Thermodynamic and electrokinetic perspectives of wet/dry photovoltaic cell and photo-electrochemical cathodic protection short-circuited cell

The concept of splitting a single equilibrium Fermi level (\({E}_{\text{F}}\)) into two distinct quasi-Fermi levels under illumination—namely, \({nE}_{\text{F}}\) for majority charge carriers (electrons) and \({pE}_{\text{F}}\) for minority charge carriers (holes)—was first proposed by Shockley in 1950 [1]^{Footnote 1} and later elaborated by Gerischer in 1990 [2]. In the dark, these quasi-Fermi levels converge into a single Fermi level, where the chemical potentials of both types of carriers are aligned. From this foundational concept, two critical parameters emerge that quantify the deviation from thermodynamic equilibrium: (1) the difference between \({nE}_{\text{F}}\) and \({pE}_{\text{F}}\) and (2) the corresponding photovoltage generated across the external terminals of a photovoltaic cell.

These over-voltages have been fundamentally understood as originating from light quantum energy absorbed by n-type and p-type semiconducting electrodes, with each overvoltage becoming quantitatively meaningful when multiplied by the electronic charge [3,4,5]. Conceptually, the negative or positive shift of the equilibrium electrode potential to the flat band potential (expressed in volts [V]) is directly correlated with the separation between the quasi-Fermi levels (expressed in energy units [eV per particle]). Thus, the photo-induced electromotive force (photo emf) can be directly interpreted as a macroscopic manifestation of the microscopic separation of charge carrier chemical potentials under steady-state illumination.

Motivated by these principles, we conceptually design two hypothetical regenerative photovoltaic cells—one for oxygen and the other for hydrogen—each producing no net chemical change in the electrolyte. The photo I–V curves of these cells resemble those of a classical self-driven Daniell cell (a type of galvanic cell) operating in the dark. When observed in particular from the aspect of diffusion and migration of a CB (conduction band) electron or VB (valence band) hole or redox electron at the internal interface between the electrode and electrolytic solution, such regenerative photovoltaic systems being emphasized from the perspective of the outside electrode inclusive the external circuit are sometimes referred to as photogalvanic cells. This naming reflects their ability to generate a photovoltage without consuming or altering the electrolyte, akin to a galvanic system that operates reversibly.

This article is a continuation of our earlier review work [6], and here we delve deeper into the working mechanisms of regenerative wet photovoltaic cells—including those with electrolytic solutions—relative to n–p junction dry photovoltaic cells and photo-electrochemical cathodic protection short-circuited cells. Additionally, we propose, for the first time to our knowledge, two hypothetical regenerative hydrogen/oxygen photovoltaic cells and analyze their thermodynamic and electrokinetic behaviors.

In the context of terminology, semiconductor physicists often define the Fermi level (\({E}_{\text{F}}\)) as the partial molar free energy (chemical potential) of CB electrons and VB holes. In contrast, electrochemists define it as the electrochemical potential of both semiconductor charge carriers and redox electrons in a redox couple in contact with an aqueous electrolytic solution. This article aims to bridge these disciplinary terminologies by offering a unified conceptual framework and consistent language to describe photo-generated electrochemical potentials across different systems.

From this foundational concept, the theoretical maximum efficiency of regenerative wet photovoltaic cells can be compared to that of dry n–p junction cells, based on their respective over-voltages, quasi-Fermi level splitting, and recombination mechanisms.

This naming convention is consistently applied throughout the article to maintain clarity in distinguishing between photo-induced and equilibrium electrochemical potentials.

By offering a unified framework that compares dry and wet cell operations under photo-illumination, this article aims to deepen understanding of charge carrier behavior in photo-electrochemical systems, fostering new designs for efficient regenerative energy devices.

Finally, this paper includes a set of quizzes with detailed answers to inspire beginning researchers. These pedagogically motivated questions are designed to deepen the reader’s understanding of the physical and electrochemical concepts discussed.

Figure 1 shows a schematic diagram of energy band bending across the space charge transition region of an n-type electrode in contact with an electrolytic solution containing a relatively reducing redox couple with a redox Fermi level below the n-type Fermi level. For simplicity, we do not consider the change of the space charge region's thickness, which is inversely proportion to its capacitance, throughout this article in all ranges considered.

If the equilibrium of any redox couple is established in aqueous solution in the absence of no externally applied potential at RT (room temperature), its thermodynamic equilibrium electrode potential exactly corresponds to the well-defined fixed Fermi level of the redox couple (being expressed more exactly, the electrochemical potential of its redox electron). From Gerischer’s treatment, based on the Frank-Condon principle, it is well known [7, 8] that the distribution of the electronic energy levels in the oxidized species (state) being characterized as the empty state density, \({D}^{\text{ox}}\), and the reduced species (state) being characterized as the occupied state density, \({D}^{\text{red}}\), will differ slightly from the equilibrium potential (a deviation of either level from the Fermi level of the redox couple, \({E}_{\text{F}}^{\text{redox}}\) is termed the rearrangement [reorientation] energy, \(\lambda \approx \text{about }0.2\text{ up to }2\text{ eV}\)) because of differing degrees of solvation resulting from differing ion charges, as indicated in Figs. 1a and 2a.

It is important to mention in advance that the equality of the electrochemical potential of the majority carrier CB electron and minority carrier VB hole within the same phase, either an n-type or intrinsic semiconductor or p-type, reduces to the equality of the chemical potential of the CB electron and VB hole because their electrical potential is canceled (that is to say, the chemical potential of CB electron equals minus of the chemical potential of the VB hole like a mirror-image character). Both chemical potentials are namely overlapped at the Fermi level of either the n-type or intrinsic semiconductor or p-type at equilibrium [6], as detailed later in the section on the model of the mass action law and Quiz 4.

Quiz 1

1. (a)

   Assuming the electrolytic solution contains a relatively oxidizing redox couple, derive the band bending over the space charge region in the p-type electrode at equilibrium and at cathodic/anodic overvoltage from the state of separated p-type and electrolytic solution and discuss the electrochemical reasoning.

2. (b)

   Starting from the flat band situation in the redox couple-free electrolytic solution, derive the band bending across the space charge region in the p-type electrode and discuss the electrochemical reasoning.

Answer 1

Adequate answers to these quizzes are included in the text as well as in Fig. 2.

If the Fermi level of the n-type electrode is higher than that of the relatively reducing redox couple as assumed, the Fermi level at which both chemical potentials are overlapped is lowered until it equals that of the redox couple to fulfill the equilibrium constraint of the equality of electrochemical potential of CB electron and redox electron on either side.

As the Fermi level is depressed, CB electrons are injected into the surface toward the redox couple, leaving behind uncompensated donor positively charged ions over the space charge region, hence creating an electron-depleted space charge region and eventually causing a downward concave band bending in the direction of the interior of the n-type bulk or a concave-up band bending toward the surface. This bending geometric form is certainly attributed to the negative second-order derivative of the potential regarding the x-coordinate in the solution to the combined Poisson equation and Gauss law [6]. Notably, the potential of the CB and VB edges are fixed at the surface, as shown in Fig 1b.

It is important to note here that the maximum possible photo emf, \({V}_{\text{oc}}^{\text{l}}\), never exceeds the contact (or diffusion) potential [9], which means it is the limiting photo emf and is similar to the electrochemical back emf characteristic of equilibrium in the dark between the electrode and electrolyte, as shown in Fig. 2b as well as Fig. 1b.

By definition the contact potential, multiplied by the electronic charge, equals the difference between the Fermi level of the n-type or p-type electrode and the redox couple before contact. It remains nearly constant, regardless of whether it is under illumination or in the dark, for the following two reasons: first, the Fermi level of the n-type never goes above the Fermi level corresponding to the flat band potential or the quasi-Fermi level, even during illumination. Second, the Fermi level of the p-type never falls below it, even during illumination. This implies the contact (diffusion) potential is one of well-defined unique equilibrium parameters.

Figure 2 delineates a schematic diagram of the energy band bending over the space charge transition region of a p-type electrode in contact with an electrolytic solution containing a relatively oxidizing redox couple with a redox Fermi level above the p-type Fermi level. Additional energy band bending is needed at the p-type/solution interface to complete the whole spectra of energy band bending across the space charge transition region, which are of great electrochemical significance in the application to such energy conversion systems as dry/wet photovoltaic/galvanic cells and photoelectrochemical cathodic protection cells.

In contrast, if the Fermi level of the p-type electrode is less than that of the relatively oxidizing redox couple as assumed, the Fermi level at which both chemical potentials overlap is raised until it equals that of the redox couple to satisfy the equilibrium constraint of the equality of electrochemical potential of the CB electron and redox electron on either side.

As the Fermi level is elevated, redox electrons are injected into the surface toward the p-type electrode from the redox couple, leaving behind uncompensated acceptor negatively charged ions over the space charge region, hence creating a hole-depleted space charge region and eventually causing an upward convex band bending toward the interior of the p-type bulk or a concave-down band bending toward the surface. This bending geometry is caused by the positive second-order derivative of the potential regarding the x-coordinate [6]. Notice that the potential of the CB and VB edges are fixed at the surface, as shown in Fig 2b. These two concave-up and concave-down band bending geometric forms [6] are based upon the simple solution to the combined Poisson equation and Gauss law.

Now, let us consider the flat band situation, assuming the electrolytic solution containing no redox couples, as shown in Figs. 1c and 2c. The flat band potential, \({V}_{\text{fb}}\), implies the minimum possible cathodic potential/maximum possible anodic potential applied externally, necessary for the band flattening, where there is no space charge region. At the same time, it is indicative of the onset of the photosensitized anodic oxidation by the VB holes and cathodic reduction by the CB electrons, as discussed in detail in the section on negative/positive inverse overvoltage (see Quizzes 7(b) and 8(b) below). Even though its physical origin is still unknown, fortunately \({V}_{\text{fb}}\) is a well-defined fixed photoelectrochemical potential showing almost no frequency dispersion [8].

If an applied potential is less/higher than the flat band potential, i.e., a cathodic/anodic overvoltage is given, the Fermi level is correspondingly raised/lowered, implying that electrons are injected into the surface toward the n-type electrode from the electrolytic solution, creating an electron-enriched space charge region on the n-type side and eventually causing a concave-down band bending across the corresponding space charge transition region to the direction of the surface, as indicated in Fig 1d. Complementarily, it indicates that electrons are injected into the surface toward the electrolytic solution from the p-type electrode, creating a hole-enriched space charge region on the p-type side and eventually causing a concave-up band bending across the corresponding space charge transition region to the direction of the surface, as indicated in Fig 2d.

Symmetrically, if an applied potential is higher/less than the flat band potential, i.e., an anodic/cathodic overvoltage is given, the Fermi level is correspondingly lowered/raised, implying that electrons are injected into the surface toward the electrolytic solution from the n-type electrode, creating an electron-depleted space charge region on the n-type side and eventually causing a concave-up band bending across the corresponding space charge transition region to the direction of the surface, as indicated in Fig 1e. In comparison, it means that electrons are injected into the surface toward the p-type electrode from the electrolytic solution, creating a hole-depleted space charge region on the p-type side, and eventually causing a concave-down band bending across the corresponding space charge transition region to the direction of the surface, as indicated in Fig 2e.

Similarly, if an applied potential is much higher/less than the flat band potential, i.e., a relatively greater anodic/cathodic overvoltage is given, the Fermi level is correspondingly lowered/raised close to the edge of VB/CB, implying that electrons are injected into the surface toward the electrolytic solution from the n-type electrode, creating an electron-depleted space charge region on the n-type side and eventually causing a concave-up band bending across the corresponding space charge transition region to the direction of the surface, as indicated in Fig 1f. In comparison, it means that electrons are injected into the surface toward the p-type electrode from the electrolytic solution, creating a hole-depleted space charge region on the p-type side and eventually causing a concave-down band bending across the corresponding space charge transition region to the direction of the surface, as indicated in Fig 2f.

Such two-space charge regions are called the inversion layer: the surface region of the former inversion layer behaves as a p-type character, while the bulk still shows the n-type character. In contrast, the surface region of the latter inversion layer behaves as an n-type character while the bulk still shows the p-type character.

If the contact is made with an oxidizing redox couple having a redox Fermi level at the valence band (VB) edge of the n-type or close to it, instead of externally applied potential, an inversion layer is formed by an electrochemical operation of injection of electrons into the surface toward the redox couple, which is physically equivalent to another electrochemical operation of injection of holes into the surface of the n-type electrode from the redox couple [3]. Such a former inversion layer was observed at a passivating Ti oxide film , indicating structural transition of an n-type character in the bulk to a p-type character at the surface [10]. To date, to our knowledge, there is no report about the latter inversion layer originating from the p-type bulk in the literature.

Quiz 2

1. (a)

   What actions do you take to generate an electron-enriched space charge region in the n-type electrode in contact with an electrolytic solution?

2. (b)

   Characterize the nature of the band bending across the electron-enriched space charge region to discuss the electrochemical significance.

Answer 2

1. (a)

   There are two approaches to electrochemically create an electron-enriched space charge region in the n-type electrode: one is giving a cathodic applied potential to the surface of the n-type electrode, less (more negative) than the flat band potential, as shown in Fig. 3a and b. Another is attaining equilibrium from the separated n-type and electrolyte in the presence of a relatively reducing redox couple, as shown in Fig. 3c and d.

2. (b)

   Electrons are injected into the surface toward the n-type electrode from the electrolytic solution, therefore creating an electron-enriched space charge region on the n-type side and eventually causing a concave-down band bending toward the surface (Fig. 3b), the same as written in the previous text regarding Fig. 1d. Since the Fermi level of the n-type is below the Fermi level of the redox couple before contact, electron injection occurs into the surface toward the n-type electrode from the electrolytic solution, therefore creating an electron-enriched space charge region on the n-type side and eventually causing a concave-down band bending across the enriched transition region toward the surface (Fig. 3d).

Quiz 3

1. (a)

   What actions do you take to generate a hole-enriched space charge region in the p-type electrode in contact with an electrolytic solution?

2. (b)

   Characterize the nature of the band bending over the hole-enriched space charge region to discuss electrochemical significance.

Answer 3

1. (a)

   There are two approaches to electrochemically create a hole-enriched space charge region in the p-type electrode: one is giving an anodic applied potential to the surface of the p-type electrode, more positive than the flat band potential, as shown in Fig. 4a and b. Another is attaining the equilibrium from the separated p-type and electrolyte in the presence of a relatively oxidizing redox couple, as shown in Fig. 4c and d.

2. (b)

   Electrons are injected into the surface toward the electrolytic solution from the p-type electrode, therefore creating a hole-enriched space charge region on the p-type side and eventually causing a concave-up band bending toward the surface (Fig. 4b), the same as written in the previous text regarding Fig. 2d. Since the Fermi level of the p-type is above the Fermi level of the redox couple before contact, electron injection occurs into the surface toward the electrolytic solution from the p-type electrode, therefore creating a hole-enriched space charge region on the p-type side and eventually causing a concave-up band bending across the enriched transition region in the direction of the surface (Fig. 4d).

Judging from the appearance of the band bending over the electron-depleted space charge region of the n-type given in Fig. 1b in the dark, it is deduced that the majority CB electron coming down along the downward concave bending from the n-type through the external load toward another counter cathode metal electrode drives a cathodic reduction to reduce the oxidized species of the redox couple concerned. In contrast, the minority VB hole going upward along the concave-up bending toward the electrolytic solution in principle can drive an anodic oxidation to oxidize the reduced species on the energy band diagram, but the occurrence of anodic oxidation is actually implausible because of an insufficient concentration of minority holes.

From the appearance of the band bending over electron-enriched space charge region of the n-type given in Fig. 3d in the dark, it is deduced that the majority CB electron coming downward along the concave-down bending from the n-type toward the electrolytic solution drives a cathodic reduction to reduce the oxidized species of the redox couple concerned. In contrast, the minority VB hole going upward along the upward convex bending from the n-type through the external load toward another counter anode metal electrode can drive an anodic oxidation there to oxidize the reduced species on the energy band diagram, but the occurrence of anodic oxidation is actually implausible because of the insufficient concentration of minority holes.

A characteristic of the n-type electrode is that only reductive current is allowed to flow but little oxidative current flow. An oxidative current blockade-rectifying behavior is characteristic of the n-type having the electron-enriched as well as electron-depleted space charge region in the dark. By the same argument, we conclude that a reductive current blockade-rectifying behavior is characteristic of the p-type having the hole-enriched as well as hole-depleted space charge region in the dark, judging from Figs. 2b and 4d.

In short, the insufficient concentration of the minority carriers is responsible for the current blockade-rectifying behavior of the interface between n-type or the p-type and electrolytic solution, regardless of the enriched and depleted space charge region, characteristic of the n-type and p-type in contact with electrolytic solution.

By comparison, under illumination, the presence of a depleted space charge region is characterized as a boundary region that is adequate and plausible for generation of photocurrent/voltage, irrespective of the n-type and p-type, because an injection of the photo-enhanced electrons/holes into the n-type/p-type causes the equilibrium \({E}_{\text{F}}\) (Figs. 1b and 2b) to go upward/come downward to approximate to the highest/lowest possible critical threshold of the flat band energy level (\(-e{V}_{\text{fb}}^{\text{n}}/e{V}_{\text{fb}}^{\text{p}}\)),^{Footnote 2} to finally make the concave-up/concave-down band bend into the flat band bending state (zero band bending or 100% band flattening), respectively. That is, the flat band potential of the n-type electrode/p-type electrode by definition is indicative of the onset of the generation of the anodic/cathodic photocurrent and photovoltage, respectively.

As suggested in Fig. 1b, the majority electron and minority hole in the n-type enhanced by illumination present in the concave-up band bending across the electron-depleted region migrate along the band bending into the n-type interior and the surface toward the redox couple in the electrolytic solution, respectively, before driving the cathodic reduction and anodic oxidation. This causes an increase in the Fermi level of the n-type electrode relative to that Fermi level of the redox couple. This separation of the Fermi level \({E}_{\text{F}}^{\text{n}}\) on either side of the n-type electrode and the redox couple in the solution eventually produces a negative photo emf (the maximum photovoltage)^{Footnote 3} given by \(\left\{-\frac{\left({E}_{\text{F},\text{fb}}^{\text{n}}-{E}_{\text{F}}^{\text{redox}}\right)}{e}=\left({V}_{\text{fb}}^{\text{n}}-{V}_{\text{eq}}^{\text{redox}}\right)<0\right\}\), which is the driving force for producing a photocurrent/voltage.

The same is true of the separation of the \({E}_{\text{F}}^{\text{p}}\) on either side of the p-type electrode and the redox couple, eventually delivering a positive photo emf ³ given by \(\left\{-\frac{\left({E}_{\text{F},\text{fb}}^{\text{p}}-{E}_{\text{F}}^{\text{redox}}\right)}{e}=\left({V}_{\text{fb}}^{\text{p}}-{V}_{\text{eq}}^{\text{redox}}\right)>0\right\}\), caused by the photoexcited majority hole and minority electron present in the concave-down band bending across the hole-depleted region of the p-type electrode, as shown in Fig. 2b. Notice that the sign of electronic energy (difference in the Fermi level) is always opposite to the sign of the potential (photo emf) and/or the energy of hole.

In contrast, the presence of an enriched space charge region is characterized as a boundary region, being inadequate and implausible for generating photocurrent/voltage, whether the n-type or p-type. This is because Fermi levels of both the n-type and p-type are above and below the flat band energy level, respectively; an injection of the photo-enhanced electrons and holes into the n-type and p-type makes it impossible to shift the equilibrium \({E}_{\text{F}}\) (Figs. 3d and 4d) to the flat band energy levels (\(-e{V}_{\text{fb}}^{\text{n}}/e{V}_{\text{fb}}^{\text{p}}\)) of the n-type and p-type, respectively. Eventually the photoexcited electrons and holes will simply enhance the degree of the concave-down and concave-up band bendings to an even greater extent, respectively.

The mass action law is valid at chemical equilibria in wet and solid state chemistry, and it is requisite for understanding the quasi-Fermi level. Applying it specifically to the generation and recombination of EHPs (electron-hole pairs) in semiconducting electrodes, it states that the product of concentrations of electrons and holes remains constant at constant temperature, regardless of such external parameters as external doping, applied potential, and degree of band bending.

We briefly derive it in the three cases of extrinsic n-type, intrinsic semiconductor, and extrinsic p-type by using both thermodynamic and kinetic approaches to equilibrium, which can be extended to all cases in equilibrium with any redox couple in an electrolytic solution to complete a simple model to illustrate it.

First, consider the creation and recombination of equal numbers of EHPs by thermal excitation from the thermodynamic point of view, which can be written as

where e and h represent the electron and hole, which are regarded as a charged chemical species; the forward reaction, the generation of EHP, leaving a broken bond, and the reverse reaction lead to a recombination, re-forming the broken bond or returning a free electron to the hole in the VB on the band diagram. At equilibrium, the condition of the equality of the electrochemical potential of all chemical species is always satisfied on either side of Eq. (1) such that the sum of the electrochemical potential of the CB electron and VB hole should be zero within a single phase, either an n-type semiconductor or intrinsic semiconductor, or p-type as follows:

with the boundary conditions (equal generation rate and recombination rate [EHPs \({\text{cm}}^{-3} \; {\text{s}}^{-1}]\)). Here, \({\mu }_{\text{e}}\) and \({\mu }_{\text{h}}\) [eV] are the chemical potential of the CB electron and VB hole, respectively; e, electronic charge, \(1.602\times {10}^{-19}[{\text{C}}\; {\text{particle}}^{-1}]\); \(\phi\) [V] means the electrostatic potential of the respective phase. Eventually, there is a complementary relationship between the two chemical potentials of the CB electron and VB hole within a single phase such that \({\mu }_{\text{e}}=-{\mu }_{\text{h}}\) always holds true.

Two electrostatic potentials can cancel each other out, for instance, [\({\phi }^{\text{in}}\left(\text{e}\right)={\phi }^{\text{in}}\left(\text{h}\right)\)], because the potential is actually not associated with the chemical species, but it belongs to the phase concerned, the n-type or intrinsic semiconductor, or p-type phase. It states that the chemical potential of the CB electron is equal to the negative chemical potential of the VB hole (\({\mu }_{\text{e}}=-{\mu }_{\text{h}}\)), implying that the two chemical potentials of the CB and the VB hole are aligned or converge at the uniquely defined equilibrium single Fermi level, \({E}_{\text{F}}\). Therefore, the Fermi energy \({E}_{\text{F}}\) indicates the chemical potential of the conduction electron, as defined by a semiconductor physicist.

Notably, the equality of the electrochemical potential of the CB electron and the negative electrochemical potential of the VB hole, such that the relation \({\eta }_{\text{e}}=-{\eta }_{\text{h}}\) for the equilibrium condition within the single phase, reduces to the equality of the chemical potential of the CB electron and the negative chemical potential of the VB hole, giving the relation \({\mu }_{\text{e}}=-{\mu }_{\text{h}}\). This is because a VB hole behaves like a positively charged electron vacancy in the VB, effectively reflecting the absence of an electron in the VB.

Applying the condition of the equality of the chemical potential of CB electron and the negative chemical potential of the VB hole, consideringing the concentration dependence of the chemical potential and finally selecting the standard chemical potential arbitrarily as specific energy level, for instance, the lower edge of the conduction band, \({E}_{\text{c}} \;\text{in} \;[\text{eV}]\), the upper edge of the valence band, \({E}_{\text{v}}\; \text{in }[\text{eV}]\), Eq. (2) becomes to

for the n-type, intrinsic semiconductor, and p-type, respectively, with the boundary constraint, \({n}_{\text{e}}^{\text{in}}={p}_{\text{h}}^{\text{in}}\). Here, \({E}_{\text{F}}^{\text{in}}\), \({E}_{\text{F}}^{\text{n}}\), and \({E}_{\text{F }}^{\text{p}}\) are the Fermi level of the respective phase in \([\text{eV}]\); \({N}_{\text{c}}\text{ in }\;[{\text{cm}}^{-3}]\), the effective state density at \({E}_{\text{c}}\); \({N}_{\text{v}}\text{ in }[{\text{cm}}^{-3}]\), the effective state density at \({E}_{\text{v}}\); \({n}_{\text{e}}^{\text{n}}\), \({n}_{\text{e}}^{\text{in}}\), and \({n}_{\text{e }}^{\text{p}}\), the equilibrium concentration of CB electron in the respective phase in \([{\text{cm}}^{-3}]\); \({p}_{\text{h}}^{\text{n}}\), \({p}_{\text{h}}^{\text{in}}\), and \({p}_{\text{h}}^{\text{p}}\), the equilibrium concentration of VB hole in the respective phase in \([{\text{cm}}^{-3}]\); k, the Boltzmann’s constant, \(8.62\times {10}^{-5} \left[\text{eV }{\text{particle}}^{-1}{\text{ K}}^{-1}\right]\); T means the absolute temperature in \(\left[\text{K}\right]\).

Finally, Eqs. (3) to (5) yield

where \({E}_{\text{g}}\) means the band gap energy [eV].

Using the kinetic approach, the recombination rate of EHP is proportional to the concentration of electrons, n, available for recombination times the concentration of holes, p, of empty states available for occupancy, in the three cases of extrinsic n-type, intrinsic semiconductor, and extrinsic p-type, with the constraint of the equality of the generation rate G in [\({\text{EHPs cm}}^{-3}\;{\text{s}}^{-1}]\) and recombination rate \({R}_{\text{rec}}\) of EHP at equilibrium, giving

where r in [EHPs \({\text{cm}}^{3}{\text{s}}^{-1}]\) is a proportionality constant which depends upon the impurities of the semiconductor and temperature, but the thermal generation rate G is a function of temperature only.

Equations (6) and (7) are termed the mass action law originating from wet- and solid-state chemistry. Both equations obviously imply that as the concentration of majority carriers increases, for instance, by two or three times, etc., by foreign atom doping or external negative/positive biasing (cathodic/anodic overvoltage), following the Fermi level in the energy band diagram, the concentration of minority carriers should accordingly decrease necessarily by 1/2 times, 1/3 times, etc., starting from the Fermi level at which the concentrations of e and h are equal (intrinsic semiconductor). Maintaining the product of concentrations constant means that chemical potentials of majority and minority carriers are equal to the opposite sign at the Fermi level on the band diagram.

Figure 5 gives a model pictorially illustrating the mass action law, which is valid at equilibrium between electrons and holes in an extrinsic n-type semiconductor, an intrinsic semiconductor, and an extrinsic p-type semiconductor, as similarly shown in [11]

Quiz 4

Semiconductor physicists claim the Fermi level is the chemical potential of CB electron and VB hole, while electrochemists claim it as the electrochemical potential of the redox electron.

1. (a)

   Do the two claims contradict each other?

2. (b)

   Justify this unless one claim excludes the other.

Answer 4

1. (a)

   The two definitions never contradict each other.

2. (b)

   Considering the transition of an electron from the valence band (VB) to the conduction band (CB) within the same phase, either n-type or intrinsic semiconductor or p-type and following the Schottky-Wagner notation, relative to the Kroeger-Vink notation, we get the following equilibrium formula and equilibrium condition for CB electron and VB hole in an intrinsic semiconductor, for instance, as in Eqs. (1) and (2) above. There their respective electrical potentials cancel each other out. The equality of chemical potentials of the electron and hole with an opposite sign indicates that both chemical potentials converge at the thermodynamically well-defined equilibrium single Fermi level, \({E}_{\text{F}}\).

The same is true for other extrinsic semiconductors. This one extreme limiting case of equality of electrical potentials within the same phase, for instance, [\({\phi }^{\text{in}}\left(\text{e}\right)={\phi }^{\text{in}}\left(\text{h}\right)\text{ or }{\phi }^{\text{n}}\left(\text{e}\right)={\phi }^{\text{n}}\left(\text{h}\right)\text{ or }{\phi }^{\text{p}}\left(\text{e}\right)={\phi }^{\text{p}}\left(\text{h}\right)\)], justifies why semiconductor physicists define \({E}_{\text{F}}\text{ being expressed in energy unit of }\left[\text{eV}{\text{ particle}}^{-1}\right]\) as the chemical potential of the CB electron and VB hole.

Next, consider the electrochemical equilibrium of a fuel cell, for instance, a hydrogen-oxygen fuel cell. The equality of the electrochemical potential of all the chemical species including the redox electron participating before and after operation at the hydrogen oxygen fuel cell gives

where superscripts A and C mean anode and cathode, respectively; \(\phi\), electrical potential \(\left[\text{V}\right]\); \(\eta\), electrochemical potential \(\left[\text{J }{\text{mol}}^{-1}\right]\); \(F=e{N}_{\text{L}}\), the Faraday constant, 96,500 \(\left[\text{C}{\text{ mol}}^{-1}\right]\); \({N}_{\text{L}}\) is the Loschmidt constant or Avogadro’s number, \(6.03\times {10}^{23} [\text{particles }{\text{mol}}^{-1}]\). The difference in the electrochemical potentials (the sum of chemical and electrical potentials) of the redox electron at both electrodes equals the difference between two terminal voltages at the hydrogen oxygen fuel cell, assuming \(\left({\mu }_{\text{e}}^{\text{Pt}\left(\text{C}\right)}={\mu }_{\text{e}}^{\text{Pt}\left(\text{A}\right)}\right)\) is the same chemical potential of the redox electrons in either electrode made of the same material, for instance, Pt. This, another extreme limiting case of equality of chemical potentials, justifies why electrochemists define \({E}_{\text{F}}\text{ being expressed in energy unit of }\left[\text{J }{\text{mol}}^{-1}\right]\) as the electrochemical potential of the redox electron.

Notice that, strictly speaking, the number of particles does not count as a physical dimension but is simply a dimensionless quantity.

The \({E}_{\text{F}}\) in equilibrim represents the chemical potential of the majority and minority carriers and their equilibrim concentrations. Performing the separation of the chemical potentials of the electron and hole from the equality constraints, Eqs. (3) and (5) give

respectively. Both are useful forms of the equilibrium concentrations of electrons and holes whether the material is intrinsic or doped, provided thermal equilibrium is maintained. Thus, \({n}_{\text{e}}^{\text{in}}\), the equilibrium concentration of CB electron in \([{\text{cm}}^{-3}]\), and \({p}_{\text{h}}^{\text{in}}\), the equilibrium concentration of VB hole in \([{\text{cm}}^{-3}]\) at the intrinsic Fermi level \({E}_{\text{F}}^{\text{in}}\) near the middle of the band gap, are given by

respectively; \({E}^{\text{in},\text{ n}}\) and \({E}^{\text{in},\text{ p}}\) are the same as the intrinsic Fermi level \({E}_{\text{F}}^{\text{in}}\). Another convenient way of writing Eqs. (9) and (10) is obtained by applying Eqs. (11) and (12)

respectively, where \(k=\frac{R}{{N}_{\text{L}}}\); R is the gas constant \(8.314 \left[{\text{Jmol}}^{-1}{\text{K}}^{-1}\right]\); kT is the thermal energy [kT = 0.0259 \(\left[\text{eV}\right]\) at RT(300 K)].

If the steady-state light quanta illumination is shone on the semiconducting electrode with no trapping (\(\delta n=\delta p\)) (excess charge carrier concentration of electron equals that concentration of hole), indicating that the rate of optical generation of EHPs, \({G}^{\text{l}} \left[\text{number of particles }{\text{cm}}^{-3}{\text{s}}^{-1}\right]\), equals the recombination rate \({R}_{\text{rec}}^{\text{l}}\), the excess concentration of the majority and minority charge carriers will increase to new state values as well. Under this condition, the excess concentrations of electrons and holes (which are excess EHPs), i.e., the departures from thermodynamic equilibrium \(\delta n\text{ and }\delta p\) are given by

where \({\tau }_{\text{n}}\) and \({\tau }_{\text{p}}\), are the lifetime to recombination of electrons and holes, respectively. Notice that the optical generation of EHPs by quanta photon energy \(h\nu\) differs from the generation by thermal phonon energy due to lattice vibration kT, which is in thermodynamic equilibrium.

Assuming excess charge carrier concentration of electrons and holes, \(\delta n=\delta p\) is relatively small compared to equilibrium concentration of the majority carrier, \({n}_{\text{e}}^{\text{n}}\) and \({p}_{\text{h}}^{\text{p}}\), (\(\delta n\ll {n}_{\text{e}}^{\text{n}}\text{ for n}-\text{type};,円 \delta p\ll {p}_{\text{h}}^{\text{p}}\text{ for p}-\text{type})\), then the steady-state charge carrier concentrations of the majority and minority carrier in the n-type accordingly are under illumination given by

respectively. Similarly, the steady-state carrier concentrations of the majority and minority carrier in the p-type accordingly are under illumination given by

respectively. These are termed low-level injections (excitations) of charge carriers by light quanta photon illumination, indicating that a given concentration of excess EHPs causes a large shift in the minority carriers compared to that majority carriers.

Under these circumstances, it is appropriate that under light quanta illumination, the equilibrium Fermi levels \({E}_{\text{F}}^{\text{n}}\) and \({E}_{\text{F}}^{\text{p}}\) can be split into two quasi-Fermi levels \({nE}_{\text{F}}^{\text{n}}\) (majority) and \({pE}_{\text{F}}^{\text{n}}\) (minority), \({nE}_{\text{F}}^{\text{p}}\) (minority) and \({pE}_{\text{F}}^{\text{p}}\) (majority), representing the steady-state chemical potentials including concentrations of electrons and holes in the n-type and of two in the p-type, respectively.

The separation of the equilibrium \({E}_{\text{F}}\) between the two quasi-Fermi levels \({nE}_{\text{F}}\) and \(p{E}_{\text{F}}\), namely, the difference \(\left({nE}_{\text{F}}-p{E}_{\text{F}}\right)\), which is the difference in chemical potential between the majority and minority carriers, is a direct measure of the departure from the thermodynamic equilibrium. Conversely, both quasi-Fermi levels \({nE}_{\text{F}}\) and \(p{E}_{\text{F}}\) will get back on track in the dark. This means both are aligned (overlapped) at \({E}_{\text{F}}\) at equilibrium [\(\left({E}_{\text{F}}^{\text{n}}={nE}_{\text{F}}^{\text{n}}={pE}_{\text{F}}^{\text{n}}\right)\) and \(\left({E}_{\text{F}}^{\text{p}}={nE}_{\text{F}}^{\text{p}}={pE}_{\text{F}}^{\text{p}}\right)\)]. In some textbooks the quasi-Fermi level is termed IMREF, which is Fermi spelled backward.

For simplicity, such a steady state can be considered close to equilibrium, holding the main framework of all the equilibria concentration formulae for electrons and holes. In this article, such a system is considered in quasi-equilibrium for convenience’s sake. Thus, we can write expressions for the steady-state concentrations of the majority and minority carriers in the n-type and p-type in the same form as the equilibrium concentration formulae in Eqs. (9) to (14), substituting the quasi-Fermi levels, \({nE}_{\text{F}}^{\text{n}}\) and \({pE}_{\text{F}}^{\text{n}}\), \({nE}_{\text{F}}^{\text{p}}\) and \({pE}_{\text{F}}^{\text{p}}\), for the equilibrium Fermi level, \({E}_{\text{F}}^{\text{n}}\) and \({E}_{\text{F}}^{\text{p}}\): first,

for the majority electron and minority hole in the n-type, respectively. Next, similarly,

for the majority hole and minority electron in the p-type, respectively.

Conversely, the deviation from equilibrium resulting from the two separate quasi-Fermi levels in the n-type and p-type can be regarded as light quanta photon energy gained and stored in the photoexcited hole and electron during illumination, which may serve as a driving force for the occurrence of photosensitized anodic oxidation and cathodic reductions, respectively.

As suggested later in the section on wet/dry photovoltaic cells, the separation between two quasi-Fermi levels in the n-type anode and that separation in the p-type cathode, divided by electronic charge e, nearly equals the shift of the electrode potential of the given redox couple to the flat band potential, respectively, which are again the limiting photo emfs (negative emf at the anode and positive emf at the cathode) regarding their respective counter metal electrode, i.e., \(-\frac{\left({nE}_{\text{F}}^{\text{n}}-{pE}_{\text{F}}^{\text{n}}\right)}{e}\approx -\frac{\left({E}_{\text{F},\text{ fb}}^{\text{n}}-{E}_{\text{F}}^{\text{redox}}\right)}{e}=\left({V}_{\text{fb}}^{\text{n}}-{V}_{\text{eq}}^{\text{redox}}\right)={\eta }_{\text{h}}^{\text{n},\text{ l}}(\text{negative inverse overvoltage})\le {V}_{\text{oc}}^{\text{n},\text{ l}}<0\), being a negative value for the n-type anode regarding the counter metal cathode; \(\frac{\left({nE}_{\text{F}}^{\text{p}}-{pE}_{\text{F}}^{\text{p}}\right)}{e}\approx \frac{\left({E}_{\text{F}}^{\text{redox}}-{E}_{\text{F},\text{ fb}}^{\text{p}}\right)}{e}=\left({V}_{\text{fb}}^{\text{p}}-{V}_{\text{eq}}^{\text{redox}}\right)={\eta }_{\text{e}}^{\text{p},\text{ l}}\left(\text{positive inverse overvoltage}\right)\ge {V}_{\text{oc}}^{\text{p},\text{ l}}>0\), being a positive value for the p-type cathode regarding the counter metal anode.

The opposite sign of the difference in electronic energy level expressed in \(\left[\text{eV}\right]\) and inverse overvoltage inclusive photo emf expressed in \(\left[\text{V}\right]\) is simply attributable to the fact that as the Fermi level is raised/lowered, the potential is accordingly lowered/raised in the energy band diagram.

More widening of the photo emf implies that the photoexcited minority hole in the n-type anode plays a larger role, relative to the majority carrier electron, in the formation of the driving force for the photosensitized anodic oxidation. The same is true of photo-enhanced minority electron in the p-type cathode. Notice that the limiting photo emf means the maximum possible photovoltage at the open circuit.

This certainly provides a thermodynamic basis for the transition of the electrode potential to the flat band potential and also a kinetic basis for predicting the photo-I vs V polarization characteristics during illumination. This important relationship among the separation between the two quasi-Fermi levels, the inverse overvoltage, and the photo emf during illumination has been applied to the present cases of the three kinds of wet regenerative photovoltaic cell, dry photovoltaic cell at the n/p junction, and photo-electrochemical cathodic protection cell. It is applicable to other photo-electrolytic cells.

Notably, the term "equilibrium," including thermal excitation in the absence of any external light quanta photon excitation, must be distinguished from the term "steady state" of even the low-level injection (excitation). We come back to the issue in the following Quiz 5 and 6.

Quiz 5

Consider that EHPs are optically created with a rate of \({G}^{\text{l}}={10}^{19 }\left[\text{EHPs }{\text{cm}}^{-3}{\text{s}}^{-1}\right]\) in a Si specimen doped with As atoms with a concentration \({10}^{14 } \left[\text{atoms }{\text{cm}}^{-3}\right]\) (n-type) and a lifetime of electron/hole to recombination, \({\tau }_{\text{n}}={\tau }_{\text{p}}=2\) [μs] at RT(300 K).

1. (a)

   What is the equilibrium concentration of minority carrier holes?

2. (b)

   What are the steady-state concentrations of the majority electrons and minority holes?

3. (c)

   The position of the equilibrium Fermi level of the electron, \({E}_{\text{F}}^{\text{n}}\), two quasi-Fermi levels of the electron \({nE}_{\text{F}}^{\text{n}}\) and hole \(p{E}_{\text{F}}^{\text{n}}\), relative to the reference levels, such as \({E}^{\text{in},\text{ n}}\), \({E}_{\text{c}}\), and \({E}_{\text{v}}\) on the energy band diagram.

4. (d)

   Does the mass action law hold? Explain why and discuss its electrochemical significance.

5. (e)

   What photo emf is expected to be developed regarding the counter metal cathode when an inert single redox couple is thought to be present in the electrolytic solution?

The given data are as follows: \({n}_{\text{e}}^{\text{in}}={p}_{\text{h}}^{\text{in}}=1.5\times {10}^{10}\left[{\text{cm}}^{-3}\right]\); \({N}_{\text{c}}=2.8\times {10}^{19}\left[{\text{cm}}^{-3}\right]\); \({N}_{\text{v}}=1.04\times {10}^{19}\left[{\text{cm}}^{-3}\right]\); kT \(\cong 0.0259\left[\text{eV}\right]\) at RT(300 K).

Answer 5

(a), (b), (c):

\({n}_{\text{e}}^{\text{n}}={10}^{14}\left[{\text{cm}}^{-3}\right]\), \(\delta n=\delta p=2.0\times {10}^{13}\left[{\text{cm}}^{-3}\right]\),

\(\therefore ,円{n}_{\text{e}}^{\text{n},\text{ l}}={n}_{\text{e}}^{\text{n}}+\delta n=1.2\times {10}^{14}\left[{\text{cm}}^{-3}\right]\), following the mass action law,

\({p}_{\text{h}}^{\text{n}}=\frac{\left\{{\left({n}_{\text{e}}^{\text{in}}\right)}^{2}\right\}}{{n}_{\text{e}}^{\text{n}}}=2.25\times {10}^{6}\left[{\text{cm}}^{-3}\right]\), \(\therefore,円 {p}_{\text{h}}^{\text{n},\text{ l}}={p}_{\text{h}}^{\text{n}}+\delta p=2.0\times {10}^{13}\left[{\text{cm}}^{-3}\right]\)

1. 1.

   \(\left({nE}_{\text{F}}^{\text{n}}-{E}^{\text{in},\text{ n}}\right)=kT\text{ln}\frac{{n}_{\text{e}}^{\text{n},\text{ l}}}{{n}_{\text{e}}^{\text{in}}}=0.0259\text{ln}\frac{{n}_{\text{e}}^{\text{n},\text{ l}}}{{n}_{\text{e}}^{\text{in}}}=0.2327[\text{eV}]>0\)

2. 2.

   \(\left({E}_{\text{c}}-{nE}_{\text{F}}^{\text{n}}\right)=-kT\text{ln}\frac{{n}_{\text{e}}^{\text{n},\text{ l}}}{{N}_{\text{c}}}=-0.0259\text{ln}\frac{{n}_{\text{e}}^{\text{n},\text{ l}}}{{N}_{\text{c}}}=0.32[\text{eV}]>0\)

3. 3.

   \(\left({E}^{\text{in},\text{ p}}-p{E}_{\text{F}}^{\text{n}}\right)=kT\text{ln}\frac{{p}_{\text{h}}^{\text{n},\text{ l}}}{{p}_{\text{h}}^{\text{in}}}=0.0259\text{ln}\frac{{p}_{\text{h}}^{\text{n},\text{ l}}}{{p}_{\text{h}}^{\text{in}}}=0.186[\text{eV}]>0\)

4. 4.

   \(\left(p{E}_{\text{F}}^{\text{n}}-{E}_{\text{v}}\right)=-kT\text{ln}\frac{{p}_{\text{h}}^{\text{n},\text{ l}}}{{N}_{\text{v}}}=-0.0259\text{ln}\frac{{p}_{\text{h}}^{\text{n},\text{ l}}}{{N}_{\text{v}}}=0.34[\text{eV}]>0\)

5. 5.

   \(\left({E}_{\text{F}}^{\text{n}}-{E}^{\text{in},\text{ n}}\right)=kT\text{ln}\frac{{n}_{\text{e}}^{\text{n}}}{{n}_{\text{e}}^{\text{in}}} =0.0259\text{ln}\frac{{n}_{\text{e}}^{\text{n}}}{{n}_{\text{e}}^{\text{in}}}=0.228\left[\text{eV}\right]>0\text{ in the dark}\)

All the data calculated are shown in Fig. 6a.

1. (d)

   Regarding the mass action law: \({n}_{\text{e}}^{\text{n},\text{ l}}{p}_{\text{h}}^{\text{n},\text{ l}}=2.4\times {10}^{27}\left[{\text{cm}}^{-6}\right]\gg {n}_{\text{e}}^{\text{in}}{p}_{\text{h}}^{\text{in}}={\left({n}_{\text{e}}^{\text{in}}\right)}^{2}=2.25\times {10}^{20}\left[{\text{cm}}^{-6}\right]\). From the data obtained, first, the mass action law seriously breaks. Second, the mass action law obviously violates the prerequisite of the low-level injection (excitation) used to derive Eqs. (20) to (23), valid for the quasi-Fermi levels.

2. (e)

   According to \(\left\{-\frac{\left({nE}_{\text{F}}^{\text{n}}-{pE}_{\text{F}}^{\text{n}}\right)}{e}\right\}\le {V}_{\text{oc}}^{\text{n},\text{ l}}<0\), the maximum possible photo emf is – 0.419 [V vs the counter metal cathode].

Quiz 6

An Si specimen is doped with B atoms with concentration \({10}^{15 }\left[\text{atoms }{\text{cm}}^{-3}\right]\) (p-type). Assume that \({10}^{20 }\left[\text{EHPs }{\text{cm}}^{-3}\right]\) are optically generated every second in the B-doped p-type Si and the lifetime of electron/hole to recombination, \({\tau }_{\text{n}}={\tau }_{\text{p}}=1\) [μs], at RT(300 K).

1. (a)

   What is the equilibrium concentration of minority carrier electrons?

2. (b)

   What are the steady-state concentrations of majority holes and minority electrons?

3. (c)

   The position of the equilibrium Fermi level of the electron, \({E}_{\text{F}}^{\text{p}}\), two quasi-Fermi levels of hole \(p{E}_{\text{F}}^{\text{p}}\), and electron \({nE}_{\text{F}}^{\text{p}}\) relative to the reference levels such as \({E}^{\text{in},\text{ p}}\), \({E}_{\text{c}}\) and \({E}_{\text{v}}\) on the energy band diagram.

4. (d)

   What, if any, mass action law holds? Reply why and discuss its electrochemical significance.

5. (e)

   What photo emf is expected to be developed regarding the counter metal anode when an inert single redox couple is thought to be present in the electrolytic solution?

The given data are as follows: \({n}_{\text{e}}^{\text{in}}={p}_{\text{h}}^{\text{in}}=1.5\times {10}^{10} \left[{\text{cm}}^{-3}\right]\); \({N}_{\text{c}}=2.8\times {10}^{19} \left[{\text{cm}}^{-3}\right]\); \({N}_{\text{v}}=1.04\times {10}^{19} \left[{\text{cm}}^{-3}\right]\); kT \(\cong 0.0259 \left[\text{eV}\right]\) at RT (300 K).

Answer 6

(a), (b), (c):

\({p}_{\text{h}}^{\text{p}}={10}^{15} \left[{\text{cm}}^{-3}\right]\), \(\delta n=\delta p={10}^{14} \left[{\text{cm}}^{-3}\right]\), \(\therefore,円 {p}_{\text{h}}^{\text{p},\text{ l}}={p}_{\text{h}}^{\text{p}}+\delta p=1.1\times {10}^{15} \left[{\text{cm}}^{-3}\right]\), following the mass action law, \({n}_{\text{e}}^{\text{p}}=\frac{\left({\left({n}_{\text{e}}^{\text{in}}\right)}^{2}\right)}{{p}_{\text{h}}^{\text{p}}}={10}^{14} \left[{\text{cm}}^{-3}\right]\), \(\therefore,円 {n}_{\text{e}}^{\text{p},\text{ l}}={n}_{\text{e}}^{\text{p}}+\delta n=2.0\times {10}^{13} \left[{\text{cm}}^{-3}\right]\)

1. 1.

   \(\left({E}^{\text{in},\text{ p}}-p{E}_{\text{F}}^{\text{p}}\right)=kT\text{ln}\frac{{p}_{\text{h}}^{\text{p},\text{ l}}}{{p}_{\text{h}}^{\text{in}}}=0.0259\text{ln}\frac{{p}_{\text{h}}^{\text{p},\text{ l}}}{{p}_{\text{h}}^{\text{in}}}=0.290[\text{eV}]>0\)

2. 2.

   \(\left(p{E}_{\text{F}}^{\text{p}}-{E}_{\text{v}}\right)=-kT\text{ln}\frac{{p}_{\text{h}}^{\text{p},\text{ l}}}{{N}_{\text{v}}}=-0.0259\text{ln}\frac{{p}_{\text{h}}^{\text{p},\text{ l}}}{{N}_{\text{v}}}=0.237[\text{eV}]>0\)

3. 3.

   \(\left({nE}_{\text{F}}^{\text{p}}-{E}^{\text{in},\text{ n}}\right)=kT\text{ln}\frac{{n}_{\text{e}}^{\text{p},\text{ l}}}{{n}_{\text{e}}^{\text{in}}}=0.0259\text{ln}\frac{{n}_{\text{e}}^{\text{p},\text{ l}}}{{n}_{\text{e}}^{\text{in}}}=0.228[\text{eV}]>0\)

4. 4.

   \(\left({E}_{\text{c}}-{nE}_{\text{F}}^{\text{p}}\right)=-kT\text{ln}\frac{{n}_{\text{e}}^{\text{p},\text{ l}}}{{N}_{\text{c}}}=-0.0259\text{ln}\frac{{n}_{\text{e}}^{\text{p},\text{ l}}}{{N}_{\text{c}}}=0.325[\text{eV}]>0\)

5. 5.

   \(\left({E}^{\text{in},\text{ p}}-{E}_{\text{F}}^{\text{p}}\right)=kT\text{ln}\frac{{p}_{\text{h}}^{\text{p}}}{{p}_{\text{h}}^{\text{in}}} =0.0259\text{ln}\frac{{p}_{\text{h}}^{\text{p}}}{{p}_{\text{h}}^{\text{in}}}=0.287\left[\text{eV}\right]>0\text{ in the dark}\)

All the data calculated are demonstrated in Fig. 6b.

1. (d)

   Regarding the mass action law: \({n}_{\text{e}}^{\text{p},\text{ l}}{p}_{\text{h}}^{\text{p},\text{ l}}=1.1\times {10}^{29}\left[{\text{cm}}^{-6}\right]\gg {n}_{\text{e}}^{\text{in}}{p}_{\text{h}}^{\text{in}}={\left({n}_{\text{e}}^{\text{in}}\right)}^{2}=2.25\times {10}^{20}\left[{\text{cm}}^{-6}\right].\) From the data obtained, first, it is recognized that the mass action law breaks. Second, unfortunately, there is no better solution to this issue. Nevertheless, we are of the opinion that Eqs. (20) to (23), used to determine the quasi-Fermi levels, urgently need to be modified by employing other assumptions or taking other boundary constraints.

2. (e)

   According to \(\left\{\frac{\left({nE}_{\text{F}}^{\text{p}}-{pE}_{\text{F}}^{\text{p}}\right)}{e}\right\}\ge {V}_{\text{oc}}^{\text{p},\text{ l}}>0\), the maximum plausible photo emf is 0.518 [V vs the counter metal anode].

Quiz 7

Conceptually construct the layout of the anode, electrolytic solution, and cathode of an ideal regenerative photovoltaic oxygen cell from relatively oxidizing single redox couple \({\text{O}}_{2}/{\text{H}}_{2}\text{O}\), using a negative inverse overvoltage for photosensitized anodic oxidation, driven by a photoexcited minority carrier hole in the n-type anode in combination with a counter metal cathode.

1. (a)

   Distinguish the anode and cathode and briefly clarify their related electrochemical reactions. Give the best candidate material for the counter cathode.

2. (b)

   Predict the photocurrent I vs photovoltage V curve using the concept of the quasi-Fermi level underlying the negative overvoltage, \({\eta }_{\text{h}}^{\text{n},\text{ l}}<0\), for the photoexcited anodic oxidation, caused by the VB minority holes in the n-type anode.

Answer 7

1. (a)

   The term "anode" originally comes from "anion." A cation is usually newly created as an oxidant at the anode/solution interface during oxidation in the dark by producing the redox electron left behind in the metal anode, and hence another partner "anion" there must compensate for the newly created cation, diffusing through the electrolyte from the counter cathode/electrolyte interface toward the anode/solution interface, to maintain the charge neutrality condition as a whole.

As an alternative, an anion is newly consumed as a reductant at the anode/solution interface during oxidation under illumination by consuming the photoexcited VB hole coming from the n-type anode interior. Hence, another partner "cation" must simultaneously be consumed at the counter metal cathode by the redox electron to compensate for the anion consumed.

Special attention needs to be paid to the two facts: (1) an oxidative operation of consuming a photoexcited VB hole during illumination is physically equivalent to another oxidative operation of producing a redox electron left behind in the metal anode in the dark; (2) on the whole, in any redox system, an electron or hole lost by the anode is simultaneously gained by the counter cathode.

In contrast, the term, "cathode" is named after "cation." A cation is usually consumed at the cathode/solution interface during reduction by either injecting the photoexcited CB electron coming from the p-type cathode surface or injecting the redox electron from the metal cathode surface in the dark. Therefore, the newly created cation must compensate for the "depleted cation," diffusing through the electrolyte from the counter anode/electrolyte interface to maintain the charge neutrality condition as a whole.

The overall cell reaction is divided into a partial anodic oxidation at the n-type anode/electrolyte interface and a partial cathodic reduction at the metal cathode/electrolyte interface, given by

respectively, giving the total overall reaction

The n-type anode under illumination is from the beginning compensated by the metal Ni anode usually used in the dark, having the best positive catalyst for electrolytic oxygen evolution, while the classical raney-Ag (an Ag–Al alloy) [12] is one of the best candidates for the counter metal cathode, having a relatively low overvoltage for cathodic oxygen reduction, following Eq. (25).

1. (b)

   Refer to Fig. 7b and related texts.

The redox couple \({\text{O}}_{2}/{\text{H}}_{2}\text{O}\) is shown to have moderate/poor reversibility [the charge transfer resistance, \({R}_{\text{ct}}=0\), an extremely low value, \({R}_{\text{ct}} \left[\text{Ohm }{\text{cm}}^{2}\right]=\frac{RT}{zF{i}_{0}}\), z \(=\) the oxidation number \(\left[\text{dimensionless}\right]\), \(F=\text{the Faraday constant}=\text{96,500 }\left[{\text{Cmol}}^{-1}\right]\) and the exchange current density (the rate of charge transfer is referred to as an idle rate at zero overvoltage), \({i}_{0}=\infty\), a considerably large value, \({i}_{0} \left[\text{A }{\text{cm}}^{-2}\right]=zF{k}_{\text{el}}\), \({k}_{\text{el}}=\text{the rate of electron transfer }\left[\text{mol }{\text{cm}}^{-2}{\text{ s}}^{-1}\right]\) are responsible for ideal reversibility], and it is expected to give a small exchange current and relatively high charge transfer resistance, accordingly delivering relatively high anodic/cathodic overvoltages on either side. Therefore, it cannot be used as a good reversible redox couple, unlike the reversible redox couple \({\left[{\text{Fe}}^{\text{III}}{\left(\text{CN}\right)}_{6}\right]}^{-3}\left(\text{ferricyanide as an oxidized form}\right)/{\left[{\text{Fe}}^{\text{II}}{\left(\text{CN}\right)}_{6}\right]}^{-4}\left(\text{ferrocyanide as a reduced form}\right)\left(\text{Pt}\right)\) on the Pt electrode, \({\text{Cl}}_{2}\)(the chlorine gas as an oxidant)/\({2\text{Cl}}^{-}\)(chloride as a reductant) couple, and the reversible hydrogen ion/gas redox couple. It is usual practice to write the oxidized species (oxidant) of the couple first [13].

We have conceptually designed a hypothetical regenerative photovoltaic oxygen cell, relative to an ideal regenerative hydrogen cell, like the Daniell cell, only for instructive use, not for a practical application.

To avoid any confusion, we will think of the reference potential of the negative (cathodic) and positive (anodic) overvoltages delivered or measured regarding \({V}_{\text{eq}}^{\text{redox}}\) [14] during any spontaneous electrochemical oxidation/reduction, respectively, in the dark; in comparison, that reference potential of the negative and positive inverse overvoltages (better called ‘photoexcited hole and electron overvoltages’) delivered regarding \({V}_{\text{fb}}^{\text{n}}/{V}_{\text{fb}}^{\text{p}}\) during photosensitized oxidation/reduction, respectively. Both reference potenials should be taken as the well-defined electrode potential, \({V}_{\text{eq}}^{\text{redox}}\), and the flat band potential, \({V}_{\text{fb}}^{\text{n}}/{V}_{\text{fb}}^{\text{p}}\), respectively, by definition.

The negative and positive overvoltages regarding their reference potential usually give the driving force for the occurrence of the cathodic reduction and anodic oxidation, respectively, in the dark. In comparison, the negative and positive inverse overvoltages occur during illumination as consequences of the single shift of their electrode potential to their flat band potential, which must necessarily be performed only by the photo-enhanced minority hole in the n-type and minority electron in the p-type, respectively, based on the band diagram. In this respect, neither any further negative overvoltage regarding \({V}_{\text{fb}}^{\text{n}}\), caused by the photo-enhanced minority hole in the n-type, nor any further positive overvoltage regarding \({V}_{\text{fb}}^{\text{p}}\), driven by the photo-enhanced minority electron in the p-type, which can make sense during illumination.

From our previous work [6] dealing with the relationship of a power producing dry photovoltaic cell at n/p junction to a power consuming n/p junction, it is recognized that the electric field strength present over the depleted space charge transition region is responsible for simultaneous migration (drift) of photoexcited minority EHP (electron and hole) across the transition region into the n-type and p-type regions, respectively, before recombining there and finally delivering a photocurrent and a photovoltage to the external load.

As shown in Fig 7a, a photosensitized anodic oxidation current at the n-type anode/solution interface, driven by a minority VB hole enhanced by illumination, simultaneously occurs only when an appreciable electric field exists across the transition region, along with a photo-sensitized cathodic reduction current at the counter metal cathode/solution interface, caused by the majority CB electron flowing from the n-type anode through the external load toward another counter metal cathode.

The same is true of a photo-sensitized cathodic reduction current at the p-type cathode/solution interface, as depicted in Fig 8a.

No photocurrent appears at the flat band potential, \({V}_{\text{fb}}\), because the electric field that is necessarily required to separate the photoexcited EHP, finally delivering a photo reaction current and a photovoltage, is absent. The photocurrent appears at any other potentials up/down to \({V}_{\text{eq}}^{\text{redox}}\) compared with the \({V}_{\text{fb}}\); hence, the \({V}_{\text{fb}}\) can be interpreted as the unique potential for photocurrent onset.

Therefore, it follows that photosensitized anodic oxidation, driven by the minority VB hole of the n-type anode, enhanced by illumination, may appear at potentials from \({V}_{\text{fb}}^{\text{n}}\) up to \({V}_{\text{eq}}^{\text{redox}}\text{ or }{V}_{\text{corr}}\) in a more positive direction of potential (Figs. 7b, 9b and 13c), relative to photosensitized cathodic reduction, caused by the minority CB electron of the p-type cathode, enhanced by illumination at potentials from \({V}_{\text{fb}}^{\text{p}}\) down to \({V}_{\text{eq}}^{\text{redox}}\) (Figs. 8b and 10b) at which the occurrence of those anodic oxidations and cathodic reductions is thermodynamically impossible in the dark.

The photo-enhanced minority hole necessarily needs to shift the equilibrium potential of the redox couple to the flat band potential in the negative direction (the negative inverse overvoltage, \({\eta }_{\text{h}}^{\text{n},\text{ l}}<0\), for photosensitized anodic oxidation) based on the band diagram as suggested in Answer 5(e) to Quiz 5 and in Fig. 6a. This shift makes it thermodynamically possible for oxygen evolution to proceed above the flat band potential by anodic transfer of the photoexcited hole, whose occurrence is thermodynamically impossible in the dark. Thus, the photo-I vs V curve in this cell behaves as those of the self-driven Galvanic cell, for instance, the Daniell cell and hydrogen/oxygen fuel cell, etc., as indicated in Fig. 7b. The same is true of the positive inverse overvoltage, \({\eta }_{\text{e}}^{\text{p},\text{ l}}>0\), for cathodic reduction, driven by a photoexcited minority electron, as mentioned in Fig. 8b.

Notably, there are two kinds of sign conventions concerning the anodic oxidative current/cathodic reductive current: one is IUPAC (European) convention 1953, saying that an oxidative current is negative owing to the loss of a redox electron, while a reductive current is positive owing to the gain of a redox electron. In this respect, remember the acronym 'OIL RIG', which spells out the initial letters of 'oxidation is loss, reduction is gain' of electrons. Another is the non-IUPAC convention, saying that following the traditional definition of the anodic overvoltage being positive regarding the electrode potential \(\left[{\eta }_{\text{anod}}=\left({V}_{\text{delivered}}-{V}_{\text{eq}}^{\text{redox}}\right)>0\right]\), and the cathodic overvoltage being negative regarding the electrode potential \(\left[{\eta }_{\text{cathod}}=\left({V}_{\text{delivered}}-{V}_{\text{eq}}^{\text{redox}}\right)<0\right]\); the oxidative current and reductive current should be positive and negative, respectively, so that their product, power \(\left(W=\eta \times I>0\right)\), never should always be negative (the second principle of electrochemical thermodynamics [15]). The latter non-IUPAC convention is adopted throughout the article.

In Fig. 7a at equilibrium, namely after dipping the electrode into the electrolytic solution containing the single redox couple, \({\text{O}}_{2}/{\text{H}}_{2}\text{O}\), the alignment of the Fermi level on the n-type anode, electrolyte, and the counter metal cathode gives rise to a downward concave band bending in the direction of the interior of the n-type. Under illumination, photo-enhanced minority holes having quasi-Fermi level,\({pE}_{\text{F}}^{\text{n}}\), going upward along the concave-up band bending from the n-type toward the interface between the n-type and electrolyte, drive an anodic oxidation at the n-type electrode on the left side like in Eq. (24).

In contrast, the majority electrons enhanced by illumination with their quasi-Fermi level, \({\text{nE}}_{\text{F}}^{\text{n}}\), coming downward along the concave-up band bending from the n-type through the external load toward another counter metal cathode, drive a cathodic reduction at the counter metal electrode on the right side like in Eq. (25).

This simultaneous oxidation/reduction processes produce a photocurrent as well as a photovoltage despite zero electrochemical emf (Gibbs free energy change, \(\Delta G=-zF\times \text{emf}=0\), \(\text{emf}=0 \left[\text{V}\right]\)) with no net chemical change, thus providing the photocurrent for the external load. The separation of the two quasi-Fermi levels, \(\left({nE}_{\text{F}}^{\text{n}}-{pE}_{\text{F}}^{\text{n}}\right)\), divided by the electronic charge is the limiting photo emf.

Notably, the separation of the quasi-Fermi levels, \(\left(n{E}_{\text{F}}^{\text{n}}-p{E}_{\text{F}}^{\text{n}}\right)\text{ in }\left[\text{eV}\right]\), in the n-type anode under illumination is responsible for generation of a negative photovoltage (compared to the counter cathode, \({V}_{\text{oc}}^{\text{n},\text{ l}}<0 \text{ in }\left[\text{V }vs\text{ cathode}\right]\)). At equilibrium in the dark, this photovoltage is zero. This negative photovoltage provides the driving force for the photo-generated current. At the same time, this quasi-Fermi level separation also causes a shift in the negative inverse overvoltage, \({\eta }_{\text{h}}^{\text{n},\text{ l}}<0 \text{ in }\left[\text{V}\right],\) at the n-type anode/solution interface. Like the photovoltage, this overvoltage is zero at equilibrium in the dark. Under illumination, however, it enables photosensitized anodic oxidation, driven by photoexcited minority holes. This behavior is a fundamental feature of n-type-based photovoltaic cells and photo-electrochemical cathodic protection cells, as illustrated in Fig. 7, and further discussed in Figs. 9 and 13.

Quiz 8

Conceptually construct the layout of the anode, electrolytic solution, and cathode of an ideal regenerative photovoltaic hydrogen cell from a relatively reducing single redox couple \({2\text{H}}^{+}/{\text{H}}_{2}\), aided by a positive inverse overvoltage for photosensitized cathodic reduction, caused by a photoexcited minority carrier electron in the p-type combined with a counter metal electrode.

1. (a)

   Distinguish the anode and cathode and clarify their related electrochemical reactions briefly. Give the best candidate material for the counter anode.

2. (b)

   Predict the photocurrent I vs photovoltage V curve using the concept of the quasi-Fermi level underlying the positive overvoltage, \({\eta }_{\text{e}}^{\text{p},\text{ l}}>0\) for the photoexcited cathodic reduction, driven by the VB minority electron in the p-type.

Answer 8

1. (a)

   The overall cell reaction is divided into a partial anodic oxidation at the metal anode/electrolyte interface and a partial cathodic reduction at the p-type anode/electrolyte interface, given by

   $${\text{H}}_{2}+2\text{h}\left({E}_{\text{F}}^{\text{M}}\right)={2\text{H}}^{+ },\text{ anodic oxidation}$$

   (27)
   $${2\text{H}}^{+ }+2\text{e}\left(\text{CB}\right)={\text{H}}_{2},\text{ cathodic reduction},$$

   (28)

   respectively, giving the total overall reaction

   $$\text{e}\left(\text{CB}\right)+\text{h}\left({E}_{\text{F}}^{\text{M}}\right)=0,\text{ recombination }\left(\text{overall cell rx}\right),$$

   (29)

   where the electron acts as a minority \(\text{e}\left(\text{CB}\right)\) in the p-type during the migration (drift) over the space charge transition region, and it also functions as a redox electron during cathodic reduction; the hole \(\left({E}_{\text{F}}^{\text{M}}\right)\) usually acts as a conduction electron in the current collector, and it also acts as a redox electron (flow of the hole in the opposite direction of the redox electron) during anodic oxidation.

From the beginning, the p-type cathode under illumination compensated for the metal Pt cathode usually used in the dark, having the best positive catalyst for electrolytic hydrogen evolution, while the classical raney-Ni (an Ni–Al alloy) [12] is one of the best candidates for the counter metal anode, having a relatively low overvoltage for anodic hydrogen ion oxidation, following Eq. (27).

1. (b)

   Refer to Fig. 8b and related texts.

The redox couple \({2\text{H}}^{+}/{\text{H}}_{2}\) has excellent reversibility (\({R}_{\text{ct}}=0\) and \({i}_{0}=\infty\) are responsible for ideal reversibility), and it is expected to give a large exchange current \({i}_{0}\) and an extremely low charge transfer resistance \({R}_{\text{ct}}\), accordingly delivering extremely low anodic/cathodic overvoltages on either side. Therefore, it can be used as a good reversible redox couple, like the reversible redox couple, ferricyanide/ferrocyanide \(\left(\text{Pt}\right)\) on Pt electrode, as well as the chlorine gas/chloride redox couple in electrochemical perspectives, but it is not usually utilized for a practical application for economical reasons [13].

The photo-enhanced minority electron necessarily needs to shift the equilibrium potential of the redox couple to the flat band potential in the positive direction (the positive inverse overvoltage, \({\eta }_{\text{e}}^{\text{p},\text{ l}}>0\), for photosensitized cathodic reduction) based on the band diagram, as suggested in Answer 6(e) to Quiz 6 and in Fig. 6b. This shift makes it thermodynamically possible for hydrogen evolution to proceed below the flat band potential by cathodic transfer of photoexcited electrons; the occurrence is thermodynamically impossible in the dark. Thus, the photo-I vs V curve in this cell behaves as those of the self-driven Galvanic cell, for instance, the Daniell cell and hydrogen/oxygen fuel cell, etc., as indicated in Fig. 8b.

In Fig. 8a at equilibrium, namely after dipping the electrode into the electrolytic solution containing the single redox couple,\({2\text{H}}^{+}/{\text{H}}_{2}\), the alignment of the Fermi level on the counter metal electrode, electrolyte, and p-type cathode gives rise to an upward convex band bending in the direction of the interior of the p-type. Under illumination, photo-enhanced minority electrons having quasi-Fermi level, \({nE}_{\text{F}}^{\text{p}}\), coming downward along the concave-down band bending from the p-type toward the interface between the p-type and electrolyte, drive a cathodic reduction at the p-type cathode on the right side like in Eq. (28).

In contrast, majority holes enhanced by illumination with their quasi-Fermi level, \({pE}_{\text{F}}^{\text{p}}\), going upward along the concave-down band bending from the p-type through the external load toward another counter metal anode, drive an anodic oxidation at the counter metal anode on the left side like in Eq. (27).

These simultaneous oxidation/reduction processes produce a photocurrent and a photovoltage despite zero electrochemical emf (Gibbs free energy change, \(\Delta G=-\text{zF}\times \text{emf}=0\), \(\text{emf}=0 \left[\text{V}\right]\)) with no net chemical change, thus providing the photocurrent for the external load. The separation of the two quasi-Fermi levels, \(\left\{\left({nE}_{\text{F}}^{\text{p}}-{pE}_{\text{F}}^{\text{p}}\right)>0\right\}\), divided by electronic charge is the limiting photo emf.

It seems adequate to mention that the separation of the quasi-Fermi levels, \(\left\{\left({nE}_{\text{F}}^{\text{p}}-{pE}_{\text{F}}^{\text{p}}\right)>0\right\}\text{ in }\left[\text{eV}\right]\), in the p-type cathode under illumination is responsible for providing the positive photovoltage regarding the counter anode, \({V}_{\text{oc}}^{\text{p},\text{ l}}>0 \text{ in }\left[\text{V }vs\text{ anode}\right]\), essential for the driving force of the photocurrent, and it is also responsible for the shift of positive inverse overvoltage, \({\eta }_{\text{e}}^{\text{p},\text{ l}}>0 \text{ in }\left[\text{V}\right]\), for photosensitized cathodic reduction at the p-type cathode/solution interface, driven by a photoexcited minority electron, which is a common characteristic of p-type-based photovoltaic cells, as shown in Fig. 8, mentioned below in Figs. 10 and 11 as well.

In comparison, the migration (drift) of minority EHP enhanced by illumination is responsible for the separation of the Fermi levels on either side, \(\left({E}_{\text{F}}^{\text{n}}-{E}_{\text{F}}^{\text{p}}\right)\approx \left(n{E}_{\text{F}}^{\text{n}}-p{E}_{\text{F}}^{\text{p}}\right)>0\text{ in }\left[\text{eV}\right]\), along the concave-up bending/concave-down bending across the depleted space charge transition region into the n-type and p-type regions, respectively, before recombination there, without any electrochemical reaction, eventually delivering a photocurrent and a positive photovoltage regarding the n-type negative pole region, \({V}_{\text{oc}}^{\text{p},\text{ l}}>0 \text{ in }\left[\text{V }vs\text{ n type negative pole}\right]\), which is characteristic of the dry photovoltaic cell at the n/p junction, as shown in Fig. 12.

The system or device is called a photovoltaic cell when it converts light quanta photon energy into electrical energy, which is a kind of energy producer. It usually involves a single redox reaction of such a single redox couple as \({\text{Fe}}^{3+}\left(\text{ferricyanide}\right)/{\text{Fe}}^{2+}\left(\text{ferrocyanide}\right)\) or \({\text{Ce}}^{4+}/{\text{Fe}}^{3+}\), or \({\text{I}}_{3}^{-}/{3\text{I}}^{-}\), having good reversibility and being composed of a partial photosensitized anodic oxidation by a photoexcited minority VB hole at the n-type anode/electrolytic solution interface and another partial photosensitized cathodic reduction of photoexcited minority CB electron at the p-type cathode/electrolytic solution interface, as shown in Eqs. (30) and (31), respectively,

respectively, giving the total overall reaction

where \(\text{Red}\) means the reduced species of the redox couple, \(\text{Ox}\) is the oxidized species of the redox couple, and no net chemical change occurs in the electrolytic solution.

Notice that the electron acts as a minority CB electron in the p-type during the migration (drift) over the space charge transition region, and it also functions as a redox electron during cathodic reduction. The hole acts as a minority VB hole in the n-type during the migration across the space charge transition region, and it also acts as redox electron (flow of the hole in the opposite direction of the redox electron) during the anodic oxidation. In this cell, photovoltage is generated between two electrodes, and the produced photocurrent is accordingly provided through the external circuit.

Figure 9 shows the working mechanisms of a photovoltaic cell based on the energy band diagram from the thermodynamic point of view, and based on photo-I vs V polarization characteristic in the kinetic aspect. Formally replenishing one single redox couple, \({\text{O}}_{2}/{\text{H}}_{2}\text{O}\), given in Fig. 7 (photovoltaic oxygen cell) with another redox couple in general, for instance, ferric ion/ferrous ion couple, gives another regenerative photovoltaic cell composed of the n-type anode and counter metal cathode, shown in Fig. 9.

The common characteristic feature of the origin of photo emf generation on the energy band diagram in both n-type-based photovoltaic cells is that splitting the Fermi level into two quasi-Fermi levels is responsible for development of the photo emf. This provides the driving force for photosensitized anodic oxidation by the minority photoexcited hole at the interface between the n-type anode and electrolytic solution containing the redox couple concerned. The other common I-V polarization characteristics of both cells are that the negative shift of the equilibrium potential of the redox couple to the flat band potential, caused by the negative inverse overvoltage, makes it possible for photosensitized anodic oxidation to proceed in the interface between the n-type and redox couple concerned in the electrolytic solution.

In particular, as seen in Fig. 9b, the photovoltage \({V}^{\text{l}}\) operates between \({V}_{\text{fb}}^{\text{n}}\) and \({V}_{\text{eq}}^{\text{redox}}\) in the closed cell, ranging between the open circuit voltage \(\left({V}_{\text{eq}}^{\text{redox}}-{V}_{\text{fb}}^{\text{n}}\right)\) on open circuit \(\left(\left|{I}_{\text{e}}^{\text{l}}\right|={I}_{\text{h}}^{\text{l}}=0\right)\) and the short circuited voltage \({V}_{\text{sc}}^{\text{l}}=0\) at the short-circuit \(\left({I}_{\text{sc}}^{\text{l}}=\left|{I}_{\text{e}\left(\text{max}\right)}^{\text{l}}\right|={I}_{\text{h}\left(\text{max}\right)}^{\text{l}}\right)\), which is different from the short-circuit current, \(\left({I}_{\text{sc}}^{\text{l}}=\left|{I}_{\text{e}\left(\text{max}\right)}^{\text{l}}\right|+{I}_{\text{h}\left(\text{max}\right)}^{\text{l}}\right)\), which is valid for the dry photovoltaic cell, as given in Fig. 12a–d. The higher the photocurrent is to finally approach the short-circuited current, the narrower the operating photovoltage becomes to eventually approach the short-circuited voltage \({V}_{\text{sc}}^{\text{l}}=0\). A finite photovoltage is developed between the two electrodes by a common finite current \(\left(\left|{I}_{\text{e}}^{\text{l}}\right|={I}_{\text{h}}^{\text{l}}\ne 0\right)\), mainly externally delivered, but an extremely small portion is consumed to cause both anodic oxidation and cathodic reduction at either electrode surface.

The finite current produced nearly equals either the anodic oxidation current or cathodic reduction current, having a negative sign regarding the oxidative current, being chosen as the reference direction, as given in Fig. 9b. This predicted I–V characteristic is qualitatively justified by I–V polarization characteristics measured by different authors, which are systematically summarized in a regenerative photovoltaic cell [16]. In contrast, the dry n/p junction photovoltaic cell externally produces the total current, consisting mainly of both migration currents by photoexcited minority electron and hole, instead of either oxidative current or reductive current, as indicated in Fig. 12b–d.

The same cell configuration as Fig. 9 may be employed as a photo-electrolytic cell (optical energy conversion into chemical free energy for an endoergic reaction), which is labeled a Schottky type photochemical diode [17,18,19].

The working mechanisms of other photovoltaic cell are shown in Fig. 10 based on the energy band diagram from the thermodynamic point of view, and based on the photo-I vs -V polarization characteristic in the kinetic aspect. Formally compensating another redox couple in general, for instance, ferric ion/ferrous ion couple for one single redox couple, \({2\text{H}}^{+}/{\text{H}}_{2}\), given in Fig. 8 (photovoltaic hydrogen cell), gives other regenerative photovoltaic cell composed of the counter metal anode and p-type cathode, shown in Fig. 10.

Concerning the origin of the developed photo emf, the energy band diagram in both p-type-based regenerative cells does not change thermodynamically. The common I–V polarization characteristics of both cells include the positive shift of the equilibrium potential to the flat band potential, caused by the positive inverse overvoltage, allows photosensitized cathodic reduction to proceed in the interface between the p-type and redox couple concerned in the electrolytic solution.

Figure 11 presents the working principle of other photovoltaic cells based on the energy band diagram from the thermodynamic perspective, and based on the photo-I vs -V polarization characteristic in the kinetic aspect. Formally substituting another p-type cathode for one counter metal cathode, given in Fig. 9 (photovoltaic cell), gives a fourth regenerative photovoltaic cell composed of the n-type anode and p-type cathode, shown in Fig. 11.

One specific characteristic of both the n-type and p-type-based regenerative photovoltaic cell types is that the photo emf becomes much wider than in the n-type-based cell type (Fig. 9), attributable to the combined separations of the Fermi level in both n-type and p-type between the two respective quasi-Fermi levels. This provides the driving force for photosensitized anodic oxidation by the minority photoexcited hole at the interface between the n-type anode and the redox couple concerned, and simultaneously for the photosensitized cathodic reduction by the minority photoexcited electron at the interface between the p-type cathode and the redox couple concerned.

Another common I–V polarization characteristic of this cell type is that the negative shift of the equilibrium potential to the flat band potential, made by the negative inverse overvoltage and simultaneously the positive shift, made by the positive inverse overvoltage, make it possible for photosensitized anodic oxidation and cathodic reduction to proceed in one interface between the n-type anode and a redox couple in the electrolytic solution, in another interface between the p-type cathode and the same redox couple as at the anode/solution interface.

The same cell configuration as Fig. 11 may be employed as a photo-electrolytic cell (optical energy conversion into chemical free energy for an endoergic reaction), which is labeled a double semiconductor type (an n-p type) photo-chemical diode [17,18,19].

Figure 12 shows the working principle of a dry photovoltaic cell at the n/p junction [20, 21] based on the energy band diagram from the thermodynamic point of view and based on the non-linear photo-I vs V characteristic of the illuminated n/p junction diode derived from the modified Fick’s second law [6] in the kinetic aspect.

Judging from the migration (or drift) of minority carrier EHPs photo-enhanced during illumination across the electron- and hole-depleted region into the n-type and p-type regions, respectively (Fig. 12a and b), it is conceivable that the photoexcited electron and hole (acting as a thermodynamic system) can convert the light quanta photon energy of the difference between two quasi-Fermi energies without an intermediate stage of participating in the photosensitized electrochemical (anodic/cathodic) reactions with any redox couple directly into the electrical work on the surroundings (external circuit).

As shown in Fig. 12a–d, one conspicuous characteristic feature of this dry cell is that the photoexcited minority electron and hole contribute to the overall current \(\left(\left|{I}_{\text{e}}^{\text{l}}\right|+{I}_{\text{h}}^{\text{l}}\right)\) (chosen hole migration direction as the reference) delivered externally, without participating in any anodic oxidation and cathodic reduction, instead of either reductive partial current, \(\left|{I}_{\text{e}}^{\text{l}}\right|\) by photoexcited CB electron or oxidative partial current, \({I}_{\text{h}}^{\text{l}}\) by photoexcited VB hole, under the presence of any redox couple, possibly resulting in any of other wet photovoltaic cells.

At equilibrium in the dark, the alignment of the Fermi level \({E}_{\text{F}}\) on either side no longer gives any driving force for the diffusion and migration of the minority carrier electrons and holes (Fig. 12a). Figure 12b shows that, under illumination, the minority electron and hole, enhanced by illumination in the depleted space transition region, migrate (or drift) along the downward concave and upward convex band bending, into the n-type and p-type sides, respectively, before recombining there. This migration process causes the rise and reduction in \({E}_{\text{F}}\), on the n-type and p-type sides, respectively, meaning the separation of \({E}_{\text{F}}\) between the two phases, i.e., the n-type and p-type sides.

This must be distinguished from that separation of \({E}_{\text{F}}\) between the two quasi-Fermi levels within a single phase, either the n-type or p-type side under illumination, but fortunately, the two separations produce nearly the same photo emf. Therefore, this separation of \({E}_{\text{F}}\) on either the n-type or p-type side, made by migration of the minority electron and hole into the n-type and p-type, respectively, delivers a photocurrent as well as a photovoltage at the open circuit, this is given as \(\frac{\left({nE}_{\text{F}}^{\text{n}}-{pE}_{\text{F}}^{\text{p}}\right)}{e}\approx \frac{\left({E}_{\text{F}}^{\text{n}}-{E}_{\text{F}}^{\text{p}}\right)}{e}\ge {V}_{\text{oc}}^{\text{p},\text{ l}}>0\).

Namely, the difference in \({E}_{\text{F}}\) will get back on track between the n-type and p-type before contact, which approximately equals the difference between the quasi-Fermi level of the majority electron in the n-type and the quasi-Fermi level of the majority hole in the p-type, divided by electronic charge (see also Fig. 11a). The positive sign of the open-circuit photovoltage of the cell system counts regarding the n-type negative pole region. The photo emf cannot exceed the reversible contact potential [9], being similar to the electrochemical back emf. Therefore, the contact potential behaves as "the hypothetical potential reservoir (source)."

There are two kinds of junctions: one is a power-consuming rectifying n-p junction that includes two types: first, the negative bias; second, the positive bias; third is a power-producing photovoltaic cell. The non-linear I vs V characteristic curve for negative bias, positive bias being in contact with external current sources, and the photovoltaic cell being in contact with external load, can be assigned to the third, first, and fourth quadrant, respectively, as shown in Fig. 12c.

Notably, the delivered photocurrent flows in the same direction as in the negative biased current, namely, from the n-type region through the transition region to the p-type, while the delivered photovoltage runs in the same direction as in the positive bias, namely from the p-type through the external load to the n-type. Therefore, we call such an n-p junction a power-producing dry photovoltaic cell.

In particular, inverting the I–V curve in the fourth quadrant for convenience of the demonstration, we get a typical non-linear I vs V curve (Fig. 12d) for all the photovoltaic cells, from which such characteristic parameters as the open circuit voltage (photo emf), \({V}_{\text{oc}}^{\text{l}}\) and the short-circuit current, \(\left({I}_{\text{sc}}^{\text{l}}=\left|{I}_{\text{e}\left(\text{max}\right)}^{\text{l}}\right|+{I}_{\text{h}\left(\text{max}\right)}^{\text{l}}\right)\), the maximum power, and efficiency of the gained electrical energy can be determined for a given light illumination level. Notice that the short-circuit current given in this dry cell (derived from Fig. 12b, c and d) differs from that short-circuit current in the wet cell mentioned in Fig. 9b.

The junction at \({I}_{\text{sc}}^{\text{l}}\) is said to be short-circuited, while the junction at \({V}_{\text{oc}}^{\text{l}}\) is said to be on an open circuit. Thus, the photo-driven voltage, \({V}^{\text{l}}\), operates between \({V}_{\text{sc}}^{\text{l}}=0\) and \({V}_{\text{oc}}^{\text{l}}={V}_{\text{emf}}^{\text{l}}\) in the external closed circuit. A finite photovoltage is developed between the two electrodes by a finite sum of migration current of the photoexcited minority electron and hole mainly delivered externally (the direction of the migration current of the hole being chosen as the reference). The maximum power, \({I}_{\text{max}}^{\text{l}}{V}_{\text{max}}^{\text{l}}\), is absolutely less than the \({I}_{\text{sc}}^{\text{l}}{V}_{\text{oc}}^{\text{l}}\) product. The ratio of \({I}_{\text{max}}^{\text{l}}{V}_{\text{max}}^{\text{l}}\) to \({I}_{\text{sc}}^{\text{l}}{V}_{\text{oc}}^{\text{l}}\) is called the fill factor, implying a figure of merit for a solar cell design.

Notably, the presence of a depleted space charge region at an n/p junction is responsible for the rectifying behavior, which is characteristic of the semiconducting electrode used for all the photovoltaic cells for generation of photocurrent/voltage in contrast to the Ohmic behavior, being valid for the presence of an enriched space charge region as well as metal-metal contact.

The system or device is termed a photo-electrolytic cell when it converts light quanta photon energy into chemical free energy, i.e., a kind of chemical material producer. The photo-electrochemical cathodic protection cell is a kind of photo-electrolytic cell, where the overall cell reaction consists of a partial anodic oxidation by a photo-excited minority VB hole at the n-type anode and another partial photo-sensitized cathodic reduction of the redox electron at the counter steel cathode, as shown in Eqs. (33) and (34), respectively,

respectively, giving the total overall reaction

where the two electrodes are short-circuited and the real operating photovoltage is so small that no significant electric power may be produced.

Figure 13 explains the working principle of a photo-electrochemical cathodic protection short-circuited cell based on the energy band diagram and photo-I vs -V polarization curve. Formally replacing the counter metal cathode, a single redox couple containing solution, and external load with a relatively high inner resistance, given in Fig. 7 (photovoltaic cell) with the counter steel cathode, an electrolytic solution composed of both the anolytic solution and catholyic solution, separated by a semi-permeable membrane, and finally a short circuit, respectively, gives a photo-electrochemical cathodic protection cell, shown in Fig. 13.

However, both energy conversion systems have different characteristics, namely, from the thermodynamic view point, a photo-enhanced electron and hole acting as a system in the former perform useful, power-producing work in the surroundings (through external load), providing an electric energy producer. However, these photo-enhanced electrons and holes in the latter perform useful, power-consuming work on the surroundings (via redox couple), giving a chemical material producer.

When the cell circuit is closed in the dark, as shown in Fig. 13a, the equality of the Fermi level between the n-type anode and counter steel cathode develops the downward concave band bending toward the interior of the n-type bulk (or concave-up band bending toward the surface) over the electron-depleted space charge region. At equilibrium in the dark, however, neither the oxygen evolution associated with the anodic hole transfer at n-type anode nor hydrogen evolution (cathodic reduction of hydrogen ion), including the reduction of ferrous ions to metallic steel (cathodic protection of steel) associated with the cathodic electron transfer at the counter steel cathode, is thermodynamically expected to occur for three reasons. First, the Fermi level of the n-type is higher than the Fermi level of the redox couple, \({\text{O}}_{2}/{\text{H}}_{2}\text{O}\). Second, the concentration of the minority hole is too small. Finally, the Fermi level of the counter steel cathode, \({E}_{\text{F}}^{\text{steel}}\), is lower than the Fermi level corresponding to the corrosion potential or mixed potential, \({V}_{\text{corr}}\).

Notably, the corrosion potential is an irreversible, non-equilibrium potential at steady state unlike the thermodynamically well defined unique equilibrium electrode potential. The mixed potential is thought to be formed by "mixing" the respective electrode potentials of the two redox couples, coupled by a couple of common currents and potential, which are called the corrosion current and potential; the latter is named the "mixed potential" [14]. As in the present case, an anodic dissolution of steel associated with the redox couple, \({\text{Fe}}^{2+}/\text{Fe}\), coupled with a cathodic hydrogen reduction associated with the redox couple, \({2\text{H}}^{+}/{\text{H}}_{2}\), is thought to be short-circuited on the common surface of steel. The corrosion potential can be treated as a quasi-equilibrium potential close to the electrode potential.

We notice that a clear distinction has been made by such German electrochemists as Vetter [22] and Kortuem [23] between the terms "overvoltage," translated from German "die Ueberspannung," and "polarization," translated from German "die Polarisation" [14]: the former refers to a deviation from the equilibrium electrode potential, while the latter refers to a departure from the corrosion (mixed) potential, in an attempt to reveal the physical significance of the mixed potential. According to German electrochemistry convention, the cathodic overvoltage means a correspondingly negative departure from \({V}_{\text{eq}}^{{2\text{H}}^{+}/{\text{H}}_{2}}\) on the photo- I vs V curve, i.e., a region above \({E}_{\text{F}}^{{2\text{H}}^{+}/{\text{H}}_{2}}\) on the energy band diagram, while cathodic polarization implies a negative deviation from \({V}_{\text{corr}}\), i.e., a region above \({E}_{\text{F}}^{\text{corr}}\) as indicated in Fig. 13b and c. In comparison, the three terms, "overvoltage," "overpotential," and/or "polarization," are used in common without any differentiation in the Anglo-American corrosion and electrochemistry community, defined as the deviation from the equilibrium state or steady-state.

When the n-type anode is photoexcited, as shown in Fig. 13b, the Fermi level of the anode goes upward (the potential of the anode is lowered) because of the negative inverse overvoltage being equivalent to light quanta photon energy gain, i.e., photo emf, and at the same time the Fermi level of the counter steel cathode, being short-circuited with the n-type anode, also goes upward to be higher than that Fermi level of corrosion as well as that of the redox couple, \({2\text{H}}^{+}/{\text{H}}_{2}\).

This shift of the Fermi level makes it thermodynamically possible for hydrogen evolution as well as cathodic protection of steel to proceed by cathodic transfer of the redox electron at the steel cathode [14]. When the Fermi level of steel, \({E}_{\text{F}}^{\text{steel}}\), is below the Fermi level \({E}_{\text{F}}^{\text{corr}}\) corresponding to corrosion potential, \({V}_{\text{corr}}\), it is implausible for cathodic electron transfer to occur. In contrast, if the Fermi level of steel, \({E}_{\text{F}}^{\text{steel}}\), is above the Fermi level \({E}_{\text{F}}^{\text{corr}}\) corresponding to corrosion potential, \({V}_{\text{corr}}\), it is possible for cathodic electron transfer to proceed to generate hydrogen and simultaneously optically to protect steel from corrosion attack.

Furthermore, photo-excitation increases the concentration of the minority hole at the n-type anode and hence shifts the quasi-Fermi level of the hole to a level lower than that of the redox couple, \({\text{O}}_{2}/{\text{H}}_{2}\text{O}\). Thus, the quasi-Fermi level provides the thermodynamic backgrounds for the oxygen evolution by anodic hole transfer at the n-type anode and hydrogen evolution including cathodic protection of steel by the cathodic electron transfer at the counter steel cathode/NaCl-containing solution interface.

It seems worthwhile to give a short comment on the sign of photovoltage. The real photovoltage is given by \(-\frac{\left({nE}_{\text{F}}^{\text{n}}-{E}_{\text{F}}^{\text{corr}}\right)}{e}\le {V}_{\text{oc}}^{\text{n},\text{ l}}<0\), as shown in Fig. 13b. The negative sign counts regarding the n-type anode as suggested in answer 5(e) to quiz 5 and in Fig. 6a. Concerning the negative sign of photovoltage, it is exactly the same as for the n-type-based photovoltaic cell, but this material producer system is by nature quite different from that energy producer system.

Subtracting all the overvoltages including iR drops from the negative inverse overvoltage, driven by the photoexcited minority hole, i.e., the photo emf, \({V}_{\text{oc}}^{\text{l}}\), we get the real operating photovoltage \({V}^{\text{l}}\), ranging between the short circuited voltage \({V}_{\text{sc}}^{\text{l}}=0\) at short circuit and the open circuit voltage \(\left({V}_{\text{corr}}-{V}_{\text{fb}}^{\text{n}}\right)\) on open circuit \(\left(\left|{I}_{\text{e}}^{\text{l}}\right|={I}_{\text{h}}^{\text{l}}=0\right)\). A finite photovoltage develops between the two electrodes by a common finite current externally, being mainly consumed to drive both anodic oxidation and cathodic reduction at either electrode surface for production of chemical materials, such as oxygen and hydrogen, and protection of steel from the corrosion attack (reduction of steel), as shown in Fig. 13b and c.

One characteristic feature of photo I vs V polarization curves, predicted in Fig. 13c, is that the lower the photocurrent is, the wider the operating photovoltage in value. This characteristic is qualitatively evidenced by some data of the photocurrent/voltage measured in the photoelectrochemical cathodic protection cell [24, 25]. This is also justified by the cell performance characteristics of photo-electrolytic cells measured in most of papers, in particular in the cell composed of the n-type \({\text{TiO}}_{2}\) anode and the p-type \(\text{GaP}\) cathode [26].

From our thoughts and deductions derived from this article, we briefly summarize that an important relationship underlying the working principles of the photovoltaic cell and photo-electrochemical cathodic protection cell has been well established of the two separate quasi-Fermi levels ( aligned the two quasi-Fermi levels at equilibrium in the dark), \(\left(n{E}_{\text{F}}^{\text{n}}-p{E}_{\text{F}}^{\text{n}}\right)>0\text{ in }\left[\text{eV}\right]/\left(n{E}_{\text{F}}^{\text{p}}-p{E}_{\text{F}}^{\text{p}}\right)>0\text{ in }\left[\text{eV}\right]\), to the generation of the negative/positive photo emf (zero photo emf at equilibrium in the dark), \({V}_{\text{oc}}^{\text{n},\text{ l}}<0 \text{ in }\left[\text{V}\right]/{V}_{\text{oc}}^{\text{p},\text{ l}}>0 \text{ in }\left[\text{V}\right]\), and the occurrence of the negative/positive inverse overvoltage (zero inverse overvoltage at equilibrium in the dark), \({\eta }_{\text{h}}^{\text{n},\text{ l}}=\left({V}_{\text{fb}}^{\text{n}}-{V}_{\text{eq}}^{\text{redox}}\right)<0 \text{ in }\left[\text{V}\right]/{\eta }_{\text{e}}^{\text{p},\text{ l}}=\left({V}_{\text{fb}}^{\text{p}}-{V}_{\text{eq}}^{\text{redox}}\right)>0 \text{ in }\left[\text{V}\right]\) for photosensitized anodic oxidation/cathodic reduction caused by photoexcited minority holes/electrons, respectively, which is closely associated with one another among the three parameters, behaving as cause and effect.

The separation of the quasi-Fermi levels by illumination, \(\left(n{E}_{\text{F}}^{\text{n}}-p{E}_{\text{F}}^{\text{n}}\right)>0\text{ in }\left[\text{eV}\right]/\left(n{E}_{\text{F}}^{\text{p}}-p{E}_{\text{F}}^{\text{p}}\right)>0\text{ in }\left[\text{eV}\right]\), which can be regarded as the light quanta photon energy stored in photoexcited minority holes/electrons, is physically almost equivalent in value to the development of the negative/positive photo emf, \({V}_{\text{oc}}^{\text{n},\text{ l}}<0 \text{ in }\left[\text{V}\right]/{V}_{\text{oc}}^{\text{p},\text{ l}}>0 \text{ in }\left[\text{V}\right]\) necessary for the driving force of the photocurrent and simultaneously to the shift of negative/positive inverse overvoltage, \({\eta }_{\text{h}}^{\text{n},\text{ l}}<0 \text{ in }\left[\text{V}\right]/{\eta }_{\text{e}}^{\text{p},\text{ l}}>0 \text{ in }\left[\text{V}\right]\), for photosensitized anodic oxidation/cathodic reduction at the n-type anode/solution interface/at the p-type cathode/solution interface, driven by photoexcited minority holes/electrons, respectively.

The photoexcited minority hole present at the quasi-Fermi level of the n-type anode, \(p{E}_{\text{F}}^{\text{n}} \text{ in }\left[\text{eV}\right]\), gives the negative photovoltage regarding the counter cathode, \({V}_{\text{oc}}^{\text{n},\text{ l}}<0 \text{ in }\left[\text{V }vs\text{ cathode}\right]\); it also drives the shift of negative inverse overvoltage, \({\eta }_{\text{h}}^{\text{n},\text{ l}}<0 \text{ in }\left[\text{V}\right]\), for photosensitized anodic oxidation.

The same is true of photoexcited minority electrons existing at the quasi-Fermi level, \(n{E}_{\text{F}}^{\text{p}}\text{ in }\left[\text{eV}\right]\), of the p-type cathode, delivering the positive photovoltage regarding the counter anode, \({V}_{\text{oc}}^{\text{p},\text{ l}}>0 \text{ in }\left[\text{V }vs\text{ anode}\right]\), and simultaneously causing the shift of positive inverse overvoltage, \({\eta }_{\text{e}}^{\text{p},\text{l}}>0 \text{ in }\left[\text{V}\right]\), for photosensitized cathodic reduction.

The single shift (negative at the n-type; positive at the p-type) of the redox Fermi level to the flat band Fermi level in the negative/positive direction provides thermodynamic backgrounds for the generation of photo emf on the energy band diagram and a kinetic basis for photocurrent I vs photovoltage V polarization characteristics on photosensitized anodic oxidation at the n-type/solution interface/cathodic reduction at the p-type/solution interface at the regenerative wet photovoltaic cell inclusive two hypothetical regenerative hydrogen/oxygen photovoltaic cells, conceptually derived from a thought experiment, and the photo-electrochemical cathodic protection short-circuited cell. The negative/positive shift gives rise to the generation of the photo emf (electromotive force) and the occurrence of negative/positive inverse overvoltage.

This contrasts with the n-p-type dry photovoltaic cell, where the diffusion (contact) potential replacing the negative/positive single shift is the limiting photo emf.

One equilibrium Fermi level \({E}_{\text{F}}\), at which chemical potentials of majority and minority carriers converge in the dark, has been split into the two quasi-Fermi levels of majority and minority carriers under illumination, for instance, \({nE}_{\text{F}}\) and \({pE}_{\text{F}}\), wherein the mass action law accordingly is no longer obeyed. The separation of \({E}_{\text{F}}\) between the two \(\left({nE}_{\text{F}}-{pE}_{\text{F}}\right)\) is a direct measure of the deviation from thermodynamic equilibrium.

The ideas of the negative inverse overvoltage, \({\eta }_{\text{h}}^{\text{n},\text{ l}}<0\), for the photosensitized anodic transfer of the VB minority holes, compared to the positive inverse overvoltage, \({\eta }_{\text{e}}^{\text{p},\text{ l}}>0\), for the photosensitized cathodic transfer of the CB minority electrons make it possible to qualitatively predict the resulting photo I vs V polarization curves. These polarization curves reduce to those electrochemical I–V curves of any self-driven Galvanic cell in the dark for two key reasons. First, the photoexcited VB minority holes in the n-type cause the negative shift of the equilibrium potential, \({V}_{\text{eq}}^{\text{redox}}\), of a redox couple to the flat band potential, \({V}_{\text{fb}}^{\text{n}}\), in the negative direction of potential. Second, the photoexcited CB minority electrons in the p-type drive the positive shift of \({V}_{\text{eq}}^{\text{redox}}\) of the redox couple to \({V}_{\text{fb}}^{\text{p}}\) in the positive direction of potential. The expected photo-I-V curves have been evidenced by experimental data published in other literature.

The negative/positive inverse overvoltage, i.e., negative/positive single shift of \({V}_{\text{eq}}^{\text{redox}}\) of the redox couple to the flat band potential \({V}_{\text{fb}}\) is recognized to be the limiting photo emf. The negative/positive inverse overvoltage, \({\eta }_{\text{h}}^{\text{n},\text{ l}}<0\)/\({\eta }_{\text{e}}^{\text{p},\text{ l}}>0\), multiplied by electronic charge, can be well understood as the light quanta energy gained and stored in the photoexcited hole/electron, which provides the thermodynamic affinity (driving force) necessary for photosensitized anodic oxidation/cathodic reduction, respectively. This can never violate the second law of thermodynamics.

In short, \(\left\{-\frac{\left({nE}_{\text{F}}^{\text{n}}-{pE}_{\text{F}}^{\text{n}}\right)}{e}\approx -\frac{\left({E}_{\text{F},\text{ fb}}^{\text{n}}-{E}_{\text{F}}^{\text{redox}}\right)}{e}=\left({V}_{\text{fb}}^{\text{n}}-{V}_{\text{eq}}^{\text{redox}}\right)={\eta }_{\text{h}}^{\text{n},\text{ l}}(\text{negative inverse overvoltage})\le {V}_{\text{oc}}^{\text{n},\text{ l}}<0\right\}\) for the n-type anode regarding the counter metal cathode is effective at the n-type based wet photovoltaic cell; \(\left\{\frac{\left({nE}_{\text{F}}^{\text{p}}-{pE}_{\text{F}}^{\text{p}}\right)}{e}\approx \frac{\left({E}_{\text{F}}^{\text{redox}}-{E}_{\text{F},\text{ fb}}^{\text{p}}\right)}{e}=\left({V}_{\text{fb}}^{\text{p}}-{V}_{\text{eq}}^{\text{redox}}\right)={\eta }_{\text{e}}^{\text{p},\text{ l}}\left(\text{positive inverse overvoltage}\right)\ge {V}_{\text{oc}}^{\text{p},\text{ l}}>0\right\}\) for the p-type cathode regarding the counter metal anode is valid for the p-type based wet photovoltaic cell.

The negative inverse overvoltage caused by the photoexcited minority VB hole makes it thermodynamically possible for photosensitized anodic oxidation to proceed at the n-type/solution interface of both the wet photovoltaic cell and the photosensitized cathodic protection cell. The same is true of the positive inverse overvoltage driven by the photoexcited minority CB electron for cathodic transfer to proceed at the p-type/solution interface.

In contrast, the separation of the Fermi level between the n-type and p-type regions constituting the dry photovoltaic cell delivers a photocurrent and a photovoltage, caused by migration (drift) of the photo-excited minority EHP (electron-hole pair) across the space charge transition region into the n-type and p-type, respectively, prior to recombination there.

Under illumination, a depleted space-charge region forms at the semiconductor/electrolyte interface. This region acts as an effective transition zone, enabling the separation and transport of photoexcited charge carriers—both majority and minority—regardless of whether the semiconductor is n-type or p-type. As a result, it facilitates photosensitized reduction and oxidation currents, as well as the generation of photovoltages, in both wet photovoltaic cells and photo-electrochemical cathodic protection cells.

In comparison, generating a depleted space charge region and accompanying rectifying behavior of the n/p junction is well suited for photoexcited minority EHP to deliver a photocurrent and a photovoltage to external circuit in the dry photovoltaic cell inclusive the n/p junction. In contrast, the presence of an enriched space charge region in the n-type and p-type should be absolutely excluded at the outset in all three types of cells.

Finally, some quizzes are presented with their answers, so the readers working on the problems are motivated to solve them with instructive perspectives.

Due to space limitations, we will go into the equilibrium and dynamic electrochemistry aspects of the photo-assisted electrolytic cell and photo-catalytic short-circuited cell using rectifying n- and p-type semiconducting electrodes in the following article.