DFT predictions and experimental confirmation of mechanical behaviour and thermal properties of the Ga-bilayer Mo2Ga2C


 Mo2Ga2C is a new MAX phase with a stacking Ga bilayer as well as possible unusual properties. To understand this unique MAX-phase structure and promote possible future applications, the structure, chemical bonding, mechanical and thermodynamic properties of Mo2Ga2C were investigated by first principles. Using the "bond stiffness" model, the strongest covalent bonding (1162 GPa) were formed between Mo and C atoms in Mo2Ga2C, while the weakest Ga-Ga (389 GPa) bonding were formed between two Ga-atomic layers, different from other typical MAX phases. Of interest, the ratio of the bond stiffness of the weakest bond to the strongest bond (0.33) was lower than 1/2, indicating the high damage tolerance and fracture toughness of Mo2Ga2C, which was confirmed by indentation without any cracks. The high-temperature heat capacity and thermal expansion of Mo2Ga2C were calculated in the framework of quasi-harmonic approximation from 0 K to 2000 K. Because of the metal-like electronic structure, the electronic excitation contribution became more significant with increasing temperature above 300 K.


Introduction
Over the past two decades, a class of ternary transition metal carbides or nitrides known as MnAXn+1 phases (where M is an early transition metal, A is a group IIIA or IVA element, X is C and/or N, and n = 1-3), formed by inserting A-group atoms into the corresponding binary carbides or nitrides, have attracted growing attention due to their unique combination of metallic and ceramic properties [1]. Having this layered structure, the MAX phases exhibit a combination of the beneficial properties of both ceramic and metallic compounds, e.g. low density [2], low thermal expansion coefficient, high modulus, high strength [3], high temperature oxidation resistance [4], good thermal conductivity [5], electrical conductivity, easy processing [6], plastic and thermal impact resistance [5].
The M2AX (211) phases including solid solutions with M = Ti, V, Cr, Nb, Ta, Zr, Hf, A = Al, S, Sn, As, In, Ga, and X = N, C, have been studied extensively both experimentally and theoretically [7][8][9][10][11][12][13]. Among them, Mo2GaC [14] is an important MAX phase showing superconducting characteristics with Tc ∼ 4.0 K. Mo2GaC is one of few having Mo as the sole occupant of the M-site [15] and was predicted to have a high bulk and low shear moduli [16]. Recent studies on the MAB phase had shown that MoAlB with Al bilayer had better damage tolerance and antioxidant properties [17,18].
Interestingly, the discovered Mo2Ga2C in 2015 [19] also has a Ga bilayer, which provides us a chance to examined the effect of the A-group bilayer on the properties of the MAX phases.
Mo2Ga2C had been mainly used to produce MXenes materials since it was discovered by Hu et al in 2015 [20][21][22]. At present, the research on Mo2Ga2C mainly focused on its mechanical properties, Hadi et al [23] calculated elastic properties including Debye temperature and theoretical Vickers hardness and Wang et al [24] predicted the stability of another structure of Mo2Ga2C under pressure. However, there was little research on Mo2Ga2C in the field of high temperature. He et al [25] found Mo2Ga2C will decompose when the temperature is higher than 700 o C and new study of Jin et al found that Mo2Ga2C's thermal conductivity and electrical resistivity were lower than those of most MAX phases probably due to the extra Ga layer [26].
For structural applications, the mechanical properties are critical, especially the damage tolerance and fracture toughness [27]. Well known, the crystal structure has an important influence on the macroscopic mechanical properties of materials. For example, it is generally believed that the high damage tolerance and high fracture toughness in MAX phase come from its layered structure and weak chemical bonding [28,29]. Moreover, as a candidate for high-temperature materials, it is important to understand the thermal properties, including their heat capacities, and thermal expansion coefficients among others [30]. In addition to experiments, the recent progress in computational methods reminds that First principles can be used to accurately predict the mechanical behaviour by the "bond stiffness" theoretical model [28,31], and thermal properties by quasi harmonic approximation (QHA) [32], with some significant advantages, such as low cost and high efficiency [33].
The present work is to investigate the mechanical and thermal properties as well as phase stability of Mo2Ga2C with the Ga bilayer by First principles, which would provide a theoretical guidance for further understanding the influence of crystal structure on macro behaviour of MAX phases, and inspire future experimental research.

Density functional theory settings
All first principles calculations were performed within the framework of density functional theory (DFT) as implemented in VASP (the Vienna Ab initio Simulation Package) [34]. A selection of exchange-correlation functional was used, including those within generalized gradient approximation (GGA) [35], comprising PBE, RPBE [36] and PW91 [37], and also a local density approximation (LDA) functional [38]. The projector-augmented wave (PAW) method was used, with 4p 6 4d 5 5s 1 , 4s 2 4p 1 and 2s 2 2p 2 electrons included as valence states for Mo, Ga and C respectively, with the cut-off energy of 350 eV and k-points sampled using Monkhorst-Pack meshes of 12×12×2 for the geometry optimization and the elastic constant determination, and 36×36×6 when calculating the electronic density of states, resulting in that total energies were converged to within ±1 meV/atom, with atomic forces less than 1 meV/Å. The relaxation of atomic geometries was achieved using the conjugate-gradient method.
The supercell approach and the force-constant method were used for the phonon dispersion and density of states. The real space force constants of the supercells were calculated using density functional perturbation theory (DFPT), and the phonon modes were calculated based on force constants using the PHONOPY package [39]. A 2×2×1 supercell with an energy cut-off of 350 eV and a k-points mesh of 6×6×2 was constructed for the resulting forces on the perturbed atoms [40], with the phonon dispersion along the four high-symmetry points (Г(0 0 0), F(0 0.5 0), Q (0 0.5 0.5), Z (0 0 0.5)) spanning the whole Brillouin zone. The convergence condition for the calculation is that the change in total energy is less than 10 -8 eV/atom.

Thermodynamic analysis on phase stability
To assess the phase stability of a particular MAX phase it is essential to account for the stability of all other competing compounds. The phase stability of Mo2Ga2C was examined by "linear optimization procedure" developed by Dahlqvist et al [41] for MAX phases, to determine the most stable combination of competing phases for a given chemical composition b M , b A and b X . The optimization problem can be expressed by the following equation: where xi and Ei are the amount and energy of competing compound i, respectively.
Notably, the minimization has to be subjected to the constraints where E is the total energy of Mo2Ga2C.

Model of bond stiffness
In the previous work on the MAX phases, Bai et al proposed a theoretical model to quantificationally calculate the bond stiffness from the DFT-simulated results for characterizing the bond strength [28,31], and based on the calculated bond stiffness further established a criteria to assess the damage tolerance and fracture toughness: high damage tolerance and fracture toughness without indentation cracks are observed when the ratio is lower than 1/2, but above this value cracks are present in the Vickers' indentation [28]. This model assumed that a second-order polynomial relation exists between the applied hydrostatic pressure (P) and deformation of chemical bonds in a solid. In practice, the interatomic distance (bond length, d) as a function of P can be estimated from the lattice parameters and internal coordinates. Because the bond strength varies with increasing d, the relative bond lengths d/d0 (d0 is the bond length at 0 GPa) as a function of P should be then fitted by a quadratic curve, whose slope is defined as 1/k, where k is the bond stiffness [31].

Heat capacity and thermal expansion
Here In the present work, F(V, T) is calculated from first principles within the framework of DFT and QHA [32], where phonon frequencies are volume-dependent, but at a fixed volume they are independent of temperature. In practice, the free energy is expressed as the sum of the following separate parts: where Etot(V) is the total energy at T = 0 K, Fel(V, T) is the finite temperature electronic free energy as a result of electronic excitation, and Fqha(V, T) is the free energy due to atomic vibrations (phonon), computed within QHA with the help of PHONOPY package [39]. The volume at each temperature must be calculated by minimization of the free energy at constant temperature. This removes the explicit temperature dependence in the frequencies: ωi = ωi(V). The phonon contribution to free energy is given by where ℏ is the reduced Planck constant and kB is the Boltzmann constant. Furthermore, CP(T) can be estimated from the calculated F(V, T) using the standard thermodynamic relation. The electronic free energy Fel(V, T) is usually divided into the energy Eel(V, T) due to electronic excitations and a remaining part as [42] el el el where Sel(V, T) is the electronic entropy, given by ideal mixing as where g is 1 or 2 for collinear spin polarized and non-spin polarized systems, and the fi(V, T) is sum runs over all electronic states with Fermi-Dirac occupation weights where εi(V) is obtained by integrating the density of states and Eel(V, T) is given by

Experimental details
The fabrication details of Mo2Ga2C were described elsewhere [25,26]. In brief,

Crystal structures
The previous work [18,29] indicates that the exchange-correlation functional usually has a significant influence on the accuracy of DFT-calculated results. In the present work, four exchange-correlation functionals (LDA, GGA-PBE, GGA-RPBA and GGA-PW91) were used for the equilibrium lattice constant of Mo2Ga2C (Table 1).
The LDA results over-bind as are often the case with /this approximation, and are the least accurate of the considered functional [44]. The bond lengths between adjacent atoms (marked in Figure 1) of Mo2Ga2C are given in

Intrinsic and thermodynamic phase stability
The intrinsic stability of Mo2Ga2C is investigated by examining its lattice dynamics (phonon). The phonon dispersion and density of state of Mo2Ga2C are showed in Figure 2. Of importance, no imaginary frequencies in the phonon spectrum mean the

Electronic structure and bond stiffness
Total and partial density of states of Mo2Ga2C are showed in Figure 3, including It is well established that weak M-A bond plays a key role in its unique mechanical properties of the MAX phases, such as high fracture toughness and damage tolerance, low hardness, micro-scale plastic deformation and so on [52,53]. The previous work indicates that the ratio kmin/kmax of the lowest bond stiffness to the highest bond stiffness determines the macroscopic mechanical properties of the ternary layered ceramics including MAX and MAB phases [18,28,29]. When kmin/kmax is more than 1/2, this class of ceramics exhibit low damage tolerance and fracture properties of typical brittle ceramics with indentation cracks, such as Ti2SC [54], (MC)nAl3C2 and (MC)nAl4C3 [55]. When the ratio is less than 1/2, these ceramics show high damage tolerance and high fracture toughness without indentation cracks, such as MoAlB and typical MAX phase ceramics [44].  Figure 5). Furthermore, it shows the model of "bond stiffness" is reliable.

Thermodynamic properties
The isobaric heat capacities CP of Mo2Ga2C as well as Mo2GaC and experimental value of MoC [56] against increasing the temperature are illustrated in 1.14 10 Interestingly, the CP of Mo2Ga2C is 1.26-1.29 times of that of Mo2GaC in the high-temperature range above 300 K, which is also observed in the previous studies that there is a certain relationship between the heat capacity of MAX phases and the corresponding binaries [1]: In fact, the CP of MAX phases is determined approximately by their constituent elements, which is applicable for the present Mo2Ga2C. However, due to the double Ga layer, the CP of Mo2Ga2C is 2.3 times of that of MoC.
The thermal expansion of Mo2Ga2C and Mo2GaC in the temperature range of 300K-2000 K are shown in Figure 6b. Notably, the average linear thermal expansion coefficient (TEC) with electronic excitations for Mo2Ga2C from 300 K to 2000 K is 24.6×10 -6 K -1 , where the electronic excitations act to decrease the thermal expansion as a function of temperature. As the temperature rises above 1000 K, the slope of thermal expansion increasing speed is more and more slow, close to its high temperature limit.
The thermal expansion of these two ternary carbides increases gradually due to the successively excitation of vibration modes.
In addition, in the whole temperature range, the coefficient of thermal expansion of Mo2Ga2C is larger than that of Mo2GaC, which is mainly due to the weaker Ga-Ga bond of the former and the downward shift of its phonon mode. To improve the prediction, and given the metal-like electronic structure, electronic excitations were included in addition to the standard phonon contribution. The average TECs of the MAX phases fall into the range of ≈5-15×10 -6 K -1 [5], which means the present TEC of Mo2Ga2C is higher than other MAX phases. In general, TECs are intimately related to the interatomic bond strengths. Compared with typical MAX phases Ti2AlC, Mo2Ga2C has the weaker Ga-Ga bond (Table 2), resulting the higher TEC than Ti2AlC (8.8×10 -6 K -1 ) [57]. This means the more serious deformation of Mo2Ga2C at high temperature.

Conclusion
(1) The structure of Mo2Ga2C could be described as a layered structure similar to MAX phase: the strong binding Mo-C atomic layer and the Mo-Ga layer with weak Al-Al bonds were staggered. The structure of Ga atoms on the same side had lower energy. Inserting Ga layers only increased the direction of c axis, but did not change the bond length significantly.
(2) The phase stability of Mo2Ga2C is studied by lattice dynamics theory and linear optimization method based on the first principle respectively, which shows the intrinsic and thermodynamic stability of Mo2Ga2C.         Heat capacity (a) and thermal expansion (b) of Mo2Ga2C, while "exp" means the data from ref. [56].