Electronic structure, bonding characteristics, and mechanical behaviors of a new family of Si-containing damage-tolerant MAB phases M5SiB2 (M = IVB—VIB transition metals)

MAB phases are layered ternary compounds with alternative stacking of transition metal boride layers and group A element layers. Until now, most of the investigated MAB phases are concentrated on compounds with Al as the A element layers. In this work, the family of M5SiB2 (M = IVB—VIB transition metals) compounds with silicon as interlayers were investigated by density functional theory (DFT) methods as potential MAB phases for high-temperature applications. Starting from the known Mo5SiB2, the electronic structure, bonding characteristics, and mechanical behaviors were systematically investigated and discussed. Although the composition of M5SiB2 does not follow the general formula of experimentally reported (MB)2zAx(MB2)y (z = 1, 2; x = 1, 2; y = 0, 1, 2), their layered structure and anisotropic bonding characteristics are similar to other known MAB phases, which justifies their classification as new members of this material class. As a result of the higher bulk modulus and lower shear modulus, Mo5SiB2 has a Pugh’s ratio of 0.53, which is much lower than the common MAB phases. It was found that the stability and mechanical properties of M5SiB2 compounds depend on their valence electron concentrations (VECs), and an optimum VEC exists as the criteria for stability. The hypothesized Zr and Hf containing compounds, i.e., Zr5SiB2 and Hf5SiB2, which are more interesting in terms of high-temperature oxidation/ablation resistance, were found to be unfortunately unstable. To cope with this problem, a new stable solid solution (Zr0.6Mo0.4)5SiB2 was designed based on VEC tuning to demonstrate a promising approach for developing new MAB phases with desirable compositions.


Introduction 
In the last two decades, a category of ternary-layered Nb, Mo, W), also known as the T2 phase, were proven to exist by Nowotny et al. [28] in 1957. These compounds crystallize in a D8 l structure, which can be described by a body-centered tetragonal unit cell (I32, space group I4/mcm) with Si as interlayers [29]. Other ternary borides with the D8 l structure have also been found experimentally in the Mo-Si-B [30][31][32], V-Si-B [33], Nb-Si-B [34], W-Si-B [35], Fe-Si-B [36], and Ta-Ge-B [37] systems. Therefore, it is clearly of interest to investigate whether M 5 SiB 2 can be considered a new family of the MAB phases by its nature and exhibits the desirable damage-tolerant and thermally shock-resistant properties. First-principles studies of V 5 SiB 2 , Nb 5 SiB 2 , and Ta 5 SiB 2 by Li et al. [29] have shown that they exhibit intrinsic brittleness. However, it is noted that the above three M 5 SiB 2 compositions have the same valence electron concentration (VEC). Considering the significant effect of VEC on the elastic properties of binary borides [38], more work is needed to investigate the M 5 SiB 2 compounds with different VECs.
In this work, first-principles calculations within the framework of DFT were carried out to systematically evaluate the thermodynamic stability, electronic structure, bonding characteristics, elastic properties, and damage tolerance of the known stable compounds M 5 SiB 2 (M = Mo, W, V, Nb), and the effects of VEC on these aspects were discussed. We then extended the selection of M atoms to other IVB-VIB transition metals for potential high-temperature applications. Among them, the candidates containing desirable M elements for oxidation/ ablation resistance, i.e., Zr and Hf, however, were found thermodynamically unstable. An approach based on the VEC tuning is then demonstrated by the stability of a newly designed composition of (Zr 0.6 Mo 0.4 ) 5 SiB 2 .

Calculation methods
The first-principles calculations were performed by the CASTEP code with the plane-wave method within the framework of DFT [39]. The exchange-correlation function was utilized with the generalized gradient approximation (GGA) in the scheme of Perdew-Burkey-Ernzerhof (PBE) [40]. The interaction between electrons and ion cores was treated in the reciprocal space by ultrasoft pseudopotentials [41]. The planewave basis set cutoff was 700 eV, and the k-points mesh separation was 0.04 Å −1 according to the Monkhorst Pack method in the Brillouin zone. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme [42] was used in the geometry optimization with the tolerance of difference on total energy within 5 × 10 −6 eV/atom, maximum ionic Hellmann-Feynman force within 0.01 eV/Å, maximum ionic displacement within 5 × 10 −4 Å, and maximum stress within 0.02 GPa. The calculations were accomplished under zero pressure, allowing all atomic sites, lattice constants, and angles to fully relax. The spin-orbit coupling effect, the employment of GGA+U approach, and the spinpolarization effect have been checked with representative compositions and found to have little effect on the calculation results for the M 5 SiB 2 phases investigated in this work (Figs. S1-S3 in the Electronic Supplementary Material (ESM)).
The band structure, density of states (DOS), and elastic constants of the M 5 SiB 2 phases were extracted from the fully geometry optimized state. The elastic constants were determined from a linear fit of the calculated stress as a function of strain. Bulk modulus (B) and shear modulus (G) were calculated from the second-order elastic constants (c ij ) according to Voigt [43], Reuss [44], and Hill [45] (VRH) approximations. Meanwhile, Young's modulus (E) was calculated from Hill's bulk modulus (B H ) and shear modulus (G H ).
Thermodynamic stability assessed by formation energy (ΔE) and formation enthalpy (ΔH) was calculated by Eqs. (1) and (2) [46]: where E MAB , E bin , and E ES are the total energy of the system, the energy of the atom in the bulk phase, and the energy of the equilibrium simplex, respectively. The equilibrium simplexes were selected from the stable compounds in the M-Si-B ternary system as competing phases, such as binary or ternary borides and silicides as listed in Table 1. Phonon calculations were performed with the supercell approach by the finite displacement method. The supercell was defined by the cutoff radius of 5 Å to ensure that the lattice constants of the supercell are larger than 10 Å. The k-points of 4 × 4 × 2 mesh was used for the phonon calculations. The bond stiffness was calculated using the model proposed by Bai et al. [15] at hydrostatic pressures of 0, 10, 20, 30, and 40 GPa.  Figure 1 illustrates the crystal structure and composition of Mo 5 SiB 2 . As shown in Fig. 1 [47] or networks as in Cr 3 AlB 4 [4]. The Mo1 atoms are all on the different (001) planes with B, as shown in Fig. 1(e), and the Mo1-B bond length is 2.347 Å. The first nearest neighbor of Si is the Mo1 atom at a distance of 2.553 Å. The Si atomic layer is the most loosely arranged layer, and the atoms are relatively far apart, as shown in Fig. 1(f), i.e., the distance between the Si atoms is as far as 4.241 Å in Mo 5 SiB 2 , which gives us a hint that the Si-Si in-plane interaction might be very weak in the Mo 5 SiB 2 structure. The above analyses of the crystal structure indicate that the nanolaminated structure of the Mo 5 SiB 2 phase is similar to those of the MAX and MAB phases.  Table 2. The agreement between the calculated and experimental data validates that the adopted calculation method is reliable. When M changes from 4d (Mo) to 5d (W), the c-axis decreases slightly due to the lanthanide contraction effect of the 5d transition metals, while along the same period Nb 5 SiB 2 has a larger lattice constant than Mo 5 SiB 2 since Nb has a larger atomic radius [48]. ΔE and ΔH are also calculated for these phases, which are all negative, indicating that these existing phases are indeed thermodynamically stable. Table 3 shows the Mulliken bond population and bond lengths in these M 5 SiB 2 phases. The positive Mulliken bond populations of B-B, M1-B, M2-B, and M1-Si bonds indicate their covalent bonding nature. The B-B bonds have a short bond length with a large bond population, indicating a strong bonding. However, they are relatively scarce, especially compared to M-B bonds, which is consistent with the absence of the B chains or networks in the structure. The M-B bonds are considered to be ionic-covalent based on their positive populations and the difference in electronegativity, which results in the electron transfer from the transition metal to boron, as can be seen in Table 4. Meanwhile, the lower bond population and less significant electron transfer from M1 to Si indicate that the ionic-covalent interaction between M1 and Si is very weak. The anisotropic bonds with significant differences in bond population within the crystal structure of M 5 SiB 2 are a clear indication of anisotropic bonding nature, which is also reflected in their electronic structure and chemical bonding, as discussed in Section 3.2.

2 Electronic structure and chemical bonding
The band structure and projected density of states (PDOS) of Mo 5 SiB 2 are shown in Fig. 2, and the Fermi level is set to 0 eV. The overlap of the energy bands with a high DOS at the Fermi energy level indicates a metal-like conductivity, and the anisotropy of the energy bands between high symmetry points shows the anisotropic electrical conductivity. The Fermi energy level lies above the bottom of the conduction bands, indicating that the excess valence electrons fill in the conduction bands. The DOS of Mo and B atoms overlap from −8.9 eV and persist until near −2.0 eV, in contrast to Mo1 atoms and Si atoms, which overlap only insignificantly between −6.0 and −3.0 eV. This phenomenon suggests that there is a stronger electronic interaction between Mo and B inside the BMo 8 sublattice, while that between Mo1 and Si across the layers is weaker.
The charge density ( Fig. 3) was employed to describe the charge accumulation in different energy ranges. Carbon-like electronic configuration (2s 2 2p 1 →2s 2 2p 2 ) was generated in boron through the transition of electrons from Mo to B. Figures 3(a) and 3(b) show that the states from −10.9 to −8.9 eV are mainly from the overlap of B sp orbitals, which form the strong σ-type covalent bonds. The states from −7.0 to −2.0 eV below the Fermi level are mainly from B 2p z , Mo1 4d(t 2g ), and Mo2 4d(e g ) orbitals, as shown in Figs. 3(c) and 3(d). This phenomenon confers boron diversity in hybridization, which is prevalent in the MAB phases. According to Figs. 3(e) and 3(f), the states near the Fermi energy level are mainly from 4d(t 2g ) orbitals of Mo1 and 4d(e g ) orbitals of Mo2, indicating that the main contribution to the conductivity is from Mo. Figure 3(g) shows the charge density of the plane containing Mo1, Si, and B atoms. It can be seen that the charge accumulation between Mo1 and B is closer to the B atom, which is typical of ionic-covalent bonding nature. However, the charge accumulation between Mo1 and Si is not obvious, suggesting a weaker interaction between them. In addition, no bonding is formed between Si atoms indicated by the absence of charge accumulation.     Figure 4 shows the band dispersion curves along with the high symmetry directions and PDOS of V 5 SiB 2 . Similar to Mo 5 SiB 2 , metallic bonding and electrical conductivity are confirmed by the overlap (crossing) of valence and conduction bands at the Fermi energy level. The DOS shifts to a higher energy than that of Mo 5 SiB 2 due to the lower VEC of V 5 SiB 2 . As a result, the Fermi energy level lies closer to the pseudogap, which indicates a more stable electronic structure [38]. Similar to Mo 5 SiB 2 , the interaction energy between V and B is lower than that between V1 and Si, implying a stronger V-B bond and a weaker V1-Si bond.
Above we have analyzed the electronic structure based on two typical compounds Mo 5 SiB 2 and V 5 SiB 2 . The variations of the electronic structure mainly derive from the valence electron of the M atom. V 5 SiB 2 has a lower VEC = 4.375 compared to Mo 5 SiB 2 with VEC = 5.000. By comparing the relative positions of the pseudogap and the Fermi energy level in the DOS of Mo 5 SiB 2 and V 5 SiB 2 , a more stable electronic structure with the VEC near 4.375 is obtained for V 5 SiB 2 . To further verify the effect of VEC on the electronic stability, the PDOS of W 5 SiB 2 (VEC = 5.000) and   with the Fermi energy level of W 5 SiB 2 being located in the conduction band and Nb 5 SiB 2 close to the pseudogap. Thus, a simple criterion about the stability of the electron is obtained. A VEC value close to 4.375 is expected to enable a more stable electronic structure.

3 Lattice dynamics
The lattice vibrations were identified by phonon dispersion and DOS, and the lattice dynamics also reflect the anisotropy of the chemical bonds. The phonon dispersion curves are shown along the high-symmetry direction of the Brillouin zone (Figs. 6 and 7), where no negative vibrational frequencies appear, thereby confirming that Mo 5 SiB 2 and V 5 SiB 2 are dynamically stable to mechanical perturbations. It can also be seen that there are five band gaps in the range of optical phonons for both Mo 5 SiB 2 and V 5 SiB 2 due to the mass differences among the atoms and the weak interactions of the atoms.  The projected phonon density of states for different atoms M1, M2, Si, and B were calculated separately to investigate the contribution of different bonds to the total phonon density of states. In general, the lowfrequency acoustic branch is mainly from M1 and M2, and the high-frequency optical branch is mainly from B-B and M-B. Also, there are mid-frequency optical branches between the high-frequency optical branch and the low-frequency acoustic branch. The midfrequency optical branches are mainly from M-Si, which indicates a weaker interaction between them [19]. The combination of anisotropic bonds will be shown to affect the elastic properties as well as the mechanical behavior of the compounds in the next section.

4 Elastic properties and mechanical behaviors
The six independent second-order elastic constants (c 11 , c 33 , c 44 , c 66 , c 12 , and c 13 ) and anisotropic Young's modulus (E x and E z ) of M 5 SiB 2 with the tetragonal system are listed in Table 5. Born-Huang criteria [49] suggested that the following conditions are necessary for tetragonal solids to be mechanically stable: c 11 > |c 12 |; c 33 (c 11 + c 22 ) -2c 2 13 > 0; c 44 > 0. All the selected M 5 SiB 2 satisfy the above-mentioned criteria, demonstrating their mechanical stability. The calculated elastic constants (c 11 and c 33 ) representing the stiffness against compression/tension deformation in main crystalline directions are very large compared to other elastic constants, indicating that the M 5 SiB 2 system is very rigid under uniaxial stress along with the directions of the a-axis and c-axis. Furthermore, the value of c 11 is higher than that of c 33 , which signifies that the bonding along the crystallographic a-axis is stiffer than that along the c-axis. The stiffness along the a-axis is largely contributed by the B-B bonds. In contrast, the interlayered M-Si bonds along the c-axis are relatively weaker, which leads to a lower c 33 . Therefore, the anisotropy of chemical bonds also leads to that E x being higher than E z . The c 44 and c 66 reflect the resistance of tetragonal crystals to shear deformation at the (100)/ (010) and (001) planes, respectively, which are all smaller than c 11 and c 33 , demonstrating a low shear deformation resistance of these new MAB phases. The anisotropic mechanical properties are like the wellinvestigated MAX and MAB phases, which are underpinned by the anisotropic chemical bonding.
The calculated elastic moduli including the bulk  [26] and MoAlB (137 GPa) [47], and lower than Cr 2 AlB 2 (197 GPa) and Cr 3 AlB 4 (182 GPa) [52]. As a result of the combined higher B and lower G, a lower G/B is achieved for Mo 5 SiB 2 . The G/B has been long used as a criterion to distinguish between ductility/ brittleness of compounds [15,38,53]. The material with a ratio larger than a critical value of 0.571 is considered brittle, and a lower ratio normally indicates improved ductility. The G/B of Mo 5 SiB 2 and W 5 SiB 2 were found to be 0.53 and 0.50, respectively, even lower than those of MoAlB [54] and Ti 3 SiC 2 [55], which have been considered to possess a promising damage tolerance. In contrast, V 5 101]. The low G min indicates a metal-like shear stress resistance and is the source of the local ductility of compounds [56]. Furthermore, the projections of the shear modulus deviate significantly from the round shape (sphere), confirming the presence of significant shear anisotropy.

Table 6 Bulk modulus (B), shear modulus (G), Young's modulus (E), Cauchy pressure (p), Pugh's ratio (G/B), and Poisson's ratio (v) of M 5 SiB 2 and several typical MAB and MAX phases
Compound Moreover, Mo 5 SiB 2 and W 5 SiB 2 show consistently positive p and large v in comparison to other MAB and MAX phases, as listed in Table 6. Positive p means that the resistance to the shear stress is weaker than the normal stress and the large v also indicates that the shape deformation is easier.
Based on the above results, some of the M 5 SiB 2 materials, e.g., Mo 5 SiB 2 have the potential to exhibit higher damage tolerance than most of MABs already investigated. Using Mo 5 SiB 2 as an example, the bonding characteristics are further discussed to understand their effects on the mechanical properties.
There are three types of covalent bonds in Mo 5 SiB 2 from strong to weak: (i) strongest B-B bonds distributed in the (001) plane; (ii) sublattice composed of BMo 8 face-shared tetrakaidecahedron with second strongest ionic-covalent bonds between the B and Mo atoms; and (iii) interlayer ionic-covalent bonds composed of Mo1 and Si, which are the weakest. In addition, no bonding is formed between the Si atoms. The weak Mo1-Si bonds contribute to low shear modulus and a lower c 33 (compared to c 11 ) due to that the Si layers are interspersed along the c-axis.
However, bond anisotropy is also present in other MAB phases and cannot fully explain the very low Pugh's ratio value of Mo 5 SiB 2 . A more quantitative perspective in terms of the ratios among different types of bonding within a unit cell in Mo 5 SiB 2 is given in Table 7 and compared to that of other typical MAB phases. It can be seen that, the number ratio of Mo-B to B-B bonds is the highest in Mo 5 SiB 2 , reaching 16, while it is only 2 in MoAlB. As a result of the small  , also support this idea as they coincide with the directions of the channels between the B-B bonds.

1 Damage tolerance and fracture toughness of Mo 5 SiB 2
In Section 3.4, Mo 5 SiB 2 was judged to exhibit promising damage tolerance due to the low Pugh's ratio, positive Cauchy pressure, and high Poisson's ratio. On the other hand, Zhang et al. [57] showed that the fracture toughness of polycrystalline Mo 5 SiB 2 was not particularly high (3.34 MPam 1/2 ). To shed more light on this discrepancy, the criterion based on the bond stiffness model [58], which characterizes the chemical bond strength indirectly by measuring the elastic response of the chemical bonds between adjacent atoms under hydrostatic pressure, was adopted to assist the evaluation of the tough/brittle properties of Mo 5 SiB 2 .
Bai et al. [15,47] have shown that MAX and some MAB phases are considered to have a higher fracture toughness when the ratio of the lowest bond stiffness (k min ) to the highest bond stiffness (k max ) is less than 0.5. The bond stiffness of Mo 5 SiB 2 is calculated and listed in Table 8. The k max occurs in the Mo2-B bonds, and the k min is found for Mo1-Si bonds. The k min /k max of Mo 5 SiB 2 is greater than 0.5, which indicates that the bond stiffnesses within the crystal are similar, implying a low fracture toughness. Therefore, the experimentally measured fracture toughness appears to be consistent with the bond stiffness model for Mo 5 SiB 2 . However, it is worth noting that the damage tolerance characterizes the sensitivity of the material to contact damage, which is not equivalent to the fracture toughness. Polycrystalline graphite is an important example. The fracture toughness was only 0.73-0.85 MPa·m 1/2 measured experimentally [59], but graphite exhibits insensitivity to notch geometry and is therefore considered a high damage tolerance material [60]. Therefore, in-depth studies are still needed in the future to study the damage tolerance and its correlation with fracture toughness in the MAB phase materials.

2 Extension to other M 5 SiB 2 phases
When the choice of M is extended to the transition metals of group IVB-VIB, the effect of VEC on stability and elastic properties is more clearly demonstrated. We calculated and compared the enthalpies of formation ΔH of M 5 SiB 2 (M = Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W), which were naturally divided into three groups based on VEC, as shown in Fig. 9. Among them, Ti 5 SiB 2 , Zr 5 SiB 2 , and Hf 5 SiB 2 have VEC = 3.750, which are thermodynamically unstable as predicted by the positive ΔH. The PDOS of Ti 5 SiB 2 , Zr 5 SiB 2 , and Hf 5 SiB 2 are plotted in Figs. 10, S1, and S2 in the ESM. The Fermi energy level lies below the peak of the bonding state, indicating incomplete filling of the valence band consistent with the positive ΔH. With a higher VEC = 4.375, the three compounds of V 5 SiB 2 , Nb 5 SiB 2 , and Ta 5 SiB 2 are thermodynamically stable in accordance with the location of the Fermi energy level close to the pseudogap (Figs. 3, 5(b), and S3 in the ESM). When VEC increased to 5.000, the filling of the d-orbital electron to the conduction band decreases the electronic stability (Figs. 2, 5(a), and S4 in the ESM). Although Mo 5 SiB 2 and W 5 SiB 2 remain thermodynamically stable, metastable phases like Cr 5 SiB 2 [49] appear. The above results follow the VEC criterion for electronic structure stability described in Section 3.2. The calculated lattice constants and elastic modulus of M 5 SiB 2 (M = Ti, Zr, Hf, Ta, Cr) are listed in Table 9. Combined with the data in Table 6, it can be seen that when VEC = 4.375, V 5 SiB 2 , Nb 5 SiB 2 , and Ta 5 SiB 2 exhibit peak values of Pugh's ratio among all M 5 SiB 2 phases. The shear modulus is closely related to the M d-B p(sp 2 ) bonding [38]. As the VEC decreases to 3.750, the reduction of the d-orbital electron leads to the decrease in the shear modulus and thus a reduction in G/B of Ti 5 SiB 2 , Zr 5 SiB 2 , and Hf 5 SiB 2 . On the other hand, when VEC increases to 5.000, additional electrons saturate the p(sp 2 )-d bonding state and fill the M d(t 2g ) and d(e g ) orbitals near the Fermi level, again reducing G and correspondingly G/B in Cr 5 SiB 2 , Mo 5 SiB 2 , and W 5 SiB 2 . It is found that the former effect is more significant, and therefore Ti 5 SiB 2 , Zr 5 SiB 2 , and Hf 5 SiB 2 phases possess a low Pugh's ratio of 0.52, 0.46, and 0.49, respectively. Considering Young's modulus of 207, 158, and 192 GPa, these compounds would exhibit good damage tolerance and thermal shock resistance if they could be stabilized.
Overall, we confirmed the effect of VEC on the stability and elastic properties of the Mo 5 SiB 2 -type compounds. A relationship between the thermodynamic stability and the electronic stability was found, and it is

3 Composition optimization for M 5 SiB 2
Our calculation shows that Mo 5 SiB 2 is expected to be a damage-tolerant material with quasi-ductile characteristics. However, oxidation is a necessary consideration for high-temperature applications. Since the oxidation products of Mo are sparse and volatile, which is not favorable for oxidation resistance [51], the M elements forming high stability oxidation products, such as Zr and Hf, would be more ideal options. Unfortunately, as demonstrated in Section 4.2, although Zr 5 SiB 2 and Hf 5 SiB 2 show even lower Pugh's ratios of 0.46 and 0.49 and adequate Young's moduli of 158 and 192 GPa respectively, they are thermodynamically unstable since their ΔH are 1.348 and 1.579 eV/atom, respectively. As discussed in Section 4.2, the instability of Zr 5 SiB 2 and Hf 5 SiB 2 is likely to be originated from the lack of sufficient valence electrons. Therefore, it is possible to stabilize them by optimizing the VEC to near 4.375, which is the minimum VEC value for the stabilization of M 5 SiB 2 as found in Section 3.2. Thus, a crystal model of (Zr 0.6 Mo 0.4 ) 5 SiB 2 was built to represent the disordered solid solution using the Atomsk code [61]. As the ratio of Mo to Zr increases to 2 : 3, the VEC reaches 4.250, which is very close to 4.375 and expected to render a stable electronic structure. Depending on the designed chemical composition, 8 of the 20 Zr atoms were randomly replaced with the Mo atoms without human intervention. To ensure the reliability of the calculation, we constructed five random models, as shown in Fig. 11, and used the average values of the 5 models. ΔH was calculated according to Eq. (2), and the equilibrium simplexes are shown in Table 1. The calculated ΔH of −1.226±0.009 eV/atom indicates that the solid solution of (Zr 0.6 Mo 0.4 ) 5 SiB 2 is thermodynamically stable at 0 K. The small error among the five models also indicates that the results are reliable. The calculated lattice constants are a = 6.3338±0.0119 Å, b = 6.3349±0.0141 Å, and c = 12.0485±0.0647 Å, which lie between those of Zr 5 SiB 2 and Mo 5 SiB 2 . The PDOS of (Zr 0.6 Mo 0.4 ) 5 SiB 2 is depicted in Fig. 12(a), where the Fermi energy level is located near the pseudogap, indicating a stable electronic structure of this solid solution. In addition, there is no imaginary frequency in phonon dispersion of this material, as shown in Fig. 12(b), suggesting that the lattice dynamics are stable. The elastic constants and moduli of the elasticity of (Zr 0.6 Mo 0.4 ) 5 SiB 2 are listed in Tables 10 and 11, respectively. Firstly, the solid solution is mechanically stable according to the Born-Huang criteria [49]. Secondly, the elastic constants exhibit similar anisotropy as analyzed in Section 3.2. Finally, the large increase in the shear modulus is attributed to the solid solution effects, leading to an increase in the Pugh's ratio to 0.61. However, it is much lower than that of V 5 SiB 2 (Pugh's ratio = 0.78) and lower than the Five randomly generated unit cells of (Zr 0.6 Mo 0.4 ) 5 SiB 2 . typical MAB phases such as MoAlB (Pugh's ratio = 0.66). An attempt to evaluate the damage tolerance of (Zr 0.6 Mo 0.4 ) 5 SiB 2 was done using the bond stiffness model (Tables S2, S3,    In summary, the calculated results in ΔH, electronic DOS, and phonon dispersion all consistently show that the proposed (Zr 0.6 Mo 0.4 ) 5 SiB 2 could be a stable solid solution with potentially good damage tolerance properties. Similarly, the study on the M site ordered and disordered T2 phases by Dahlqvist et al. [46] has also shown that the VEC of the stabilizing phases is all above 4.000, such as Ti 4 MoSiB 2 (VEC = 4.000), Ti 4 MnSiB 2 (VEC = 4.125), and (V 0.8 Mo 0.2 ) 5 SiB 2 (VEC = 4.500). In combination with our calculations, it appears that M 5 SiB 2 is most likely to be unstable when VEC is smaller than 4.000. On the other hand, the study by Dahlqvist et al. found that the ordered Zr 4 MoSiB 2 and disordered (Zr 0.8 Mo 0.2 ) 5 SiB 2 are thermodynamically unstable even with a VEC value of 4.125, and the current work shows that further increasing VEC to 4.250 with the composition of (Zr 0.6 Mo 0.4 ) 5 SiB 2 is sufficient to stabilize the Zr containing compound.

Conclusions
1) M 5 SiB 2 consists of alternating layers of four faceshared BM 8 tetrakaidecahedrons and Si atoms. The covalent B-B bonds with sp hybridization form a discrete distribution on the (001) plane, and large amounts of M 4d-B 2p bonds formed by the electron transfer are present in the BM 8 tetrakaidecahedron. A weak M 4d-Si 3p bonding is present between the BM 8 and Si layers. The anisotropy of the chemical bond and the slip channel between the B-B bonds combine to result in a particularly low Pugh's ratio of 0.53 in Mo 5 SiB 2 . It is lower than other common MAB phases reported in the literature [5,19,47,52] and is therefore considered to potentially have a better damage tolerance.
2) The VEC plays a dominant role in the electronic stability and thermodynamic stability of M 5 SiB 2 . The M 5 SiB 2 compounds containing the group VB M elements (V, Nb, and Ta) have an optimum VEC of 4.375, resulting in the most stable structure, while partial filling of the valence band or conduction band reduces the stability for M 5 SiB 2 , represented by Zr 5 SiB 2 and Cr 5 SiB 2 .
To stabilize Zr 5 SiB 2 , which has an even lower Pugh's ratio of 0.46 and the desirable M element for potential high-temperature applications, the solid solution (Zr 0.6 Mo 0.4 ) 5 SiB 2 was designed with the principle of VEC optimization and is predicted to be stable based on the calculated thermodynamic, electronic, mechanical, and lattice dynamics.