Theoretical study of thermodynamic and magnetic properties of transition metal carbide and nitride MAX phases

We systematically perform density-functional theory (DFT) calculations for all possible ${\mathrm{M}}_{n+1}{\mathrm{AX}}_n$ (MAX) phases with transition metal M=Sc to Au (excluding Tc), A in group IIIA-IVA, X=C, N, and $n=1,2,3$, a total of about 1200 systems. The thermodynamic stability is determined by comparing the formation enthalpy (at 0 K) against all possible combinations of unary, binary, and ternary boundary phases (available from online DFT databases). Thereby, we identify 124 so far unknown phases (in terms of both experimental synthesis and other theoretical predictions), of which 54 are carbides and 70 are nitrides. Among all stable MAX phases, we identify nine with magnetic properties. In addition to already known and synthesized magnetic phases $({\mathrm{Cr}}_2\mathrm{AlC},$ ${\mathrm{Cr}}_2\mathrm{GeC},$ ${\mathrm{Cr}}_2\mathrm{GaN},$ and ${\mathrm{Mn}}_2\mathrm{GaC})$, we predict five more MAX phases with magnetic ordering $\left[{\mathrm{Mn}}_2\mathrm{A}\right(=\mathrm{Ge}, \mathrm{Sn})\mathrm{C}, {\mathrm{Cr}}_3\mathrm{A}(=\mathrm{Ga}, \mathrm{Ge}){\mathrm{N}}_2, \mathrm{and} {\mathrm{Cr}}_4\mathrm{A}(=\mathrm{Ge}){\mathrm{N}}_3]$. Evaluating previously suggested descriptors for the stability of MAX phases [valence electron concentrations (VECs), differences in atomic radius difference $\mathrm{\Delta }{R}_{\mathrm{at}}$, and differences in electronegativities $\mathrm{\Delta }\chi$], we find that $\mathrm{\Delta }{R}_{\mathrm{at}}$ does not correlate with stability and stable phases are characterized by $\text{VEC}<5.5$, $\mathrm{\Delta }\chi >1.5$. The reverse is, however, not true; for example, a MAX phase with $\text{VEC}<5.5$ and $\mathrm{\Delta }\chi >1.5$ is not necessarily stable.

Figure 1

Thermodynamic (meta)stability of individual MAX phases is inferred by the sign and magnitude of the calculated maximum formation enthalpies, relative to all possible combinations of known competing phases, including unaries, binaries, and ternaries, that are present in the M-A-X phase diagram [top panel; see Eq. (1)]. Boundary phases are retrieved from MP [28] and OQMD [21, 22]. The bottom panel contains crystal structures of ($3\times 3\times 1$) supercells of $n=1,2$, and 3 MAX phases along with other exemplary unary, binary, and ternary phases.

Figure 2

All symmetrically unique spin configurations of (a) ${\mathrm{M}}_2\mathrm{AX}$- and (b) ${\mathrm{M}}_3{\mathrm{AX}}_2$-type MAX phases considered that are possible within a primitive cell. The unique configurations of a ${\mathrm{M}}_4{\mathrm{AX}}_3$-type MAX phase are provided in Fig. S7 in the Supplemental Material [29].

Figure 3

Validation of the maximum formation enthalpy ${\mathrm{\Delta }}_{r}H$ as a descriptor for thermodynamic stability. For 65 experimentally reported MAX phases, only 7 MAX phases show a positive value of ${\mathrm{\Delta }}_{r}H$. Thermodynamic (meta)stability is indicated by ${\mathrm{\Delta }}_{r}H<+30$ meV/atom (marked by the vertical dotted line). The compositions in the bottom part are known to synthesize (taken from Refs. [27, 35, 36, 37]), while the two at the top could not be synthesized despite several experimental attempts [38].

Figure 4

Maximum formation enthalpies ${\mathrm{\Delta }}_{r}H$ for various MAX carbides and nitrides of type $n=1,2,3$ composed of different M and A elements; ${\mathrm{\Delta }}_{r}H<+30$ meV/atom indicates potential (meta)stability. The red (turquoise) shaded background of individual entries indicates a negative (positive) maximum formation enthalpy; the intensity increases with increasing (absolute) magnitude. MAX phases that have been reported experimentally [27, 35, 36, 37] and theoretically by others [26, 39] are further highlighted by green and black frames, respectively. We note that only the M and A elements for which stable combinations exist are listed here; the full list of all considered MAX phases is provided in Sec. S2. of the Supplemental Material [29].

Figure 5

(a) The effect of varying the number of MX layers as defined by n on formation enthalpy with respect to unary constituents ($\mathrm{\Delta }{E}_f$) of Al-containing MAX carbides of $3d$ transition metals. Two different behaviors are visible: (1) an increase in $\mathrm{\Delta }{E}_f$ for late transition metals, e.g., Cr, Fe, etc., and (2) a decrease or negligible change in $\mathrm{\Delta }{E}_f$ with n. The color coding marks the respective trend. (b) Calculated formation enthalpies with respect to unary constituents of the C- and Al-centered octahedron (O) and trigonal prism (T), respectively, which are the building blocks of a MAX phase (see the inset).

Figure 6

(a) Correlation of the valence electron concentration (VEC) per atom with the maximum formation enthalpy ${\mathrm{\Delta }}_{r}H$ of all 1200 compositions studied in this work. (b) Usage and validation of VEC as a potential instability descriptor for identifying unstable ordered and disordered double ${\mathrm{MM}}^{\prime }\mathrm{AX}$ phases that have been studied by others [19, 26]. In (b) ${\mathrm{\Delta }}_{r}H$ (at 0 K) and ${\mathrm{\Delta }}_r{G}_{\mathrm{sol}}$ (at 2000 K) are plotted for ordered and disordered phases, respectively. ${\mathrm{\Delta }}_r{G}_{\mathrm{sol}}$ is the free energy of a disordered phase (solid solution) at a particular temperature (solid solution) and is defined in detail in Ref. [19]. The vertical line marks the (meta)stability limit at (a) 30 meV/atom and (b) 60 meV/atom as used in Ref. [19] for (dis)ordered phases. All the unstable phases in (a) have VEC above $\sim 5.5$ electrons/atom (horizontal dashed line), which is verified by (b), demonstrating VEC as a general thermodynamic instability descriptor. The color gradient of the scatter is the same as in Fig. 4.

Figure 7

Relationship of (a) radius differences $\mathrm{\Delta }{R}_{\mathrm{at}}$ and (b) electronegativity differences $\mathrm{\Delta }\chi$ of M, A, and X atoms, with maximum formation enthalpy ${\mathrm{\Delta }}_{r}H$ and VEC. $\mathrm{\Delta }{R}_{\mathrm{at}}$ values are scattered throughout the ${\mathrm{\Delta }}_{r}H$ range. Below $\mathrm{\Delta }\chi$ of $\sim 1.5$ (horizontal dashed line), the majority of the phases are unstable; an indirect correlation with ${\mathrm{\Delta }}_{r}H$ and VEC exists. The vertical line marks the (meta)stability limit at $+30$ meV/atom. The electronegativities and atomic radii are taken from data available in pymatgen [46].

Figure 8

Calculated magnetic moments of numerous MAX phases that are found to be magnetic, out of 1200 compositions studied here. The experimentally synthesized compositions are denoted by a check pattern. FM, AFM, and FiM represent ferromagnetic, antiferromagnetic, and ferrimagnetic order, respectively. The label at the top of each bar is the total magnetization. All the symmetrically unique spin configurations of a ${\mathrm{M}}_2\mathrm{AX}$-and ${\mathrm{M}}_3{\mathrm{AX}}_2$-type MAX phase are shown in Fig. 2, whereas that of a ${\mathrm{M}}_4{\mathrm{AX}}_3$-type MAX phase are provided in Fig. S7 in the Supplemental Material [29].