Chemically stable new MAX phase V₂SnC: a damage and radiation tolerant TBC material

Abstract

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.

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1. Introduction

MAX phases are a family of more than 80 ternary carbides, nitrides and borides in hexagonal crystal symmetry.^1,2 This family is chemically represented as M_n+1AX_n, where M is an early transition metal, A is an A-group element, X is C or N or B, and the integer ‘n’ ranges from 1 to 3.³ Depending on the value of layer index n, MAX phases are categorized as 211, 312 and 413 phases for n = 1, 2, 3, respectively. MAX phases crystallize in the hexagonal space group P6₃/mmc (194). In their crystal structures, M₆X-octahedra with the X-elements fill the octahedral positions between the M-elements as do in the corresponding MX binaries. The octahedra exchange with the A-atomic layers placed at the centers of trigonal prisms, which are larger, and thus more accommodating of the larger A-atoms. The interposing pure metallic A-atomic planes are mirror planes to the meandering ceramic M_n+1X_n slabs. Due to alternating metallic and ceramic layers in MAX phases they possess a unique set of metallic and ceramic properties.⁴ The common metallic properties are electrical and thermal conductivities, high fracture toughness, machinability, damage tolerant and thermal shock resistance. The typical ceramic properties are lightweight, oxidation and corrosion resistance, elastic stiffness, resistant to fatigue and ability to maintain the strength to high temperature.⁵ The MAX phases also experience plastic-to-brittle transitions at high temperatures, and can resist high compressive stresses at ambient temperature.⁶ These exceptional properties of MAX phases make them suitable for potential uses as tough and machinable, coatings for electrical contacts, thermal shock refractories, and heating elements at high temperature. Additionally, their neutron irradiation resistance makes them suitable in nuclear applications. A common recent use of MAX phases is as precursors for producing two-dimensional MXenes.⁷ The MAX phases are attractive due to their unique combination of structural characteristics, wide range of properties, and many prospective uses.

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.

2. Methodology

All calculations are performed with the DFT method as implemented in CASTEP.¹⁴ The non-spin polarized Perdew–Burke–Ernzerhof (PBE) functional within generalized gradient approximation (GGA) is chosen to describe the electronic exchange-correlation potential.¹⁵ Ultra-soft pseudo-potential developed by Vanderbilt is used to model the interactions between electrons and ion cores.¹⁶ A Γ-centered k-point mesh of 15 × 15 × 3 grid in the Monkhorst-Pack (MP) scheme is used to integrate over the first Brillouin zone in the reciprocal space of the MAX phase hexagonal unit cell.¹⁷ To expand the eigenfunctions of the valence and closely valence electrons in terms of a plane-wave basis, a cutoff energy of 700 eV is chosen. Total energy and internal forces are minimized during the geometry optimization with the BFGS minimization technique named after by the first letter of the name of Broyden–Fletcher–Goldfarb–Shanno.¹⁸ The self-consistence convergence is achieved with the difference in the total energy within 5 × 10⁻⁶ eV per atom, the maximum ionic Hellmann–Feynman force within 0.01 eV Å⁻¹, maximum ionic displacement within 5 × 10⁻⁴ Å, and maximum stress within 0.02 GPa. For self-consistent field calculations, the tolerance is chosen as 5 × 10⁻⁷ eV per atom.

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.

3. Results and discussion

3.1. Structural aspects and phase stability

The newly synthesized V₂SnC phase crystalizes in the hexagonal MAX phase crystal structure with space group P6₃/mmc (no. 194). The position of each atom is found as V at 4f (1/3, 2/3, 0.0744), Sn at 2d (2/3, 1/3, 1/4), and C at 2a (0, 0, 0). The V and Sn atoms stack along the z-direction (c-axis) (see Fig. 1, where M represents V, A refers to Sn and X is C). There are two layers of V atoms in each V–C slab, and every two layers of V atoms and one layer of Sn atoms are consecutively arranged along the z-direction. The optimized lattice constants a and c and internal parameter z_M are in agreement with the experimental and theoretical values.¹³ The present values (a = 3.121 Å, c = 12.947 Å, z_M = 0.0759) are closer to the experimental results (a = 2.981 Å, c = 13.470 Å, z_M = 0.0776) compared with the previous theoretical values (a = 3.134 Å, c = 12.943 Å, z_M = 0.0751). The reason may be the use of coarse k-point mesh (9 × 9 × 2) and low cutoff energy (400 eV) in the previous theoretical study. In the M₂SnC systems, we observed that the unit cell parameters show a better relationship with the crystal radius of M atoms.¹² In this relation, the lattice parameters exhibit increasing trend with the increase of crystal radius of transition metal M. The newly synthesized V₂SnC also obeys this relationship (refer to Fig. 2) (Table 1).

Fig. 1 (a) Crystal structure and (b) 2D view in yz-plane of 211 MAX phase.

Fig. 2 Lattice parameters of M₂SnC as a function of crystal radius of M atoms.³³

Table 1 Lattice parameters (a, c, z_M in Å), hexagonal ratio (c/a) and cell volume (V in ų) of M₂SnC

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

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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 ΔH_cp by comparing its energy to the energy of the equilibrium simplex according to

ΔH_cp = E(V₂SnC) − E(equilibrium simplex) (1)

The phase V₂SnC is considered to be stable if ΔH_cp < 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

ΔH_cp = E(V₂SnC) − E(V₂C) − E(Sn) (2)

where E(V₂SnC), E(V₂C), and E(Sn) are the ground state total energies of V₂SnC, V₂C and Sn, respectively. Using eqn (2), we find ΔH_cp = −7 meV per atom. The negative value of ΔH_cp is indicative of the chemical (thermodynamic) phase stability of V₂SnC. In the next Sections 3.2 and 3.5, we will find that V₂SnC is mechanically and dynamically stable compound as well.

Table 2 Lattice parameters, unit cell volume and total energies of V₂SnC and its competing phases

Phase	Prototype structure	Pearson symbol	Space group	a (Å)	b (Å)	c (Å)	V (Å^3)	E (eV fu^−1)
C	C (graphite)	hP4	P63/mmc (194)	2.462744	2.462744	8.985673	47.2	−155.088
C	Diamond	cF8	Fd3̄m (227)	3.567776	3.567776	3.567776	45.4	−154.901
V	W	cI2	Im3̄m (229)	3.011843	3.011843	3.011843	27.3	−1976.398
α-Sn	Diamond	cF8	Fd3̄m (227)	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	F4̄3m (216)	5.134950	5.134950	5.134950	135.4	−248.722
SnC	NaCl	cF8	Fm3̄m (225)	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	P3̄1m (162)	5.017396	5.017396	4.546570	99.1	−4109.186
V2C	Inverse CdI2	hP3	P3̄m1 (164)	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	P3̄m1 (164)	5.003269	5.003269	4.535382	98.3	−4109.191
VC2	MoB2	tR18	R3̄m (166)	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	Pm3̄n (223)	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	Pm3̄m (221)	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

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3.2. Elastic properties

Elastic constants estimate the response of the crystalline solids to external stresses and measure the strength of the materials. Elastic constants also provide a fundamental insight into the nature of bonding character between adjacent atomic planes and the anisotropic character of the bonding and structural stability. They can link between a material's dynamical behaviour and its mechanical and thermal properties. For hexagonal MAX phases, five nonzero independent elastic constants, namely C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄ are obtained.³⁶ Table 3 lists the elastic constants of the newly synthesized V₂SnC calculated at zero pressure and zero temperature along with the values found in literatures for existing M₂SnC phases for comparison. For Ti₂SnC, Nb₂SnC, Hf₂SnC, and Zr₂SnC, we have listed C_ij calculated with either different codes or different functionals. The CASTEP-GGA results are consistent to the VASP-GGA values. Fig. 3a presents C_ij calculated with CASTEP-GGA, in which the M-elements are shown along the x-axis according to the order of their groups for seeking a trend. All C_ij show a tendency of monotonic increase when the M-element moves from left to right across the group-3 to −5, though C₁₃ shows almost linear increase. The constant C₆₆ is not independent as C₆₆ = (C₁₁ − C₁₂)/2. The elastic constants of M₂SnC including newly synthesized V₂SnC fulfills the mechanical stability criteria for hexagonal crystals:³⁷

C₁₁, C₃₃, C₄₄ > 0; C₁₁ > |C₁₂| and (C₁₁ + C₁₂)C₃₃ > 2C₁₃C₁₃ (3)

Table 3 Elastic properties of M₂SnC (M = V, Ti, Zr, Nb and Hf) MAX phases

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-GGA^13
Lu2SnC	172	173	56	64	46	36	82	61	147	0.199	1.33	CASTEP-GGA^12
Ti2SnC	268	265	100	95	79	74	139	97	236	0.217	1.43	CASTEP-GGA^12
	253	254	93	79	91	74	138	87	217	0.238	1.57	VASP-GGA^46
	337	329	169	126	86	102	176	138	329	0.188	1.27	FP-L/APW + lo^47
	303	308	121	109	84	88	160	114	275	0.212	1.40	CASTEP-LDA^48
							152	83.9	207.4	0.24		Experimental^39,41
Zr2SnC	230	232	94	84	62	91	131	83	206	0.237	1.57	CASTEP-GGA^12
	225	227	87	77	72	90	131	78	196	0.251	1.68	VASP-GGA^46
	269	290	148	94	81	107	157	110	268	0.215	1.42	FP-L/APW + lo^47
	279	272	111	104	70	89	147	104	252	0.215	1.42	CASTEP-LDA^48
									178			Experimental^39
Hf2SnC	251	238	101	90	71	107	145	87	218	0.250	1.67	CASTEP-GGA^12
	249	252	99	85	73	101	144	87	218	0.247	1.65	VASP-GGA^46
	330	292	167	138	54	126	173	132	316	0.195	1.30	FP-L/APW + lo^47
	311	306	119	109	92	97	167	112	275	0.225	1.49	CASTEP-LDA^48
							169		237			Experimental^40
Nb2SnC	255	236	94	77	102	122	160	78	202	0.290	2.05	CASTEP-GGA^12
	253	250	98	74	103	120	160	80	206	0.286	2.00	VASP-GGA^46
	341	321	183	118	106	169	209	126	315	0.250	1.67	FP-L/APW + lo^47
	315	309	124	108	99	141	189	107	189	0.262	1.77	CASTEP-LDA^48
							180		216			Experimental^40

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Fig. 3 Elastic constants and moduli of M₂SnC as a function of M-elements.

Furthermore, for all M₂SnC phases, it is observed that the principal elastic constants C₁₁ and C₃₃ are larger than all other C_ij. 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, C_ij 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:

and

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 Hf₂SnC^46–48 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 A_X of each elastic modulus X is defined as follows:

Table 4 Minimal and maximal values of each modulus and elastic anisotropy obtained from them

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

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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 A_X 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 C_ij that can be determined using the succeeding expressions:⁵³

which is associated with the {100} shear planes between the 〈011〉 and 〈010〉 directions;which is related to the {010} shear planes between the 〈101〉 and 〈001〉 directions; and finally,which signifies shear anisotropy in the {001} shear planes between the 〈110〉 and 〈010〉 directions. For isotropic crystals, all A_i's (i = 1, 2, 3) have unit value. A value other than unity quantifies the anisotropic state of crystals. The deviation of A_i from unity (ΔA_i) measures the level of elastic anisotropy in shear.

The calculated values of A_i for V₂SnC are listed in Table 5 along with reported values for other M₂SnC phases and the anisotropy level ΔA_i 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 C_ij, i.e. k_c/k_a = (C₁₁ + C₁₂ − 2C₁₃)/(C₃₃ − C₁₃) is used to quantify the elastic anisotropy upon compression; where k_a and k_c are the linear compressibility coefficients along the a- and c-axis, respectively.⁵⁴ Deviation of k_c/k_a from unity (Δ(k_c/k_a)), 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.

Table 5 Elastic anisotropy factors for M₂SnC (M = Lu, Ti, Zr, Hf and Nb) MAX phases

Phases	A1	A2	A3	kc/ka	AB%	AG%	A^U	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-GGA^12
Ti2SnC	0.9683	1.0582	1.0247	1.0419	0.0088	0.0284	0.0030	CASTEP-GGA^12
Zr2SnC	0.6950	1.1190	0.7778	0.7801	0.2082	0.9355	0.0986	CASTEP-GGA^12
Hf2SnC	0.6106	1.1222	0.6852	0.8244	0.1093	1.6814	0.1732	CASTEP-GGA^12
Nb2SnC	0.6046	1.2288	0.7429	0.9912	0.0020	1.6962	0.1726	CASTEP-GGA^12

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Fig. 5 Shear anisotropy level in M₂SnC MAX phases.

According to Hill, the difference between B_V and B_R as well as G_V and G_R is proportional to the degree of elastic anisotropy of crystals, which leads to define the percentage anisotropy factors A_B and A_G with the succeeding equations:⁵⁵

The percentage anisotropy factors A_B and A_G 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

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.

3.3. Electronic properties

Electronic structure plays a significant role in understanding of material properties at the microscopic level. Electronic energy band structure calculated along high symmetry points of the Brillouin zone for V₂SnC is shown in Fig. 6a. Similar to other M₂SnC and remaining MAX phases, the band structure of V₂SnC reveals the metallic characteristics as a large number of its valence bands cross the Fermi level E_F and overlap with the conduction bands. The position of Fermi level in V₂SnC is just below the valence band maximum near the Γ-point as in Ti₂SnC.⁵ In Nb₂SnC, the Fermi level is above the valence band maximum at the Γ-point.⁵ The Γ-point, where the maximum of the valence bands accumulate, lies above the Fermi levels of Sn-based other MAX phases Hf₂SnC, Zr₂SnC and Lu₂SnC (see Fig. 6 in ref. 5). The band structure of V₂SnC is very similar to that of Nb₂SnC rather than other M₂SnC phases.⁵ Nb₂SnC is a superconducting phase. Resemblance of two band structures indicates the possibility of superconductivity in V₂SnC. The main characteristic of the band structure is the significantly anisotropic nature with less energy dispersion along the c-axis. It is evident from the reduced dispersion along the short H–K and M–L directions. The anisotropic band structure near and below the Fermi level implies that the electrical conductivity is also anisotropic for the new MAX phase studied here.

Fig. 6 Electronic structures of V₂SnC, (a) band structure and (b) density of states; E_F 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 E_F, whereas the total DOS at E_F 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.

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.

3.4. Thermal properties

In this section, we have reported the elastic Debye temperature, melting point, lattice thermal conductivity and minimum thermal conductivity of V₂SnC. Debye temperature θ_D is a characteristic temperature of solid materials that can be calculated from the elastic moduli using Anderson method.⁵⁹ This method is simple and rigorous, which determines θ_D using average sound velocity calculated from the shear and bulk moduli via the equation:

In this equation, ℏ and k_B are respectively the Plank and Boltzmann constants, N_A 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 v_m is obtained from the longitudinal and transverse sound velocities v_l and v_t by the equation:

With the bulk modulus B and shear modulus G, v_l and v_t can be determined as:

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 θ_D 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 θ_D for Nb₂SnC.^60,61 The remaining two sets of θ_D 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.

Table 6 Sound velocities in km s⁻¹, Debye temperature and melting point in K, minimum and lattice thermal conductivity in W m⁻¹ K⁻¹ of M₂SnC (M = Lu, Ti, Zr, Hf and Nb) MAX phases

Phases	ρ	vl	vt	vm	θD	Tm	κmin	κph^b	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-GGA^12
Ti2SnC	6.346	6.503	3.910	4.325	525	1556	0.99	29.98	CASTEP-GGA^12
	6.346	6.327	3.703	4.106	498	1494	1.23	22.24	VASP-GGA^46^,^a
	6.473	7.113	4.337	4.790	585	1859	1.45	49.51	FP-L/APW + lo^47
	6.76	6.783	4.099	4.532	561	1725	1.08	36.87	CASTEP-LDA^48
Zr2SnC	7.313	5.749	3.369	3.735	426	1392	0.76	20.61	CASTEP-GGA^12
	7.313	5.669	3.266	3.627	414	1370	0.73	17.22	VASP-GGA^46^,^a
	7.280	6.357	3.831	4.236	483	1596	0.86	34.67	FP-L/APW + lo^47
	7.75	6.111	3.683	4.073	472	1599	0.86	31.82	CASTEP-LDA^48
Hf2SnC	11.796	4.704	2.716	3.015	348	1464	0.63	15.92	CASTEP-GGA^12
	11.796	4.695	2.716	3.015	348	1479	0.63	16.15	VASP-GGA^46^,^a
	11.828	5.228	3.118	3.446	398	1782	0.72	32.92	FP-L/APW + lo^47
	12.06	5.121	3.050	3.376	393	1746	0.71	26.94	CASTEP-LDA^48
Nb2SnC	8.369	5.616	3.053	3.469	412	1473	0.76	12.38	CASTEP-GGA^12
	8.369	5.645	3.092	3.448	410	1488	0.76	12.61	VASP-GGA^46^,^a
	8.388	6.358	3.626	4.030	480	1859	0.89	29.96	FP-L/APW + lo^47
	8.53	6.150	3.493	3.883	469	1763	0.87	22.92	CASTEP-LDA^48

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Fig. 8 Elastic sound velocities and Debye temperature of M₂SnC phases.

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

In this formulation, M_av is the average atomic mass in kg mol⁻¹, θ_D 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

The factor A(γ) due to Julian⁶⁵ can be obtained as

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.

Fig. 9 Variation of lattice thermal conductivity of M₂SnC phases with temperature T.

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 κ_min at high temperature, as Clarke described in his model:⁶⁷

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.

3.5. Vibrational properties

To verify the dynamical stability of the newly synthesized V₂SnC MAX phase, the phonon dispersion and phonon density of states are investigated. The phonon dispersion curve is shown in the left panel of Fig. 10. There is no negative phonon frequency in the whole Brillouin zone. The absence of negative phonon frequency ensures the absence of soft phonon modes, indicating that the phase V₂SnC is dynamically stable against the mechanical perturbation at ambient state like the other existing M₂SnC phases.¹² 211 MAX phases have eight atoms in their unit cell, which lead to 24 vibrational modes including three acoustic and 21 optical modes. The lower branches correspond to the acoustic modes (orange) and the upper branches with frequencies greater than 2 THz correspond to the optical modes (light blue). Lower optical branches overlap with the acoustic branches and consequently there is no phononic band gap between the acoustic and optical branches. The zero phonon frequency of the acoustic modes at the Γ point is another indication of dynamical stability of the V₂SnC MAX phase. The phonon DOS shown in the right panel of Fig. 10, reveals that the acoustic and lower optical modes arise due to the vibration of heavier atoms Sn and V. The higher optical modes mainly originate from the vibration of lighter atom C. Acoustic phonon is caused by the coherent vibrations of atoms in a lattice outside their equilibrium position. On the other hand, when an atom moves to the left and its neighbour to the right, the optical phonon is originated due to the out-of-phase oscillation of the atom in a lattice. Optical phonons control the most of the optical properties of crystals.

Fig. 10 Phonon dispersion and phonon DOS of V₂SnC.

3.6. Vickers' hardness

Herein, the theoretical method based on Mulliken population developed by Gou et al.⁶⁹ is used to calculate the Vickers' hardness of partial metallic compounds. Within this method, the bond hardness H^μ_v is calculated as:where P^μ is the Mulliken overlap population of the μ-type bond, is the metallic population and can be calculated with the unit cell volume V and the number of free electrons in a cell, n_free as follows: and E_P and E_F are the energy at the pseudogap and at the Fermi level, respectively, v^μ_b is the volume of μ-type bond and is calculated from the bond length d^μ of μ-type and the number of bonds N^v_b of v type per unit volume using the equation . Then, the theoretical Vickers hardness for complex multiband crystal can be determined as a geometric average of all bond hardness values as follows:where n^μ represents the number of μ-type bonds. The Vickers' hardness calculated for M₂SnC including new phase V₂SnC is listed in Table 6. The new phase V₂SnC has highest Vickers' hardness in the M₂SnC family. There are two sets of experimental values for Ti₂SnC, Zr₂SnC, Hf₂SnC, and Nb₂SnC.^32,70 The determined values show deviations from one set to another, except in the case of Ti₂SnC. Indeed, the determined values depend on the purity of the sample, instrumental set up and error. The present theoretical values (refer to Table 7) also differ from the experimental values. The temperature of the sample may be an additional reason. The theoretical H_V of M₂SnC ranges from 0.2 to 2.9 GPa. It is worth mentioning that the measured values of H_V for MAX phases range from 2 to 8 GPa.

Table 7 Bond number n^μ, bond length, d^μ (Å), bond population P^μ, bond volume v^μ_b (ų), bond hardness H^μ_ν (GPa), metallic population , and hardness H_V (GPa) of M₂SnC MAX phases

Compound	Bond	n^μ	d^μ	P^μ		v^μb	H^μν	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 3.5^70
Zr2SnC	Zr–C	4	2.3118	1.05	0.01302	36.18	1.9	1.9	3.5,^32 3.9^70
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 4.5^70
Nb2SnC	Nb–C	4	2.2014	0.99	0.00139	31.98	2.3	2.3	3.8,^70 3.5^70

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The theoretical H_V 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.

3.7. Defect processes

Frenkel defect energies provide the information regarding nuclear applications of a material as the low pair formation energy is linked to a higher content of more persistent defects. These in turn cause the loss of ordering in the structure of a crystal. An accumulation of defects in a crystal that are formed by the displacement cascades are indicative of radiation tolerance of the material.⁷¹,⁷² In Table 8, the relations (1–3) are the key reactions for the Frenkel defects in Kröger–Vink notation⁷³ for M₂SnC phases.

Table 8 The defect reaction energies as calculated for V₂SnC and existing M₂SnC [5] MAX phases

	Reaction (V′ denotes vacancy)	Defect energy (eV)
		V2SnC	Lu2SnC	Ti2SnC	Zr2SnC	Hf2SnC	Nb2SnC
1		6.40	6.61	8.75	8.66	9.34	8.70
2		7.95	3.57	8.97	6.63	7.51	7.56
3		5.12	2.23	6.10	5.34	4.68	5.18
4	MM + SnSn → MSn + SnM	4.67	3.67	4.92	4.83	4.72	5.12
5	MM + CC → MC + CM	9.37	11.79	12.81	15.40	16.37	12.64
6	SnSn + CC → SnC + CSn	8.64	7.75	9.98	9.64	10.07	10.05
7		−5.17	−3.61	−6.86	−4.71	−5.17	−4.34
8		−0.80	−0.13	−1.07	0.12	1.47	−0.48
9		−4.51	−2.90	−5.94	−5.75	−6.96	−6.79
10		0.03	1.56	−0.19	0.22	0.89	−0.10
11		−1.35	3.08	−0.97	1.28	0.88	−0.76
12		−4.46	0.39	−4.91	−2.55	−3.01	−2.58
13	Mi + SnSn → MSn + Sni	3.44	0.67	3.03	0.88	0.55	0.76
14	Mi + CC → MC + Ci	3.77	5.31	5.13	6.62	5.56	4.42
15	Sni + MM → SnM + Mi	1.24	3.01	1.89	3.95	4.17	4.36
16	Sni + CC → SnC + Ci	0.66	2.62	1.19	2.79	1.67	2.60
17	Ci + MM → CM + Mi	5.60	6.49	7.69	8.78	10.81	8.22
18	Ci + SnSn → XSn + Sni	7.98	5.13	8.79	6.85	8.40	7.46
	Schottky reaction	5.83	9.99	7.97	9.69	8.57	6.70

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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 M_A antisites for all the M₂SnC MAX phases studied here, there will be very little concentration of M_i 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 M_i will recombine with to produce M_A antisites. Moreover, the creation of C_Sn through the reaction (8) should be anticipated for Ti₂SnC. After irradiation, these processes may only be relevant, given that the formation energies of the M_i 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.

4. Conclusions

In summary, the density functional theory is employed to investigate the phase stability and physical properties of a newly synthesized 211 MAX phase, V₂SnC for the first time. The calculated results are compared with those of other existing M₂SnC (M = Ti, Zr, Hf, Nb, and Lu) phases. The newly synthesized compound V₂SnC has passed the mechanical, dynamic and thermodynamic stability tests. The new phase V₂SnC is the second most capable phase to resist the deformation in M₂SnC systems following the order: Nb₂SnC > V₂SnC > Hf₂SnC > Ti₂SnC > Zr₂SnC > Lu₂SnC. It also has the second highest value of B and θ_D and an intermediate value of G and E in the M₂SnC family. V₂SnC is softer than Ti₂SnC and stiffer than other existing M₂SnC MAX phases. V₂SnC has potential to be etched into 2D MXene like the other M₂SnC phases. V₂SnC and Nb₂SnC are ductile and damage tolerant and the remaining phases are brittle in nature. The directional dependence of E, β, G and ν of M₂SnC is calculated. All M₂SnC phases show directional dependence of E, G, and ν in the xz and yz planes. Ti₂SnC and Lu₂SnC show almost directional independency on β. Elastic anisotropy in V₂SnC is higher than Ti₂SnC and less than the other M₂SnC phases. The band structure and Fermi surface are indicative of possible superconductivity of V₂SnC. V₂SnC is anticipated to be a promising TBC material as Lu₂SnC among M₂SnC phases. V₂SnC is more radiation tolerant than Lu₂SnC and less than the remaining other M₂SnC phases.

Authors contribution

M. A. Hadi: conceptualization, data curation, investigation, methodology, formal analysis, writing – original draft. M. Dahlqvist: software, formal analysis, review & editing; S.-R. G. Christopoulos: investigation, data curation; S. H. Naqib: project administration, review & editing; A. Chroneos: formal analysis, writing, review & editing. A. K. M. A. Islam: formal analysis, review & editing.

Data availability

Supplementary data will be made available on request.

Conflicts of interest

There are no conflicts of interest to declare.

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