High-Entropy Oxides (HEOs)

• High-entropy oxides (HEOs) are compositionally complex ceramic materials where five or more cations in near-equimolar ratios stabilize single-phase structures through maximized configurational entropy.
• They are synthesized using diverse methods such as laser-driven, high-pressure, and sol–gel routes, yielding materials with ultralow thermal conductivity, colossal dielectric constants, and tunable band gaps.
• Control over defect chemistry and local structural disorder in HEOs enables enhanced catalytic, magnetic, and electronic functionalities for energy, optoelectronic, and other advanced applications.

High-entropy oxides (HEOs) are compositionally complex ceramic solid solutions in which five or more cation species occupy crystallographic lattice sites in approximately equimolar fractions. The principal scientific rationale for HEOs is to exploit maximized configurational entropy ($S_{\rm conf}$) to stabilize single-phase structures that are otherwise destabilized or immiscible, yielding emergent properties such as ultralow thermal conductivity, colossal dielectric constants, tunable band gaps, defect engineering, and enhanced catalytic activity. HEOs have been demonstrated in a wide structural palette, including rock-salt, spinel, perovskite, fluorite, and more complex silicate frameworks, encompassing up to 20 distinct cationic elements (Aamlid et al., 2023, Wei et al., 11 Apr 2025, Aamlid et al., 2024, Barber et al., 15 Jun 2025).

1. Thermodynamic Principles: Configurational Entropy and Phase Stabilization

A defining feature of HEOs is the large positive configurational (mixing) entropy from the statistical distribution of multiple cations over equivalent sites. For $N$ total site occupations and $n_i$ cations of species $i$,

$S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$

which, under Stirling's approximation and for molar quantities, reduces to

$S_{\rm conf} = -R\sum_i x_i\ln x_i$

where $x_i=n_i/N$ and $R$ is the gas constant (Aamlid et al., 2023). For $M$ equimolar species, $S_{\rm conf} = R\ln M$; e.g., $N$0 gives $N$1 (Aamlid et al., 2023, Berardan et al., 2016, Aamlid et al., 2024).

The impact of $N$2 is encoded in the temperature-dependent free energy:

$N$3

where $N$4 is the enthalpic penalty or benefit from mixing unlike cations. High $N$5 can overcome a positive $N$6, rendering $N$7 and stabilizing a single disorder phase (Aamlid et al., 2023, Kumar et al., 10 Oct 2025, Barber et al., 15 Jun 2025). Calorimetric and theoretical analyses confirm that for archetypal (Mg,Co,Ni,Cu,Zn)O, $N$8, but entropy stabilization renders the rock-salt structure stable above $N$9 (Aamlid et al., 2023).

Empirically, a threshold of $n_i$0 is often cited for robust entropy stabilization, but recent work demonstrates that intermediate $n_i$1 is sufficient to enforce single-phase reversible behavior under favorable kinetic conditions (Kumar et al., 10 Oct 2025).

2. Synthesis Strategies and Structural Diversity

HEOs have been synthesized via solid-state, sol–gel, molten-salt, hydrothermal, combustion, high-pressure, high-pressure-torsion, and laser-driven solid-state (LSS) routes, enabling control over microstructure, defect concentration, and phase selection (González-Rivas et al., 2024, Wei et al., 11 Apr 2025, Aamlid et al., 2024, Akrami et al., 2023, Edalati et al., 2023). Laser-driven methods have enabled single-phase HEOs with up to 20 cationic elements, rapidly accessing extreme temperatures ($n_i$2C) and cooling rates ($n_i$3), which are critical for trapping high-entropy microstates in complex structures such as silicates and pyrochlores (Wei et al., 11 Apr 2025).

Across all methods, microstructural uniformity and cationic homogeneity prove sensitive to the synthesis pathway. Combustion yields nearly ideal cation mixing, whereas conventional solid-state approaches risk micron-scale clustering and chemosegregation; these differences directly affect magnetic and dielectric responses (González-Rivas et al., 2024).

High-pressure synthesis (up to 15 GPa) grants access to high-density and high-coordination-number HEO polymorphs (e.g., rock-salt ZnO at $n_i$4 GPa, modified ludwigite (Cr,Mn,Fe,Co,Ni)$n_i$5O$n_i$6), not accessible via ambient routes. The balance of pressure–volume ($n_i$7), thermal, and configurational entropy effects defines novel stability windows and can suppress or promote phase decomposition (Aamlid et al., 2024).

3. Local Structure, Disorder, and Characterization Modalities

While HEOs retain average crystal symmetry (by XRD or neutron diffraction), profound local and intermediate-range structural disorder is encoded in bond-length distributions, local environments, and site-specific occupancy preferences. Extended X-ray absorption fine-structure (EXAFS), X-ray absorption near-edge structure (XANES), electron energy-loss spectroscopy (EELS), total diffractive pair distribution function (PDF), atom-probe tomography (APT), and STEM-based mapping are required to fully characterize cation/site disorder, short-range order, valence variation, and lattice strain (Barber et al., 15 Jun 2025, Sarkar et al., 2021, González-Rivas et al., 2024).

A key controversy is the degree of configurational disorder. In spinel HEOs, ideal entropy-maximized models (random cation allocation) are seldom realized; site-preference enthalpy terms (e.g., crystal-field stabilization energies) induce strong cation partitioning and locally reduce $n_i$8 (e.g., from $n_i$9 ideal to observed $i$0 in (Co,Cr,Fe,Mn,Ni)$i$1O$i$2) (Sarkar et al., 2021). The balance is always between $i$3 and $i$4 (Sarkar et al., 2021, Barber et al., 15 Jun 2025).

4. Electronic, Magnetic, and Transport Properties

Configurational disorder and correlated defect chemistry give rise to multiple emergent properties in HEOs inaccessible to single-cation analogs, including:

• Thermal transport: Increasing $i$5 systematically suppresses lattice thermal conductivity ($i$6), via mass, force-constant, and strain-field disorder. In (NiCuZnCoMg)$i$7-based HEOs, $i$8 drops from 5.9 to 2.0 W m$i$9 K$S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$0 as $S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$1 rises from 0.5 R (binary) to 1.77 R (quinary+aliovalent), with further reductions via K$S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$2 doping (Kumar et al., 10 Oct 2025). Entropy-induced phonon scattering is central to this effect (Aamlid et al., 2023, Barber et al., 15 Jun 2025).
• Ionic transport: Spatially variable local environments and aliovalent doping yield broad distributions of oxygen-vacancy formation and migration energies. Li-doped rock-salt HEOs reach oxygen-conductivity $S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$3 S cm$S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$4 at $S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$5C (Barber et al., 15 Jun 2025).
• Dielectric behavior: Colossal dielectric constants ($S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$6) and moderate loss tangents are realized in (Mg,Co,Ni,Cu,Zn)O-based HEOs. Charge compensation via partial oxidation (Co$S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$7$S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$8Co$S = k_B \ln\Omega = k_B \ln\frac{N!}{\prod_i n_i!}$9) or oxygen vacancies is essential to sustaining the rock-salt phase when introducing aliovalent cations (e.g., Li$S_{\rm conf} = -R\sum_i x_i\ln x_i$0, Ga$S_{\rm conf} = -R\sum_i x_i\ln x_i$1) (Berardan et al., 2016).
• Magnetism: HEOs exhibit a wide range of magnetic ground states, with antiferromagnetic, ferrimagnetic, and spin-glass behaviors observed depending on composition, structure, and the degree of site disorder. In (Co,Cr,Fe,Mn,Ni)$S_{\rm conf} = -R\sum_i x_i\ln x_i$2O$S_{\rm conf} = -R\sum_i x_i\ln x_i$3, a significant entropy suppression is revealed by site preference, yet bulk and element-specific magnetic moments remain strongly affected by local order (Sarkar et al., 2021). In high-entropy perovskites, antiferromagnetic transitions persist despite A-site or B-site mixing, with $S_{\rm conf} = -R\sum_i x_i\ln x_i$4 and coercivity tunable via cation selection and tolerance factor (Witte et al., 2019).
• Electronic tunability: Control of oxygen chemical potential ($S_{\rm conf} = -R\sum_i x_i\ln x_i$5) enables stabilization of multivalent cations in desired oxidation states within the HEO phase field, as evidenced by the formation of Fe$S_{\rm conf} = -R\sum_i x_i\ln x_i$6 and Mn$S_{\rm conf} = -R\sum_i x_i\ln x_i$7-containing single-phase rock-salt oxides in reduced atmospheres (Almishal et al., 10 Mar 2025). Effective descriptors such as the oxygen chemical potential overlap ("Hoverlap" descriptor) provide rapid screening for compositional viability.

5. Defect Chemistry, Band Structure, and Photofunctionality

HEOs display versatile defect landscapes—oxygen vacancies, mixed valence cations, interstitials, and phase boundaries—that can be harnessed for catalytic, electronic, and optical function:

• Tunable band gaps and defect states: In rare-earth fluorite HEOs, intermediate 4f levels (e.g., Pr$S_{\rm conf} = -R\sum_i x_i\ln x_i$8/Pr$S_{\rm conf} = -R\sum_i x_i\ln x_i$9, Ce$x_i=n_i/N$0/Ce$x_i=n_i/N$1) introduce electronic states within the gap, allowing reversible band-gap tuning between 1.9 and 3.2 eV by redox-induced occupation of these levels and crystallographic transitions between fluorite and bixbyite (Sarkar et al., 2020). Control over the occupancy and distribution of such intermediate bands is key for optical and optoelectronic applications.
• Photocatalysis: HEOs with engineered cationic electronic configurations (e.g., mixed d$x_i=n_i/N$2 and d$x_i=n_i/N$3 cations in TiZrNbTaGaO$x_i=n_i/N$4) show intrinsic donor/acceptor sites and visible light absorption (E$x_i=n_i/N$5~2.5 eV) without the need for precious-metal cocatalysts (Hidalgo-Jiménez et al., 16 Jan 2025). Multiphasic HEOs (e.g., TiZrNbTaWO$x_i=n_i/N$6) exploit multiple heterojunctions for efficient electron–hole separation, delivering oxygen evolution under visible light with AQE and O$x_i=n_i/N$7 rates comparable to leading OER catalysts (Edalati et al., 2023). Defect- and strain-rich dual-phase HEOs dramatically improve CO$x_i=n_i/N$8 conversion rates, outperforming classical anatase TiO$x_i=n_i/N$9 and BiVO$R$0 (Akrami et al., 2023).
• Switchable and transport states: Oxygen vacancy ($R$1) engineering in perovskite HEO films produces nontrivial, sometimes "Janus-faced," effects on the electronic phase diagram, inducing transitions from metallic to weakly-localized, variable-range-hopping, and ultimately Mott–Anderson insulating states as a function of $R$2 (Joshi et al., 8 Jul 2025).

6. Design Principles, Limitations, and Future Perspectives

A rational design workflow for HEOs integrates ideal-entropy rules with enthalpy management, charge neutrality, geometrical tolerance factors, and kinetic trapping:

• Maximize $R$3 via $R$4 cations in (near-)equiatomic proportions on large, high-symmetry sublattices (rock-salt, fluorite, perovskite, spinel) (Aamlid et al., 2023, Barber et al., 15 Jun 2025).
• Select cations for comparable ionic radius and compatible valence, using Goldschmidt/Pauling geometric criteria ($R$5 for perovskites) and enforce average charge balance (Tang et al., 2020).
• Employ first-principles screening of $R$6, bond-length variance, and $R$7 windows (e.g., Hoverlap descriptor) to predict viable chemistries (Almishal et al., 10 Mar 2025).
• Alloying, aliovalent doping, and fast quenching or non-equilibrium processing (PLD, LSS) can trap desired high-entropy phases and defect configurations inaccessible at equilibrium (Wei et al., 11 Apr 2025, Yang et al., 3 Dec 2025).
• The practical phase stability boundary is composition-, structure-, and temperature-dependent, with kinetic factors essential for retaining single-phase disorder at room temperature (Kumar et al., 10 Oct 2025, González-Rivas et al., 2024).

Notably, the suppression of ideal configurational entropy by site preference, strain, and local order must be explicitly considered in predictive models—HEOs are not universally maximally disordered (Sarkar et al., 2021). Synthesis method exerts a profound influence on functional properties, making precise process specification essential for reproducibility (González-Rivas et al., 2024).

HEOs continue to expand the design space for functional ceramics, enabling bespoke tuning of thermal, dielectric, magnetic, catalytic, and electrochemical properties. The integration of high-throughput computational and experimental protocols, defect and interface engineering, and advanced spectroscopic diagnostics will accelerate the discovery of novel HEOs for energy, electronics, catalytic, and optoelectronic applications (Barber et al., 15 Jun 2025, Aamlid et al., 2023, Wei et al., 11 Apr 2025).