M. A. Hadi*a,
M. Dahlqvist
b,
S.-R. G. Christopoulosc,
S. H. Naqib
a,
A. Chroneoscd and
A. K. M. A. Islamae
aDepartment of Physics, University of Rajshahi, Rajshahi 6205, Bangladesh. E-mail: hadipab@gmail.com
bThin Film Physics Division, Department of Physics (IFM), Linköping University, SE-581 83 Linköping, Sweden
cFaculty of Engineering, Environment and Computing, Coventry University, Priory Street, Coventry CV1 5FB, UK
dDepartment of Materials, Imperial College, London SW7 2AZ, UK
eInternatinal Islamic University Chittagong, Kumira, Chittagong 4318, Bangladesh
First published on 8th December 2020
Using density functional theory, the phase stability and physical properties, including structural, electronic, mechanical, thermal and vibrational with defect processes, of a newly synthesized 211 MAX phase V2SnC are investigated for the first time. The obtained results are compared with those found in the literature for other existing M2SnC (M = Ti, Zr, Hf, Nb, and Lu) phases. The formation of V2SnC is exothermic and this compound is intrinsically stable in agreement with the experiment. V2SnC has potential to be etched into 2D MXene. The new phase V2SnC and existing phase Nb2SnC are damage tolerant. V2SnC is elastically more anisotropic than Ti2SnC and less than the other M2SnC phases. The electronic band structure and Fermi surface of V2SnC indicate the possibility of occurrence of its superconductivity. V2SnC is expected to be a promising TBC material like Lu2SnC. The radiation tolerance in V2SnC is better than that in Lu2SnC.
Early studies on M2SnC phases by Jeitschko et al. in mid 1960s resulted in synthesized phases with M = Ti, Zr, Hf, and Nb.8–10 Kuchida et al.11 synthesized the first Lu-based MAX phase Lu2SnC in the M2SnC family. Lu is the last element in the lanthanide series although it is sometimes mentioned as the first member in the 6th-period transition metals. Lu replaced the common early transition metals “M” in the M2SnC MAX phases. Theoretical investigations reveal that Lu2SnC is softer and more easily machinable than the other existing M2SnC phases. It is also a promising candidate as a thermal barrier coating (TBC) material owing to its high thermal shock resistance, low minimum thermal conductivity, high melting temperature and characteristically good oxidation resistance.12 Xu et al.13 focused their attention on M2SnC phases and synthesized V2SnC by sintering V, Sn, and C powder mixture at 1000 °C. They identified the crystal structure of V2SnC as 211 MAX phases through X-ray diffraction, first-principles calculation, and high-resolution transmission scanning electron microscopy.
M2SnC compounds show diversity in their properties. Nb2SnC and Lu2SnC show superconducting transition, Nb2SnC has also better radiation tolerance in the M2SnC family, Hf2SnC is highly dense, and Zr2SnC is highly elastically anisotropic in this group.5,12 Additionally, the lattice constants of Sn-containing 211 MAX phases show an increasing trend with the crystal radius of M-elements.12 The diverse properties of M2SnC motivated the use of density functional theory (DFT) calculations to explore the physical properties of V2SnC and the analysis of the trend of physical properties in M2SnC MAX phases. In this study, the mechanical, lattice dynamical and thermodynamic phase stability is examined for the new phase V2SnC. Mechanical behaviors, elastic anisotropy, Debye temperature, melting point, lattice thermal conductivity, minimum thermal conductivity, lattice dynamics and defect processes of V2SnC are investigated for the first time.
The elastic stiffness constants and moduli are calculated from the first-principles investigations using finite-strain method implemented in the CASTEP code.19 This method involves setting the deformation to a predetermined value, relaxing all free parameters and computing the stress. The convergence criteria for elastic calculations are chosen as: the difference in total energy within 10−6 eV per atom, the maximum ionic Hellmann–Feynman force within 2 × 10−3 eV Å−1, and the maximum ionic displacement within 10−4 Å. Elastic calculations with CASEP code have been successful for all kind of crystal systems.20–31 The lattice dynamic properties such as phonon dispersion and phonon density of states are calculated by means of the finite displacement supercell method executed with a 3 × 3 × 1 supercell within the code.
Defect calculations are performed with a 72-atomic site (36M, 18A, and 18C) supercell using a 3 × 3 × 1 MP k-point mesh under constant pressure. Allowing for all possible interstitial sites, an intensive computational search is carried out to identify the potential interstitial sites. The defect energies are defined as effectively energy differences between the isolated defects.
![]() | ||
Fig. 2 Lattice parameters of M2SnC as a function of crystal radius of M atoms.33 |
Compound | a | c | c/a | zM | V | Remarks |
---|---|---|---|---|---|---|
V2SnC | 3.121 | 12.947 | 4.148 | 0.0759 | 109.2 | Calc. (This work) |
2.9792 | 13.4441 | 4.513 | 0.0744 | 103.3 | Expt.13 | |
Ti2SnC | 3.172 | 13.772 | 4.342 | 0.0806 | 120.0 | Calc.12 |
3.1635 | 13.675 | 4.323 | — | 118.5 | Expt.32 | |
Nb2SnC | 3.258 | 13.918 | 4.272 | 0.0820 | 128.0 | Calc.12 |
3.2408 | 13.802 | 4.259 | — | 125.5 | Expt.32 | |
Hf2SnC | 3.367 | 14.548 | 4.320 | 0.0865 | 142.9 | Calc.12 |
3.3199 | 14.388 | 4.334 | — | 137.3 | Expt.32 | |
Zr2SnC | 3.367 | 14.730 | 4.374 | 0.0849 | 144.7 | Calc.12 |
3.3576 | 14.568 | 4.339 | — | 142.2 | Expt.32 | |
Lu2SnC | 3.546 | 15.323 | 4.320 | 0.0850 | 166.9 | Calc.12 |
3.514 | 15.159 | 4.314 | — | 162.1 | Expt.32 |
Phase stability of MAX phases with respect to the constituent elements cannot be used to predict whether a material is thermodynamically stable. Instead, all competing phases need to be included in the analysis. The thermodynamic stability of the recently synthesized V2SnC MAX phase is examined at 0 K with respect to decomposition into any combination of competing phases. The most competitive set of competing phases, designated as equilibrium simplex, is identified using a linear optimization procedure.34 This procedure has already been successful to confirm the experimentally known MAX phases in addition to predicting the existence of new ones.35 The stability of V2SnC is quantified in terms of formation enthalpy ΔHcp by comparing its energy to the energy of the equilibrium simplex according to
ΔHcp = E(V2SnC) − E(equilibrium simplex) | (1) |
The phase V2SnC is considered to be stable if ΔHcp < 0. The considered competing phases for V2SnC are listed in Table 2. V2C and Sn are identified as the most competing phases according to linear optimization procedure. The eqn (1) can be rewritten as
ΔHcp = E(V2SnC) − E(V2C) − E(Sn) | (2) |
Phase | Prototype structure | Pearson symbol | Space group | a (Å) | b (Å) | c (Å) | V (Å3) | E (eV fu−1) |
---|---|---|---|---|---|---|---|---|
C | C (graphite) | hP4 | P63/mmc (194) | 2.462744 | 2.462744 | 8.985673 | 47.2 | −155.088 |
C | Diamond | cF8 | Fd![]() |
3.567776 | 3.567776 | 3.567776 | 45.4 | −154.901 |
V | W | cI2 | Im![]() |
3.011843 | 3.011843 | 3.011843 | 27.3 | −1976.398 |
α-Sn | Diamond | cF8 | Fd![]() |
8.596983 | 8.596983 | 8.596983 | 635.4 | −94.217 |
β-Sn | β-Sn | tI4 | I41/amd (141) | 5.973130 | 5.973130 | 3.183849 | 113.6 | −95.465 |
SnC | ZnS | cF8 | F![]() |
5.134950 | 5.134950 | 5.134950 | 135.4 | −248.722 |
SnC | NaCl | cF8 | Fm![]() |
4.921133 | 4.921133 | 4.921133 | 119.2 | −247.793 |
VC | CrB | oC8 | Cmcm (63) | 2.784524 | 7.500027 | 3.378178 | 70.5 | −2131.336 |
VC | α-MoB | tI16 | I41/amd (141) | 2.875169 | 2.875169 | 19.390061 | 160.3 | −2131.692 |
V2C | β-V2N | hP9 | P![]() |
5.017396 | 5.017396 | 4.546570 | 99.1 | −4109.186 |
V2C | Inverse CdI2 | hP3 | P![]() |
2.908853 | 2.908853 | 4.555253 | 33.4 | −4109.146 |
α-V2C | ζ-Fe2N (Fe2N0.94) | oP12 | Pbcn (60) | 4.563895 | 5.751247 | 5.040818 | 132.3 | −4109.208 |
β-V2C | W2C | hP3 | P63/mmc (194) | 2.908607 | 2.908606 | 4.555192 | 33.4 | −4109.147 |
β′-V2C | ε-Fe2N | hP9 | P![]() |
5.003269 | 5.003269 | 4.535382 | 98.3 | −4109.191 |
VC2 | MoB2 | tR18 | R![]() |
2.599774 | 2.599774 | 24.095326 | 141.0 | −2285.036 |
VC2 | AlB2 | hP3 | P6/mmm (191) | 2.554138 | 2.554138 | 4.407399 | 24.9 | −2284.212 |
VC3 | Ni3Ti | hP16 | P63/mmc (194) | 4.471182 | 4.471182 | 7.239285 | 125.3 | −2439.254 |
V6C5 | V6C5 | hP33 | P3112 (151) | 5.122689 | 5.122689 | 14.387631 | 327.0 | −12639.362 |
V8C7 | cP60 | P4332 (212) | 8.328377 | 8.328377 | 8.328377 | 577.7 | −16905.598 | |
VSn2 | Mg2Cu | cF48 | Fddd (70) | 5.523603 | 9.500626 | 18.914470 | 992.6 | −2167.631 |
V3Sn | Cr3Si | cP8 | Pm![]() |
5.003758 | 5.003758 | 5.003758 | 125.3 | −6024.943 |
V3Sn | Mg3Cd | hP6 | P63/mmc (194) | 5.664976 | 5.664976 | 4.517371 | 125.5 | −6025.165 |
V3Sn2 | Cr3Si2 | tP10 | P4/mbm (127) | 7.107742 | 7.107742 | 3.555283 | 179.6 | −6118.656 |
VSnC | MoAlB | oC12 | Cmcm (63) | 2.966421 | 22.230689 | 2.858936 | 188.5 | −2226.001 |
V2SnC | Cr2AlC | hP8 | P63/mmc (194) | 3.136333 | 3.136333 | 13.011838 | 110.8 | −4204.700 |
V2Sn2C | Mo2Ga2C | hP10 | P63/mmc (194) | 3.185703 | 3.185703 | 18.846330 | 165.6 | −4299.833 |
V3SnC | CaTiO3 | cP5 | Pm![]() |
4.081427 | 4.081427 | 4.081427 | 68.0 | −6180.677 |
V3SnC2 | Ti3SiC2 | hP12 | P63/mmc (194) | 3.055478 | 3.055478 | 18.163485 | 146.9 | −6336.856 |
V4SnC3 | Ta4AlN3 | hP16 | P63/mmc (194) | 3.009399 | 3.009399 | 23.156747 | 181.6 | −8469.124 |
C11, C33, C44 > 0; C11 > |C12| and (C11 + C12)C33 > 2C13C13 | (3) |
Phases | C11 | C33 | C44 | C66 | C12 | C13 | B | G | E | v | B/G | Remarks |
---|---|---|---|---|---|---|---|---|---|---|---|---|
V2SnC | 243 | 300 | 87 | 84 | 76 | 124 | 156 | 82 | 209 | 0.276 | 1.91 | CASTEP-GGA (This work) |
336 | 304 | 85 | 105 | 126 | 122 | 190 | 95 | 244 | 0.286 | 2.00 | CASTEP-GGA13 | |
Lu2SnC | 172 | 173 | 56 | 64 | 46 | 36 | 82 | 61 | 147 | 0.199 | 1.33 | CASTEP-GGA12 |
Ti2SnC | 268 | 265 | 100 | 95 | 79 | 74 | 139 | 97 | 236 | 0.217 | 1.43 | CASTEP-GGA12 |
253 | 254 | 93 | 79 | 91 | 74 | 138 | 87 | 217 | 0.238 | 1.57 | VASP-GGA46 | |
337 | 329 | 169 | 126 | 86 | 102 | 176 | 138 | 329 | 0.188 | 1.27 | FP-L/APW + lo47 | |
303 | 308 | 121 | 109 | 84 | 88 | 160 | 114 | 275 | 0.212 | 1.40 | CASTEP-LDA48 | |
152 | 83.9 | 207.4 | 0.24 | Experimental39,41 | ||||||||
Zr2SnC | 230 | 232 | 94 | 84 | 62 | 91 | 131 | 83 | 206 | 0.237 | 1.57 | CASTEP-GGA12 |
225 | 227 | 87 | 77 | 72 | 90 | 131 | 78 | 196 | 0.251 | 1.68 | VASP-GGA46 | |
269 | 290 | 148 | 94 | 81 | 107 | 157 | 110 | 268 | 0.215 | 1.42 | FP-L/APW + lo47 | |
279 | 272 | 111 | 104 | 70 | 89 | 147 | 104 | 252 | 0.215 | 1.42 | CASTEP-LDA48 | |
178 | Experimental39 | |||||||||||
Hf2SnC | 251 | 238 | 101 | 90 | 71 | 107 | 145 | 87 | 218 | 0.250 | 1.67 | CASTEP-GGA12 |
249 | 252 | 99 | 85 | 73 | 101 | 144 | 87 | 218 | 0.247 | 1.65 | VASP-GGA46 | |
330 | 292 | 167 | 138 | 54 | 126 | 173 | 132 | 316 | 0.195 | 1.30 | FP-L/APW + lo47 | |
311 | 306 | 119 | 109 | 92 | 97 | 167 | 112 | 275 | 0.225 | 1.49 | CASTEP-LDA48 | |
169 | 237 | Experimental40 | ||||||||||
Nb2SnC | 255 | 236 | 94 | 77 | 102 | 122 | 160 | 78 | 202 | 0.290 | 2.05 | CASTEP-GGA12 |
253 | 250 | 98 | 74 | 103 | 120 | 160 | 80 | 206 | 0.286 | 2.00 | VASP-GGA46 | |
341 | 321 | 183 | 118 | 106 | 169 | 209 | 126 | 315 | 0.250 | 1.67 | FP-L/APW + lo47 | |
315 | 309 | 124 | 108 | 99 | 141 | 189 | 107 | 189 | 0.262 | 1.77 | CASTEP-LDA48 | |
180 | 216 | Experimental40 |
Furthermore, for all M2SnC phases, it is observed that the principal elastic constants C11 and C33 are larger than all other Cij. While for the three systems with M = V, Lu, or Zr, the principal elastic constants are classified as C33 > C11, and the remaining three systems with M = Hf, Ti, or Nb, exhibit C11 > C22 within the same code and functional (CASTEP-GGA). It implies that the former group is more incompressible along the c-axis. Either C11 > C33 or C33 > C11 is the evident of elastic anisotropy of M2SnC MAX phases. The phases Ti2SnC, Zr2SnC and Lu2SnC are elastically less anisotropic than other phases as their C11 and C33 values are very close to each other. The shear elastic constants C12 and C13 lead mutually to a functional stress component in the crystallographic a-axis with a uniaxial strain along the crystallographic b- and c-axis, respectively. This stress component measures the resistance of shear deformation of a material along the crystallographic b- and c-axis, when stress is applied along the a-axis. The phase Nb2SnC is most capable to resist such deformation, while Lu2SnC will deform easily under the same stress along the a-axis. The new compound V2SnC is the next most capable phase to resist the deformation in M2SnC systems if a rank is made one obtains: Nb2SnC > V2SnC > Hf2SnC > Ti2SnC > Zr2SnC > Lu2SnC.
We have obtained the bulk modulus B and shear modulus G of polycrystalline aggregates from individual elastic constants, Cij by the Hill approximations as implemented within the code.38 Using B and G, the average Young's modulus E and the Poisson ratio ν can be obtained:
![]() | (4) |
![]() | (5) |
All the elastic moduli B, G, and E and Poisson's ratio v are also listed in Table 3, from which we observe that the results obtained with the CASTEP-GGA and VASP-GGA are consistent to each other, while the other results deviate considerably. Fig. 3b presents elastic moduli calculated with the CASTEP-GGA scheme. B is highly correlated to the chemical composition and structure, while G is linked to chemical bonding. On the other hand, E measures the response under uniaxial tension averaged over all directions. Within the results of this scheme, B is highest for Nb2SnC (160 GPa) and lowest for Lu2SnC (82 GPa). The new phase V2SnC has second highest value of 156 GPa. Ti2SnC has highest G (97 GPa) and E (236 GPa) values. The lowest values of G (61 GPa) and E (147 GPa) are found for Lu2SnC. The new phase V2SnC has an intermediate value of G (80 GPa) and E (209 GPa). When we move from left to right in the Fig. 3b, we cross the group-3 element (Lu) to group-5 elements (V, Nb) via the group-4 elements (Ti, Zr, Hf). The elastic moduli G and E show the almost similar trend. B of the phases containing groups 3 and 4 elements as M-atom show the similar trend of G and E but the phases containing the group-5 elements show the reverse trend. For comparison, we have found 152 ± 3, 180 ± 5, and 169 ± 4 GPa as the measured values of B for Ti2SnC, Nb2SnC, and Hf2SnC, respectively.39,40 These values are larger than the values calculated with GGA within CASTEP and VASP codes by 9–14% and smaller than the other values by 2–16%. The experimental shear modulus is found for Ti2SnC, which is comparable with the GGA-value and much smaller than the other theoretical values listed in Table 3.41 The experimental Poisson's ratio for Ti2SnC is 0.24, which is also very close to the GGA-values rather than other values.41 The experimental value of E for Ti2SnC, Nb2SnC, Zr2SnC, and Hf2SnC are 207.4, 216, 178, and 237 GPa, respectively.40,41 For Nb2SnC and Hf2SnC, the experimental E is larger than the theoretical E calculated with both the CASTEP-GGA and VASP-GGA by 7–8%, while the experiment E of Ti2SnC and Zr2SnC is smaller than the theoretical E derived with GGA within CASTEP and VASP codes by 5–16%. From other theoretical values listed in Table 3, the experimental E deviates within 13–51%. Therefore, the GGA values of B and E obtained with the CASTEP and VASP codes deviates from experimental values within a reasonable range. The larger the E value, the stiffer the system, and therefore the larger the exfoliation energy.42 Amongst the productively etched MAX phases into two-dimensional (2D) MXenes, V2AlC has the largest theoretical exfoliation energy, whose E is reported 311 and 316 GPa.43,44 Accordingly, the exfoliation energy of new MAX phase V2SnC and previously observed M2SnC have lower exfoliation energy than V2AlC. It is evident that all M2SnC (M = V, Ti, Zr, Hf, Nb, and Lu) phases have potential to etch into 2D MXenes.
Poisson's ratio ν provides the information regarding the bonding forces and reflects the stability of a material against shear. The M2SnC MAX phases including newly synthesized V2SnC have Poisson's ratio within 0.195–0.290. As the obtained values with CASTEP-GGA for V2SnC, Hf2SnC and Nb2SnC fall in the range of 0.25–0.5; their interatomic forces can be considered as central forces.45 Literature values of v for Hf2SnC46–48 lie on the lower side of this range, while a literature value for Zr2SnC46 falls within this range. The values outside this range indicate that the interatomic force is non-central. The low value of v for Lu2SnC indicates that it is more stable against shear than other M2SnC phases including the new phase V2SnC.1 Additionally, a pure covalent crystal has a Poisson's ratio of 0.1 and a totally metallic compound has a value of 0.33. As the Poisson's ratio for M2SnC MAX phases lies between these two characteristic values their atomic bonding is expected to be a mixture of covalent and metallic in nature. Furthermore, Poisson's ratio can classify the solid materials as either brittle or ductile with a value of 0.26.49,50 Brittle materials have values less than 0.26 and ductile materials have values larger than this value. Accordingly, the new phase V2SnC and Nb2SnC are ductile and the remaining phases are brittle. Therefore, V2SnC and Nb2SnC are predicted to be damage tolerant.
Bulk modulus to shear modulus ratio (B/G), known as Pugh's ratio can serve as a tool for measuring the ductile/brittle nature of a material.51 If the Pugh's ratio is greater than 1.75 the material is expected to be ductile, otherwise it is brittle in nature. Accordingly, V2SnC and Nb2SnC are ductile in nature as predicted from their Poisson's ratio.
Indeed, it is essential to analyze and visualize the directional dependence of elastic properties – such as Young's modulus (E), linear compressibility (β), shear modulus (G) and Poisson's ratio (ν) of anisotropic materials – rather than their averages. For the M2SnC phases, the directional dependency of E, β, G and ν are calculated using the ELATE suit program52 and the 2D presentations are shown in Fig. 4 for V2SnC, indicating that there is no directional dependence in the xy plane as the plots are uniformly circular but in both xz and yz planes there are directional dependences and which are similar in nature as the hexagonal crystal symmetry of V2SnC. The linear compressibility for some compounds can be negative in some directions, which is represented with an additional curve of red color. For V2SnC, no negative value is found for any direction. The shear modulus G and Poisson's ratio ν are not so straightforward to represent, as they depend on two orthogonal unit vectors a and b, which respectively represent the direction of the stress applied and the direction of measurement. For G and ν there are two curves: translucent blue curve represents the maximal values and green curve represents the minimal positive values. There are a lot of compounds having negative Poisson's ratio in some directions. In this case, an additional curve of translucent red represents the negative values. For V2SnC, no negative Poisson's ratio is found for any directions. For other M2SnC phases, the above discussions are applicable. The directional dependence of E, β, G and ν in xz and yz planes is almost identical for Nb2SnC, Hf2SnC and Zr2SnC. Ti2SnC shows almost isotropic nature of E, β, G and ν in xz and yz planes also. The directional dependence of E, β, G and ν in Lu2SnC is different compared to other M2SnC phases. Linear compressibility in Ti2SnC and Lu2SnC is almost directional independent.
![]() | ||
Fig. 4 Directional dependence of Young's modulus (E), linear compressibility (β), shear modulus (G) and Poisson's ratio (ν) of V2SnC. |
ELATE also provides a quantitative analysis by reporting the minimal and maximal values of each modulus as well as the directions along which these extrema occur. This allows the determination of directions of particular interest in the elastic properties, which are not necessarily along the crystallographic axes of the material. Minimal and maximal values of each modulus are listed in Table 4. A measure of the anisotropy AX of each elastic modulus X is defined as follows:
![]() | (6) |
Phases | Young's modulus (GPa) | Linear compressibility (TPa−1) | Shear modulus (GPa) | Poisson's ratio | ||||
---|---|---|---|---|---|---|---|---|
Emin | Emax | βmin | βmax | Gmin | Gmax | νmin | νmax | |
V2SnC | 188.79 | 223.85 | 1.0964 | 2.7112 | 71.355 | 86.673 | 0.12849 | 0.38828 |
Hf2SnC | 168.97 | 236.44 | 1.9579 | 2.4813 | 66.846 | 99.802 | 0.12198 | 0.38969 |
Lu2SnC | 143.59 | 167.39 | 3.8307 | 3.9590 | 56.841 | 70.092 | 0.16641 | 0.26442 |
Nb2SnC | 168.47 | 237.13 | 1.7785 | 2.1490 | 66.303 | 97.202 | 0.15262 | 0.41665 |
Ti2SnC | 233.72 | 239.43 | 2.4154 | 2.4509 | 95.408 | 100.210 | 0.19446 | 0.22531 |
Zr2SnC | 174.63 | 222.77 | 2.1404 | 2.7889 | 68.418 | 94.736 | 0.13615 | 0.33803 |
Elastic anisotropy AX | ||||
---|---|---|---|---|
AE | Aβ | AG | Aν | |
V2SnC | 1.186 | 2.4729 | 1.215 | 3.0219 |
Hf2SnC | 1.399 | 1.2674 | 1.493 | 3.1947 |
Lu2SnC | 1.166 | 1.0335 | 1.233 | 1.5889 |
Nb2SnC | 1.408 | 1.2083 | 1.466 | 2.7300 |
Ti2SnC | 1.024 | 1.0147 | 1.050 | 1.1586 |
Zr2SnC | 1.276 | 1.3030 | 1.385 | 2.4828 |
The obtained elastic anisotropy AX for each elastic modulus is also listed in Table 4. It is observed that Young's modulus shows maximum anisotropy for Nb2SnC and minimum for Ti2SnC. Anisotropy in linear compressibility is maximum for V2SnC and minimum for Ti2SnC. Anisotropy in shear modulus is highest for Hf2SnC and lowest for Ti2SnC. Maximum anisotropy of Poisson's ratio is observed in Hf2SnC and minimum in Ti2SnC. Considering all parameters Ti2SnC is the least anisotropic in M2SnC family.
Here, we want to quantify the degree of elastic anisotropy of V2SnC and compare with previously synthesized M2SnC phases. For hexagonal M2SnC crystals, there are three shear anisotropy factors linked to Cij that can be determined using the succeeding expressions:53
![]() | (7) |
![]() | (8) |
![]() | (9) |
The calculated values of Ai for V2SnC are listed in Table 5 along with reported values for other M2SnC phases and the anisotropy level ΔAi is shown in Fig. 5, suggesting that all M2SnC phases including V2SnC are elastically anisotropic in shear. Shear anisotropy level is highest in Nb2SnC and lowest in Ti2SnC in all respects. The anisotropy level in the new phase V2SnC is higher than that in Ti2SnC and lower than those in other M2SnC phases. It is observed that the shear anisotropy level within a group of M atoms of M2SnC phases increases in the descending order. A different anisotropy factor for hexagonal crystals depending on Cij, i.e. kc/ka = (C11 + C12 − 2C13)/(C33 − C13) is used to quantify the elastic anisotropy upon compression; where ka and kc are the linear compressibility coefficients along the a- and c-axis, respectively.54 Deviation of kc/ka from unity (Δ(kc/ka)), determines the anisotropy level upon linear compression. The calculated value reveals that the compressibility along the c-axis is smaller than that along the a-axis for the new phase V2SnC as well as for Zr2SnC, Hf2SnC, and Nb2SnC. For Lu2SnC and Ti2SnC the compressibility along the c-axis is greater than that along the a-axis.
Phases | A1 | A2 | A3 | kc/ka | AB% | AG% | AU | Remarks |
---|---|---|---|---|---|---|---|---|
V2SnC | 0.8103 | 1.0419 | 0.8443 | 0.4034 | 1.8476 | 0.7283 | 0.1110 | CASTEP-GGA (This work) |
Lu2SnC | 1.2500 | 0.8889 | 1.1111 | 1.0657 | 0.0256 | 0.3781 | 0.0385 | CASTEP-GGA12 |
Ti2SnC | 0.9683 | 1.0582 | 1.0247 | 1.0419 | 0.0088 | 0.0284 | 0.0030 | CASTEP-GGA12 |
Zr2SnC | 0.6950 | 1.1190 | 0.7778 | 0.7801 | 0.2082 | 0.9355 | 0.0986 | CASTEP-GGA12 |
Hf2SnC | 0.6106 | 1.1222 | 0.6852 | 0.8244 | 0.1093 | 1.6814 | 0.1732 | CASTEP-GGA12 |
Nb2SnC | 0.6046 | 1.2288 | 0.7429 | 0.9912 | 0.0020 | 1.6962 | 0.1726 | CASTEP-GGA12 |
According to Hill, the difference between BV and BR as well as GV and GR is proportional to the degree of elastic anisotropy of crystals, which leads to define the percentage anisotropy factors AB and AG with the succeeding equations:55
![]() | (10) |
![]() | (11) |
The percentage anisotropy factors AB and AG calculated for V2SnC are also listed in Table 5 together with the literature value of other M2SnC phases. These two factors assign zero values for completely isotropic crystals in view of compressibility and shear, respectively. A positive value quantifies the level of anisotropy. It is evident that the new phase V2SnC is more anisotropic in compression, whereas Nb2SnC is more anisotropic in shear. Nb2SnC is less anisotropic in compression and Ti2SnC is less anisotropic in shear. An anisotropy factor named “universal anisotropy index” is recently proposed for an appropriate universal measure of elastic anisotropy of crystals and defined as:56
![]() | (12) |
This index has either zero or positive value. Zero value signifies the completely isotropic nature and positive value indicates the anisotropy level in elastic properties of crystals. According to this index (see Table 5), the new phase V2SnC is more anisotropic than Lu2SnC, Ti2SnC and Zr2SnC and less anisotropic than Hf2SnC and Nb2SnC. It is evident that the universal anisotropy level follows the trend of shear anisotropy level. That is, the universal anisotropy level within a group of M atoms of M2SnC phases increases in the descending order.
![]() | ||
Fig. 6 Electronic structures of V2SnC, (a) band structure and (b) density of states; EF denotes the Fermi level. |
To realize the bonding nature, the total and partial density of states (DOS) are calculated for V2SnC and presented in Fig. 6b. The Fermi level of V2SnC lies far from a pseudogap at the left instead of near a pseudogap as found for other existing M2SnC MAX phases.5 Consequently, the new phase V2SnC is not as stable as other M2SnC phases. In fact, the Fermi level of V2SnC lies at the wall of a large peak and as a result V2SnC has a large total DOS of 6.12 states per eV per unit cell at EF, whereas the total DOS at EF for other M2SnC phases ranges from 2.35–3.93 states per eV per unit cell.5 The valence band of V2SnC contains two main parts. An intermediate low flat-type valence band arises due to Sn-s orbitals in similar to in other M2SnC and M3SnC2 compounds.5,27 The lower valence band consists of a single peak arising owing to the hybridization between V-3d and C-2s states, which indicate strong covalent V–C bond in V2SnC similar to the M–C bonds in M2SnC. The higher valence band contains three distinct peaks similar to those of Nb2SnC.5 The small peak at the left of the higher valence band arises due to the interaction between V-3d and C-2p-orbitals. The middle peak is the highest peak and arises owing to the hybridization between V-3d and C-2p electrons. The third peak corresponds to the interaction between V-3d and Sn-5p states. This interaction results in weaker covalent V–Sn bonding due to closeness of the peak to the Fermi level. It is clear that the V–C bond is stronger than V–Sn bond as M–C bonds are stronger than M–A bonds. Weaker M–A bond favours the exfoliation of M2SnC MAX phases to 2D MXenes.59 The overall bonding nature in the new phase V2SnC is a combination of metallic, covalent, and, due to the difference in electronegativity between the constituent atoms, ionic like other MAX phase compounds.1,3,5,27,57
We have calculated the electron charge density map and Fermi surface to understand the nature of chemical bonding in V2SnC. In the contour map of electron charge density (Fig. 7a) it is seen that the charge distributions around V atoms are practically spherical and its intensity specifies the amount of charge accumulation. The charge accumulated around the V atom is 0.32e, whereas the charge accumulation around the M atoms in other M2SnC systems ranges from 0.28–0.45e.5 The highest charge is deposited around the Lu atom (0.45e) and lowest charge around the Hf atom (028e). The V-charge overlaps with the C-charge and slightly edges with the Sn-charge, which indicates the strong V–C and weak V–Sn bonds, respectively. Analogous bonds are also seen in the contour maps of other M2SnC compounds.5 The spherical charge distributions around the atoms also indicate some ionic nature in chemical bonds in V2SnC as well as in other M2SnC MAX phases.
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Fig. 7 Electronic structures of V2SnC; (a) charge density map, (b) Fermi surface and (c) fourth Fermi sheet. |
The Fermi surface (FS) calculated for the V2SnC MAX phase is shown in Fig. 7b, which contains four different sheets. All sheets are seen to be centered along the Γ–A direction. The first and second sheets are cylindrical. They have an extra part like a half-folded plain sheet along each L–M direction. The third sheet shows a lot of nesting nature. It has also an additional part along H–K directions, whose shape is like a cylinder cutting parallel to its axis. The fourth sheet consists of two identical parts along the H–K directions. The lower part is just the mirror inversion of the upper part (Fig. 7c). As seen from Fig. 5a, near the Γ (0,0,0) point, two hole-like bands appear, giving rise to two hole-like Fermi surface sheets near the Γ point (first and second sheets). Conversely, the calculated band structure near the H(1/3, 1/3, 1/3) point consists of two electron-like bands, corresponding to two electron-like Fermi surface sheets near the H point (third and fourth sheets). The FS of V2SnC has a lot of similarities to that of superconducting Nb2SnC among all M2SnC MAX phases.5 Nb2SnC is a superconducting phase having nesting nature in its FS. This nesting plays a role in strong electron–phonon interactions and is ultimately able to enhance the superconducting order of the material.58 Nesting nature is an indication of superconductivity of V2SnC. We hope that the experimentalists will be stimulated to confirm the prediction.
![]() | (13) |
In this equation, ℏ and kB are respectively the Plank and Boltzmann constants, NA is Avogadro's number, ρ is the mass density, n is the number of atoms in a molecule, and M is the molecular weight. The average sound velocity vm is obtained from the longitudinal and transverse sound velocities vl and vt by the equation:
![]() | (14) |
With the bulk modulus B and shear modulus G, vl and vt can be determined as:
![]() | (15) |
The obtained sound velocities and Debye temperature of V2SnC is listed in Table 6 along with the literature values for existing M2SnC phases and the CASTEP-GGA values are given in Fig. 8. There are several sets of literature values of θD for M = Ti, Zr, Hf, and Nb. It is observed that the values derived with the GGA functional using the CASTEP and VASP codes are consistent as we have a close measured value (380 K) and a theoretical value (412 K) of θD for Nb2SnC.60,61 The remaining two sets of θD values show large deviations from the former sets as well as from the available experimental and theoretical values. It is evident from the Fig. 8 that the sound velocities and Debye temperature follow the reverse trend of shear and universal anisotropy level. That is, the sound velocities and Debye temperature within a group of M atoms of M2SnC phases decrease in the descending order.
Phases | ρ | vl | vt | vm | θD | Tm | κmin | κphb | Remarks |
---|---|---|---|---|---|---|---|---|---|
a Calculated from published data.b Calculated at 300 K. | |||||||||
V2SnC | 7.073 | 6.125 | 3.405 | 3.792 | 472 | 1533 | 1.20 | 14.38 | CASTEP-GGA (This work) |
Lu2SnC | 9.847 | 4.073 | 2.489 | 2.748 | 300 | 1130 | 0.51 | 14.91 | CASTEP-GGA12 |
Ti2SnC | 6.346 | 6.503 | 3.910 | 4.325 | 525 | 1556 | 0.99 | 29.98 | CASTEP-GGA12 |
6.346 | 6.327 | 3.703 | 4.106 | 498 | 1494 | 1.23 | 22.24 | VASP-GGA46,a | |
6.473 | 7.113 | 4.337 | 4.790 | 585 | 1859 | 1.45 | 49.51 | FP-L/APW + lo47 | |
6.76 | 6.783 | 4.099 | 4.532 | 561 | 1725 | 1.08 | 36.87 | CASTEP-LDA48 | |
Zr2SnC | 7.313 | 5.749 | 3.369 | 3.735 | 426 | 1392 | 0.76 | 20.61 | CASTEP-GGA12 |
7.313 | 5.669 | 3.266 | 3.627 | 414 | 1370 | 0.73 | 17.22 | VASP-GGA46,a | |
7.280 | 6.357 | 3.831 | 4.236 | 483 | 1596 | 0.86 | 34.67 | FP-L/APW + lo47 | |
7.75 | 6.111 | 3.683 | 4.073 | 472 | 1599 | 0.86 | 31.82 | CASTEP-LDA48 | |
Hf2SnC | 11.796 | 4.704 | 2.716 | 3.015 | 348 | 1464 | 0.63 | 15.92 | CASTEP-GGA12 |
11.796 | 4.695 | 2.716 | 3.015 | 348 | 1479 | 0.63 | 16.15 | VASP-GGA46,a | |
11.828 | 5.228 | 3.118 | 3.446 | 398 | 1782 | 0.72 | 32.92 | FP-L/APW + lo47 | |
12.06 | 5.121 | 3.050 | 3.376 | 393 | 1746 | 0.71 | 26.94 | CASTEP-LDA48 | |
Nb2SnC | 8.369 | 5.616 | 3.053 | 3.469 | 412 | 1473 | 0.76 | 12.38 | CASTEP-GGA12 |
8.369 | 5.645 | 3.092 | 3.448 | 410 | 1488 | 0.76 | 12.61 | VASP-GGA46,a | |
8.388 | 6.358 | 3.626 | 4.030 | 480 | 1859 | 0.89 | 29.96 | FP-L/APW + lo47 | |
8.53 | 6.150 | 3.493 | 3.883 | 469 | 1763 | 0.87 | 22.92 | CASTEP-LDA48 |
The Debye temperatures of M2SnC MAX phases follow the order of Lu2SnC < Hf2SnC < Nb2SnC < Zr2SnC < V2SnC < Ti2SnC. High average sound velocity corresponds to a high Debye temperature of Ti2SnC. The new phase V2SnC has the second highest Debye temperature. Lu2SnC has a low Debye temperature because of its low average sound velocity. Generally, the higher the Debye temperature the stiffer the material. Therefore, V2SnC is softer than Ti2SnC and stiffer than the other existing M2SnC MAX phases. A low Debye temperature results in low thermal conductivity of a material, which favors it for being a promising thermal barrier coating (TBC) material.62 The Debye temperature of a promising TBC material, Y4Al2O9 is 564 K,63 which is larger than those of existing M2SnC phases. Therefore, M2SnC phases including new phase V2SnC have possibility to be potential TBC materials if they have low thermal conductivity, high thermal expansion coefficient, high melting point and oxidation resistance. For comparison, we have experimental Debye temperature only for Nb2SnC (380 K),60 which is comparable to the theoretical value (412 K).13
Lattice thermal conductivity is one of the most fundamental properties of solids. As the MAX phases have dual characters of metals and ceramics, therefore, to determine their lattice thermal conductivity, the Slack model is appropriate as it deals with materials having partial ceramic nature.64 The model considers the average of the atoms (M/n) in a “molecule” (or the atoms in the formula unit of the crystal) and their average atomic weight. This model is useful to determine the temperature-dependent lattice thermal conductivity of materials. On the other hand, Clarke's model is very advantageous for calculating the temperature-independent minimum thermal conductivity of compounds.1 Slack's equation for calculating the lattice thermal conductivity is
![]() | (16) |
In this formulation, Mav is the average atomic mass in kg mol−1, θD is the Debye temperature in K, δ is the cubic root of average atomic volume in m, n is the number of atoms in a conventional unit cell, T is the temperature in K, and γ is the Grüneisen parameter, which is calculated from the Poisson's ratio with the equation
![]() | (17) |
The factor A(γ) due to Julian65 can be obtained as
![]() | (18) |
The lattice thermal conductivity of V2SnC calculated at room temperature (300 K) is listed in Table 6 and its temperature dependence is shown in Fig. 9. Table 5 also lists the literature values for other existing M2SnC phases. Lattice thermal conductivity is highly sensitive to the Debye temperature. As the Debye temperature agrees fairly with the experimental value for the theoretical results with the GGA functional within the CASTEP and VASP codes, the room temperature lattice thermal conductivity calculated with the same functional within the same codes is expected to be consistent with the experiment if it is done in future. The reliability of Slack model has been established for MAX phases as their calculated lattice thermal conductivity agrees fairly well with the experimental values. For instance, the calculated (experimental) lattice thermal conductivity at 1300 K for Ta4AlC3 and Nb4AlC3 are 5 (6) W m−1 K−1 and 7 (7) W m−1 K−1, respectively.61 The lattice thermal conductivity at room temperature for M2SnC MAX phases ranges from 14 to 30 W m−1 K−1 within the CASTEP-GGA calculations, which does not exceed the typical range for MAX phases.66 Fig. 9 exhibits the gradual decrease of lattice thermal conductivity of M2SnC with the increase of temperature. The new phase V2SnC has lattice thermal conductivities very close to those of Lu2SnC for the whole range of temperatures. Lu2SnC is already predicted as better TBC materials among M2SnC (M = Lu, Ti, Nb, Zr, Hf) phases.12 Therefore, the new phase V2SnC is expected to be a promising TBC material as Lu2SnC.
The theoretical lower limit of intrinsic thermal conductivity of a material at high temperature is defined as its minimum thermal conductivity. The phonons become unpaired at high temperature and hence the heat energy is transferred to the adjacent atoms. In this situation, the mean free path of phonons is supposed to be the average interatomic distance. According to this approximation, different atoms can be substituted within a molecule with an equivalent atom having average atomic mass of M/n (n is the number of atoms in a primitive cell). A single “equivalent atom” within the cell never exhibits optical modes and hence it can be used to derive a formulation to determine the minimum thermal conductivity κmin at high temperature, as Clarke described in his model:67
![]() | (19) |
The symbols used in this expression carry the same meanings of those used in eqn (4). The minimum thermal conductivity calculated for the new MAX phase V2SnC is listed in Table 6 along with literature values for other M2SnC MAX phases. In the similar fashion of other properties, the minimum thermal conductivity calculated with GGA functional within CASTEP and VASP codes show more consistency than other results listed in Table 6. For comparison, we have another theoretical result of 0.755 W m−1 K−1 for Nb2SnC,61 which is identical to 0.76 W m−1 K−1 obtained in the present and a previous46 calculations with GGA within CASTEP and VASP codes. The new phase has the highest value of 1.20 W m−1 K−1 among M2SnC phases considering same functional within same code, which is very close to 1.13 W m−1 K−1 of a promising TBC material, Y4Al2O9.63 Additionally, the ultralow minimum thermal conductivity of 1.25 W m−1 K−1 is used for selecting appropriate materials for TBC applications.68 Therefore, M2SnC phases including new phase V2SnC have the possibility to be promising TBC materials.
![]() | (20) |
![]() | (21) |
Compound | Bond | nμ | dμ | Pμ | vμb | Hμν | HV | HV (expt.) | |
---|---|---|---|---|---|---|---|---|---|
V2SnC | V–C | 4 | 2.0526 | 1.02 | 0.05432 | 27.30 | 2.9 | 2.9 | |
Ti2SnC | Ti–C | 4 | 2.1414 | 1.08 | 0.01525 | 30.00 | 2.7 | 2.7 | 3.5,32 3.570 |
Zr2SnC | Zr–C | 4 | 2.3118 | 1.05 | 0.01302 | 36.18 | 1.9 | 1.9 | 3.5,32 3.970 |
Lu2SnC | Sn–C | 4 | 4.3478 | 0.12 | 0.00348 | 41.82 | 0.2 | 0.2 | |
Hf2SnC | Hf–C | 4 | 2.3158 | 1.39 | 0.00541 | 35.73 | 2.6 | 2.6 | 3.8,32 4.570 |
Nb2SnC | Nb–C | 4 | 2.2014 | 0.99 | 0.00139 | 31.98 | 2.3 | 2.3 | 3.8,70 3.570 |
The theoretical HV of Lu2SnC is very small compared to the lower limit of measured value for MAX phases. The reason may be the absence of typical M–C bond in the structure of Lu2SnC. This also reduces the elastic constants, elastic moduli and melting and Debye temperature in Lu2SnC. Consequently, Lu2SnC is the most soft and easily machinable compound in M2SnC as well as in MAX family. Indeed, the hardness of MAX phases is very small compared to their corresponding binary phases. Low hardness of MAX phases makes them machinable compounds. All phases in M2SnC family are easily machinable compared to many other MAX phases.
Antisite defects are point defects formed due to either recombination or occupation of atoms at alternative lattice sites during radiation damage.71 Low energy antisite formation energy indicates that a major population of residual defects will persist in a material, as a net reduction of defect mobility arises due to change of an interstitial into an antisite,71,74 The antisite formation mechanisms are given by the reactions (4)–(6) in Table 8.
Displacive radiation causes an athermal concentration of Frenkel pairs, as it is assumed that the radiation tolerance of materials depends on the resistance to form persistent populations of Frenkel (and antisite) defects.72 In this context, high defect energy is indicative of radiation tolerance. In a previous study of M2SnC (M = Lu, Ti, Zr, Hf, and Nb) phases, Nb2SnC is predicted as most radiation tolerant MAX phase in these systems.5 If the new phase V2SnC is included in these systems Nb2SnC remains at the same position. Comparing with other M2SnC phases, the radiation tolerance in V2SnC is better than Lu2SnC and lower than remaining ones.
Although the M interstitials, according to reaction (9), will recombine with to form MA antisites for all the M2SnC MAX phases studied here, there will be very little concentration of Mi in the first place due to the very high reaction energies of reaction (1) listed in Table 8. Under equilibrium conditions, this will effectively render reaction (7) practically irrelevant. Similar arguments are also applicable for the other antisite reactions (8) and (9). These reactions may become pertinent in the case of a non-equilibrium environment (i.e., under irradiation) where an increased defect concentration is feasible. In this environment, it is expected that Mi will recombine with
to produce MA antisites. Moreover, the creation of CSn through the reaction (8) should be anticipated for Ti2SnC. After irradiation, these processes may only be relevant, given that the formation energies of the Mi defects according to the Frenkel reaction (relation-(1)) are high for all the M2SnC MAX phases studied here (6.40–9.34 eV, refer to Table 8). The processes considered for the displacement of lattice atoms by interstitials (reactions (13)–(18)) are all positive in energy. The new phase requires lowest energy in Schottky reaction. From an experimental point of view, the radiation tolerance and oxidation resistance of M2SnC phases have to be determined at high temperature. A detailed understanding of the radiation tolerance of V2SnC requires systematic experimental work and simulation over a range of timescales and system sizes.
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