Microscopic origin of MXenes derived from layered MAX phases

Abstract

Two-dimensional transition metal carbides/nitrides M_n+1X_ns labeled as MXenes derived from layered transition metal carbides/nitrides referred to as MAX phases attract increasing interest due to their promising applications as Li-ion battery anodes, hybrid electro-chemical capacitors and electronic devices. To predict the possibility of forming various MXenes, it is necessary to have a full understanding of the chemical bonding and mechanical properties of MAX phases. In this work, we investigated the chemical bonding changes of MAX phases in response to tensile and shear stresses by ab initio calculations using M₂AlC (M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W) as examples. Our results show that the M₂C layer is likely to separate from the Al layer during the tensile deformation, where the failure of M₂AlC is characterized by an abrupt stretch of the M–Al bonds. While under shear deformation, the M₂C and Al layers slip significantly relative to each other on the (0001) basal planes. It is found that the ideal strengths of M₂AlC are determined by the weak coupling of the M₂C and Al layers, closely related to the valence-electron concentration. Our results unravel the possibility as well as the microscopic mechanism of the fabrication of MXenes through mechanical exfoliation from MAX phases.

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1. Introduction

Recently, a new family of two-dimensional (2-D) transition metal carbides/nitrides labeled as MXenes has attracted growing interest due to their great potential applications as Li-ion battery (LIB) anodes, hybrid electro-chemical capacitors and electronic devices.^1–9 More information about MXenes is referred to in review articles published very recently.^10,11 The MXenes are produced by selective chemical etching of the “A” metal from MAX phases, which are a family of layered hexagonal transition metal carbides/nitrides. On the other hand, an attempt to mechanically exfoliate MXenes from MAX phases has also been tried, which however, led to multilayer structures.¹² The MAX phases having a formula of M_n+1AX_n, where M is an early transition metal, A is a group IIIA or IVA element, X is either C and/or N, and n = 1–3, have the combined properties of both metals and ceramics,^13,14 such as unusual high stiffness, plastic deformability, good electrical and thermal conductivity. So far, more than 70 MAX phases are known to exist, among which there are roughly 50 M₂AX phases (space group P63/mmc, prototype Cr₂AlC).¹³ The structure of M₂AX can be described as M₂X layers interleaving with A layers.^15–19 Therefore, it is possible to form many types of MXenes derived from the M₂AX phases through an elegant exfoliation approach. However, up to now, only four M₂C-stoichiometry MXenes (Ti₂C, Nb₂C, V₂C and (Ti_0.5Nb_0.5)₂C) have been synthesized.¹⁰ Furthermore, the microscopic origin of exfoliating MXenes from the MAX phases is not clear, which is important for predicting the possibility of forming various MXenes. In this work, to provide an insight into the microscopic mechanism of forming MXenes, we have extensively investigated the mechanical property and the chemical bonding changes of MAX phases in response to tensile and shear strains by ab initio calculations using M₂AlC (M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W) as examples.

Many theoretical works have been done focusing on the mechanical properties of M₂AlC compounds in the past decades.^{15,16,20–29} In our previous studies, we have investigated the elastic properties of nanolayered M₂AlC/M₂AlN,^15,16,25,26 with M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W, by ab initio calculations. We have also classified the M₂AlC phases into two groups by whether the bulk modulus of binary MC is conserved or changed after the incorporation of Al,¹⁶ which can be understood in terms of the coupling between the MC and Al layers. On the other hand, the M₂AlC phases (M = Sc, Y, La, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W) could also be classified into two groups based on the valence-electron concentration induced changes in C₄₄.²² On the other hand, the elastic properties of MXenes have also been calculated recently.³⁰ However, these works are mainly focused on elastic properties, which are the secondary derivative of the total energy with respect to a small strain. To gain a deeper understanding of the micro mechanism for the exfoliation of M₂C from M₂AlC that is achieved by the cleavage of M₂C and Al layers, it is essential to investigate the mechanical properties under deformation, i.e. complete stress–strain relations, from which the first maximum stress is ideal strength.^31–33 Ideal strength is the first derivative of total energy with respect to the critical point where instability occurs under deformation, which can be considered as the lowest stress required for cleavage or fracture of perfect crystal.^34,35 Furthermore, the macroscopic mechanical properties can be interpreted by studying the atomic displacement and the chemical bonding changes at the microscopic scale when a large strain is applied. In this paper, we have calculated the stress–strain curves of M₂AlC with M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W at a large strain and investigated the deformation mechanism by ab initio calculations. We have also discussed the relationship between the valence-electron concentration of M and the ideal strength of M₂AlC.

2. Theoretical methods

Our calculations are based on Density Functional Theory (DFT) in conjunction with projector augmented wave (PAW) potentials, as implemented in the Vienna ab initio simulation package.³⁶ For the exchange-correlation functional the generalized gradient approximations (GGA)³⁷ of Perdew–Burke–Ernzerhof (PBE)³⁸ was employed. The PAW potentials with the semi-core electrons treated as valence states were used: 3p3d4s for Ti, V and Cr; 4p4d5s for Zr, Nb and Mo; 5p5d6s for Hf, Ta and W; 3s3p for Al; 2s2p for C. The coulomb effect (U) for “d” electrons of the transition metals has been tested to be negligible, and hence it is not considered in the present work. The tetrahedron method with Blöchl corrections³⁹ was used for cohesive energy calculations. The convergence with respect to self-consistent iterations was achieved when the total energy difference between cycles was less than 10⁻⁵ eV. The k-points of 9 × 9 × 9 and 11 × 11 × 11 automatically generated with Monkhorst–Pack scheme⁴⁰ were used for structure optimization and static self-consistent calculations for the M₂AlC unit cell, respectively. While for the 3 × 3 × 1 supercell with 72 atoms, the k-point mesh of 2 × 2 × 2 and 4 × 4 × 4 were employed for structure relaxation and static self-consistent calculation, respectively. The cut-off energy was set as 500 eV. The chosen of these computational parameters are based on the convergence tests for the total energy of the systems. The electron localization function (ELF)⁴¹ was analyzed by using the VESTA code.⁴²

For the layered hexagonal structure of M₂AlC phases, the [0001] direction has the lowest strength,^13,16 therefore tension along the [0001] direction of the unit cell was used to investigate the interlayer interactions and the overall strength of the compounds. For layered hexagonal ternary compounds, the activated slip systems were quite limited on the (0001) basal-plane. Furthermore, perfect dislocations on the (0001) basal planes were detected by transmission electron microscopy after a deformation at room temperature.^43,44 To understand the micro-scale shear deformation mechanism of M₂AlC, shearing along the (0001)[12̄10] direction was studied. Supercell with 72 atoms was employed in the shear deformation, which has been tested large enough to avoid the periodic limit. To ensure that the material is under uniaxial tension or pure shear, the lattice vectors and internal atomic positions were simultaneously relaxed using a conjugate-gradient scheme until the Hellmann–Feynman stress tensor components orthogonal to the pre-set strain are less than 0.2 GPa.^33,45 Then, the fully relaxed structures under deformation were used to calculate the variations of bonds length, lattice parameter, cohesive energy, DOS and ELF versus strains in the present work.

Finally, it is worth to address the van der Waals effect on the interlayers interactions during tension along the [0001] direction. For this purpose, we calculated the stress–strain curves for Ti₂AlC and Nb₂AlC by the DFT-D2 method which includes the van der Waals corrections.⁴⁶ The results show that the van der Waals effect only slightly increases the absolute value of the ideal tensile stress for both Ti₂AlC and Nb₂AlC, but it does not change the relative trend of the ideal strengths and hence should not have any influence on the conclusion. Therefore, we do not include the van der Waals correction in the present results.

3. Results and discussion

To investigate interlayer interactions of the M₂AlC compounds, a tensile strain along the [0001] direction and a shear deformation along the (0001)[12̄10] direction were performed as schematically illustrated in Fig. 1.

Fig. 1 The schematic view of M₂AlC under (a) the [0001] tension strain, (b) no-strain and (c) the (0001)[12̄10] shear strain.

Fig. 2a shows the stress–strain relations for M₂AlC under uniaxial tension along the [0001] direction. For all the considered compounds, the stress increases monotonically with increasing the applied strain until reaching a critical point, after which the stress starts to fall. The critical point at the strain–stress curve of M₂AlC corresponds to a strain of larger than 20%, indicating good resistance to the tensile deformation, which is in good agreement with the experimental results.¹³ Since all these nine M₂AlC compounds considered here show similar stress–strain curves, we use Nb₂AlC as the model system to investigate the deformation mechanism under tensile strains. Fig. 2b displays the lattice parameters (c and a) and bond lengths versus the applied strain for Nb₂AlC. As the tensile strain was applied along the [0001] direction, c increases monotonically with strain, while the lattice parameter a decreases slightly. On the other hand, from the initial state to the critical tensile strain, the change in the Nb–Al length with strain is relatively slow and the inter-layer Nb–Al bonds are stretched without breaking, resulting in a monotonic increase of the stress. While after the critical tensile strain, the slope of the curve is larger compared to that before the critical point, indicating that the change in the Nb–Al length with strain is faster and the Nb–Al bonds are finally stretched to be unstable, resulting in a monotonic decrease of the tensile stress. Finally, it is obviously seen that the uniaxial tension along the [0001] direction predominantly elongates the inter-layer Nb–Al bonds, while the stronger Nb–C bonds remain almost unchanged. Fig. 2b reveals variation of the cohesive energy of the relaxed unit cell under various tensile strains. The monotonically increase in cohesive energy with the applied strain is consistent with the gradual stretching of the Nb–Al bonds. It also indicates that extra energy is required to elongate or compress a chemical bond from its initial stable state.

Fig. 2 (a) The calculated stress–strain curves of M₂AlC, (b) the variations of lattice parameter (a, c) and the bond lengths (Nb–Al, Nb–C) of Nb₂AlC, (c) the variations of cohesive energy of Nb₂AlC under tensile strain along the [0001] direction. The lattice parameter and bond lengths are normalized by the equation of V_strain/V₀, where V₀ and V_strain represent the values in structures under zero and various strains, respectively. The positions of the labeled atoms (Nb₂) can be found in Fig. 1.

Further analysis on the electron localization functions (ELF)⁴¹ visualizes the chemical bonding changes as a series of tensile strains were applied. The topological analysis of ELF is a very useful tool for the determination of chemical bonding strength. To gain a directly observation of the bond variation and atom distortion during the tensile deformation, Fig. 3 displays the ELF contour plots projected on the (112̄0) planes for Nb₂AlC at various tensile strains (0, 10%, 20%, 28%, 35% and 49%). It is clearly seen in Fig. 3 that with increasing the engineering tensile strain, the Nb–C bond strength remains unchanged, while the Nb–Al bonds are weakened significantly, resulting in the electrons being localized at around Al atoms. All the other eight M₂AlC compounds display similar characters during applying the tensile strain. Given the analysis above, the deformation mechanism of M₂AlC under the uniaxial tensile strain along the [0001] direction can be concluded as the stretch and breaking of M–Al bonds, and finally the compounds are separated into Al and M₂C layers. Meanwhile, though the anisotropy of chemical bonding of MAX phase has been commonly accepted, the present results directly verify the assertion and simulate the bonds variations under deformation with detailed calculations. Therefore, the ideal tensile strengths of M₂AlC are determined by the coupling strength of M₂C and Al layers. In other words, two dimensional M₂Cs could be synthesized through mechanical exfoliation, if the external applied tensile stress could exceed the ideal tensile strength of M₂AlC.

Fig. 3 The ELF contour plots projected on the (112̄0) planes of Nb₂AlC under the [0001] tensile strains of 0, 10%, 20%, 28%, 35% and 49%.

Fig. 4 displays the total and projected densities of states (DOS) for the strain-free Nb₂AlC and that under tensile strains of 10%, 28% and 49%. As seen in Fig. 4, for unstrained Nb₂AlC, the Al 3p and Nb 4d states, as well as the C 2p and Nb 4d states are hybridized at the energies of ∼−1.5 eV and ∼−4 eV, respectively. With increasing the tensile strain, it is found that the Al 3p and Nb 4d states are significantly shifted towards the Fermi level and the quantitative number of total DOS at Fermi level correspondingly increases in Table 1, indicating the instability of the stretched Nb–Al bonds. Meanwhile, the hybridization of the Al 3p and Nb 4d states are much weaker compared with the initial states, suggesting that the Nb–Al bonds are softened and can finally be broken by tensile strain. However, the Nb–C bonds do not show any reduction, thus the Nb–C bonds remain stable under all the investigated tensile strains. These conclusions on bonding picture are in good agreement with the above analysis on the basis of ELF results.

Fig. 4 The calculated total and partial density of states for Nb₂AlC under the [0001] tensile strains of (a) 0, (b) 10%, (c) 28% and (d) 49%.

Table 1 The quantitative numbers of total density of states at the Fermi level for Nb₂AlC under the [0001] tensile strains of 0, 10%, 28% and 49%

Tensile strain	0	10%	28%	49%
Numbers of TDOS at EF	3.64	4.41	4.78	5.68

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In case of shear deformation of M₂AlC, the stress–strain relations under pure shear strain along the (0001)[12̄10] direction is illustrated in Fig. 5a. For all the 9 investigated compounds, the stresses reach the peak position of the curve at a shear strain of around 12%, and then decrease to zero when the strain is about 23%. Afterwards, the stresses go through a negative minimum and reach to about zero again at a strain of around 49%. To gain a full understanding of these interesting stress–strain curves of M₂AlC along the (0001)[12̄10] direction, we have plotted the bond length change of Nb₂AlC under various shear strains in Fig. 5b. It is seen that when Nb₂AlC is under shear deformation, Nb₂–Al bond is stretched and softened gradually, while Nb₁–Al is being compressed up to the critical strain of 12%, where the stress reaches the maximum. Above the critical strain, Nb₁–Al bond is recovered to the initial bond length and Nb₂–Al bond is completely broken at a strain of 23%, which leads to a large stress decrease to be about zero. The positive shear stresses during this shear stage (from 0 to 23%) are coincided with the increase of cohesive energy versus strains in Fig. 5c. With further increasing the shear strain (from 23% to 49%), the distance between Nb₂ and Al atoms increases monotonously and the stresses undergo a negative minimum and reach to another zero again. Meanwhile, the cohesive energy decreases significantly, which could be used to interpret the negative values of stress, indicating that this shear deformation stage (from 23% to 49%) is energetically favorable. Finally, it is interesting to see in Fig. 5b that the Nb–C bond lengths remain nearly unchanged during the whole shear process, indicating that the Nb₂C is stable under the investigated shear deformation.

Fig. 5 (a) The shear stress–strain curves of M₂AlC, (b) the variations of bond length (Nb₁–Al, Nb₂–Al and Nb–C) of Nb₂AlC, (c) the variations of cohesive energy of Nb₂AlC under shear strain along the (0001)[12̄10] direction. The bond lengths are normalized by the equation of V_strain/V₀, where V₀ and V_strain represent the values in structures under zero and various strains, respectively.

To unravel the mechanism of shear deformation of M₂AlC, the variation of crystal structure is also analyzed for the case of Nb₂AlC. Fig. 6 illustrates the side view of crystal structure for Nb₂AlC under the shear strains of 0, 23% and 49% to investigate the atom displacement during the shear deformation. For Nb₂AlC at the zero-strain state (Fig. 6a), the labeled Al atom locates on the top left of Nb₃ atom, where the Nb₂–Al bond has the normal bond length. At a shear strain of 23%, the labeled Al atom slips to the position right above Nb₃ atom (Fig. 6b), resulting in the distance between Nb₂ and Al atoms increases significantly. Since Nb–C bonds remain stable during the shear deformation, this displacement mostly origins from the slipping between Al and the Nb₂C layers. Meanwhile, due to the highest cohesive energy, the structure under the shear strain of 23% is unstable and could be considered as the middle state during the slip displacement of Al atoms. Hence with further increasing the shear strain to be around 49%, the labeled Al atom slips to the top right of Nb₃ atom in Fig. 6c and the structure reaches to a stable state, corresponding to a accomplishment of a one-atom-step relatively displacement between Al and Nb₂C layer. The present simulated slipping between Al and Nb₂C layers is consistent with the experimentally observed perfect dislocations on the (0001) basal planes by transmission electron microscopy after deformation at room-temperature.^43,44 In a word, the shear deformation process of M₂AlC can be described as the relative slipping between the Al and M₂C layers on the (0001) basal planes.

Fig. 6 The side view of crystal structures for Nb₂AlC under the (0001)[12̄10] shear strains of (a) 0, (b) 23% and (c) 49%, where the dashed lines surrounded Nb₂C layer are used for guiding the eyes.

The first maximum in a tensile (shear) stress–strain curve is defined as the ideal tensile (shear) strength. As seen in Fig. 7, for the 9 individual compounds, a general rule can be concluded that the ideal tensile strength increases with the valence electron concentration, i.e., elements M in group IVB leads to the lowest strength while those in group VIB are the strongest when they form the compound M₂AlC. However, the variance trend of the ideal shear strength is relatively complex, and the strength of the M₂AlC containing elements in group IVB is quite small considering the downward curve from group VB to group VIB. Apparently, both the tensile and shear strength of the compound is very sensitive to the valance electron concentration.

Fig. 7 The calculated ideal strengths and the ratio of tensile to shear strength for M₂AlC (M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W).

To all the considered M₂AlC phases, their ideal tensile strengths are much larger than the ideal shear strength as can be seen in Fig. 7. Meanwhile, the corresponding critical points in tensile deformation are all much larger compared to those in shear deformation. The crack formation needs a corresponding local tensile stress, which is in the direction perpendicular to the cleavage plane and is larger than the ideal tensile strength. Therefore, for M₂AlC, the dislocation nucleation after loading a force larger than the ideal shear stress will be activated before crack formation which needs a force larger than the ideal tensile stress. This means that 2-D M₂Cs are much easier to obtain through shear rather than tensile deformation. According to the black solid line in Fig. 7, it is expected that producing 2-D M₂Cs is much more easily when M is an element from group IVB and VIB than that from group VB. Finally, among the nine investigated compounds, it can be predicted that two dimensional W₂C and Zr₂C are the relative easy to obtain because of their low shear strength and high ratio of tensile/shear strength.

4. Conclusions

In summary, we have systematically investigated the tensile and shear properties of layered M₂AlC with M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W using the ab initio calculations in order to investigate the possibility of fabricating MXenes via mechanical exfoliation. Using Nb₂AlC as an example, we unraveled the deformation mechanism of M₂AlC. During both the tensile and shear deformation, the M–C bonds remain stable, while the bond strength of M–Al varies with the engineering strain, which could be used to understand the experimental exfoliation of 2-D M₂Cs from M₂AlC. In the case of tensile deformation, all the M–Al bonds will be stretched gradually, resulting in the stress increasing to the ideal tensile strength. With further increasing the tensile strain, all the M–Al bonds are broken and the stress is relaxed, leading to M₂AlC separated into M₂C and Al layers. While for the shear deformation, half of the M–Al bonds are compressed while the other half ones are stretched up to be broken, leading to the slip displacement of Al layer with respect to M₂C layers in the (0001) basal planes. Finally, we have also shown that the variation of ideal strength of the M₂AlC compounds can be understood in terms of the valence-electron concentrations. It is worth to mention that the present conclusions based on the ground state calculations at 0 K are applicable to describe the nature of exfoliating 2-D M₂C from M₂AlC phases at room temperature or above before the phase separation of the MAX phases. More importantly, the quantitative and systematic calculations in the present study demonstrate the possibility and microscopic process of mechanically exfoliating 2-D M₂C from M₂AlC phases, and hence benefit to the synthesis of novel MXenes.

Acknowledgements

This work is supported by National Natural Science Foundation for Distinguished Young Scientists of China (51225205) and the National Natural Science Foundation of China (61274005).

References

1. M. R. Lukatskaya, O. Mashtalir, C. E. Ren, Y. Dall'Agnese, P. Rozier, P. L. Taberna, M. Naguib, P. Simon, M. W. Barsoum and Y. Gogotsi, Science, 2013, 341, 1502–1505.
2. Q. Tang, Z. Zhou and P. Shen, J. Am. Chem. Soc., 2012, 134, 16909–16916.
3. X. Zhang, M. Xue, X. Yang, Z. Wang, G. Luo, Z. Huang, X. Sui and C. Li, RSC Adv., 2015, 5, 2762–2767.
4. M. Naguib, O. Mashtalir, J. Carle, V. Presser, J. Lu, L. Hultman, Y. Gogotsi and M. W. Barsoum, Two-Dimensional Transition Metal Carbides, ACS Nano, 2012, 6, 1322–1331.
5. Y. Xie, M. Naguib, V. N. Mochalin, M. W. Barsoum, Y. Gogotsi, X. Yu, K. Nam, X. Yang, A. I. Kolesnikov and P. R. C. Kent, J. Am. Chem. Soc., 2014, 136, 6385–6394.
6. L. Yate, L. E. Coy, G. Wang, M. Beltrán, E. Diaz-Barriga, E. M. Saucedo, M. A. Ceniceros, K. Załeski, I. Llarena, M. Möller and R. F. Ziolo, RSC Adv., 2014, 4, 61355–61362.
7. X. Zhang, Z. Ma, X. Zhao, Q. Tang and Z. Zhou, J. Mater. Chem. A 2015, 3, 4960–4966.
8. Y. Lee, Y. Hwang, S. B. Cho and Y. Chung, Phys. Chem. Chem. Phys., 2014, 16, 26273–26278.
9. Z. Ma, Z. Hu, X. Zhao, Q. Tang, D. Wu, Z. Zhou and L. Zhang, J. Phys. Chem. C, 2014, 118, 5593–5599.
10. M. Naguib, V. N. Mochalin, M. W. Barsoum and Y. Gogotsi, Adv. Mater., 2014, 26, 992–1005.
11. Q. Tang and Z. Zhou, Prog. Mater. Sci., 2013, 58, 1244–1315.
12. X. Zhang, J. Xu, H. Wang, J. Zhang, H. Yan, B. Pan, J. Zhou and Y. Xie, Angew. Chem., Int. Ed., 2013, 52, 4361–4365.
13. M. W. Barsoum and M. Radovic, Annu. Rev. Mater. Res., 2011, 41, 195–227.
14. Z. Sun, Int. Mater. Rev., 2011, 56, 143–166.
15. Z. Sun, R. Ahuja, S. Li and J. M. Schneider, Appl. Phys. Lett., 2003, 83, 899–901.
16. Z. Sun, D. Music, R. Ahuja, S. Li and J. M. Schneider, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 70, 092102–092105.
17. Z. Sun, M. Denis, R. Ahuja and M. S. Jochen, J. Phys.: Condens. Matter, 2005, 17, 7169–7176.
18. Y. Zhou and Z. Sun, Mater. Res. Innovations, 2000, 3, 286–291.
19. M. Khazaei, M. Arai, T. Sasaki, M. Estili and Y. Sakka, Sci. Technol. Adv. Mater., 2014, 15, 014208–014214.
20. Z. Sun, J. Zhou, D. Music, R. Ahuja and J. M. Schneider, Scr. Mater., 2006, 54, 105–107.
21. Z. Sun and R. Ahuja, Appl. Phys. Lett., 2006, 88, 161913–161915.
22. D. Music, Z. Sun, A. A. Voevodin and J. M. Schneider, Solid State Commun., 2006, 139, 139–143.
23. Z. Sun, Y. Zhou and J. Zhou, Philos. Mag. Lett., 2000, 80, 289–293.
24. D. Music, Z. Sun and J. M. Schneider, Solid State Commun., 2006, 137, 306–309.
25. D. Music, Z. Sun, R. Ahuja and J. M. Schneider, J. Phys.: Condens. Matter, 2006, 18, 8877–8881.
26. Z. Sun, D. Music, R. Ahuja and J. M. Schneider, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 71, 193402–193405.
27. Z. Sun, S. Li, R. Ahuja and J. M. Schneider, Solid State Commun., 2004, 129, 589–592.
28. Z. Sun, R. Ahuja and J. M. Schneider, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 68, 224112–224115.
29. D. Music, Z. Sun and J. M. Schneider, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 71, 092102–092104.
30. M. Kurtoglu, M. Naguib, Y. Gogotsi and M. W. Barsoum, MRS Comm., 2012, 2, 133–137.
31. L. Qi and D. C. Chrzan, Phys. Rev. Lett., 2014, 112, 115503–115507.
32. X. Li, S. Schönecker, J. Zhao, B. Johansson and L. Vitos, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 214203–214214.
33. C. Jiang and S. G. Srinivasan, Nature, 2013, 496, 339–342.
34. D. M. Clatterbuck, C. R. Krenn, M. L. Cohen and J. W. Morris, Phys. Rev. Lett., 2003, 91, 135501–135504.
35. D. M. Clatterbuck, D. C. Chrzan and J. W. Morris Jr, Acta Mater., 2003, 51, 2271–2283.
36. J. Hafner, J. Comput. Chem., 2008, 29, 2044–2078.
37. J. P. Perdew and Y. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 45, 13244–13249.
38. J. P. Perdew, K. Burke and Y. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 16533–16539.
39. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953–17979.
40. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188–5192.
41. B. Silvi and A. Savin, Nature, 1994, 371, 683–686.
42. K. Momma and F. Izumi, J. Appl. Crystallogr., 2011, 44, 1272–1276.
43. L. Farber, M. W. Barsoum, A. Zavaliangos, T. El-Raghy and I. Levin, J. Am. Ceram. Soc., 1998, 81, 1677–1681.
44. M. W. Barsoum, L. Farber and T. El-Raghy, Metall. Mater. Trans. A, 1999, 30, 1727–1738.
45. M. Zhang, M. Lu, Y. Du, L. Gao, C. Lu and H. Liu, J. Chem. Phys., 2014, 140, 174505–174510.
46. S. Grimme, J. Comput. Chem., 2006, 27, 1787–1799.

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