Temperature dependence of phase stabilities of hexagonal hemicarbides from first principles

Transition metal hemicarbides, specifically M2C (where M = V, Nb, Ta, Mo and W), have received considerable attention in the fields of catalysis and metallurgy. However, the determination of the exact phase of each compound can still be challenging due to the close energetic proximity of its various polymorphs. This study uses first-principles calculations to carefully consider the subtle differences between different polymorphs, with temperature as a key factor. It is found that the energies of each polymorph of M2C are close, but can be distinguished when temperature effects are considered. This means that temperature plays an important role in the polymorph transformations. In addition, these phases exhibit dynamic stability at both zero and finite temperatures, in part due to the particular ordering of C atom occupancy in the metal lattice interstices. The study can provide calculations of a range of properties that help to identify stable structures and deepen the understanding of these materials in terms of chemical bonding, structural changes, lattice thermal vibrations and molecular dynamics.


Introduction
Hexagonal close-packed (hcp) transition metal carbides are mainly the Group V and VI transition metal carbides of the composition M 2 C. Though they exhibit physical features resembling those of ceramics, they possess electronic characteristics similar to those of metals [1,2].These properties make them useful in various fields.They are commonly used as precipitation phases in metallurgy [3][4][5], have significant applications in catalysis [6][7][8], and are expected to be utilized as topological superconductors [9].
Despite their importance in applications, their polymorphs, which refer to alternative crystal structures at fixed composition, are often misinterpreted.For instance, W 2 C is often considered to crystallize in the CdI 2 antitype structure (belonging to the P3̅ m1 space group), whereas its ordered phase should be the ε-Fe 2 N type (the space group P3̅ 1 m form) [10].Structures belonging to the space group P3̅ m1 are believed to be present in the hexagonal M 2 C phases.However, this polymorphic form is unambiguously observed in only one of them, i.e., in Ta 2 C [11].
The identification of polymorphic phases in experiments is usually achieved through methods such as neutron refinement and high-resolution electron microscopy observations, which require extensive and meticulous work.Computational approaches are used to predict the structure of the polymorphic forms by comparing their thermodynamic energy and/or dynamic stability [12].This method can conveniently and comprehensively evaluate the polymorphic forms of these materials, and thus minimize phase misinterpretation for technological applications.Additionally, computational considerations are important in the study of phase transitions.While experimental data on the precipitation reaction and direct synthesis of these phases have been extensively studied [1,2,13], a systematic computational evaluation is required to reveal the energetic aspects of their (trans)formation.
Computational methods can effectively distinguish between phases.However, for the different polymorphs of M 2 C, the energy separation between them is expected to be extremely small.This is because the polymorphs of hexagonal M 2 C are all based on an hcp metallic sublattice.The structural difference [1,2,13,14] only lies in the occupation of carbon atoms within the metallic sublattice, resulting in different non-metallic sublattice orderings.Due to the small energy differences between competing structures, a high level of accuracy and comprehensive consideration is required.In such cases, the first principles approach appears to be the only viable option.Additionally, synthesizing M 2 C often requires high temperatures.Therefore, it is crucial to understand the effects of temperature on the relative energies of different polymorphs.While ab initio studies typically do not take temperature into account, temperature-dependent free energies provide useful trends and lead to plausible agreement with experiments.They are therefore a key factor of interest in the prsent work.
The temperature-dependent energies of the solid materials aforementioned were calculated using the Quasi-Harmonic Approximation (QHA) method, which has been deemed reliable [12,15].This method enables the calculation of vibrational entropy, which is the primary factor contributing to the temperature dependence of free energy.This contribution can lower the energy of one polymorph relative to the others as a material is heated.Additionally, we examined the effects of electron excitation on the QHA phonon calculation, as the electronic thermal contribution of the Fermi energy level becomes important at high temperatures.This study also investigated the dynamic stabilities of the phases using Density Functional Perturbation Theory (DFPT) and Ab Initio Molecular Dynamics (AIMD).These estimations can help interpret experimental results and explore the relationship between structure and energy of the M 2 C phases.

Electronic-structure calculations
At zero temperature, the thermodynamic potential is equivalent to the total internal energy (U), which comprises the energy of a static lattice (E static ) and the Zero-Point Energy (ZPE).The E static terms can be determined by examining all competing phases in each relevant chemical space using open materials databases, here the Automatic FLOW (AFLOW) [16]; and the ZPE terms can be calculated by phonon calculations, see section 2.2.
When M 2 C is formed in its parent metal M, interstitial atoms occupy available interstices in the parent metal lattice, deforming these lattices.The calculation of the deformation energy (E def ) provides a qualitative measure for the strain in the structure of hemicarbide M 2x C x .This value indicates the degree to which the hexagonal hemicarbide structure deforms from an ideal body-centred cubic metal (M bcc ) structure [17].The value is normalized per carbon atom and can be determined using Here, E total ref denotes the total energy of the various hexagonal hemicarbide structures without any carbon atoms.The bcc M atom is used as a reference because it is the common structure of the transition metals involved.Note that the hemicarbide structures are derived from an hcp lattice.Therefore, the E def calculated here includes an energy contribution for the rearrangement of the bcc to the hcp lattice [17].
For each calculation case, the total energy convergence was confirmed using different k-point sampling numbers and cut-off energies.Brillouin zone integration was performed using a Monkhorst-Pack scheme.Electron-ionic core interactions were represented by a norm-conserving pseudopotential, and the plane-wave cut-off energy was set at 700 eV to ensure convergence of the total energies within 1 × 10 -7 eV per atom.All calculations were conducted using DFT and GGA-PBEsol [18] for the exchange-correlation functional, as implemented in CASTEP [19,20].

Phonon calculations
At finite temperature, the appropriate thermodynamic potential is the Gibbs energy, which can be calculated from the ground-state of the vibration.The QHA method can reasonably approximate the vibrational properties by assuming that lattice vibrations are harmonic in each volume and that phonon frequencies are functions of temperature only through their volume dependence.
To calculate vibrational thermodynamics from first principles, we can accurately separate the Helmholtz free energy (F) as where F vib (V, T) is the phonon contribution, and F elec (V, T) is the thermal free energy due to electronic excitations The first term represents the ZPE, while the second term denotes the vibrational energy.The Phonon Density of States (PDOS) at the given volume is represented by g(ω(V )).
F elec (V, T) is generated by exciting electrons from their ground states and can be expressed as The electronic entropy S elec (T) can be calculated based on the Electronic Density of States (EDOS) as ò e e e e e e = -+ -- Here, n elec (ε) indicates the EDOS, and f (ε where β is equal to 1/k B T and μ elec (T) represents the electron chemical potential at temperature T. This study only considers the zero-pressure case.Consequently, the Gibbs free energy (G) is equivalent to the Helmholtz free energy (F).
The phonon calculations were computed using the DFPT method.The adjustable parameters were optimized to achieve an error in the total energies of 1 × 10 −5 eV/atom.This led to a plane-wave cut-off of 700 eV and a k-point separation of at least 0.04 Å −1 .

AIMD calculations
To compare with the results of phonon calculations, it is required to simulate the dynamics at a given temperature to determine the power spectrum G(ω).The power spectrum can be obtained by calculating the Fourier transform of the velocity autocorrelation function Ψ(t), which is expressed as The Ψ(t) is defined as The atomic velocities v i (i = 1, K, N) are obtained from AIMD with N atoms per supercell.The dynamics simulation is also being used in further calculation of Mean Square Displacements (MSD).
The microcanonical ensemble was used to carry out AIMD simulations.Instead of wavefunction extrapolation schemes, the extended Lagrangian Born-Oppenheimer MD [21,22] was employed.An ultrasoft pseudopotential with a cut-off energy of 280 eV was utilized.The total energy convergence was set at 2 × 10 -5 eV per atom, and the k-point mesh only sampled the Γ point.The phases with supercells containing at least 72-atom underwent dynamic simulation for 2 ps, with a time step of 0.4 fs and temperatures of 300 K, 1000 K, and 2000 K.

Results and discussion
3.1.Ground-state energy 3.1.1.Total internal energy The thermodynamic understanding of the phase stability of transition metal hemicarbides is still limited compared to the most well-known transition metal monocarbides.Specifically, there has been no systematic comparison of the energy differences between these mostly hexagonal compounds with their ordered structures (or polymorphs).These ordered structures stem from the various forms of small C-atom occupation in the hcp interstices of the transition metals.This results in structures such as the CdI 2 antitype, ε-Fe 2 N type, ζ-Fe 2 N type, ξ-Nb 2 C type, and CaCl 2 antitype.The symmetries of these structures correspond to the five space groups P 3̅ m1, P 3̅ 1 m, Pbcn, Pnma, and Pnnm, respectively.
Figure 1 schematically displays the crystal structures of the P 3̅ m1 and P 3̅ 1 m types.The carbon atoms are arranged in an ordered fashion to fill the octahedral interstices between the close-packed planes {0001} hcp , which are perpendicular to the z-axis in figure 1, of the hcp structure formed by the transition metal atoms.Table 1 lists the identified polymorphs from the experiments [1,3,10,11,13] and summarizes how carbon atoms are filled in the ordered M 2 C polymorphs.
For the energy of a static lattice, since there is no entropy contribution at 0 K, the reaction energy is equivalent to the formation enthalpy, ΔH f .As seen in table 2, the ΔH f values of different polymorphs in each M 2 C are typically similar.Thus, a periodic table trend in the affinity of the transition metals for carbon can be observed from this.The affinity increases from right to left within each period, that is, from Mo to Nb and from W to Ta.However, there is no discernible pattern in the variation within a group, i.e., when moving vertically.
To obtain the total internal energy of a lattice at zero temperature, it is necessary to consider zero-point vibrational effects.Table 2 displays these thermodynamic results (ΔU), in which the polymorphic forms of Nb 2 C, Ta 2 C, and Mo 2 C agree with the experimentally determined stable phases.This means that for these compounds, the experimentally determined stable polymorphs are also the lowest energy ones in table 2.
For V 2 C, the lattice energies with space groups Pbcn, Pnma, and Pnnm are very similar, differing by only 1 meV.A first principles study also showed that the Pnnm type structure has the lowest formation energy among several polymorphic types of V 2 C [23].However, besides the V 2 C-Pbcn phase, the other two have not yet been reported experimentally.
The larger discrepancy between the calculated and experimental results comes from the W 2 C system.The ΔH f values of W 2 C were close to zero, after the zero-point energy correction, then positive.The formation of these tungsten hemicarbides seems to be thermodynamically unsupported.However, W 2 C-P 3̅ 1 m is often observed experimentally [10].
A possible explanation for these facts is that the chemical potential effect [24] has not been factored in.However, the composition of a set of polymorphs remains fixed, making the polymorph energies insensitive to changes in chemical potential [12].A comparison of the relevant values for carbides with different stoichiometry (refer to figure 2) illustrates this.Thus, the inconsistency in the results may be due to the lack of consideration of the entropic contribution.The internal energy contribution dominates the free energy at lower temperatures, while at higher temperatures, the entropy term may become more important.This will be discussed in section 3.2.2.

Deformation energy
As aforementioned, the relative stability of the different polymorphic structures is subtle.One way to account for small energy differences between competing phases is to calculate the deformation energy.In the hexagonal M 2 C, the carbon atoms are placed in available interstitial sites in a distorted hcp structure of metal atoms, while the pure metal phases often adopt bcc crystal structures.Thus, hemicarbide formation alters the metal crystal structure, and the deformation energy required for this change can be calculated.
Figure 3 illustrates the deformation energy with respect to a bcc M lattice in the different hemicarbide structures.The deformation energy values are significant, indicating that the rearrangement from bcc to hcp lattice is energetically challenging.In addition, the Pnma and Pnnm structures have relatively low deformation   energies, but they are not commonly found in hemicarbides [2].In contrast, the more common structures such as P 3̅ 1 m and Pbcn exhibit higher deformation energies.One possible explanation is that if the phase change occurs partially through lattice deformation (rather than a pure diffusional mechanism), it must be an interfacial reaction aided by a specific interfacial structure that can offer partial shear deformation.This implies that the specific interfacial structure is a stepped interface, and a moving step can produce shear.In this context, the shear on the atomic habit plane (i.e. the terrace within a stepped structure), which controls the 'pattern advance' of the structural step, is therefore significant.This context concerns the mechanism of the transformation rather than the reasons why a particular phase change occurs (as discussed in this study).For further details, please refer to [25].
3.2.Finite temperature effects 3.2.1.Electronic free energy At low temperatures, the zero-point vibrational effect plays a critical role, whereas at high temperatures, entropy takes over [26,27].Before discussing this major contribution, another effect on the free energy related to temperature-electron excitation-is presented.
Table 3 reveals that the Fermi energy (E F ) between the different ordered structures is almost the same, indicating that they have comparable chemical stability.This is in line with the previous analysis.Upon examining figure 4, it is evident that the electronic free energy, resulting from electron excitations, makes a significant contribution to the free energy differences for each M 2 C polymorph.These differences can be  understood since the electronic free energy is approximately proportional to the EDOS at the Fermi energy n elec (ε).(Assuming the free-electron gas model and not at high temperatures, the free energy can be approximated by the Sommerfeld expression.)That is, variations in the covalent bonding (or metallic bonding) component of each polymorph cause these differences in the free energy.Further, this electronic free energy contribution reflects the presence of notable metallic bonding features.
As is well-known for transition metal carbides, the chemical bonds are dominated by two terms: a covalent bond between the metal d and carbon p orbitals, and a metallic bond resulting from the metal's d electrons [1,2].According to the literature [14], the metallic bonding of the d electrons becomes increasingly important for transition metal hemicarbides compared to transition metal monocarbides.The increase in the number of metal atoms in the M 2 C compounds results in a greater contribution to the metallic bonding.As a result, the transition between bonding and anti-bonding states shifts toward the bonding behavior of pure metals in the periodic table.Experimental findings suggest that the M 2 C compounds are typically hemicarbides of Group V and VI transition metals, which can be explained by this bonding consideration.

Vibrational free energy
The calculation of the vibrational free energy involves integrating the constant volume specific heat from 0 K to T. This information can be obtained from the phonon density of states, which is calculated from first principles.This density-of-states represents the distribution of phonon frequencies and is responsible for the vibrational entropy in solids.
Figure 5 presents the vibrational entropy results for the ordered hexagonal M 2 C up to a temperature limit of 2000 K.The upper limit is due to the potential loss of accuracy of the current method at higher temperatures [26].The present calculations did not include the P6 3 /mmc M 2 C type with a disordered hexagonal structure for the comparison of competing phase stability.This is because the P6 3 /mmc polymorph stabilises in temperature ranges higher than the upper temperature limit of the figures, except in V 2 C. Based on figure 5, the polymorphic forms with the highest vibrational entropy in each compound are the Pbcn form in V 2 C, Ta 2 C, and Mo 2 C, and the P 3̅ 1 m form in W 2 C. The lower vibrational entropies in V 2 C, Ta 2 C, and Mo 2 C are all P 3̅ m1 forms.In addition, the vibrational entropies of several polymorphs in Nb 2 C are very close to each other.
Figure 6 displays the temperature-dependent free energies for the ordered hexagonal M 2 C. Interesting results were obtained for each hemicarbide.
In V 2 C, the polymorphic form Pbcn has a lower free energy.Though the formation enthalpies of the Pnma and Pnnm forms are lower than that of the Pbcn form at 0 K, experimental reports typically indicate the Pbcn form [13].This is due to the lattice vibrational contributions at finite temperature.
In Nb 2 C, the free energies of the P 3̅ 1 m, Pbcn, Pnma, and Pnnm phases are very similar.This explains why the structural forms of these phases are different in the literature, e.g. they are summarized as P 3̅ 1 m and Pbcn forms [3] and Pbcn and Pnma forms [13].The vibrational entropy of these structures is also close, so even at high temperatures, there may not be a significant energy difference between them in the case of polymorphic forms.
For Ta 2 C and Mo 2 C, the P 3̅ m1 type of the former and the Pbcn type of the latter have lower free energies in the temperature ranges.This is in agreement with the experimental confirmation [11,13] of both compounds.For W 2 C, the Pbcn form has an energetic advantage in the low to medium temperature range, while the P 3̅ 1 m form is comparable to the Pbcn one at high temperatures.Experimental reports often indicate mutual transitions in the ordering of both non-metallic sublattices [28].Though the lattice energy of the Pbcn form is lower than that of the P 3̅ 1 m one, the contributions of lattice vibrational entropy and electronic thermal effects at high temperatures effectively stabilise the P 3̅ 1 m form.It should be noted that the transition temperatures between the P 3̅ 1 m and Pbcn types cannot be precisely determined due to the subtlety of the energy changes involved.
The free energy calculations above demonstrate the current stability trends with temperature for all possible structures of this materials family.Additionally, they highlight the primary contribution of vibrational entropy to the temperature dependence of the free energy.This is illustrated for the P 3̅ 1 m form in W 2 C, and for the Pbcn form in V 2 C and in Mo 2 C. It is worth noting that the vibrational entropies of the Pbcn and P 3̅ 1 m forms of Mo 2 C are very similar, resulting in significant kinetic competition between the formation of the two carbides at high temperatures [27].However, Ta 2 C is an exception to this picture.The lattice vibrational entropy of the P 3̅ m1 form of the compound is low, but so is the free energy of this form.This is partially attributed to the fact that the P 3̅ m1 form has the lowest static lattice energy.These findings suggest that thermodynamic considerations must be multifactorial.
In addition to vibrational entropy, configurational entropy also contributes significantly to the free energies of crystalline solids.In some cases, both types of entropy can occur on the same scale [12].The face-centred cubic (fcc) monocarbides MC 1−x , which deviate significantly from the stoichiometric composition [1], are expected to have significant energy contributions due to configurational entropy.In contrast, the contribution of configurational free energy in hemicarbides of the V and VI groups is relatively low.Since they exist over a narrow homogeneity range of M 2 C 1−x .

Dynamic stability 3.3.1. Vibrational stability
When studying the stability of a polymorph, it is necessary to consider both its energetic and dynamic stability.The Phonon Density of States (PDOS) calculations include both perspectives.These calculations determine the vibrational entropy, which is the main factor contributing to energy reduction at finite temperatures.They can also indicate stability in terms of dynamic stability, which refers to energy-lowering collective displacements.
Phonons are quantized lattice waves with time-dependent amplitudes.Imaginary phonon energies indicate the unphysical growth of the wave amplitude over time, which is a signature of a dynamically unstable system [29].In a stabilised solid material, all vibrational normal modes should be stable with positive frequencies.The crystal structures in figure 7 are dynamically stable, as evidenced by the absence of imaginary phonon energies (negative frequencies) for any polymorphic form of each compound.
The PDOS calculations do not consider anharmonic effects, which are phonon interactions resulting from higher-order terms of the ion-ion interaction potential.Though introducing an anharmonic approximation [30][31][32] could further improve the description further, these effects are significant only when the materials are near their melting point [26].Furthermore, anharmonicity mainly affects the thermal conductivity about phonon transport theory, which is not related to this study.

Molecular dynamics stability
The AIMD simulation enables the examination of the dynamic stability of inorganic solids at finite temperatures.This simulation is relatively low-throughput.Only the phases listed in table 1 underwent dynamic simulation.Following the AIMD simulations, we utilized sample configurations at 10 random time-points as inputs for standard DFT geometry relaxations.The results indicate that all of these configurations converge to their initial space group forms.This conclusion is further supported by the dynamic stability of the crystal structures.
Two representative structures of these polymorphs, Ta 2 C-P 3̅ m1 and W 2 C-P 3̅ 1 m, were chosen to demonstrate the results of the molecular dynamics.The power spectrum and MSD results were analysed as indicators and are displayed in figures 8 and 9, respectively.The motions in the directions [11 2̅ 0] hcp , [11̅ 00] hcp , and [0001] hcp are represented by the components x, y, and z.The motion frequencies in the power spectrum at finite temperatures agree to some extent with the high-frequency phonon part of Ta 2 C-P 3̅ m1 in figure 7(c) and W 2 C-P 3̅ 1 m in figure 7(e) at zero temperature.It is worth noting that the linewidths in figure 8 with temperature are consistent with the broadening atomic distribution functions in the real space with temperature.This corresponds to the findings of the literature [33].
For the time-evolution of displacement, the rate of molecular motion decreases with decreasing temperature.For Ta 2 C-P 3̅ m1, the z-component is dominant for motion at all three temperatures; for W 2 C-P 3̅ 1 m, the x-and y-components dominate, with the z-component contributing little.This can be attributed to the distinction between the two structures (and their bonding), where for the P 3̅ m1 type (refer to figure 1(b)), the carbon atoms fill every other plane of the octahedral spaces between the close-packed layers; and for the P 3̅ 1 m type (refer to figure 1(a)) and the other three types (outlined in table 1), every layer of the octahedral interstices between the metallic layers is filled with an order.Despite the component differences, the common feature is that the apparent directions of the displacement time-evolution are those of the close-packed and near close-packed directions, as well as the direction of the close-packed plane.This demonstrates the contribution of the arrangement of both metal and non-metal atoms to the dynamic stability.That is, the stability of the hemicarbides is partially due to their uniquely ordered nature.
In general, all of these ordered structures have in the common feature that carbon atoms and interstices occupy alternate octahedral sites on interstitial rows parallel to the [0001] hcp direction.This filling rule ensures that no interstitial atoms are stacked when they are arranged into two octahedral interstitial planes.The alternate planes are usually either ' ' filled by small atoms, the last two of which are illustrated in figure 1.The dynamic stability of these filling modes is demonstrated in figure 7. The energetic advantage of adopting such structures for these hemicompounds is that the metal atoms maintain the close-packed form of an hcp, in which the octahedral interstices are symmetric and the small interstitial atoms fill only half of these interstices, i.e., the stoichiometry is represented as 2-to-1.Furthermore, the rule for interstitials filling implies that the nearest neighbouring interstitial positions of an interstitial atom along the close-packed plane are always empty.It is readily seen that interstitial filling yields the (probably) largest energy barrier-the distortion of the parent metal lattice-to be effectively relieved.As noted in figure 3, the deformation energies are close among the different polymorphs of each compound.This suggests that there is more than just geometric intrinsic consistency in the filling rule.This is further supported by the minimal energy discrepancies between polymorphs in the total energy at zero temperature (see table 2) and the free energy at finite temperature (see figure 6).
Overall, under these circumstances, the filling of interstitial atoms follows the same rule, while at the same time the polymorphic forms are diverse.As such, the hemicarbides can choose the kind of polymorphs they prefer.

Conclusions
The thermodynamic and dynamic of the ordered hexagonal transition metal hemicarbides have been calculated for all known polymorphic forms.The calculated results are consistent with experimental findings.The main conclusions can be drawn from the above investigations.
• Temperature effects play a dominant role in the phase stability of inorganic solids at higher temperatures.
Vibrational entropic stabilisation can affect metastable phases, especially at higher temperatures, as the role of entropy in controlling equilibrium increases linearly.However, at lower temperatures the high-temperature phases will not stabilise due to the critical factor of zero-point vibrational energy.This is in accordance with well-established thermodynamic principles.
• The ground-state energies and deformation energies of each polymorph of M 2 C are very similar, partially due to the rule of ordered interstitial-filling, which induces relatively small lattice distortions.The arrangement of both metal and non-metal atoms contributes to the molecular dynamics stability, it is therefore crucial to preserve their uniquely ordered feature when working on doping modifications of such materials.• The electronic free energy, resulting from electron excitations, makes a significant contribution to free energy differences for each M 2 C polymorph.This significant metallicity leads to a lowering of free energy and potentially higher catalytic activity.For these materials, however, covalent bonding plays a fundamental role in maintaining stability.Thus, it is necessary to balance these components in catalytic applications.
• If the transformation of M 2 C from its M matrix occurs partially through lattice deformation, it must be an interfacial reaction aided by a specific interfacial structure that can offer a relatively large shear on the atomic habit plane.This is because the calculations have shown that the energy required for deforming the lattice is usually large compared to the energy reduction of the thermodynamic aspects of the system.

Figure 1 .
Figure 1.The ordering of carbon atoms in the transition metal hcp sublattice, (a) P 3̅ 1 m and (b) P 3̅ m1 forms.The metal atoms are represented in grey and the carbon atoms in blue.
W 6 C 3[10], Nb 6 C 3[3] A sequence in which successive layers are by one-third and two-thirds filled Pbcn V 8 C 4[13], Nb 8 C 4[13], Mo 8 C 4[1] Interstitials are distributed in zig-zag chains parallel to the ortho-hexagonal b axis Pnma Nb 8 C 4[13] Interstitials pack most closely in the ortho-hexagonal a axis Pnnm -Alternating single strings of interstitials along the pseudo-hexagonal a axis

Figure 2 .
Figure 2. The enthalpies of formation as a function of the elemental chemical potentials in the Ta-C system.

Figure 3 .
Figure 3.The deformation energy of the ordered hexagonal M 2 C at zero temperature.

Figure 4 .
Figure 4. Temperature dependence of electronic free energy of the ordered hexagonal M 2 C.

Table 1 .
The structure features of the ordered hexagonal transition metal hemicarbides (M 2 C).

Table 2 .
The formation enthalpy (ΔH f ), zero-point energy (ZPE, for U at 0 K), and relative internal energy (ΔU) of the ordered hexagonal M 2 C in eV per atom.The structures observed in the experiments are represented by italicised values, with relevant references provided in table 1.

Table 3 .
The Fermi energy (E F ) and EDOS (n elec (ε)) at the Fermi level of the ordered hexagonal M 2 C.