Band Diagram Insights into the Kinetic and Thermodynamic Engineering of Tandem Photocatalytic Cells

Abstract

In this work, we theoretically investigate the impact of kinetic and thermodynamic properties on the performance of photocatalytic cells operating in an unassisted tandem configuration, including electron affinity and ionization energies, recombination rates, and reaction rates. To this end, we present general rules and metrics for identifying and isolating the origin of an observed shift in the onset potential at either the photoanode or the photocathode of these devices. The correlation between kinetic and thermodynamic shifts in the onset potential is demonstrated through the use of band diagrams and key comparable features within readily accessible characterization tools: current-voltage plots are taken both under illumination and in the dark and further coupled with Mott-Schottky plots. To illustrate this conceptual framework, a model system comprised of a p-type doped BiVO₄ photocathode and an n-type doped BiVO₄ photoanode is employed. By varying each of the aforementioned kinetic and thermodynamic parameters in isolation, the manner in which these various mechanisms shift the onset potential is demonstrated. This work intends to showcase how kinetic and thermodynamic effects are distinctly manifested in these commonly used characterization tools and further proposes thermodynamic band-edge engineering as a potentially useful and largely unexplored avenue for possibly improving tandem cell performance, in addition to the conventional approach of optimizing kinetics.

Figure 1

(a) Schematic of electron–hole pair generation in a PEC operating in a tandem configuration. Electrons are denoted by solid red circles and holes are denoted as unfilled red circles. The ionization energy is depicted by E_IE, the electron affinity by E_EA, and the band gap by E_G. The space charge region is denoted by SCR. The conduction bands E_C, the valence bands E_V, and the electron and hole quasi-Fermi levels E_Fn and E_Fp are shown for both the photoanode and photocathode, denoted by the PA and PC subscripts. (b) Schematic of typical current (J) versus voltage (V) curves for a photoelectrochemical cell with an n-type photoanode (red) and p-type photocathode (green). The flat band and saturation regions for each photoelectrode are shown as well as the onset potentials, V_PA^onset and V_PC^onset, as indicated by circle markers. The operating current (J_OP) occurs at the intersection of the photoanode and photocathode photocurrent curves. To achieve a higher operating photocurrent, one seeks to move the photoanode curve to the left (cathodic) and the photocathode curve to the right (anodic).

Figure 2

Modified self-consistent semiconductor continuity equations and Poisson’s equation employed by the model to arrive at rapidly converged solutions and band diagram insights. Hole generation and recombination are conserved within the depletion region. (a) SLJCompact model for the photoanode. Beyond the depletion region, the continuity equation is solved for a fixed hole diffusion length L_p. The electron (n) quasi-Fermi level (E_Fn) is held flat outside the SCR. The hole quasi-Fermi level is approximated as flat within the depletion region, though with an offset as determined by the surface hole density p_s. (b) SLJCompact model of the photocathode. Beyond the depletion region, the continuity equation is solved for a fixed electron diffusion length L_n. The hole (p) quasi-Fermi level (E_Fp) is held flat outside the SCR. The electron quasi-Fermi level is approximated as flat within the depletion region, though with an offset as determined by the surface electron density n_s.

Figure 3

(a) Example BiVO₄ photoanode and photocathode light and dark JV curves for tandem PEC cell operation with poor performance. The bias is with respect to the photoanode flat band. The photocurrent is normalized with respect to qΦ, given the incoming light and band gap restrictions. Here, τ = 10^–11 s for both the photoanode and photocathode, k_p0 = k_n0 = 0.1 s^–1 and χ = 0.5. Under illumination, the operating current is depicted as J_OP = 0.04 (normalized to qΦ), and the operating applied bias is depicted as V_OP = 1.05 V. (b) Photoanode and photocathode band diagrams in the light JV curves for the system in (a) at the operating junction bias V_OP.

Figure 4

Effects of carrier lifetime and χ on JV behavior. (a) Photoanode light and dark JV curves for various values of τ at k_p0 = 0.1 s^–1 and χ = 0.5. The photocurrent is normalized with respect to qΦ, given the incoming light and band gap restrictions. (b) Photocathode light and dark JV curves for various values of τ at k_p0 = 0.1 s^–1 and χ = 0.5, analogous to (a). (c) Photoanode onset voltage shifts for various values of τ and χ at k_p0 = 0.1 s^–1, normalized by the onset at τ = 10^–11 for each value of χ. Negative shifts indicate a left (cathodic) shift in the JV curve. (d) Photocathode onset voltage shifts for various values of τ and χ at k_n0 = 0.1 s^–1, normalized by the onset at τ = 10^–11 for each value of χ. Positive shifts indicate a right (anodic) shift of the JV curve. (e) Photoanode Mott–Schottky plot for the entire range of τ from (a) and χ from (c). (f) Photocathode Mott–Schottky plot for the entire range of τ from (b) and χ from (d).

Figure 5

Effect of carrier transfer rates on J–V behavior. (a) Photoanode light and dark J–V curves for various values of k_p0 at τ = 10^–10 s and χ = 0.5. The photocurrent is normalized with respect to qΦ, given the incoming light and band gap restrictions. (b) Photocathode light and dark J–V curves for various values of k_n0 at τ = 10^–10 s and χ = 0.5, analogous to (a). (c) Photoanode onset voltage shifts for various values of k_p0 and χ at τ = 10^–10 s, normalized by the onset at k_p0 = 0.01 s^–1 for each value of χ. Negative shifts indicate a left (cathodic) shift in the J–V curve. (d) Photocathode onset voltage shifts for various values of k_n0 and χ at τ = 10^–10 s, normalized by the onset at k_n0 = 0.01 s^–1 for each value of χ. Positive shifts indicate a right (anodic) shift on the JV curve. (e) Photoanode Mott–Schottky plot for the entire range of k_p from (a) and χ from (c). (f) Photocathode Mott–Schottky plot for the entire range of k_n from (b) and χ from (d).

Figure 6

JV behavior as a function of increasing conduction band minima via decreasing electron affinity (E_EA) at a fixed E_V relative to vacuum. (a) Photoanode light and dark JV curves for various values of E_EA at τ = 10^–11 s, χ = 0.5, and k_p0 = 0.1 s^–1. The photocurrent is normalized with respect to qΦ, given the incoming light and band gap restrictions. (b) Photoanode light and dark JV curves for the same conditions as (a) with bias with respect to the NHE. (c) Illuminated band diagram for data in (b) with E_EA = 5.0 eV at a 1.4 V bias vs NHE. The Fermi level splitting is ΔV_F = 1.66 eV. (d) Dark band diagram for the same electron affinity and photocurrent as (c). (e) Illuminated band diagram for data in (b) with E_EA = 4.6 eV at a 1.4 V bias vs NHE. The quasi-Fermi level splitting is ΔV_F = 2.06 eV. (f) Dark band diagram for the same electron affinity and photocurrent as (d). (g) Photoanode Mott–Schottky plot for decreasing electron affinities.

Figure 7

JV behavior as a function of decreasing valence band maxima via increasing ionization energy (E_IE) at fixed E_C relative to vacuum. (a) Photocathode light and dark JV curves for various values of E_G at τ = 10^–11s, χ = 0.5, and k_n0 = 0.1 s^–1. Bias is with respect to the flat band. The photocurrent is normalized with respect to qΦ, given the incoming light and band gap restrictions. (b) Photocathode light and dark JV curves for the same conditions as (a) with bias with respect to NHE. (c) Illuminated band diagram for data in (b) with E_IE = 7.0 eV at a 1.4 V bias vs NHE. The Fermi level splitting is ΔV_F = 1.66 eV. (d) Dark band diagram for the same ionization energy and photocurrent as (c). (e) Illuminated band diagram for data in (b) with E_IE = 7.4 eV at a 1.4 V bias vs NHE. The quasi-Fermi level splitting is ΔV_F = 2.06 eV. (f) Dark band diagram for the same ionization energy and photocurrent as (d). (g) Photocathode Mott–Schottky plot for increasing ionization energies.

Figure 8

Effect of pH on tandem JV behavior. Biases are with respect to NHE. The photocurrent is normalized with respect to qΦ. (a) Photoanode and photocathode light JV curves under increasing pH conditions. The intersections of the tandem curves for systems at each pH value are marked by black circles. (b) Photoanode and photocathode dark JV curves for increasing pH conditions. (c) Photoanode Mott–Schottky plot for various pH conditions. (d) Photocathode Mott–Schottky plot for various pH conditions. (e) Band diagram for tandem operation for various pH conditions at the operating voltage. The vacuum and NHE potentials shift for both the photoanode and the photocathode. The point of zero charge is assumed to be at pH = 7.

Figure 9

Decision tree outlining the procedure to isolate contributing factors to an observed onset shift for photoanode–photocathode tandem systems. Mott–Schottky and JV plots are typically taken with respect to a reference electrode. Other characterization methods should also be applied to verify such interpretations.⁻ This simplified decision tree is not always sufficient, particularly when each of these kinetic and thermodynamic factors is acting together and must be disentangled.