
defect clusters of any complexity. The methodology is general enough to apply
to other B1-ordered carbides, nitrides, or borides in the future.

##  2 Methodology

The point defect energies were obtained by DFT calculations using the Vienna
Ab initio Simulation package (VASP) [17, 18]. The calculations were carried
out using six different cubic or orthorhombic supercells containing 96, 144,
216, 288, 384, and 512 atoms, obtained by replicating the conventional 8-atom
unit cell of the B1 structure. Using large supercell sizes with up to 512
atoms ensured size convergence, which was especially important for large
defect clusters creating strong elastic strain fields. The convergence plots
are presented in the Supplementary Information file accompanying this article.
The projected augmented wave [19] method was utilized with exchange-
correlation interactions treated in the generalized gradient approximation in
the Perdew-Burke-Ernzerhof formalism [20, 21]. The semi-core $p$ electrons
were included as valence states for both Ta and Hf. Following convergence
test, the energy cutoffs of 780 eV and 860 eV were applied to HfC and TaC,
respectively. The k-point grid convergence tests were conducted for all
supercells; see the Supplementary Information file. A Gaussian smearing of
width 0.05 eV was chosen to perform electron minimization with the convergence
criterion of $10^{-8}$ eV/atom. For ionic relaxations, we used a conjugate-
gradient algorithm with the 0.001 eV/Å force criterion, followed by a quasi-
Newton algorithm to further improve accuracy. Before introducing defects, each
supercell was subjected to volumetric relaxation while preserving its original
shape. The point defects were created by adding, removing, or changing the
species of an atom or a group of atoms, and the structures were relaxed with
respect to local atomic displacements.

Prior to the point-defect calculations, perfect lattice properties of both
carbides were computed to demonstrate the reliability of the DFT methodology.
Table 2 shows that the lattice parameter $a$ and elastic constants $c_{ij}$
obtained by our calculations compare well with the available experimental [22,
23, 24] and theoretical [25, 11, 14, 26] values. The elastic constants were
computed by the energy-strain method using second order polynomial fits for
strains below 6%.

The obtained energies of the perfect and defected supercells served as input
to the statistical-mechanical model for calculating the equilibrium point-
defect concentrations as explained below.

##  3 Point-defect energies

The B1 structure of the TaC and HfC carbides consists of two penetrating face-
centered cubic (FCC) sublattices occupied by metallic and carbon atoms. In the
perfectly stoichiometric carbides at zero temperature, the two sublattices are
filled with the respective atoms without vacancies. At finite temperatures, a
stoichiometric carbide develops thermal disorder in the form of vacancies on
both sublattices, antisite defects, interstitial atoms, and clusters of these
defects. Deviations from the perfect stoichiometry are accommodated by
additional point defects called constitutional (or structural). Depending on
the chemical composition and temperature, the disorder is dominated by either
thermal or constitutional defects. At low temperatures, the defects are
primarily constitutional and can be different on either side of the
stoichiometric composition. They can also be different between TaC and HfC.

To analyze the point defects in both carbides in a unified manner, we will
consider a generic carbide AB, in which the element A is either Ta or Hf and
the element B is carbon. The respective FCC sublattices are denoted $\alpha$
and $\beta$. This generalized notation will allow us to apply the present
analysis to other binary carbides or any ordered compound with the AB
stoichiometry in the future.

There can be six types of elementary (single) point defects:

  * $V_{\alpha}$ = vacancy on sublattice $\alpha$

  * $V_{\beta}$ = vacancy on sublattice $\beta$

  * $A_{\beta}$ = antisite atom A on sublattice $\beta$

  * $B_{\alpha}$ = antisite atom B on sublattice $\alpha$

  * $I_{A}$ = interstitial metallic atom

  * $I_{B}$ = interstitial carbon atom

An antisite defect is obtained by replacing a metallic atom with carbon on the
metallic sublattice ($B_{\alpha}$) or a carbon atom with a metallic atom on
the carbon sublattice ($A_{\beta}$). Interstitial atoms are inserted in a
tetrahedral position.

The elementary defects can form dynamic clusters. The simplest cluster is a
pair of elementary defects separated by a nearest-neighbor distance $r_{0}$.
Examples include divacancies $V_{\alpha}V_{\beta}$, antisite pairs
$A_{\beta}B_{\alpha}$, and vacancy-antisite pairs such as
$V_{\alpha}B_{\alpha}$ and $V_{\beta}A_{\beta}$. In the divacancy and the
antisite pairs, the elementary defects are nearest neighbors on different
sublattices ($r_{0}=a/2$), whereas in the vacancy-antisite pairs
$V_{\alpha}B_{\alpha}$ and $V_{\beta}A_{\beta}$, they are nearest neighbors on
the same sublattice ($r_{0}=a/\sqrt{2}$). Another defect pair, called a
Frenkel pair, is composed of a vacancy and interstitial atom. This defect is
obtained by moving an atom from its sublattice to an interstitial position.
Defect clusters can be composed of three or more elementary defects and can
have several geometric configurations.

DFT calculations produce a set of “raw” energies [27, 28, 29, 30] of point
defects, both elementary and clustered. The “raw” energy of a point defect is
_defined_ as the energy difference between a relaxed supercell containing the
defect and a perfect supercell containing the same number of sites.
Specifically, the “raw” energy $\varepsilon_{d}$ of a defect $d$ is calculated
by the formula

$\varepsilon_{d}=E_{d}(N)-E_{0}(N),$ (1)

where $E_{d}(N)$ and $E_{0}(N)$ are the total energies of an $N$-site
supercell with and without the defect, respectively. Since the two supercells
can have different chemical compositions, the “raw” energy generally depends
on the reference atomic energy used in the DFT calculations. Exceptions
include the antisite pairs $A_{\beta}B_{\alpha}$, Frenkel defects such as
$V_{\alpha}I_{A}$ and $V_{\beta}I_{B}$, and other defects obtained by
displacing atoms from their perfect lattice positions without changing the
chemical composition. In all other cases, the “raw” energy of the defect is
not a physically meaningful quantity by itself. However, it can be shown [27,
28, 29, 30] that a complete set of “raw” energies, together with the cohesive
energy $\varepsilon_{0}$ of the perfect crystal (potential energy per atom
relative to ideal gas) uniquely defines the equilibrium point-defect
concentrations. Such concentrations are calculated using the statistical-
mechanical model discussed in section 4, which uses the “raw” energies as
input.

Table 3 summarizes the “raw” energies of the elementary defects and several
defect clusters in both carbides. The “raw” energy of each defect was
determined by linear extrapolation to zero of the values obtained in different
supercells when plotted against the reciprocal of the number of sites, as
proposed in Ref. [31]. The plots are shown in the Supplementary Information
file. Some of the geometrically possible defect clusters are mechanically
unstable and are not included in Table 3. For example, the vacancy-antisite
pair $V_{\alpha}A_{\beta}$ is unstable in both carbides: during the
relaxation, the antisite atom $A_{\beta}$ fills the vacancy and the pair
transforms into a carbon vacancy $V_{\beta}$. The vacancy-antisite pair
$V_{\beta}B_{\alpha}$ in TaC is also unstable and relaxes into a metallic
vacancy $V_{\alpha}$. However, in HfC, the same pair survives relaxation and
transforms into a linear structure consisting of a divacancy
$V_{\alpha}V_{\alpha}$ and a carbon interstitial dumbbell $I_{B}$ aligned
parallel to the [110] direction (Fig. 1(a)). As another example, the metallic
Frenkel pair $V_{\alpha}I_{A}$ comprising a vacancy and a tetrahedral
interstitial is unstable in both carbides: during the relaxation, the
interstitial atom fills the vacancy, recovering the perfect crystal. The
carbon Frenkel pair $V_{\beta}I_{B}$ is likewise unstable in TaC but remains
stable in HfC.

In some of the defect pairs, the relaxation is accompanied by small atomic
displacements preserving the initial elementary defects. For example, the
relaxed divacancies $V_{\alpha}V_{\beta}$ in both carbides are composed of two
distinct vacancies. In other cases, the defect pair undergoes a significant
reconstruction. For example, during the relaxation of the
$V_{\alpha}B_{\alpha}$ pair, the antisite carbon atom $B_{\alpha}$ shifts into
an interstitial position, leaving a metallic vacancy behind.111This
interstitial position is at the midpoint of the nearest-neighbor C-C bond in
the B1 structure and cannot be classified as either tetrahedral or octahedral.
It is unique to this reconstructed structure.  As a result, the initial defect
pair relaxes into a linear structure comprising a metallic divacancy
$V_{\alpha}V_{\alpha}$ and a carbon interstitial $I_{B}$ in between (Fig.
1(b)). The antisite pair $A_{\beta}B_{\alpha}$ also reconstructs upon
relaxation: the antisite carbon atom $B_{\alpha}$ relaxes toward a nearby
carbon site while the antisite metallic atom $A_{\beta}$ relaxes toward a
metallic site, creating a linear structure consisting of a metallic vacancy
and a carbon interstitial dumbbell with the [110] orientation (Fig. 1(c)). As
yet another example, the $V_{\beta}A_{\beta}$ pair in TaC relaxes into a
carbon divacancy $V_{\beta}V_{\beta}$ and a chain of three metallic atoms in a
criss-cross configuration (Fig. 1(d)).

In addition to the divacancy $V_{\alpha}V_{\beta}$, Table 3 includes the “raw”
energies of vacancy clusters $V_{\alpha}V_{\beta}^{n}$, $n=2,...,6$, obtained
by adding up to six carbon vacancies as nearest neighbors of the metallic
vacancy. The structures with $n=2$, $3$ and $4$ can have several symmetrically
non-equivalent configurations with different energies. The
$V_{\alpha}V_{\beta}^{2}$ cluster can have a triangular (T) or linear (L)
configuration described in Ref. [15]. In the $V_{\alpha}V_{\beta}^{3}$
cluster, the carbon vacancies can form an in-plane (IP) or off-plane (OP)
configuration [15]. In the $V_{\alpha}V_{\beta}^{4}$ cluster, the two occupied
nearest-neighbor carbon sites of $V_{\alpha}$ can be in either L or T
configuration. Accordingly, the four carbon vacancies can form an IP or an OP
configuration [15]. By contrast, all configurations of the
$V_{\alpha}V_{\beta}^{5}$ and $V_{\alpha}V_{\beta}^{6}$ clusters are
symmetrically equivalent. In the latter case, all nearest-neighbor sites of
$V_{\alpha}$ are vacant.

The binding energies of the elementary defects into the clusters were also
calculated. The binding energy is defined as the “raw” energy difference
between the cluster and a system of isolated elementary defects forming the
cluster. This energy can be calculated using a supercell containing the
cluster and a set of supercells containing the elementary defects. The
respective supercell energies must be appropriately scaled to ensure the
conservation of the total number of sites [16]. In contrast to the “raw”
energies, the binding energy is a well-defined physical quantity independent
of reference energies. A negative binding energy indicates that the elementary
defects attract each other when forming the cluster.

The binding energies obtained by the DFT calculations are reported in Table 4.
Note that the divacancy and the antisite pair are bound much stronger in HfC
than in TaC. In TaC, the vacancies are weakly bound into
$V_{\alpha}V_{\beta}^{n}$ clusters when $n\leq 3$ and unbound (positive
binding energy) in $V_{\alpha}V_{\beta}^{4}$, $V_{\alpha}V_{\beta}^{5}$ and
$V_{\alpha}V_{\beta}^{6}$. The most stable vacancy cluster is
$V_{\alpha}V_{\beta}^{2}$ with the linear configuration (binding energy
$-0.26$ eV). In other words, a metallic vacancy in TaC is most likely bound to
two carbon vacancies but this binding is relatively weak. The same conclusion
was previously reached in Refs. [32, 16]. In contrast, in HfC, the vacancy
binding into $V_{\alpha}V_{\beta}^{n}$ clusters is strong and increases in
magnitude with $n$, reaching the most negative value of $-5.49$ eV in
$V_{\alpha}V_{\beta}^{6}$. Thus, the metallic vacancies in HfC are likely to
be surrounded by up to six carbon vacancies as first neighbors. This
conclusion is consistent with previous reports [32, 16].

As discussed by Razumovskiy et al. [32, 16], the vacancy binding into
$V_{\alpha}V_{\beta}^{n}$ clusters results from competition between the
attraction of carbon vacancies to the metallic vacancy and their repulsion
from each other. Indeed, Fig. 2 shows the interaction energy between carbon
vacancies as a function of separation. In both carbides, the interaction
between the first and second neighbors is repulsive, and the repulsion between
second neighbors is much stronger in HfC than in TaC. Nevertheless, this
repulsion is overpowered by the attraction to the metallic vacancy, resulting
in the strongly bound $V_{\alpha}V_{\beta}^{n}$ clusters in HfC.

It was shown [15] that in the $V_{\alpha}V_{\beta}^{n}$ clusters, a nearby C
atom can jump into an interstitial position and leave a new carbon vacancy
behind. In other words, a Frenkel pair $V_{\beta}I_{B}$ can form next to the
$V_{\alpha}V_{\beta}^{n}$ cluster. In TaC, the energy increases in this
process, indicating that the new Frenkel pair is energetically unfavorable.
But in HfC, the energy decreases for certain Frenkel pair orientations. Thus,
the $V_{\alpha}V_{\beta}^{n}$ clusters in HfC are unstable with respect to
Frenkel pair formation in their vicinity. Further investigation of the
$V_{\alpha}V_{\beta}^{n}-I_{B}$ structures in HfC was not pursued in this
work.

##  4 Point-defect concentrations

This section discusses a statistical model of point defects in binary carbides
with the B1 structure. As before, we consider a general carbide AB, where
element A is a transition metal and element B is carbon. Off-stoichiometric
carbides are described by the formula A1+xB1-x, where $x$ measures the
deviation from the perfect stoichiometry. We consider slight deviations
accommodated by small concentrations of point defects. Under this assumption,
interactions among the point defects can be neglected.

Several methods were proposed for calculating the equilibrium point-defect
concentrations in ordered compounds [27, 28, 29, 30, 33]. Here, we follow the
quasi-chemical method [27, 29], which gives the same results as all other
methods but is more straightforward and transparent. The method was previously
applied to compounds dominated by antisite disorder [27, 29] and vacancy
disorder [16]. Here, we provide a general treatment without the presumption of
a particular disorder mechanism.

The point defects are treated as an ideal gas mixture of several “chemical
components” representing the different types of elementary point defects or
their clusters. The equilibrium defect concentrations at a temperature $T$ are
calculated from two conditions:

  * •

Equilibrium with respect to chemical reactions in the gas mixture

  * •

Material balance preserving the given off-stoichiometry $x$

If the model considers $M$ defect types, then $(M-1)$ independent reactions
must be chosen, with the material balance condition providing another
equation.

The defects are assigned the chemical potentials

$\mu_{d}=\varepsilon_{d}+k_{B}T\ln\dfrac{X_{d}}{\sigma_{d}},$ (2)

while the formula unit AB is assigned the chemical potential
$\mu_{AB}=2\varepsilon_{0}$ ($k_{B}$ is Boltzmann’s constant). For single
vacancies and antisite defects, $X_{d}$ denotes their fraction of the
respective sublattice. For single interstitial atoms, $X_{d}$ is the occupied
fraction on the available interstitial positions (interstices). Each defect
cluster is assumed to contain at least one vacancy or one antisite. Let us
call this vacancy/antisite the cluster center. The configurational entropy of
the cluster considers all possible locations of its center on the respective
sublattice. The cluster concentration is then defined as the fraction $X_{d}$
of the sublattice sites occupied by the cluster centers. In addition, for a
given center location, the relaxed cluster structure can have $\sigma_{d}$
different orientations relative to the lattice with the same energy. This
additional degeneracy of the micro-states contributes the orientational
entropy $k_{B}\ln\sigma_{d}$ per cluster. This explains the appearance of the
symmetry factor $\sigma_{d}$ in Eq.(2). For example, the vacancy-antisite pair
$V_{\beta}B_{\alpha}$ in HfC shown in Fig. 1(a) can have six different
orientations with equal energy, thus $\sigma_{d}=6$. In this case, either
$V_{\beta}$ or $B_{\alpha}$ can be taken as the cluster center. For elementary
defects $\sigma_{d}=1$. The symmetry factors for all defect clusters
considered in this work are summarized in Table 4.

Although the chemical potentials of the defects are expressed through the
“raw” energies, it can be shown [29] that all reference energies cancel out
and do not affect the defect concentrations predicted by this method. We
emphasize that the model only includes the configurational and orientational
micro-states and neglects the vibrational, electronic, and all other forms of
free energy.

Although any choice of the $(M-1)$ reactions is equally legitimate, we find
the following set of reactions most intuitive. For reactions among the six
elemental defects, we choose

$V_{\alpha}+V_{\beta}=-AB,$ (3) $B_{\alpha}+A_{\beta}=0,$ (4)

$V_{\alpha}=B_{\alpha}+V_{\beta},$ (5)

$V_{\alpha}+I_{A}=0,$ (6)

$V_{\beta}+I_{B}=0.$ (7)

In reaction (3), a vacancy pair is created by removing one formula unit of the
carbide. In reaction (4), a pair of antisite defects is created without adding
or removing atoms. In reaction (5), an atom B fills the vacancy $V_{\alpha}$,
creating an antisite defect $B_{\alpha}$ and leaving a vacancy $V_{\beta}$
behind. Again, the system remains closed. Finally, reactions (6) and (7)
describe the Frenkel pair formation by atoms A and B, respectively. For every
cluster $d$ composed of elementary defects $d_{i}$, we write the dissociation-
recombination reaction

$d=\sum_{i}d_{i}.$ (8)

It is assumed that the point defects participating in the reactions are
separated well enough to neglect their interaction.

The equations describing dynamic equilibrium with respect to the defect
reactions are obtained by replacing the defect symbols by the respective
chemical potentials. The equations obtained have the form of the mass action
law known from chemistry. In particular, reactions (3)-(7) yield the equations

$X_{V_{\alpha}}X_{V_{\beta}}=\exp\left(-\dfrac{\varepsilon_{V_{\alpha}}+\varepsilon_{V_{\beta}}+2\varepsilon_{0}}{k_{B}T}\right),$
(9)
$X_{B_{\alpha}}X_{A_{\beta}}=\exp\left(-\dfrac{\varepsilon_{B_{\alpha}}+\varepsilon_{A_{\beta}}}{k_{B}T}\right),$
(10)
$\dfrac{X_{V_{\alpha}}}{X_{B_{\alpha}}X_{V_{\beta}}}=\exp\left(-\dfrac{\varepsilon_{V_{\alpha}}-\varepsilon_{B_{\alpha}}-\varepsilon_{V_{\beta}}}{k_{B}T}\right),$
(11)

$X_{V_{\alpha}}X_{I_{A}}=\exp\left(-\dfrac{\varepsilon_{V_{\alpha}}+\varepsilon_{I_{A}}}{k_{B}T}\right),$
(12)

$X_{V_{\beta}}X_{I_{B}}=\exp\left(-\dfrac{\varepsilon_{V_{\beta}}+\varepsilon_{I_{B}}}{k_{B}T}\right).$
(13)

Similarly, each cluster dissociation-recombination reaction (8) gives the
equation

$\dfrac{X_{d}}{\prod_{i}X_{d_{i}}}=\sigma_{d}\exp\left(-\dfrac{\varepsilon_{d}-\sum_{i}\varepsilon_{d_{i}}}{k_{B}T}\right),$
(14)

where the numerator $\varepsilon_{d}-\sum_{i}\varepsilon_{d_{i}}$ in the
right-hand side has the meaning of the binding energy of the cluster. For
example, for a vacancy cluster $V_{\alpha}V_{\beta}^{n}$ we have

$\dfrac{X_{V_{\alpha}V_{\beta}^{n}}}{X_{V_{\alpha}}\left(X_{V_{\beta}}\right)^{n}}=\sigma_{V_{\alpha}V_{\beta}^{n}}\exp\left(-\dfrac{\varepsilon_{V_{\alpha}V_{\beta}^{n}}-\varepsilon_{V_{\alpha}}-n\varepsilon_{V_{\beta}}}{k_{B}T}\right).$
(15)

It is easy to see that the number of equations obtained is $(M-1)$.

The mass balance equation is generally nonlinear [29] with respect to the
defect concentrations. However, a linear approximation can be applied
considering that the defect concentrations are small. This approximation
neglects all terms quadratic in the defects concentrations, i.e., terms
representing products of different concentrations or their squares. The
following linear equation can be derived:

$\displaystyle x$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}\left(X_{V_{\beta}}-X_{V_{\alpha}}\right)+\dfrac{1}{2}\left(X_{A_{\beta}}-X_{B_{\alpha}}\right)-\dfrac{1}{2}\nu\left(X_{I_{B}}-X_{I_{A}}\right)$
(16) $\displaystyle+$
$\displaystyle\dfrac{1}{4}\sum_{d}X_{d}\left[L_{V_{\beta}}^{d}-L_{V_{\alpha}}^{d}+2\left(L_{A_{\beta}}^{d}-L_{B_{\alpha}}^{d}\right)+\left(L_{I_{B}}^{d}-L_{I_{A}}^{d}\right)\right].$

Here, the first line represents the deviation from the stoichiometry due to
the elementary defects, $\nu$ being the number of interstitial positions per
lattice site. We only consider tetrahedral interstitials, for which $\nu=8$.
The second line is the contribution of the defect clusters $d$, where
$L_{d_{i}}^{d}$ is the number of elementary defects $d_{i}$ in the cluster
$d$. For example, for the vacancy cluster $V_{\alpha}V_{\beta}^{n}$ we have
$L_{V_{\alpha}}^{V_{\alpha}V_{\beta}^{n}}=1$,
$L_{V_{\beta}}^{V_{\alpha}V_{\beta}^{n}}=n$, and
$L_{B_{\alpha}}^{V_{\alpha}V_{\beta}^{n}}=L_{A_{\beta}}^{V_{\alpha}V_{\beta}^{n}}=0$.
The complete list of the $L-$numbers is provided in the Appendix, along with
Eq.(16) specialized for the chosen set of defect clusters.

Note that the divacancy, the antisite pair, and the Frenkel pairs do not
affect the material balance. Accordingly, the terms representing these pairs
mutually cancel in Eq.(16). This is true for any composition-conserving defect
cluster. The formation of such clusters is accompanied by addition or removal
of $m$ formula units AB or does not require any addition or removal ($m=0$).
It is easy to show that the concentration of composition-conserving clusters
is

$X_{d}=\sigma_{d}\exp\left(-\dfrac{\varepsilon_{d}+2m\varepsilon_{0}}{k_{B}T}\right).$
(17)

For example, $m=1$ for divacancies and $m=0$ for antisite pairs and Frenkel
pairs. This concentration is independent of the off-stoichiometry and can be
immediately calculated at any given temperature without solving any equations.

Eqs.(9)-(14) and (16) (less the equations for the composition-conserving
clusters) constitute a complete set of equations that must be solved for the
defect concentrations numerically.

Figures 3(a) and (b) present the computed composition dependencies of the
defect concentrations in TaC and HfC, respectively. The temperature is fixed
at 2500 K, but results for other temperatures are qualitatively similar. The
Supplementary Information file shows the plots for 3500 K. The plots in Fig. 3
only include the stoichiometric and metal-rich compositions ($x\geq 0$)
because carbon-rich compositions ($x<0$) are unstable according to the phase
diagram [34, 35]. In the plots, the chemical compositions are measured by the
fraction of metallic atoms $c_{A}=1+x$ (A = Ta or Hf).

From Figs. 3(a) and (b), it is evident that thermal disorder in the
stoichiometric carbides is dominated by Ta and C vacancies in TaC and by
carbon vacancies and carbon interstitials in HfC. This difference reflects the
different energetics of the point defects and, ultimately, the difference in
the chemical bonding in the two carbides. The Hf vacancy concentration in
stoichiometric HfC is extremely small relative to the carbon defects. As the
off-stoichiometry increases, so does the carbon vacancy concentration, showing
that the carbon vacancies are the constitutional defects on the metal-rich
side in both carbides. As expected from the mass action law, the carbon
vacancies suppress the metallic vacancies and carbon interstitials. In off-
stoichiometric Ta1+xC1-x with $x<0.04$, the metallic sublattice is dominated
by single vacancies $V_{\mathrm{Ta}}$. At larger deviations from the
stoichiometry ($x>0.04$), the Ta vacancies start forming divacancies
$V_{\mathrm{Ta}}V_{\mathrm{C}}$ and tri-vacancies
$V_{\mathrm{Ta}}V_{\mathrm{C}}^{2}$, whose concentrations eventually become
comparable to that of single vacancies. In off-stoichiometric Hf1+xC1-x, the
dominant metallic defects likewise depend on the chemical composition. At
small deviations from the stoichiometry ($x<0.001$), the leading metallic
defects are single vacancies $V_{\mathrm{Hf}}$. As the off-stoichiometry
increases, the concentrations of vacancy clusters
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{n}$ rapidly grow and at $x>0.001$ they exceed
the $V_{\mathrm{Hf}}$ concentration. The six-vacancy clusters
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{5}$ have the highest concentration, although
the clusters with smaller $n$ come close.

##  5 Effective defect formation energies

Figures 4 and 5 present the Arrhenius diagrams, $\log X_{d}$ versus $1/T$, of
the defect concentrations in TaC and HfC, respectively. The chemical
compositions are fixed at $x=0$ (stoichiometry) and $x=0.02$ (representative
off-stoichiometric state). Recall that the composition-conserving defect
clusters strictly follow the Arrhenius law, see Eq.(17). As a result, their
plots are represented by perfect straight lines on the Arrhenius diagrams and
perfect horizontal lines in the composition plots (Fig. 3). The single defects
and non-conserving defect clusters do not follow the Arrhenius law exactly but
their plots still look fairly straight. As a good approximation, all defect
concentrations can be represented in the Arrhenius form

$X_{d}=X_{d}^{0}\exp\left(-\dfrac{\overline{\varepsilon}_{d}}{k_{B}T}\right)$
(18)

with appropriate prefactors $X_{d}^{0}$ and effective formation energies
$\overline{\varepsilon}_{d}$. Although not exact, Eq.(18) is useful for
applications and comparison with experiments.

As discussed in Ref. [29], Eq.(18) works best in two limiting cases: when one
defect concentration is much higher than all other concentrations and when the
compound is stoichiometric with thermal disorder strongly dominated by two
defect types. In the first case, the right-hand side of Eq.(16) can be
represented by a single term, so the respective defect concentration is
proportional to $x$. In the second case, the right-hand side of Eq.(16) is
represented by two terms while the left-hand side is zero; thus the two defect
concentrations are equal up to a numerical factor. In both cases, the material
balance equation is simplified, and Eqs.(9)-(14) can be solved analytically
with solutions in the form of Eq.(18).

The first case is realized in the off-stoichiometric carbides dominated by
constitutional carbon vacancies. In our notation, $X_{V_{\beta}}$ is much
greater than all other concentrations $X_{d}$. Accordingly, the balance
equation (16) is simplified to $x=\frac{1}{4}X_{V_{\beta}}$, from which
$X_{V_{\beta}}=4x$. Thus, $X_{V_{\beta}}$ is temperature-independent and
proportional to the off-stoichiometry parameter $x$. Inserting this
$X_{V_{\beta}}$ in Eq.(9), we obtain the metal vacancy concentration

$X_{V_{\alpha}}=\dfrac{1}{4x}\exp\left(-\dfrac{\varepsilon_{V_{\alpha}}+\varepsilon_{V_{\beta}}+2\varepsilon_{0}}{k_{B}T}\right).$
(19)

Next, we find the concentrations of the vacancy clusters
$V_{\alpha}V_{\beta}^{n}$ using Eq.(15):

$X_{V_{\alpha}V_{\beta}^{n}}=\sigma_{V_{\alpha}V_{\beta}^{n}}(4x)^{n-1}\exp\left(-\dfrac{\varepsilon_{V_{\alpha}V_{\beta}^{n}}-(n-1)\varepsilon_{V_{\beta}}+2\varepsilon_{0}}{k_{B}T}\right).$
(20)

For the interstitial concentrations we use Eqs.(12) and (13) to obtain

$X_{I_{A}}=4x\exp\left(-\dfrac{\varepsilon_{I_{A}}-\varepsilon_{V_{\beta}}-2\varepsilon_{0}}{k_{B}T}\right).$
(21)
$X_{I_{B}}=\dfrac{1}{4x}\exp\left(-\dfrac{\varepsilon_{I_{B}}+\varepsilon_{V_{\beta}}}{k_{B}T}\right).$
(22)

This chain of calculations can be continued to obtain all other defect
concentrations. Note that they all have the Arrhenius form (18). The
respective effective formation energies and prefactors are summarized in Table
5. Equations (19) and (22) confirm the trend mentioned above: deviations from
stoichiometry with $x>0$ suppress the metallic vacancies and carbon
interstitials. At the same time, such deviations promote the formation of the
vacancy clusters $V_{\alpha}V_{\beta}^{n}$ and metallic interstitials $I_{A}$.

The second case is realized in stoichiometric carbides. The leading thermal
defects in the two carbides are different. In TaC, such defects are the
vacancies $V_{\alpha}$ and $V_{\beta}$. The balance equation (16) gives
$V_{\alpha}=V_{\beta}$, which we combine with Eq.(9) to obtain

$X_{V_{\alpha}}=X_{V_{\beta}}=\exp\left(-\dfrac{\varepsilon_{V_{\alpha}}+\varepsilon_{V_{\beta}}+2\varepsilon_{0}}{2k_{B}T}\right).$
(23)

The interstitial concentrations are then obtained by inserting Eq.(23) into
Eqs.(12) and (13):

$X_{I_{A}}=\exp\left(-\dfrac{2\varepsilon_{I_{A}}-2\varepsilon_{0}+\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}}}{2k_{B}T}\right),$
(24)
$X_{I_{B}}=\exp\left(-\dfrac{2\varepsilon_{I_{B}}-2\varepsilon_{0}+\varepsilon_{V_{\beta}}-\varepsilon_{V_{\alpha}}}{2k_{B}T}\right).$
(25)

The concentrations of the vacancy clusters $V_{\alpha}V_{\beta}^{n}$ are
readily calculated by inserting the single vacancy concentrations from Eq.(23)
into Eq.(15), which gives

$X_{V_{\alpha}V_{\beta}^{n}}=\sigma_{V_{\alpha}V_{\beta}^{n}}\exp\left(-\dfrac{2\varepsilon_{V_{\alpha}V_{\beta}^{n}}+(n-1)(\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}})+2(n+1)\varepsilon_{0}}{2k_{B}T}\right).$
(26)

The remaining defect concentrations are calculated similarly, and the
Arrhenius parameters obtained are summarized in Table 6.

In stoichiometric HfC, the leading thermal defects are carbon vacancies
$V_{\beta}$ and carbon interstitials $I_{B}$. The balance equation (16) gives
$X_{V_{\beta}}=2\nu X_{I_{B}}$ (recall that $\nu=8$ in the B1 structure).
Using Eq.(13) we have

$X_{V_{\beta}}=\sqrt{2\nu}\exp\left(-\dfrac{\varepsilon_{I_{B}}+\varepsilon_{V_{\beta}}}{2k_{B}T}\right).$
(27)
$X_{I_{B}}=\dfrac{1}{\sqrt{2\nu}}\exp\left(-\dfrac{\varepsilon_{I_{B}}+\varepsilon_{V_{\beta}}}{2k_{B}T}\right).$
(28)

Next, the metal vacancy concentration is obtained from Eq.(9):

$X_{V_{\alpha}}=\dfrac{1}{\sqrt{2\nu}}\exp\left(-\dfrac{2\varepsilon_{V_{\alpha}}+\varepsilon_{V_{\beta}}-\varepsilon_{I_{B}}+4\varepsilon_{0}}{2k_{B}T}\right),$
(29)

while Eq.(12) gives

$X_{I_{A}}=\sqrt{2\nu}\exp\left(-\dfrac{2\varepsilon_{I_{A}}-\varepsilon_{V_{\beta}}+\varepsilon_{I_{B}}-4\varepsilon_{0}}{2k_{B}T}\right).$
(30)

The vacancy cluster concentrations are obtained from Eq.(15), which gives

$X_{V_{\alpha}V_{\beta}^{n}}=\sigma_{V_{\alpha}V_{\beta}^{n}}(2\nu)^{(n-1)/2}\exp\left(-\dfrac{2\varepsilon_{V_{\alpha}V_{\beta}^{n}}+(n-1)(\varepsilon_{I_{B}}-\varepsilon_{V_{\beta}})+4\varepsilon_{0}}{2k_{B}T}\right).$
(31)

Continuing the calculations, we derive the Arrhenius parameters of all other
defects summarized in Table 6.

Table 7 reports the numerical values of the Arrhenius parameters of the point
defects in TaC and HfC obtained from the first-principles raw energies.
Inserting these values in Eq.(18), the Arrhenius lines obtained closely
approximate the numerical solutions shown in Figs. 4 and 5.

##  6 Discussion

The goal of this work was to understand and predict the point defects in the
binary carbides TaC and HfC by DFT calculations. To this end, a statistical-
mechanical model has been developed capable of predicting point-defect
concentrations in binary compounds with the AB stoichiometry. The model
applies to both single defects and defect clusters of any complexity. Although
our primary interest is in the particular carbides TaC and HfC, the the model
is general enough to be applied to other binary carbides, nitrides, and
borides with the AB stoichiometry. It can also be generalized to ordered
compounds with different crystal structures and stoichiometries [29].

The model generalizes the previous treatment of point defects in intermetallic
compounds [29]. Thermal and compositional disorder in intermetallic compounds
is also governed by point defects and has been studied by similar DFT
statistical mechanics methods [27, 28, 29, 30]. However, intermetallics do not
usually support interstitial defects, and vacancies display a weak clustering
trend. Thus, the point-defect system is much simpler than in the carbides.

In the present model, the free energy of the point defects includes the
configurational and orientational effects but neglects other contributions,
such as atomic vibrations. This is not a severe limitation. As discussed
previously [29], the effect of vibrations can be included by replacing the
“raw” defect energy $\varepsilon_{d}$ by the “raw” free energy
$f_{d}=\varepsilon_{d}-Ts_{d}$, where $s_{d}$ is the change in the vibrational
entropy due to the defect formation in a perfect crystal. DFT-based
calculations of $s_{d}$ are challenging, especially for defect clusters, but
calculations with interatomic potentials (traditional or machine-learning type
[36]) are straightforward. The effect of applied pressure on the point-defect
concentrations can also be included, provided the defect formation volumes can
be computed.

As mentioned in Section 1, our primary motivation for calculating the point-
defect concentrations is that they constitute a required ingredient for
diffusion calculations. We will briefly discuss the possible diffusion
mechanisms suggested by the present results, focusing on the most realistic
case of carbon-deficient chemical compositions Ta1+xC1-x ($x>0$). As evident
from the calculations, deviations from the stoichiometry towards carbon-
deficient compositions are accommodated by carbon vacancies. The carbon
vacancy concentration is virtually temperature-independent and can reach a few
percent (recall that $X_{V_{\beta}}\approx 4x$), dominating over all other
point defects. The fraction of carbon vacancies bound into clusters is orders
of magnitude smaller. Thus, it is highly probable that carbon diffusion is
mediated by single vacancy jumps on the carbon sublattice. Interactions among
the carbon vacancies can hardly affect their diffusion given their repulsion
as first and second neighbors (Fig. 2). To a good approximation, carbon
diffusion can be treated as occurring by the simple vacancy mechanism on the
FCC sublattice with the geometric correlation factor. The same reasoning
applies to carbon diffusion in Hf1+xC1-x ($x>0$).

Diffusion of the metallic atoms is more complex. In TaC, single Ta vacancies
have a slightly higher concentration than the divacancies
$V_{\mathrm{Ta}}V_{\mathrm{C}}$ and trivacancies
$V_{\mathrm{Ta}}V_{\mathrm{C}}^{2}$. However, the literature data [14, 7]
indicate that Ta vacancies have a lower jump barrier when bound with carbon
vacancies. Thus, divacancies and possibly
$V_{\mathrm{Ta}}V_{\mathrm{C}}^{2}$(T) clusters (triangular configuration)
should also be considered. The linear clusters
$V_{\mathrm{Ta}}V_{\mathrm{C}}^{2}$(L) have a lower concentration and are in a
locked configuration: the Ta vacancy cannot make a jump without breaking away
from this cluster. One complication is that atomic diffusion mediated by
divacancies and trivacancies is accompanied by jump correlation effects, which
can be significant. Approximate semi-analytical methods exist for calculating
the jump correlation factors in the B1 structure under the divacancy mechanism
[37, 38]. However, more accurate calculations utilize kinetic Monte Carlo
(KMC) simulations [38]. For trivacancies, KMC is the most viable option.

In HfC, Hf diffusion is mediated by the $V_{\mathrm{Hf}}V_{\mathrm{C}}^{n}$
clusters. The size of the dominant cluster depends on temperature and chemical
composition. Fig. 5 shows that at the off-stoichiometry of $x=0.02$, the
clusters having the highest concentration are
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{3}$(IP) above $\sim 2500$ K and
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{5}$ below $\sim 2500$ K. The transition
temperature depends on the off-stoichiometry. Note, however, that in both
cases, the Hf vacancy cannot make a jump without destroying the cluster. In
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{3}$(IP), such a jump causes one of the
dissociation reactions
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{3}\mathrm{(IP)}\rightarrow
V_{\mathrm{Hf}}V_{\mathrm{C}}^{2}\mathrm{(T)}+V_{\mathrm{C}}$ or
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{3}\mathrm{(IP)}\rightarrow
V_{\mathrm{Hf}}V_{\mathrm{C}}+2V_{\mathrm{C}}$. The isolated carbon vacancies
produced by these reactions repel each other and are unlikely to recombine
into the original cluster. In $V_{\mathrm{Hf}}V_{\mathrm{C}}^{5}$, a Hf
vacancy jump causes one of the dissociations
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{5}\rightarrow
V_{\mathrm{Hf}}+5V_{\mathrm{C}}$,
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{5}\rightarrow
V_{\mathrm{Hf}}V_{\mathrm{C}}+4V_{\mathrm{C}}$, or
$V_{\mathrm{Hf}}V_{\mathrm{C}}^{5}\rightarrow
V_{\mathrm{Hf}}V_{\mathrm{C}}^{2}\mathrm{(T)}+3V_{\mathrm{C}}$, which again
produces isolated carbon vacancies repelling each other. This behavior
reflects the general trend that vacancy cluster migration by single jumps is
subject to severe geometric constraints and strong correlation effects.

Three alternatives can be considered. One is to allow the vacancy clusters to
evolve by a chain of dissociation-recombination reactions among all possible
clusters [7]. One should then resort to KMC simulations as analytical
treatment of the atomic transport caused by such chains of reactions could be
impractical. A second approach is to consider collective (simultaneous) jumps
of a group of atoms filling the vacant sites and shifting the entire vacancy
cluster to a new position. Such mechanisms were discussed in Ref. [32] for the
TiC and ZrC carbides and seem plausible. For simple clusters such as
divacancies, calculating the minimum-energy path is straightforward, but
multi-vacancy clusters present a challenge. One can test a set of _a priori_
chosen atomic trajectories and select one with the lowest barrier [32]. A more
general treatment should use a saddle-point search algorithm not relying on _a
priori_ assumptions about which atom will land in which position. Yet another
approach is to use molecular dynamics to discover the migration mechanisms.
This approach is appealing but will likely require a surrogate model such as a
machine-learning potential. It should also be noted that comparison of
different vacancy clusters cannot be made solely from their equilibrium
concentration and migration barrier. The attempt frequency also matters. The
latter depends on the total mass $m$ of the collectively jumping atoms
(approximately as $1/\sqrt{m}$) and cluster-specific vibrational modes.

##  7 Conclusions

We have combined DFT calculations with a statistical-mechanical model to
predict point-defect concentrations $X_{d}$ in the TaC and HfC carbides as a
function of temperature and chemical composition. For each defect type $d$,
the function $X_{d}(c_{A},T)$ can be conveniently and accurately represented
by two parameters: an Arrhenius prefactor $X_{d}^{0}$ and effective formation
energy $\overline{\varepsilon}_{d}$. Our results complement the previous work
[39, 7, 15, 32, 16] and can be summarized as follows:

  * •

Atomic mechanisms of thermal and constitutional disorder in TaC and HfC are
complex and involve multiple types of point defects occurring simultaneously.

  * •

The strong short-range binding among the elementary point defects, especially
in HfC, leads to persistent point-defect clusters. The dominant type of defect
cluster depends on the chemical composition and temperature and is different
between TaC and HfC.

  * •

The presence of relatively large concentrations of defect clusters is one of
the hallmarks of these carbides distinguishing them from other ordered phases
such as intermetallic compounds.

  * •

The diversity of the point defects suggests that the mechanisms of metal atom
diffusion are complex and may involve chains of point-defect reactions and
collective atomic rearrangements. Their computational studies require new
methods.

The mode developed here is general enough to be applied to other binary
B1-ordered compounds and can be further generalized to compounds with other
crystal structures and different stoichiometries.

Acknowledgments

This research was supported by the Office of Naval Research under Award No.
N00014-22-1-2645.

## Appendix: The $L$-numbers and the material balance equation

The table below presents the complete list of the $L$-numbers for all defect
clusters considered in this work. These numbers appear in Eq.(16) for the
material balance.

Table 1: $L$-numbers for the defect clusters considered in this work.
Cluster | Elementary defect
---|---
| $V_{\alpha}$ | $V_{\beta}$ | $A_{\beta}$ | $B_{\alpha}$ | $I_{A}$ | $I_{B}$
$V_{\alpha}V_{\beta}$ | 1 | 1 | 0 | 0 | 0 | 0
$A_{\beta}B_{\alpha}$ | 0 | 0 | 1 | 1 | 0 | 0
$V_{\alpha}B_{\alpha}$ | 1 | 0 | 0 | 1 | 0 | 0
$V_{\beta}A_{\beta}$ | 0 | 1 | 1 | 0 | 0 | 0
$V_{\beta}B_{\alpha}$ | 0 | 1 | 0 | 1 | 0 | 0
$V_{\beta}I_{B}$ | 0 | 1 | 0 | 0 | 0 | 1
$V_{\alpha}V_{\beta}^{2}\leavevmode\nobreak\ (T)$ | 1 | 2 | 0 | 0 | 0 | 0
$V_{\alpha}V_{\beta}^{2}\leavevmode\nobreak\ (\mathrm{L})$ | 1 | 2 | 0 | 0 | 0 | 0
$V_{\alpha}V_{\beta}^{3}\leavevmode\nobreak\ (\mathrm{IP})$ | 1 | 3 | 0 | 0 | 0 | 0
$V_{\alpha}V_{\beta}^{3}\leavevmode\nobreak\ (\mathrm{OP})$ | 1 | 3 | 0 | 0 | 0 | 0
$V_{\alpha}V_{\beta}^{4}\leavevmode\nobreak\ (\mathrm{IP})$ | 1 | 4 | 0 | 0 | 0 | 0
$V_{\alpha}V_{\beta}^{4}\leavevmode\nobreak\ (\mathrm{OP})$ | 1 | 4 | 0 | 0 | 0 | 0
$V_{\alpha}V_{\beta}^{5}$ | 1 | 5 | 0 | 0 | 0 | 0
$V_{\alpha}V_{\beta}^{6}$ | 1 | 6 | 0 | 0 | 0 | 0

For the set of defect clusters considered in this work, the material balance
equation is

$\displaystyle x$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}\left(X_{V_{\beta}}-X_{V_{\alpha}}\right)+\dfrac{1}{2}\left(X_{A_{\beta}}-X_{B_{\alpha}}\right)-\dfrac{1}{2}\nu\left(X_{I_{B}}-X_{I_{A}}\right)$
(32) $\displaystyle+$
$\displaystyle\dfrac{1}{4}\left(3X_{V_{\beta}A_{\beta}}-3X_{V_{\alpha}B_{\alpha}}-X_{V_{\beta}B_{\alpha}}\right)$
$\displaystyle+$
$\displaystyle\dfrac{1}{4}\left(X_{V_{\alpha}V_{\beta}^{2}(T)}+X_{V_{\alpha}V_{\beta}^{2}(L)}+2X_{V_{\alpha}V_{\beta}^{3}(IP)}+2X_{V_{\alpha}V_{\beta}^{3}(OP)}\right)$
$\displaystyle+$
$\displaystyle\dfrac{1}{4}\left(3X_{V_{\alpha}V_{\beta}^{4}(IP)}+3X_{V_{\alpha}V_{\beta}^{4}(OP)}+4X_{V_{\alpha}V_{\beta}^{5}}+5X_{V_{\alpha}V_{\beta}^{6}}\right).$

This equation contains 17 defect concentrations. Their calculation requires
solving this equation simultaneously with 16 equations representing the
reactions among the elementary defects and the formation of non-conserving
defect clusters.

## References

  * Cedillos-Barraza et al. [2016] O. Cedillos-Barraza, D. Manara, K. Boboridis, T. Watkins, S. Grasso, D. D. Jayaseelan, R. J. M. Konings, M. J. Reece, W. E. Lee,  Investigating the highest melting temperature materials: A laser melting study of the TaC-HfC system,  Scientific Reports 6 (2016) 37962.
  * Viñes et al. [2005] F. Viñes, C. Sousa, P. Liu, J. A. Rodriguez, F. Illas,  A systematic density functional theory study of the electronic structure of bulk and (001) surface of transition-metals carbides,  The Journal of Chemical Physics 122 (2005) 174709.
  * Resnick et al. [1966] R. Resnick, R. Steinitz, L. Seigle,  Diffusion of carbon in tantalum monocarbide,  Trans. Metall. Soc. AIME 236 (1966) 1732–1738.
  * Brizes [1968] W. F. Brizes,  Diffusion of carbon in the carbides of tantalum,  Journal of Nuclear Materials 26 (1968) 227–231.
  * Rafaja et al. [1998] D. Rafaja, W. Lengauer, H. Wiesenberger,  Non-metal diffusion coefficients for the Ta–C and Ta–N systems,  Acta Materialia 46 (1998) 3477–3483.
  * Wallace and Butt [1996] T. C. Wallace, D. P. Butt, Review of diffusion and vaporization of Group 4 and 5 transition metal carbides, Springer Netherlands, Dordrecht, 1996, pp. 53–90. URL: https://doi.org/10.1007/978-94-009-1565-7.
  * Tang et al. [2020] X. Tang, R. Salehin, G. B. Thompson, C. R. Weinberger,  Statistical study of vacancy diffusion in TiC and TaC,  Physical Review Materials 4 (2020) 093602.
  * Jubair et al. [2019] M. Jubair, A. M. M. T. Karim, M. Nuruzzaman, M. A. K. Zilani,  Comparison of structural, mechanical and optical properties of tantalum hemicarbide with tantalum monocarbide: Ab initio calculations,  Journal of Physics Communications 3 (2019) 055017.
  * Sun et al. [2020] W. Sun, X. Kuang, H. Liang, X. Xia, Z. Zhang, C. Lu, A. Hermann,  Mechanical properties of tantalum carbide from high-pressure/high-temperature synthesis and first-principles calculations,  Physical Chemistry Chemical Physics 22 (2020) 5018–5023.
  * De Leon et al. [2015] N. De Leon, X. Yu, H. Yu, C. Weinberger, G. Thompson,  Bonding effects on the slip differences in the B1 monocarbides,  Physical Review Letters 114 (2015) 165502.
  * Yu et al. [2014] X. Yu, C. R. Weinberger, G. B. Thompson,  Ab initio investigations of the phase stability in tantalum carbides,  Acta Materialia 80 (2014) 341–349.
  * Yan et al. [2016] W.-L. Yan, M. Sygnatowicz, G.-H. Lu, F. Liu, D. K. Shetty,  First-principles study on surface stability of tantalum carbides,  Surface Science 644 (2016) 24–28.
  * Rimsza et al. [2019] J. M. Rimsza, S. Foiles, J. Michael, W. Mackie, K. Larson,  Role of defects on the surface properties of HfC,  Applied Surface Science 495 (2019) 143500.
  * Yu et al. [2015] X.-X. Yu, G. B. Thompson, C. R. Weinberger,  Influence of carbon vacancy formation on the elastic constants and hardening mechanisms in transition metal carbides,  Journal of the European Ceramic Society 35 (2015) 95–103.
  * Salehin et al. [2021] R. Salehin, X. Tang, G. B. Thompson, C. R. Weinberger,  Vacancy-cluster and off-lattice metal-atom diffusion mechanisms in transition metal carbides,  Computational Materials Science 199 (2021) 110713.
  * Razumovskiy et al. [2015] V. I. Razumovskiy, M. N. Popov, H. Ding, J. Odqvist,  Formation and interaction of point defects in group IVB transition metal carbides and nitrides,  Computational Materials Science 104 (2015) 147–154.
  * Kresse and Furthmüller [1996] G. Kresse, J. Furthmüller,  Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,  Comput. Mat. Sci. 6 (1996) 15.
  * Kresse and Furthmueller [1996] G. Kresse, J. Furthmueller,  Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,  Phys. Rev. B 54 (1996) 11169–11186.
  * Blochl [1994] P. E. Blochl,  Projector augmented-wave method,  Phys. Rev. B 50 (1994) 17953–17979.
  * Perdew et al. [1992] J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, C. Fiolhais,  Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation,  Phys. Rev. B 46 (1992) 6671–6687.
  * Perdew et al. [1996] J. P. Perdew, K. Burke, M. Ernzerhof,  Generalized gradient approximation made simple,  Phys. Rev. Lett. 77 (1996) 3865–3868.
  * Brown et al. [1966] H. L. Brown, P. E. Armstrong, C. P. Kempter,  Elastic properties of some polycrystalline transition-metal monocarbides,  The Journal of Chemical Physics 45 (1966) 547–549.
  * Weber [1973] W. Weber,  Lattice dynamics of transition-metal carbides,  Physical Review B 8 (1973) 5082.
  * Zhang et al. [2017] C. Zhang, A. Gupta, S. Seal, B. Boesl, A. Agarwal,  Solid solution synthesis of tantalum carbide-hafnium carbide by spark plasma sintering,  Journal of the American Ceramic Society 100 (2017) 1853–1862.
  * He et al. [2008] L. He, Z. Lin, J. Wang, Y. Bao, Y. Zhou,  Crystal structure and theoretical elastic properrty of two new ternary ceramics Hf3Al4C6 and Hf2Al4C5,  Scripta Materialia 58 (2008) 679–682.
  * Guo et al. [2021] Y. Guo, J. Chen, W. Song, S. Shan, X. Ke, Z. Jiao,  Electronic, mechanical and thermodynamic properties of ZrC, HfC and their solid solutions studied by first-principles calculation,  Solid State Communications 338 (2021) 114481.
  * Mishin and Farkas [1997a] Y. Mishin, D. Farkas,  Atomistic simulation of point defects and diffusion in B2 NiAl .1. Point defect energetics,  Philosophical Magazine A 75 (1997a) 169–185.
  * Mishin and Farkas [1997b] Y. Mishin, D. Farkas,  Atomistic simulation of point defects and diffusion in B2 NiAl .2. Diffusion mechanisms,  Philosophical Magazine A 75 (1997b) 187–199.
  * Mishin and Herzig [2000] Y. Mishin, C. Herzig,  Diffusion in the Ti-Al system,  Acta Mater. 48 (2000) 589–623.
  * Lozovoi and Mishin [2003] A. Y. Lozovoi, Y. Mishin,  Point defects in NiAl: The efffec of lattice vibrations,  Phys. Rev. B 68 (2003) 184113.
  * Mishin et al. [2001] Y. Mishin, M. R. Sørensen, A. F. Voter,  Calculation of point defect entropy in metals,  Philos. Mag. A 81 (2001) 2591–2612.
  * Razumovskiy et al. [2013] V. I. Razumovskiy, A. V. Ruban, J. Odqvist, P. A. Korzhavyi,  Vacancy-cluster mechanism of metal-atom diffusion in substoichiometric carbides,  Physical Review B 87 (2013) 054203.
  * Woodward et al. [2001] C. Woodward, M. Asta, G. Kresse, J. Hafner,  Density of constitutional and thermal point defects in ${L}1_{2}$ Al3sc,  Physical Review B 63 (2001) 094103.
  * Mehdikhan et al. [2017] B. Mehdikhan, G. H. Borhani, S. R. Bakhshi, H. R. Baharvandi,  Effect of milling and sintering temperature of TaC–TaB2 composite on lattice parameter and C/Ta ratio,  Refractories and Industrial Ceramics 57 (2017) 507–512.
  * Shabalin [2019] I. L. Shabalin,  Hafnium monocarbide,  in: I. L. Shabalin (Ed.), Ultra-High Temperature Materials II: Refractory Carbides I. (Ta, Hf, Nb and Zr Carbides), Springer Netherlands, Dordrecht, 2019, pp. 145–248. URL: https://doi.org/10.1007/978-94-024-1302-1.
  * Mishin [2021] Y. Mishin,  Machine-learning interatomic potentials for materials science,  Acta Mater. 214 (2021) 116980.
  * Howard [1966] R. E. Howard,  Random-walk method for calculating correlation factors: Tracer diffusion by divacancy and impurity-vacancy pairs in cubic crystals,  Phys. Rev. 144 (1966) 650–661.
  * Belova et al. [2009] I. V. Belova, D. Shaw, G. E. Murch,  Limits of the ratios of tracer diffusivities for diffusion by vacancy pairs: Application to compound semiconductors,  Journal of Applied Physics 106 (2009) 113707.
  * Smith et al. [2018] C. J. Smith, M. A. Ross, N. De Leon, C. R. Weinberger, G. B. Thompson,  Ultra-high temperature deformation in TaC and HfC,  Journal of the European Ceramic Society 38 (2018) 5319–5332.
  * He et al. [2022] R. He, L. Fang, T. Han, G. Yang, G. Ma, J. Liu, X. Chen, L. Xie, L. Liu, Q. Li, et al.,  Elasticity, mechanical and thermal properties of polycrystalline hafnium carbide and tantalum carbide at high pressure,  Journal of the European Ceramic Society 42 (2022) 5220–5228.
  * Smith et al. [2018] C. J. Smith, X.-X. Yu, Q. Guo, C. R. Weinberger, G. B. Thompson,  Phase, hardness, and deformation slip behavior in mixed HfxTa1-xC,  Acta Materialia 145 (2018) 142–153.
  * Singh and Wiedemeier [1997] M. Singh, H. Wiedemeier,  Estimation of thermal expansion behaviour of some refractory carbides and nitrides,  Journal of materials science 32 (1997) 5749–5751.

Table 2: Lattice parameter $a$, cohesive energy $\varepsilon_{0}$, elastic
constants $c_{ij}$, and bulk modulus $B$ obtained by the present DFT
calculations. The experimental and calculated values from the literature are
shown in the round and square brackets, respectively.
Property | TaC | HfC
---|---|---
$a$ (Å) | 4.478 [4.471]a, (4.450)b | 4.646 [4.642]a, [4.647]f, (4.631)g
$\varepsilon_{0}$ (eV) | $-11.10$ | $-10.53$
$c_{11}$ (GPa) | 759.8 [737]a,c, [674]d | 509.9 [577]h, [540]a,d, (500)i
$c_{12}$ (GPa) | 118.0 [141]a,c, [172]d | 111.5 [117]h, [112]a,d
$c_{44}$ (GPa) | 170.8 [175]a,c, [167]d | 159.2 [171]h, [171]a,d
$B$ (GPa) | 331.9, (332)e, (355.9)b, [340]a,c, [339]d | 244.3, (242)j, (272)b, [253]a,d
aRef. [14], bRef. [40], cRef. [11] , dRef. [41], eRef. [24], fRef. [26]
gRef. [42], hRef. [25], iRef. [23], jRef. [22]
Table 3: “Raw” energies (in eV) of point defects in TaC and HfC obtained by
DFT calculations. The defect pairs are considered to have the shortest defect
separation. L, T, IP, and OP are vacancy cluster configurations explained in
the main text. *Unstable configuration.
Defect type | Symbol | TaC | HfC
---|---|---|---
Metal vacancy | $V_{\alpha}$ | $15.30$ | $19.30$
Carbon vacancy | $V_{\beta}$ | $9.45$ | $10.20$
Antisite on carbon sublattice | $A_{\beta}$ | $7.59$ | $9.13$
Antisite on metal sublattice | $B_{\alpha}$ | $11.78$ | $13.80$
Metal interstitial | $I_{A}$ | $-0.27$ | $-0.31$
Carbon interstitial | $I_{B}$ | $-2.71$ | $-3.99$
Divacancy | $V_{\alpha}V_{\beta}$ | $24.61$ | $27.90$
Antisite pair | $A_{\beta}B_{\alpha}$ | $5.45$ | $5.00$
Antisite-vacancy pair on metal sublattice | $V_{\alpha}B_{\alpha}$ | $22.63$ | $25.86$
Antisite-vacancy pair on carbon sublattice | $V_{\beta}A_{\beta}$ | $15.62$ | $18.27$
Antisite-vacancy pair on different sublattices | $V_{\beta}B_{\alpha}$ | * | $20.46$
Nearest-neighbor Frenkel defect | $V_{\beta}I_{B}$ | * | $4.30$
Vacancy cluster (T) | $V_{\alpha}V_{\beta}^{2}$ | $34.08$ | $36.80$
Vacancy cluster (L) | $V_{\alpha}V_{\beta}^{2}$ | $33.94$ | $37.03$
Vacancy cluster (IP) | $V_{\alpha}V_{\beta}^{3}$ | $43.60$ | $46.01$
Vacancy cluster (OP) | $V_{\alpha}V_{\beta}^{3}$ | $43.62$ | $46.17$
Vacancy cluster (IP) | $V_{\alpha}V_{\beta}^{4}$ | $53.16$ | $55.39$
Vacancy cluster (OP) | $V_{\alpha}V_{\beta}^{4}$ | $53.13$ | $55.75$
Vacancy cluster | $V_{\alpha}V_{\beta}^{5}$ | $62.6$ | $65.19$
Vacancy cluster | $V_{\alpha}V_{\beta}^{6}$ | $72.13$ | $75.01$
Table 4: Binding energies and symmetry factors $\sigma$ of defect clusters in
TaC and HfC obtained by DFT calculations. L, T, IP, and OP are vacancy cluster
configurations explained in the main text. *Unstable configuration. The values
in square brackets refer to previous calculations.
Cluster type | TaC | HfC
---|---|---
| $\sigma$ | Energy (eV) | $\sigma$ |  | Energy (eV)
$V_{\alpha}V_{\beta}$ | 6 |  $-0.14$ [$-0.16$]a | 6 |  |  $-1.60$ [$-1.54$]a, [$-1.49$]b,
$A_{\beta}B_{\alpha}$ | 6 | $-13.92$ | 6 |  | $-17.93$
$V_{\alpha}B_{\alpha}$ | 6 | $-4.45$ | 6 |  | $-7.24$
$V_{\beta}A_{\beta}$ | 6 | $-1.42$ | 12 |  | $-1.06$
$V_{\beta}B_{\alpha}$ |  | * | 6 |  | $-3.54$
$V_{\beta}I_{B}$ |  | * | 8 |  | $-1.91$
$V_{\alpha}V_{\beta}^{2}\leavevmode\nobreak\ (T)$ | 12 | $-0.12$ | 12 |  | $-2.90$
$V_{\alpha}V_{\beta}^{2}\leavevmode\nobreak\ (\mathrm{L})$ | 3 | $-0.26$ | 3 |  | $-2.67$
$V_{\alpha}V_{\beta}^{3}\leavevmode\nobreak\ (\mathrm{IP})$ | 12 | $-0.05$ | 12 |  | $-3.89$
$V_{\alpha}V_{\beta}^{3}\leavevmode\nobreak\ (\mathrm{OP})$ | 8 | $-0.03$ | 8 |  | $-3.73$
$V_{\alpha}V_{\beta}^{4}\leavevmode\nobreak\ (\mathrm{IP})$ | 3 | $0.06$ | 3 |  | $-4.71$
$V_{\alpha}V_{\beta}^{4}\leavevmode\nobreak\ (\mathrm{OP})$ | 12 | $0.03$ | 12 |  | $-4.35$
$V_{\alpha}V_{\beta}^{5}$ | 6 | $0.05$ | 6 |  | $-5.11$
$V_{\alpha}V_{\beta}^{6}$ | 1 | $0.13$ | 1 |  |  $-5.49$ [$-5.52$]b
aRef. [15], bRef. [16] |  |  |
Table 5: Equations for the Arrhenius parameters (prefactors $X_{d}$ and
effective formation energies $\overline{\varepsilon}_{d}$) of point defects in
metal-rich carbides ($x>0$) with constitutional defects $V_{\beta}$.
Defect | $X_{d}^{0}$ | $\overline{\varepsilon}_{d}$
---|---|---
$V_{\alpha}$ | $\dfrac{1}{4x}$ | $\varepsilon_{V_{\alpha}}+\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
$V_{\beta}$ | $4x$ | $0$
$A_{\beta}$ | $(4x)^{2}$ | $\varepsilon_{A_{\beta}}-2\varepsilon_{V_{\beta}}-2\varepsilon_{0}$
$B_{\alpha}$ | $\dfrac{1}{(4x)^{2}}$ | $\varepsilon_{B_{\alpha}}+2\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
$I_{A}$ | $4x$ | $\varepsilon_{I_{A}}-\varepsilon_{V_{\beta}}-2\varepsilon_{0}$
$I_{B}$ | $\dfrac{1}{4x}$ | $\varepsilon_{I_{B}}+\varepsilon_{V_{\beta}}$
$A_{\beta}B_{\alpha}$ | $\sigma_{A_{\beta}B_{\alpha}}$ | $\varepsilon_{A_{\beta}B_{\alpha}}$
$V_{\alpha}B_{\alpha}$ | $\dfrac{\sigma_{V_{\alpha}B_{\alpha}}}{(4x)^{3}}$ | $\varepsilon_{V_{\alpha}B_{\alpha}}+3\varepsilon_{V_{\beta}}+4\varepsilon_{0}$
$V_{\beta}A_{\beta}$ | $\sigma_{V_{\beta}B_{\beta}}(4x)^{3}$ | $\varepsilon_{V_{\beta}B_{\beta}}-3\varepsilon_{V_{\beta}}-2\varepsilon_{0}$
$V_{\beta}B_{\alpha}$ | $\dfrac{\sigma_{V_{\beta}B_{\alpha}}}{4x}$ | $\varepsilon_{V_{\beta}B_{\alpha}}+\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
$V_{\beta}I_{B}$ | $\sigma_{V_{\beta}I_{B}}$ | $\varepsilon_{V_{\beta}I_{B}}$
$V_{\alpha}V_{\beta}$ | $\sigma_{V_{\alpha}V_{\beta}}$ | $\varepsilon_{V_{\alpha}V_{\beta}}+2\varepsilon_{0}$
$V_{\alpha}V_{\beta}^{2}$ | $\sigma_{V_{\alpha}V_{\beta}^{2}}(4x)$ | $\varepsilon_{V_{\alpha}V_{\beta}^{2}}-\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
$V_{\alpha}V_{\beta}^{3}$ | $\sigma_{V_{\alpha}V_{\beta}^{3}}(4x)^{2}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{3}}-2\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
$V_{\alpha}V_{\beta}^{4}$ | $\sigma_{V_{\alpha}V_{\beta}^{4}}(4x)^{3}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{4}}-3\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
$V_{\alpha}V_{\beta}^{5}$ | $\sigma_{V_{\alpha}V_{\beta}^{5}}(4x)^{4}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{5}}-4\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
$V_{\alpha}V_{\beta}^{6}$ | $\sigma_{V_{\alpha}V_{\beta}^{6}}(4x)^{5}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{6}}-5\varepsilon_{V_{\beta}}+2\varepsilon_{0}$
Table 6: Equations for the Arrhenius parameters (prefactors $X_{d}$ and
effective formation energies $\overline{\varepsilon}_{d}$) of point defects in
stoichiometric carbides ($x=0$). Two mechanisms of thermal disorder are
considered with the leading defects $V_{\alpha}$ & $V_{\beta}$ and $V_{\beta}$
& $I_{B}$.
| Leading defects $V_{\alpha}$ & $V_{\beta}$ | Leading defects $V_{\beta}$ & $I_{B}$
---|---|---
Defect | $X_{d}^{0}$ | $\overline{\varepsilon}_{d}$ | $X_{d}^{0}$ | $\overline{\varepsilon}_{d}$
$V_{\alpha}$ | 1 | $\dfrac{\varepsilon_{V_{\alpha}}+\varepsilon_{V_{\beta}}+2\varepsilon_{0}}{2}$ | $\dfrac{1}{\sqrt{2\nu}}$ | $\varepsilon_{V_{\alpha}}+2\varepsilon_{0}+\dfrac{\varepsilon_{V_{\beta}}-\varepsilon_{I_{B}}}{2}$
$V_{\beta}$ | 1 | $\dfrac{\varepsilon_{V_{\alpha}}+\varepsilon_{V_{\beta}}+2\varepsilon_{0}}{2}$ | $\sqrt{2\nu}$ | $\dfrac{\varepsilon_{V_{\beta}}+\varepsilon_{I_{B}}}{2}$
$A_{\beta}$ | 1 | $\varepsilon_{A_{\beta}}-\varepsilon_{V_{\beta}}+\varepsilon_{V_{\alpha}}$ | $2\nu$ | $\varepsilon_{A_{\beta}}-\varepsilon_{V_{\beta}}+\varepsilon_{I_{B}}-2\varepsilon_{0}$
$B_{\alpha}$ | 1 | $\varepsilon_{B_{\alpha}}+\varepsilon_{V_{\beta}}-\varepsilon_{V_{\alpha}}$ | $\dfrac{1}{2\nu}$ | $\varepsilon_{B_{\alpha}}+\varepsilon_{V_{\beta}}-\varepsilon_{I_{B}}+2\varepsilon_{0}$
$I_{A}$ | 1 | $\varepsilon_{I_{A}}-\varepsilon_{0}+\dfrac{\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}}}{2}$ | $\sqrt{2\nu}$ | $\varepsilon_{I_{A}}-2\varepsilon_{0}+\dfrac{\varepsilon_{I_{B}}-\varepsilon_{V_{\beta}}}{2}$
$I_{B}$ | 1 | $\varepsilon_{I_{B}}-\varepsilon_{0}+\dfrac{\varepsilon_{V_{\beta}}-\varepsilon_{V_{\alpha}}}{2}$ | $\dfrac{1}{\sqrt{2\nu}}$ | $\dfrac{\varepsilon_{V_{\beta}}+\varepsilon_{I_{B}}}{2}$
$A_{\beta}B_{\alpha}$ | $\sigma_{A_{\beta}B_{\alpha}}$ | $\varepsilon_{A_{\beta}B_{\alpha}}$ | $\sigma_{A_{\beta}B_{\alpha}}$ | $\varepsilon_{A_{\beta}B_{\alpha}}$
$V_{\alpha}B_{\alpha}$ | $\sigma_{V_{\alpha}B_{\alpha}}$ | $\varepsilon_{V_{\alpha}B_{\alpha}}+\varepsilon_{0}+\dfrac{3(\varepsilon_{V_{\beta}}-\varepsilon_{V_{\alpha})}}{2}$ | $\dfrac{\sigma_{V_{\alpha}B_{\alpha}}}{(2\nu)^{3/2}}$ | $\varepsilon_{V_{\alpha}B_{\alpha}}+4\varepsilon_{0}+\dfrac{3(\varepsilon_{V_{\beta}}-\varepsilon_{I_{B})}}{2}$
$V_{\beta}A_{\beta}$ | $\sigma_{V_{\beta}B_{\beta}}$ | $\varepsilon_{V_{\beta}B_{\beta}}+\varepsilon_{0}+\dfrac{3(\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}})}{2}$ | $(2\nu)^{3/2}\sigma_{V_{\beta}B_{\beta}}$ | $\varepsilon_{V_{\beta}A_{\beta}}-2\varepsilon_{0}-\dfrac{3(\varepsilon_{V_{\beta}}-\varepsilon_{I_{B})}}{2}$
$V_{\beta}B_{\alpha}$ | $\sigma_{V_{\beta}B_{\alpha}}$ | $\varepsilon_{V_{\beta}B_{\alpha}}+\varepsilon_{0}+\dfrac{\varepsilon_{V_{\beta}}-\varepsilon_{V_{\alpha}}}{2}$ | $\dfrac{\sigma_{V_{\beta}B_{\alpha}}}{\sqrt{2\nu}}$ | $\varepsilon_{V_{\beta}B_{\alpha}}+2\varepsilon_{0}+\dfrac{\varepsilon_{V_{\beta}}-\varepsilon_{I_{B}}}{2}$
$V_{\beta}I_{B}$ | $\sigma_{V_{\beta}I_{B}}$ | $\varepsilon_{V_{\beta}I_{B}}$ | $\sigma_{V_{\beta}I_{B}}$ | $\varepsilon_{V_{\beta}I_{B}}$
$V_{\alpha}V_{\beta}$ | $\sigma_{V_{\alpha}V_{\beta}}$ | $\varepsilon_{V_{\alpha}V_{\beta}}+2\varepsilon_{0}$ | $\sigma_{V_{\alpha}V_{\beta}}$ | $\varepsilon_{V_{\alpha}V_{\beta}}+2\varepsilon_{0}$
$V_{\alpha}V_{\beta}^{2}$ | $\sigma_{V_{\alpha}V_{\beta}^{2}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{2}}+3\varepsilon_{0}+\dfrac{\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}}}{2}$ | $\sqrt{2\nu}\sigma_{V_{\alpha}V_{\beta}^{2}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{2}}+2\varepsilon_{0}+\dfrac{\varepsilon_{I_{B}}-\varepsilon_{V_{\beta}}}{2}$
$V_{\alpha}V_{\beta}^{3}$ | $\sigma_{V_{\alpha}V_{\beta}^{3}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{3}}+(\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}})+4\varepsilon_{0}$ | $2\nu\sigma_{V_{\alpha}V_{\beta}^{3}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{3}}+2\varepsilon_{0}+\varepsilon_{I_{B}}-\varepsilon_{V_{\beta}}$
$V_{\alpha}V_{\beta}^{4}$ | $\sigma_{V_{\alpha}V_{\beta}^{4}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{4}}+5\varepsilon_{0}+\dfrac{3(\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}})}{2}$ | $(2\nu)^{3/2}\sigma_{V_{\alpha}V_{\beta}^{4}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{4}}+2\varepsilon_{0}+\dfrac{3(\varepsilon_{I_{B}}-\varepsilon_{V_{\beta}})}{2}$
$V_{\alpha}V_{\beta}^{5}$ | $\sigma_{V_{\alpha}V_{\beta}^{5}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{5}}+2(\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}})+6\varepsilon_{0}$ | $(2\nu)^{2}\sigma_{V_{\alpha}V_{\beta}^{5}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{5}}+2\varepsilon_{0}+2(\varepsilon_{I_{B}}-\varepsilon_{V_{\beta}})$
$V_{\alpha}V_{\beta}^{6}$ | $\sigma_{V_{\alpha}V_{\beta}^{6}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{6}}+7\varepsilon_{0}+\dfrac{5(\varepsilon_{V_{\alpha}}-\varepsilon_{V_{\beta}})}{2}$ | $(2\nu)^{5/2}\sigma_{V_{\alpha}V_{\beta}^{6}}$ | $\varepsilon_{V_{\alpha}V_{\beta}^{6}}+2\varepsilon_{0}+\dfrac{5(\varepsilon_{I_{B}}-\varepsilon_{V_{\beta}})}{2}$
Table 7: Arrhenius parameters (prefactors $X_{d}$ and effective formation
energies $\overline{\varepsilon}_{d}$) of point defects in TaC and HfC with
stoichiometric ($x=0$) and metal-rich ($x>0$) compositions. The values in
square brackets were computed from the literature reports.
| TaC | HfC
---|---|---
|  $x=0$, $V_{C}$ & $V_{Ta}$ |  $x>0$, $V_{C}$ |  $x=0$, $V_{C}$ & $I_{C}$ |  $x>0$, $V_{C}$
Defect | $X_{d}^{0}$ |  $\overline{\varepsilon}_{d}$ (eV) | $X_{d}^{0}$ |  $\overline{\varepsilon}_{d}$ (eV) | $X_{d}^{0}$ |  $\overline{\varepsilon}_{d}$ (eV) | $X_{d}^{0}$ |  $\overline{\varepsilon}_{d}$ (eV)
$V_{\alpha}$ | 1 | 1.28 | $\dfrac{1}{4x}$ | 2.55 [$2.65$]b | $\dfrac{1}{\sqrt{2\nu}}$ | 5.34 | $\dfrac{1}{4x}$ | 8.44 [$8.57$]a
|  |  |  |  |  |  |  | [$8.64$]b
$V_{\beta}$ | 1 | 1.28 | $4x$ | 0.00 | $\sqrt{2\nu}$ | 3.11 | $4x$ | 0.00
$A_{\beta}$ | 1 | 13.44 | $16x^{2}$ | 10.90 | $2\nu$ | 16.00 | $16x^{2}$ | 9.79
$B_{\alpha}$ | 1 | 5.93 | $\dfrac{1}{16x^{2}}$ | 8.48 | $\dfrac{1}{2\nu}$ | 6.93 | $\dfrac{1}{16x^{2}}$ | 13.14 [$13.51$]a
$I_{A}$ | 1 | 13.76 | $4x$ | 12.48 | $\sqrt{2\nu}$ | 13.66 | $4x$ | 10.55
$I_{B}$ | 1 | 5.47 | $\dfrac{1}{4x}$ | 6.74 | $\dfrac{1}{\sqrt{2\nu}}$ | 3.11 | $\dfrac{1}{4x}$ | 6.21
$A_{\beta}B_{\alpha}$ | 6 | 5.45 | 6 | 5.45 | 6 | 5.00 | 6 | 5.00
$V_{\alpha}B_{\alpha}$ | 6 | 2.76 | $\dfrac{6}{64x^{3}}$ | 6.58 | $\dfrac{6}{(2\nu)^{3/2}}$ | 5.03 | $\dfrac{6}{64x^{3}}$ | 14.34
$V_{\beta}A_{\beta}$ | 6 | 13.30 | $384x^{3}$ | 9.47 | $12(2\nu)^{3/2}$ | 18.05 | $384x^{3}$ | 8.73
$V_{\beta}B_{\alpha}$ | * | * | * |  | $\dfrac{6}{\sqrt{2\nu}}$ | 6.50 | $\dfrac{6}{4x}$ | 9.60
$V_{\beta}I_{B}$ | * | * | * |  | 8 | 4.30 | 8 | 4.30
$V_{\alpha}V_{\beta}$ | 6 | 2.41 | 6 | 2.41 [$2.49$]b | 6 | 6.84 | 6 | 6.84 [$7.08$]a
$V_{\alpha}V_{\beta}^{2}$ (T) | 12 | 3.71 | $48x$ | 2.43 [$2.50$]b | $12\sqrt{2\nu}$ | 8.65 | $48x$ | 5.54 [$5.85$]b
$V_{\alpha}V_{\beta}^{2}$ (L) | 3 | 3.57 | $12x$ | 2.29 [$2.36$]b | $3\sqrt{2\nu}$ | 8.88 | $12x$ | 5.77 [$6.05$]b
$V_{\alpha}V_{\beta}^{3}$ (IP) | 12 | 5.05 | $192x^{2}$ | 2.50 | $24\nu$ | 10.76 | $192x^{2}$ | 4.55 [$4.31$]b
$V_{\alpha}V_{\beta}^{3}$ (OP) | 8 | 5.07 | $128x^{2}$ | 2.52 [$2.55$]b | $16\nu$ | 10.92 | $128x^{2}$ | 4.71 [$5.00$]b
$V_{\alpha}V_{\beta}^{4}$ (IP) | 3 | 6.44 | $192x^{3}$ | 2.61 [$2.65$]b | $3(2\nu)^{3/2}$ | 13.05 | $192x^{3}$ | 3.73
$V_{\alpha}V_{\beta}^{4}$ (OP) | 12 | 6.41 | $768x^{3}$ | 2.58 | $12(2\nu)^{3/2}$ | 13.41 | $768x^{3}$ | 4.10 [$4.18$]b,
$V_{\alpha}V_{\beta}^{5}$ | 6 | 7.70 | $1536x^{4}$ | 2.60 [$2.70$]b | $6(2\nu)^{2}$ | 15.75 | $1536x^{4}$ | 3.33 [$3.45$]b,
$V_{\alpha}V_{\beta}^{6}$ | 1 | 9.06 | $1024x^{5}$ | 2.68 [$2.78$]b | $(2\nu)^{5/2}$ | 18.47 | $1024x^{5}$ | 2.95 [$2.95$]b
|  |  |  |  |  |  |  | [$3.05$]a
aRef. [16] , bRef. [15] |  |  |

Figure 1: Initial (left column) and relaxed (right column) structures of
selected point-defect pairs undergoing structural reconstructions. (a)
$V_{\beta}B_{\alpha}$ pair in HfC (b) $V_{\alpha}B_{\alpha}$ pair in TaC and
HfC. (c) $A_{\beta}B_{\alpha}$ antisite pair in TaC and HfC. (d)
$V_{\beta}A_{\beta}$ pair in TaC . The structures are viewed along a
$\left\langle 100\right\rangle$ direction. The metal atoms, carbon atoms,
metal vacancies, and carbon vacancies are shown in green, yellow, purple, and
light gray, respectively. Some atoms are encircled in red for tracking. Figure
2: Interaction energy between carbon vacancies in the TaC and HfC carbides as
a function of vacancy separation. The literature data from aRef. [15] shows
similar trends.

(a)(b)

Figure 3: Composition dependence of point defect concentrations in (a) TaC and
(b) HfC at 2500 K.

(a)

(b)

Figure 4: Arrhenius plots of point defect concentrations in TaC at (a)
stoichiometric composition and (b) Ta-rich composition of
$c_{\mathrm{Ta}}=0.52$ ($x=0.02$).

(a)

(b)

Figure 5: Arrhenius plots of point defect concentrations in HfC at (a)
stoichiometric composition and (b) Hf-rich composition of
$c_{\mathrm{Hf}}=0.52$ ($x=0.02$).

SUPPLEMENTARY INFORMATION

First-principles prediction of point defect energies and concentrations in the
tantalum and hafnium carbides

> I. Khatri, R. K. Koju, and Y. Mishin

Department of Physics and Astronomy, MSN 3F3, George Mason University,
Fairfax, Virginia 22030, USA

Figure 1: Initial (left column) and relaxed (right column) structures of
point-defects undergoing structural reconstructions. (a) $V_{\alpha}$ (b)
$V_{\beta}$, (c) $V_{\alpha}V_{\beta}$, and (d) $A_{\beta}$ in TaC and HfC.
The structures are viewed along a $\left\langle 100\right\rangle$ direction.
The metal atoms, carbon atoms, metal vacancy and carbon vacancy are shown in
green, yellow, purple, and light gray, respectively. Some atoms are encircled
in red for tracking.  Figure 2: Initial (left column) and relaxed (right
column) structures of point-defects undergoing structural reconstructions. (a)
$B_{\alpha}$ in TaC and HfC, (b) $I_{B}$ in TaC and HfC, (c) $I_{A}$ in TaC
and HfC, and (d) $V_{\beta}I_{B}$ in HfC. The structures are viewed along a
$\left\langle 100\right\rangle$ direction. The metal atoms, carbon atoms,
metal vacancy and carbon vacancy are shown in green, yellow, purple, and light
gray, respectively. Some atoms are encircled in red for tracking.  Figure 3:
Initial (left column) and relaxed (right column) structures of point-defect
clusters undergoing structural reconstructions. (a)
$V_{\alpha}V_{\beta}^{2}(L)$, (b) $V_{\alpha}V_{\beta}^{2}(T)$ (c)
$V_{\alpha}V_{\beta}^{3}(OP)$, and (d) $V_{\alpha}V_{\beta}^{3}(IP)$ in TaC
and HfC. The structures are vie wed along a $\left\langle 100\right\rangle$
direction. The metal atoms, carbon atoms, metal vacancies and carbon vacancies
are shown in green, yellow, purple, and light gray, respectively. Figure 4:
Initial (left column) and relaxed (right column) structures of point-defect
clusters undergoing structural reconstructions. (a)
$V_{\alpha}V_{\beta}^{4}(IP)$, (b) $V_{\alpha}V_{\beta}^{4}(OP)$, (c)
$V_{\alpha}V_{\beta}^{5}$, and (d) $V_{\alpha}V_{\beta}^{6}$ in TaC and HfC.
The structures are viewed along a $\left\langle 100\right\rangle$ direction.
The metal atoms, carbon atoms, metal vacancy and carbon vacancy are shown in
green, yellow, purple, and light gray, respectively. Figure 5: Raw energy of
metal vacancy ($\epsilon_{V_{\alpha}}$) as the function of reciprocal of the
number of atoms prior to any defects for TaC and HfC in rocksalt crystal
structure. Figure 6: Raw energy of antisite on metal sublattice
($\epsilon_{B_{\alpha}}$) as the function of reciprocal of the number of atoms
prior to any defects for TaC and HfC in rocksalt crystal structure. Figure 7:
Raw energy of carbon vacancy ($\epsilon_{V_{\beta}}$) and antisite on carbon
sublattice ($\epsilon_{A_{\beta}}$) as a function of the reciprocal of the
number of atoms prior to any defects for TaC and HfC in rocksalt crystal
structure. Open and closed circles of carbon vacancy ($V_{\beta}$) are for HfC
and TaC respectively. Figure 8: Raw energy of metal interstitial
($\epsilon_{I_{\alpha}}$) in tetrahedral position as a function of the
reciprocal of the number of atoms prior to any defects for TaC and HfC in
rocksalt crystal structure. Figure 9: Raw energy of carbon interstitial
($\epsilon_{I_{\beta}}$) in tetrahedral position as a function of the
reciprocal of the number of atoms prior to any defects for TaC and HfC in
rocksalt crystal structure. Open and closed triangles are for HfC and TaC
respectively. Figure 10: Raw energy of Frenkel pair
($\epsilon_{V_{\beta}I_{\beta}}$) in HfC and antisite pair
($\epsilon_{B_{\alpha}A_{\beta}}$) in HfC and TaC as a function of the
reciprocal of the number of atoms prior to any defects in rocksalt crystal
structure. Figure 11: Raw energy of divacancy
($\epsilon_{V_{\alpha}V_{\beta}}$) as a function of the reciprocal of the
number of atoms prior to any defects for TaC and HfC in rocksalt crystal
structure. Figure 12: Raw energy of three vacancies cluster
($\epsilon_{V_{\alpha}V_{\beta}^{2}}$) in Linear (L) and Triangular (T)
configurations as a function of the reciprocal of the number of atoms prior to
any defects for TaC and HfC in rocksalt crystal structure. Figure 13: Raw
energy of four vacancies cluster ($\epsilon_{V_{\alpha}V_{\beta}^{3}}$) with
three carbon vacancies are off-plane (OP) and in-plane (IP) configurations, as
a function of the reciprocal of the number of atoms prior to any defects for
TaC and HfC in rocksalt crystal structure. Figure 14: Raw energy of five
vacancies cluster ($\epsilon_{V_{\alpha}V_{\beta}^{4}}$) with four carbon
vacancies are off-plane (OP) and in-plane (IP) configurations, as a function
of the reciprocal of the number of atoms prior to any defects for TaC and HfC
in rocksalt crystal structure. Figure 15: Raw energy of six vacancies cluster
($\epsilon_{V_{\alpha}V_{\beta}^{5}}$) as a function of the reciprocal of the
number of atoms prior to any defects for TaC and HfC in rocksalt crystal
structure. Figure 16: Raw energy of seven vacancies cluster
($\epsilon_{V_{\alpha}V_{\beta}^{6}}$) as a function of the reciprocal of the
number of atoms prior to any defects for TaC and HfC in rocksalt crystal
structure. Figure 17: Raw energy of vacancy antisite pair on metal sublattice
($\epsilon_{V_{\alpha}B_{\alpha}}$) as a function of the reciprocal of the
number of atoms prior to any defects for TaC and HfC in rocksalt crystal
structure. Figure 18: Raw energy of vacancy antisite pair on carbon sublattice
($\epsilon_{V_{\beta}A_{\beta}}$) and carbon vacancy and antisite pair on
metal sublattice ($\epsilon_{V_{\beta}B_{\alpha}}$) in HfC as a function of
the reciprocal of the number of atoms prior to any defects in rocksalt crystal
structure. Figure 19: Raw energy of vacancy antisite pair on carbon sublattice
($\epsilon_{V_{\beta}A_{\beta}}$) in TaC as a function of the reciprocal of
the number of atoms prior to any defects in rocksalt crystal structure.

(a)(b)

Figure 20: Composition dependence of point defect concentrations in (a) TaC
and (b) HfC at 3500 K. Table 8: The k-points used for raw energy calculations
of various defects within the supercells of HfC and TaC obtained through the
repetition of the relaxed conventional 8-atom unit cell of the B1 crystal
structure.
Supercell Size (atoms) | K-points in TaC | K-points in HfC
---|---|---
$2\times 2\times 3$ (96) | $11\times 11\times 7$ | $2\times 2\times 1$
$3\times 3\times 2$ (144) | $7\times 7\times 11$ | $1\times 1\times 2$
$3\times 3\times 3$ (216) | $7\times 7\times 7$ | $3\times 3\times 3$
$3\times 3\times 4$ (288) | $6\times 6\times 4$ | $2\times 2\times 1$
$4\times 4\times 3$ (384) | $4\times 4\times 6$ | $1\times 1\times 2$
$4\times 4\times 4$ (512) | $5\times 5\times 5$ | $1\times 1\times 1$