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 reduction4 so that the STCH cycle can proceed under viable conditions for the temperatures and partial pressures. In the prototypical STCH oxide CeO2, 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,

MxOMxO1δ+δ2O2,
(1)

should ideally occur at a temperature Ttr ≤ 1400 °C and O2 partial pressures pO2 ≥ 10−3 atm, whereas the gas splitting (GS) reaction

MxO1δ+δH2OMxO+δH2
(2)

should occur at Tgs ≥ 850 °C and at pH2 ≥ 10−1 atm.7,8 For the oxygen chemical potential, we obtain the corresponding inequalities ΔμO,TR ≥ −2.46 eV and ΔμO,GS ≤ −2.94 eV, when taking pH2O = 1 atm and solving ideal gas equations9 using tabulated values10 for the standard enthalpies and entropies of gaseous O2, H2, and H2O. For these two values of ΔμO, the pO2 and pH2 vs T relationships are shown in Fig. 1. We see that chemical potentials ΔμO < −2.46 eV would result in undesirably high temperatures for the TR step, and that ΔμO > −2.94 eV would result in undesirably low H2 pressures in the GS step. Thus, in order to reconcile the two conditions, ΔμO must increase by 0.48 eV over the temperature interval between 850 and 1400 °C. The temperature dependence of the O chemical potential ΔμO 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

ΔμOT=δΔGMxO1δΔGMxOδΔHredTΔSred,
(3)

and, hence,

TΔμOT=δΔSred.
(4)

Thus, the desirable gain in ΔμO 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., CeO2/Ce2O3). In fact, when neglecting vibrational and magnetic free energies, ΔSred/∂δ = 0 and ΔμO(T) = −ΔHred/∂δ = const. for such reactions with no gain in ΔμO 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.

FIG. 1.

Ideal gas law calculation of pO2 and pH2 vs temperature for two values of the O chemical potential, ΔμO,TR = −2.46 eV and ΔμO,GS = −2.94 eV, corresponding to desirable conditions for the TR and GS steps (dots), respectively. pH2 is determined from the H2 + 12O2 ↔ H2O equilibrium with pH2O = 1 atm.

FIG. 1.

Ideal gas law calculation of pO2 and pH2 vs temperature for two values of the O chemical potential, ΔμO,TR = −2.46 eV and ΔμO,GS = −2.94 eV, corresponding to desirable conditions for the TR and GS steps (dots), respectively. pH2 is determined from the H2 + 12O2 ↔ H2O equilibrium with pH2O = 1 atm.

Close modal

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 EF.13,14 The equilibrium defect concentration follows from minimization of Gibb’s free energy associated with defect formation,

ΔGxi,T=ixiΔgincf(T)TΔScfxi.
(5)

Here, ΔG is separated into the non-configurational (ncf) contributions and the configurational (cf) entropy ΔScf. The ncf contributions are proportional to the fractional concentrations 0 ≤ xi ≤ 1, which include both defect (e.g., VO) and non-defect (e.g., OO) site-occupations. The Δgincf are usually taken as the defect formation enthalpy Δhi, with Δhi = 0 for the non-defect sites, but vibrational enthalpy and entropy contributions for each defect15 can be included in this step (lowercase symbols are used to denote energies per defect, e.g., Δgincf=/xiΔGncf). Minimizing Eq. (5) with respect to xi, subject to the condition ixi1=0 via a Lagrange multiplier, and using the regular solution configurational entropy in Stirling’s approximation, gives the equilibrium defect concentrations

xi=exp(Δhi/kT)iexp(Δhi/kT).
(6)

Taking the O vacancy as example, the defect formation energy can be written as16 

ΔhqΔμO,ΔEF=Δhrefq+ΔμO+qΔEF.
(7)

Here, q is the charge state of the defect (e.g., q = 0 or q = +2 for VO) and ΔEF 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., ΔμO = μO − ½E(O2). 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,

ne(T)=EcbmDcbEfFDE,TdE,
(8)

where fFD is the Fermi-Dirac distribution, Ecbm is the energy of the conduction band minimum (cbm), and Dcb 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 Δhrefq (which are T-independent constants), the band gap energy Eg, 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 Δhq for VOq 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,

TΔμO(T)=TΔhqTqΔEF(T),
(9)

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, /δΔGδ,T=0, substitute Δhq from Eq. (7) for Δgincf, and take the T partial derivative, yielding

TΔμO(T)=δΔScfδqTΔEF(T).
(10)

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 Δgincf, 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 ΔHred 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 ΔSred/∂δ [cf. Eq. (4)], the desired gain of ΔμO of 0.48 eV over 550 °C (cf. Fig. 1) can be expressed as an average entropy of about 10 kB. Since the regular solution entropy ΔScf/∂δ varies between, say, 6.9 kB for δ = 0.1% and 2.9 kB 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 × 1022 cm−3. The enthalpy of reduction ΔHred/∂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 ΔSred/∂δ = ΔμO/∂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 EF [cf. Eq. (9)], and the amount of the increase depends on δ and the conduction band effective mass. To elucidate this dependence, ΔEF(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/m0 ≥ 2 (with m0 being the rest mass of the electron). With increasing T, EF must move to lower energies deeper in the gap such that the negative charge due to the electron density ne(T) [Eq. (8)] remains equal to the positive charge due to VO+2 defects. The larger m*e, the more pronounced this shift, and, hence, the larger the contribution to the entropy of reduction.

FIG. 2.

(Left column) The enthalpy of reduction ∂Hred/∂δ required to establish an O off-stoichiometry of δ = 0.3% and 3.0%, as a function of the temperature T. (Center column) The entropy of reduction ∂Sred/∂δ, for the case of neutral (n) O vacancies VO0 (dashed line), and that of charged VO+2 defects (solid lines) for various values of the conduction band effective mass. (Right column) The Fermi level relative to the conduction band minimum (Ecbm). In the upper and middle rows, Ecbm was taken as T independent, whereas in the bottom row, a linear T dependence was assumed with dEcbm/dT = −3 × 10−4 eV/K.

FIG. 2.

(Left column) The enthalpy of reduction ∂Hred/∂δ required to establish an O off-stoichiometry of δ = 0.3% and 3.0%, as a function of the temperature T. (Center column) The entropy of reduction ∂Sred/∂δ, for the case of neutral (n) O vacancies VO0 (dashed line), and that of charged VO+2 defects (solid lines) for various values of the conduction band effective mass. (Right column) The Fermi level relative to the conduction band minimum (Ecbm). In the upper and middle rows, Ecbm was taken as T independent, whereas in the bottom row, a linear T dependence was assumed with dEcbm/dT = −3 × 10−4 eV/K.

Close modal

For the larger off-stoichiometry δ = 3.0% and the corresponding larger defect charge, EF stays close to or even above Ecbm 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 Ecbm with temperature will further reduce the Fermi energy that produces [via Eq. (8)] the electron density required for charge balance and therefore contribute to ΔSred/∂δ. Temperature coefficients of the band gap energy in the order of dEg/dT = −5 × 10−4 eV/K are common for oxides.22 The contribution dEcbm/dT from the conduction band may be difficult to determine experimentally but is readily accessible in first principles calculations.23 Taking dEcbm/dT = −3 × 10−4 eV/K as an example, we see in Fig. 2 that reduction entropies in excess of 10 kB 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 TiO2,18 or similarly in CeO2,25 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 VO defect level lies above about Ecbm(0) − 0.4 eV for the larger values of m*e. If the defect level lies lower, EF will drop below the defect level towards the lower end of the temperature interval between TGS and TTR, causing a corresponding reduction of the entropy ΔSred/∂δ due to neutral vacancy formation.

It is notable that CeO2 fulfills the criteria discussed here rather well: The pO2 dependence of the stoichiometry determined from thermogravimetric experiments shows that vacancies form in a doubly charged state.19 A recent first principles calculation26 reported VO 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. CeO2 has also a large conduction band density of states due to the localized Ce-4f orbitals,27 with m*e/m0 ≈ 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.

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