Compositionally Complex Carbides

• Compositionally complex carbides are multicomponent transition-metal ceramics that extend high-entropy concepts to include non-equimolar and ordered phases for enhanced performance.
• Recent research employs tailored phonon engineering to modulate bandgaps and scattering, directly influencing thermal conductivity in extreme environments.
• Data-driven workflows combining machine learning and DFT screening efficiently optimize synthesis, stability, and hardness across diverse structural motifs.

Compositionally complex carbides (CCCs) are multicomponent carbide ceramics situated within the broader category of compositionally complex ceramics, a framework that extends high-entropy ceramics to include medium-entropy and non-equimolar compositions, as well as ordered, defect-rich, and dual-phase states (Wright et al., 2020, Wright et al., 2019). In current carbide research, the term is used primarily for multicomponent transition-metal carbides in rocksalt, hexagonal, or related structural prototypes, where compositional complexity is exploited to tune thermodynamic stability, phonon transport, elasticity, hardness, and, more generally, performance in extreme environments (Malakkal et al., 6 Aug 2025, Liu et al., 28 May 2026). The field has progressively shifted from an emphasis on maximizing configurational entropy to a broader design philosophy in which chemical identity, local order, phase constitution, and microstructure are treated as equally consequential variables (Luo, 7 Oct 2025).

1. Definition and conceptual scope

The conceptual origin of CCCs lies in the expansion of high-entropy ceramics (HECs) into a broader umbrella class. In the ceramic literature, HECs are typically described as ceramics with five or more principal cations, usually in equimolar or near-equimolar fractions, whereas CCCs encompass HECs, medium-entropy ceramics (MECs), and non-equimolar compositions with additional structural or defect complexity (Wright et al., 2020, Wright et al., 2019). The 2020 perspective explicitly framed CCCs as including non-equimolar compositions, multiple cation sublattices, aliovalent cations, anion vacancies, and ordered structures, thereby decoupling the notion of useful complexity from the narrower requirement of maximal entropy (Wright et al., 2020).

For ideal mixing, the configurational entropy is written as

$\Delta S_{\mathrm{mix}} = -R \sum_i x_i \ln x_i,$

with $x_i$ the mole fraction of component $i$ (Wright et al., 2020, Nakahira et al., 2021). In this formulation, high entropy is a special case rather than a universal design target. Later perspective work sharpened this point by asking whether the entropy in HECs is truly high and whether maximizing entropy should always be the goal; the proposed answer was negative, because non-equimolar compositions and correlated disorder can enhance properties beyond those of higher-entropy counterparts (Luo, 7 Oct 2025).

For carbides, the classical reference point is the rocksalt-structured equimolar high-entropy carbide $$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$, cited in the perspective literature as a quintessential HEC (Luo, 7 Oct 2025). CCCs retain this multication carbide core but enlarge the accessible design space to include medium-entropy, non-equimolar, ordered, and multiphase variants. This broader scope is central to current work on synthesizability, hardness, and phonon engineering in multicomponent carbides (Liu et al., 28 May 2026, Malakkal et al., 6 Aug 2025).

2. Structural motifs, disorder, and thermodynamic stability

Recent screening studies have made the structural landscape of CCCs substantially more explicit. A machine-learned-potential study screened over 1500 equiatomic multicomponent carbide compositions composed of group 4, 5, and 6 transition metals across rocksalt ($Fm\bar{3}m$), hexagonal (WC-type, $P\bar{6}m2$), and hcp ($P6_3/mmc$) prototypes (Liu et al., 28 May 2026). In that work, synthesizability at $1500^\circ\mathrm{C}$ was predicted using a free-energy model with configurational entropy but without vibrational entropy, and compositions were classified as synthesizable when

$\Delta g^{\rm hull}_\mathrm{disorder}<0.$

The same study reported that predictions at $1500^\circ\mathrm{C}$ agree well with experimental reports for both single-phase and multiphase carbides, and the detailed analysis summarized agreement for about $x_i$0 of 42 experimentally reported quaternary and quinary rocksalt CCCs (Liu et al., 28 May 2026).

A central thermodynamic result is that the group number of the constituent metals governs both stability and hardness (Liu et al., 28 May 2026). Increasing the fraction of group 4 and 5 metals stabilizes rocksalt carbides, whereas adding group 6 metals destabilizes rocksalt and favors hexagonal structures. The free energy of a disordered carbide was represented as

$x_i$1

and short-range-order contributions were found to be small, typically a few meV/atom. As a consequence, a perfectly disordered solid solution was judged a reasonable approximation for high-throughput screening, except for compositions very close to the convex hull (Liu et al., 28 May 2026).

At the same time, the same screening work showed that disorder is not the only relevant structural descriptor. For compositions mixing group 4 or 5 metals with group 6 metals, a new family of stacking-ordered phases was identified with formation energies well below those of disordered rocksalt or hexagonal structures (Liu et al., 28 May 2026). The detailed summary reported energy reductions up to about $x_i$2 meV/atom relative to random rocksalt or hexagonal solutions, and specifically noted that the ABCAABC stacking of $x_i$3 is about $x_i$4 meV/atom lower than either pure rocksalt or hexagonal stacking (Liu et al., 28 May 2026). DFT calculations corroborated these predictions and suggested that stacking-ordered phases should be experimentally accessible.

This body of work modifies a common simplifying picture of CCCs as merely disordered rocksalt solid solutions. In practice, CCCs span disordered, short-range-ordered, and stacking-ordered states, and the distinction can be thermodynamically decisive near phase boundaries (Liu et al., 28 May 2026).

3. Phonon engineering and thermal transport

Thermal transport in CCCs is now understood as a nontrivial consequence of cation disorder rather than a monotonic function of configurational complexity. In rock-salt-structured CCCs containing Zr, Ti, Ta, Nb, and Hf, ab initio calculations were used to predict phonon band structures and to explore the influence of mass variance and force-constant variance on the phonon spectral function from binary to five-metal carbides (Malakkal et al., 6 Aug 2025). The force constants were written as

$x_i$5

and disorder was represented explicitly through 216-atom special quasi-random structures (SQS), with DFT calculations performed in VASP within the local density approximation and second-order force constants obtained by finite displacement with PHONOPY (Malakkal et al., 6 Aug 2025).

The central result is that the selection and concentration of constituent elements can be strategically utilized to tune the phonon band structure, phonon bandgap, and phonon scattering in CCCs (Malakkal et al., 6 Aug 2025). Pronounced cation disorder introduces mass and force-constant variations that often cause strong phonon scattering, especially in the mid- and high-frequency ranges, together with broadening of phonon modes. However, the study showed that this tendency is not universal. Among binaries, $x_i$6 was reported to have no phonon bandgap and significant broadening, while $x_i$7 maintained a large bandgap and minimal broadening (Malakkal et al., 6 Aug 2025). In the quinary $x_i$8, a wider and more defined bandgap appeared and phonon broadening was suppressed relative to less complex compositions (Malakkal et al., 6 Aug 2025).

These phononic trends were linked directly to measured thermal conductivity using spatial-domain thermoreflectance. At low temperatures, where phonon contributions dominate, the quinary carbide exhibited higher thermal conductivity than the binary $x_i$9 and the ternary $i$0 (Malakkal et al., 6 Aug 2025). This finding was explicitly described as contrary to the expectation that greater cation disorder should always increase scattering and reduce thermal transport. The proposed explanation was a combination of wider phonon bandgaps, suppressed phonon broadening, and effective filtering of destructive scattering channels by compositional selection (Malakkal et al., 6 Aug 2025).

The implication is not that disorder is unimportant, but that CCCs permit disorder to be engineered spectrally. For nuclear and other extreme-environment applications, this means that thermal stability, elasticity, thermal conductivity, and thermodynamic behavior can be adjusted by targeted control of phonon structure rather than by treating compositional complexity as a one-parameter proxy for scattering (Malakkal et al., 6 Aug 2025).

4. Hardness, elastic response, and compositional rules

The most systematic carbide-wide hardness results currently come from high-throughput screening with machine-learned interatomic potentials. In the 2026 study, a fine-tuned MACE potential trained on approximately 28,000 DFT calculations was used to predict thermodynamic stability and elastic properties of multicomponent carbides spanning the full nine-metal space of Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, and W with carbon (Liu et al., 28 May 2026). For thermodynamically relevant structures, formation-energy errors of about $i$1 meV/atom were achieved using only $i$2 of the training data (Liu et al., 28 May 2026).

Elastic constants were converted to Vickers hardness through several surrogate models, with Teter’s model judged to show the best agreement with both DFT and experiment: $i$3 where $i$4 is the shear modulus (Liu et al., 28 May 2026). The hardest synthesizable compositions were reported to be mixtures of three to five metals, primarily in the rocksalt structure, with $i$5 exceeding $i$6 GPa (Liu et al., 28 May 2026).

The decisive design rule was again group identity rather than compositional complexity alone. In rocksalt CCCs, hardness increases with group 4 and 5 content, while in hexagonal CCCs hardness is maximized with group 6 content, although those hexagonal phases are difficult to stabilize unless the composition is group-6-rich (Liu et al., 28 May 2026). Equally important, the study found no systematic increase in hardness with the number of metal elements: binary and ternary carbides can rival or exceed the hardness of more compositionally complex phases (Liu et al., 28 May 2026).

This result addresses a recurrent misconception in the CCC literature. “High-entropy” does not imply “hardest” by default. The screening results instead indicate that hardness is governed primarily by the chemical character of the constituent transition-metal groups and by prototype stability, with the number of principal elements acting only indirectly (Liu et al., 28 May 2026). Earlier perspective work had already pointed in a related direction by noting that, in high-entropy carbides, Sarker et al. identified mass and size disorder parameters as key predictors for hard, tough, single-phase carbides, not configurational entropy alone (Wright et al., 2020).

5. Discovery workflows, synthesis optimization, and kinetic modeling

Because CCCs occupy extremely large composition spaces, the field has adopted data-efficient discovery strategies rather than exhaustive enumeration. The MACE-based carbide study already exemplifies one branch of this approach: universal machine-learned interatomic potentials combined with SQS generation, free-energy modeling, and elasticity-based surrogates to screen synthesizable compositions with target properties (Liu et al., 28 May 2026). In select compositions, hybrid MC/MD was used to relax chemical order at elevated temperature, confirming that short-range order usually changes free energies by only a few meV/atom and that precise SQS decoration is often not critical for screening (Liu et al., 28 May 2026).

A second branch comes from closed-loop optimization developed in adjacent compositionally complex systems and explicitly described as transferable to CCCs. In compositionally complex refractory alloys, a “two-shot” workflow reduced a space of about 9.37 million Ti-V-Nb-Mo-Hf-Ta-W alloys to about 1,000 representative candidates using configurational entropy thresholds, single-phase BCC stability at $i$7, solidus/liquidus criteria, density, solidification range, and $i$8-medoids clustering (Paramore et al., 2024). Two experimental batches of 24 alloys were then selected, with the second batch guided by batch Bayesian optimization using Gaussian-process regressors and expected hypervolume improvement. The Pareto hypervolume increased by $i$9 between the first and second iterations, and 10 of the 24 alloys in the second iteration dominated all alloys from the first (Paramore et al., 2024). The detailed analysis stated that this computational/experimental workflow is directly applicable to CCCs, with constraints adapted to carbide phase stability, oxidation resistance, thermal conductivity, or related objectives (Paramore et al., 2024).

A complementary route addresses synthesis-parameter optimization rather than composition ranking. Active learning for magnetron sputtering of thin-film compositionally complex alloys compared Gaussian-process and random-forest models and reported that the best-performing models discovered synthesis parameters for a target quinary alloy in 14 iterations (Johnson et al., 2024). The same summary reported prediction accuracy within $$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$0 of target composition after only 10 samples for a quinary system, as well as immediate improvement when models trained on ternary or quaternary systems were transferred to quinary targets (Johnson et al., 2024). The work was described as model-agnostic with respect to the target material system; this suggests that CCC thin-film synthesis with tunable processing parameters and measurable feedback is a natural extension of the same active-learning logic.

Kinetics-aware methods address a third bottleneck: the evolution of local order under diffusion. The neural network kinetics framework introduced for Nb-Mo-Ta predicts path-dependent migration barriers from an on-lattice “neuron map” representation and uses kinetic Monte Carlo to simulate diffusion-induced chemical evolution (Xing et al., 2023). For each vacancy, eight path-dependent barriers are predicted in bcc materials, with jump probabilities

$$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$1

Although demonstrated in a refractory alloy, the framework was presented as broadly applicable to compositionally complex materials, including carbides (Xing et al., 2023). For CCCs, this provides a route to studying how local kinetic heterogeneity, short-range order, or precipitate formation emerge from the same compositional complexity that governs thermodynamic stability.

6. Broader design principles, misconceptions, and adjacent developments

Several conclusions now recur across the CCC literature. First, maximizing configurational entropy is not synonymous with maximizing performance. In fluorite and pyrochlore oxides, Wright et al. reported that medium-entropy and non-equimolar compositions can outperform equimolar high-entropy counterparts, and that size disorder can be a better predictor of thermal conductivity suppression than configurational entropy itself (Wright et al., 2020, Wright et al., 2019). The carbide literature is converging on an analogous position: group identity, local order, band-structure effects in the phonon spectrum, and stacking sequence can outweigh the nominal “entropy level” (Malakkal et al., 6 Aug 2025, Liu et al., 28 May 2026).

Second, more disorder does not invariably mean lower thermal conductivity. The quinary carbide results from phonon engineering showed higher thermal conductivity than some binary and ternary carbides, precisely because a wider phonon bandgap and reduced phonon broadening can offset the additional cation disorder (Malakkal et al., 6 Aug 2025). Third, a single-phase disordered rocksalt description is often useful but not exhaustive. Mixed group 4/5/6 compositions can instead prefer stacking-ordered phases with lower formation energy than either disordered rocksalt or disordered hexagonal states (Liu et al., 28 May 2026).

Fourth, multiphase and interfacial complexity need not be treated merely as defects. The 2025 perspective discussed dual-phase CCCs and cited Qin et al. for boride-plus-carbide systems derived from equimolar starting mixtures that partitioned into non-equimolar equilibrium phases; these dual-phase CCCs displayed hardness exceeding both the individual single-phase high-entropy counterparts and rule-of-mixtures predictions (Luo, 7 Oct 2025). The same perspective emphasized ultrafast reactive sintering as a processing route for high-entropy borides and noted that such approaches are also possible for carbides (Luo, 7 Oct 2025). This shifts attention from equilibrium single-phase synthesis alone to microstructural architectures that may be inaccessible or impractical under slower processing.

A related conceptual development came from outside the carbide family. In A15 superconducting intermetallics, the compositionally complex alloy concept combined high configurational entropy and phase multiplicity to increase the upper critical field $$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$2 of $$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$3 by about $$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$4–$$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$5 relative to $$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$6 at similar $$(\mathrm{Ti}_{1/5}\mathrm{Zr}_{1/5}\mathrm{Hf}_{1/5}\mathrm{Nb}_{1/5}\mathrm{Ta}_{1/5})\mathrm{C}$$7, and the study explicitly suggested that compositionally complex carbides and related intermetallics may similarly benefit from engineered inhomogeneity and high configurational entropy (Nakahira et al., 2021). This suggests that CCCs are increasingly viewed not simply as disordered solid solutions, but as a broader design space in which disorder, order, and phase coexistence can all be functional variables.

Taken together, these results define CCCs as a field organized less by a single entropy threshold than by a set of coupled design questions: which elements determine the favored prototype, which local arrangements govern stability near the hull, which phonon features control transport, which stacking sequences compete with random disorder, and which synthesis workflows can reach the relevant regions of composition-processing space efficiently (Luo, 7 Oct 2025, Liu et al., 28 May 2026).