Cerium-Based Lanthanide High-Entropy Oxides

• Cerium-based LN-HEOs are multicomponent ceramics with a disordered fluorite or bixbyite lattice that enable tunable defect chemistry and redox behavior.
• Synthesis strategies and thermodynamic analyses reveal key phase boundaries and oxygen vacancy regimes that critically affect ionic and electronic transport.
• Controlled redox cycling and precise synthesis protocols allow dynamic band-gap engineering, optimizing these materials for solid oxide electrolytes and electronic devices.

Cerium-based lanthanide high-entropy oxides (LN-HEOs) are multicomponent ceramics comprised of a fluorite or bixbyite-derived lattice, typically formulated as Ce$_x$(YLaPrSm)$_{1-x}$O%%%%2%%%%. These materials feature pronounced chemical disorder on the cation sublattice and are stabilized by configurational entropy, allowing for unique defect chemistries and tunable electronic and ionic properties. LN-HEOs leverage the redox versatility of Ce and Pr, high oxygen-vacancy tolerance, and multi-cation mixing, making them promising for applications such as solid oxide electrolytes and tunable electronic materials. The following sections detail the computational, experimental, and thermodynamic formalism underpinning the structure–property relations, phase stability, synthesis strategies, and electronic structure of these oxides.

1. Structural Phases and Compositional Landscape

Ce-based LN-HEOs form a pseudo-binary system, Ce$_x$(YLaPrSm)$_{1-x}$O$_{2-\delta}$, whose equilibrium and metastable phases depend critically on the Ce fraction ($x$), oxygen nonstoichiometry ($\delta$), and synthesis temperature. The dominant polymorphs are:

• Cubic Fluorite (Fm–3m): Disordered cation/anion lattice, fully disordered O-vacancy sublattice stabilizing up to $\delta \approx 0.33$ for high $x$. Lattice parameter $a$ rises from 5.40 to 5.41 Å at $x=0.5–0.8$ (1500 °C).
• Cubic Bixbyite (Ia–3): Ordered 25% O-vacancy sublattice (8b Wyckoff positions), preferred at low $x$ or high $\delta$. Lattice parameter $a \approx 10.47–10.55$ Å for $x=0.2$.
• Vacancy-Ordered Fluorite (Intermediate): Nanoscale vacancy ordering with average Fm–3m symmetry, observed for $x \approx 0.325–0.35$ at 1300–1500 °C.

Single-phase fluorite is observed only for $x \gtrsim 0.35$ at elevated temperatures (typically >1300 °C for $x=0.35$; down to 1200 °C for $x \geq 0.50$). For $x\leq0.30$, only bixbyite forms under equilibrium bulk synthesis (Yang et al., 3 Dec 2025). The oxygen vacancy fraction $\delta$ is set by electroneutrality:

$$\delta(x) = 2 - \frac{1}{2} [4x + 3.67(0.2) + 3(0.8 - x)]$$

This yields $\delta \approx 0.33$ for $x=0.20$, comparable to vacancy concentrations in $\delta$-Bi$_2$O$_3$, and drops as $x$ increases. Thin-film syntheses via pulsed-laser deposition can kinetically trap the fluorite phase even at $x=0.20$, resulting in metastable materials with exceptionally high vacancy fractions (Yang et al., 3 Dec 2025).

2. Thermodynamic Stability and Phase Diagram

Phase stability is governed by the competition between formation enthalpy and configurational entropy. DFT and free-energy analysis reveal:

• Zero-K Enthalpy: Bixbyite is always lower in enthalpy than fluorite by $\Delta H_f \approx 0.05–0.10$ eV/atom for any $x,\,\delta$ (Caucci et al., 28 Dec 2025, Yang et al., 3 Dec 2025).
• Configurational Entropy:
  • Cation mixing: $S_{\text{mix}} = -R\sum x_i \ln x_i$; for five equimolar cations, $S_{\text{mix}} \approx 13.4$ J mol$^{-1}$ K$^{-1}$.
  • Anion–vacancy mixing: $$S_{\text{vac}} = -R[(1-\delta)\ln(1-\delta) + \delta\ln\delta]$$; for $\delta=0.33$, $S_{\text{vac}}\approx 5.8$ J mol$^{-1}$ K$^{-1}$.
• Finite-Temperature Free Energy: The fluorite phase is stabilized by the entropy term at high $T$, with the phase boundary defined by:

$$\Delta G(x, \delta, T) = [H_{\text{bixbyite}} - H_{\text{fluorite}}] + T S_{\text{config}}^{\text{anion}}(\delta)$$

The critical $\delta_c(x,T)$ for the fluorite–bixbyite transition fits approximately:

$\delta_c(x, T) \simeq 0.25 – 0.6\,x – 0.0001\,T$

• At $T<1000$ K, bixbyite is favored unless $\delta<0.05$.
• At $T=1750$ K, fluorite is stable for $x\geq0.30$, $\delta<0.22$ (Caucci et al., 28 Dec 2025).

3. Local Structure, Defects, and Disorder

In fluorite, oxygen vacancies are randomly distributed, causing local variation in RE–O bond lengths ($\sim$0.02–0.05 Å larger than CeO$_2$). The polyhedral distortion index $A_d$ increases with $\delta$ up to $2\times10^{-2}$, maximized at Pr and Ce sites due to their redox flexibility. In bixbyite, vacancies are strictly ordered, yielding well-defined LnO$_6$ octahedra with bond-length distributions collapsing as $\delta\to0.25$ (Caucci et al., 28 Dec 2025, Yang et al., 3 Dec 2025).

Mixing enthalpies ($\Delta H_f$) are sensitive to Ce content at low $\delta$, but less so at high $\delta$ where bixbyite is fully vacancy-ordered. Bixbyite binding energies for V$_\text{O}$ are more negative (–1.8 eV) than for fluorite (–1.5 to –1.2 eV as $\delta$ increases), indicating stronger vacancy–lattice coupling in the ordered phase (Caucci et al., 28 Dec 2025).

Raman spectroscopy distinguishes these environments: Bixbyite F$_g$ modes (~355 cm$^{-1}$) disappear above $x\approx0.50$; defect bands (~560 cm$^{-1}$) show intensity ratios (D/F$_{2g}$) tracking $\delta$ and $x$. Transmission electron microscopy documents nanoscale vacancy planes at $x\approx0.325$, confirming intermediate regimes (Yang et al., 3 Dec 2025).

4. Electronic Structure and Band-Gap Tuning

Electronic properties are set by the occupation and energy of intermediate $4f$ states (primarily Ce and Pr), with the following aspects:

• Valence States: La, Sm, Y remain trivalent; Pr is mixed 3+/4+ (average valence $\sim3.55$); Ce is predominantly 4+ (>90%) with a minor Ce$^{3+}$ fraction ($\sim$10%) at all $x$ (Bejger et al., 12 May 2025).
• Band Gap Modulation:
  • For fluorite, $E_g$ decreases from $\sim1.6$ eV ($\delta<0.1$) to $\sim0.5$ eV ($\delta>0.20$) as Pr$^{3+}$ and then Ce$^{3+}$ levels emerge at the VBM.
  • Bixbyite displays $E_g$ peaking near $\delta=0.15$ and rising at higher $\delta$ due to full redox saturation.
  • Optical band gaps can be reversibly tuned via redox cycling: 1.93 eV (as-synthesized) $\rightarrow$ 2.47 eV (vacuum, reduced Pr$^{3+}$) or 3.21 eV (full reduction with H$_2$ anneal), all returning to 2.0 eV in air (Sarkar et al., 2020).
• Density of States and Hybridization: The formation of in-gap unoccupied $4f$ bands leads to additional absorption edges (Pr 4$f$: $\sim$1.9 eV, Ce 4$f$: $\sim$2.5 eV, RE 5$d$: $\sim$5.5 eV above O 2$p$) (Sarkar et al., 2020). XANES O $K$-edge and RE $M_{4,5}$, $L_3$-edges reveal reversible occupancy of these $4f$ states.

A plausible implication is that precise control of oxygen partial pressure or targeted redox protocols enables dynamic band-gap engineering in LN-HEOs, expanding their functionality as memristors or optoelectronic devices (Sarkar et al., 2020, Caucci et al., 28 Dec 2025).

5. Synthesis Approaches and Control of Phase Stability

Bulk LN-HEOs are synthesized via solid-state reaction from CeO$_2$, Pr$_6$O$_{11}$, La$_2$O$_3$, Sm$_2$O$_3$, and Y$_2$O$_3$ with high-energy milling and sintering at 1200–1500 °C (10 h, air), followed by air quenching. Pulsed laser deposition enables non-equilibrium trapping of the high-symmetry, vacancy-rich fluorite with as little as 20 % Ce at low substrate temperatures ($\leq450$ °C) and high laser fluence ($\geq2.5$ J/cm$^2$) (Yang et al., 3 Dec 2025).

Practical guidelines for phase targeting:

Target Phase	Synthesis Ce Content ($x$)	Sintering $T$ (°C)	Notes
Bixbyite	$x\leq0.30$	$>1300$	Order, low vacancy
Ordered Fluorite	$x=0.32$–0.35	1300–1500	Quench to freeze VO planes
Disordered Fluorite	$x\geq0.35$	$\geq1200$ (bulk)/PLD	Maximal entropy, high vacancy

Control of oxygen partial pressure during synthesis directly tunes $\delta$ and, thus, governs entry into the desired phase domain. Rapid cooling “freezes” high-temperature cation/anion disorder, stabilizing defect-fluorite with elevated ionic conductivity (Caucci et al., 28 Dec 2025, Yang et al., 3 Dec 2025).

6. Ionic and Electronic Transport Properties

LN-HEOs exhibit high oxygen-ion conductivity, attributed to extensive disordered V$_\text{O}$ networks in fluorite-rich phases:

• Bulk σ at 600 °C: $3\times10^{-3}$–$6\times10^{-3}$ S cm$^{-1}$ ($x$=0.20–0.80), activation energies $E_a=0.6$ eV (bixbyite-rich) to $0.8$ eV (fluorite-rich) (Yang et al., 3 Dec 2025).
• Ionic migration in fluorite is facilitated by wide distributions of V$_\text{O}$ arrangements and variable local RE–O environments, resembling or exceeding conventional stabilized ceria (Caucci et al., 28 Dec 2025).
• The persistence of a finite band gap in vacancy-ordered bixbyite suggests lower electronic conductivity but potentially robust O$^{2-}$ transport channels.

A plausible implication is that the high configurational and vacancy entropy in the fluorite domain is essential for achieving both high oxygen vacancy concentration and ionic mobility without promoting detrimental electronic conduction due to Ce$^{3+}$ percolation (Caucci et al., 28 Dec 2025, Yang et al., 3 Dec 2025, Bejger et al., 12 May 2025).

7. Design Principles and Application Guidelines

Key design strategies for Ce-based LN-HEOs:

• Stabilize Disordered Fluorite for Fast Ionic Transport:
  • $x\geq0.35$, $T\geq1500$ °C with quenching; target $\delta\sim0.15–0.20$ for maximal O$^{2-}$ mobility.
• Promote Vacancy Order (Bixbyite) for Electronics:
  • $x<0.25$, $T<1250$ °C, or controlled annealing to optimize $\delta\to0.25$ and ordered vacancies.
• Band Gap Engineering via Redox and Processing:
  • Use redox-active atmospheres (vacuum, H$_2$, O$_2$) and moderate $T$ ($\leq750$ °C) to switch $4f$ occupancy and reversibly tune E$_g$.
• Anion Entropy and Alloying:
  • Five-cation mixing maintains single-phase stability over wide $\delta$ while mitigating excessive redox activity in Ce/Pr (Bejger et al., 12 May 2025).

These principles permit systematic control over phase, defect, and band-gap states, enabling tailored ionic and electronic properties for solid-state devices including fuel cells, oxygen sensors, and programmable resistive electronics (Caucci et al., 28 Dec 2025, Sarkar et al., 2020, Yang et al., 3 Dec 2025).

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Ce-based LN-HEOs, by their capacity for vacancy engineering, multivalent cation chemistry, and high configurational entropy, provide a versatile materials platform. Their design is underpinned by well-defined thermodynamic, electronic, and synthesis mappings, offering robust tunability for advanced energy and electronic technologies (Caucci et al., 28 Dec 2025, Bejger et al., 12 May 2025, Yang et al., 3 Dec 2025, Sarkar et al., 2020).