Chapter 10

CHAPTER 10

Photoelectrochemical WaterSplitting: A First PrinciplesApproach

ANDERS HELLMAN

Applied Physics and Competence Centre for Catalysis, Chalmers Universityof Technology, SE-41296, SwedenEmail: [email protected]

10.1 Introduction

In the novel, L�ıle mysterieuse, Jules Verne writes, ‘‘water will be the coal of thefuture’’. This is the passing conclusion of the main characters after a lengthydiscussion on whether the known coal reserves of that time would be depletedwithin the near future. The discussion in the novel is still very relevant today.Crude oil is the backbone of the energy system in modern society. However,owing to increased global energy consumption, expectation of peak oil pro-duction and in conjunction with increasing concerns regarding environmentalimpact, there is a growing awareness that we need to find a renewable source ofenergy, and thereby decrease our dependence on fossil fuels. Although Vernewill never be right about water being used as fuel (due to thermodynamicsconsiderations), water still is a key element in the envisioned future hydrogen-based energy infrastructure. With enough supply of energy, water can be sep-arated into its elementary components, hydrogen and oxygen, and the hydrogencan be used, for example, as fuel in a fuel cell. As a matter of fact, the use ofhydrogen and oxygen is also what was envisioned in the story of Jules Verne.

RSC Energy and Environment Series No. 9

Photoelectrochemical Water Splitting: Materials, Processes and Architectures

Edited by Hans-Joachim Lewerenz and Laurence Peter

r The Royal Society of Chemistry 2013

Published by the Royal Society of Chemistry, www.rsc.org

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Continuing this line of thought, the challenge remains to find a source ofenergy that is abundant, free and sustainable. The most obvious idea is toharvest the energy of the sun, which is constantly bombarding the Earth withenough energy to sustain society over any foreseeable future.1–4 Energy fromthe sun can be harvested in many different ways. Sunlight can be transformedinto chemical energy by means of photosynthesis, which is crucial for life onearth. For fuel production, plants with low water and fertilizer requirementscan be used to produce large quantities of biomass that can be used at a laterstage to produce power or fuel. Furthermore, sunlight can be used to drivephotovoltaic cells, which produce electricity that can be converted into fuels byelectrocatalytic processes. The electricity can also be produced from wind orwave power originating from the energy of the sun. Finally, the absorption ofsunlight can be coupled directly with electrochemical processes in photoelec-trocatalytic devices that produce fuel directly from electrical currents obtainedfrom absorbed sunlight.

Although Bequerel, with his measurement of photovoltaic effect from anilluminated silver chloride electrode (ca.1839) can be viewed as the founder ofphotoelectrochemistry, it is the measurements by Honda and Fujishima5

published in 1972 that really showed the potential of photoelectrochemicalsystems to harvest and store solar energy as chemical energy. In the Honda-Fujishima experiment, titanium dioxide (TiO2) was used as the photoanodematerial to produce hydrogen from water using the energy of light.

In order to make use of water oxidation from a sustainable perspective,stable and inexpensive photoanode materials are required.6,7 To date, titaniumdioxide is the benchmarking material.8,9 It is stable over a range of pHs andpotentials, and under favourable circumstances requires no additional bias torun the water splitting reaction. However, the downside of TiO2 is its largeband gap, which limits the photon absorption to only a fraction of the solarspectrum in the ultraviolet. This corresponds to only a few percent of photonsthat actually reach the surface of the Earth, which limits the long-term potentialof TiO2. Other metal oxides, such as tungsten oxide and hematite have a morefavorable band gap, and both are considered as promising candidates forphotoanode material. Several other semiconductor photoanodes, such as Si,have even more favorable band gaps, but are not stable under the aqueousconditions required for water oxidation.

The ideal photoanode material6,7 should meet the following criteria; (i) aband gap ranging between 1.8 and 2.4 eV, (ii) band edge positions that bracketsthe water redox potentials, (iii) electron-hole mobility and lifetimes that allowthe electron-hole pair to reach the active site, (iv) the rate for water-splittingshould be faster than any competing recombination reaction. Finally, thematerial needs to be stable in an aqueous environment under illumination. Sofar no material has met all these criteria.

The actual processes involved in photoelectrochemistry are many,10 seeFigure 10.1. The first is the capture of photons; the second is the creation ofelectron-hole pairs that need to be separated. The separated electrons and/orholes then need to be transported to sites, preferable catalytically active, at

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which the transformation to chemical energy can occur. The multitude ofprocesses has made research in photoelectrochemistry truly cross-disciplin-ary.11 For instance, photon absorption and the transport of charge carriers aretopics of interest to semiconductor physics, but the chemical transformationsoccur at surface sites, so surface science and heterogeneous catalysis are alsorelevant areas. The close connection with electrochemistry is perhaps the mostobvious and important area.

This chapter describes some of the processes involved in photoelec-trochemistry and demonstrates how first-principles methods can be utilized toprovide further understanding of these processes (first-principles are used inconjunction with density functional theory throughout the chapter). Thechapter is not intended to be a complete overview, and some aspects do not getthe attention they deserve, for which the author can only apologize. Further-more, the chapter may deviate from the normal description of the processesinvolved, given that the author’s background is not photoelectrochemistry, butrather theoretical surface science12 and computational aspects of femtosecondspectroscopy,13 non-adiabatic surface processes14 and heterogeneous cata-lysis.15 It is hoped that this different perspective will provide new angles toapproach what is already a very interesting area of research.

10.2 Capture of the Photon

As the photon penetrates a material, the electronic structure of the substanceinteracts with the propagating electromagnetic wave.13,16,17 The interactionmight result in an electron accruing the photon energy, i.e. there is an electronicexcitation in which an electron in an occupied band is transferred to an

Figure 10.1 A simplified energy diagram for a photoanode (n-type semiconductor).Several important steps are illustrated, namely: (i) light absorption;(ii) charge transfer; (iii) charge transport; and (iv) surface chemical reactions.Reprinted with permission from ref. 10. Copyright 2011 Elsevier.

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unoccupied band. If the perturbation is small, the process can be described byFermi’s golden rule, where the probability for a transition from state i to j isexpressed as follows,

Ti!j ¼2p�h

i���H:

0��� j

D E���

���rirj ð1Þ

Here, ri and rj are the corresponding densities of states of the initial and finalstates. From equation (1), it is clear that a good light-absorbing material needsto have not only a large optical cross-section, but also a favorable density ofstate distribution. Here metals, of course, stand out. However, as the goal is tobe able to utilize the photon energy to generate something other than heat, thematerial needs to have a band gap.

In semiconductor and insulating materials, the valence and conductionbands are separated by a band gap. The band gap efficiently hinders the dis-sipation of energy, as the lack of accessible electronic states quenches many ofthe common dissipation channels (carrier-carrier interaction and the phononcoupling). Generally one can classify materials with band gaps into two classes;direct and indirect band gap materials. The difference lies in the prerequisitesfor photon adsorption. In a material with a direct band gap, the momentumof electrons and holes is conserved during the transition. In a material withan indirect band gap, the conservation of momentum requires the creation/annihilation of one (or many) lattice phonon(s). Focusing on the process ofcapturing the photon, two main (first-principles) research directions can berecognized. The first is to focus on the band gap, i.e. how to optimize thepositions and densities of states, whereas the second focuses on the photonabsorption process.

10.2.1 Band Gap Design

In its simplest form, the power conversion efficiency in a semiconductor de-pends only on the band gap and the incident light spectrum,7 As the solarspectrum is fixed, only the band gap can be varied to optimize the conversionefficiency. Density-functional theory (DFT) is supposedly a predicting theory,and there exist several successful examples in the literature. However, in thecase of band gap design, there is a well-known problem. The most frequentlyused approximations for the exchange-correlation functional fail at calculatingthe band gaps of even the simplest materials. For instance, Si is calculated tohave a band-gap of 0.52 eV, whereas the experimental value is 1.17 eV.18 Thereare many extensions to DFT designed to circumvent this problem, such as,DFTþU, hybrid functionals and random-phase approximation.19 However,work has also been done on the simple semi-local functionals20,21 thatdescribe PBEsol (PBE¼Perdew-Burke-Ernzerhof) and GLLB-SC (GLLB¼Gritsenko, van Leeuwen, van Lenthe, Baerends). All of these extensions arecomplicated and lie beyond the scope of this chapter. In many cases the use offirst-principles for trend studies are is still valid without the use of theseextensions.

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Doping has a strong influence on the fundamental and optical band gaps ofsemiconductors, and there are numerous examples in the literature where first-principles have been used to guide how to include dopant atoms in order toimprove the band gap of a given material. Asahi et al.22 showed that nitrogendoped into substitutional sites of TiO2 resulted in a band-gap narrowing, sincethe p states of N contribute to the narrowing by mixing with O 2p states.Teeffelen et al.23 investigated the shifts of the fundamental and optical bandgap energies as functions of dopant concentration in heavily n-type and p-typedoped Si1�xGex in order to improve solar cell efficiency. The calculated bandgap narrowing of Si and of Si0.82Ge0.18 was found to be in good agreement withvalues derived from photoluminescence measurements. The above examplesshow that first-principles methods can be used to provide explanations of whatis happening at the atomic level. However, first-principles methods should alsobe able to provide guidelines for the search for new materials with improvedband gaps.

The next example indicates what we can expect in a near future. Recently,Castelli et al.24 demonstrated the power of computational screening for thediscovery of new light harvesting materials for water splitting. More than 2700oxides with the cubic perovskite structure where investigated with respect tostability and band-gap. In the end, 15 potential candidates were identified(see Figure 10.2). Unfortunately, these candidates are already known to besuitable for water splitting, but the successful outcome of the theoretical workappears to promise that identification of new material formulations will be re-liable if more complex structures and compounds are included in the screening.

Figure 10.2 (Right) Calculated and measured band-gap of a number of oxides. Thefirst-principles are based on two different state-of-the-art functionals,namely, PBESol20 and GLLB-SC.21 (Left) The identified oxides andoxynitrides in the cubic perovskite structure with potential for splittingwater in visible light. The figure shows the calculated band edges for boththe direct (red) and indirect (black) gaps. The levels for hydrogen andoxygen evolution are also indicated.Reprinted with permission from ref. 24. Copyright 2011 Royal Society ofChemistry.

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10.2.2 Plasmon-Assisted Photon Absorption

When light interacts with a (metallic) surface, resonant collective oscillations inthe electronic system can be excited. If these oscillations are localized in a nano-sized entity such as a noble metal nanoparticle, they are called localized surfaceplasmon resonances (LSPR).25 The frequency of the LSPR can be tuned acrossthe electromagnetic spectrum through the choice of metal size, shape and di-electric environment. LSPR results in a strong coupling between the photonand the nanoparticle, which gives rise to an enhanced photon absorption andlight scattering and strongly enhances electromagnetic fields around the metalnanoparticle.

All of these effects are interesting from a solar harvesting perspective, also forphotoelectrochemical devices.26 Besides the obvious benefit of enhanced pho-ton absorption, light scattering can be utilized to engineer light management inorder to decrease the amount of material required.27–29 The near-field canstimulate light absorption in the proximity of the metal nanoparticle. This isparticularly interesting if the minority carriers have a short lifetime, since theinclusion of LSPR particles at the interface should lead to carrier generationcloser to the active site for water splitting. If the LSPR particles were placed atthe bottom of the photoanode instead, this would help in the generation of themajority carriers. Since they are situated at the photoanode/electrolyte inter-face, special consideration must be given to the influence of the nanoparticleson (i) stability, (ii) Fermi level and (iii) band-bending. Furthermore, catalyticeffects and the formation of trap states may also be critical.

There are several recent studies reported in the literature that provide evi-dence for plasmon-assisted enhancement of light-driven reactions. Linicet al.30,31 measured an increase in reaction rate in the case of Ag nanoparticles,where the enhancement was attributed to energy transfer from the metal to thesemiconductor arising from overlap with the LSPR frequency of the Ag (seeFigure 10.3). Overall, the work of Ingram and Linic provides a strong indi-cation that there needs to be an absorption overlap between the semiconductorand LSPR for energy transfer to result in preferential excitation of the semi-conductor in the vicinity of the metal nanoparticle. This is in agreement withtheoretical studies of similar systems.32

Plasmon-enhanced photon absorption is particularly interesting if the pho-toanode material has a weak optical absorption or if the intrinsic properties ofthe material make some of the necessary processes to slow. For instance,hematite suffers from a short hole diffusion length.33–35 Here, the idea is thatthe plasmon particle will enhance photon absorption, creating the electron-holepair closer to the anode-electrolyte interface. Several successful measurementsdemonstrating this effect have appeared in the recent literature.36–38

In principle, the collective motion of electrons that constitutes the LSPRphenomenon can be described theoretically by time-dependent density func-tional theory.39 However, owing to the high computational cost of such anapproach, a linear response formalism is much more favorable. Therefore, theidea here is to introduce a small perturbation that can be evaluated as a Dyson

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equation. The poles of the solution then basically yield the plasmonic responseof the considered system. Thus, there exists a theoretical framework with rootsin DFT that can be developed and used to evaluate the plasmonic responsefrom various materials at the atomic level.40–42

From a photoelectrochemical point of view, the decay of plasmons is inter-esting because the decay of plasmons at finite momentum transfer is dominatedby Landau damping, i.e. the decay into electron–hole (e–h) pairs. The dampingof surface plasmons is purely a quantum mechanical process and is governed bythe coupling between the surface plasmons and e–h pairs. Unfortunately, thisprocess is not well understood, even for the simplest crystalline surfaces. In arecent study, Gao and coworkers43 presented a semiclassical model ofplasmon–electron coupling and Landau damping for metal thin films andsurfaces, based on the quantization of the plasmon hybridization (PH) modelof Nordlander and coworkers,44,45 which describes this process qualitatively.Desirable developments for the near future are (i) to connect first-principlesresults with the nanostructure of the photoactive material, and (ii) to investi-gate how the plasmon decay can be incorporated into first-principles studies.

10.3 Electron–Hole Separation

Once the electron–hole pair is created, it needs to be separated, but this is not astraightforward process. Figure 10.4 shows some of the different fundamental

Figure 10.3 (Left) Mechanism of plasmon induced charge transfer with approximateenergy levels on the NHE scale. Dashed red lines refer to the water-splittingredox potentials. As the plasmon decays the energy is transferred to anelectron–hole pair where the electron can transfer to a nearby semicon-ductor particle. Depending on the position of the plasmon and the valanceand conduction band this process can drive the water oxidation reaction.(Right) Photocurrent as a function of broadband visible-light intensity forsamples of TiO2 , with and without the plasmon active particles.Reprinted with permission from ref. 31. Copyright 2011 American Chem-ical Society and from ref. 30. Copyright 2011 Nature Materials.

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processes that can happen after the photon is absorbed. Depending on thematerial at hand, some of the pathways will be more useful than others.

If the photon energy is larger than the minimum band gap, states deeperdown in the valence band and/or higher in the conduction band are available tocreate the electron–hole pair. As a consequence, the newly created electron–hole pair is effectively in a non-equilibrium state, but the additional energy isdissipated rapidly owing to carrier-carrier interaction and phonon coupling.The cooling process normally goes through the following steps.46–48 Firstelectrons and the holes reach equilibrium via their respective carrier-carriercollisions, resulting in two different temperatures defined by the distribution ofenergy of the respective carrier distributions. This temperature is always higherthan the phonon temperature, giving rise to the terms ‘‘hot electrons’’ and ‘‘hotholes’’. This initial relaxation process is very rapid (1–10 fs). Next, the hotcarriers equilibrate with phonons via carrier-phonon interactions, transferringexcess energy to the heating of the photoanode material. This second relaxationprocess occurs on the time scale of 1–100 ps. The last step involves electron-holerecombination, which can be either radiative (luminescence) or non-radiative(heat). Clearly, the last step is undesirable in photoelectrochemistry.

The electron–hole pair feels a coulomb attraction, and if it is strong enough,the electron–hole pair can be referred as an exciton. The attraction will affectthe spatial distribution and transport properties of both the electron and thehole. However, as the wavefunction of the electron–hole pair becomes un-correlated, electrons and holes can be viewed as free carriers. The charge carrierwith the lowest effective mass (as determined by the second derivative of thedispersion relation) will exhibit a larger root-mean-square motion as compared

Figure 10.4 The relaxation process of a hot electron–hole pair. There exist severalpathways for dissipation of the energy from the electron–hole pair. Moredetails of this are given in the text.Reprinted by permission from Annual Review: Annual Review of PhysicalChemistry, copyright 2001.46

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to the heavier charge carrier. This implies that the lighter carrier will effectivelycreate a charge separation, known as the photo-Dember effect.10 This is themain effect that determines carrier transport under zero-field conditions.

In semiconductor physics, an interface is often used to separate electron–holepairs. At such an interface, a space charge layer is formed owing to the dif-ference in the electrochemical potentials of electrons in the two phases. As thesystem equilibrates, electron flows from higher free energy to lower free energyuntil the compensating field is sufficient to stop the flow. This is the physicsbehind p-n junctions. In a photoelectrochemical system, the interface is createdbetween the semiconductor and the liquid phase (see Figure 10.5). Owing to the

Figure 10.5 Schematic showing the electronic energy levels at the interface betweenan n-type semiconductor and an electrolyte containing a redox couple.The four cases indicated are: (a) flat band potential, where no space-charge layer exists in the semiconductor; (b) accumulation layer, whereexcess electrons have been injected into the solid producing a downwardbending of the conduction and valence band towards the interface;(c) depletion layer, where electrons have moved from the semiconductorto the electrolyte, producing an upward bending of the bands; and(d) inversion layer where the electrons have been depleted below theirintrinsic level, enhancing the upward band bending and rendering thesemiconductor p-type at the surface.49

Reprinted by permission from Macmillan Publishers Ltd: Nature 414338, copyright 2001.

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mobile charge carriers in the liquid, a Schottky junction is formed that involvesa redistribution of the charge on the electrolyte side that corresponds to for-mation of the Helmholtz layer. The field in the semiconductor can extend overseveral hundred nm, depending on the doping level of the semiconductor.

The field in the space charge region will assist in the separation of the pho-togenerated electron–hole pairs and drive the minority carrier to the surface.The minority carriers will pick up excess energy from the field. Furthermore,carrier-carrier interaction is much less in the space-charge layer, so that only thephonon channel is open for dissipation of energy. This implies that the carrierscan arrive at the surface with an excess energy corresponding to the degree ofbend banding, i.e. hot carriers. In contrast, the majority carrier will experienceboth the carrier-carrier and phonon-scattering processes during its propagationthrough the bulk semiconductor material.

The inclusion of plasmon-active metal nanoparticles can have pronouncedeffects on bend bending because the work functions of metals generally differfrom the electrochemical potentials of electrolyte solutions. This implies thatthe band bending is different in different regions of the semiconductor. Themost common plasmon-active nanoparticles used are gold and silver, where themetal Fermi level is positioned at a higher energy than the redox Fermi levelcorresponding to water oxidation. This results in less band bending at themetal-semiconductor contact, thereby hindering charge separation. Further-more, the metal-semiconductor interface can generate surface states and causeFermi level pinning, which will have a deleterious effect on photoelec-trochemical performance. However, if surface states are removed or if the workfunction of the metal is positioned more suitably, improvements in chargeseparation can be observed.26,49

As the electron or hole is transported through the lattice, the lattice respondsto this charge by delocalizing it over many atoms (a large polaron) or by lo-calizing it over just a few atoms (a small polaron). As the extra charge carrierwill fill or empty states with the bonding or antibonding characteristics of thelattice atoms, the charge influences bond lengths and angles in the material. Ifthe charge is delocalized, the influence is small because it is spread over manyatoms, while a highly localized charge distorts the lattice significantly. One wayto view the process is to consider a carrier dragging a cloud of phonons along asit propagates through the lattice. This gives rise to an understanding of thephenomenon of self-trapping. A small polaron will shift the surrounding latticeto its new position within a few lattice vibrations. This increases the stability ofthe state, creating a deeper potential well that must be overcome to transfer thepolaron to a neighboring lattice site, i.e. there is a large reorganization energyassociated with small polarons.

Deskins et al.50 calculated the electronic structure of one excess electron inbare and singly hydroxylated rutile (110) surfaces. According to their calcula-tions, the excess electron behaves as a small polaron with its spin density andassociated lattice distortion localized around a single site. The study alsoshowed that the surface hydroxyl group only perturbs the electronic potentialslightly and that both clean and hydroxylated surfaces exhibit similar polaron

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stability. This is an important conclusion because hydroxyls and photoinducede-polarons give rise to excess charges, which can affect the reactivity of surfaceabsorbates and other photochemical processes. As for the formation and mi-gration of hole polarons, Deskins et al.51 used DFT in combination with theMarcus/Holstein theory of electron/polaron transfer52–54 to show that holeswere formed by removal of an O(2p) valence electron, and that hole hopping(where reorganization energy and electronic coupling was taken into account)in most directions in rutile and along one direction in anatase was adiabatic incharacter, i.e. thermal processes coupled to phonons. Lattice distortionsaround hole polarons were found to be larger than around electron polarons.The study also showed that holes are thermodynamically more stable in therutile phase, while electrons are more stable in the anatase phase.51 Further-more, Pacchioni et al.55,56 showed that in order to get the correct description ofthe localized defect states on reduced and hydroxylated TiO2(110), it is neces-sary to go beyond semi-local description of the exchange-correlation functionaland use hybrid exchange functionals instead. Even though the electron trappingnature of Ti(OH) groups was verified, no evidence that these defects also act ashole traps was found. The results show that the Ti(OH) defect can be a goodelectron trap, and that the lattice distortion is essential in the electron trappingprocess.

A similar approach was used by Sicolo et al.57 to describe the electronicstructure and spectral properties of self-trapped holes in SiO2, where twoclassical variants of self-trapped holes were studied, namely, (i) a hole trappedat the 2p nonbonding orbital of an O atom bridging two Si atoms, and (ii) ametastable defect where the hole is delocalized over the 2p orbitals of twobridging O atoms. The first-principles results showed that the ground state ofthe first type of self-trapped hole allows an unpaired electron to occupy anonbonding 2p level of a bridging oxygen, where the 2p level is normal to theSi-O-Si plane. This results in a strong elongation of one Si-O bond, therebyclassifying this center as a small polaron. The description of the second self-trapped hole required modification of the normal lattice structures to make thestructure flexible enough to allow the O-Si-O angle to shrink from 1101 to801–901. This latter condition actually reflects the fact that the electronicstructure of the second-type of self-trapped hole uses a bonding combinationbetween the 2p levels on two O atoms, resulting in a net bonding interactionthat closes the O-Si-O angle and decreases the O-O distance. Other anglesaround the defect also assume values that deviate substantially from those ofquartz, while the Si-O distances are only moderately elongated. In this respect,the center has the typical characteristics of a molecular polaron.57

Kleiman-Shwarsctein et al.58 introduced strain in a Fe2O3 film by substi-tutional doping of Al, which resulted in a 2- to 3-fold increase of the incidentphoton conversion efficiency. By means of first-principles, it was shown thatthere was no substantial change to the electronic structure. Instead, the dopingbenefits small polaron migration, resulting in an improvement in conductivitycompared to the undoped sample. In a similar study (although based on un-restricted Hartree-Fock calculations), Liao and Carter59 investigated how the

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activation energies for hole diffusion is affected by different dopants. The studysuggests that hole hopping occurs via oxygen anions for hematite, and holecarriers are predicted to be attracted to O anions near the dopants.

The examples above indicate that DFT is able to characterize electron/holepolarons, and that the first-principles results can also be used as input intomodeling of polaron transport. However, further developments will be neces-sary before DFT can be used to calculate all the details involved in the electron-hole transport process. The difficulty in obtaining an adequate description ofthe energy difference between a localized and a delocalized state is particularlycrucial, and the predictive power of DFT is hampered by the need to comparewith experiment. Furthermore, in the case of issues such as non-equilibriumsystems and nonadiabatic transitions, the correct handling of different lengthand time scales are all important.60 In a recent review, Shluger et al.61 give moreexamples of modeling electron and hole trapping in metal oxides and alsodiscussion concerning the different challenges involved.

10.4 Charge Transfer

Assuming a four-electron transfer water oxidation mechanism, the photo-generated holes that have reached the photoanode surface need to transferelectrons from the water molecule (or the intermediate products). Transfer ofan electron in one direction is equivalent to transfer of a hole in the other,which is convenient because it allows ET and HT (hole transfer) to be treatedwithin the same theoretical framework. Now electron transfers (ET) are verycommon and important for chemical reactions, but are of course crucial for allelectrochemical reactions. It is very common to classify an ET into either anadiabatic or a non-adiabatic process (sometimes equivalently termed diabatic).

Both adiabaticity and non-adiabticity are sometimes described in quantum-mechanics where the of atomic and electronic motions are separated in theadiabatic approximation, also called the Born–Oppenheimer approximation(BOA).62 An early description of the terms adiabatic and non-adiabatic wasgiven by O’Malley:63 the adiabatic states are simply the eigenstates of theelectronic Hamiltonian, and the adiabatic PESs are the corresponding eigen-values of the same Hamiltonian defined for each nuclear configuration, R. Thediabatic representation provides an alternative description that includes manyof the so-called non-adiabatic transitions in a natural and straightforward way,particularly for processes in which fast electronic transitions either occur withina spatially localized region of configuration space or they do not occur at all. Inmany areas of physics, there are real highlights in the breakdown of the BOA.Recent experiments64 and calculations65 have shown the importance ofelectron–hole pair excitation in gas–surface dynamics by analyzing the chemi-current – a current due to direct transformation of chemical into electricalenergy – of electrons and holes in a Schottky diode induced by the adsorptionof molecules. For more details, see reference 14.

Electron transfer plays a prominent role in key processes in all areas ofphysics, chemistry, and biology. For instance, bond making and bond breaking

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involve electron transfer from one electronic state to another. In adiabatic re-action rates described by transition-state theory, the potential energy surfacefor nuclear motion is well separated from higher potential energy surfaces sothat the transitions to the latter surfaces are negligible. However, sometimesthis approximation comes into question.66 Even though electron transfer is acommon phenomenon, its modeling is often primitive and intended more tomake use of simple analytical models than to represent the system accurately.However, such models do have considerable interpretive value. The Landau–Zener model is widely used to describe non-adiabatic processes occurring in thegas and liquid phases. It was originally constructed to calculate the probabilityof non-adiabatic transitions in a two-level system, and as such it is a goodstarting point to study non-adiabticity in electron transfer from one electronicstate to another. However, the continuum of states that are present in thevalence and conduction bands of semiconductor materials may cause the two-state approximation to fail, but nevertheless – with proper modification – themodel can still be very useful for photoelectrochemical processes.67 For furtherdetails, see Figure 10.6.

In the chemistry and electrochemistry community, ET processes are usuallydescribed in the context of the Marcus theory.54 In short, the basic idea ofthe classical electron transfer theory of Marcus can be summarized as (i) thecomplete system starts off from the equilibrium state of the donor state, (ii) thedonor and acceptor states are described by two separate potential energy sur-faces, (iii) electron transfer occurs at the intersection of both potential energycurves. Thermal fluctuations are needed for the system to reach the intersection.Fluctuations in the vibrational coordinates and the orientational coordinatesare important ingredients. The rate constant for electron transfer then dependson the probability of reaching the intersection, a frequency factor, and theprobability of crossing the surface. The analogy with the Landau–Zener model

Figure 10.6 (Left) A simple picture of Marcus theory for symmetric polaron transfer.The potential energy surfaces of the initial state and final state are shown.The adiabatic energy curves are shown as dashed lines, with the elec-tronic coupling matrix element, given as half the energy differencebetween the two adiabatic states. Reprinted with permission from refer-ence 51.51 Copyright 2008 American Chemical Society. (Right) Theordinary and modified Landau-Zener model for single and multiplecrossing of potential energy surface.Reprinted with permission from ref. 67. Copyright 1997 AmericanPhysical Society.

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is clear. The direct application of the simple models to photoelectrochemistry isnot without problems. For instance, the processes of transferring charge(electrons, protons, etc.) across the interface between the semiconductor andthe electrolyte are difficult to model correctly. There are often significant dipolemoments, which affect the description of the donor and acceptor states.Moreover, the dynamics of screening in the semiconductor and electrolytecomplicate the picture further.60

10.5 Surface Reaction (Electrochemical Conversion)

Although the holes that drive the water oxidation reaction are generated byphotons and the driving force comes from the valence band position and notfrom an external potential, the surface reaction is the same as in an electro-chemical cell. This is a real advantage, since the recent developments in com-putational electrochemistry have been quite remarkable. The frameworksformulated by, e.g. the Anderson group,68–72 the Neurock group,73–77 and theRossmeisl/Norskov group78–84 have provided a molecular-level insight into theatomic-scale processes that occur at the vicinity of the anode/cathode surface.

Recently, Valdes et al.85–87 suggested a novel, theoretical framework in whichthe photo-oxidation of water can be described by first-principles methods. It isan extension of the electrochemical framework suggested by Rossmeislet al.78–84 Although the framework only treats the thermodynamics of the re-action mechanism, it provides a methodology for a detailed atomistic under-standing of photoelectrochemical water splitting. The framework assumes thatthe driving force for the reaction at the anode originates from the photo-generated hole at the edge of the valence band. Hence, in the scale obtained byaligning the energy levels of oxide semiconductors with the redox level of thestandard hydrogen electrode, a deeper valence band edge energy level will resultin a larger thermodynamic driving force.

Within the computational electrochemical framework, the potential of thestandard hydrogen electrode (SHE) is used as reference point. The SHE is zeroby definition, where the chemical potential of the H1(aq)þ e� pair is equal tothat of 1/2 H2 in the gas-phase. This solves the problem of calculating theenergy of solvated protons and electrons, and instead the gas-phase value of theenergy of H2, which is easily described by first-principles, can be used. Fur-thermore, the effect of the electrode potential on the adsorption energies issimple to include by addition of a stabilization energy of þ eU when appro-priate. In principle, the adsorption energy of reaction intermediates can dependon the electrode potential. However, first-principles studies indicate that thiseffect is small, e.g. the adsorption energies of *O, *H, and *OH are onlychanged slightly when an electric field in the range of � 0.3 V/A to 0.3 V/A isused. Assuming a double layer thickness of 3 A, the range corresponds to apotential of �0.9 V and 0.9 V with respect to the zero potential.88 Therefore,the primary effect of the electrode potential is to change the (free) energy of theelectrons.

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At the photoanode/electrolyte interface, the solvent water molecules mayplay an important role. Contributions from the liquid phase are normally ap-proximated by including several of water layers in the simulation.89,90 Thewater network, with its many hydrogen bonds, will have a large effect on thestability of the reaction intermediates, especially the ones that can contributewith more hydrogen bonds. First-principles results show that OH-containingreaction intermediates, such as, *OH and *OOH, can be stabilized up to 0.6 eVon Pt (111), whereas those of H and O are affected to a lesser extent, ca.0.1 eV.91 Furthermore, entropy contributions from the liquid phase are ap-proximated by reference to the equilibrium pressure in contact with liquidwater at 298.15K and vapor pressure 0.0317 bar, where the free energy of gasphase water is equal to the free energy of liquid water. This permits the use ofgas-phase water to calculate the binding energies and its transformation intoliquid-phase water when adding the entropy correction.82 The free energy changeof the reaction step involving the formation of O2 is set to the experimentallyobtained value of 4.92 eV per O2 molecule. The free energy of the reactionintermediates is calculated via DFT by also including the zero-point energy(ZPE) and vibrational contributions. Normally the entropy contribution is lowas the temperatures are not so high under photoelectrochemical conditions.

The main difference between the frameworks for electrochemistry and pho-toelectrochemistry is the origin of the driving potential.85–87,92 In electro-chemistry, the driving potential can be varied externally, whereas thephotoelectrochemical driving force is the redox potential originating from thephotoinduced hole in the valence band. It should be noted that the energyposition of the valence band of oxide semiconductors depends on pH, but thesame dependence applies to the free energy of each reaction step for wateroxidation. Thus, to a first approximation, the thermodynamics of the reactionis unchanged by changes in the pH.

10.5.1 Pourbaix Surface Diagrams

From the photoelectrochemical framework it is now possible to establish asurface phase diagram, which gives an estimate of which reaction intermediatesare adsorbed on the surface for given pH values and under dark or light con-ditions, see Figure 10.7. In electrochemistry these phase diagrams are calledPourbaix diagrams.83,93,94 Although Pourbaix diagrams were originally con-structed for bulk transitions, the use of first-principles has shown that they canaccurately describe which reaction intermediates are present and which areunstable under electrochemical conditions. The phase diagrams are only ap-plied under stable conditions, which implies that a photostationary concen-tration of holes builds up at interface in order to generate a driving force for thephotoanode reaction. Furthermore, it should be noted that the potential ex-perienced by a metal nanoparticle at the surface of a photoelectrode duringillumination differs from the externally applied potential because the quasiFermi level of holes shifts to positive potentials, leading to a shift in the Fermilevel of the metal particle.

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10.5.2 Reaction Mechanism

A four proton and electron transfer water oxidation mechanism could consistof the following elementary steps,

H2Oþ � ! OH� þHþ þ e�

OH� ! O� þHþ þ e�

O� þH2O! OOH� þHþ þ e�

OOH� ! O2 þ � þHþ þ e�

Once the reaction mechanism has been proposed, the photoelectrochemicalframework allows the calculation of the reaction landscape of all reactionintermediates from first-principles. Since all steps only involve one proton andelectron transfer, the height of the different steps scales linearly with the redoxpotential. See Figure 10.8 for an example of the thermodynamics of water-oxidation at different potentials.

The described photoelectrochemical framework has been used to studyphoto induced water-oxidation on rutile TiO2(110),

86 WO3 (various facets),87

and Fe2O3(0001).92 In most cases, the redox potential arising from the valence

band edge is sufficiently positive to make the reaction thermodynamicallyfavorable. Only in the case of Fe2O3 was, water oxidation predicted to beprohibited on some surface terminations. However, these terminations were notthe most stable ones, which implies that they do not play a part under normaloperation conditions.

The electrochemical framework that has been described so far does not lenditself easily to the calculation of activation barriers; hence, information on thekinetics of any reaction is still missing. However, there exist extensions that dealwith this issue. For instance, by varying the number of protons/electrons in the

Figure 10.7 The relative stability of all considered surface terminations as a functionof applied potential and at two different pH, namely, pH¼ 0 (left) andpH¼ 14 (right).Reprinted with permission from ref. 92. Copyright 2011 AmericanChemical Society.

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electrolyte, Skulasson et al.80,90 were able to determine the activation energy forthe hydrogen evolution reaction as a function of electrode potential. One im-portant conclusion from the study is the manifestation of a Brønsted–Evans–Polanyi-type relationship between activation energy and reaction energy foundthroughout surface chemistry.95–97 This result implies that theoretical electro-chemistry can use a lot of the knowledge gained in heterogeneous catalysis.98,99

Since the driving force in photoelectrochemistry comes – at least as a firstapproximation – from the hole (electron) at the edge of the valence (con-duction) band, it still remains to be confirmed whether the same technique canbe used to find new catalyst formulations in photoelectrochemistry. However,with the envisioned development of band gap design it might still be possible.

10.5.3 Overpotential

Recently, Abild-Pedersen et al.100 demonstrated the existence of approximatelylinear relationships between the adsorption energy of hydrogen and non-hydrogen containing species, e.g. OH and O, over many different materials.In combination with Brønsted-Evans-Polanyi relations,95 this finding hasprovided a breakthrough in the computational screening of heterogeneouscatalysts. In electrochemistry, the same linear relations have been used to showthat in the oxygen evolution reaction (OER) and the oxygen reduction reaction(ORR) there exists a fundamental overpotential.101–103 This was done by cal-culating the difference in Gibbs free energy for each reaction step, and by use of

Figure 10.8 The water splitting reaction at different potentials. At potentials between0 and 0.78V all steps in the oxygen reduction are exothermic. Forpotentials beyond 2.55V all water splitting reaction steps becomeexothermic. This variation is obtained by varying the term eU in thefree energy per electron transferred to the electrode.Reprinted from Chemical Physics 319 (2005) 178, Copyright 2005, withpermission from Elsevier.95

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the linear scaling relations, expressing each reaction intermediate as a functionof one of these differences or a linear combination. Especially the combinationDGO*�DGOH* has proved universal for the description of OER and ORRactivities of a large set of data.102 The OER volcano plots for different sub-strates are shown in Figure 10.9. Furthermore, the fundamental overpotential,which originates front the fixed distance between the binding of OH and OOH,is also shown. This observation provides an upper limit on how good an OERelectrocatalyst can be expected to be.

The origin of the fundamental overpotential provides directions of how toovercome this limitation. If a catalyst is able to stabilize *OOH with respect to*OH (i.e. make the free energy difference between *OOH and *OH to valuescloser to 2.46 eV), it will circumvent the fundamental overpotential. One pos-sible pathway would be to use 3D structures that are able to differentiatebetween the intermediates by, e.g. confining the OOH group. Another sug-gestion is the use of the so-called Hangman porphyrins,103 which have indeed

Figure 10.9 The Gibbs free energy of adsorbed HOO*, O*, HO* on rutile surfaces(110) and (101) and for WO3 (200,020,002). Filled symbols represent theadsorption energies on the surfaces with a high coverage of oxygen. Thehollow symbols represent adsorption energies on the clean surfaces withno nearest neighbors. Triangles are for HOO* and HO* species, whilethe circles are for O* species. The difference between the two red dashedhorizontal lines is the standard free energy for oxygen molecule to beformed. (Left) The activity trends for oxygen evolution (OER) on therutile surfaces (black line). The negative value of theoretical overpoten-tial is plotted against the descriptor for OER (the standard free energy ofHO* oxidation). Solid black triangles include the effect of the interactionwith the oxygen from the neighboring sites, while the red trianglesinclude the effect of the interaction with the HO* species. The diamondsymbols represent the overpotentials for WO3. The minimum possibleoverpotential for any oxide is shown by the red arrow (the differencebetween the peak of the volcano and the zero line).Reprinted from ref. 11, Copyright 2012, with permission from RoyalSociety of Chemistry.

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been shown to be good catalysts for OER. However, there is no support of thisclaim from first-principles as of yet.104

10.6 Conclusion and Outlook

This chapter has discussed a selection of the many processes involved inphotoelectrochemistry and shown how first-principles methods can help ininterpretation and future improvements. The many success stories reveal apromising future, i.e. soon first-principles calculations will not only be used forreproducing the experimentally known facts about a given photoelec-trochemical reaction on a given photoanode (cathode) material, but could alsobecome the standard starting point when a new photoelectrocatalyst for aknown reaction is desired, or even when an unknown reaction to obtain a givenproduct is needed. However, some issues remain that should be addressed andsolved before this promise can become reality. The use of more accurateexchange-correlation functionals seems to be able to push first-principles frombeing explanatory to predicative. This development will certainly continue, andhopefully band gap design will soon be reality. As for the plasmon-assistedphoton capture process, the clear link between the microscopic descriptionprovided by first-principles and the nanostructured plasmon active particleremains to be settled. The problem associated with charge transfer and theconnection between DFT calculations of photoelectrocatalysis and dynamicalsimulations of electron transfer has not yet fully matured. The use of thestandard hydrogen electrode as the reference point has opened up the possi-bility of using first-principles methods for some straightforward applications toelectrochemical and photoelectrochemical systems. However, the approach canonly be used for electrochemical steps, i.e. elementary reactions in which aproton and an electron are simultaneously transferred. In the case of reactionswhere only a proton or only an electron is transferred, the standard hydrogenelectrode approximation cannot be applied. In spite of these remaining un-resolved issues, the future looks bright for first-principles and its impact onphotoelectrochemistry.

Acknowledgements

The author wants to acknowledge the financial support of the SwedishResearch Council. Furthermore, Chris Cornwall is acknowledged for proof-reading the chapter.

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