Calculation of Energy Band Diagram of a Photoelectrochemical Water Splitting Cell

A physical model is presented for a semiconductor electrode of a photoelectrochemical (PEC) cell, accounting for the potential drop in the Helmholtz layer. Hence both band edge pinning and unpinning are naturally included in our description. The model is based on the continuity equations for charge carriers and direct charge transfer from the energy bands to the electrolyte. A quantitative calculation of the position of the energy bands and the variation of the quasi-Fermi levels in the semiconductor with respect to the water reduction and oxidation potentials is presented. Calculated current-voltage curves are compared with established analytical models and measurement. Our model calculations are suitable to enhance understanding and improve properties of semiconductors for photoelectrochemical water splitting.

Introduction Research on hydrogen production by photoelectrochemical (PEC) cells is propelled by the worldwide quest for capturing, storing and using solar energy instead of decreasing fossil energy reserves. Hydrogen is widely considered as a key solar fuel of the future. 1 Hydrogen is also part of power to gas conversion systems developed to resolve intermittency in the wind and solar energy production. 2 Although a PEC/photovoltaic cell with 12.4% efficiency was demonstrated with GaInP 2 /GaAs, 3 decreasing its cost and increasing its lifetime are still under way. An alternative approach often pursued is to use abundant and cheap metal oxides as a viable class of semiconductor materials for PEC electrodes. [4][5][6] However, their recombination losses, charge carrier conduction and water oxidation properties need to be understood and optimized both by measurement and numerical simulation. 7 Several approaches for a mathematical analysis of semiconductor electrodes can be found in the literature, including analytical 8,9 and numerical models 10,11 of PEC cells. An extensive numerical study of PEC behavior of Si and GaP nanowires was recently conducted with commercial software. 12 Since surface states play a major role for many semiconductors, corresponding models were also developed to analyze their effect on electrochemical measurements. [13][14][15] On the PEC system level, models of the coupled charge and species conservation, fluid flow and electrochemical reactions were recently developed. 16,17 The latter studies revealed how PEC systems should be designed with minimal resistive losses and low crossover of hydrogen and oxygen by use of a non-permeable separator.
Almost every publication on PEC cells features a schematic energy band diagram of a PEC cell, mostly sketched by hand from basic physical understanding described in textbooks on electrochemistry. 7,18,19 Although such sketches might be qualitatively correct, numerical calculations of the charge carrier transport might reveal additional features not captured by the sketches. We are aware that the development of numerical calculations is frequently hindered by the complicated physical processes in the actual materials and lack of measurements of parameter values for these processes. 20 In spite of these obstacles, we think that the recent advent of user-friendly numerical software and advanced measurement techniques could fill the gap between experimental and numerical approaches if experimentally validated models are developed.
Model In this work, we present calculation of an energy band diagram of a PEC electrode from a physical model with clearly formulated assumptions. 21 The model is based on charge carrier continuity equations with direct charge transfer from the valence or conduction band to the electrolyte.
We consider a PEC cell consisting of an n-type semiconductor with bandgap energy E g , and an electrolyte which can easily accept a single electron or hole (such as H 2 We reserve subscript 0 for equilibrium values in the dark in the following. To derive our model, we use and repeat some of the general definitions introduced in our previous work 24 shown in 1. Note that we use notation of subscript sc for semiconductor, s for surface quantity, b for a bulk semiconductor quantity (where electrons and hole remain at equilibrium in the dark).
Bulk equilibrium properties of the isolated semiconductor are denoted with a subscript 0i. The bulk of the semiconductor is electrically neutral, hence the concentration of electrons in the bulk Figure 1: Scheme of a n-type semiconductor electrode, with electron energy indicated in the absolute energy scale (with respect to vacuum level), and potentials in the electrochemical scale, with respect to SHE. Reprinted with permission of ACS.
n 0i must be equal to the number of fully ionized donors N D , n 0i = N D . Thus, the concentration of holes is p 0i = n 2 i /n 0i , where n i denotes intrinsic carrier concentration. An isolated unbiased semiconductor before contact to an electrolyte has a conduction band edge E c,0i and a Fermi level E F,0i related to the vacuum level E vac and electron affinity χ by where k B is the Boltzmann constant, T is the temperature, q is the elementary charge and N C the effective density of states in the conduction band, and ζ nb is the distance of the conduction band edge to Fermi level. In the following, we use E vac =0 eV as usual. The potential drop in the Helmholtz layer in the dark V H is calculated from the local vacuum level (LVL) at the surface of the semiconductor (−qφ s ) and LVL of the electrolyte (−qφ el ), 1, Note that the potential drop in the Helmholtz layer can be a different value at flatband situation H ) than at other measurable voltage (denoted V H ). We measure the voltage V r of the semiconductor electrode with respect to a reference electrode, which means the difference of the Fermi level of electrons in the semiconductor back contact E Fn,b and Fermi level of the reference In this article, we use both the Standard Hydrogen Electrode (SHE) energy and Reversible Hydrogen Electrode (RHE) as reference electrodes and scale of energy. Measured voltage with respect to the SHE is denoted V r (without subscript SHE) and measurable voltage with respect to the RHE where V th = k B T q is thermal voltage and pH denotes pH value of the solution. We draw attention to the fact that negative bias versus RHE brings the energy closer to the vacuum level E vac . The position of the electron Fermi level at the semiconductor back contact is calculated as (see 1) where V sc denotes the potential drop in the semiconductor. What is usually reported in the literature is the value of flatband potential, which is the measurable voltage when the bands are flat (V sc = 0) The value of V Then from eqs. ??, ??, ?? follows The second option is to refer the voltage to the equilibrium of semiconductor-electrolyte interface (SEI) and this value is denoted V app where built-in voltage is denoted V bi and potential drop across the Helmholtz layer in dark equilibrium V H0 . Equilibrium of SEI means V app = 0 V.
On the semiconductor side of the junction, the electrostatic potential φ is obtained by solving where ε 0 is the permittivity of vacuum, ε r is the relative permittivity of the semiconductor, N D is the concentration of fully ionized donors, n(x) is the concentration of free electrons and p(x) is the concentration of free holes (p(x) n(x) for n-type semiconductor in the dark). We can write for the conduction and the valence band edge energies E c and E v in the electrostatic potential φ (x) H for any measurable voltage (assumed in the following), otherwise the band edges become unpinned.
A simple approximation to solve Poisson's equation, Eq. ??, is to assume that the total space charge is uniformly distributed inside the space charge region (SCR) of width w (also called deple-tion region approximation) The boundary conditions for the electrostatic potential φ follow directly from the definitions on 1 The concentration of free electrons and holes in the dark n dark (x) and p dark (x) can be written as The value of electrostatic potential in the semiconductor bulk φ b appears in the above expressions because we have made general definition of electrostatic potential including the potential drop in the Helmholtz layer. Therefore, φ b is not zero unlike recent textbook definition. 7 The approximate solution of Poisson's eq. φ a is then When the measurable voltage V r is positive of the flatband potential V f b , the n-type semiconductor is in the depletion regime. When the measurable voltage is negative of the flatband potential, the semiconductor is in the accumulation regime (due to the sign of V sc ).
Upon illumination, we assume low-injection conditions with the number of photogenerated electrons lower than the donor concentration. Thus electron concentration is roughly equal to the dark electron concentration n(x) = n dark (x). The hole continuity equation is solved to obtain free hole concentration p inside of the semiconductor of thickness d We consider the generation rate of charge carriers from the simple Lambert-Beer law G h (x) = αP e −αx , where P = λ g λ min Φ(λ )dλ is number of photons with energy above E g = hc λ g which are absorbed in the semiconductor, Φ(λ ) is the spectral photon flux of standard AM15G spectrum with intensity 100 mW/cm 2 , 25 α is the absorption coefficient of the semiconductor. The hole current density j h is expressed using the analytical solution of Poisson's equation where µ h = qD h k B T is the hole mobility, and D h is the hole diffusion constant. A direct band-to-band nonlinear recombination is assumed We assume that charge transfer under illumination occurs exclusively from the valence band to the electrolyte. We do not include charge transfer from surface states in the current analysis. The transfer current density of valence band holes at the SEI is described by a first-order approxima- where k trh is the rate constant for hole transfer, and a linear dependence on the deviation of the interfacial hole concentration p(0) from its dark value p dark (0) at the interface is assumed. Since the thickness of the semiconductor is in the order of the penetration length of light α −1 for the hematite parameters listed in 1, we consider the hole current at the back contact of the semicon-ductor to depend on a surface recombination velocity r s We use r s = 10 5 m/s for numerical calculations throughout this article. 12 In order to obtain convergence of the numerical solution procedure, the continuity equation was solved in a non-dimensional form after applying the usual normalization of the variables of the drift-diffusion equations. 27 The quasi-Fermi energies E Fn , E F p under the influence of an electrostatic potential φ (x) are calculated by the Boltzmann distribution Results We numerically solved the hole (electron) continuity equation Eq. 22 for a n-type (ptype) semiconductor by using the depletion region approximation of the electrostatic potential   The energy band diagram is shown for a three-electrode setup in 3. A measurable voltage of V r,RHE = 1.23 V is assumed, which is the standard voltage used for comparison of different PEC electrodes. 28,29 The measurable voltage V r,RHE is indicated in 3a) with an arrow on the energy scale, −qV r,RHE . This is also explained in our previous work. 24 The band edges of the semicon- polarization in the following. 33 In the electrolyte, we plot two reference electrode energies E SHE 0 and E RHE 0 , standard water reduction and oxidation energy E red (0 eV vs RHE) and E ox (1.23 eV vs RHE). Note that the relation of E red and E ox to E redox depends on the concentrations (activities) of oxidizing and reducing species in the solution. 34 The energy band diagram in the semiconductor for different values of the measurable voltage V r,RHE is plotted in 3b). For increasing V r,RHE the band bending increases and the electron quasi- Fermi level E Fn shifts down on the RHE scale. Interestingly, the hole quasi-Fermi level E F p (0) at the SEI remains nearly constant for increasing V r,RHE (see Figure S1 in Supporting Information) and thus the splitting of the quasi-Fermi levels (photovoltage) approaches zero. In the neutral region w < x < d, the hole quasi-Fermi level E F p (x) is more negative for increasing V r,RHE and the photovoltage is nearly constant. When the hole diffusion length L h = √ D h τ h is increased, the flat region of the hole quasi-Fermi level E F p near the SEI is enlarged, 3c), and the hole concentration in the neutral region decreases (see Figure S2 in Supporting Information).
We simulated current-voltage curves with our numerical model j h (0) (eq. 25) and compared the results with published models of Gartner 8 and Reichmann, 10 4. According to Gartner, the minority charge carrier concentration is calculated from the diffusion equation, neglecting recombination in the SCR and assuming that every hole in SCR contributes to the photocurrent. Photocurrent density of Gartner is Therefore, j G overestimates the minority carrier photocurrent in comparison to our numerical model j h (0). Recombination in the SCR by Sah-Noyce-Shockley formalism was incorporated into the model by Reichmann 10 with resulting photocurrent j R (the detailed expression is given in the Supporting Information). For small V r,RHE , j R is much smaller than j G because the SCR recombination loss is included in j R . The onset of the photocurrent calculated by Reichmann j R starts when γ = j s e − V app V th qk trh p dark (0) ≈ 1 ( j s is the saturation current density as defined in the SI). Therefore, if we consider faster charge transfer kinetics (larger k trh ), we need a smaller value of the onset potential V r,RHE (and thus V app ) to obtain a similar value γ ≈ 1. For increasing V r,RHE , j R approaches j G because the SCR recombination becomes negligible in j R , 10 but the numerical photocurrent j h (0) is smaller than j G since SCR recombination is included in j h (0). The numerical photocurrent j h (0) onsets when V r,RHE is more positive than V f b,RHE and it is larger than j R for small V r,RHE .
Increasing the rate constant k trh represents a faster exchange rate of holes with the solution. This also shifts the numerical j-V curve to the left as predicted by the Reichmann model, decreasing the onset potential of the photocurrent. We checked that the maximum photocurrent obtainable from the hematite electrode based purely on the number of absorbed photons is qP = 12.5 mA/cm 2 for AM15G illumination. This value is also obtained for the Gartner photocurrent eq. ?? when the bracket term is close to one and also for the Reichmann photocurrent (which recovers the Gartner photocurrent in regime of large voltages). The plateau of numerical photocurrent j h (0) cannot be computed here, because our model cannot be used to predict photocurrents at voltages higher than V r,RHE > V inv r,RHE . At such voltages inversion layer is formed as described in the previous text and this would need degenerate statistics to be included in the model. In the case of p-type Cu 2 O, the majority carriers are holes, and thus the counter electrode carries out the oxidation reaction (including the associated overpotential η). Although the electron quasi-Fermi level E Fn is negative with respect to E red making it suitable for hydrogen evolution, 5, corrosion prevents hydrogen evolution in the experiment unless the Cu 2 O is protected by overlayers. 37 So far, our model does not consider corrosion; here we aimed at showing the general energetic configuration of a p-type PEC photoelectrode.

Conclusion
We presented a physical model for minority charge carrier transport in semiconductor PEC electrodes in contact with an electrolyte. Direct charge transfer to the electrolyte from 0 d 325.    The difference between the semiconductor conduction band energy and the electron Fermi level φ V Local electrostatic potential φ a V Approximate solution for local electrostatic potential φ el V Local electrostatic potential of the electrolyte φ s V Local electrostatic potential at SEI φ b V Local electrostatic potential in the semiconductor bulk