The ideal material for solar thermochemical water splitting, which has yet to be discovered, must satisfy stringent conditions for the free energy of reduction, including, in particular, a sufficiently large positive contribution from the solid-state entropy. By inverting the commonly used relationship between defect formation energy and defect concentration, it is shown here that charged defect formation causes a large electronic entropy contribution manifesting itself as the temperature dependence of the Fermi level. This result is a general feature of charged defect formation and motivates new materials design principles for solar thermochemical hydrogen production.

Solar thermochemical hydrogen (STCH) production is one of the most advanced technologies for sustainable fuel generation,^{1} but further material design advances are needed for commercial viability.^{2,3} Suitable oxides must satisfy inequalities for the enthalpy and entropy of reduction^{4} so that the STCH cycle can proceed under viable conditions for the temperatures and partial pressures. In the prototypical STCH oxide CeO_{2}, the entropy contribution has been shown to originate from configurational and vibrational entropies associated with oxygen vacancy defects,^{5} as well as from a “giant” onsite electronic entropy in the Ce *f* orbitals.^{6} However, most STCH related studies have so far considered only charge-neutral vacancy formation. With special attention to the case of charged defect formation, this work makes a connection between STCH thermodynamics and the calculation of defect equilibria, where the latter are determined from a numerical solution of a thermodynamic detailed balance model. It is shown that charged defect formation generally implies large entropies that depend on the density of states (DOS) in the conduction band, thereby capturing similar physics as the electronic on-site entropy in lanthanide *f* orbitals.^{6} This finding points towards new design criteria for STCH materials that can be screened for by means of first principles calculations.

In the STCH redox cycle, the thermal reduction (TR) of a metal (M) oxide,

should ideally occur at a temperature *T*_{tr} ≤ 1400 °C and O_{2} partial pressures *p*O_{2} ≥ 10^{−3} atm, whereas the gas splitting (GS) reaction

should occur at *T*_{gs} ≥ 850 °C and at *p*H_{2} ≥ 10^{−1} atm.^{7,8} For the oxygen chemical potential, we obtain the corresponding inequalities Δμ_{O,TR} ≥ −2.46 eV and Δμ_{O,GS} ≤ −2.94 eV, when taking *p*H_{2}O = 1 atm and solving ideal gas equations^{9} using tabulated values^{10} for the standard enthalpies and entropies of gaseous O_{2}, H_{2}, and H_{2}O. For these two values of Δμ_{O}, the *p*O_{2} and *p*H_{2} vs *T* relationships are shown in Fig. 1. We see that chemical potentials Δμ_{O} < −2.46 eV would result in undesirably high temperatures for the TR step, and that Δμ_{O} > −2.94 eV would result in undesirably low H_{2} pressures in the GS step. Thus, in order to reconcile the two conditions, Δμ_{O} must increase by 0.48 eV over the temperature interval between 850 and 1400 °C. The temperature dependence of the O chemical potential Δμ_{O} can be related to the enthalpy and entropy of reduction, as introduced in Ref. 4. Specifically, considering the free energy of reaction (1), we obtain

and, hence,

Thus, the desirable gain in Δμ_{O} must result from the entropy of reduction. In Ref. 4, it was shown that the required constraints are generally not fulfilled for the reduction/oxidation between ordered stoichiometric phases (e.g., CeO_{2}/Ce_{2}O_{3}). In fact, when neglecting vibrational and magnetic free energies, *∂*Δ*S*_{red}/*∂δ* = 0 and Δμ_{O}(*T*) = −*∂*Δ*H*_{red}/*∂δ* = *const.* for such reactions with no gain in Δμ_{O} with *T*. When considering the tabulated thermochemical properties (which include these effects), the resulting entropies are negative for most materials, making the STCH reactions even more unfavorable. Thus, positive solid-state entropies of considerable magnitude are essential for successful STCH materials.

A route towards larger entropies is available via off-stoichiometric reactions resulting from defect formation. In semiconductors or insulators, neutral and charged defects assume an equilibrium that involves also the intrinsic charge carriers, i.e., electrons and holes.^{11,12} The charged species interact with each other through a self-regulation mediated by the Fermi energy *E*_{F}.^{13,14} The equilibrium defect concentration follows from minimization of Gibb’s free energy associated with defect formation,

Here, Δ*G* is separated into the non-configurational (ncf) contributions and the configurational (cf) entropy Δ*S*^{cf}. The ncf contributions are proportional to the fractional concentrations 0 ≤ *x*_{i} ≤ 1, which include both defect (e.g., *V*_{O}) and non-defect (e.g., O_{O}) site-occupations. The Δ*g*_{i}^{ncf} are usually taken as the defect formation enthalpy Δ*h*_{i}, with Δ*h*_{i} = 0 for the non-defect sites, but vibrational enthalpy and entropy contributions for each defect^{15} can be included in this step (lowercase symbols are used to denote energies per defect, e.g., $\Delta gincf=\u2202/\u2202xi\Delta Gncf$). Minimizing Eq. (5) with respect to *x*_{i}, subject to the condition $\u2211ixi\u22121=0$ via a Lagrange multiplier, and using the regular solution configurational entropy in Stirling’s approximation, gives the equilibrium defect concentrations

Taking the O vacancy as example, the defect formation energy can be written as^{16}

Here, *q* is the charge state of the defect (e.g., *q* = 0 or *q* = +2 for *V*_{O}) and Δ*E*_{F} is the Fermi energy relative to a reference (ref), usually taken as the valence band maximum. For the atomic reservoir, the reference is the zero-temperature enthalpy, e.g., Δμ_{O} = μ_{O} − ½*E*(O_{2}). The defect equilibrium is then obtained by a self-consistent solution of Eqs. (6) and (7) under the condition of overall charge balance between charged defects and electronic carriers.^{13,17} For a given Fermi level, the electron and hole concentrations are determined by, e.g., for the case of electrons,

where *f*_{FD} is the Fermi-Dirac distribution, *E*_{cbm} is the energy of the conduction band minimum (cbm), and *D*_{cb} is the conduction band density of states (DOS), which is here approximated by using a DOS effective mass.^{18}

For a given material, all input parameters required for solving the defect equilibrium can be calculated from first principles, i.e., the defect formation energies $\Delta hrefq$ (which are *T*-independent constants), the band gap energy *E*_{g}, and the effective masses *m**_{e} and *m**_{h}. For the purpose of the present work, however, we invert the problem and determine the formation energy Δ*h*^{q} for *V*_{O}^{q} as a function of the O off-stoichiometry *δ* [cf. Eq. (1)] for a range of temperatures *T*. In practice, this is done by numerically searching within the self-consistent loop for a formation energy that results in the targeted value for *δ*. From Eq. (7), we then obtain the temperature dependence of the O chemical potential,

thereby gaining access to information about the requirements for the STCH reactions discussed above. On the other hand, we can consider the minimum condition of Eq. (5) for the defect equilibrium, $\u2202/\u2202\delta \Delta G\delta ,T=0$, substitute Δ*h*^{q} from Eq. (7) for Δ*g*_{i}^{ncf}, and take the *T* partial derivative, yielding

The first term on the rhs of Eq. (10) is simply the differential configurational entropy, which is widely recognized as an important factor for meeting the STCH conditions.^{5,19} If vibrational entropy is included in Δ*g*_{i}^{ncf}, it will contribute here a corresponding additional term. The last term in Eq. (10), i.e., the temperature derivative of the Fermi level, enters as the electronic contribution to the entropy of reduction, and it is the consequence of charged defect formation. Within the detailed balance model, the charged defect formation energy [Eq. (7)] is coupled via the Fermi level to the Fermi Dirac distribution [cf. Eq. (8)], which is the statistical origin of the electronic entropy as it describes the thermal occupation of the conduction band density of states. Further, by comparing Eqs. (9) and (10), we see that the temperature derivative of the enthalpy is equal to the configurational entropy. Thus, it is worth mentioning that the approximate *T* independence of Δ*H*_{red} in the case of stoichiometric redox reaction does not apply to off-stoichiometric (defect mediated) reactions.

Considering that the temperature derivative of the O chemical potential equals the differential entropy of reduction *∂*Δ*S*_{red}/*∂δ* [cf. Eq. (4)], the desired gain of Δμ_{O} of 0.48 eV over 550 °C (cf. Fig. 1) can be expressed as an average entropy of about 10 *k*_{B}. Since the regular solution entropy *∂*Δ*S*^{cf}/*∂δ* varies between, say, 6.9 *k*_{B} for *δ* = 0.1% and 2.9 *k*_{B} for *δ* = 5.0%, it is clear that configurational entropy alone is insufficient. The electronic entropy contribution due to charged defect formation therefore deserves special attention. So far, materials discovery strategies for STCH based on high throughput calculation have been limited to the case of charge neutral O vacancies.^{20,21}

Figure 2 shows the results of numerical defect equilibria calculations, assuming an O site density of 5 × 10^{22} cm^{−3}. The enthalpy of reduction *∂*Δ*H*_{red}/*∂δ* for a given off-stoichiometry *δ* simply follows Eq. (6), irrespective of the question whether the O vacancies form as charge neutral (*q* = 0) or fully charged (*q* = +2) defects. More interestingly, the entropy of reduction *∂*Δ*S*_{red}/*∂δ* = *∂*Δμ_{O}/*∂T* increases dramatically in the case of charged defects, when compared to the purely configurational entropy obtained for the neutral defects. This increase is a result of the temperature dependence of *E*_{F} [cf. Eq. (9)], and the amount of the increase depends on *δ* and the conduction band effective mass. To elucidate this dependence, Δ*E*_{F}(*T*) is shown in the right column of Fig. 2. In the case of the smaller off-stoichiometry *δ* = 0.3%, the Fermi energies remain mostly inside the band gap for effective masses *m**_{e}/*m*_{0} ≥ 2 (with *m*_{0} being the rest mass of the electron). With increasing *T*, *E*_{F} must move to lower energies deeper in the gap such that the negative charge due to the electron density *n*_{e}(*T*) [Eq. (8)] remains equal to the positive charge due to *V*_{O}^{+2} defects. The larger *m**_{e}, the more pronounced this shift, and, hence, the larger the contribution to the entropy of reduction.

For the larger off-stoichiometry *δ* = 3.0% and the corresponding larger defect charge, *E*_{F} stays close to or even above *E*_{cbm} and exhibits a smaller temperature dependence and correspondingly smaller entropies. So far, we have, however, not considered that the band gap is temperature dependent. A lowering of the conduction band edge *E*_{cbm} with temperature will further reduce the Fermi energy that produces [via Eq. (8)] the electron density required for charge balance and therefore contribute to *∂*Δ*S*_{red}/*∂δ*. Temperature coefficients of the band gap energy in the order of *dE*_{g}/*dT* = −5 × 10^{−4} eV/K are common for oxides.^{22} The contribution *dE*_{cbm}/*dT* from the conduction band may be difficult to determine experimentally but is readily accessible in first principles calculations.^{23} Taking *dE*_{cbm}/*dT* = −3 × 10^{−4} eV/K as an example, we see in Fig. 2 that reduction entropies in excess of 10 *k*_{B} should be attainable, provided that the O vacancies do form in the +2 charged state instead of the charge-neutral state. At this point, it is useful to recall that O vacancies often form deep negative-U centers in main group oxides,^{17,24} whereas in transition metal oxides, such as TiO_{2},^{18} or similarly in CeO_{2},^{25} electrons can often form a localized small polaron state that is bound to the vacancy. In either case, the localized, deep states can be expected to be relatively independent on perturbations like temperature or pressure,^{26} compared to the band gap and band edge energies. Therefore, the condition of the charged vacancy formation is likely to be fulfilled if the temperature dependent Fermi energy remains below the defect level of the vacancy. For the example shown in Fig. 2, this would be the case if the *V*_{O} defect level lies above about *E*_{cbm}(0) − 0.4 eV for the larger values of *m**_{e}. If the defect level lies lower, *E*_{F} will drop below the defect level towards the lower end of the temperature interval between *T*_{GS} and *T*_{TR}, causing a corresponding reduction of the entropy *∂*Δ*S*_{red}/*∂δ* due to neutral vacancy formation.

It is notable that CeO_{2} fulfills the criteria discussed here rather well: The *p*O_{2} dependence of the stoichiometry determined from thermogravimetric experiments shows that vacancies form in a doubly charged state.^{19} A recent first principles calculation^{26} reported *V*_{O} defect levels within less than 0.5 eV from the energy of the conduction band minimum at low temperature, which is consistent with the experimental observations. CeO_{2} has also a large conduction band density of states due to the localized Ce-4*f* orbitals,^{27} with *m**_{e}/*m*_{0} ≈ 20. Thus, the fact that this prototypical STCH material in retrospect fulfills the criteria for entropy gain through charged defect formation suggests that considering the possibility of charged defect formation is an important next step in the search and design of novel STCH oxides.

In conclusion, the temperature dependence of the O chemical potential was calculated from a materials-agnostic defect equilibrium model, and related to the entropy of reduction in thermochemical redox cycles. The detailed balance condition in this model couples the defect formation and the electronic free energy, i.e., the Fermi level, thereby incorporating the electronic entropy. It was shown that a large “giant” electronic entropy is not exclusive to *f*-orbital elements but occurs quite generally for the case of *charged* defect formation. In addition to the commonly used neutral defect formation energy, the present finding suggests new screening criteria for STCH materials that can be addressed by first principles calculations, i.e., the defect level of the O vacancy, the conduction band effective mass, and the temperature dependence of the conduction band minimum.

This work was supported by the U.S. Department of Energy (DOE) under Contract No. DE-AC36-8GO28308 to the National Renewable Energy Laboratory (NREL) and was made possible through funds from both the Office of Science, Basic Energy Sciences (BES), Energy Frontier Research Centers, and the Office of Energy Efficiency and Renewable Energy (EERE), Fuel Cell Technologies Office. This work used EERE sponsored computational resources located at NREL.

## REFERENCES

_{2}O or CO

_{2}splitting materials

_{2}