Hematite (α-Fe 2 O 3 ) is a promising material for several important applications, such as photoelectrochemical (PEC) water splitting, energy conversion and storage, [1][2][3] receiving greater attention due to its high abundance, remarkable stability, non-toxicity, and moderate bandgap. 4,5 However, its intrinsic low carrier concentration (primarily small electron polarons in hematite) has created a major bottleneck, which limits the usage of hematite for these applications. [6][7][8] The most common strategy to overcome this limitation is by atomic doping wherein the substituted dopant may generate electron polarons and enhance electron polaron concentrations, thereby improving efficiency. [8][9][10][11][12][13] Yet, outstanding questions remain in the pursuit of highly efficient Fe 2 O 3 -based devices with atomic doping. For example, the identity of the intrinsic electron polaron donor in undoped hematite remains under debate. Specifically, various experimental works have claimed that oxygen vacancies (V O ) are the source of extra electron polarons in n-type Fe 2 O 3 , 14 whereas theoretical works have shown that V O have an extremely large ionization energy compared to k B T, 15 which suggests that they are not supposed to be the primary contributor of electron polarons. Meanwhile, some other theoretical works have supported that iron interstitials (Fe i ) are the major electron polaron donors due to a significantly smaller ionization energy than that of V O . 16 Another important question, which is more general to oxides than particularly to Fe 2 O 3 , is how to determine dopants which will yield the highest carrier concentration? Insights into the design of efficient oxide based devices by simple yet practical prediction of atomic doping are highly desired. 2 There have been several theoretical studies on intrinsic defects and atomic doping in Fe 2 O 3 ; for example, defect formation energy and charge transition levels have been computed for Fe 2 O 3 , which can help discover dopants with low ionization energy, such as Sn, Ge, and Ti. 15,17 However, these works cannot yet address the above questions since direct evaluation of electron polaron concentrations requires knowledge of the combined effects of dopant solubility and ionization energy from all intrinsic defects and extrinsic dopants in the system at the charge neutrality condition. Intrinsic defects should be considered simultaneously as they may compensate the electrons or holes from dopants, and yet, the effects of intrinsic defects with or without external doping are not well-established in Fe 2 O 3 so far, as mentioned above.
In this work, we answer these questions by calculating electron polaron concentrations in Fe 2 O 3 under various conditions from first-principles. By careful evaluation of defect concentrations in the presence of small electron polarons, we can reliably predict the concentrations of electron polarons in Fe 2 O 3 , in excellent agreement with experiments at similar conditions. Further detailed computational analysis of dopant solubility, ionization energy, chemical condition, and synthesis temperature is provided in order to answer outstanding questions. What are the intrinsic major electron polaron donors in undoped Fe 2 O 3 ? Which dopants are the best at raising electron polaron concentrations? What makes a dopant effective in raising electron polaron concentrations (e.g., high solubility or low ionization energy)? The work is organized as follows. First, we discuss intrinsic defects in undoped Fe 2 O 3 . Second, we systematically study tetravalent and pentavalent dopants, identifying the best dopants for this system. Third, we analyze the importance of solubility against ionization energy in enhancing electron polaron concentrations in hematite as a function of synthesis conditions. Fourth, we reveal the general trends of formation energy/ionization energy within each group with respect to ionic radius. Finally, we investigate the effect of dopant clustering on electron polaron concentration. The contribution of entropy to the formation free energy is also rigorously considered. This work provides an in-depth and comprehensive study of the interactions among intrinsic defects, extrinsic dopants, and small polarons and their effects on carrier concentrations in polaronic oxides.
Density functional theory (DFT) calculations were performed in the open-source plane-wave code QuantumESPRESSO 18 using ultrasoft GBRV pseudopotentials 19 and an effective Hubbard U 20 value of 4.3 eV for Fe 3d orbitals. 15,21 Plane-wave cutoff energies of 40 and 240 Ry were used for wavefunctions and charge density, respectively. For all calculations except phonon frequency ones, we employed a 2 Â 2 Â 1 supercell (120 atoms) of the hexagonal unit cell with a 2 Â 2 Â 2 k-point mesh for the integration over the Brillouin zone.
The chemical potentials of Fe and O were evaluated from firstprinciples following the approach employed in Ref. 16. Particularly, for Fe 2 O 3 in thermodynamic equilibrium growth conditions, the chemical potentials μ Fe and μ O must satisfy Eqs. ( 1
In an O rich environment, Fe 2 O 3 will be in equilibrium with O 2 as in Eq. ( 2), whereas in an O poor environment (Fe rich), Fe 2 O 3 will be in equilibrium with Fe 3 O 4 as in Eq. ( 3). In order to correct the well-known overbinding problem of O 2 by DFT, we used the experimental value of 5.23 eV for the binding energy of O 2 . 22 Meanwhile, the chemical potential of each dopant X was computed as in Ref. 17. Specifically, the chemical potentials of dopants μ X are limited by
to avoid formation of secondary phases between dopants and oxygen [Eq. ( 4) for group IV elements and Eq. ( 5) for group V elements]. After obtaining the chemical potential of O in the rich/ poor limit, we will obtain the corresponding dopant chemical potentials through Eqs. ( 4) and (5). The chemical potentials of dopant X are also restricted by their elemental solids: μ X E X . In order to show the effect of oxygen partial pressure on defect and electron polaron concentrations, we relate chemical potential of oxygen to oxygen partial pressure at finite temperature. The details of the conversion can be found in the supplementary material.
Charged defect formation energies and defect concentrations, including carrier concentrations, were evaluated from firstprinciples. Note that unless specified, defects include both intrinsic defects and extrinsic dopants in this work. First, the formation energies for each defect (X) at a charge state q were obtained according to
where E q (X) is the total energy of the defect system (X) with charge q, E prist is the total energy of the pristine system, μ i and ΔN i are the elemental chemical potential and change in the number of atomic species i, and ε F is the Fermi energy. A charged defect correction Δ q to remove spurious interactions of a charged defect with its periodic images and background counter-charge was computed with techniques developed in Refs. 23 and 24 as implemented in
the JDFTx code. 25 The elemental chemical potentials were carefully evaluated against the stability of by-product compounds as detailedly discussed in Sec. II B. The corresponding charge transition levels (CTLs) of the defects were obtained from the value of ε F where the stable charge state transitions from q to q 0 ,
The ionization energies are computed by referencing the CTLs to the free polaron state. [26][27][28][29] Specifically, in Fe 2 O 3 , it has been experimentally observed that photoexcited carriers relax on the picosecond timescale to form small polarons, 7 which have been measured to form at energies 0:5 eV below the conduction band minimum (CBM). 6,8 Theoretically, this free polaron level is computed as the charge transition level from q ¼ 0 to q ¼ À1 in the pristine system,
. By this method, we obtain that the free polaron level is positioned at 0.497 eV below the CBM in excellent agreement with experimental observation. 6,8 By referencing to this state instead of the CBM, ionization energy defined in Eq. ( 7) reflects the energy that it takes to form a free polaron for conduction from a defect-bound polaron. The CTLs of all surveyed dopants in this work are listed in Fig. S3 of the supplementary material. Finally, charged defect concentration (c q ) can be computed directly from the charged defect formation energies,
where g is the degeneracy factor accounting for the internal degrees of freedom of the point defect, k B is the Boltzmann factor, and T is the temperature. Two temperatures are introduced during the evaluation of defect concentrations. First, a synthesis temperature (T S ) simulating an experimental synthesis condition is employed to determine the concentration of each defect at different charge states, satisfying the charge neutrality condition, which is expressed as
where n h and n e are free delocalized hole and electron concentrations. Second, while keeping the concentration of each defect (sum of all charge states) the same as that at synthesis temperature, charge neutrality condition is reinforced at operating temperature (T O : room temperature in this work). This procedure will change the relative concentration of different charge states for a defect at T O (although defect concentrations with summing up all charge states are kept unchanged as the ones at T S ), with the ratio between different charge states as 16
By evaluating charge neutrality, the defect concentrations can be uniquely determined at a particular oxygen partial pressure and synthesis temperature without external parameters.
The effect of entropy is also rigorously considered in this work, which includes two parts: configurational entropy and vibrational entropy. The configurational entropy is computed based on the assumption of an ideal solution (reasonable at the dilute doping limit), 30,31 while vibrational entropy is calculated with firstprinciples phonon frequencies and an entropy expression as detailed in the supplementary material. We compute phonon frequencies of 60 atom hexagonal supercells at the Γ-point by using density functional perturbation theory at the DFT+U level as implemented in the Vienna Ab initio Simulation Package (VASP). [32][33][34] Further details about the computational methods of entropy can be found in the supplementary material.
The intrinsic source of electron carriers in undoped Fe 2 O 3 has been the subject of debate for a long time. Here, the electron carriers include both free small electron polarons and free delocalized electrons in principles; however, the latter has negligible concentrations in Fe 2 O 3 . In undoped hematite, intrinsic defects, such as vacancies (V O , V Fe ) and interstitials (O i , Fe i ), may form within the lattice along with the generation of carriers, mainly small electron polarons (EPs). Their formation energy plots at different conditions (O rich, one atmosphere, and O poor environment) are shown in Fig. S1 of the supplementary material. Since V O and Fe i are n-type defects, they introduce small electron polarons into the lattice with the corresponding EP wavefunctions shown in Figs. 1(a) and 1(b). In Fig. 1(c), intrinsic defect concentrations (including vacancies and interstitials) are provided at room temperature (300 K) as a function of oxygen partial pressure ( p O2 ) in undoped hematite for three synthesis temperatures (T S ¼ 873, 1073, and 1373 K, corresponding to 600, 800, and 1100 C commonly used in experiment). First, we note that the defect with the highest concentration is V O (sum of all charged states of oxygen vacancies, lowest formation energy in Table I), which should be chiefly responsible for the nonstoichiometry observed in Fe 2 O 3 . 35 Second, we find that intrinsically excess electrons can form into free electron polarons [dashed red line labeled as EP in Fig. 1(c)], whereas the concentrations of free delocalized electrons and holes are negligible (less than 10 8 cm À3 ), consistent with experimental measurements showing EPs being the majority of photoexcited electrons in Fe 2 O 3 . 7 Also, the lower formation energy of V O and Fe i compared with p-type defects (V Fe and O i ) in Table I is consistent with the intrinsic n-type nature of Fe 2 O 3 .
In terms of identifying the primary donor of these EPs, the conclusions are dependent on the synthesis conditions and cannot be determined from formation energy or ionization energies alone. At T S ¼ 873 K, it is the case that ionized oxygen vacancies [V þ O , dashed purple line in Fig. 1(c)] are the primary donor to free electron polaron concentrations (overlaps with the dashed red line labeled as EP). Interestingly, as the synthesis temperature is elevated, for example, to T S ¼ 1373 K, free electron polaron concentrations are not just more abundant; they are also generated from a Journal of Applied Physics different source, i.e., Fe interstitials [Fe þ i , solid light blue line in Fig. 1(c)]. The switch of the primary donor to electron polarons and their concentrations as a function of synthesis temperature are shown in Fig. 1(d) (at p O2 ¼ 1 atm). We find that for synthesis temperatures below a critical temperature of 1104 K, V þ O is the primary donor, whereas above this threshold, Fe þ i will become the primary donor. To simultaneously show the effect of oxygen partial pressure, we plot heat maps of electron polaron concentrations and the difference between Fe þ i and V þ O concentrations in Figs. 1(e) and 1(f ). The close resemblance between the two figures reveals the importance of forming Fe i in achieving higher electron polaron concentrations in undoped Fe 2 O 3 .
Our results suggest that previous debate over the primary donor in pristine hematite can be explained by the transition from V þ O to Fe þ i while increasing synthesis temperature, which has not been identified before. Furthermore, this transition highlights the varying importance of defect solubility vs ionization energy. While the ionization energy of V O is as high as 0.7 eV, it has the highest
TABLE I. The formation energy (E f ) at the neutral state and ionization energies (electron affinities) of intrinsic defects at p O2 = 1 atm in undoped Fe 2 O 3 , where IE represents ionization energy and EA is electron affinity. IE references to a free electron polaron level, while EA references to VBM. E f IE(0/+1) IE(+1/+2) IE(+2/+3) EA(-1/0) EA(-2/-1) EA(-3/-2) Defect (eV) (eV) (eV) (eV) (eV) (eV) (eV) V O 2.06 0.70 0.81 … … … … Fe i 3.46 -0.01 0.52 1.40 … … … V Fe 4.14 … … … 0.18 0.41 0.56 O i 3.15 … … … … … …
) at the room temperature operating condition as a function of synthesis temperature (T S ) and oxygen partial pressure ( p O2 ). In the atomic plots, gold ¼ Fe, red ¼ O, and blue ¼ Fe i . The yellow cloud is an isosurface of the polaron wavefunction with an isosurface level of 1% of its maximum.
solubility among intrinsic defects (Table I). At lower synthesis temperatures (e.g., below 1 100 K), the formation of Fe i is sparse [less than 10 10 cm À3 at T S ¼ 873 K as shown in Fig. 1(c)], and by consequence, V O is the primary source of electron polarons. In this situation, the electron polaron concentrations are extremely low, 10 12 cm À3 , because the Fermi level is pinned at the first charge transition level of V O at 0:70 eV below the free electron polaron level. This observation is in good agreement with recent measurements of undoped Fe 2 O 3 , which exhibit Fermi level positions between 0.8 and 1.2 eV referenced to CBM. 8 When the synthesis temperature is increased or the oxygen partial pressure is decreased, the formation of Fe i is more achievable and eventually, it can act as the major electron polaron donor in Fe 2 O 3 . In this situation, Fe i is always ionized to Fe þ i due to a negative ionization energy, À0:01 eV, and therefore, the electron polaron concentrations of hematite can be dramatically increased [blue regions in Figs. 1(d)-1(f ) where Fe þ i is the primary donor, and EP concentrations can reach 10 18 cm À3 ]. In this situation, the Fermi level will approach the free polaron limit as experimentally observed. 8
In order to achieve higher electron polaron concentrations and optimize the efficiency of Fe 2 O 3 -based devices, extrinsic doping will be necessary. Here, we broadly investigate potential dopants and identify optimal doping strategies by considering all group IV and V elements as substitutional dopants. Intuitively substituting trivalent Fe ions by tetravalent or pentavalent ions will donate electrons due to the increased valence electron count. In Fig. 2, we only show the atomic structures and electronic structures of two representative extrinsic dopants (Ti and Nb) that we find enhancing electron polaron concentration of Fe 2 O 3 significantly, under typical synthesis conditions, e.g., p O2 ¼ 1 atm and T S ¼ 1073 K. 36 Since the two electron polarons of an Nb dopant (group V) could occupy different Fe sites, different possible configurations (Fig. S4 and Table S1 in the supplementary material) are rigorously considered to find the most stable one [Fig. 2(d)]. Although the energy levels of the two electron polaron states are very close to each other, they are actually not degenerate [Figs. 2(e) and 2(f )]. The atomic structures and electronic structures of all doped hematite are shown in Figs. S5-S10 of the supplementary material. Our predictions of Ti, Ge, Sb, Nb, and Sn as effective dopants in raising elecron polaron concentrations to 10 19 -10 20 cm À3 are consistent with experimental measurements as well (Table S2 in the supplementary material). [10][11][12][36][37][38][39][40][41][42][43] The excellent agreement with experiments on various dopants highlights the robustness of our first-principles prediction of equilibrium dopant and carrier concentrations. An important note from our calculations is that although group V in principles can provide two free electrons from each dopant, we did not see that they are guaranteed to give larger electron polaron concentrations compared to group IV dopants as shown in Table S2 of the supplementary material. The reason is that the second ionization energy of group V elements can be much larger and difficult to contribute to electron polaron concentrations.
Next, we study the effect of synthesis conditions (synthesis temperature and oxygen partial pressure) on polaron and defect concentrations in extrinsically doped hematite. These two factors affect polaron and defect concentrations via different mechanisms. For example, the decrease of oxygen partial pressure lowers oxygen chemical potentials, which increases the dopants' chemical potentials [set by Eqs. (4) or ( 5)] and lowers their formation energy. As a result, the dopant concentration and the polaron concentration can be increased. On the other hand, polaron and dopant concentrations can be tuned by the synthesis temperature through the Boltzmann distribution. We pick four dopants (Ti, Si, Nb, and P) as examples (Fig. 3). Complete results of all dopants that we have studied can be found in Figs. S11-S14 of the supplementary material. It is clear that some dopants raise electron polaron concentrations more than other dopants, for example, Ti over Si and Nb over P (Fig. 3). A natural question is raised: what is the most important factor for a dopant or an intrinsic defect to increase electron polaron concentrations, e.g., low formation energy or low ionization energy?
In order to directly answer this question, we performed a linear regression on the larger data sets obtained from dopant calculations to analyze the importance of dopant formation energy or solubility against that of ionization energy to contributing to electron polaron concentrations at different synthesis temperatures. Figure 4(b) shows the predictive score (e.g., the coefficient of determination R 2 ) of modeling electron polaron concentration over either formation energy (blue line) or ionization energy (red line) or both (black line) at different synthesis temperatures. Figures 4(c)-4(e) give the fitting of electron polaron concentrations over dopant formation energies at three synthesis temperatures (873, 1073, and 1373 K). For example, at T S ¼ 873 K [Fig. 4(c)], the R 2 of a fitting electron polaron concentration over the formation energy corresponds to the first data point in blue line in Fig. 4(b), while the data points in red line correspond to R 2 of fitting over ionization energy (not shown). When fitting with both dopant formation energies and ionization energies (black), the predictive score typically exceeds 0.85, which signifies that these two properties fully determine the electron polaron concentrations since intrinsic defect contribution to electron polaron concentrations is negligible in comparison as shown in Fig. 3. We note that the deviation of the predictive score from 1 could be due to ignoring the second ionization energy of group V dopants in the fitting in Fig. 4(b).
For lower temperatures, the solubility of the dopant (formation energy, R 2 0:8) almost determines how well the dopant is able to raise electron polaron concentrations, while ionization energy is significantly less important (R 2 0:1). This explains why V O , despite a significantly larger ionization energy (0.7 eV) than Fe i (À0:01 eV), is still the major donor in undoped Fe 2 O 3 at lower synthesis temperatures due to much lower formation energy of V O 2.06 eV vs Fe i 3.46 eV (Table I). This can also explain the observation in Table II and Fig. 4(a) that Si underperforms P at low synthesis temperature (T S ¼ 873 K) due to its slightly higher formation energy (Si: 1.946 eV vs P: 1.926 eV), despite a significant difference in their ionization energy (Si: 0.165 eV and P: 0.348). As the synthesis temperature is elevated, poor solubility can be Journal of Applied Physics overcome, and dopants' ability to be ionized is weighted equally [blue and red lines approach R 2 0:5 in Fig. 4(b)]. This explains the dramatic increase in the polaron concentration under Si doping by increasing the synthesis temperature shown in Fig. 4(a), as well as the transition from V O to Fe i above T S ¼ 1104 K as the primary electron donor in undoped Fe 2 O 3 shown in Figs. 1(c)-1(f ). Additionally, we conclude that less soluble dopants, such as Si, require a higher synthesis temperature to reach its optimal electron polaron concentrations compared to more soluble dopants, such as Ti, Ge, and Sb (Table S2 in the supplementary material).
On the other hand, we also analyze the importance of dopant formation energy/ionization energy to electron polaron concentrations as a function of oxygen partial pressure (Fig. S15 in the supplementary material), among which formation energy is always more dominant than ionization energy across the entire oxygen partial pressure range. This is expected since the oxygen partial pressure affects the dopants' formation energy through its relation to the dopants' elemental chemical potential but does not correlate with ionization energy directly.
Moreover, some elemental-specific trends between the formation energy/ionization energy and the ionic radius of dopants within each group are observed. The formation energies of dopants in each group have a parabolic shape with respect to their ionic radius [Fig. 5(a)]. Some dopants, such as Ti and Sb, have small formation energies (high solubility), while some others, such as P and Pb, have much higher formation energies (lower solubility). A radius around 60 pm seems to correspond to the minimum formation energy, which is smaller than the ionic radius of Fe 3þ (64.5 pm). The reason is that the formation of a small electron polaron expands the crystal lattice locally; therefore, a smaller ionic radius of dopants are desired to mitigate the expansion strain from electron polarons. On the other hand, in Fig. 5(b), the trends of ionization energies as a function of ionic radius for group IV and group V elements are different. For group V elements, ionization energies generally get smaller with increasing ionic radius (blue and green dots). However, the trend is opposite for group IV FIG. 3. The changes of polaron and defect concentrations in doped hematite with respect to synthesis conditions (oxygen partial pressure and synthesis temperature). Room temperature intrinsic defects, dopants, and electron polaron concentrations of Ti, Si, Nb, and P doped hematite at (a) T S ¼ 1073 K as a function of p O2 partial pressure and (b) p O2 ¼ 1 atm as a function of synthesis temperature T S .
elements (red and orange dots). Specifically, ionization energies generally increase with ionic radius. These trends could provide useful guidance to experimentalists on what dopants to choose based on simple known constants, such as an elemental ionic radius.
In this section, we discuss the consequence of dopant clustering by forming electric multipoles in hematite, recently shown to be responsible for the doping bottleneck of hematite. 47 Dopant clustering at various synthesis temperatures and partial pressures was calculated using the method described in our previous work. 47 We found that clustering binding energy and formation energy of dopants together determine how much dopant clustering affects the EP concentration. We picked three representative dopants, Ti, Si, and P, as examples to discuss the effect of clustering to EP (Fig. 6). Owing to the very TABLE II. Summary of the four representative dopants with their formation energy at the neutral state (E f ) at p O2 = 1 atm, first ionization energy [IE (0/+1)], second ionization energy [IE (+1/+2)], electron polaron concentration (ρ EP , computed at room temperature with synthesis at 873, 1073, 1373 K, and p O2 = 1 atm), and corresponding experimentally measured electron polaron concentrations. Both IE reference to a free electron polaron level as before. Dopant E f (eV) IE(0/+1) (eV) IE(+1/+2) (eV) ρ EP (cm -3 ) at T S = 873 K ρ EP (cm -3 ) at T S = 1073 K ρ EP (cm -3 ) at T S = 1373 K ρ exp EP (cm -3 ) Si 1.949 0.165 … 1.35 × 10 15 1.4 × 10 17 7.24 × 10 18 … P 1.926 0.348 0.528 6.57 × 10 15 1.8 × 10 17 1.19 × 10 18 … Nb 1.461 0.153 0.287 2.04 × 10 18 2.2 × 10 19 1.54 × 10 20 ∼10 19 (Ref. 44) Ti (w/o clustering) 0.884 … 1.39 × 10 19 7.4 × 10 19 2.80 × 10 20 10 19 -10 20 (Refs. 37-39, 41, 45, and 46) Ti (w/ clustering) 1.535 0.308 0.361 1.34 × 10 19 3.5 × 10 19 2.04 × 10 19 10 19 -10 20 (Refs. 37-39, 41, 45, and 46) Group V elements also have inconsequential clustering effects but for a different reason. They were found to have positive binding energies, Table S3 in the supplementary material, indicating that these elements do not favor cluster formation in hematite. Unlike electric multipole formation with group IV elements, group V elements with two donated electrons cannot form such stable multipoles; therefore, we do not need to consider a similar type of dopant clustering with group V elements. P is also plotted in Fig. 6 as a representative case for group V elements. Since most dopants' EP concentration is not significantly affected by dopant clustering, our conclusions from Secs. III A-III D are unchanged. For dopants that are easy to
cluster such as Ti and Ge, we suggest using quenching (such as cooling in liquid N 2 ) to "freeze" dopants and avoid the diffusion and clustering of them in the cooling process.
At last, we carefully examined the effect of entropy at finite temperature on the formation free energy, as well as polaron and defect concentrations. Entropy contributes to formation free energy from two aspects: configurational entropy and vibrational entropy. Configurational entropy depends on the number of different possible configurations for the defect to be placed in a hematite lattice, which always stabilizes the dopant formation (lower formation free energy). It can again be separated into two parts, configurational entropy from an ideal solution, in which all constituent atoms are treated equal in size and randomly placed in space and excess entropy taking into account of the atomic size difference, the atomic packing fraction, and the number of elements. Since the latter part is much smaller than the former for defects at the dilute limit, 30,31 in this work, the configurational entropy of an ideal solution is used to approximate the total configurational entropy. On the other hand, the vibrational entropy is computed for each system entering formation energy definition in Eq. ( 6). 48,49 We choose Sn and Nb as two representatives for group IV and V elements and find their entropy contribution to the formation free energy to be 0:1-0:2 eV, which does not affect polaron and defect concentrations significantly (see Figs. S17 and S18 in the supplementary material). Therefore, formation energy without entropy contributions is mostly used in this work unless specified since calculating entropy for all dopants is computationally intensive.
In summary, this work demonstrates the interplay among intrinsic defects, dopants, and small polarons in determining carrier concentrations in a prototypical oxide, Fe 2 O 3 . This work identifies the critical role of synthesis conditions, such as synthesis temperature and oxygen partial pressure on carrier concentrations (primarily small electron polarons) in hematite, both undoped and doped, from first-principles calculations. For undoped hematite, the major electron donor switches from V O to Fe i with ramping up synthesis temperature. For doped hematite, the increase of synthesis temperature or lowering of oxygen partial pressure both can increase the electron polaron concentration. However, the increased magnitude is affected by clustering with a form of electric multipoles. For example, without considering the effect of clustering, Ti is the best dopant in boosting electron polaron concentrations among all dopants we surveyed. Nonetheless, the high tendency of clustering for Ti dramatically lowers the electron polaron concentration, while for other dopants, such as Si and group V elements, which have low tendency of such clustering, it has negligible effects on electron polaron concentrations. Although formation energy is more dominant in determining EP concentrations than ionization energy in hematite for most cases, tuning synthesis conditions, such as increasing synthesis temperature, could overcome the poor solubility issue of certain dopants, which suggests that less soluble dopants, such as Si, will require elevated synthesis temperatures to boost EP concentrations. Quenching would be one possible approach to mitigate the clustering of dopants such as Ti and Ge. This work answers several outstanding questions for hematite, which are also applicable to other polaronic oxides. Therefore, our work deepens the fundamental understanding on tuning carrier concentration and provides important guidelines for material design and synthesis, required by efficient energy conversion and storage devices.
See the supplementary material that includes the following contents: defect formation energy, charge transition levels of all dopants, different configurations of doped hematite, electronic structures, defect concentrations of all dopants, the effect of oxygen partial pressure on defect concentration, and the effect of entropy to defect concentrations.
J. Appl. Phys. 130, 245705 (2021); doi: 10.1063/5.0074698