(171d) Estimation of Electronic Entropy Contributions to Redox Thermodynamics in Ceria

The primary motivation of this work is to develop a method through which the entropy change
of oxygen vacancy formation may be computed from first principles for strongly correlated metal
oxides which exhibit intrinsic spin polarization. Namely, a methodology via which the electronic
structure may be utilized to predict electronic entropy contributions to the overall free energy change.
Leveraging electronic structure properties facilitates the process by circumventing the need for
extensive combinatorics and statistical methods which are conventional approaches for entropy
calculations within the realm of statistical physics. Nonstoichiometric cerium dioxide (CeO2) and the
manganite perovskites (AMnO3) form a prototypical composition space meeting the criteria above and
possessing a high degree of versatility with applications ranging from electrocatalysis and solar
thermochemical energy storage to electronic and magnetic devices. The acquisition of a fundamental
understanding of the mechanisms governing entropy in these materials would enhance the rate at which
they may be optimized and tailored for current applications and potentially enable the discovery of
novel applications for which they are ideal candidates.

It is commonly assumed that finite temperature electronic contributions to the free energy of
insulators and semiconductors may be neglected due to the lack or absence of electron density near the
Fermi level. This may be the case when treating ideal, pristine systems since there are no electronic
levels available for population due to thermal excitation of electrons near the valence band maximum.
This sizable bandgap in medium to wide gap materials limit the number of electronic configurations
accessible to the highest occupied levels. The validity of this assumption is highly questionable
however, especially when charged defect levels are present within the energy gap increasing the density
of states near the Fermi level. Shallow defect levels with unpaired electrons near the VBM would allow
for an increase in the number of microstates for thermal population of electrons contributing to the
electronic configurational free energy. The electronic density-of-states data from a structural relaxation
at the HSE06 level of theory with a 12-atom Ce4O8 bulk supercell was implemented to calculate the
Fermi level (electron chemical potential) as a function of temperature. The Fermi-Dirac distribution
along with the normalized DOS were utilized to estimate the majority carrier concentration (assuming
only electronic charge carriers) via the difference between the conduction band and valence band
integrals.

A greedy algorithm is utilized to evaluate the Fermi level at which the error between the
calculated carrier concentration (via above parameters) at a given temperature and the
predicted/estimated carrier concentration is minimized. The corresponding electronic entropy agrees
relatively well with results from work by Naghavi et al. in magnitude and variation with respect to
temperature. Further validating the significance of electronic entropy contributions to the energetics of
oxygen vacancy formation in ceria and other similar mixed conducting oxides which may be
comparable to if not greater than structural configurational and vibrational entropic terms. Recent work
by Lany highlighted this relationship between temperature, Fermi level and charged defect formation
energies specifying that electronic entropy is the physical or thermodynamic consequence of this
phenomenon. Specifically, the energy level of the oxygen vacancy, the conduction band effective mass
and the temperature dependence of the conduction band minimum were claimed to be the electronic
structure properties that govern the electronic entropy contribution to vacancy formation free energies.
The work presented herein provides numerical evidence to support the claims made by Lany in a purely
analytical deduction.

1. Naghavi, S. Shahab, et al. “Giant Onsite Electronic Entropy Enhances the Performance of Ceria for
Water Splitting.” Nature Communications, vol. 8, no. 1, 2017, doi:10.1038/s41467-017-00381-2.
2. Lany, Stephan. “Communication: The Electronic Entropy of Charged Defect Formation and Its
Impact on Thermochemical Redox Cycles.” The Journal of Chemical Physics, vol. 148, no. 7, 2018, p.
071101., doi:10.1063/1.5022176.
3. Kolodiazhnyi, Taras, et al. “Disentangling Small-Polaron and Anderson-Localization Effects in
Ceria: Combined Experimental and First-Principles Study.” Physical Review B, American Physical
Society, 23 Jan. 2019, journals.aps.org/prb/abstract/10.1103/PhysRevB.99.035144.
4. Sun, Lu, et al. “Disentangling the Role of Small Polarons and Oxygen Vacancies in $\Mathrm{Ce}{\
Mathrm{O}}_{2}$.” Physical Review B, American Physical Society, 1 June 2017, journals.aps.org/prb/
abstract/10.1103/PhysRevB.95.245101.