Thermodynamics-inspired high-entropy oxide synthesis

Abstract

High-entropy oxide (HEO) thermodynamics transcend temperature-centric approaches, spanning a multidimensional landscape where oxygen chemical potential plays a decisive role. Here, we experimentally demonstrate how controlling the oxygen chemical potential coerces multivalent cations into divalent states in rock salt HEOs. We construct a preferred valence phase diagram based on thermodynamic stability and equilibrium analysis, alongside a high throughput enthalpic stability map derived from atomistic calculations leveraging machine learning interatomic potentials. We identify and synthesize seven equimolar, single-phase rock salt compositions incorporating Mn, Fe, or both, as confirmed by X-ray diffraction and fluorescence. Energy-dispersive X-ray spectroscopy confirms homogeneous cation distribution, whereas X-ray absorption fine structure analysis reveals predominantly divalent Mn and Fe states, despite their inherent multivalent tendencies. Ultimately, we introduce oxygen chemical potential overlap as a key complementary descriptor for predicting HEO stability and synthesizability. Although we focus on rock salt HEOs, our methods are chemically and structurally agnostic, providing a broadly adaptable framework for navigating HEOs thermodynamics and enabling a broader compositional range with contemporary property interest.

Fig. 1. Composition and thermodynamic landscape governing stability in rock salt high-entropy oxides.

a Rock salt HEO composition map with ΔH_mix and σ_bonds. The map includes all equimolar four-, five- and six-component compositions drawn from the cation cohort: Mg, Ca, Mn, Fe, Co, Ni, Cu and Zn as gray points. Experimental synthesis results for single and multi-phase are indicated with green circles and red crosses, respectively. Region predicted as single-phase are shown in a light green shade. All 5-component compositions in this study are labeled and indicated with blue circles. Gray points with descriptor values lower than those of the five-component systems in blue circles correspond to four-component compositions (majority without Zn), as reported in Table S3. Figure 1a and its datasets are adapted with permission from J.T., Sivak, S.S.I. Almishal et al., Phys. Rev. Lett. 134, 216101 (2025). Copyright (2025) by the American Physical Society. b Temperature and oxygen partial pressure phase diagram illustrating each cation preferred oxidation state in its stable binary oxide phase. In region (1) all cations in Mg_1/5Co_1/5Ni_1/5Cu_1/5Zn_1/5O are stable in 2+ oxidation state in their A²⁺O^2- binary oxide form; in region (2) all cations in Mg_1/5Mn_1/5Co_1/5Ni_1/5Zn_1/5O are stable in 2+ oxidation state in their A²⁺O^2- binary oxide form; and in region (3) Fe is stable in the 2+ oxidation state in its A²⁺O^2- binary oxide form, while Ni and Co reduce to their metallic states. Orange shading indicates cations reduced to oxidation states lower than 2 + . Because Ti, V, and Cr require substantially lower oxygen partial pressures to maintain their 2+ oxidation states, we excluded them from this analysis (see Supplementary Information Note 6). The uncolored version of (b), with legends, is included in the Source Data file.

Fig. 2. Mn- and Fe-containing rock salt compositions evolution with temperature.

The flowchart illustrates the phase progression in the Mn- and Fe-containing compositions with temperature, emphasizing that stabilizing these compositions as single-phase rock salt requires synthesis under a controlled atmosphere. White circles represent oxygen, while colored circles indicate cations (shown with reduced connectivity in projection for simplicity; e.g., corundum cations are octahedrally coordinated though only four bonds are immediately visible here). We highlight that in addition to wurtzite ZnO, spinel phases (Co₃O₄, Mn₃O₄ and Fe₃O₄) and corundum phases (Mn₂O₃ and Fe₂O₃) should be explicitly considered in low temperature processes as competing phases. Note Mn₂O₃ can also form bixbyite Ia$\overline{3}$ phase and Mn_xFe_yO_δ sintered in air forms a spinel and bixbyite mixture, as detailed in Supplementary Information Note 1. Δμ is the change in chemical potential, Δh is the change in molar enthalpy, T is temperature, and Δs is the change in molar entropy. This figure is expanded from our original figure in G.N.K Kotsonis, S.S.I. Almishal et al., Journal of the American Ceramic Society, 106(10), 5587–5611 [1].

Fig. 3. Structural and chemical characterization of Mn- and Fe-containing rock salt high-entropy oxides.

a X-ray diffraction scans of prototypical single-phase MgCoNiCuZnO and the six five-component systems containing Mn, Fe or both after firing in ambient pO₂. All systems containing Mn and Fe form a spinel and rock salt phase mixture. ‘RS’ denotes the peaks corresponding to the rock salt structure, while ‘S’ denotes those corresponding to the cubic spinel structure. Red crosses indicate compositions that did not form the desired rock salt phase, while green check marks denote successful stabilization of the rock salt structure. b X-ray diffraction scans of the same systems under processing in a reducing environment with Ar gas flow. All systems containing Mn and Fe form a single-phase rock salt, while MgCoNiCuZnO forms secondary phases as a result of excess reduction. c X-ray fluorescence spectra of each composition highlighting all detected cations except Mg. Thick colored spectra represent the measured data, while thin dotted lines indicate the corresponding fits generated by the calibrated quantification software. Fitting details and elemental concentrations are provided in Supplementary Information Table S1. Source data are provided as a Source Data file.

Fig. 4. Local electronic structure analysis confirms 2+ oxidation state in Mn- and Fe-containing compositions.

a X-ray absorption near edge structure (XANES) measurements of Mn K-edge spectra in MgNiZnMnFeO and CoNiZnMnFeO in comparison to Mn^x+ reference spectra from standards. b Mn K-edge photon energy versus valence state with best-fit line confirming a predominance of Mn²⁺ within both high entropy compositions, the 2+ reference value is indicated by the orange star. c XANES measurements of Fe K-edge spectra in MgNiZnMnFeO and CoNiZnMnFeO in comparison to Fe^x+ reference spectra from standards. d Fe K-edge photon energy vs valence state with best-fit line confirming a predominance of Fe²⁺ within both high entropy compositions, the 2+ reference value is indicated by the orange star. Source data are provided as a Source Data file.

Fig. 5. Phase evolution, structural and chemical characterization, and elemental homogeneity in the six-cation MgCoNiZnMnFeO parent high-entropy oxide.

a X-ray diffraction patterns of the 6-component high-entropy composition Mg_1/6Co_1/6Ni_1/6Zn_1/6Mn_1/6Fe_1/6O (MgCoNiZnMnFeO) sintered for 5 h under 100SCCM of Ar at different temperatures, suggesting that the transition to single phase occurs between 850–900 °C with the disappearance of the wurtzite peaks. ‘RS’ denotes peaks from the rock salt structure, and ‘W’ indicates the wurtzite structure. b X-ray diffraction patterns of MgCoNiZnMnFeO sintered at 1100 °C under varying oxygen partial pressures. A single-phase rock salt structure forms after 5 h under a 100 SCCM Ar flow. In contrast, sintering in air results in the emergence of a spinel phase, while introducing a small percentage of H₂ leads to the formation of a reduced metallic phase. ‘RS’ denotes peaks from the rock salt structure, ‘S’ denotes those from the cubic spinel structure, and ‘FCC’ indicates the face-centered cubic metal structure. c–e are obtained by characterizing the parent composition MgCoNiZnMnFeO sintered for 5 h under 100SCCM Ar. c Selected area electron diffraction (SAED) along the [110] zone axis of MgCoNiZnMnFeO sintered for 5 h under 100SCCM Ar, with the inset showing the selected area (yellow circle ~700 nm in radius) corresponding to the diffraction pattern. The electron diffraction pattern is consistent with the rock salt crystal structure. d Energy-dispersive spectroscopy (EDS) maps showing a homogeneous distribution of elements at the 50 nm scale. e Mn K-edge and Fe K-edge photon energy vs valence state with best-fit line confirming a predominance of Mn²⁺ and Fe²⁺ within MgCoNiZnMnFeO composition, with the 2+ reference values are indicated by orange stars. Source data for (a), (b) and (e) are provided as a Source Data file.

Fig. 6. Chemical potential overlap as a descriptor for phase stability and synthesizability in rock salt high-entropy oxides.

a Chemical potential diagrams for Mg-O, Cu-O and Mn-O extracted from the Materials Project database, demonstrating stability windows of each cation in their A_xO_y binary oxides. A green color denotes oxygen chemical potential regions in which A²⁺O²⁻ compositions are stable. Note the significant overlap MgO has with CuO and MnO, but the separation between CuO and MnO. b Chemical potential overlap descriptor (μ_overlap) for all two-cation AO combinations. Cu, Mn, and Fe consistently have the smallest μ_overlap with negative values corresponding to large separation, while Mg and Zn have the largest positive overlap. c Introducing chemical potential overlap as a third descriptor (color overlay) along with ΔH_mix and σ_bonds reveals the uniqueness of prototypical MgCoNiCuZnO as the only 5- or 6-cation combination with significant overlap in A²⁺O^2- binary oxide stability windows. This diagram also demonstrates the difficulty of stabilizing compositions with Cu, Mn and Fe. The combinations are colored with the same scale in part (b). Source data for (b) and (c) are provided as a Source Data file.