Theoretical maximum photogeneration efficiency and performance characterization of InxGa1−xN/Si tandem water-splitting photoelectrodes

InxGa1−xN is a promising material for flexible and efficient water-splitting photoelectrodes since the bandgap is tunable by modifying the indium content. We investigate the potential of an InxGa1−xN/Si tandem used as a water-splitting photoelectrode. We predict a maximum theoretical photogeneration efficiency of 27% for InxGa1−xN/Si tandem photoelectrodes by computing electromagnetic wave propagation and absorption. This maximum is obtained for an indium content between 50% and 60% (i.e., a bandgap between 1.4 eV and 1.2 eV, respectively) and a film thickness between 280 nm and 560 nm. We then experimentally assess InxGa1−xN photoanodes with the indium content varying between 9.5% and 41.4%. A Mott–Schottky analysis indicates doping concentrations (which effectively represent defect density, given there was no intentional doping) above 8.1 × 10 cm (with a maximum doping concentration of 1.9 × 10 cm for an indium content of 9.5%) and flatband potentials between −0.33 VRHE for x = 9.5% and −0.06 VRHE for x = 33.3%. Photocurrent–voltage curves of InxGa1−xN photoanodes are measured in 1M H2SO4 and 1M Na2SO4, and the incident photon-to-current efficiency spectra in 1M Na2SO4. The incident photon-to-current efficiency spectra are used to computationally determine the diffusion length, the diffusion optical number, as well as surface recombination and transfer currents. A maximum diffusion length of 262 nm is obtained for an indium content of 23.5%, in part resulting from the relatively low doping concentration (9.8 × 10 cm at x = 23.5%). Nevertheless, the relatively high surface roughness (rms of 7.2 nm) and low flatband potential (−0.1 VRHE) at x = 23.5% cause high surface recombination and affect negatively the overall photoelectrode performance. Thus, the performance of InxGa1−xN photoelectrodes appears to be a tradeoff between surface recombination (affected by surface roughness and flatband potential) and diffusion length (affected by doping concentration/defect density). The performance improvements of the InxGa1−xN photoanodes are most likely achieved through modification of the doping concentration (defect density) and reduction of the surface recombination (e.g., by the deposition of a passivation layer and co-catalysts). The investigations of the ability to reach high performance by nanostructuring indicate that reasonable improvements through nanostructuring might be very challenging. © 2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0007034., s


I. INTRODUCTION
InxGa 1−x N layers of high structural and optical quality grown on Si have been successfully synthesized in the last few years, 1,2 opening a new perspective for inexpensive and efficient solar harvesting devices. The theoretical maximum efficiency for such tandem photoelectrochemical (PEC) water-splitting systems could reach over 22.5%, using Si as the bottom cell and InxGa 1−x N with a bandgap between 1.6 eV and 1.8 eV 3 (corresponding to an indium content of 37%-44%, respectively 2 ) as the top cell. A PEC watersplitting device utilizing concentrated irradiation and with a solarto-hydrogen (STH) efficiency above 20% could, theoretically, lead to a hydrogen prize below 3 $ kg −1 , a prize competitive with the price of gasoline for the same amount of energy. 4,5 Notwithstanding the high theoretical efficiency of InxGa 1−x N for solar hydrogen production, previous attempts to fabricate InxGa 1−x N water-splitting photoelectrodes have led to very poor performance with photocurrents below 0.1 mA cm −2 under AM1.5G irradiation, i.e., a performance even inferior to pure n-GaN with a bandgap of 3.4 eV. 6 This low performance was attributed to the low crystalline quality of InxGa 1−x N without further investigating the reasons for the gap between theoretically predicted and experimentally observed efficiencies. An in-depth investigation of InxGa 1−x N was performed based on finite element simulations of p-GaN/n-InxGa 1−x N/n-In 0.5 Ga 0.5 N/p-Si/n-Si solar cells, where the InxGa 1−x N layer was graded. These simulations predicted 28.9% efficiency and showed that the thickness and the doping concentration of the graded region substantially affected the performance. 7 The modeling of InGaN/Si tandem solar cells (ignoring space charge recombination) predicted a theoretical maximum efficiency of 31%. 8 These models were based on InxGa 1−x N solar cells and not on InxGa 1−x N water-splitting photoelectrodes, which are somewhat different physical systems due to the presence of the semiconductor-electrolyte interface. 9 Experimental and computational studies of InxGa 1−x N/Si tandem water-splitting photoelectrodes, to identify and quantify their main losses, are yet to be completed.
In this work, we first quantify, through theoretical evaluations of InxGa 1−x N water-splitting photoelectrodes with varying In contents, the maximum theoretical photogeneration efficiencies, before we experimentally assess their performance as photoelectrodes. We report measured photocurrent-voltage (I-V) curves and incident photon-to-current efficiencies (IPCEs) of InxGa 1−x N photoelectrodes with indium contents varying between x = 9.5% and 41.4%. The diffusion length, diffusion optical number, internal and surface losses, and a nanostructuring opportunity factor of the InxGa 1−x N water-splitting photoelectrodes were also calculated, utilizing the measured IPCE and the analytical method proposed by Gaudy and Haussener. 10 The combined computational and experimental characterization allowed us to identify main reasons for the low performance of the current InxGa 1−x N photoanodes and to suggest approaches for their improvements.

II. METHODOLOGY
A. Theoretical photogeneration efficiency of Inx Ga 1−x N/Si photoelectrodes Electromagnetic waves (EMWs), propagating perpendicular to thin InxGa 1−x N layers deposited on a 350 μm thick Si substrate, give rise to interferences between the forward-and backwardpropagating waves due to reflection at the InxGa 1−x N/Si interface. These interferences can lead to "wavy" generation rates inside the InxGa 1−x N semiconductor film and can be predicted by solving Maxwell's equation 11,12 (Beer-Lambert's law [13][14][15] or ray-tracing methods 16 are unable to properly address these effects). Thus, the generation rate and the photogeneration current density in InxGa 1−x N and Si semiconductors were calculated by solving the Maxwell's curl equations for each frequency, ν, 17 where E(z, ν) is the local frequency-dependent electric vector field, n(ν) = n(ν)−jk(ν) is the complex refractive index, and k 0 is the freespace wavenumber. The optical power absorbed per unit volume is given by where I{ε(z, ω)} is the imaginary part of the material's complex permittivity that can be calculated from the material's complex refractive index (n + jk) and the vacuum permittivity (ε 0 ): ε = εrε 0 = (n + jk) 2 ε 0 . The generation rate along the film thickness, G(z), is calculated by integrating the spectral generation rate over the considered spectrum, where the upper integration boundary νmax is νmax ≥ Egap/h and h is Planck's constant. The photogeneration current, i l , is calculated by integrating the generation rate over the thickness of the semiconductor where q is the elementary charge and d is the thickness of the semiconductor. The interface between the InxGa 1−x N and the Si layers was assumed to be perfect, and no additional interlayer (such as an electrical coupling layer) was considered. Such interlayers might in practice be required and-if well engineered-even further improve the performance. 18,19 The EMW propagation was calculated in both the electrolyte and the semiconductors, assuming perpendicular AM1.5G front illumination (i.e., 0 ○ incident angle) and a reflecting back boundary (refractive index for Si from Edwards 20 ), as depicted in Fig. 1. The reflection loss at the semiconductor-electrolyte interface and the absorption in the electrolyte was accounted for by considering a 2 μm thick layer of water in front of the InxGa 1−x N layer (the absorption loss by the electrolyte was very small since the water extinction coefficient is below 10 −5 , k < 4 × 10 −6 in the visible range 21 ). Indeed, the same photogenerated current density was observed using an electrolyte layer with a thickness of 100 μm or 2 μm (consistent

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scitation.org/journal/apm with absorption calculations reported in Ref. 22 or absorption effects discussed in Ref. 23), but the latter could save computational time. EMW propagation was modeled in a 2D domain (with the corresponding boundaries), enabling simulations with incident angles different from 0 ○ .The current for an in-series tandem cell such as the InxGa 1−x N/Si photoelectrode is limited by the lowest subcell's current. Thus, the highest photogeneration efficiency, η, of a tandem cell is obtained when the photogenerated current in all subcells is equal (i.e., subcells are current matched), where i l,In x Ga 1−x N is the photogenerated current density in the InxGa 1−x N subcell, i l,Si is the photogenerated current density in the Si subcell, and i 0 is the incident photon flux at AM1.5G converted into a photocurrent density (i.e., 59.7 mA cm −2 ). We neglected the incident UV light (λ ph < 400 nm, i.e., ∼10% of the incident sunlight) because the complex refractive data of InxGa 1−x N were available from 400 nm only 24 and because the absorption of the UV light in the electrolyte was larger than that of the visible light. 21 The subscript l is used to specify that the current is a photogenerated current and not a photocurrent (the latter also accounting for separation and injection losses). The photogeneration efficiency of InxGa 1−x N/Si tandem photoelectrodes was investigated for varying thicknesses and indium contents (i.e., varying bandgap) of the InxGa 1−x N layer. The thickness of the InxGa 1−x N layer was varied from 10 nm to 1000 nm with the indium content varying from 17% to 59%, corresponding to an optical bandgap variation between 2.6 eV and 1.2 eV. Such bandgaps have the potential for theoretical solar-to-hydrogen (STH) efficiencies above 5% for tandem water-splitting cells. 4 The films below 50 nm were investigated to determine if the ultrathin InxGa 1−x N films could benefit from resonant light trapping. Indeed, the ultrathin films of InxGa 1−x N would greatly reduce the amount of indium and gallium, two relatively expensive and rare materials, and reduce losses due to poor minority charge conductivity or high bulk recombination. 11 The theoretical photogeneration efficiency was also calculated without any reflection losses at the InxGa 1−x N/electrolyte interface and absorption in the electrolyteby simply removing the electrolyte layer in front of the InxGa 1−x N film ( Fig. 1)-to account for an anti-reflective coating.
The complex refractive index of InxGa 1−x N, required in the EMW propagation model, was linearly interpolated from the data of Hazari et al. 24 The complex refractive index of InxGa 1−x N with varying indium contents of the photoelectrodes is depicted in Fig. 2 (samples with indium contents of x = 9.5%, 16.5%, 23.5%, 33.3%, and 41.4% were used for experimental studies). The data of the complex refractive index of the single crystalline silicon layer were taken from Edwards. 20

Computational details
The wavelengths used for the EMW model were varied from 400 nm to 1116 nm, corresponding, respectively, to the smallest visible wavelength and the bandgap of Si (1.11 eV i.e., 1116 nm). The ultraviolet light (λ ph < 400 nm) was not considered in this work because the complex refractive index of InxGa 1−x N was only available between 400 nm and 1687 nm 24 and the UV light's partial absorption by the electrolyte layer. 22 The light was considered as transverse electric, and therefore, only the out-of-plane electric field was calculated. Bloch-Floquet theory was assumed for the periodicity on both sides of the computational domain ( Fig. 1), which is typically used for infinite stab models where no boundary effects appear and where the phase shift is determined by a wave vector and the distance between the source and the destination. 25,26 The Maxwell's curl equations and the generation rate given by Eqs. (1)-(3) were solved with a commercial solver. 27 The convergence was obtained with a direct MUMPS solver. A relative tolerance of 10 −4 in the electric field was used as a convergence criterion. 28 Mesh convergence was obtained for linear mesh discretization, i.e., a piecewise linear finite element basis function, with a size ratio of 4 and element numbers, n el , depending on the irradiation wavelength, λ, and the layer thickness, d, namely n el = d ⋅ β/λ with β = 150 for Si, β = 200 for InxGa 1−x N, and β = 30 for water. The number of mesh elements perpendicular to the direction of light propagation was fixed to 5 with a width of 5 μm. We observed that for planar problems, the solver

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was less robust using cubic or quadratic discretization orders, leading to fluctuations of the electric field in the perpendicular direction of propagation. Linear discretization needed finer meshes, but the calculation time was still reduced compared to quadratic or cubic discretization with a coarser mesh.

B. Performance and characteristics of Inx Ga 1−x N photoelectrodes
Experimentally, we investigated only the performance and characteristics of the InxGa 1−x N photoanodes, given that Si watersplitting photocathodes have widely been studied elsewhere. [29][30][31][32][33][34] Linear sweep voltammetry with chopped light was conducted in solutions of 1M H 2 SO 4 and 1M Na 2 SO 3 as a hole scavenger. The hole scavenger was used to investigate photocurrents with highly increased reaction kinetics at the InxGa 1−x N/electrolyte interface. The diffusion length, L, the diffusion optical number, α 500 L, the ratio of current, R S,VIPCE , the nanostructuring opportunity factor, fnano, the surface recombination loss, the reflection loss, and the bulk loss of InxGa 1−x N photocathodes, were determined by using the method described in Ref. 10 and the corresponding in-house software POPe. 35 However, in order to fully characterize the electrode, the IPCE spectra of the photoelectrode and the following parameters are required: the complex refractive index, the bandgap, the flatband potential, the doping concentration, the relative permittivity, and the thickness of the photoelectrode. The complex refractive index of InxGa 1−x N was linearly interpolated from the data of Hazari et al., 24 presented in Fig. 2. The bandgap of InxGa 1−x N was calculated by Vegard's law with a bowing parameter of 2.5 eV and bandgaps of 0.7 eV and 3.4 eV for InN and GaN, 2 respectively, as follows: The flatband potential and the doping concentration were obtained from the Mott-Schottky analysis. As our samples were not intentionally doped, our reported doping concentration represents more accurately an impurity and defect density. The equivalent circuit depicted in Fig. 6 was used to fit the measured impedance spectra. In the equivalent circuit, Rs is the series resistance of the electrolyte and the semiconductor, Rsc and Csc are the resistance and capacitance in the space charge region (SCR), respectively, Rss and Css are the resistance and capacitance caused by interface states at the semiconductor-electrolyte interface, respectively. This equivalent circuit is usually used for photoelectrodes with species adsorption at the semiconductor-electroylte interface and was previously used for n-type GaN. 36 Here, we added the Warburg diffusion element, W, an extension typically added to account for the diffusion of charges to a large planar electrode, as for InxGa 1−x N photoelectrodes 37 or as previously used for pyrite photoelectrodes. 38 Indeed, the presence of a very high doping concentration in InxGa 1−x N photoelectrodes 39 reduces the SCR to few nanometers only-below 1 nm for InxGa 1−x N (Sec. III B) at an applied potential of 1.23 V RHEand therefore, most of the photogenerated charges are transported by diffusion. The indium content-dependent relative permittivity of InxGa 1−x N was linearly interpolated from the relative permittivity of GaN and InN 7 as The thicknesses of the InxGa 1−x N photoelectrodes were measured by scanning electron microscopy (SEM) (Sec. II B 1).
Here, we briefly introduce the parameters obtained by the semianalytical method of Gaudy and Haussener 10 used to characterize the performance of InxGa 1−x N photoelectrodes. The diffusion optical number is given by The product of the absorption coefficient, α (given as α = 4πk/λ) at a wavelength of 500 nm and the diffusion length. Note that wave interferences were not observed in the IPCE measurements, n of InGaN for x = 9.5%-23.5% are similar to n of the GaN substrate, and therefore, the sample appears quite homogeneous and wave interferences are expected to be small, and k of InGaN was large enough to promote an exponential absorption behavior in the initial ∼100 nm of the layer. For all these reasons, the exponential decrease in the generation rate seemed to be a good approximation and the procedure of Ref. 10 is applicable.
The ratio of current, given as the ratio of the surface charge transfer velocity that contributes to the water-splitting reaction, S T , and the sum of the surface recombination velocity, S R and S T , is given as The ratio of currents is given for a fixed potential, the potential at which the IPCE is measured, V IPCE . For R S,V IPCE = 1, no surface recombination occurs at the semiconductor-electrolyte interface, and for R S,V IPCE = 0, no photocurrent is observed and electron-hole pairs fully recombine at the interface. The nanostructuring opportunity factor (which is valid only for flat photoelectrodes and not for nanostructured ones) is given by where α 500 L 0.95 is defined as the product of the absorption coefficient at 500 nm and the diffusion length, L 0.95 that provides internal quantum efficiency (IQE) ≥95% at 500 nm without surface recombination A photoelectrode with fnano > 2 is expected not to reach high performance even when nanostructured. If fnano ≤ 2, it might be possible to reach higher performance when nanostructuring. If fnano < 0, high performance is expected even without nanostructuring. More details on these parameters can be found in our previous work. 10 1. Experimental details a. InxGa1−xN photoelectrode preparation. To prepare InxGa 1−x N photoelectrodes, first, InxGa 1−x N crystalline layers with varying In contents (from 9.5% to 41.4%) and similar thicknesses (close to 500 nm) were grown on commercial KYMA templates by plasma-assisted molecular beam epitaxy. 2 The template hosts a highly n-type doped GaN layer grown on the sapphire (0006) substrate by metal-organic chemical vapor deposition (a thin insulating AlN buffer layer is deposited on the sapphire substrate, previous to the GaN growth, to improve crystal quality of the GaN epilayer grown above). Second, the ohmic contacts were made with a scratch to put indium in contact with the highly doped GaN and the InxGa 1−x N layers. Third, the copper wires were fixed to indium using conductive silver paint. Finally, the photoelectrode edges and ohmic contacts were covered by white epoxy (Fig. 3). The fully exposed area for each sample is given in Table I. Typically, about 8% of the surface was covered by the epoxy (Loctite EA 9466). This epoxy is not fully opaque and (even though applied generously) transmittance in the range of ∼10% in the wavelength range of interest is still possible. 40 As the diffusion length of the InxGa 1−x N layers was in the range of 11 nm-262 nm only (see Table III and analysis below), we believe that the inaccuracy with respect to epoxy transparency and relevant active area is negligible.
b. Characterization of the InxGa1−xN layers: In content, layer thickness, surface roughness, and ohmic contact. The indium content of the InxGa 1−x N photoelectrodes, measured by x-ray diffraction (XRD), was found at 9.5%, 16.5%, 23.5%, 33.3%, and 41.4% (more information in the supplementary material, Fig. S1). The layer thickness was assessed by taking the average thickness of three measurements using a scanning electron microscope (supplementary material, Fig. S2). The surface topology and the surface roughness, expressed as the root mean square (rms) of the surface variation, were measured by atomic force microscopy (AFM). More details can be found in the supplementary material, Sec. S1. The InxGa 1−x N photoelectrodes for the five indium contents exhibit different surface features and different surface roughnesses (see Figs. S2 and S3 in the supplementary material). Their thicknesses, on the other hand, are found similar (Table I). The ohmic contacts of the photoelectrodes were tested by measuring the resistance between two ohmic contacts separated by a distance of 3 ± 0.2 mm. The resistance was always below 16 Ω (Table I), ensuring a negligible contact resistance since the measured photocurrent was always below 0.25 mA (maximum 4 mA cm −2 as depicted in Fig. 5), which provides a potential shift of only 4 mV. The indium content, the layer thickness, the surface's rms, the ohmic contact resistance between two points separated by 3 ± 0.2 mm, and the surface area of all the InxGa 1−x N photoelectrodes are summarized in Table I. c. PEC experimental setup and measurements. Electrochemical experiments were carried out in a three-electrode setup to refer to the potential of our measurements to the reversible hydrogen electrode. The reference electrode was Ag/AgCl (sat. KCl) and the counter electrode was Pt. The aqueous electrolyte solution was 1M H 2 SO 4 and 1M Na 2 SO 4 . The I-V curves were obtained by using linear sweep voltammetry with a varying voltage rate of 10 mV s −1 in the range of 0.5 V RHE -1.5 V RHE . The voltage rate of 10 mV s −1 gave a stable steady-state current without any photocurrent hysteresis. To ensure that the InxGa 1−x N photoelectrodes were stable with a potential step of 50 mV and covering a frequency range between 10 Hz and 20 kHz, with ten measurements per frequency decade. IPCE spectra were measured at 1.23 V RHE from 300 nm to 900 nm in 1M Na 2 SO 4 as a hole scavenger to achieve measurable photocurrents. However, the lowest wavelength used by the semi-analytical model was 420 nm because it provided better Rsquare values of the numerical and experimental IPCEs fitting. Indeed, the R-square was 0.5 when starting from a wavelength of 420 nm for x = 9.5% and below 0 when starting from a wavelength of 400 nm.

A. Photogeneration efficiency of Inx Ga 1−x N/Si tandem photoelectrodes
The calculated photogeneration efficiencies for varying InxGa 1−x N film thicknesses and bandgaps are depicted in Fig. 4 (the photogeneration efficiencies for varying InxGa 1−x N film thicknesses and indium contents are shown in the supplementary material, Fig. S6a). The theoretical maximum efficiency of 27% is slightly lower than the previous prediction of 31% for InGaN/Si solar cells. 8 The photogenerated current density at the maximum efficiency of 27% is 16.8 mA cm −2 (compared to 59.7 mA cm −2 at 100%). Two regions with maximum efficiency can be observed: a region with a lower bandgap 1.2 eV-1.4 eV (indium content of 59%-51%) and a lower film thickness 300 nm-550 nm, and a region with a higher bandgap ≈1.5 eV (x = 47%) and a higher film thickness 700 nm-900 nm. The lower bandgap region corresponds to a photogeneration efficiency limited by the photogenerated current in the Si subcell. Namely, a top cell bandgap of 1.2 eV-1.4 eV can absorb most of the light, thus, its thickness must be reduced to let enough light reach the bottom Si subcell. By contrast, the efficiency of the 1.5 eV bandgap region is limited by the InxGa 1−x N subcell, and therefore, its thickness must be increased up to 700 nm to enhance the photogenerated current in the InxGa 1−x N subcell itself.
As can be seen from our calculations, the maximum theoretical efficiency can only be obtained for InxGa 1−x N bandgaps between 1.2 eV and 1.5 eV, that is, slightly lower than the computationally predicted ideal and optimum photoelectrochemical full tandem device with 1.6 eV-1.8 eV bandgap top subcell (in the case of the InxGa 1−x N film, it corresponds to an indium content of 37%-44%) and the Si bottom subcell. 3 The maximum photogeneration efficiency of InxGa 1−x N bandgaps between 1.6 eV and 1.8 eV would reach up to 23.5% with a bandgap of 1.6 eV and a thickness of 1 μm. Reducing the thickness will reduce the efficiency to 6% at a bandgap of 1.8 eV and a thickness of 100 nm.
The photogeneration efficiency, considering a perfect antireflective coating without any reflection loss at the InxGa 1−x N/ electrolyte interface and no electrolyte absorption, is depicted in Fig. S6c. The electrolyte absorption is negligible for our 100 μm water layer. However, the absence of the air-electrolyte and electrolyte-InxGa 1−x N interfaces omits the overall reflectance losses in the electrolyte layer of 12%. The maximum efficiency, therefore, increases to 31% (∼3% more than the efficiency including the reflection losses) and a current density of 18.7 mA cm −2 . In addition, in that case, two maximum efficiency regions can be observed, a first region with low bandgaps between 1.2 eV and 1.4 eV and low film thicknesses of 300 nm-550 nm, and a second region with a bandgap of 1.5 eV and film thicknesses between 700 nm and 900 nm.
If the water or electrolyte layer absorption was more significant (i.e., the layer was thicker), the peak efficiency would be further reduced. Additionally, since the water absorption is more significant for larger wavelengths, the bottom cell (the Si cell) would be more significantly affected by the water absorption and the location of the second photogeneration peak (limited by the Si cell, see Fig. 4) would be shifted, requiring a thinner InxGa 1−x N layer and a smaller In content (or larger bandgap).
The generation rates of the ultrathin InxGa 1−x N film on the Si film showed clear interference between forward-and backwardpropagating waves (Fig. S7) but no resonant light trapping (Fig. S8). Indeed, resonant light trapping occurs for the ultrathin film deposited on a highly reflective material, such as gold or silver. 11,12 The real part of the refractive index of Si is n = 3.55-5.59 for visible light and does not yet lead to resonant light trapping (i.e., the reflectivity is still relatively small), in contrast to gold with n = 0.04-0.09 or silver with n = 0.1-1.67 (both leading to significantly higher reflectivity).
The obtained photogeneration rates, together with bandgap information, can already be used in detailed balance limits and electrochemical load models 3,41 to give initial indications if the possible open circuit potential will suffice (i.e., is at least few 100 mV larger than the equilibrium potential of the reaction). B. Performance characteristics of Inx Ga 1−x N photoelectrodes

I-V curves of InxGa 1−x N photoanodes
The I-V curves of InxGa 1−x N photoanodes with chopped light in 1M H 2 SO 4 and 1M Na 2 SO 4 are depicted in Fig. 5. The fact that the photocurrent density at 1.23 V RHE 1M H 2 SO 4 decreases with an increase in the indium content is in conflict with the generated charge carriers, the density of which increases with an increase in the indium content [since the bandgap decreases, Fig. 5(a)], indicating that light absorption is not the dominant mechanism to limit efficiency. The photocurrent density at 1.23 V RHE in 1M Na 2 SO 4 [ Fig. 5(b)] does not decrease with the indium content but neither does it follow a continuous increase. Indeed, the photocurrent density increases from x = 9.5% to 16.5%, then decreases at x = 23.5% and then increases again, reaching the maximum photocurrent density of 1.9 mA cm −2 at x = 41.4%. The photocurrent dropped from 1.1 mA cm −2 to 0.6 mA cm −2 (at 1.23 V RHE ) with the In content increasing from 16.5% to 23.5%, while the doping concentration decreased from 1.4 × 10 22 cm −3 to 9.8 × 10 20 cm −3 ( Fig. 6 and Table II). These results were unexpected, as a lower doping concentration (or defect density) usually leads to a reduction of the bulk 42 and surface 43 recombinations, as well as a higher minority charge mobility 44 and a larger SCR width, all factors that should positively affect the photocurrent 45 (see Fig. S12).
The decrease of the photocurrent in 1M H 2 SO 4 with an increase in the indium content and the irregular variation of the photocurrent depending on the indium content in 1M Na 2 SO 4 appeared to be correlated with the surface roughness (expressed in root mean square, rms) of the photoelectrode. The rms of the surface roughness of our InxGa 1−x N films increased continuously with the indium content (expected for x = 41.4%, where it stagnated), as depicted in Table I. Thus, we hypothesize that the photocurrent decrease in 1M H 2 SO 4 electrolyte is caused by dominant surface recombination resulting from the increased surface roughness (see Fig. S12). The situation is different when using a hole scavenger, since the surface recombination is decreased by the enhanced charge transfer kinetics (thus, the surface roughness' effect on surface recombination is reduced). Nevertheless, a weak correlation between the surface roughness and the photocurrent at 1.23 V RHE in the case of operation with a hole scavenger is still observed: rms at x = 23.5% is 7.2 nm and depicts stronger surface features (see the supplementary material, Fig. S3) than at x = 16.5% with an rms of 5.9 nm, accompanied by a photocurrent drop when increasing x from 16.5% to 23.5% [ Fig. 5(b)]; when the In content is increased from 33.3% to 41.4%, the rms is slightly reduced (from 12.2 nm to 11.1 nm), while the photocurrent is doubled [ Fig. 5(b)]. More detailed investigations supporting our hypothesis are presented in the next sections based on the doping concentration, the flatband potential, the diffusion length, the diffusion optical number, and the ratio of currents.

FIG. 6.
Flatband potential (left y-axis) and doping concentration (right y-axis) of InxGa 1−x N water-splitting photoelectrodes determined by the Mott-Schottky analysis (Fig. S10 in the supplementary material). The equivalent circuit for the electrochemical impedance spectrum fit is also indicated.

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scitation.org/journal/apm The rather strong (and nearly linear) increase in the photocurrent with an increase in the potential (potentials beyond 1.23 V vs RHE) seems to indicate that-at least for the samples that have low surface recombination (see the ratio of the current in Sec. III B 3 and the IPCE simulations in Fig. S11)-the bulk recombination dominates their performance. The reasons why for such cases the current keeps increasing is not fully resolved.
Note that we utilize 1.23 V vs RHE for the comparison and later for the IPCE measurements as it allows for large enough photocurrents that the measurements are more accurate while avoiding the range in which the dark current exponentially increases.

Flatband and doping concentration of InxGa 1−x N photoanodes
The flatband potential and the doping concentration for varying indium contents are depicted in Fig. 6. More details on the Mott-Schottky analysis can be found in the supplementary material (Sec. S4). The largest flatband potential is −0.33 V RHE and is obtained at x = 9.5%, while the smallest flatband potential is −0.06 V RHE for x = 33.3%. The doping concentrations (or defect density) are very high for all our samples and In contents and range between 8.1 × 10 20 cm −3 (x = 33.3%) and 1.9 × 10 22 cm −3 (x = 9.5%). The high doping concentration (or defect density) of InxGa 1−x N is probably the main cause for the low efficiency of these photoelectrodes. As previously mentioned, a high doping concentration causes a reduction in the minority charge mobility 44 and an increase in the bulk 42 and surface recombinations. 43 Moreover, the SCR width being below 1 nm at a doping concentration ≥8.1 × 10 20 cm −3 , the charge carrier separation is driven by diffusion and not drift, which further reduces the performance of the photoelectrodes. Thus, we estimate that lowering the doping concentration (or defect density) could be a fast approach for increasing the efficiency of InxGa 1−x N photoelectrodes. Lowering the doping concentration could also prolong the charge carriers' lifetime, as observed for GaN. 46 3. Diffusion length, diffusion optical number, ratio of currents, and nanostructuring opportunity factor Table II (the layer's thickness is given in Table I) summarizes the different material parameters required to determine the diffusion length, the diffusion optical number, the ratio of currents, and the nanostructuring opportunity factor [Sec. II B, Eqs. (8)- (11)]. 10 The calculated performance parameters of InxGa 1−x N photoelectrodes are given in Table III. The highest diffusion length, 262 nm, is obtained for x = 23.5%, then for x = 33.3% with a diffusion length of 158 nm, 32 nm for x = 9.5%, 16 nm for x = 41.4%, and 11 nm for x = 16.5%. The diffusion length correlates with the doping concentration (or defect density), as expected. For x = 23.5% and x = 33.3%, the diffusion lengths are above 100 nm, while the doping concentrations are below 10 21 cm −3 , whereas for x = 9.5%, 16.5%, and 41.4% the diffusion lengths are below 100 nm and the doping concentrations are above 10 21 cm −3 . Thus, the higher the doping concentration, the lower the diffusion length.
The diffusion optical number and the ratio of transfer to total currents are key parameters to understand the performance of InxGa 1−x N photoelectrodes. The diffusion optical number is maximized at x = 23.5%, in accordance with the diffusion length (Table III) and the lower doping concentration (Fig. 6). However, the flatband potential is smaller (−0.1 V RHE for x = 23.5% compared to −0.25 V RHE for x = 16.5%), which reduces the space charge region potential, and thus, increases surface recombination. Moreover, the surface roughness is also increased (rms = 5.9 nm for x = 23.5% to rms = 7.2 nm for x = 23.5%), which can further increase the surface recombination. The combined negative effects of the lower flatband potential and the higher surface roughness are reflected in the ratio of current that is significantly decreased from R S,V IPCE = 0.96 at x = 16.5% to R S,V IPCE = 0.07 at x = 23.5%. In other words, the surface recombination loss is significantly increased from x = 16.5% to x = 23.5% (see Fig. S11). The experimental and numerically predicted IPCE spectra of all InxGa 1−x N photoelectrodes are featured in Fig. S11, together with the numerically predicted internal, reflection, and surface recombination losses.
The increase in the surface recombination loss appears to be uncompensated by the increase in the diffusion optical number. Thus, the overall performance is reduced when the In content is increased from x = 16.5% to x = 23.5%, as observed by the photocurrent drop depicted in Fig. 5(b), even if the photogeneration current and the diffusion length are increased. It appears that the hole scavenger (Na 2 SO 4 ) is unable to fully suppress the surface recombination. In our opinion, the combined effect of the surface roughness and high doping concentration (or defect density) gives rise to fast surface recombination kinetics that competes with the hole scavenger oxidation reaction kinetics (see Fig. S12).
Finally, the nanostructuring opportunity factors, fnano, of InxGa 1−x N photoelectrodes were predicted, indicating that only InxGa 1−x N photoelectrodes at x = 23.5% and x = 33.3% show some performance improvement potential through nanostructuring. As fnano remains only slightly lower than 2 (1.6 for x = 33.3% and 1.9 for x = 23.5%), performance improvements are expected to be more successful through reducing the doping concentration (or defect density) rather than working on nanostructuring.

IV. CONCLUSION
Tunable bandgap photoelectrode materials can be a route to highly efficient and inexpensive PEC water-splitting devices. We calculated a theoretical maximum efficiency of 27% for InxGa 1−x N/Si tandem photoelectrodes for an indium content between 50% and 60% (bandgap between 1.2 eV and 1.4 eV) and a film thickness between 280 nm and 560 nm. A maximum efficiency of 31% was obtained when neglecting surface reflection losses (i.e., by neglecting the electrolyte and incorporating an anti-reflecting coating). Despite the high theoretical efficiency of InxGa 1−x N/Si tandem photoelectrodes, our fabricated InxGa 1−x N photoelectrodes (with an indium content varying from 9.5% to 41.4%) led to a maximum photocurrent density of only 1.9 mA cm −2 at 1.23 V RHE for x = 41.4% and when using a hole scavenger (1M Na 2 SO 4 ) electrolyte. This photocurrent density represents only 8% of the maximum theoretical photocurrent density of 23.8 mA cm −2 for a semiconductor with a bandgap of 1.7 eV (corresponding to InxGa 1−x N with x = 41.4%). We observed that the surface roughness, the flatband potential, and the doping concentration played key roles in the performance of InxGa 1−x N photoanodes. As the samples were not intentionally doped, the doping concentration represents more precisely an impurity and defect density. The diffusion length and the electron/hole generation are higher for x = 23.5% (bandgap of 2.3 eV and diffusion length of 262 nm), but the photocurrent density in 1M Na 2 SO 4 was smaller for x = 23.5% than that for x = 16.5% (bandgap of 2.6 eV and diffusion length of 11 nm). Indeed, the higher surface roughness and the lower flatband potential for x = 23.5% are causing a higher surface recombination loss and an overall lower photocurrent compared to the x = 16.5% case. Aside of the InxGa 1−x N photoelectrodes with x = 23.5% and x = 33.3% that showed not only higher surface recombination loss but also higher diffusion length (thanks to their relative lower doping concentration or defect density), the performance of InxGa 1−x N photoelectrodes in 1M Na 2 SO 4 was mainly driven by the bandgap, i.e., the lower the bandgap, the more light is absorbed and converted to electron/hole pairs, leading to the higher performance. In 1M H 2 SO 4 , the performance of InxGa 1−x N photoelectrodes is limited by the surface recombination that is extremely high and hinders comparison between the photoelectrodes of different indium contents. Our Mott-Schottky analysis showed that the doping concentration or defect density for InxGa 1−x N photoelectrodes was extremely high, i.e., a doping concentration higher than 8.1 × 10 20 cm −3 for all indium contents (Table II) were measured. Thus, we mostly attribute the low performance of InxGa 1−x N photoanodes to their high doping concentration (or defect density).
We estimated that nanostructuring could improve the performance of these photoanodes, especially InxGa 1−x N photoanodes with x = 23.5% and x = 33.3% (Table III). However, efforts on reducing the doping and defect concentration should be prioritized to increase the efficiency of InxGa 1−x N photoanodes. Indeed, the doping concentration of 4.1 × 10 21 cm −3 for x = 41.4% leads to a diffusion length of only 16 nm and a diffusion optical number of 0.074. A reduction in doping concentration and defect density might be achieved by optimizing the molecular beam epitaxy growth of InxGa 1−x N layers, or by using an alternative synthesis method. Generally, we believe that research progress in the III-V photovoltaics community in recent years might be translated also to such photoelectrodes, especially with respect to reduction of surface recombination and broadly the electrode-electrolyte interface engineering.
We showed that the combined usage of an analytical IPCE model 10 and common experimental methods, such as SEM images, AFM surface topography, experimental IPCE, and the Mott-Schottky analysis, appears to be a powerful way to investigate the ARTICLE scitation.org/journal/apm performance of InxGa 1−x N photoanodes and is general so that it can be applied to any emerging photoelectrode materials.

SUPPLEMENTARY MATERIAL
See the supplementary material for details on the sample growth and In-content quantification, SEM and surface roughness measurements, photocorrosion measurements, photogeneration calculations, the Mott-Schottky analysis, IPCE measurements and calculations, and an overview plot of In-content on performance and correlations between relevant parameters.