Intrinsic defects, Mo-related defects, and complexes in transition-metal carbide VC: A first-principles study

Intrinsic defects and Mo-related defects in vanadium carbide VC, as well as the defect complexes between vacancies and Mo defects were investigated by means of first-principles calculations within the framework of density functional theory. In addition, Mo diffusion in VC was also studied using LST/QST method. The formation energies of defects have clearly shown that except C vacancy (V C ) all other point defects are not energetic favorable compared to perfect VC. V C can exist in the lattice forming nonstoichiometric carbide VC x (x < 1), and also can stabilize the Mo-related defects (S Mo-V , S Mo-C , and T Mo ). Free Mo atoms have the strong tendency to enter the already formed V V and occupy the lattice position of V atoms. Meanwhile, Mo atom in C lattice (S Mo-C ) and interstitial Mo (I Mo ) atom can also enter the V V position stabilizing the lattice structure. S Mo-C + V V will transform into S Mo-V + V C and I Mo + V V will transform into S Mo-V during optimization, and large binding energy makes Mo atom tend to exist in the interstitial position. From the perspective of energy, Mo atom tends to diffuse through the interstitial position.


| INTRODUCTION
][3] Among all kinds of TMCs, vanadium carbide VC has attracted more attention all over the world due to its relatively low price of element V compared with W/Mo and relatively low formation energy, which is beneficial to control the precipitation of carbides. 4,57][8][9] Ultra-high hardness and strength can also be obtained if VC is applied into hard alloys, which are commonly used as tool materials to cut some difficult-to-machine materials. 10In terms of material modification, VC can be used not only as coating materials but also as additive materials in coatings. 11,12ecause of the extremely high melting point, VC normally precipitates at high temperature in metal materials, 13 the hard alloys which are made of VC need to be fabricated by high-temperature sintering, 14 and some special coatings containing VC will be manufactured by methods like thermal | 7227 GUO et al.
spraying and thermal diffusion. 15So some of the temperature-related point defects, such as vacancy, interstitial atoms, and impurity atoms, will exist inevitably in VC who has already gone through high-temperature treatment. 16,179][20] Therefore, it is of great significance to investigate the behavior of point defects in crystals.
Recently, the researches about point defects in TMCs have been reported from time to time.It is well-known that the point defects are the ones at the atomic scale, and it is difficult to study their effect in detail by conventional experimental methods.So first-principles calculations are commonly used by many researchers around the world.Burr et al 21 studied the formation and migration of point defects in tungsten carbide WC.Kong et al 22 revealed that the carbon vacancy is the dominant defect in hexagonal WC.Carbide TiC was chosen by Sun et al 23 to calculate the self-diffusion of Ti interstitial-based point defects and complexes.Razumovskiy et al 24 focused on a description of a complex vacancy behavior of the group IVb substoichiometric TMCs TiC, ZrC, and HfC carbides and found a strong tendency toward vacancy clustering in the carbides.DP Daroca et al 25 performed a study of the stability of different point defects in thorium carbide ThC, and demonstrated that C isolated vacancies are the most likely defects in ThC, whereas Th vacancies are highly unfavorable as compared to C ones.However, nearly all studies about the point defects in VC mainly focus on the nonstoichiometry caused by vacancy as well as the order and disorder problem, [26][27][28][29] and the analyses and discussion of different types of defects are rarely reported.So the first-principles method is adopted in this work to calculate the VC systems with different intrinsic defects (vacancy, substitution, and tetrahedral interstitial) and analyze their formation energy and electronic structure.In addition, other metal elements can exist in VC unavoidably during formation/fabrication process and diffuse into VC lattice at high temperature.For example, element Mo is often used as the binder in the preparation of hard alloys, 30 and Mo content is always found relatively high in VC in a self-designed novel Febased alloy by our group. 31So Mo-related defects and the influence of impurity Mo atoms on the defects in VC, as well as Mo diffusion are also studied in this paper.

| CALCULATION DETAILS
First-principles calculations based on density functional theory (DFT) 32,33 with ultrasoft pseudopotentials were used to evaluate the intrinsic defects and Mo-related defects in VC supercell.The calculations were performed using Cambridge Sequential Total Energy Package (CASTEP).The valence electrons of V atom 3d 3 4s 2 and C atom 2s 2 2p 2 orbitals were treated by projector-augmented wave theory.The generalized gradient approximation (GGA) with Perdew-Burke-Ernzerho (PBE) functional was employed as exchange-correlation energy. 34,35The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm was applied in relaxation process of models to optimize the structures.The energy, the maximum force, and maximum displacement were set as 5 × 10 -6 eV/atom, 0.01 eV/Å, and 5 × 10 −4 Å for the convergence tolerances, respectively.Based on convergence tests, which are shown in Figure S1, plane-wave cutoff energy was set at 380eV and Brillouin zone sampling was set at 6 × 6 × 6 Monkhorst-Pack mesh for VC unit cell.LST/QST method was used to perform the migration barrier calculations to investigate the diffusion behavior of Mo atom.
The determination of supercell size was tested by calculating the formation energy of one V vacancy (V V ) or one C vacancy (V C ) existed in 8-atom, 16-atom, 32-atom, 64-atom, 96-atom, 144-atom, and 216-atom VC supercells, respectively (see Table S1 and Figure S2).Defect calculations were performed in a prerelaxed 64-atom supercell, consisting of 2 × 2 × 2 replica of VC unit cell.The models of three different intrinsic defects that may exist in VC supercell are established, which are (a) vacancies, (b) substitutions (antisite), and (c) interstitials (tetrahedral interstitials are considered here).Each intrinsic defect contains two different types considering the existence of V and C atom in the supercell.So first-principles calculations about intrinsic defects are performed in VC supercell with V vacancy (V V ), C vacancy (V C ), V substitution (Sv), C substitution (S C ), V interstitial (Iv), and C interstitial (I C ), respectively.Only one atom defect is considered among the intrinsic defects throughout.Figure 1 shows the structures of VC with different intrinsic defects.Note that only unit cells are displayed here for clarity.

3
Mo-related defects in VC supercell contain two different types, one is Mo substitutions and the other is Mo interstitial.Mo substitutions refer to the defects that one V or C atom is replaced by one Mo atom in the structure, which are represented by S Mo-V and S Mo-C in this paper.While Mo interstitial defect (I Mo ) is similar to I V and I C , which means one Mo atom enters into the tetrahedral interstitial site in crystal structure.The corresponding structure models are shown in Figure 2.
The distances of atoms in the supercell near defect in supercell are collected and listed in Table 1.For perfect VC supercell, D 1N , D 2N , and D 3N represent the first, second, and third nearest distance between atoms, which can be seen in Figure 1A.For VC supercell with V V , V C , S V , S C , S Mo-V , and S Mo-C defects, D 1N , D 2N , and D 3N represent the distance from the defect to the first, second, and third nearest atom.For VC supercell with I V , I C , and I Mo , D 1N , D 2N , and D 3N represent the first, second, and third nearest distance between atoms in the cube (the part surrounded by dashed lines in Figures 1E,F, and 2C) where the interstitial atom locates in the center.Detailed figures explaining D 1N , D 2N , and D 3N for different defect situations that listed in Table 1 can also be found in Figures 1 and 2.
From Table 1, although V V and V C both belong to vacancy defect, the effect of existence of V and C vacancy on the structure lattices differs from each other.V vacancy mainly affects D 2N .In this case, the second nearest distance D 2N is the distance from V V to V atoms nearby, which is indicated by (V V -V) after the value in Table 1.The distance change compared to perfect system ∆D 2N /D per 2N for V V is the largest, −2.38%, but still much smaller than the largest distance change ∆D 1N /D per 1N for V C , 7.17%.Interestingly, the distance change values for V V distribute more evenly and only ∆D 2N /D per 2N is larger than values for V C , but the volume change (−0.548%) for VC supercell with V V is larger than that (0.119%) with V C .In addition, when V vacancy exists in the supercell, D 1N , D 2N , and D 3N all become smaller compared to the perfect system resulting in the shrinkage of the crystal structure, which obviously seems reasonable.However, after one C atom is removed from the lattice, D 1N from V C to the nearest V atoms increases instead, which directly leads to slight lattice expansion.
For substitution defect Sv and S C , one larger V atom substitutes smaller C atom resulting in the increase in supercell volume, whereas one smaller C atom substitutes larger V atom resulting in the decrease in supercell volume, both of which are understandable.Only minor change in D 2N and D 3N can be seen compared to the perfect system, especially for S V .D 1N become larger than that of perfect system for S V , and the distance change ∆D 1N /D per 1N reaches 12.37%, whereas D 1N for S C decreases compared to 605.825 5.428 perfect system with the distance change in −9.53%, which shows that substitution atom mainly influence its nearest atom position.
When a new V atom is added in the tethahedral site of the lattice (cubic part represented by dashed lines in Figure 1E), eight vertices of the cube where V atom is located in the center all expand outward, which make D 1N (V-C) and D 3N (V-C) both increase.Additionally, the displacement of V atoms at the vertex is larger than that of C atoms, so the value of D 2N (V-V) is greater than that of D 2N (C-C).When a new C atom is added into the lattice, four vertices containing V atoms expand outwards, whereas four C atoms contract inwards, which make D 2N (V-V) increase and D 2N (C-C) decrease.Finally, the combined result of V and C movement leads to slight increase in both D 1N (V-C) and D 3N (V-C).Overall, when V atom is added into the tetrahedral site, the lattice changes more obviously than C addition, mainly because of the larger radius of V atom resulting in the severe lattice distortion.The larger volume change for I V also confirms the above conclusion.
Mo is also a transition-metal element with an atomic radius similar to but slightly larger than V, as well as same bonding between Mo and C atom with V-C bond, which results in a little larger D 1N , D 2N , and D 3N values than perfect system if one V atom is replaced by Mo atom with volume change of only 0.565%.When one Mo atom substitutes C atom in the structure, the situation is totally different from S Mo-V .V-Mo-V atoms instead of V-C-V atoms arrange in a row along x, y, and z directions and the nearest V atoms around Mo atom move outwards finally causing the volume expansion.The introduction of Mo atom into the tethahedral position has the similar result to I V .D 1N , D 2N , and D 3N values all become much larger than the perfect structure, and the displacement of C atoms at the vertex is smaller than that of V atoms.Due to the larger radius of Mo than V, all the values for I Mo listed in Table 1 are larger than I V , the volume change is the largest among all the supercells with defects.

| Formation energy
In this paper, two kinds of formation energy-(a) formation energy for intrinsic defects, (b) formation energy for bulk material with defects-are considered to investigate the stability of VC structure after the defects being introduced.
The formation energy for intrinsic defects can be expressed by Eq.( 1)-(3) 36,37 : For vacancy defect: For substitution defect: For interstitial defect: where E def is formation energy for intrinsic defects, the sub- script V, S, I here represent vacancy, substitution, and interstitial defect; E(N − 1), E(N), and E(N + 1) are the relaxed total energy of supercell including one vacancy, substitution, or interstitial defect; E per (N) is the relaxed total energy of the corresponding ideal supercell crystal; E(V) and E(C) are the energy per V atom in pure cubic V crystal and C atom in graphite crystal, respectively.The formation energy of defects is also close related to the chemical potential that may change in different chemical environments.If chemical potential is considered here, the formation energy of intrinsic defects in VC can be computed as following expression: where E tot (defect) is the relaxed total energy of supercell in- cluding intrinsic defects; E per is the relaxed total energy of the perfect supercell; n j represents the atom number of type j removed from (Δn j > 0) or added (Δn j < 0) to crystal lattice when defect is formed, and μ j is corresponding chemical potential for alloy element.
In order to make VC compound stable, Δµ V and Δµ C should satisfy the following equation: where ΔH(VC) in Equation ( 5) is the formation enthalpy of VC per unit cell in solid phase; Δµ V and Δµ C are the relative chemical potentials for V and C, which can be denoted as: where solid V and solid C are the energies of pure V and C in their standard states, respectively.
To avoid precipitation of element solid V and C, the relative chemical potentials should be restricted by the following equations: From the binary phase diagram of V-C system, the VC phase equilibrates with its neighboring V 2 C phase under (1) V-rich condition.To avoid the formation of V 2 C, chemical potentials must also be bounded by the following equation: The chemical potential ranges for V and C are determined according to above equations and shown in Figure 3.
The formation energy for bulk material with defects is presented as follows: where E bulk is formation energy for bulk material with defects; E(VxCy) is the relaxed total energy of supercell including defects; x and y are the number of V and C atom in corresponding supercell.
The formation energy for Mo-related defects can be expressed as follows: For Mo substitution defect: For Mo interstitial defect: The formation energy for bulk material with Mo-related defects is presented as follows: where E bulk Mo is formation energy for bulk material with Morelated defects; E(VxCyMo) is the relaxed total energy of supercell including Mo-related defects.
From Figure 3, it can be seen that if chemical potentials are considered, the formation energies for different intrinsic defects are not a fixed value, but within a certain range.In general, the formation energies for vacancy defect are the smallest, then followed by interstitial and substitution defects, and no obvious overlapping can be observed for their value ranges.So the formation energies of defects that calculated by Equations ( 1)-( 3), ( 12)-( 13) without consideration of chemical potentials are analyzed in this section.Table 2 lists the results of formation energy for all supercells with different kinds of defects.The results of formation energy for intrinsic defects are in agreement with the results of formation energy for bulk material with defects.For a specific point defect, larger formation energy for the defect corresponds to larger formation energy for bulk material with defect.The more negative the energy, the easier it is to form such a defect.Among all intrinsic and Mo-related defects, V C has the lowest formation energy values indicating that V C is the most stable defect in the structure, which is already demonstrated by the researchers around the world, 38,39 and all other point defects are not energetic favorable compared to perfect VC.
As the extraneous defects, Mo-related defects are less stable than intrinsic ones in VC structure in general, such as S Mo- C and I Mo , but S Mo-V is an exception.The formation energy of S Mo-V is only larger than that of V C and perfect cell showing relatively higher probability to exist in the crystal cell.This (10)   indicates that when element Mo is present in the alloy, Mo atoms seems more likely to enter into VC carbide and occupy the lattice position of V atoms to form extraneous Mo-related defects, but the premise is that V V should be formed firstly.It is known from the formation energies for V V in Table 2 that the formation ability of V V is only at an intermediate level, and the structure with S Mo-V still has higher formation energy than perfect cell, so the existence of S Mo-V is just a matter of probability, which does not reveal that S Mo-V must exist in the system.

| Electronic structures
Figure 4 shows the electron density difference distribution for VC supercell with intrinsic defects along specific direction.The value of electron density difference around V atoms in the structure is negative, whereas the value is positive for C atoms, which means that V atoms lose the electron and C atoms gain electron.When V V exists in the supercell (see Figure 4A), the charge around C atoms in the nearest position mainly transfers to V atoms in the second nearest position, that isthe nearest C atoms form bonds with the second nearest V atoms around V V , but the electron-gaining capacity of the nearest C atoms around the vacancy is slightly weaker than that of other normal C positions in the structure (see Table 3).If there is V C in the supercell shown in Figure 4B, the V atoms in the nearest position mainly bond with the fourth nearest C atoms rather than the second nearest C atoms around V C , resulting in the larger D 1N (2.227Å) than that of perfect structure.Meanwhile, a circular region on the side of V C exists close to the nearest V atoms, which has strong ability to gain electrons.The existence of S V directly leads to the appearance of the circular area with positive differential charge density along ±x and ±y around the antisite V atom showing a very strong tendency to gain electrons according to Figure 4C.Similar to the situation of V C , there also exists a circular area with strong ability to gain electrons for four nearest V atoms, which are found to bond with the fourth nearest C atoms.The above-mentioned strong electron-gaining areas may result in the smaller atomic charge around defect V and its nearest V atoms in VC supercell.From Table 3, the atomic charge value −0.05 of substitutional C atom is much smaller than normal C atoms in the supercell showing very small tendency to gain electrons, which can be easily observed through the color difference between defect atom and normal C atoms in Figure 4D.Meanwhile, the nearest C atoms are mainly bonded with the second nearest V atoms after original V atom is substituted by C atom.
When V atom is added in the interstitial position, the interstitial V atom itself nearly shows no electron transfer (charge 0.01), which seems not consistent with electron density difference distribution in Figure 4E V-V bonds causes the tendency of sharing electrons.Maybe the combination of losing electrons in V-C bonds and sharing electrons in V-V bonds leads to unobvious electron transfer for interstitial V atom.The addition of interstitial C atom is the only condition that makes V atoms nearby lose more electrons than perfect structure among all intrinsic defects.This is because the nearest V atoms not only bond with the normal C atoms but also bond with the interstitial C atom (see Figure 4F).Figure 5 shows the electron density difference distribution for VC supercell with Mo-related defects along specific direction.When one Mo atom substitutes V atom in VC structure, it can be found in Figure 5A that Mo atom not only bonds with all nearest C atoms, which gain electrons with atomic charge of −0.6, but also bonds with the second nearest V atoms and shares electrons with them.So higher atomic charge of V atoms (0.63) results in the lower value of 0.23 for substituted Mo.After one C atom is substituted by Mo atom (see Figure 5B), Mo atom forms bond with the second nearest C atom with longer bond length than Mo-C bond in situation of S Mo-V , which indicates that the ability of Mo losing electrons is getting worse.Meanwhile, Mo atom shares electrons with the nearest V atoms showing the capability of gaining electrons to some extent.So Mo atom shows no obvious electron transfer (charge 0.03) in Table 3. From Figure 5C, electron density difference distribution for I Mo is similar to that of I V , but the atomic charges for interstitial atom of V (I V ) and Mo (I Mo ) in Table 3 are totally different.Interstitial Mo atom mainly bond with the nearest C atoms forming strong Mo-C bonds with bond length of 1.96nm, which is similar to the reaction of interstitial V atom and nearest C atoms in Figure 4E.However, Mo and V are both metallic atoms, but with different abilities of losing electrons, which make Mo atom exhibit the characteristic of gaining electrons.

| Defects complexes
When two defects exist in the crystal structure at the same time, the equation for binding energy of defect-defect interaction can be expressed as follows 40 : where E (N, defect 1) and E (N, defect 2) are the relaxed total energy of supercell containing one defect in the system; E (N, 2 defects) is the relaxed total energy of supercell includ- ing two defects at the same time.
After calculation according to Equation ( 15), the negative value represents the exothermic reaction between two defects indicating that these two defects repel each other, whereas the positive value shows the reaction of two defects is endothermic, which means the two defects attract each other.
It is well-known that vacancy-type defects are commonly seen in crystal structures, especially V C defect, which has ( 15) already been demonstrated in this paper.Moreover, vacancy-assisted diffusion is one of the most basic diffusion mechanisms, and alloy element atoms can leave their previous site and migrate to a nearby vacant lattice site to achieve diffusion inside the crystal.So in this section, only the combination of Mo-related defects and vacancy defect is considered and analyzed here, which can lay a theoretical foundation for the subsequent diffusion research in next section.For S Mo-V + V V and S Mo-V + V C , the atoms on the (001) plane where both substituted Mo atom and V/C vacancy locate do not change too much after structure optimization (see Figure 6A,B), and the volume expansion caused by Mo substitution with larger atomic radius and volume shrinkage caused by V/C vacancy both contribute to unobvious volume change of supercell with S Mo-V + V V and S Mo-V + V C defects in Table 4.As a metal atom, Mo occupies the place where another metal atom V locates in supercell making S Mo-V defect very stable, so the values of binding energy between S Mo-V and V V /V C are both very small showing the weak attraction between them.However, due to the existence of the most stable V C defect in the supercell, the formation energies for S Mo- V + V C is much smaller than those of S Mo-V + V V and almost the same with perfect structure.
For S Mo-C + V V , the whole optimization process consists of 34 steps, and Mo atom has completely entered into V vacancy at Step 13.Meanwhile, Mo atom can perfectly fit into the crystal lattice even undergoing obvious displacement but still doesn't affect the positions of other atoms in the supercell observed from (001) plane in Figure 6C.This indicates the S Mo-C + V V complex is not stable and does not exist in the lattice, so it is meaningless to calculate the corresponding binding energy between S Mo-C and V V .When S Mo-C and V C defects both exist in the structure (see Figure 6D), Mo atom moves from original substituted position to the interstitial position in the lattice resulting in the formation of two C vacancies existed simultaneously.After Mo atom enters into the interstitial position, the nearest V atoms around Mo move outward and the displacement of atoms in other positions has also changed significantly, which lead to the volume increase in structure.When both of I Mo and V V exist in the supercell (see Figure 7A), as the optimization proceeds, interstitial Mo atom moves in the direction of V V and almost completely enters into the vacancy at Step 15 forming a perfect crystal lattice without interstitial defect anymore.During this process, the neighboring atoms around Mo first deviate from the original lattice position, and as Mo atom enters the lattice they gradually move back to the original position again.Finally, no obvious displacement can be observed for each atom in the lattice.The situation of I Mo + V V is found similar to that of S Mo-C + V V , which means no such defect complex exists stably in VC structure, so no binding energy value was calculated and listed in Table 4.

Mo interstitial-Vacancy defects
If one of the nearest C around interstitial Mo was deleted instead of V atom, I Mo + V C defect can be formed shown in Figure 7B.During the optimization process, Mo atom moves toward C vacancy but never enters into the lattice completely, and the surrounding atoms around Mo have shifted significantly until the final step, especially the surrounding V atoms near Mo.After Step.16, nearly all atoms have no obvious change in position.Eventually, interstitial Mo atom locates near to C vacancy instead of staying in the center of tetrahedral interstitial position.Moreover, relatively small binding energy in Table 4 manifests a certain level of attraction between I Mo and V C , but the attraction is not strong enough to make the interstitial Mo totally enter into the lattice, which leads to the largest structure volume among all the defect complexes and larger formation energies indicating relatively unstable existence of I Mo + V C defect.

| Mo atom diffusion
In this section, two diffusion ways which are commonly seen in the study of atomic diffusion behavior are considered here: (a) Interstitial diffusion and (b) Vacancy-assisted diffusion.
When interstitial Mo defect exists in the crystal alone, Mo atom diffuses through the interstitial site.In this section, we only consider one interstitial Mo atom jumps into the nearest interstitial site.Figure 8A shows the diffusion path and obtained energy profile for Mo interstitial diffusion.The energy profile is in symmetry and saddle point locates at halfway position from initial interstitial site to the final interstitial site.The migration barrier is 0.68 eV, which represents the energy difference between the initial point and saddle point.Due to the lower migration barrier value obtained here, Mo atom is easier to move by interstitial diffusion.
When vacancy defect exists (V V or V C ) in the lattice, atoms can diffuse through the defect by occupying vacant site from their original position.From the analyses of defect complexes in Section 3.2, the following types of vacancy-assisted diffusion are unlikely to exist: (a) I Mo → V V , (b) I Mo → V C , (c) S Mo-V → V C , (d) S Mo-C → V V , and (e) S Mo-C → V C .So the situation of substitution Mo atom (S Mo-V ) moving into the nearest V V is calculated and shown in Figure 8B.The energy file for V V -assisted Mo diffusion is similar to that of interstitial Mo diffusion, which is symmetric as well.However, the energy barrier of 5.07 eV is much larger than the value in Figure 8A, which means that the interstitial Mo diffusion path is energetically more favorable than the V V -assisted one. 4| DISCUSSION Among all intrinsic defects in VC, V C has the smallest formation energy for bulk material and only negative value for formation energy of defect, which means that V C definitely exists in VC carbide, whereas all other point defects are not energetic favorable compared to perfect VC.In general, vacancy defect has the highest stability, then followed by interstitial and substitution defects, which are consistent with the results for ThC carbide with the same cubic structure, 25 but not with the stability trend in hexagonal WC. 22 Except for the substitution defects, C defects are accommodated much more easily than V defects for vacancy and interstitial defects, and many first-principles calculations for other TMCs have the exactly same conclusion with the results in this paper. 22,25,41t is already known that the formation energy for V V is relatively large, so its stability is not so high.When V V already exists in the crystal, free atoms have the tendency to enter the vacancy and fill the lattice as long as they can lower the formation energy and stabilize the structure.Free C atom is unlikely to fill V V position forming S C defect due to the higher formation energy than the structure only with V V .While free Mo atom has high probability to enter into V V position to form S Mo-V , because the formation energy for S Mo-V is significantly lower than that for V V , which indicates that V V in VC cell is easily occupied by free Mo atom.When extraneous Mo atom has already occupied V/C lattice or interstitial position, such  When V C is present in the supercell, due to its extremely high stability, neither the free V or Mo atoms nor the existing Mo atoms in the lattice can occupy V C position to form a more stable structure.So after considering the situation about S V , S Mo-C and S Mo-V + V C (S Mo-C + V V ), VC carbide is found to tend to always exist in the form of nonstoichiometric VC x (x < 1).In addition, the existence of V C can stabilize the Morelated defects (S Mo-V , S Mo-C , and I Mo ) in the lattice.It is found that the formation energies for structures with defect complexes S Mo-V + V C , S Mo-C + V C , and I Mo + V C are all smaller than those without V C .
For Mo-related defect complexes in VC system, the atom evolution during optimization process can be generally deduced according to the binding energy between two defects.Small binding energy (<1eV) shows the weak attraction between the defects, so Mo atom barely move in the lattice (S Mo-V + V V and S Mo-V + V C ).Large binding energy indicates that there is a certain attraction between two defects resulting in the displacement of Mo atom, so Mo atom tends to exist in the interstitial position (S Mo-C + V C and I Mo + V C ) after optimization.However, the binding energy values between defects discussed earlier are not fully matched with the formation energy values of defect complexes.Binding energy reveals the interaction between the defects and the trend of atom movement, whereas formation energy represents the stability of the final optimized structure.For example, both of the binding energies for S Mo-V + V V and S Mo-V + V C are small, which means the attraction between two defects in both of the complexes is not strong and the energy decrease in the system after the combination of two defects is not obvious.However, their formation energy values differ a lot from each other, and S Mo-V + V C is found more stable than S Mo-V + V V .
VC is able to resist high temperature, high mechanical stress and intense radiation no matter as ceramic material or as precipitates in other metal-based materials in its applications, and a variety of point defects is inevitably formed in VC crystals.Among the three intrinsic defects studied in this paper, vacancy defects have the best stability.Through the analysis of Mo atom in VC, vacancy can provide sites for the existence and diffusion of other metal atoms at high temperature.When other alloy elements enter in the lattice and gradually accumulate, the performance of the entire VC crystal will surely change.In addition, metal atoms can stabilize the whole structure if they occupy the V sites in VC, so microalloying treatment can be considered for VC ceramics in production process.However, once the compositions are formed, the treatments under extreme conditions such as high temperature need to be adopted to achieve the interstitial diffusion for metal atoms in VC.Moreover, there must be V C in VC crystal, but V C cannot bind metal atoms in the lattice positions stably.Once metal atoms exist in the interstitial position, large lattice distortion will be generated due to the size effect improving the performance of ceramic materials to some extent.

| CONCLUSION
1.The existence of V V leads to the shrinkage of the crystal structure, whereas V C causes slightly volume expansion after optimization.Substitution defects S V and S C mainly influences the nearest atom position D 1N in the lattice, and I V is found to have more obvious lattice change than I C .In addition, Mo-related defect S Mo-V will not affect lattice constants markedly, but S Mo-C will due to the similar character for Mo to the neighboring V atoms.The volume change for I Mo is the largest among all the supercells with defects.2. V C is the most stable defect that can exist in the structure and all other point defects are not energetic favorable compared to perfect VC.Moreover, C defects are accommodated much more easily than V defects for vacancy and interstitial defects.3.In general, Mo-related defects are less stable than intrinsic ones expect S Mo-V .Free Mo atoms have the tendency to occupy the lattice position of V atoms and exist more stable than already formed V V .4. Both vacancy defect and substitution defect weaken the electron-gaining/losing capacity of the nearest atoms.For substitution defect and interstitial defect, the defect atom gains/losses less electrons than normal atoms, among which atomic charge of C defect atom for S C is only −0.05 and the interstitial V atom for I V nearly shows no electron transfer.I C is the only condition that make V atoms nearby lose more electrons among all intrinsic defects.For Mo-related defects, substitution Mo atom exhibits much worse ability of losing electron for both S Mo-V and S Mo-C , whereas interstitial Mo atom even shows the characteristic of gaining electrons.5. Due to the stability of S Mo-V , the values of binding energy for S Mo-V + V V and S Mo-V + V C defect complexes are very small showing the weak attraction between S Mo-V and V V / V C .Large binding energy makes Mo atom in S Mo-C + V C and I Mo + V C tends to exist in the interstitial position after optimization.6.The possible diffusion path for Mo atom includes interstitial diffusion and V V -assisted diffusion.The energetically most favorable migration path for Mo atom is through the interstitial one.

F I G U R E 1
Schematic diagram of VC structure with (A) V vacancy (V V ), (B) C vacancy (V C ), (C)V substitution (Sv), (D) C substitution (S C ), (E) V interstitial (Iv), and (F) C interstitial (I C ). Green and gray balls represent V and C atoms; Green and gray dashed circles in (A) and (B) indicate V V and V C ; Green and gray semitransparent balls in (C) and (D) are Sv and Sc, whereas in (E) and (F) are Iv and Iv [Color figure can be viewed at wileyonlinelibrary.com] | 7229 GUO et al.

F I G U R E 2
Schematic diagram of VC structure with Mo-related defects: (A) Mo substitutes V (S Mo-V ), (B) Mo substitutes C (S Mo-C ), and (C) Mo interstitial (I Mo ). Green and gray balls represent V and C atoms; Purple semi-transparent balls in (A), (B), and (C) are S Mo-V , S Mo-C , and I Mo [Color figure can be viewed at wileyonlinelibrary.com] T A B L E 1 Distance (D 1N , D 2N , and D 3N , Å) from the atom/defect to the first, second, and third nearest atom in VC supercells with/without defects after optimization; V 0 and V (Å 3 ) are the volume of perfect VC supercell and supercells with defect; ∆V/V 0 (%) is the volume change between VC supercell with defect and perfect system, ∆V = V − V 0 where the subscript Mo-S, Mo-I in E def represent Mo substitu- tion and Mo interstitial defect; E(N, Mo) and E(N + 1, Mo) are the relaxed total energy of supercell including Mo substitution or Mo interstitial defect; E(Mo) is the energy per Mo atom in pure hexagonal Mo crystal.

E 3
Formation energies of intrinsic defects as function of the chemical potentials [Color figure can be viewed at wileyonlinelibrary.com]T A B L E 2 Formation energy (eV) of intrinsic defects and bulk material . Strong V-C bonds between interstitial V atom and nearest C atoms can be observed clearly.Additionally, a circular region near I V along F I G U R E 4 Electron density difference distribution for VC supercell with intrinsic defects of (A) V V , (B) V C , (C) Sv, and (D) S C along (100) plane, as well as (E) Iv and (F) I C along (110) plane [Color figure can be viewed at wileyonlinelibrary.com]

T A B L E 3 F I G U R E 5
Atomic charge (e) of defect atom and the first (1N), second (2N), and third (3N) nearest atoms in VC supercells with defects.In perfect VC structure, atomic charge for V and C atom is 0Electron density difference distribution for VC supercell with Mo-related defects of (A) S Mo-V and (B) S Mo-C along (100) plane, as well as (C) I Mo along (110) plane [Color figure can be viewed at wileyonlinelibrary.com]

3. 2 . 1 |
Mo substitution-Vacancy defects Mo substitution-Vacancy defects, which are presented by S Mo-V + V V , S Mo-V + V C , S Mo-C + V V , and S Mo-C + V C , can obtained by replacing one V or C atom by one Mo atom to form S Mo-V and S Mo-C defect first and deleting the nearest V or C atom of S Mo-V or S Mo-C defect subsequently.
Mo interstitial-Vacancy defects I Mo + V V and I Mo + V C can be formed by introducing one Mo atom into the tetrahedral F I G U R E 6 Structure model after optimization along (001) plane of VC supercell with (A) S Mo-V + V V , (B) S Mo-V + V C , (C) S Mo-C + V V , and (D) S Mo-C + V C .The green and gray balls represent V and C atoms, the dashed green and gray circles represent the original place where V V and V C locate, and the semitransparent purple balls represents the original place where substituted Mo atom locates [Color figure can be viewed at wileyonlinelibrary.com] | 7235 GUO et al. interstitial position and then removing the nearest V or C atom around interstitial Mo.

F I G U R E 7
Structure evolution of VC supercell with (A) I Mo + V V and (B) I Mo + V C during optimization process.The green, gray, and purple balls represent V, C, and Mo atoms; the green and gray arrow in the first graph (Step 1) of (A) and (B) indicate the place where V V and V C locate; the last graphs in (A) and (B) are observed along the direction indicated by the red arrows [Color figure can be viewed at wileyonlinelibrary.com] | 7237 GUO et al. as in the case of S Mo-V , S Mo-C , and I Mo , it is suggested from formation energy values that Mo atom in C lattice and interstitial Mo atom can enter the V V position stabilizing the lattice structure.This also indicates that S Mo-C + V V transforms into S Mo-V + V C automatically during optimization process, which has been certified by Figure 7B, and these two defect complexes have almost identical formation energy values.

F I G U R E 8
Calculated diffusion energy profiles of Mo atom diffusing along different diffusion ways in VC: (A) interstitial diffusion and (B) V V -assisted diffusion [Color figure can be viewed at wileyonlinelibrary.com]