Theory of Charge Transport in the Illuminated Semiconductor/Liquid Junctions

Abstract

The field of photoelectrochemical (PEC) cells for solar water splitting or CO2 reduction has attracted intense attention of many research groups in last 15 years. Nevertheless, a cost-effective and efficient PEC cell for hydrogen production in the large scale was not yet discovered. The core functionality of the PEC cell is provided by the semiconductor/liquid junction, creating the electrostatic field to separate the photogenerated charges. This work aims to be a starting point for a newcomer in the field providing a compact knowledge about the charge transport and electrochemistry fundamentals in semiconductor/liquid junctions in the steady state. We describe charge transport within the semiconductor and electron transfer between the semiconductor and electrolyte, followed by the effect of illumination and charge recombination on charge transport. Finally, we discuss the effects due to surface trap states and the relation of the theoretical expressions and experimental results.

1. Introduction

The transition to renewable energy sources is recognized as one of the greatest societal challenges of the twenty-first century, and it will have a major impact on climate, environment, and economy [1]. The major bottleneck for broader utilization of renewable energy is missing large-scale and long-term energy storage technologies [2, 3]. In this respect, fossil fuels make up 85% of the worldwide energy consumption, and they have an order of magnitude higher energy density than lithium-ion batteries [4]. Hence, to replace fossil fuels and also enable a low-carbon economy in the long term, an alternative route to produce fuels exclusively from the renewable resources is sought for.

In the last two decades, a lot of research effort was given to produce renewable fuels from solar energy and/or CO₂ [5, 6, 7]. Hydrogen is one of the leading renewable fuels with important impact on a variety of industrial processes [8, 9], and European Union recognized its importance in 2020 and put an ambitious “Hydrogen Strategy for Climate-Neutral Europe” (https://ec.europa.eu/energy/sites/ener/files/hydrogen_strategy.pdf). From the variety of approaches for the renewable hydrogen production [6], great research activity is given to the photoelectrochemical (PEC) hydrogen production. For overview of the PEC water splitting principles, material requirements, characterization methods, and device architectures, we suggest the recent handbooks [10, 11, 12].

This chapter discusses theory related to the core functionality of the PEC hydrogen production or PEC CO₂ reduction, which lies in the separation of the photogenerated charges by the electric field created by the semiconductor/electrolyte interface. The physical processes at the typical n-type photoanode/electrolyte junction in the alkaline electrolyte are shown in Figure 1a. When a semiconductor is immersed in a solution, a charge transfer occurs at the interface because of the difference in the electrochemical potential of the two phases, creating a built-in electrostatic field, which separates electronic charges. When the light is absorbed in the semiconductor, it excites electrons and holes to their respective bands, and these are separated by the built-in electrostatic field. Holes move to the electrolyte, consuming OH⁻, and oxidize water to oxygen. Electrons move away from the electrolyte and are sent through the external wire to the metal counterelectrode with an additional bias voltage, where they reduce water to hydrogen and OH⁻. The exchange of OH⁻ between the photoanode and the counterelectrode is completed by ion migration in the electrolyte. Although with intensive research efforts, a single semiconductor material being able to do overall PEC water splitting (two half-reactions) has not yet been identified.

Figure 1.

(a) Schematic diagram of processes in the photoanode during three-electrode investigation for water splitting in alkaline electrolyte. (b) Sketch of the tandem PEC cell with large-bandgap photoanode and small-bandgap photocathode connected via an ohmic contact.

Therefore, a natural strategy is to distribute the two half-reactions to two semiconductor photoelectrodes (similar to Z-scheme in natural photosynthesis) [13], where wider-bandgap n-type semiconductor photoanode (1.8–2.4 eV) drives oxygen evolution reaction, and a smaller-bandgap semiconductor photocathode (1.0–1.5 eV) placed behind it drives hydrogen evolution reaction, Figure 1b. Such tandem configuration makes optimum usage of the full solar spectrum, leading to higher efficiencies and a better ability to cope with the fluctuating illumination conditions.

The PEC hydrogen production is currently at the lab scale and has the potential to compete if solar-to-hydrogen (STH) efficiencies over 10–15% and stability for over 10 years are considered [14, 15]. The highest reported STH efficiencies for PEC water splitting, Figure 2, are 14–19% for III–V tandems [16, 17], 10–14% for silicon-based multijunctions [18], and 1–3% for oxide-based devices [13].

Figure 2.

Reported STH efficiencies as a function of year and sorted by the number of tandem photovoltaic junctions used (2 or 3). The fill color represents the semiconductor materials used in the photovoltaic portion of the device. All STH conversion efficiencies are as reported in the original publications. Reprinted with the permission from the Royal Society of Chemistry [13].

2. Semiconductor/liquid junction

When semiconductor is immersed in liquid electrolyte containing a redox couple, electron transfer between occupied (empty) state in the semiconductor and empty (occupied) state of the ion species in the solution is possible, respectively. This charge transfer will continue until an equilibrium is reached when Fermi level of the semiconductor $E_F$ equals to the redox Fermi level in the electrolyte $E_{\mathit{\text{redox}}}$. Band edge positions of some semiconductors with respect to the redox potentials for water splitting are shown in Figure 3. Under equilibrium conditions, semiconductor region close to semiconductor/electrolyte interface (SEI) is depleted of the majority carriers, which is counterbalanced by the ions adsorbed at the semiconductor from the electrolyte side.

Figure 3.

Band edge positions of various semiconductors at pH = 0 (conduction band is shown in red, valence band in black). Voltage is reported with respect RHE and redox potentials of water splitting are shown as dashed lines. Bandgap energy is shown below the valence band position.

We start by describing the charge transfer in the semiconductor in the semiclassical approach. The resulting electrostatic potential $\varphi \left(x\right)$, electron and hole concentrations $n\left(x\right),p\left(x\right)$ in the semiconductor can be calculated by solving the Poisson equation together with the carrier continuity equations

$$\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}=-\frac{\rho \left(x\right)}{{\epsilon }_0{\epsilon }_r},$$E1

$$-\frac{1}{q}\frac{\mathrm{d}{j}_e}{\mathrm{d}x}=G\left(x\right)-R\left(x\right),$$E2

$$\frac{1}{q}\frac{\mathrm{d}{j}_h}{\mathrm{d}x}=G\left(x\right)-R\left(x\right),$$E3

where $\rho \left(x\right)=q\left(N_D-n\left(x\right)+p\left(x\right)\right)$ is the charge density, $N_D$ is the concentration of the ionized donors for n-type semiconductor (full ionization is assumed), and ${\epsilon }_0,{\epsilon }_r$ denote permittivity of vacuum and relative permittivity of the semiconductor. The generation rate of the carriers is labeled $G\left(x\right)$ and the recombination rate $R\left(x\right)$. The electron and hole currents are given by the drift and diffusion terms

$$j_e={\mathit{qD}}_e\frac{\mathrm{d}n\left(x\right)}{\mathrm{d}x}-q{\mu }_{e}n\left(x\right)\frac{\mathrm{d}\varphi \left(x\right)}{\mathrm{d}x}$$E4

and

$$j_h=-{\mathit{qD}}_h\frac{\mathrm{d}p\left(x\right)}{\mathrm{d}x}-q{\mu }_{h}p\left(x\right)\frac{\mathrm{d}\varphi \left(x\right)}{\mathrm{d}x},$$E5

where electron and hole mobility are denoted ${\mu }_e,{\mu }_h$.

Usually, a space charge region (SCR) approximation is used to solve the stand-alone Poisson equation (without continuity equations), Figure 4, by assuming that a constant charge density exists in the SCR of width $w$

Figure 4.

(a) Energy diagram of stand-alone semiconductor and electrolyte, before contacting them. (b) Energy diagram of semiconductor and electrolyte after contact and application of voltage $V_a$ with respect to reversible hydrogen electrode $E_{0,\mathit{RHE}}$.

$$\rho \left(x\right)={\mathit{qN}}_D,0<x<w$$E6

and zero charge density exists outside of the SCR. The boundary condition of the vanishing electrostatic field at the edge of SCR gives $\frac{\mathrm{d}\varphi }{\mathrm{d}x}\left(w\right)=0$, and furthermore, we choose arbitrary value $\varphi \left(w\right)=0$. In this way, the Poisson equation is integrated to give $\varphi \left(x\right)$ inside SCR

$$\varphi \left(x\right)=-\frac{{\mathit{qN}}_D}{2{\epsilon }_0{\epsilon }_r}{\left(w-x\right)}^2.$$E7

The width of the SCR is given by

$$w=\sqrt{\frac{2{\epsilon }_0{\epsilon }_r{V}_{\mathit{sc}}}{{\mathit{qN}}_D}}.$$E8

The potential drop in the SCR region can be altered by applying the voltage $V_a$ to the semiconductor back contact

$$V_{\mathit{sc}}=V_a-V_{\mathit{fb}},$$E9

where for the flatband voltage $V_a=V_{\mathit{fb}}$, the bands of the semiconductor become flat and $V_{\mathit{sc}}=0$. The semiconductor energy bands remain usually pinned at the interface to the electrolyte (with or without redox couple) as measured by the impedance spectroscopy [19], and this is also valid for $\varphi \left(x\right)$, Eq. (7). Therefore, position of the semiconductor band edges can be given on the traditional electrochemical energy scale, for example, relative to the reversible hydrogen electrode (RHE) [20]. Due to strong interaction of semiconductors with water in aqueous elecrolytes, applying the voltage bias to the semiconductor leads only to the change of the electrostatic potential across the space charge layer $V_{\mathit{sc}}$ (the potential across the Helmholtz double layer remains constant). The energies of the conduction band and valence band $E_{\mathit{cb}},E_{\mathit{vb}}$ are given by

$$E_{\mathit{cb}}\left(x\right)=-\chi -\mathit{q\varphi }\left(x\right),E_{\mathit{vb}}\left(x\right)=E_{\mathit{cb}}\left(x\right)-E_g,$$E10

and the quasi-Fermi level of electrons and holes $E_{\mathit{Fn}},E_{\mathit{Fp}}$

$$E_{\mathit{Fn}}=E_{\mathit{cb}}+\mathit{kT}ln\left(\frac{n}{N_c}\right),E_{\mathit{Fp}}=E_{\mathit{vb}}-\mathit{kT}ln\left(\frac{p}{N_v}\right),$$E11

where the electron affinity is labeled $\chi$, symbol $k$ denotes the Boltzmann constant, $T$ is the absolute temperature. We remark that in our discussion, no electronic states at the surface of the semiconductor are assumed for simplicity. The electron and hole concentrations at the SEI, $n_s$ and $p_s$, are given by the Boltzmann distribution

$$n_s=n_{0}exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)$$E12

and

$$p_s=p_{0}exp\left(+\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right).$$E13

The equilibrium electron (hole) concentration is denoted as $n_0$ ($p_0$). The equilibrium hole concentration $p_0$ is given by the expression

$$p_0=\frac{N_c{N}_{v}exp\left(-\frac{E_g}{\mathit{kT}}\right)}{N_D},$$E14

where the effective densities of states in the CB and VB are denoted $N_c$ and $N_v$ and the bandgap energy is $E_g$.

2.1 In the dark

Turning now our attention to the redox couple, the energy values of $\mathit{Red}$ ($\mathit{Ox}$) species in the solution $E_{\mathit{red}},E_{\mathit{ox}}$ can be described with the solvation state model introduced by Gerischer [21]. Redox couple is a pair of two ions ($\mathit{Red}$ is the reduced species, $\mathit{Ox}$ is the oxidized species), which can interchange electrons

$$\mathit{Red}-e^{-}->\mathit{Ox}$$E15

The Nernst equation describes the electrochemical potential of electrons in the redox couple, which is equivalent to the Fermi level of the redox couple $E_{\mathit{\text{redox}}}$ when the same reference is used for the semiconductor and redox system [22].

$$E_{\mathit{\text{redox}}}=E_{\mathit{\text{redox}}}^0+\mathit{kT}ln\left(\frac{c_{\mathit{ox}}}{c_{\mathit{red}}}\right),$$E16

where the reference redox level is denoted $E_{\mathit{\text{redox}}}^0$, and $c_{\mathit{red}},c_{\mathit{ox}}$ are the concentrations of the $\mathit{Red}\left(\mathit{Ox}\right)$ species. Herein, we assume weak interaction of the semiconductor with the redox couple and equal concentrations of $\mathit{Red}$ and $\mathit{Ox}$ species. The addition (removal) of an electron to (from) the $\mathit{Ox}$ ($\mathit{Red}$) species is accompanied by the increase (decrease) of the energy equal to the reorganization energy $\lambda$ (typically around 1 eV). Taking into account the rotation and motion of the solvent ions, probability distributions $W_{\mathit{ox}},W_{\mathit{red}}$ of the energy states around the mean values $E_{\mathit{ox}},E_{\mathit{red}}$ are obtained and shown in Figure 5

Figure 5.

N-type semiconductor in contact with a redox couple in the electrolyte with $E_{\mathit{\text{redox}}}$ close to $E_F$. (a) Energy diagram along with distribution of the occupied states in the electrolyte (painted in green color). The conduction (valence) band energy is denoted $E_{\mathit{cb}}$, $E_{\mathit{vb}}$, and the Fermi energy is labeled $E_F$. (b) The current-voltage curves in the logarithmic scale for parameters $j_v^0=1$ mA/cm${}^2$ and $j_c^0=100$ mA/cm${}^2$. The negative currents were shown with a positive value in the logarithmic plot.

$$W_{\mathit{ox}}\left(E\right)=W^{0}exp\left(-\frac{{\left(E-E_{\mathit{\text{redox}}}+\lambda \right)}^2}{4\mathit{kT\lambda }}\right),$$E17

$$W_{\mathit{red}}\left(E\right)=W^{0}exp\left(-\frac{{\left(E-E_{\mathit{\text{redox}}}-\lambda \right)}^2}{4\mathit{kT\lambda }}\right),$$E18

where $W^0={\left(4\mathit{kT\lambda }\right)}^{-1/2}$ is a constant to provide unit integrated probability over the whole energy spectrum. If $E_{\mathit{\text{redox}}}$ is closer to the conduction (valence) band edge of the semiconductor, electron exchange between the conduction (valence) band and redox couple is energetically preferred, which is a consequence of small loss of electron energy when both energy states have similar energy. The energy distribution functions are proportional to the concentration of the respective species and probability distribution

$$D_{\mathit{ox}}=c_{\mathit{ox}}{W}_{\mathit{ox}},$$E19

$$D_{\mathit{red}}=c_{\mathit{red}}{W}_{\mathit{red}}.$$E20

The rate of electron transfer from the semiconductor to the electrolyte is given by the integral of the probability of the transition between these quantum states over all electron energies

$${\int }_{E}f\left(E\right)\rho \left(E\right)D_{\mathit{ox}}\left(E\right)\mathrm{d}E,$$E21

where $f\left(E\right)$ denotes the Fermi function in the semiconductor, $\rho \left(E\right)$ denotes the energy distribution in the semiconductor, and $W_{\mathit{ox}}\left(E\right)$ is the distribution of empty states in the electrolyte. For the electron transfer from the conduction band to the electrolyte (cathodic current, superscript-), the latter expression becomes

$$j_c^{-}={\mathit{qc}}_{\mathit{ox}}{\int }_{E_c}^{\infty }{n}_{s}exp\left(-\frac{{\left(E-E_{\mathit{\text{redox}}}+\lambda \right)}^2}{4\mathit{kT\lambda }}\right),$$E22

with the electron concentration at the semiconductor-electrolyte interface $n_s$. Typically, by assuming a small overlap of the energy states in the semiconductor and electrolyte (within $1\mathit{kT}$ of the conduction band), the latter integral is simplified to

$$j_c^{-}={\mathit{qk}}_c^{-}{n}_s{c}_{\mathit{ox}},$$E23

with $k_c^{-}$ is the rate constant containing the exponential term (potential independent), and $q$ is the electronic charge. The current $j_c^{+}$ corresponding to the opposite process of the electron transfer from the electrolyte to the semiconductor (anodic current, superscript +) is calculated similarly

$$j_c^{+}={\mathit{qk}}_c^{+}{N}_c{c}_{\mathit{red}},$$E24

where we have used that the density of empty states in semiconductor equals $N_C$, and the distribution of the occupied states is now proportional to $c_{\mathit{red}}$.

In a similar way, the cathodic current from the valence band is potential-independent

$$j_v^{-}={\mathit{qk}}_v^{-}{N}_v{c}_{\mathit{ox}}$$E25

whereas the anodic current from the valence band depends on the potential through $p_s$

$$j_v^{+}={\mathit{qk}}_v^{+}{p}_s{c}_{\mathit{red}}.$$E26

Further simplification of the expressions for currents is derived by considering the balanced equilibrium charge transfer with the equilibrium conduction band current $j_c^0$

$$j_c^{+}=j_c^{-}=j_c^0,$$E27

and equilibrium valence band current $j_v^0$

$$j_v^{+}=j_v^{-}=j_v^0.$$E28

In equilibrium $n_s=n_s^0$, $p_s=p_s^0$. Henceforth, the net current through the conduction band from the semiconductor to the electrolyte is obtained by subtracting the cathodic current from the anodic current

$$j_c=j_c^{+}-j_c^{-}=-j_c^0\left(\frac{n_s}{n_s^0}-1\right)=-j_c^0\left[exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)-1\right]$$E29

and accordingly

$$j_v=j_v^{+}-j_v^{-}=j_v^0\left(\frac{p_s}{p_s^0}-1\right)=j_v^0\left[exp\left(\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)-1\right]$$E30

Our notation is compatible with the traditional electrochemical notation of the positive anodic current and the negative cathodic current, Figure 5c. We remark, however, that the conduction or valence band current is positive (negative) for the cathodic (anodic) bias voltage.

To illustrate the profile of the current-voltage curves, we have taken a small $j_v^0$ and large $j_c^0$, which is expected for $E_{\mathit{\text{redox}}}$ positioned close to the conduction band of n-type semiconductor. For a positive voltage (reverse bias voltage), the anodic current $j_c^{+}$ of electrons from electrolyte to the conduction band is approximately constant. For the increasing negative voltage (forward bias voltage), more electrons (majority carriers) become available, and hence, the cathodic current $j_c^{-}$ of electrons from the conduction band to the electrolyte rises exponentially (rectification). The valence band currents are much smaller in magnitude and follow an exponential rise of $j_v^{+}$ for the reverse bias and a constant value of $j_v^{-}$ for the forward bias (opposite trend as compared with the conduction band current). Generally, a redox reaction at the semiconductor-eletrolyte interface can either receive electrons from the semiconductor or send electrons to the semiconductor. Electrons are majority (minority) carriers in the n-type (p-type) semiconductor, respectively, and hence, the situation with the relative magnitude of the currents across the interface becomes interesting when minority carriers are created by light excitation and can completely alter the current-voltage behavior. The total current at the semiconductor electrode with the Tafel slope 60 mV/dec

$$j=j_c+j_v=-j_c^0\left[exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)-1\right]+j_v^0\left[exp\left(\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)-1\right].$$E31

partially resembles the Butler-Volmer equation for the current at the metal electrode with Tafel slope 120 mV/dec

$$j=j_0\left[exp\left(\frac{\mathit{qV}}{2\mathit{kT}}\right)-exp\left(-\frac{\mathit{qV}}{\mathit{2kT}}\right)\right].$$E32

The difference between the metal and semiconductor current is caused by the potential drop falling over the Helmholtz layer in the metal, while the potential drops over SCR in the semiconductor. Furthermore, the exchange current density is much smaller for the semiconductor than for a metal since metals have much higher density of states near Fermi level than semiconductors.

For the n-type semiconductor and electrons (majority carriers) being transferred from the electrolyte to the semiconductor, it is usually described in the literature that holes (minority carriers) are transported from the semiconductor to the electrolyte and minority carrier reaction proceeds at the interface. In this situation, the anodic current is dominated by $j_v^{+}$ since $j_c^{-}$ is essentially constant (due to a constant number of the empty states in the conduction band $N_C$).

If electrons (majority carriers for n-type) are transferred from the n-type semiconductor to the electrolyte, it is usually termed majority carrier reaction, and hence, the cathodic current is dominated by $j_c^{-}$ since $j_v^{-}$ is essentially constant (due to a constant number of occupied states in the valence band $N_V$).

2.2 Upon illumination

In the preceding section, we presented anodic and cathodic current exchange with the electrolyte in the dark, and we neglected the carrier recombination. Magnitude of the first-order recombination is governed by the population of very few available minority carriers. Additionally, population of the minorites cannot be altered by the voltage bias, as only majority carriers can be injected by the voltage bias [23].

When a light of sufficient energy strikes the semiconductor, the generated electron-hole pairs contribute to the electron exchange with the redox couple or recombine. Due to carrier recombination, the currents calculated in the preceding section cannot be used without further considerations. Here we sketch the development due to Reichman [24], which is instructive but simple enough for tractability. The total current under illumination is composed of the conduction band and valence band currents

$$j=j_c+j_v.$$E33

The conduction band current, Eq. (30), remains the same under illumination as the electron population is not substantially altered upon illumination. The valence band current, Eq. (30), holds when the hole concentration at the surface $p_s$ is calculated under illumination

$$j_v=j_v^0\left(\frac{p_s}{p_s^0}-1\right).$$E34

To obtain $p_s$, it is assumed that quasi-equilibrium of holes is valid even under illumination

$$p\left(x\right)=p_{w}exp\left(+\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)$$E35

and the stand-alone hole continuity equation, Eq. (3), is solved in the neutral region of the semiconductor, where electric field can be neglected. This simplification leads to the diffusion equation valid when hole diffusion length $L_h$ is much smaller than the neutral region thickness

$$D_h\frac{{\mathrm{d}}^{2}p}{\mathrm{d}{x}^2}-\frac{p-p_0}{{\tau }_h}+I_0\alpha exp\left(-\mathit{\alpha x}\right)=0.$$E36

The hole diffusion constant is written as $D_h$, hole recombination lifetime ${\tau }_h$ (first-order recombination), absorption coefficient $\alpha$, and incoming monochromatic photon flux from the electrolyte side $I_0$ (assuming Lambert-Beer conditions). The analytic solution with the boundary conditions $p\left(\infty \right)=p_0$ and $p\left(w\right)=p_w$ is

$$p=p_0+\frac{\alpha I_0{\tau }_{h}exp\left(-\mathit{\alpha x}\right)}{1-{\alpha }^2{L}_h^2}+Kexp\left(-x/L_h\right),$$E37

where $L_h^2=D_h{\tau }_h$ and the constant $K$

$$K=exp\left(w/L_h\right)\left(p_w-p_0-\frac{\alpha I_0{\tau }_{h}exp\left(-\mathit{\alpha w}\right)}{1-{\alpha }^2{L}_h^2}\right).$$E38

The diffusion current at the edge of the neutral region $x=w$ is

$$j_{\mathit{\text{diff}}}=-j_0\left(\frac{p_w}{p_0}-1\right)+\frac{{\mathit{qI}}_0\alpha L_{h}exp\left(-\mathit{\alpha w}\right)}{1+\alpha L_h},$$E39

with the saturation current density denoted $j_0=\frac{{\mathit{qp}}_0{D}_h}{L_h}$.

In the space-charge region, we simply assume all photogenerated charges are separated by the electric field of the SEI, and there is no recombination, hence, the hole current to the electrolyte is

$$j_{\mathit{sc}}={\mathit{qI}}_0\left[1-exp\left(-\mathit{\alpha w}\right)\right].$$E40

The valence band current is then sum of the currents in the neutral and the space-charge region for both anodic and cathodic bias voltage

$$j_v=j_{\mathit{\text{diff}}}+j_{\mathit{sc}}=\frac{j_g-j_{0}exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)}{1+\frac{j_0}{j_v^0}exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)}.$$E41

The term

$$j_g=j_0+{\mathit{qI}}_0\left[1-\frac{exp\left(-\mathit{\alpha w}\right)}{1+\alpha L_h}\right]$$E42

represents the generation current. We remark that the Gartner photocurrent $j_G$ corresponds to the second term in the last equation

$$j_G={\mathit{qI}}_0\left[1-\frac{exp\left(-\mathit{\alpha w}\right)}{1+\alpha L_h}\right]$$E43

and was derived by assuming the ideal collection of holes within the SCR, no recombination in SCR and $p_w=0$ [25]. The photocurrent onset voltage for Gartner equation overlaps with the flatband voltage $V_{\mathit{fb}}$ of the SEI, Figure 6b. There are two current constants $j_0,j_v^0$ in Reichman photocurrent Eq. (41), which govern the profile of JV curve.

Figure 6.

(a) Schematic of the illuminated SEI. (b)–(d) Current-voltage Eq. (41) for baseline parameters in Table 1 and several values of $j_v^0={10}^{-15},{10}^{-12},{10}^{-9}$ mA/cm² (direction of arrow, placed at the corresponding critical voltage $V_c$) in (b) linear plot and (c) semi-logarithmic plot and (d) in the dark (without illumination). For the cathodic (negative) bias voltage, sign of the current is reversed to enable plot in the semilogarithmic scale.

Parameter	Value	Description
$E_g$ (eV)	1.2	Bandgap energy
${\epsilon }_r$	10	Relative permittivity
$N_D$ (cm^−3)	10^17	Donor concentration
$N_c$ (cm^−3)	10^20	Effective density of states in CB
$N_v$ (cm^−3)	10^20	Effective density of states in CB
$L$ (nm)	100	Semiconductor thickness
$\alpha$ (cm^−1)	10^3	Absorption coefficient of the semiconductor
$L_h$ (nm)	10	Hole diffusion length
${\tau }_h$ (ps)	1	Hole recombination lifetime
$I_0$ (cm^−2/s)	5$\cdot$10${}^{16}$	Incoming monochromatic photon flux
$V_{\mathit{fb}}$ (V vs. RHE)	0	Flatband voltage

Table 1.

Baseline material properties of the N-type semiconductor photoelectrode used in this chapter.

Without illumination ($I_0=0$), the Reichman dark current includes recombination in neutral region and equals

$$j_v^{\mathit{\text{dark}}}=\frac{j_0-j_{0}exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)}{1+\frac{j_0}{j_v^0}exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)}.$$E44

Photocurrent $j_v$ has onset delayed more anodic with respect to Gartner photocurrent $j_G$ for decreasing (slower) rate of hole transfer to electrolyte $j_v^0$, see linear-scale current-voltage plot, Figure 6a. This is expected behavior as holes queue at SEI due to their slow transfer to electrolyte. For large anodic bias, Reichman photocurrent $j_v$ approaches Gartner photocurrent $j_G$ as recombination becomes the limiting process. Under large anodic bias voltage ($V_{\mathit{sc}}>0$), the dark current approaches value $j_0$ and the photocurrent reaches $j_G$, Figure 6. Under large cathodic bias voltage ($V_{\mathit{sc}}<0$), both dark current and photocurrent approach $j_v^0$.

For intermediate anodic bias voltage, both the dark current and photocurrent scale as $\sim exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)$ (Tafel slope 60 mV/dec) and minority carrier recombination and transfer govern the kinetics similar to the pn-junction or semiconductor/metal contact. The range of the intermediate anodic bias ($0<V_{\mathit{sc}}<V_c$) with approximate slope 60 mV/dec increases with the decreasing value of $j_v^0$ (rate of hole current transfer at the SEI). The critical voltage bias $V_c$ can be derived from the condition that the terms in the denominator become equal and is given by [26, 27].

$$V_c=\frac{\mathit{kT}}{q}ln\frac{j_0}{j_v^0}.$$E45

For baseline parameters and $j_v^0={10}^{-15},{10}^{-12},{10}^{-9}$ mA/cm${}^2$, we get $V_c=\mathrm{0.49,0.31,0.13}$ V, and these correspond well with plots in Figure 6.

For completeness, we remark that SCR recombination can be easily incorporated in the treatment of Reichman, whereas we decided not to include this term in our plots [24]. The recombination current in the SCR is

$$j_{\mathit{sc}}^R=Q\sqrt{\frac{p_s}{p_{s0}}},$$E46

where these expressions are used

$$Q=\frac{\pi {\mathit{kTn}}_{i}wexp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{2\mathit{kT}}\right)}{4{\tau }_h{V}_{\mathit{sc}}},$$E47

$$\sqrt{\frac{p_s}{p_{s0}}}=\frac{-Q+\sqrt{Q^2+4\mathit{UW}}}{2U},$$E48

$$U=j_v^0+j_{0}exp\left(-\frac{{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right),$$E49

$$W=j_v^0+j_G.$$E50

For very large $j_v^0$, the cathodic current can be diffusion-controlled, and we do not discuss this point here. Around the flatband voltage, the conduction band current, Eq. (30), might dominate the total current due to dark conduction band current, Figure 5b. The presence of bulk trap states in the semiconductor [23] or catalyst layer on the semiconductor [28, 29] was recently treated in model systems, while these are not considered herein.

The relation of the theoretical and measured photocurrent-voltage curves is discussed next. The conditions assumed for the Gartner photocurrent (fast electron transfer to the electrolyte, negligible recombination in the SCR or at the surface) are valid for high-quality semiconductors such as silicon, GaAs, GaP, Figure 7a, and the measured photocurrent onset voltage for GaP is $\sim$0.1 V larger than the flatband voltage (which marks the onset voltage of the Gartner photocurrent). This is not the case for most metal oxide photoelectrodes with short diffusion length of minority carriers and slow electron transfer to electrolyte. Taking iron oxide Fe₂O₃ as an example, measured photocurrent onset voltage is $\sim$0.4 V or more anodic of the flatband voltage, and the photocurrent rises slowly in the beginning (so called S-shape) in contrast to the steep rise predicted by the Gartner equation, Figure 7b. Some efforts lead to reproducing of the photocurrent S-shape by considering the surface recombination [32], but in general, spectroscopic methods are needed to provide an additional evidence for the correctness of the model and the extracted parameters from the photocurrent response.

Figure 7.

(a) Measured photocurrent-voltage curves (green color) for p-GaP in 0.5 mol H₂SO₄ recorded for the low-level monochromatic illumination at fixed wavelengths, the data were taken from [30]. We remark that voltage is measured against the saturated calomel electrode (SCE), and negative voltage is to the right. The Gartner photocurrent is shown in dashed black line. (b) Photocurrent-voltage curve for n-Fe₂O₃ (hematite) in NaOH solution and simulated AM1.5G illumination, the data were taken from [31]. The corresponding Gartner photocurrent is shown in dashed black line.

The complications that arise due to recombination in the surface states were treated by several studies [32, 33, 34, 35], and here we briefly outline theory by Peter [36, 37]. For the surface trap at the SEI with concentration $N_s$ located $E_s$ above the Fermi level of the bulk semiconductor, the trap occupancy in the dark is

$$f_0=\frac{1}{1+exp\left(\frac{E_s-{\mathit{qV}}_{\mathit{sc}}}{\mathit{kT}}\right)}$$E51

and the trap occupancy under illumination is

$$f=\frac{-B-\sqrt{B^2-4\mathit{AC}}}{2A},$$E52

where constants $A,B,C$ are

$$A=k_n{k}_p{n}_s{N}_s^2,B=-k_n{N}_s{j}_G-A\left(1+f_0\right)-k_0{k}_p{n}_s{N}_s$$E53

and

$$C=k_n{N}_s{j}_G+{\mathit{Af}}_0+k_0{k}_p{n}_s{f}_0.$$E54

Henceforth, the photocurrent voltage is described by the relation

$$j_s=j_G-{\mathit{qk}}_n{n}_s{N}_s\left(f-f_0\right).$$E55

For the increasing concentration of the surface states, the photocurrent rises steeply as the surface state occupation reaches unity when the Fermi level of the semiconductor moves through the surface state energy level, Figure 8a.

Figure 8.

(a) Photocurrent-voltage curves for surface recombination via surface state located $E_s$ =0.3 eV above the bulk Fermi level and $N_s={10}^{11},{10}^{12},{10}^{14}$ cm⁻². (b) Potential drop in the Helmholtz layer for the same $N_s$ as in (a).

The charging of the surface states influences the distribution of electrostatic potential in the semiconductor. The potential drop in the Helmholtz layer can be written as

$$\delta V_H=\frac{{\mathit{qN}}_s\left(1-f_0\right)}{C_H},$$E56

when the vacant surface state is assumed to carry a positive charge, and $C_H$ is the capacitance of the Helmholtz layer. The increasing surface state concentration leads to more band-edge unpinning shown as flattening in Figure 8b.

3. Conclusions

This chapter provided introduction to the topic of charge transport in the dark and illuminated semiconductor/liquid junction in order to understand behavior of the ideal SEI. The level to which real semiconductor electrodes mimic such ideal SEI can be very broad. First, we introduced basic considerations of the semiclassical charge transport equations in the semiconductor. Second, the interaction of the semiconductor with redox system in the electrolyte was described, along with the energetics of the electron transfer between semiconductor and redox system. Third, the importance of recombination kinetics in the semiconductor was considered upon illumination, and classical photocurrent formulas due to Gartner and Reichman were discussed. Finally, the effect of surface state occupancy on the photocurrent-voltage curves was described.

Acknowledgments

Funding for this work was partially covered by grant of APVV-20-0528.

Conflict of interest

The authors declare no conflict of interest.