A. Structural relationship between TMDs and TiC

In Fig. 1, we show the interface between TiC(111) and WS₂. TiC crystallizes in the Fm3m (rock salt type) structure, while WS₂ has P6₃/mmc structure.^{Reference Schutte, Boer and Jellinek38} The (111) surface of TiC has Ti atoms in a hexagonal arrangement, which match the hexagonal symmetry of sulfur atoms of the W–S plane in WS₂. At this interface, sulfur atoms are situated in hollow positions above three Ti atoms. This geometry allows for the formation of three Ti–S bonds, which is the most common bonding geometry in Ti–S compounds. To achieve a “perfect” fitting of the two surfaces, i.e., zero mismatch of lattice parameters at this interface, would require that TiC has a lattice parameter of 4.4593 Å (Table I), which is 3.0% larger than the experimental one (a _TiC = 4.328 Å^{Reference Martienssen and Warlimont39}). WSe₂ has the same structure as WS₂; therefore, the interface shown in Fig. 1 represents the optimal fitting for WSe₂, as well. However, for the interface between TiC(111) and WSe₂, a larger TiC lattice parameter is required for the optimal fitting (4.6414 Å, see Table I).

FIG. 1.

Schematic representation of the TiC(111)/WS₂ interface [(a) top view, (b) side view]. In panel (a), only W–S bonds are displayed for a clear representation. Here, C atoms lie under S atoms. Ti, C, W, and S atoms are displayed by blue, brown, gray, and yellow, respectively.

TABLE I.

Required lattice parameter of TiC at the TiC(111)/TMD interface, shown in Fig. 1, to achieve optimal fitting (see text). ∆a _TiC is the deviation from the experimental lattice parameter of pure TiC.

Fitting TiC(111) and WTe₂ is more complicated, since WTe₂ is of orthorhombic Pmn2₁ structure giving a puckered layer structure.^{Reference Mar, Jobic and Ibers40} However, a similar fit as the one shown in Fig. 1 is possible, and fitting TiC (111) to the a = 3.477 Å lattice parameter of WTe₂ would require TiC having a lattice parameter of 4.9172 Å, which is 13.6% larger than the experimental one (Table I).

We note that it is possible to fit TiC(100) and TMDs when the TMD is rotated by 45° with respect to TiC(100). This fitting would require the same lattice expansion of TiC as for the interface shown in Fig. 1. At this interface, however, different Ti-chalcogen bonds are formed. Some of the chalcogen atoms are situated in bridge positions above two Ti atoms, and some are in hollow positions above four Ti atoms. Such bonds occur in different Ti-chalcogenides. Nevertheless, some chalcogen atoms at this interface possess top positions above Ti atoms, which is an unfavorable geometry.

B. Formation and properties of TiC_1−xS_x, TiC_1−xSe_x, and TiC_1−xTe_x

In a previous study, we showed that it is possible to synthesize TiC with a large fraction of S atoms on the carbon sites using the magnetron sputtering technique.^{Reference Sundberg, Nyberg, Särhammar, Kádas, Wang, Eriksson, Nyberg, Jacobson and Jansson25} This technique allows a solid solubility of S above the thermodynamic solubility limit at equilibrium and leads therefore to the formation of a metastable TiC_1−xS_x phase. Since sulfur has a larger atomic radius than carbon, the solid solution causes a notable expansion of the lattice. The theoretical lattice parameters of this system, calculated with both supercell and coherent potential approximation (CPA), agree with the experimental lattice parameter of Ti–C–S.^{Reference Sundberg, Nyberg, Särhammar, Kádas, Wang, Eriksson, Nyberg, Jacobson and Jansson25} We note that in the experiments, the Ti–C–S phase also contained vacancies on the C sites, which is commonly observed in sputtered Ti-carbides. However, the effect of vacancies on the lattice parameter was negligible compared to that of S substitution. We obtained similar lattice parameters with the supercell and the CPA method; the difference being 0.1–0.9% for 0.0625 ≤ x ≤ 0.5. In the present study, we apply the supercell approximation only.

First, we determine how Se and Te substitution influences the lattice parameter of TiC and compare that to S substitution. Figure 2 shows the theoretical lattice parameters of TiC_1−xS_x, TiC_1−xSe_x, and TiC_1−xTe_x for x values between 0.0625 and 0.4375. As can be seen, the lattice parameter increases monotonically with increasing chalcogen concentration for all the three systems. We calculate the largest lattice parameters in TiC_1−xTe_x, followed by TiC_1−xSe_x and TiC_1−xS_x at each chalcogen concentration. At the highest chalcogen concentration, x = 0.4375, S addition leads to 6.6% lattice expansion compared to pure TiC, whereas Se increases the lattice parameter by 10.1%, and Te addition leads to 16.2% expansion. This is in line with the increasing covalent atomic radii of the chalcogens (S: 104 pm, Se: 117 pm, Te: 137 pm).^{Reference Martienssen and Warlimont39} We note that all the three chalcogens have larger atomic radius than carbon (77 pm). Furthermore, comparing the calculated lattice parameters with those required for optimal fitting of the TiC(111) surface with TMDs (Table I), we conclude that optimal fitting of chalcogen-containing TiC with TMDs is indeed possible. We obtain lattice parameters required for optimal fitting for x ≈ 0.2 in TiC_1−xS_x, x ≈ 0.3 in TiC_1−xSe_x, and x ≈ 0.35 in TiC_1−xTe_x. In the whole chalcogen concentration range examined in our study, the maximal mismatch of the lattice parameters is 3.7% for TiC_1−xS_x, 5.1% for TiC_1−xSe_x, and 9.4% for TiC_1−xTe_x.

FIG. 2.

Calculated equilibrium lattice parameters in TiC_1−xS_x (black line), TiC_1−xSe_x (red line), and TiC_1−xTe_x (green line).

Substitution of chalcogens for carbon atoms in TiC leads to decreasing bulk moduli, B (Fig. 3) with increasing chalcogen concentration. The largest chalcogen, Te, causes the largest change in B, and it is followed by Se and S. At the highest chalcogen concentration (x = 0.4375), S, Se, and Te addition decreases the bulk modulus by 24.9, 34.6, and 48.1% compared to pure TiC, respectively. Introduction of larger chalcogen atoms and higher chalcogen concentrations consistently leads to a reduced bulk modulus in TiC. As the bulk modulus gives information about the average bond strength of materials, the observed trend in B also means that S, Se, and Te decrease the average bond strength in chalcogen-containing TiC, the largest decrease caused by Te. We note that in TiC_1−xS_x, a similar decreasing trend can be observed in the measured Young's modulus (not shown) with increasing sulfur content.

FIG. 3.

Calculated equilibrium bulk moduli in TiC_1−xS_x (black line), TiC_1−xSe_x (red line), and TiC_1−xTe_x (green line).

In the following, we examine the electronic structure of TiC_1−xS_x, TiC_1−xSe_x, and TiC_1−xTe_x and compare it to that of pure TiC. In Fig. 4, we show the most important partial electronic density of state (DOS) curves, i.e., Ti d, C s and p, and the chalcogen s and p states. In TiC, there is a pseudogap at the Fermi level, E _F [Fig. 4(a)]. States below this pseudogap are bonding, whereas those above it are of antibonding character.^{Reference Jansson, Lewin, Rasander, Eriksson, Andre and Wiklund41} In TiC, the bonding states are filled, and the antibonding states are empty. Figure 4(a) shows that the DOS below E _F is dominated by Ti d and C p orbitals, and these states correspond to the strong TiC bonds. The empty antibonding states above E _F are built up by Ti d orbitals that hybridize with C 2p states. In TiC_1−xS_x, TiC_1−xSe_x, and in TiC_1−xTe_x, however, the Fermi level lies above the pseudogap, i.e., some of the antibonding states are filled (see Figs. 4b–4d). In chalcogen-containing TiC, Ti d and C p orbitals corresponding to Ti–C bonds give about equal contributions to the states right below the pseudo gap. Mainly chalcogen p orbitals build up the DOS feature between −8 and −4 eV, with a smaller contribution from Ti d orbitals. These states correspond to the Ti-chalcogen (Ti–S, Ti–Se, Ti–Te) bonds, which are weaker than the Ti–C bonds. This feature moves slightly to higher energies with increasing atomic number of the chalcogen, i.e., from TiC_1−xS_x to TiC_1−xTe_x (see Figs. 4b–4d), which correlates with the Ti-chalcogen bond strength decreasing in this order. In all the three TiC_1−xA_x systems, there is a peak at the Fermi level, which is an indication of a possible instability. This peak, and accordingly the DOS at E _F, increases from TiC_1−xS_x to TiC_1−xTe_x. Experimentally, it was shown previously that Ti–C–S nanocomposite coatings can be synthesized by means of reactive sputtering,^{Reference Sundberg, Nyberg, Särhammar, Kádas, Wang, Eriksson, Nyberg, Jacobson and Jansson25} and it is known that the conditions during magnetron sputtering are far from equilibrium. It is also known that metastable solid solutions are easily obtained in TiC.^{Reference Jansson, Lewin, Rasander, Eriksson, Andre and Wiklund41} Therefore, the possible instability due to S-, Se-, and Te-addition is likely not to hinder the synthesis of Ti–C–S, Ti–C–Se, and Ti–C–Te coatings.

FIG. 4.

Electronic DOSs calculated for TiC (a), TiC_1−xS_x (b), TiC_1−xSe_x (c), and TiC_1−xTe_x (d), for x = 0.4375. Ti d, C s, C p, chalcogen s, and chalcogen p DOSs are displayed by black, green, red, cyan, and blue, respectively.

C. Reaction of TiC_1−xS_x, TiC_1−xSe_x, and TiC_1−xTe_x with W and a-C

In our previous study, we showed that the reason for the observed low friction of the Ti–C–S nanocomposite coatings against W-coated steel is the formation of WS₂ in the tribological contact.^{Reference Sundberg, Nyberg, Särhammar, Kádas, Wang, Eriksson, Nyberg, Jacobson and Jansson25} We speculate that a similar reaction can occur in Se- or Te-containing TiC. To examine how different chalcogens influence the tribochemical formation of W-dichalcogenides, we investigate the stability of TiC_1–xS_x, TiC_1–xSe_x, and TiC_1–xTe_x in contact with tungsten. To this end, we calculate the formation energies (ΔE) of the following reactions:

(1)$${\rm{Ti}}{{\rm{C}}_{1 - x}}{{\rm{S}}_x} + {x \over 2}{\rm{W}} \to {\rm{Ti}}{{\rm{C}}_{1 - x}} + {x \over 2}{\rm{W}}{{\rm{S}}_2}\quad ,$$

(2)$${\rm{Ti}}{{\rm{C}}_{1 - x}}{\rm{S}}{{\rm{e}}_x} + {x \over 2}{\rm{W}} \to {\rm{Ti}}{{\rm{C}}_{1 - x}} + {x \over 2}{\rm{WS}}{{\rm{e}}_2}\quad ,$$

(3)$${\rm{Ti}}{{\rm{C}}_{1 - x}}{\rm{T}}{{\rm{e}}_x} + {x \over 2}{\rm{W}} \to {\rm{Ti}}{{\rm{C}}_{1 - x}} + {x \over 2}{\rm{WT}}{{\rm{e}}_2}\quad ,$$

where W is of bcc structure, WS₂ and WSe₂ are of P6₃/mmc structure,^{Reference Schutte, Boer and Jellinek38} WTe₂ is of Pmn2₁ structure.^{Reference Mar, Jobic and Ibers40} A negative formation energy means that the right-hand sides of Eqs. (1)–(3) represent stable phases.

For reaction (1), negative formation energies are obtained in the x = 0.0625–0.4 range (black curve in Fig. 5), i.e., formation of WS₂ is energetically favorable. Above x ≈ 0.4, ΔE becomes positive, since the formation of TiC_x, which has more than 40% vacancies on the carbon site, becomes increasingly energetically unfavorable with increasing vacancy content. We know, however, from experiments, that the tribochemical reaction described in Eq. (1) takes place even for these high chalcogen concentrations.^{Reference Sundberg, Nyberg, Särhammar, Kádas, Wang, Eriksson, Nyberg, Jacobson and Jansson25} It is therefore possible and even likely that not all sulfur in TiC_1−xS_x form WS₂ at high S contents. Considering this scenario, we analyzed the following reaction:

(4)$${\rm{Ti}}{{\rm{C}}_{1 - x}}{{\rm{S}}_x} + {y \over 2}{\rm{W}} \to {\rm{Ti}}{{\rm{C}}_{1 - x}}{{\rm{S}}_{x - y}} + {y \over 2}{\rm{W}}{{\rm{S}}_2}\quad ,$$

and we found that the formation energies are negative even for high sulfur concentrations. For y = 0.125, we calculate ΔE = −0.024 eV/f.u. at x = 0.40625, and ΔE = −0.003 eV/f.u. at x = 0.4375. This shows that at higher S concentrations, only a part of sulfur can form WS₂.

FIG. 5.

Formation energies calculated for Eqs. (1) (black line), (2) (red line), and (3) (green line) at different S, Se, and Te contents. The blue and magenta triangles show formation energies calculated for Eq. (5) (see text).

In the above discussion, we did not take into account that in the experimentally synthesized coatings, the TiC_1−xS_x phase is embedded in an amorphous carbon matrix. Therefore, in the following, we discuss the role of the surrounding a-C matrix. With this aim, we propose that upon the tribochemical reaction, carbon atoms of the matrix enter into the carbide and occupy vacancy sites in Ti–C–S. To investigate the possibility of this scenario, we calculate formation energies for the following reaction:

(5)$${\rm{Ti}}{{\rm{C}}_{1 - x}}{{\rm{S}}_x} + {x \over 2}{\rm{W}} + y{\rm{C}} \to {\rm{Ti}}{{\rm{C}}_{1 - x + y}} + {x \over 2}{\rm{W}}{{\rm{S}}_2}\quad ,$$

where 1 − x + y ≤ 1. Here, the energy of the a-C matrix is approximated by that of graphite. For y = 0.34375, i.e., when almost all the vacancy sites are “filled up” by carbon atoms from the a-C matrix, 1 − x + y being 0.9375, we calculate ΔE = −0.564 eV/f.u. as the formation energy for reaction (5) (see the blue triangle in Fig. 5). This energy is much lower than any of the formation energies obtained for reaction (1), where the a-C matrix was not considered (black curve in Fig. 5). This indicates that an amorphous carbon matrix can strongly stabilize the reaction of TiC_1−xS_x with tungsten. This argument finds further support from the fact that we obtain negative formation energy for the reaction in Eq. (5), ΔE = −0.195 eV/f.u., even for y = 0.03125, when only ≈3% of the vacancy sites are occupied by C atoms from the a-C matrix (see the magenta triangle in Fig. 5). Consequently, the amorphous matrix can support the formation of WS₂ by providing C atoms, which occupy the vacancy sites left in the Ti–C–S phase after sulfur diffusion. In this way, all the sulfur can form WS₂, even at high sulfur concentrations.

In the following, we discuss the reaction of TiC_1−xSe_x with tungsten [Eq. (2)]. For this reaction, we obtain negative formation energies in the whole chalcogen concentration range (red curve in Fig. 5). The formation energies are found to be substantially lower than those calculated for Ti–C–S (black curve in Fig. 5), the difference being 0.601 eV/f.u. at x = 0.3125. This means that the reaction of TiC_1−xSe_x is energetically much more favorable than that of TiC_1−xS_x. Analyzing reactions (1) and (2), we establish that TiC_1−xSe_x has higher energy than TiC_1−xS_x at each y chalcogen concentration, which favors the reaction to form WSe₂. However, WS₂ has lower energy than WSe₂, which favors the reaction of TiC_1−xS_x. The energy difference of the two carbides with chalcogens, E(TiC_1−xSe_x) − E(TiC_1−xS_x), increases with increasing y, and the energy difference of the tungsten dichalcogenides [y/2 E(WSe₂) − E(WS₂)] cannot compensate it; therefore, the formation energy of reaction (2) is lower than that of reaction (1) for each chalcogen concentration. The formation of WSe₂ is so favorable energetically, that all the Se of TiC_1−xSe_x can form WSe₂, even at high chalcogen concentrations. Furthermore, the reaction can take place without the a-C matrix providing C atoms to fill up the vacancy sites in the carbide phase.

We calculate large negative formation energies for the reaction of TiC_1−xTe_x with tungsten described in reaction (3), as seen in the green curve in Fig. 5. This reaction is the most favorable energetically to such an extent that all the Te can form WTe₂ in the whole chalcogen concentration range, and the reaction can take place without the support from the surrounding a-C matrix. Comparing the three chalcogens in TiC considered in the present study, we find that the formation of WTe₂ is the most favorable, followed by WSe₂ and WS₂. This can be explained by their atomic size: the larger the chalcogen is compared to carbon, the more unfavorable a solid solution on the carbon site is, and consequently, the more favorable the TMD formation becomes.^{Reference Hugosson, Eriksson, Jansson and Abrikosov42}

D. Suggestion for experimental studies

Our calculations clearly show that solid solutions of TiC_1−xA_x (A = S, Se, Te) are suitable as “precursors” for TMD growth. This was also demonstrated experimentally for WS₂ in Ref. Reference Sundberg, Nyberg, Särhammar, Kádas, Wang, Eriksson, Nyberg, Jacobson and Jansson25. From a thermodynamic point of view, the solid solubility of the chalcogenides on the carbon sites is low (at equilibrium conditions), and all compositions calculated in this study can therefore be considered as metastable solid solutions. Magnetron sputtering at reduced temperatures is known to produce extremely high quenching rates (>10⁶ K/s), which favors the formation of such metastable solid solutions.^{Reference Voevodin, O’Neill, Prasad and Zabinski43} Our calculations show that Te (with the largest atomic radius) forms the energetically most unfavorable solid solution followed by Se and S. We suggest, however, that solid solutions of TiC_1−xSe_x as well as TiC_1−xTe_x can be deposited by magnetron sputtering, although the maximum solubility of the chalcogenides is expected to be lower in these compounds compared to TiC_1−xS_x. Furthermore, to be an acceptable source for the formation of 2D TMDs, an epitaxial growth of the TiC_1−xA_x film is required. This was demonstrated for pure TiC and solid solutions thereof, e.g., MgO(111).^{Reference Palmquist, Birch and Jansson44} We suggest that similar epitaxial films of TiC_1−xA_x can be deposited by magnetron sputtering, although we predict that the quality of the films will be reduced with increasing amount of chalcogens. To form a TMD, tungsten then has to be added in a controlled manner. We suggest that this can be achieved by e-beam evaporation or short pulses of sputtering. During a subsequent annealing process, the chalcogen will then diffuse out from the TiC_1−xA_x phase forming a TMD. If the amount of W is limited, a 2D film of the TMD should be possible to form.

The formation of 2D TMDs described above is certainly experimentally possible, albeit a challenge. From a technological point of view, however, a more obvious application of our results is the use of TiC_1−xA_x materials in low-friction coatings. This concept was already demonstrated for Ti–C–S films in Ref. Reference Sundberg, Nyberg, Särhammar, Kádas, Wang, Eriksson, Nyberg, Jacobson and Jansson25. Our calculations suggest that Ti–C–Se and Ti–C–Te films should exhibit similar and possible superior low-friction properties.