o‑MAX Phases: Structure and Properties

• o‑MAX phases are layered ternary carbides/nitrides with distinct transition-metal layers ordered along the c-axis.
• They exhibit a stable hexagonal P6₃/mmc structure with negative formation energies and metallic conductivity.
• These phases offer practical advantages in thermal barrier coatings and MXene production due to low thermal conductivity and customizable A-site engineering.

o‑MAX phases are layered derivatives of the MAX family, but the label is not uniform across the literature. In current structural usage, it most directly denotes out‑of‑plane ordered MAX phases, in which different transition metals occupy distinct layers along the $c$-axis rather than a single repeated $M$ sublattice; recent density-functional work treats $\mathrm{Mo_2A_2AlC_3}$ ($A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$) as representative 413-type o‑MAX compounds with hexagonal $P6_3/mmc$ symmetry, negative formation energies, metallic electronic structure, and mechanically and dynamically stable lattices (Tanim et al., 28 Sep 2025). Elsewhere, the same label has also been used for electrically anisotropic “optical MAX” behavior and for oxidation-optimized Al-bearing MAX systems. Any technical discussion of o‑MAX phases therefore requires explicit attention to context.

1. Terminology and scope

The underlying MAX family comprises layered ternary carbides and nitrides with general formula $M_{n+1}AX_n$, typically $n=1,2,3$, where $M$ is an early transition metal, $A$ is a p‑block element, and $X$ is C or N; structurally they are nanolaminates of $M$0 blocks separated by metallic $M$1 layers, most often in the hexagonal space group $M$2 (Shein et al., 2010).

Usage of “o‑MAX”	Defining feature	Representative source
Out-of-plane ordered MAX	Different transition metals occupy distinct layers along $M$3	(Tanim et al., 28 Sep 2025)
A-site/vacancy-ordered MAX	A layers host controlled substitution or vacancy fractions	(Wang et al., 14 Sep 2025)
Optical MAX	$M$4 and $M$5 in an anisotropic regime	(Kyriienko et al., 2011)
Oxidation-optimized MAX	Al-bearing MAX engineered to form protective $M$6 scales	(Singh et al., 2019)

In the recent crystal-chemical literature, the structural meaning dominates: o‑MAX phases are “out‑of‑plane ordered MAX phases” characterized by distinct transition-metal layers stacked along $M$7, often written as $M$8 for $M$9, with $\mathrm{Mo_2A_2AlC_3}$0 occupying the outer layers next to $\mathrm{Mo_2A_2AlC_3}$1 and $\mathrm{Mo_2A_2AlC_3}$2 the inner layers strongly coordinated by $\mathrm{Mo_2A_2AlC_3}$3 (Tanim et al., 28 Sep 2025). At the same time, A-site ordering and vacancy ordering are closely allied concepts, because controlled occupancy of the $\mathrm{Mo_2A_2AlC_3}$4 layer can also impose a nontrivial repeat sequence along the layered direction (Wang et al., 14 Sep 2025).

2. Crystal structure of out-of-plane ordered MAX phases

The clearest explicit o‑MAX case in the supplied literature is the predicted 413 family $\mathrm{Mo_2A_2AlC_3}$5 with $\mathrm{Mo_2A_2AlC_3}$6. These phases are hexagonal, belong to $\mathrm{Mo_2A_2AlC_3}$7 (No. 194), contain two formula units per unit cell, and have 16 atoms per cell. Their Wyckoff assignments are Mo at $\mathrm{Mo_2A_2AlC_3}$8, $\mathrm{Mo_2A_2AlC_3}$9 at $A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$0, Al at $A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$1, and C distributed over $A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$2 and $A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$3 sites. The ordering is explicitly out-of-plane: Mo-rich and $A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$4-rich transition-metal layers alternate along $A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$5, while carbon sustains octahedral $A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$6 environments and Al remains in trigonal-prismatic coordination between metal layers (Tanim et al., 28 Sep 2025).

Phase	$A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$7 (Å)	$A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$8 (Å)	$A=\mathrm{Zr}, \mathrm{Nb}, \mathrm{Ta}$9	$P6_3/mmc$0 (eV/atom)
$P6_3/mmc$1	3.1502	23.999	7.618	–0.506
$P6_3/mmc$2	3.1026	23.551	7.590	–0.349
$P6_3/mmc$3	3.1436	23.797	7.569	–0.386

All three formation energies are negative, indicating chemical stability within that study. The reported order is $P6_3/mmc$4 in magnitude of stability. Local polyhedral geometry was analyzed through Hug’s distortion indices: $P6_3/mmc$5 for the $P6_3/mmc$6 octahedra and $P6_3/mmc$7 for the $P6_3/mmc$8 trigonal prisms. In the interpretation given there, the octahedra are more distorted whereas the trigonal prisms remain near ideal, consistent with the negative formation energies and retained layered topology (Tanim et al., 28 Sep 2025).

This ordered architecture should be distinguished from conventional single-$P6_3/mmc$9 MAX phases such as the superconducting 211 compounds $M_{n+1}AX_n$0, $M_{n+1}AX_n$1, $M_{n+1}AX_n$2, $M_{n+1}AX_n$3, $M_{n+1}AX_n$4, and $M_{n+1}AX_n$5, which preserve the standard hexagonal $M_{n+1}AX_n$6 structure but do not exhibit out-of-plane transition-metal ordering (Shein et al., 2010).

3. Stability, bonding, and classification frameworks

For $M_{n+1}AX_n$7, stability is supported at several levels. The single-crystal elastic constants satisfy the hexagonal Born–Mouhat criteria, all $M_{n+1}AX_n$8 are positive, and the phonon dispersions contain no imaginary frequencies anywhere in the Brillouin zone. Electronic structure calculations show significant overlap of valence and conduction bands at the Fermi level, so all three compounds are metallic. The density of states near $M_{n+1}AX_n$9 is dominated by Mo‑4d states, with additional $n=1,2,3$0-site $n=1,2,3$1 character and smaller Al‑3p and C‑2p contributions; charge-density and Mulliken analyses indicate strong covalent $n=1,2,3$2 bonding with ionic contribution, whereas $n=1,2,3$3 interactions are more metallic. Within that set, the bond-strength trend inferred from DOS and bond-overlap analyses is $n=1,2,3$4 (Tanim et al., 28 Sep 2025).

A broader design framework is provided by the systematic study of 288 $n=1,2,3$5 phases, which treats MAX compounds as approximate rigid-band systems. In that analysis, valence electron concentration and a geometrical factor associated with atomic size jointly control heat of formation, directional elastic constants, and bonding. A recurring reference point is a VEC near 8.4, where the $n=1,2,3$6 $n=1,2,3$7 bonding is saturated without strongly filling $n=1,2,3$8 antibonding states; beyond that, additional electrons increasingly populate $n=1,2,3$9 $M$0 antibonds and can reduce $M$1. The same work emphasizes that $M$2 $M$3 interactions uniquely contribute to $M$4, and explicitly correlates larger $M$5 with stronger $M$6 (Wu et al., 2021). This suggests that out-of-plane ordering should be especially sensitive to layer-resolved $M$7-band filling and to any enhancement or suppression of interlayer $M$8 $M$9 bonding.

A second, more chemical, screening approach is the Hume–Rothery structure map for MAX formability. It uses an electron concentration

$A$0

and an atomic-difference ratio

$A$1

Although developed for conventional hexagonal MAX phases, that study states that the logic can be transferred, with care, to o‑MAX and other polymorphs. A plausible implication is that compositions lying within or near the known MAX formability domain in the $A$2 plane are better o‑MAX targets than compositions far outside it (Zhang et al., 2019).

At a still broader level, the ZIA-phase work proposes a unifying stoichiometric rule $A$3, with $A$4 and $A$5, for nanolaminated close-packed solids. In that formulation, all known MAX variants, including hybrid orders and the 514 order, fall within one overarching class together with the newly introduced fcc ZIA phases (Tunes et al., 2023). This places o‑MAX phases within a wider landscape of ordered nanolaminates rather than as isolated anomalies.

4. Mechanical, thermal, and optical properties

The predicted $A$6 phases are mechanically stable but brittle by all three criteria explicitly evaluated: $A$7, $A$8, and negative Cauchy pressure. Their theoretical Vickers hardness values remain in the MAX-typical range, while fracture toughness is comparable to values quoted there for 3YSZ. The work further identifies $A$9 as the stiffest member, $X$0 as having the lowest Young’s modulus and therefore better thermal-shock tolerance, and $X$1 as the easiest to machine by the reported machinability index (Tanim et al., 28 Sep 2025).

Phase	$X$2 (K)	$X$3 (W/m·K)	$X$4 (K)	$X$5 (GPa)	$X$6 (MPa·m$X$7)
$X$8	634.1	0.89	2064	6.03	2.67
$X$9	632.6	0.90	2237	6.28	2.78
$M$00	526.0	0.74	2131	6.22	2.59

Thermophysically, the most striking result is the very low minimum thermal conductivity: $M$01, $M$02, and $M$03 for the Zr, Nb, and Ta members, respectively. The calculated lattice thermal conductivities at 300 K are $M$04, $M$05, and $M$06, and decrease strongly with temperature up to roughly 1100 K before tending toward saturation. The Debye temperatures are 634.1 K, 632.6 K, and 526.0 K, and the room-temperature thermal expansion coefficients are $M$07, $M$08, and $M$09. These combinations are the basis for the proposal of these compounds as next-generation thermal-barrier-coating candidates (Tanim et al., 28 Sep 2025).

The optical response is likewise metallic. For both [100] and [001] polarizations, reflectivity reaches about $M$10 in the infrared near 1 eV, remains roughly $M$11 in the visible, and stays around $M$12 in the ultraviolet. Plasma frequencies extracted from the loss function are close to 21 eV. The same study argues that reflectivity above 44% across the visible–near-UV range makes these o‑MAX phases attractive as solar-heat-shielding coatings as well as thermal-barrier materials (Tanim et al., 28 Sep 2025).

5. Chemical engineering, vacancy order, and MXene derivation

A-site engineering is a major route by which o‑MAX-related chemistries are accessed. A vapor-phase chlorosilane strategy converts Al-based MAX precursors into Si-substituted MAX phases at $M$13, covering $M$14 with $M$15. In the Nb system this route produces $M$16 from $M$17 and $M$18 from $M$19, corresponding to approximately 25% and 50% vacancy fractions on the A site. The first product shows A-site occupancy near 0.75 by Rietveld refinement and about 0.73 by EDS; the second gives occupancies near 0.475 by XRD refinement and about 0.57 by EDS. The same work reports $M$20, and even $M$21, for the Si-substituted vacancy-rich phases, together with an EPR signal at $M$22 assigned to Si vacancy centers (Wang et al., 14 Sep 2025).

That chemistry is directly relevant to ordered MAX concepts because it demonstrates that the A layer can be driven far from full occupancy while the MAX framework persists. The authors explicitly compare these vacancy-rich phases to double-A-layer systems such as $M$23, $M$24, and $M$25, and interpret both as manifestations of A-site engineering—one with over-occupied A layers and the other with under-occupied or vacancy-rich A layers (Wang et al., 14 Sep 2025).

Molten-salt A-site replacement offers a complementary route. Reaction of $M$26 with CuI yields $M$27 with essentially complete A-site replacement, whereas $M$28 with $M$29 yields $M$30. The same work emphasizes that these are still conventional 211 MAX phases, not o‑MAX phases, because no out-of-plane A-site ordering, superstructure reflections, or symmetry lowering are observed. It nevertheless shows that extensive A-site substitution is structurally viable and that Cu-containing phases have lower cleavage energies than their Al-containing counterparts, making them promising MXene precursors (Ding et al., 2019).

The exfoliation study on 82 experimentally synthesized MAX phases extends this logic to ordered parents. It identifies force-constant and exfoliation-energy descriptors for successful MXene derivation, using $M$31 for $M$32 as an empirical upper bound for the A-site stiffness of exfoliable MAX phases, $M$33 for $M$34 as a lower bound for the X-site stiffness of robust MXene cores, and $M$35 as an empirical exfoliation criterion. Already exfoliated ordered double-$M$36 parents include $M$37, $M$38, $M$39, and $M$40, showing that chemically ordered MAX structures are fully compatible with MXene production (Khazaei et al., 2018).

6. Alternative meanings of “o‑MAX” in optical and oxidation literature

In one distinct usage, “o‑MAX” denotes optical MAX behavior arising from strong electrical anisotropy. There the dielectric tensor is taken as

$M$41

with each component described by a Drude form

$M$42

When $M$43, one has $M$44 and $M$45, so the material is metallic in plane and dielectric along $M$46. In this regime TE waves are screened, TM waves exhibit inverse total internal reflection above a critical angle, and surface plasmon polaritons show stronger confinement than in isotropic metals. For $M$47, the cited values are $M$48, $M$49, and relaxation times $M$50 (Kyriienko et al., 2011).

A second distinct usage is oxidation-optimized MAX, where the design objective is rapid Al transport to the surface and growth of a continuous protective $M$51 scale. First-principles work on chemically disordered $M$52 shows that disorder lowers A-site vacancy formation energies and can reduce the A-site vacancy migration barrier to about $M$53 in two-site disordered Cr-containing alloys, thereby easing Al diffusion at high operating temperatures and promoting passivating alumina formation (Singh et al., 2019).

The same oxidation-focused meaning appears experimentally in burner-rig oxidation of $M$54, which produces the layered stack

$M$55

After cyclic oxidation at $M$56, the $M$57 interface has a measured fracture toughness of $M$58, and the $M$59 interface gives $M$60. The excellent macroscopic adherence is attributed there to a combination of low internal strain from thermal-expansion matching and the convoluted interface morphology (Gibson et al., 2019).

Accordingly, “o‑MAX phases” is best treated as a context-sensitive term. In recent structure/property work it refers primarily to out-of-plane ordered MAX phases; in plasmonics it denotes an anisotropic optical regime; and in oxidation studies it denotes MAX systems optimized to form protective alumina scales.