Unravelling Small-Polaron Transport in Metal ... - ACS Publications

Perspective pubs.acs.org/JPCL

Unravelling Small-Polaron Transport in Metal Oxide Photoelectrodes Alexander J. E. Rettie,† William D. Chemelewski,‡ David Emin,§ and C. Buddie Mullins*,†,‡,∥ †

McKetta Department of Chemical Engineering, ‡Texas Materials Institute, and ∥Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, Texas 78712, United States § Department of Physics and Astronomy, The University of New Mexico, Albuquerque, New Mexico 87131, United States S Supporting Information *

ABSTRACT: Transition-metal oxides are a promising class of semiconductors for the oxidation of water, a process that underpins both photoelectrochemical water splitting and carbon dioxide reduction. However, these materials are limited by very slow charge transport. This is because, unlike conventional semiconductors, material aspects of metal oxides favor the formation of slowmoving, self-trapped charge carriers: small polarons. In this Perspective, we seek to highlight the salient features of small-polaron transport in metal oxides, offer guidelines for their experimental characterization, and examine recent transport studies of two prototypical oxide photoanodes: tungsten-doped monoclinic bismuth vanadate (W:BiVO4) and titanium-doped hematite (Ti:α-Fe2O3). Analysis shows that conduction in both materials is well-described by the adiabatic small-polaron model, with electron drift mobility (distinct from the Hall mobility) values on the order of 10−4 and 10−2 cm2 V−1 s−1, respectively. Future directions to build a full picture of charge transport in this family of materials are discussed.

P

hotoelectrochemical (PEC) cells are attractive devices for converting sunlight into chemical fuels. Pertinent processes are the splitting of water into hydrogen and oxygen1,2 and the reduction of CO2 to species such as methanol.3 These cells consist of one or two semiconducting electrodes, which use photogenerated charges in the half-reactions of interest, for example, evolving H2 at the cathode and O2 at the anode in a water splitting cell. This arrangement offers potential efficiency and capital cost advantages over a coupled electrolyzer− photovoltaic system.4 However, efficient water splitting and CO2 reduction both rely on the development of suitable photoanodes for the oxidation of water.

Efficient water splitting and CO2 reduction both rely on the development of suitable photoanodes for the oxidation of water.

Figure 1. Cartoon of a photoanode in contact with an electrolyte for water oxidation. WD: depletion layer width; EF: Fermi level or chemical potential; e−: electrons, h+: holes. Dashed lines indicate the quasiFermi levels under illumination. Electron small-polaron hopping is depicted as occurring in a narrow band of transport states at the bottom of the conduction band.

When the photoanode is irradiated, photons with energies greater than the band gap produce electron−hole pairs, some of which are separated by a built-in electric field (space-charge or depletion layer) and/or an applied bias (Figure 1). Holes are used to oxidize water at the photoanode surface, and electrons are collected in the external circuit for reduction processes. Potential electrode materials must meet many stringent requirements simultaneously. These include all of the desirable attributes for a photovoltaic material (e.g., efficient visible light absorption and charge transport) as well as appropriately positioned band edges and stability in aqueous electrolyte. The last two criteria have motivated research into metal oxides, which offer resistance to further oxidation and large © XXXX American Chemical Society

overpotentials due to a low-lying valence band of mainly O 2p character. Well-studied metal oxide photoelectrodes include rutile and anatase TiO2, monoclinic WO3, α-Fe2O3,5 and monoclinic BiVO4.6 The wide band gaps of TiO2 and WO3 mean that their theoretical maximum solar-to-hydrogen (STH) efficiencies are limited to ∼2 and ∼6%, respectively.7 Higher STH efficiencies are achievable by the oxides with smaller band gaps: α-Fe2O3 (2.1 eV) and BiVO4 (2.4 eV). These materials Received: September 25, 2015 Accepted: January 13, 2016

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DOI: 10.1021/acs.jpclett.5b02143 J. Phys. Chem. Lett. 2016, 7, 471−479

The Journal of Physical Chemistry Letters

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have recently emerged as the current state-of-the-art photoanodes for water oxidation8−10 and are both limited by bulk charge transport. A deep understanding of how carriers move through these materials is therefore of key importance in the development of PEC cells for water splitting and CO2 reduction. Charge carriers in many transition-metal oxides form “polarons”. A polaron results when an electronic charge carrier moves slowly enough to displace surrounding atoms in response to its presence. The potential well produced by these atomic displacements lowers the carrier’s energy, thereby fostering its confinement. The confined carrier is then said to be self-trapped (because it is trapped within the potential well that its very presence induces) and may only move when the atoms surrounding it appropriately alter their positions. The reader should note the distinction between self-trapped carriers, which can move, and trapped (localized) carriers, which cannot. There are two classes of polaron based on the spatial extent of the self-trapped carrier compared to the material’s repeating structural units. The polaron is termed “large” when it is spread over many structural units and “small” when it collapses to a single unit. Large polarons are characterized by a drift mobility that is moderate in size (>1 cm2 V−1 s−1) and generally falls with increasing temperature.11,12 In contrast, small polarons exhibit an extremely small drift mobility (≪1 cm2 V−1 s−1), which is thermally activated. The drift mobility is defined as the velocity of a carrier under an applied electric field, distinct from the mobility measured by the Hall effect, as will be discussed later. Small polarons have been observed in a myriad of compounds including metal oxides13 and amorphous14,15 and organic semiconductors.16,17 Although small-polaron theory is relatively mature,18−23 studies where solid theoretical footing is married with experimental data using model systems are lacking.15,24,25 The goal of this Perspective is to introduce the physics and experimental characterization of small polarons to researchers working in photoelectrochemistry. Recent efficiency increases in oxide photoanode systems have been rapid, and further progress will result from a better fundamental understanding of charge transport. We will be concerned with transport at around room temperature, though we note that the thermally activated nature of small-polaron hopping suggests that PEC cell operation at elevated temperatures may be beneficial in some cases.26 First, we build on our description of polaron formation, focusing on the small polaron’s thermally activated motion. We then discuss how to interpret the principal characterization techniques of these quasi-particles: dc conductivity, the Seebeck effect, and the Hall effect. These ideas will be illustrated using two important n-type metal oxide photoelectrodes: BiVO4 and α-Fe2O3. Finally, we discuss additional means to advance understanding of charge transport in metal oxide photoelectrodes. As previously discussed, if a carrier is slow-moving and the surrounding ions are easily displaceable, it may minimize its free energy by localizing and becoming self-trapped. These conditions are often satisfied in metal oxides, which tend to have narrow transport bands (and hence, carriers with high effective masses) and large dielectric constants. Alternatively stated, polaron formation is favorable when the energy gained by the lattice due to the trapping of the carrier exceeds the strain penalty caused by the displacement of the nearby ions. Let us dissect the steps involved in small-polaron hopping and, hence, why it should be thermally activated. An example

If a carrier is slow-moving and the surrounding ions are easily displaceable, it may minimize its free energy by localizing and becoming self-trapped.

Figure 2. Simplified potential energy landscape and cartoon for smallpolaron hopping between two identical metal ions in MO4 units after ref 29. Colored circles depict the initial (blue) and final (red) sites, M1 and M2, respectively, and open circles represent oxygen atoms. Note the increase in ionic radius and elongation of the M−O bonds due to electron localization. (Configuration a) The electron is located on the initial site. (Configuration b) Sites are in coincidence, and a transfer is possible. (Configuration c) The electron has completed the hop and resides fully on the final site. Ea is the noninteracting activation energy, and t is the transfer integral at the coincidence configuration. The dashed line indicates an adiabatic surface due to interactions between the two sites.

hop between two MO4 units is illustrated in Figure 2, where M is a metal cation in the 4+ oxidation state. For simplicity, the sites are identical, and a field is applied that slightly favors localization on the final site over the initial site, denoted M2 (red circle) and M1 (blue circle) in Figure 2, respectively. Initially, the electron is trapped on the blue ion, reducing it from M14+ to M13+ (Figure 2, configuration a). A hop can occur when the electronic energies of the initial and final sites are equal, a situation that is termed a coincidence event (Figure 2, configuration b). These coincidence events arise due to thermal motion of the lattice as the atoms surrounding the initial and final sites vibrate into configurations that are less disparate. The energy required to bring the system to this crossing point is the activation energy, Ea. Once in coincidence, the carrier can be thought of as belonging to both sites (Figure 2, configuration b), its fate depending on how strongly the sites interact. This interaction is represented by the “electronic transfer integral”, t, and is a measure of the amount of orbital overlap, either directly between cations or through bridging oxygen atoms. It is useful to define two limiting regimes, adiabatic and nonadiabatic.18,19 In the adiabatic case, t is large and lowers the energy barrier to hopping between sites. Here, the probability of a carrier completing a hop (Figure 2, configuration c) is unity, and transfers are limited only by the frequency of coincidence events.27 In the nonadiabatic scenario, the interaction between sites and probability of a transition is low, requiring many coincidence events before a transfer is completed. The low values of the transfer integral required in this case are unphysical for transition-metal oxides,11,28 though note that the nonadiabatic 472



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a wide variety of materials at 300 K. It follows from eqs 1 and 2 that the conductivity versus temperature may be fit to

limit is often employed in calculations for simplicity. Generally, in transition-metal oxides at around room temperature, the high-temperature adiabatic small-polaron hopping regime is applicable, and the hopping rate has an Arrhenius temperature dependence with an activation energy of (Ea − t).

⎛ −E ⎞ σ(T ) = σ0(T ) exp⎜ σ ⎟ ⎝ kBT ⎠

where σ0 is a temperature-dependent prefactor and Eσ is the conductivity activation energy. Note that Eσ is the sum of (Ea − t) and any activation energies associated with carrier generation, these being informed by the Seebeck effect (discussed below). The validity of the adiabatic model can be tested by estimating the prefactor of the conductivity data given by

Generally, in transition-metal oxides at around room temperature, the high-temperature adiabatic small-polaron hopping regime is applicable, and the hopping rate has an Arrhenius temperature dependence with an activation energy of (Ea − t).

σ0(T ) =

n0(T )e 2ga 2ν0 kBT

(4)

where n0(T) is the carrier density prefactor (see section S1 in the SI). Values for σ0 are on the order of ∼102−103 S cm−1 at 300 K for intrinsic small-polaron hopping,14,15 but n0(T) and hence σ0(T) will be much smaller for lightly doped semiconductors.35 If the measured value of σ0 is considerably larger than that predicted by eq 4, other effects such as carrier-induced softening may be at play, where a carriers’ rapid transfers affect the vibrations of the surrounding atoms.11 On the other hand, a considerably smaller value than that predicted by the adiabatic model indicates that the nonadiabatic framework may be applicable. Due to the simplicity of this model, order of magnitude agreement is generally acceptable. From eq 1, it is evident that a thermally activated conductivity could be a result of the temperature dependence of the carrier concentration or mobility. In conventional semiconductors with wide (disperse) bands, these effects are decoupled using complementary transport experiments to independently measure the carrier concentration, such as the Seebeck coefficient or Hall effect. Seebeck Effect. When a temperature gradient is applied to a metal or semiconductor, a corresponding thermoelectric voltage is generated. The ratio of this voltage to the temperature gradient is termed the Seebeck coefficient, S. It is also referred to as the thermoelectric power or thermopower. In physical terms, it measures the entropy transported with a carrier divided by its charge. Typically, the main contribution to the Seebeck coefficient is the change in the entropy of mixing due to the addition of carriers to transport states37 and thus is impacted by changes in the carrier concentration, n, or the density of thermally accessible transport states, N. The sign of this coefficient can unambiguously determine the majority carriers in the sample (positive for holes and negative for electrons). Bulk samples may be cut or polished to appropriate bar or wafer geometries, and thin films may be measured provided that the substrate contribution is accounted for.32 In the case where the Seebeck coefficient’s magnitude decreases with increasing temperature, the carrier concentration is thermally activated, and S may be fit to

The traditional experimental probes of small polarons are the dc conductivity and Seebeck and Hall effect measurements as functions of temperature. As we will only briefly touch on the experimental details of these measurements, the reader is directed to several reviews on those topics.30−32 The simple case of electron small-polaron hopping in a single, narrow transport band will be dealt with here due to n-type defects such as oxygen vacancies being the most common in metal oxides. However, hole small-polaron hopping is analogous. Conductivity. The conductivity, σ, of an n-type semiconductor is given by σ = neμd (1) where n is the carrier concentration, e is the electronic charge, and μd is the electron drift mobility. Experimentally, the conductivity is determined by passing current through a welldefined sample geometry and measuring the resulting voltage drop. Either four-point collinear or van der Pauw33 contact geometries are acceptable. The metal−semiconductor contacts used must be ohmic (current being a linear function of voltage), small relative to the sample dimensions, and inert over the temperature range to be measured.30 Contact making can be a true art, and trial and error is often needed to determine the best material for the task. Generally, the nature of the metal−semiconductor contact depends on the work functions of the two materials and the quality of the interface. Ohmic contact to n-type metal oxides is often achieved using relatively low work function metals such as In, InGa eutectic, and Ag.34 The drift mobility of small-polaron hopping at high temperature in the adiabatic regime follows an Arrhenius temperature dependence (cf. section IX in ref 18) ⎡ ega 2ν ⎤ ⎛ −(Ea − t ) ⎞ 0 ⎥ exp⎜ μd (T ) = ⎢ ⎟ ⎝ kBT ⎠ ⎣ kBT ⎦

(3)

(2)

where g is a geometric prefactor determined by vector analysis (see the Supporting Information (SI) of this article and ref 35), a is the hop distance, generally equal to the distance between transition-metal sites, ν0 is the characteristic phonon frequency (typically on order of 1013 Hz), kB is the Boltzmann constant, T is the absolute temperature, and (Ea − t) is the energy barrier to hopping between ions (illustrated in Figure 2). Raman spectroscopy, which probes optical phonon modes, can be used to estimate ν0.35,36 For back-of-the-envelope calculations, note that the pre-exponential factor, ega2ν0/kBT ≈ 1 cm2 V−1 s−1 for

S = −(kB/e)(ES/kBT + A)

(5)

where ES is the characteristic carrier-generation energy and A is the heat-of-transport constant. In a conventional, wide-band semiconductor, ES ≈ Eσ, and A is about 1−2,15 while in smallpolaron conductors, ES < Eσ, and A may approach values of 10.38 The magnitude of A gives an indication of the steepness of onset of the small-polaron band.15 (cf. Figure 4c of ref 35 for a graphical illustration of small-polaron bands with different A values.) 473



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The Seebeck coefficient becomes very simple in the hightemperature limit where all N states of the extremely narrow small-polaron band are thermally accessible. Then, if each state can be occupied by a small polaron having one of two spin states, the Seebeck coefficient is given by39 ⎛ k ⎞ ⎡ 2(1 − c) ⎤ S = ⎜ B ⎟ ln⎢ ⎥ ⎝e ⎠ ⎣ c ⎦

In small-polaron conductors where carriers are strongly localized, the hopping site geometry and local orbital arrangement play large roles in the resulting Hall voltage. Therefore, it cannot be used to determine the true carrier density or the sign of majority carriers as in the free-carrier case.

(6)

where c = n/N. This representation of the Seebeck coefficient is especially useful when the carrier density is temperatureindependent. Hall Effect. Occasionally termed the “queen” of transport measurements,40 the Hall effect experiment has been a vital probe in our understanding of semiconductors since its discovery by Edwin H. Hall in 1879.41 Most simply, it measures the deflection of charge carriers by a magnetic field. This is achieved using van der Pauw as well as five- or six-point contact geometries, with the applied magnetic field perpendicular to current flow.30 In resistive, low-mobility materials, use of the traditional static magnetic field configuration can be challenging, with the very small Hall voltage being overwhelmed by misalignment voltages and/or thermal effects. The AC field Hall effect, where an alternating magnetic field and lock-in amplification are used, has had great success in discerning these small voltages.42,43 The Hall coefficient, RHall, combines the experimentally relevant quantities RHall =

VHt iB

Perhaps the most striking feature of small-polaron hopping is the frequently observed occurrence of Hall effect sign anomalies.11,14,15,25,44 Then, n-type small polarons are deflected in the direction expected for p-type free carriers, and p-type small polarons are deflected in the direction expected for n-type free carriers. Simple rules associated with the geometrical arrangement of hopping sites and the symmetry of their electronic states explain these Hall effect sign anomalies.46 Bismuth Vanadate (BiVO4). The electronic structure of monoclinic BiVO4 (space group I2/b, a = 5.1935 Å, b = 5.0898 Å, c = 11.6972 Å, γ = 90.387°)47 comprises a conduction band of V 3d character and a valence band made up of O 2p and Bi 6s states (Figure 3a).48 The crystal lattice is best

(7)

where, VH is the Hall voltage, t is the sample thickness, i is the current, and B is the applied magnetic field. The Hall mobility, μHall is obtained using μHall = σRHall

(8)

and importantly may differ in magnitude, temperature dependence, and sign compared to the carrier’s true drift mobility in small-polaron conductors.44 To understand the origins of these differences, it must be recognized that the Hall effect measures the deflection of carriers in a magnetic field, as opposed to the velocity of carriers in an electric field, which defines the drift mobility, as mentioned previously. In conventional semiconductors (e.g., Si, Ge), carriers may be treated as free, and the effective-mass approximation may be used. Free carriers behave similarly to classical charged particles responding to the Lorentz force, and therefore, the Hall mobility is commensurate with the drift mobility. In small-polaron conductors where carriers are strongly localized, the hopping site geometry and local orbital arrangement play large roles in the resulting Hall voltage. Therefore, it cannot be used to determine the true carrier density or the sign of majority carriers as in the freecarrier case. We made this error in our previous publication when analyzing the Hall effect in doped BiVO4 single crystals.43 Calculations of the small-polaron Hall mobility are quite complex. Furthermore, such calculations are much simpler in the nonadiabatic limit than those for adiabatic hopping. Thus, square45 and triangular19 hopping site geometries have been treated theoretically for nonadiabatic hopping, corresponding to cubic and hexagonal lattices, respectively, while only the triangular site geometry for n-type carriers has been determined explicitly in the adiabatic case.18

Figure 3. (Left panels) Orbital projected density of states (DOS) for (a) pristine and (b) electron-doped tetragonal BiVO4; the gray, green, blue, and red lines represent the Bi 6s, Bi 6p, V 3d, and O 2p states, respectively, and the energy zero is set at the Fermi level, which is indicated by the vertical dashed line. In addition, the V dz2 states (shaded in blue) are shown in the inset of (a), and the electron polaron state is indicated by an arrow in (b). (Upper-right panel) Crystal structure representation of scheelite BiVO4 with indication of BiO8 dodecahedra (in purple) and VO4 tetrahedra (in gray). (Lowerright panel) Band-decomposed charge density of the localized electron state (with an isosurface value of 0.02 e Å−3); purple, silver, and red balls represent Bi, V, and O atoms, respectively, and the arrows indicate elongation of the V4+−O bonds. This instructive figure is reproduced from Kweon et al.,49 with permission from the PCCP Owner Societies. We note that the tetragonal scheelite structure was used in these calculations for simplicity but that the results for electron small-polaron hopping are applicable to the monoclinic scheelite case. 474



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thought of as a slightly distorted tetragonal structure. Relevant to small-polaron transport, excess electrons localize primarily on vanadium ions, reducing them from V5+ to V4+, with only a small fraction of the charge residing on the surrounding oxygen atoms.49,50 The localization of the electron causes elongation of the V−O bonds in the VO4 tetrahedra (Figure 3, right, analogous to the example in Figure 2); this distortion plus the electron is what comprises the small polaron. To illustrate the equations described in the preceding section, we will straightforwardly analyze the transport data of 0.3% W:BiVO4 single crystals in the ab-plane.35 These samples were synthesized using the floating zone technique in an image furnace.43 Tungsten is predicted to substitute for vanadium as a shallow electron donor and exist in the W6+ (d0) oxidation state.51 The W concentration of 0.3% is relative to the number density of vanadium sites (∼3.9 × 1022 cm−3). Nearest- and next-nearest-neighbor V sites are ∼4 Å apart in the ab-plane. First, consider the conductivity versus temperature. These data were fitted to eq 2 (Figure 4a), resulting in a conductivity activation energy of 300 meV and a prefactor of ∼10 S cm−1 at 300 K. Table 1 shows the prefactor to be in good agreement with the adiabatic small-polaron model. In our prior report on this material, a deviation from linearity of ln(σT) versus 103/T was observed at ∼250 K and assigned to the onset of a variable range hopping (VRH) process.52 The VRH processes envisioned by Mott53 involve hopping between distant dopant centers and are only realistic at very low temperatures. Thus, we revise this assignment to be the freezing out of high-temperature multiphonon processes and the onset of high-energy tunneling transfers between cations (cf. Figure 11.2, ref 11). The Seebeck coefficient was large and negative (Figure 4b), consistent with a lightly doped n-type semiconductor. The decreasing magnitude of S with increasing temperature indicated that there was an energy barrier associated with carrier generation, ES, of ∼50 meV. This can be thought of as the energy required to ionize electrons from donor states to small-polaron transport states (cf. Figures S1 in the SI and 4c in ref 35). The difference between Eσ and ES yields (Ea − t) = ∼0.25 eV and allows estimation of the drift mobility using eq 4. This operation yields a room-temperature μd value of ∼10−4 cm2 V−1 s−1 at 300 K, considerably lower than that of semiconductors with wide (disperse) bands such as singlecrystal silicon, μd = 102−103 cm2 V−1 s−1 at 300 K depending on doping level (see Figure 3-23 in ref 54). This is in excellent agreement with the room-temperature carrier mobility of W:BiVO4 polycrystalline thin films recently reported by Abdi et al.55 using time-resolved microwave conductivity (TRMC) measurements. We note that TRMC cannot decouple electron and hole contributions to carrier mobility and lifetime. It is possible that combining temperature-dependent TRMC measurements with electron-transport information could yield information about minority carriers in this material and other metal oxides. The V-site geometry in BiVO4 is more complex than the 2D triangular or square arrangements that have been treated in the literature. Still, the qualitative trends of the drift and Hall mobilities can evidence small-polaron formation, namely, a drift mobility that is smaller and more temperature-activated than the Hall mobility,11 as shown in Figure 4c. Iron Oxide (Hematite, α-Fe2O3). At ambient conditions, hematite is a canted antiferromagnet that crystallizes in a distorted hexagonal structure (R3̅ c space group, a = 5.035 Å,

Figure 4. Transport data in the ab-plane for a 0.3% W:BiVO4 single crystals. (a) Conductivity fit to eq 2, (b) Seebeck coefficient fit to eq 6, and (c) Hall and calculated drift mobilities versus reciprocal temperature. Modified from ref 35.

c = 13.75 Å),56 with basal planes of Fe atoms separated by hexagonal close-packed oxygen layers. The interpretation of hematite transport data is often confounded by polycrystalline or impure natural and synthetic samples.57,58 Of most importance, in our view, is the accurate determination of the sample composition regarding impurities or vacancies that may affect electronic properties. Further, measurements are commonly made at elevated temperatures due to the high resistivity of the material.59,60 Recently, Zhao et al. synthesized high-quality epitaxial thin films of 3−9% titanium-doped hematite (Ti:α-Fe2O3) by oxygen-assisted molecular beam epitaxy,61 where the doping level is based on Fe substitution. Titanium was shown to substitute for iron as Ti4+ (d0) by X-ray photoelectron spectroscopy (XPS) and X-ray absorption near-edge spectroscopy (XANES). Ti4+ is an n-type dopant, and the excess 475



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Table 1. Fitting Parameters and Estimated Prefactors Using the Adiabatic Modela σ0 at 300 K (S cm−1) W:BiVO4 Ti:α-Fe2O3 a

sample type

doping concentration (%)

Eσ (meV)

ES (meV)

data

estimated

single crystal epitaxial thin film

0.3 5

300 116

50

7.5 124

12 153

Doping concentrations are atomic percentages based on V- or Fe-site substitution in W:BiVO4 and Ti:Fe2O3, respectively.

electron is predicted to localize on a nearby Fe ion,62,63 reducing it to Fe2+. Therefore, transfers between Fe2+ and Fe3+ ions are presumed to be the electron small-polaron hops. The Fe sites are arranged in six-fold rings, and nearest-neighbor cation distances are ∼3 Å in-plane. The conductivity and Hall effect were measured in the basal plane from room temperature down to 50 K. Here, we extend the authors’ analysis within the adiabatic small-polaron framework. Only the data for the 5% Ti-doped sample are shown for clarity (Figure 5), but analyses of the data for 3 and 9% Ti-doped films yielded the same results (see section S2 in the SI).

hematite in this temperature range and there is no energy barrier associated with carrier generation; in other words, ES can be neglected. The hopping activation energy of ∼0.1 eV is in good agreement with ab initio calculations for adiabatic nearest-neighbor hopping in the basal plane.29 Additionally, the magnitude of the prefactor agrees well with the adiabatic model (Table 1). As above, μd can be calculated as ∼10−2 cm2 V−1 s−1 at 300 K. The estimated drift mobility is plotted alongside the measured Hall mobility from Zhao et al. in Figure 5b. The Hall mobility is effectively constant with a value of 10−1 cm2 V−1 s−1 between 190 and 290 K. Qualitatively, the trends with temperature are consistent with small-polaron hopping, with a Hall mobility that is larger in magnitude and more weakly temperature-activated than the drift mobility. The 2D basal plane hopping considered here facilitates analysis of the Hall effect’s sign, which depends on the interference processes between n sites that comprise a hopping loop. Although the crystal structure of hematite is hexagonal, the hopping geometry is not triangular, but rather, the cations form sixmember hexagonal rings (n = 6). Consider a noncovalently bonded semiconductor, such as a metal oxide. Here, electron hopping in a nearly empty band would produce a negative Hall coefficient for all values of n as the sign is given by (−1)2n+1. Indeed, this was observed in Ti:α-Fe2O3.61 However, in the case of hole small-polaron hopping in a nearly filled band, the sign of the Hall coefficient is (−1)n and thus may be anomalously signed (negative) for odd n.11 Future Opportunities and Challenges. In addition to the electronic-transport measurements discussed in this Perspective, optical measurements can reveal the distinctive absorption peaks produced by exciting a small polaron’s electronic carrier from and within its self-trapping potential well.11,64 The photoconductivity that results from those optical absorptions that free self-trapped carriers can also be observed and possibly harnessed to improve the material’s bulk conductivity. Well-defined samples such as single crystals and epitaxial thin films allow for straightforward interpretation of transport data and are vital to furthering our understanding of electron transport in metal oxides. However, high-efficiency photoelectrodes are, for the most part, polycrystalline thin films with high (nominally ≥1 atom %) doping concentrations. Bridging this divide is key to the development of higher-efficiency PEC cells, namely, what are the effects of grain boundaries and high doping concentrations on small-polaron hopping? A specific question is whether the small-polaronic hopping observed in heavily doped materials is intrinsic or associated with the dopants themselves. For example, whether the small-polaronic transport reported in heavily doped LaMnO3 is intrinsic or due to hopping between dopant-related states is addressed in section 5.2 of ref 11. Further, it should be realized that while doping levels are nominally ≥1 atom %, in reality, these dopants may not exist at these concentrations in the bulk of polycrystalline materials, possibly segregating at grain boundaries and defects. Model polycrystalline systems, theoretical

Figure 5. Transport data in the basal plane of 5% Ti:Fe2O3 epitaxial thin films.61 (a) Conductivity fit to eq 2 and (b) Hall and calculated drift mobilities versus reciprocal temperature.

Fitting of the conductivity data to eq 1 yielded the activation energies and prefactors in Table 1. Although the Seebeck coefficient was not measured on these samples, comparably doped Ti:α-Fe2O3 single crystals exhibited a Seebeck coefficient in the basal plane that was negative and increased in magnitude with increasing temperature,60 opposite behavior to that of W:BiVO4. This indicates that the Ti donors are fully ionized in 476



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guidance, and sensitive characterization techniques65 will be required to answer these questions. As mentioned previously, measurement of the Seebeck coefficient can provide valuable information about the carrier concentration and should therefore be more commonly employed, especially for high-efficiency thin-film photoanode architectures. Additionally, the Seebeck coefficient could be measured as functions of the temperature and of the density of charge injected in a field effect transistor (FET). The results of these measurements can be compared with the predictions of eq 6. This method has been utilized in some organic semiconductors to address polaron formation and to determine the fraction of injected charge that succumbs to traps.17 Finally, the dark transport characterization of metal oxides covered here should be combined with minority carrier (typically hole) information and carrier dynamics, other crucial aspects of transport in PEC cells. Density functional theory calculations suggest that while holes in α-Fe2O3 form small polarons localized on oxygen atoms,66 in BiVO4, the holes are associated with the BiO8 units and have large-polaronic behavior, with a mobility much greater than that of the electrons.67 Temperature-dependent measurements such as transient absorption spectroscopy,68,69 photoconductivity, and time-resolved microwave conductivity55 will shine much needed light on these aspects of carrier transport. To conclude, we have outlined theoretical and experimental aspects of small-polaron transport that are most pertinent to metal oxide photoanodes for water oxidation. The thermally activated nature of this process means that temperature is a vital probe in the study of these materials. Conductivity and Seebeck coefficient measurements can reveal the energy barrier to hopping and hence the drift mobility of majority carriers. While the Hall effect does not yield the drift mobility or carrier concentration as in a conventional semiconductor, it can provide valuable information about the hopping geometry and local orbital overlap. Electronic transports of W:BiVO4 single crystals and Ti:α-Fe2O3 epitaxial films are both in excellent agreement with the adiabatic small-polaron hopping model, with electron drift mobilities on the order of 10−4 and 10−2 cm2 V−1 s−1, respectively. A deeper fundamental understanding of electron−hole dynamics as well as the interplay between dopant and grain boundary effects in metal oxides will be critical in developing higher-efficiency PEC cells. Although impressive progress has been made using α-Fe2O3- and BiVO4-based systems, further material exploration is needed, and it is likely that newly discovered metal oxide photoelectrodes will also host small polarons, for example, CuWO470 and CuFeO2.71 We hope that the guidelines illustrated here aid in accelerating our understanding of charge transport in established and emerging metal oxide photoelectrodes.

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Notes

The authors declare no competing financial interest. Biographies Alex J. E. Rettie obtained his Ph.D. (Chemical Engineering) from the University of Texas at Austin in 2015 under the supervision of Prof. C. B. Mullins, where he studied metal oxides for photoelectrochemical water splitting. Currently he is a post-doctoral researcher at Argonne National Laboratory with Prof. M. G. Kanatzidis working on the exploratory synthesis of superconductors and gamma ray detection materials. See more details at https://scholar.google. com/citations?user=TV2Mo3gAAAAJ&hl=en Will Chemelewski completed his Ph.D. (Materials Science and Engineering) under the supervision of Prof. C. B. Mullins at the University of Texas at Austin in 2015. His work focused on improved materials and devices for photoelectrochemical water splitting. He is currently engaged in research on the clean energy economy in North and South Carolina. David Emin, 1968 Ph.D. in Physics from the University of Pittsburgh and 1977 Fellow of the American Physical Society, joined Sandia National Laboratories in 1969, became an inaugural Distinguished Member of Technical Staff in 1983, and retired in 1997. He is currently an Adjunct Professor in the Department of Physics and Astronomy of the University of New Mexico. http://physics.unm.edu/ pandaweb/webpages/emin/ C. Buddie Mullins received the Ph.D. in Chemical Engineering from the California Institute of Technology in 1990 and was a PostDoc at the IBM Almaden Research Center from 1989−1991, after which he joined the faculty at the University of Texas at Austin. Mullins currently holds the Z.D. Bonner and the Matthew Van Winkle Regents Professorships. http://research.engr.utexas.edu/mullins/

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ACKNOWLEDGMENTS The authors graciously acknowledge generous support from the Welch Foundation Grant F-1436 and the U.S. Department of Energy (DOE) Grant DE-FG02-09ER16119. A.J.E.R. acknowledges the Hemphill-Gilmore Endowed Fellowship for financial support. We thank B. Meekins for careful reading of the manuscript.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02143. Detailed calculations of conductivity prefactors and Hall effect analyses (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 477



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