M₂M'AlB₄ (M = Mn, Fe, Co, M' = Cr, Mo, W): Theoretical predicted ordered MAB phases with Cr₃AlB₄ crystal structure

1. Introduction

Compounds with nanolaminated crystal structure have attracted great investigation interests due to their unusual properties [[1], [2], [3]]. The well-known MAX phases (M_n+1AX_n, M is an early transition metal, A is a III_A-VI_A group element, X is carbon or nitrogen) are a distinctive family of nanolaminated compounds [[1], [2], [3], [4], [5], [6]], which are formed by alternative stacking of M_n+1X_n layers and A element layers. In MAX phases, chemical bonds within M_n+1X_n layer are stiff and hard, while bonding between M_n+1X_n layer and A element layer is soft. The diverse chemical bonds and layered nature in crystal structure lead them to display both metal-like and ceramic-like properties, including good electrical and thermal conductivities, moderate elastic properties, good thermal shock and oxidation resistances, and tolerant to damage, etc [[1], [2], [3], [4], [5], [6]]. The unique combination of these excellent properties makes them promising for applications in high and ultra-high temperatures. Moreover, it was found that etching the A element out from MAX phases can produce a new set of 2D materials, MXene [6], which stimulates enormous research interests into these materials.

Inspired by successes of MAX phases, a new set of nanolaminated compounds, MAB phases (M is a transition metal, A is Al or Si, and B is boron), have been synthesized and characterized in recent years [[7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]]. Similar to MAX phases, MAB phases are formed by alternative stacking of M-B stiff layers and A element layers, which were confirmed by direct observation using Z-contrast high resolution scanning transmission electron microscopy (HRSTEM) by Lu et al. [10]. The new MAB phases that have been investigated include (CrB₂)_nCrAl (n = 1,2,3) [7], Cr₄AlB₄ [8], Fe₂AlB₂, Mn₂AlB₂ [[9], [10], [11], [12], [13]], MoAlB, WAlB [7,9,14,15]. Experimentally, Ade and Hillebrecht [7] synthesized signal crystals of (CrB₂)_nCrAl (n = 1,2,3), Mn₂AlB₂, MoAlB and WAlB, identified their crystal structures and measured their microhardness [7,8]. Kota et al. [14,15] synthesized MoAlB and systematically measured its properties, including mechanical, thermal and oxidation properties. Other MAB phases, including Fe₂AlB₂ and Mn₂AlB₂, have also been synthesized and characterized [[9], [10], [11], [12], [13]]. Moreover, similar to MAX phases, possible 2D MBenes has been reported by selective etching Al out from Cr₂AlB₂ [16]. Theoretically, first principles calculations based on density functional theory (DFT) have been applied to investigate chemical bonding characteristic, elastic properties, thermal properties and preferred failure models of MAB phases [17,[8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]]. Except for MAB phases mentioned previously, other ternary borides that display MAX-phase-like properties have also been investigated, such as Y₅Si₂B₈ [[24], [25], [26]].

Recently, a new set of MAX phases, ordered quaternary MAX phases, have been discovered, which enriches the family of MAX phases, and opens a new window to tailor the properties of MAX phases and to develop new MXenes [[27], [28], [29], [30], [31]]. For example, Liu et al. [27] reported the ordered Cr₂TiAlC₂ by neutron diffraction, where the Cr-layers sandwiched the Ti-layers in a M₃AX₂ phase structure. Caspi et al. [28] used high-resolution neutron diffraction to show that V and Cr atoms showed a strong tendency to ordering, with V only occupying the middle layer in (Cr_0.5V_0.5)_n+1AlC_n system. Anasori et al. [29,30] reported ordered Mo₂TiAlC₂ and Mo₂Ti₂AlC₃ with the Mo layers sandwiching the Ti-layers, where the ordered structure was confirmed by direct observation by using Z-contrast HRSTEM. Except for those out of plane ordering, in plane ordering has also been found in MAX phases by Martin et al. [31]. Up to date, no study on ordered quaternary MAB phase has been reported. Since ordering not only provides the opportunity to tailor the properties of MAB phases but also opens the window for compound design, it is the scope of the present work to predict possible ordered MAB phases by first principles calculations based on DFT.

2. Calculation methods

First-principles calculations were performed using the Cambridge Serial Total Energy Package (CASTEP) code [32] with the ultrasoft pseudopotential [33] and exchange-correlation described by generalized gradient approximation (GGA) based on the Perdew-Burke-Ernzerhof (PBE) scheme [34]. The cutoff energy of plane wave basis set was set to be 400 eV. k-points mesh with a separation of 0.04 Å^-1 according to Monkhorst-Pack method [35] was adopted in the Brillouin zone. Spin polarization was taken into consideration in the calculations. Both ordered and disordered cells were optimized under zero pressure by using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) [36] minimization scheme. The convergence criteria for optimizations were set as follows: the difference in total energy within 5 × 10^-6 eV/atom, the maximum ionic Hellmann-Feynman force within 0.01 eV/Å, the maximum ionic displacement within 5 × 10^-4 Å, and maximum stress difference within 0.02 GPa. These settings for calculation have been tested in our previous work [23], with which the relaxed crystal structures are consistent with experiments.

3. Results and discussion

Among those reported MAB phases, e.g. Cr₂AlB₂, Cr₃AlB₄, Cr₄AlB₆, MoAlB etc., we chose Cr₃AlB₄ to investigate possible ordered quaternary phases. The structure of Cr₃AlB₄ is shown in Fig. 1(a), which is crystallized with space group Pmmm [7]. In this crystal structure, Cr atoms occupy 2t (0.5, 0.5, 0.188) and 1g (0, 0.5, 0.5) positions, Al atoms occupy 1a (0, 0, 0) positions, while B atoms occupy 2s (0.5, 0, 0.394) and 2q (0, 0, 0.288) positions. There are two distinct Cr sites, 2t and 1g, in the unit cell, as shown in Fig. 1(a). The 2t site locates adjacent to Al atoms, which will be referred to as M site in the ordered structure. The 1g site locates at the center of hexagonal boron rings, which will be referred to as M' site in the ordered structure. Then, different elements may prefer either 2t site or 1g site due to the differences in local chemical environments, which can result in ordered quaternary MAB phases. Fig. 1(b) and (c) respectively illustrates an ordered and a disordered M₂M'AlB₄.

3.1. Stability of ordered M₂M'AlB₄

Elements selected for M and M' sites include Cr, Mo, W, Mn, Fe, Co and Ni, most of which are experimentally reported M elements in MAB phases. To predict possible ordered M₂M'AlB₄ phases, energies of ordered and disordered M₂M'AlB₄ (M, M' = Cr, Mo, W, Mn, Fe, Co and Ni) were calculated and energies of other competition compounds, including elementary substances, binary compounds and ternary compounds, were also calculated. Disordered structures were modeled by using supercells consisting of 3 × 3×1 M₃AlB₄ unit cells. Solid solutions on M and M' sites were generated by using the spatial quasi-random structure (SQS) method implemented in ATAT package [37,38], where the distribution of elements in M and M' sites were chosen based on the criterion to mimic the spatial correlation function. The method has been broadly adopted to investigate order-disorder configurations on substitution atoms, vacancy, spin and etc. [[27], [28], [29], [30], [31],[39], [40], [41]].

From thermodynamic point of view, a stable ordered phase needs to have relatively lower energy than those of its competition phases, which can be represented by the following equation:

ΔH=Hor_{dered phase}-$\sum_{i}$n_i$H^{i}_{competition phasei}$<0

subjec to:n_i>0, $\sum_{i}$n_im_ij=M_j

where H is enthalpy, n_i is the equilibrium content of phase i without considering the ordered phase, m_ij means composition of element j in phase i and M_j is the overall composition of element j in the system. The phases competing to ordered M₂M'AlB₄ considered in this work include: (1) disordered M₂M'AlB₄, (2) elementary substances M (M = Cr, Mo, W, Mn, Fe, Co and Ni), Al and B, (3) intermetallic compounds MAl (NiAl structure, space group Pmm), MAl (MnAl structure, space group P4/mmm), MM₃Al (Mo₃Al structure, space group Pmn), M₃Al (Fe₃Al structure, space group Fmm), (4) binary borides M₃B₄ and ordered ternary borides M₂M'B₄, (5) ternary MAB phases M₃AlB₄.

For simplicity of discussion, the following enthalpy changes were calculated:

$H_{orderedM_{2}M'AlB_{4}}\sum_{i}$n_i$H^{i}_{competition phase}$ (1)

$H_{orderedM_{2}M'AlB_{4}}-\frac{2}{3}H_{M_{3}AlB_{4}}-\frac{1}{3}H_{M'_{3}AlB_{4}}$ (2)

$H_{orderedM_{2}M'AlB_{4}}-H_{disorderedM_{2} M'AlB_{4}}$ (3)

In Eq. (1) disordered M₂M'AlB₄ and ternary MAB phases M₃AlB₄ were excluded from competition compounds. The enthalpy changes of Eq. (1) to Eq. (3) are shown in Fig. 2(a) - (c), respectively. In Fig. 2, the results related to diagonal M₃AlB₄ phases were all set to 0, since here we only study possible quaternary ordered phases. Most enthalpy changes in Fig. 2(a) show large negative values except for those compounds containing Ni. It means that most of these ordered M₂M'AlB₄ phases are stable in comparison to their competition phases, including elementary substances, intermetallic compounds, binary borides and ordered ternary borides. Then, ordered M₂M'AlB₄ phases can be synthesized from these compounds in principle. Fig. 2(b) and (c) display similar characteristics, where the lower left region shows negative enthalpy changes, while the upper right region shows positive enthalpy changes. Then, compounds locate at the lower left region have great possibility to form stable ordered M₂M'AlB₄ phases instead of disordered M₂M'AlB₄ phases or separated M₃AlB₄ and M'₃AlB₄, especially when M = Mn, Fe and Co and M' = Cr, Mo and W. In addition, the ordering tendency increases when M changes from Mn to Co, which can be deduced from enthalpy changes between ordered and disordered M₂M'AlB₄ phases in Fig. 2(c). Though compounds with M = Ni also show negative values in Fig. 2(b) and (c), they do not show large negative values in Fig. 2(a). Therefore, formation of quaternary M₂M'AlB₄ phase with M (or M') = Ni from elementary substances or other compounds may not be easy. It is found that Ni-Al intermetallic compounds are always existed in the most competition phases. As is well-known, Ni-Al intermetallic compounds are stable hardening phases in superalloys, which then may show low tendency to further transform to other compounds.

To further verify whether ordered M₂M'AlB₄ (M = Mn, Fe and Co and M' = Cr, Mo and W) phases are preferred, the entropy effect is considered. The free energy change ΔG is evaluated as ΔG = ΔH_ordering - TΔS, where ΔS = -Nk_B[xlnx + (1-x)ln(1-x)] is configurational entropy change between the fully ordered phase and random disordered phase. N is the number of M and M' sites in the unit cell, k_B is Boltzmann constant, x = 1/3. Fig. 3 shows contour surface of ΔG with respect to ΔH_ordering and T. The line with ΔG = 0 represents the phase boundary between ordered and disordered M₂M'AlB₄ phases. It reveals that even when T is as high as 1500 K, enthalpy change of ordering ΔH_ordering with a value lower than -0.25 eV/cell can result in an ordered phase. Fig. 2(c) shows that ΔH_orderings of M₂M'AlB₄ (M = Mn, Fe and Co and M' = Cr, Mo and W) phases are all lower than -0.30 eV/cell, indicating strong ordering tendency of these compounds. As reported in experiments, the synthesis temperatures of MAB phases are around 1000 °C [[7], [8], [9], [10], [11], [12], [13], [14], [15]]. Then, it is highly possible that ordered M₂M'AlB₄ (M = Mn, Fe and Co and M' = Cr, Mo and W) phases can be synthesized in experiments.

3.2. Chemical bonding origin of ordering in M₂M'AlB₄

To explore the chemical bonding origin of ordering in M₂M'AlB₄, electron density of state (DOS) from constituent elements were analyzed. Fig. 4(a) and (b) show the partial DOS contributions from s, p, d orbitals of constituent elements in Co₂CrAlB₄ and Cr₂CoAlB₄, respectively. Fig. 5(a) and (b) show the partial DOS contributions from s, p, d orbitals of constituent elements in Fe₂MoAlB₄ and Mo₂FeAlB₄, respectively. By comparing the bonding peaks in Fig. 4(a) and (b), Fig. 5(a) and (b), it can be seen that there is an extra bonding peak adjacent to Fermi level in Al-p orbital in Fig. 4(a) and Fig. 5(a). This extra peak contributes to bonding between Al-Co in Co₂CrAlB₄ or Al-Fe in Fe₂MoAlB₄. It means that bonding between Al-Mn, Al-Fe or Al-Co might be stronger than bonding between Al-Cr, Al-Mo or Al-W, which may serves as the main driving force of ordering.

Fig. 6 plots partial DOS curves of Al-p orbital in Cr₃AlB₄, Mn₂CrAlB₄, Fe₂CrAlB₄, and Co₂CrAlB₄. It shows that the main bonding peaks of Al-p for Mn₂CrAlB₄, Fe₂CrAlB₄, and Co₂CrAlB₄ almost overlap with each other, while the main bonding peak of Cr₃AlB₄ locates at a slightly higher level. It indicates a stronger bonding between Al and Mn, Fe, Co in comparison to Al-Cr. This finding is consistent with the fact that Al can form stable intermetallic compounds with Mn, Fe, Co. In addition, the assembling structure between Al and Mn, Fe, Co in intermetallic compounds is close to that in ordered M₂M'AlB₄ phases, which is CsCl like. The extra bonding peak in Fig. 6 gradually shifts from high energy level to low energy level when M changes from Cr to Co. The shifting of bonding peak to lower level means that bonding strength gets stronger. It agrees well with ΔH_ordering values in Fig. 2(c), which shows that ordering tendency increases when M changes from Mn to Co.

Though element preference at 2t site can be well illustrated from DOS peaks, element preference at 1g site is hard to analyze based on DOS peaks. Here, a rough and indirect way was adopted to analyze element preference at 1g and 2t site. According to the local chemical environment of 2t and 1g site, formation enthalpies (ΔH_fs) of MAl with CsCl crystal structure and MB₂ with AlB₂ crystal structure were calculated. For M = Cr, Mo, W, Mn, Fe, Co and Ni, ΔH_fs of MAl are 0.43, 0.20, 0.70, -0.24 -1.15, -1.51, and -1.39 eV per M atom, ΔH_fs of MB₂ are -0.94, -0.90, -0.25, -0.43, -0.75, -0.21, 0.47 eV per M atom. Then, formation enthalpy differences between MB₂ and MAl are -1.37, -1.09, -0.95, -0.19, 0.39, 1.30, 1.85 eV per M atom for M = Cr, Mo, W, Mn, Fe, Co and Ni. It indicates that Cr, Mo, W, and Mn prefer to locate at 1g site instead of 2t site, while Fe, Co and Ni prefer to locate at 2t site instead of 1g site. Nevertheless, when Mn and Cr, Mo, or W are in the same compound, Mn will be forced to occupy 2t site. The remarkable formation enthalpy differences of elements at 2t and 1g sites confirm the strong ordering tendency of these compounds, which are consistent with results in Fig. 2. For example, ΔH_ordering of ordered Co₂CrAlB₄ is -0.70 eV/cell, and changes to 0.74 eV/cell for ordered Cr₂CoAlB₄. It means that swapping Co at 2t site and Cr at 1g site will results in a substantial energy increase, indicating strong element preference at specific sites. Therefore, except for the bonding strength between M and Al, bonding strength between M' and hexagonal boron ring may also play some role in promoting ordering, where Cr, Mo, W show strong tendency to occupy 1g site than 2t site.

3.3. Properties of ordered M₂M'AlB₄

The magnetic property and elastic properties of ordered M₂M'AlB₄ (M = Mn, Fe and Co and M' = Cr, Mo and W) phases were calculated and compared with those of Cr₃AlB₄. Table 1 shows lattice parameters, atomic positions (z_2t, z_2q, z_2s) and the differences between spin up and spin down elections per unit cell (Δs) of ordered M₂M'AlB₄ (M = Mn, Fe and Co and M' = Cr, Mo and W) phases. The results reveal that the ordered M₂M'AlB₄s all exhibit non-trivial Δs values for M = Mn and Fe, while Δs values of Co₂M'AlB₄s are zero. It means that ordered M₂M'AlB₄ (M = Mn, Fe) phases are ferromagnetic.

Table 2 shows elastic properties of ordered M₂M'AlB₄ (M = Mn, Fe and Co and M' = Cr, Mo and W) phases. Elastic constants c_ij were determined from a linear fit of the calculated stress as a function of strain. Both positive and negative strains were applied for each strain component, with a maximum strain value of 0.3%. Since M₂M'AlB₄ is in orthorhombic symmetry, there are nine independent elastic constants c_ij (c₁₁, c₂₂, c₃₃, c₄₄, c₅₅, c₆₆, c₁₂, c₂₃, and c₃₁) [42]. For a crystal to be stable under elastic perturbations, c_ij values should satisfy the Born-Huang criteria [43], which for orthorhombic crystals are:

c₁₁>0,c₂₂>0,c₃₃>0,c₄₄>0,c₅₅>0,c₆₆>0

c₁₁c₂₂-$c^{2}_{12}$>0

c₂₂c₃₃-$c^{2}_{23}$>0

c₃₃c₁₁-$c^{2}_{31}$>0

c₁₁c₂₂c₃₃+2c₁₂c₂₃c₃₁-c₁₁$c^{2}_{23}$-c₂₂$c^{2}_{31}$-c₃₃c$c^{2}_{12}$>0

It is checked that the Born-Huang criteria are satisfied for c_ij values in Table 2, indicating the stability of ordered M₂M'AlB₄ phases. The bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio ν of polycrystalline M₂M'AlB₄ phases can be evaluated by applying the Voigt-Reuss-Hill approximations [[44], [45], [46]]. The computed B, G, E and ν values are also listed in Table 2. As can be seen from Table 2, the elastic properties of ordered M₂M'AlB₄ display the following tendencies. c_ii (i = 1-6), B, G, E values of ordered M₂M'AlB₄ are lower than those of Cr₃AlB₄. For a given M', shear moduli (c₄₄, c₅₅, c₆₆) decrease when M changes from Mn to Co. However, moduli c₁₁, c₂₂, and c₃₃, do not show trend dependence on element M. Similarly, shear modulus G and Young's modulus E of polycrystalline decreases when M changes from Mn to Co, while bulk modulus B almost keeps constant with M. Table 2 also lists the Pugh’s ratio (G/B), which is usually adopted to predict the brittle versus ductile property of materials [47]. Due to the lower shear moduli, ordered M₂M'AlB₄ phases exhibit lower G/B ratios, indicating better ductility of these ordered M₂M'AlB₄ phases in comparison with Cr₃AlB₄. Reducing in shear modulus and Young’s modulus with increasing in valence electron concentration has been discovered in diborides [48], which is suggested due to the excessive occupation of d-d bonding between M elements. It is possible that similar mechanism also play some role in ordered M₂M'AlB₄ phases, since the local M-B structure is similar to that in diboride.

Ordering in the crystal structure not only enriches the family of MAB phases, but also provides a way to tailor the properties of MAB phases. On the one hand, based on the predicted stabilities of ordered M₂M'AlB₄ phases, people can try to synthesize new ordered quaternary MAB phases, which can significantly enrich the family of MAB phases and expand their potential applications. Till now, only eight stable MAB phases have been synthesized in experiments: (CrB₂)_nCrAl (n = 1,2,3) [7], Cr₄AlB₄ [8], Fe₂AlB₂, Mn₂AlB₂ [[9], [10], [11], [12], [13]], MoAlB, WAlB [7,9,14,15]. Nine new stable ordered quaternary MAB phases (M₂M'AlB₄ phases with M = Mn, Fe, Co and M' = Cr, Mo, W) are predicted here. On the other hand, properties of reported MAB phase, e.g. Cr₃AlB₄, can be tailored by solid solution of Mn, Fe, or Co into Cr sites. Then, the solute atoms tend to occupy M site instead of M' site. According to results in Table 1, Table 2, Cr₃AlB₄ with ferromagnetic and improved ductility may be obtained by solid solution of Mn, Fe, or Co. Tuning the properties by solid solution is a fundamental method in material science. As reported in MAX phases, the magnetic properties of Cr₂AlC can be well tuned by solid solution of Mn [41,49]. In addition, theoretical investigations suggested that ordering of Mn plays a crucial role in the development of magnetic MAX phases [41,49]. It is interesting that similar magnetic features are discovered in MAB phases. In MAX phases, most of the M elements are early transition metals, e.g. Ti, V, Cr, Nb, Ta, Mo, etc. Then, it is hard to obtain non-paramagnetic MAX phases with the first non-paramagnetic MAX phase discovered after decades of investigations on MAX phases [49]. However, M elements in MAB phases belong to middle transition metals, e.g. Cr, Mo, W, Mn, Fe, etc. It suggests that obtain non-paramagnetic MAB phases may be easier in comparison to MAX phases.

4. Conclusion

In this work, possible ordered quaternary MAB phases with Cr₃AlB₄ structure (space group: Pmmm) were predicted by first-principles calculations. The results show that M₂M'AlB₄ phases with M = Mn, Fe, Co and M' = Cr, Mo, W display strong tendency of ordering, where M locates at 2t site and M' locates at 1g site. In addition, the ordering tendency increases when M changes from Mn to Co, which can be deduced from enthalpy changes between ordered and disordered M₂M'AlB₄ phases. Analyses on chemical bonds reveal that bonding strength between M and Al may serve as the main driving force of ordering. It is found that Mn, Fe, Co can form extra bonding with Al in comparison to Cr, Mo, W. The peak position of extra bonding shifts from high energy level to low energy level when M changes from Mn to Co, being consistent with the ordering tendency derived from enthalpy changes. The magnetic property and elastic properties of predicted ordered M₂M'AlB₄ phases were calculated. It shows that ferromagnetic M₂M'AlB₄ phases can be obtained with M = Mn or Fe. In comparison to Cr₃AlB₄, ordered M₂M'AlB₄ phases display lower shear resistance and possibly better ductility. The reduction effect on shear resistance increases when M changes from Mn to Co.

Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant No. U1435206 and No. 51672064.

The authors have declared that no competing interests exist.

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