Dft Celdas Solares

Recent advances in the use of density functional theory to design efficientsolar energy-based renewable systemsRamy Nashed, Yehea Ismail, and Nageh K. Allam

Citation: J. Renewable Sustainable Energy 5, 022701 (2013); doi: 10.1063/1.4798483 View online: http://dx.doi.org/10.1063/1.4798483 View Table of Contents: http://jrse.aip.org/resource/1/JRSEBH/v5/i2 Published by the American Institute of Physics.

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Recent advances in the use of density functional theoryto design efficient solar energy-based renewable systems

Ramy Nashed,1,2 Yehea Ismail,2 and Nageh K. Allam1,a)1Energy Materials Laboratory, Physics Department, School of Sciences and Engineering,The American University in Cairo, New Cairo 11835, Egypt2Center of Nanoelectronics and Devices (CND), American University in Cairo/Zewail Cityof Science and Technology, Cairo, Egypt

(Received 18 January 2013; accepted 13 March 2013; published online 27 March 2013)

This article reviews the use of Density Functional Theory (DFT) to study the

electronic and optical properties of solar-active materials and dyes used in solar

energy conversion applications (dye-sensitized solar cells and water splitting). We

first give a brief overview of the DFT, its development, advantages over ab-initiomethods, and the most commonly used functionals and the differences between

them. We then discuss the use of DFT to design optimized dyes for dye-sensitized

solar cells and compare between the accuracy of different functionals in

determining the excitation energy of the dyes. Finally, we examine the application

of DFT in understanding the performance of different photoanodes and how it

could be used to screen different candidate materials for use in photocatalysis in

general and water splitting in particular. VC 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4798483]

I. INTRODUCTION

The energy demand is currently increasing at an unprecedented rate due to the high tech-

nology revolution. Humans used to depend on fossil fuels as their primary source of energy.

However, fossil fuel reserves are limited1 and their production is declining over the time. Also,

fossil fuels have serious adverse impacts on the environment because the gases emitted from

burning the fuels trap the solar radiation leading to global warming and threatening the lives of

humans and other creatures.2 For these reasons, scientists are investigating alternative sources

of energy. Solar energy is one of the most attractive options as the amount of solar energy

reaching the earth is four orders of magnitude greater than the current worlds energy

consumption.3

Dye-Sensitized Solar Cells (DSSCs) represent one of the most promising solar cell struc-

tures due to their low-cost, simple fabrication method, and relatively high efficiency that

reached 11.4%.4 The main problem limiting the further development of DSSCs is the fact that

dyes with high absorption coefficients have narrow bands and vice-versa. Forster Resonance

Energy Transfer (FRET) mechanism proves to be an efficient technique that combines high-

absorption dyes with wide-band ones in a donor-acceptor fashion leading to systems with wide-

strong absorption.5,6 In order for this mechanism to work, the absorption spectrum of the

acceptor should have large overlap with the emission spectrum of the donor and the two materi-

als should be within one Forster radius apart.5 However, the combination of dyes is still not

optimized and is based on guess-and-check procedures. A more systematic approach isrequired in order to find an optimum combination of dyes.

A similar approach to DSSCs is the solar-hydrogen production by the photoelectrolysis of

water. Here, water acts as the electrolyte medium. Hydrogen is particularly chosen since it

a)Author to whom correspondence should be addressed. Electronic mail: [email protected]. Fax: 202 27957565.

1941-7012/2013/5(2)/022701/27/$30.00 VC 2013 American Institute of Physics5, 022701-1

JOURNAL OF RENEWABLE AND SUSTAINABLE ENERGY 5, 022701 (2013)


represents the richest fuel in terms of energy per unit mass.7 Briefly, the system for solar hydro-

gen production consists of two electrodes: a working electrode, which is made of a semiconduc-

tor, and a counter electrode, which is mainly platinum, dipped in water. When the working

electrode is illuminated, electrons are excited from the valence band to the conduction band of

the semiconductor. The holes in the valence band diffuse to the semiconductor/water interface

where they oxidize water to oxygen gas and hydrogen ions. The excited electrons will flow

through the wire to the cathode where they reduce the hydrogen ions forming hydrogen fuel.

The main challenge to achieve efficient solar hydrogen production systems is to find a cheap

and stable material with wide absorption spectrum to be used as the working electrode. Some

oxides, such as TiO2, tend to be stable due to their large energy gap. Unfortunately, this large

band gap limits the absorption capabilities of the material. On the other hand, low band materi-

als, such as Fe2O3 and Cu2O, have wide absorption spectrum but tend to be unstable. This

incites researchers to use mixed metal oxides in which a high band gap oxide is mixed with a

low band gap one hoping to arrive at a stable, yet widely absorbing material. Different mixed

metal oxides such as Ti-Fe-O,8 Ti-Cu-O,9 Ti-W-O,10 Ti-Nb-Zr,11 and Ti-Pd12 have been tried

and showed very promising results for photocatalytic fuel production. However, as in the case

of DSSCs, the mixing between oxides is highly combinatorial13 and still needs to be optimized.

A systematic approach still needs to be devised to understand the effect of atomic composition

of metal oxides on the electronic and photocatalytic properties of the material.14

Computational science is considered a good approach to solve the guess-and-check prob-

lem involved in the design of DSSCs as well as solar hydrogen production systems.

Specifically, it can be used to study the changes in the electronic structures of the fabricated

systems. This would give a better understanding of the systems behavior and hence helps to

pinpoint the positions of weaknesses in the design, which would assist to arriving at more

optimized solutions. Density Functional Theory (DFT) is regarded as one of the most efficient

computational tools to be used in this domain because it is computationally inexpensive albeit

accurate. DFT is used in wide range of research topics such as structural materials, catalysis

and surface science, magnetism, semiconductors and nanotechnology, and biomaterials. Ikehata

et al.15 used DFT calculations to design a low-Youngs modulus high-strength titanium alloy.Ford motor company makes use of DFT to enhance the properties of aluminum cast alloys.16,17

Tripkovic et al.18 studied the oxygen reduction reaction mechanism on platinum surface forPEM fuel cell design using DFT calculations. DFT-based calculations have also been used to

suggest Fe-Co alloys as a good candidate for high-density magnetic storage.19 In semiconductor

nanotechnology, DFT has been used to gain insight into the electronic structure of carbon nano-

tubes, quantum dots, and semiconducting nanoparticles.20

It is clear that DFT has wide range of applications; however, we will limit our discussion

to solar energy applications. In the next section, we will discuss the development of DFT and

assess the accuracy of its different variations. Section III discusses the use of DFT in the design

of DSSCs and Sec. IV discusses its use in designing the working electrode for solar hydrogen

production.

II. OVERVIEW OF DFT

A. Pre-DFT attempts

The physical and chemical properties of any system can be determined exactly by solving

the many-body Schrodinger equation,

H^Wir;R EiWir;R; (1)

where Wi is the wave function of the system, Ei is the Eigen-values, which are the allowedenergy states produced by solving Eq. (1), and H^ is the Hamiltonian operator. For interactingatoms, H^ is defined as21

022701-2 Nashed, Ismail, and Allam J. Renewable Sustainable Energy 5, 022701 (2013)


H^ XPI1

h2

2MIr2I

XNi1

h2

2mr2i

e2

2

XPI1

XPJ 6I

ZIZJjRI RJj

e2

2

XNi1

XNj 6i

1

jrI rJj e2XPI1

XNi1

ZIjRI rij; (2)

where R {RI}, I 1,, P, is a set of P nuclear coordinates, and r {ri}, i 1,, N, is a setof N electronic coordinates. ZI and MI are the P nuclear charges and masses, respectively.

When interpreted physically, the first term on the right hand side of Eq. (2) is the kinetic

energy of the P nuclei, the second term is the kinetic energy of the N electrons, the third term

is the Coulomb repulsive potential between each pair of nuclei, the fourth term is the Coulomb

repulsive potential between each pair of electrons, and the fifth term is the Coulomb attraction

potential between the electrons and the nuclei in the system.

It is obvious that the Hamiltonian for such systems is very complicated and requires large

computational effort especially for large atoms and molecules. Also, the analytical expression

for the many-electron Hamiltonian is not known. For these reasons, various simplifications have

been introduced to Eq. (2). The first approximation is the Born-Oppenheimer approximation22

which is based on neglecting the kinetic energy of nuclei and treating their repulsive potential

as a constant. The plausibility of this approximation is due to the fact that the mass of the

nuclei is much greater than that of the electrons and thus the nuclei can be assumed stationary

with respect to the electrons. This gives rise to the so-called electronic Hamiltonian,

H^elec XNi1

h2

2mr2i

e2

2

XNi1

XNj 6i

1

jrI rJj e2XPI1

XNi1

ZIjRI rij: (3)

The electronic energy, Eelec, can be found by substituting Eq. (3) in Eq. (1). The total energycan then be calculated by adding Eelec to the constant nuclear repulsion term Enuc,

Enuc e2

2

XPI1

XPJ 6I

ZIZJjRI RJj: (4)

Although the Hamiltonian was greatly simplified by the Born-Oppenheimer approximation,

the second term in Eq. (3) still represents a computational problem as it involves pair-wise

Coulombic correlation between electrons and hence it is required to consider the contribution

of two electrons every time we write the wave function. This renders the wave function compli-

cated and the solution of Schrodinger equation hard. Hartree proposed a solution to this prob-

lem by assuming that each electron in the system feels an average potential energy due to the

other electrons.23 This allows for treating a single electron at a time and consequently to

express the wave function as a product of one-electron wave functions. It uses separation of

variables to solve Schrodinger equation, which greatly simplifies the calculations. To determine

the expressions for the one-electron wave functions, Hartree and Fock (HF) introduced a

method which took into account Pauli exclusion principle where the many-electron wave func-

tion is approximated by a product of anti-symmetrical one-electron wave functions in the form

of a Slater determinant,21

WHF w1r1; r1 w1r2; r2 w1rN; rNw2r1; r1 w2r2; r2 w2rN; rN

. ..

wNr1; r1 wNr2; r2 wNrN; rN

; (5)



where ri signifies the spatial position of the electron i and r signifies its spin. Wi(ri, ri) areexpressed as a Linear Combinations of Atomic Orbitals (LCAOs) to form Molecular Orbitals

(MOs). Using this approximation, the energy of the system can be calculated as

EHF XNi1

Hi 12

XNi;j1

Jij Kij; (6)

where Hi represents the kinetic energy and the electron-nucleus Coulomb attraction, Jij are thecoulomb integrals, which represent the repulsive potential that the electron feels due to an aver-

age distribution of the rest of the electrons, and Kij is the exchange integrals that are a quantummechanical effect occurring due to the overlapping of orbitals, which combines all possible per-

mutations of electron energy distribution in the system. This approximation is called HF or

Self-Consistent Field (SCF) approximation and it includes particle exchange in an exact man-

ner.24,25 The main drawback of this method is that the computational effort needed to compute

Eq. (5) scales by M3, where M is the number of atomic orbitals.

B. Development of DFT

Despite the different approximations that have been applied to the Hamiltonian and the

wave function, solving Schrodinger equation remains very hard and nearly impossible for large

atoms and molecules since the wave function is a function of 3N variables, where N is the

number of electrons in the system. Density functional theory solved this problem by reducing

the number of variables to three variables only.26 This is because DFT is based on using the

electron density, which is a function of the three spatial coordinates, to calculate the energy of

the system. This considerably reduced the computational cost and allowed for determining the

physical and chemical properties of large atoms and molecules.

The efforts of Thomas27 and Fermi,28 which date back to 1927, represent the seed of the

DFT. In Thomas-Fermi model, the energy of the system is calculated in terms of the electron

density as

ETFqr 310

3p22=3q5=3rdr Z

qrr

dr 12

qr1qr2

r12dr1dr2; (7)

where q is the electron density. The first term in Eq. (7) represents the kinetic energy of elec-trons, the second term is the nuclear attraction between nuclei and electrons, and the third term

is the Coulomb repulsion between electrons. The kinetic energy term is found by solving a par-

ticle in a box problem assuming a constant electron density. This is a very crude approximation

since the electron density is non-uniform and is actually rapidly changing near the nuclei. Also,

the exchange and correlation effects are neglected.21 In 1930, Dirac used the uniform electron

density approximation to introduce an expression for the exchange energy,29 which gave rise to

Thomas-Fermi-Dirac theory.30 Weizsacker31,32 was the first to target the non-uniform electrondensity problem in 1935 by providing an expression for the kinetic energy of electrons that

depends on the gradient of the electron density in the neighborhood. Considering the gradient

of electron density allowed for adding information about how the electron density changes in

the vicinity of each point in space. This led to two refinements to the Thomas-Fermi theory: (1)

Thomas-Fermi-Weizsacker theory,32 which corrects the kinetic energy term in Thomas-Fermitheory by considering non-uniform electron density but did not consider exchange correlation

energy, (2) Thomas-Fermi-Dirac-Weizsacker32 which not only corrects the kinetic energy termin Thomas-Fermi but also includes the exchange energy term using Dirac approximation.

However, this theory is still not accurate as it is based on Dirac approximation.

DFT started to attract great attention after the work done by Hohenberg and Kohn in 1964

who proved that the potential is a unique functional of electron density.26 This is a marvelous

achievement because it means that for each electron density distribution, there is one and only

one expression for the energy of the system. The proof of this theorem comes from the fact



that, in order to determine the Hamiltonian operator, one needs to determine the number of

electrons in the system as well as the positions of the nuclei. The electron density is very

powerful in this aspect as the integration of the electron density over the whole space gives the

number of electrons and the positions of the cusps found when plotting the electron density ver-

sus position represent the locations of the nuclei. According to Hohenberg-Kohn theorem, the

energy of the system can be expressed as26

E vrqrdr 1

2

qrqr0jr r0j drdr

0 Gq; (8)

where v(r) represents the nuclear potential. The first term represents the nuclear Coulombattraction, the second term is the electron Coulomb repulsion and G[q] is the sum of the elec-tron kinetic energy, T[q], and the exchange and correlation energy,26 Exc[q],

Gq TqExcq: (9)

In their original paper, Hohenberg and Kohn did not propose explicit forms to the kinetic

energy and exchange and correlation energies. Kohn and Sham addressed this problem in

1965,33 shortly after the publication of the original Hohenberg-Kohn theorem. Kohn and Sham

provided an exact expression for T[q] as well as a semi-exact expression for Exc[q]. The calcu-lation of the Exc[q] term depends on splitting it into two terms: Exchange term, Ex, andCorrelation term, Ec, where Ex is calculated exactly from Hartree-Fock equations and Ec isapproximated under the assumption of a uniform electron density. Although the calculation of

Exc[q] is very accurate, it requires large computational power since the calculation of Ex isbased on Hartree-Fock equations, which involve wave functions instead of electron density. For

this reason, a simpler expression for Exc[q] is suggested by Kohn and Sham assuming uniformelectron density for the whole expression of Exc[q]. From the above discussion, it is obviousthat the main challenge in DFT is to find the proper expression for T[q] and Exc[q].

C. Basis sets

The Kohn-Sham equation, which is analogous to Schrodinger equation, can be written as33

12r2 vr

qr0jr r0j dr

0

lxcqr

wir eiwir; (10)

where v(r) is the attractive Coulomb potential between the electron and the nuclei, lxc(r) is thedensity of Exc with respect to q, and wi(r) is the Kohn-Sham orbitals which are analogous towave functions in Schrodinger equation. The numerical solution of Eq. (10) requires expanding

Kohn-Sham orbitals in a set of pseudopotentials (PPs).34

The main types of basis functions are the Slater-Type Orbitals (STOs), Gaussian-Type

Orbitals (GTOs), Contracted Gaussian Functions (CGFs), and PPs. Slater-Type Orbitals35 are

functions which decay exponentially far from the origin. They closely resemble the true behav-

ior of atomic wave functions as they have cusps at the nuclei positions. However, they require

large computational efforts. On the other hand, Gaussian-Type Orbitals36 are not as accurate as

STOs but they are easier to handle numerically since the product of two GTOs located at differ-

ent atoms is another GTO located between the atoms, whereas the product of two STOs does

not lead to an STO.34 Contracted Gaussian functions37 represent a compromise between the ac-

curacy of STO and the simplicity of GTO where CGF is constructed by approximating STO by

a small number of GTOs. Pseudopotentials represent the most attractive basis functions for sys-

tems with large number of electrons.34 The idea of using pseudopotentials is based on the fact

that the binding energy of solids and molecules is dominated by the valence electrons of each

atom and hence only the valence electrons need to be considered in Eq. (10), which



tremendously reduces the number of electrons treated explicitly. This allows for performing

DFT calculations on large systems.

D. DFT functionals

As mentioned above, the main challenge in DFT is to find the proper expression for T[q]and Exc[q]. Several expressions have been proposed which are briefly described in this section.

1. Local density approximation

Local Density Approximation (LDA) is the first and simplest approximation in DFT. It is

based on decomposing the real problem of a non-uniform interacting system into two simpler

components: a spatially non-uniform non-interacting system to calculate T[q], and a uniforminteracting system to calculate Exc[q].

34 The expression of Exc[q] follows that proposed byKohn and Sham:33

Excq qrexcqrdr; (11)

where exc[q] is the exchange and correlation energy per electron of a uniform electron gas. InLDA, exc[q] is decomposed, like in Kohn-Sham, into two functionals: exchange functional (ex)and correlation functional (ec). The exchange functional is calculated from Diracs form

38 while

the correlation function is unknown and has been simulated in numerical quantum Monte Carlo

calculations assuming uniform electron density and yielded nearly exact results.39 In LDA,

Exc[q] is very-well approximated since the errors in ec tend to be cancelled by ex.34

2. Generalized gradient approximation

Generalized gradient approximation (GGA) builds on LDA by considering non-uniform dis-

tribution of electrons. In GGA, exc is a functional of electron density as well as its gradientwhich helps to take into account the way by which the electron density changes in the vicinity

of the point of interest. This is very crucial when considering points near the nuclei in which

the electron density is strongly changing. Nowadays, the most popular GGA in physics is PBE

which was proposed by Perdew et al.,40 whereas BLYP, which is a combination of Beckesexchange energy41 with Lee et al.s correlation energy,42 is the most popular GGA inchemistry.34

3. Meta-GGA

Although GAA has shown great improvements in calculations compared to LDA, the

chemical accuracy, which requires that the errors in calculations should exceed 1 kcal/mol, was

not reached yet.34 For this reason, several beyond-GGA functionals were introduced. Meta-

GGA43,44 is an example of beyond-GGA development in which the exchange energy depends

on the Laplacian of the spin density, r2q, or the local kinetic energy density, s. The incorpora-tion of Meta-GGA helped to solve some problems of the previous functional such as self-

interaction problem in the correlation functional, increasing the accuracy of calculating the

exchange functional by recovering the fourth order gradient expansion for slowly varying den-

sities, and obtaining a finite exchange potential at the nucleus.45

4. Hybrid exchange functionals

Although LDA and GGA give good approximations for many calculations, they tend to

underestimate the transition energy. This is because they do not contain the correct 1/R depend-

ence (where R is the distance between charges) in the exchange functional expression. Hybrid

functionals can remedy this problem through the incorporation of the exact Hartree-Fock

exchange functional.



Hybrid functionals are based on the exact adiabatic approach,46 which allows for the exact

representation of the exchange and correlation energy functional as

Excq 12

drdr0

1

k0

dkke2

jr r0j hqrqr0iq;k qrdr r0; (12)

where k is a coupling constant with k 0 corresponding to non-interacting system and k 1corresponding to fully interacting system. A non-interacting system is well-represented by

Hartree-Fock equations while GGA is a good representation for a fully interacting system with

a uniform electron density. A logical approximation to the integral in Eq. (12) is to consider

the extreme cases with k 0 and k 1 and use a weighted average to approximate Exc[q].Beckes hybrid functional,47 B3, employs this idea and is considered the most successful

exchange functional for chemical applications, especially when combined with LYP GGA42

functional for Ec to form B3LYP functional which is the most popular functional in quantumchemistry.34

5. Long-range corrected functionals

For hybrid functional, the asymptotic behavior in the exchange-correlation expression

decays as 0.2/R. This is accurate enough for small molecules. However, as the molecules

become more complicated, the discrepancy between the exact value and that obtained by hybrid

functional increases. The idea of long-range-corrected (LC) functionals, similar to hybrid func-

tionals, is based on using exact Hartree-Fock exchange functional. However, the fraction of

exact exchange increases with increasing the separation distances, whereas this fraction is kept

constant for hybrid functionals. This represents an optimization of both the computational cost

and accuracy. At small separation distances, where there is negligible difference between

Hartree-Fock and DFT calculations, the fraction of Hartree-Fock is kept low. On the other

hand, a large fraction of Hartree-Fock exchange functional is employed at longer distances. The

performance of LC functionals depends on the range-separation parameter, l, whose optimumvalue was found to be system-dependent.48 For example, the l-dependence of small moleculeexcitation energies (CO, H2CO, and CH3CO) is markedly different from that observed in larger

molecules (anthracene, indole, pyridazine, benzocyclobutendione, benzaldehyde, and pyrrole).48

This difference was attributed to the fact that the valence excited-state densities for these larger

molecules sample both the long- and short-range parts of the Coulomb potential, whereas the

small molecules fit more or less within the length scale described using normal TD-DFT.48

These observations stress the importance of having large-molecule excitation energies in the

training set of any LRC functional that is intended for use in TD-DFT.4850 In fact, the value

of l was found to be inversely proportional to the size of the molecules under study.49

6. Time-dependent density functional theory

This theory is considered as the Hohenberg-Kohn analog in time dependent systems in

which the electron density is a function of time and position. The theory was developed by

Runge and Gross in 1984 to study the excited state properties of the materials such as atomic

and nuclear processes, photoabsorption in atoms, and the dynamic response of systems.51 The

main advantages of TD-DFT include the balance of accuracy and efficiency as well as the wide

applicability range.51 However, TD-DFT must use an approximation for the exchange-

correlation energy. The main drawback of TD-DFT is the underestimation results for large

molecules even when used with hybrid functionals.52 However, combining TD-DFT with LC

functionals represents a promising approach towards an accurate study of large systems.4850

7. Performance of various functionals

The performance of LDA, GGA, and Meta-GGA functionals has been assessed by Kurth

et al.53 The study was made on several molecules and the mean relative error was calculated



for each approximation. The GGA functionals considered were BLYP, PBE, and HCTH,

whereas the Meta-GGA functionals considered were VS98 and PKZB. Table I illustrates the

mean relative error in the atomization energy, unit cell volume, and bulk modulus for each

functional. For atomization energy, GGA offers a great improvement to LDA reducing the error

from 22% down to 3%. Meta-GGA does not provide a significant improvement in this aspect.

On the other hand, LDA performs very well in approximating the unit cell volume producing

results comparable to that of GGA. The errors in bulk modulus tend to be large with the excep-

tion of PBE and PKZB, which show relatively small errors.

III. OPTIMIZING DYE-SENSITIZED SOLAR CELLS USING DENSITY FUNCTIONAL

THEORY

Figure 1 illustrates the schematic diagram of DSSCs. The operation of DSSCs can be sum-

marized as follows: Upon illumination, electrons are excited from the HOMO levels to the

LUMO of the dye. The excited electrons are then transported to the conduction band of the

semiconductor. The dye is regenerated with the help of the redox electrolyte, which compen-

sates the electrons that the dye loses to the semiconductor.

In order to have an efficient system, certain criteria should be met: (1) the absorption of

the dye should cover a wide range of the solar spectrum, (2) the energy of the dyes LUMO

should be less negative than the conduction band edge of the semiconductor, (3) the dye should

be well anchored to the semiconductor surface to allow for efficient charge transfer, (4) the

conduction band of the semiconductor should be located well above the HOMO of the dye to

minimize recombination, and (5) the Redox potential of the electrolyte should lie above the

HOMO of the dye to permit dye regeneration. Having these criteria in mind, it should be sim-

ple to design an efficient system if we can predict the band structure of each component of the

system.

TABLE I. Mean relative error for atomization energy, unit cell volume, and bulk modulus for different functionals.

Adapted from Ref. 55.

Functional Atomization energy (%) Unit cell volume (%) Bulk modulus (%)

LDA 22 5 19

BLYP 5 8 22

PBE 7 4 10

HCTH 3 6 20

VS98 2 8 29

PKZB 3 3 9

FIG. 1. Schematic diagram of dye-sensitized solar cell.



A lot of experimental work4,5465 has been done to obtain efficient systems. However, most

of that work focused on optimizing a certain criterion and neglecting the others. Therefore an

ultimately efficient system has not been realized yet, as the criteria mentioned above are very

interdependent. Only recently, the computational methods have been combined with experimen-

tal work in DSSCs to explain the behavior of current systems aiming at the design of more

optimized systems. Density functional theory plays a very important role in this aspect as it

provides information about the electronic and optical characteristics of each component in the

system as well as how would a change in a certain component reflect on the performance of

others.

For example, Kusama et al.66 studied the effect of adsorption of N-containing heterocycles,namely 4-t-butylpyridine (TBP) and imidazole, on the bandgap of TiO2. They used DFT to cal-

culate the density of states (DOSs) of the system before and after adsorption of the hetero-

cycles. They found that both the valence and conduction band of TiO2 move upwards with the

same value keeping the bandgap of TiO2 unchanged (Figure 2 (Ref. 66)). They also found that

the shift in case of imidazole is higher than that of TBP which led to a higher open circuit volt-

age in case of imidazole as the distance between the Fermi level of the semiconductor and the

Redox potential of the electrolyte increases. Also, assuming the position of the TBPs and imi-

dazoles LUMO is the same, this shift led to a lower short circuit current in case of imidazole

due to decreased injection rate. However, the position of the LUMO for different dyes is gener-

ally different as asserted by Hagberg et al.52 Therefore, it is more desirable to calculate the rel-ative shift of the conduction band edge with respect to the heterocycle in order to assess the

injection rate of electrons. Furthermore, the bandgap of TiO2 calculated from Figure 2 is about

2 eV which is quite far from the experimental value of 3.2 eV. This discrepancy is due to two

reasons: (a) using the conventional PBE functional,40 which suffers from errors due to elec-

tronic self-interaction energy,67 and (b) using DFT, which is limited to ground state calcula-

tions, to calculate excited state. The first problem can be alleviated by using a more exact

exchange correlation functionals such as hybrid functionals whereas the second problem can be

overcome by implementing time-dependent density functional theory (TD-DFT) proposed by

Runge and Gross51 who extended the Hohenberg-Kohn theorem to arbitrary time dependent

systems, which allowed for studying the dynamics of the systems.

FIG. 2. Calculated total density of states. Reprinted with permission from H. Kusama et al., Sol. Energy Mater. Sol. Cells92(1), 8487 (2008). Copyright 2008 Elsevier.



The ability of TD-DFT to accurately calculate the electronic properties of DSSCs was

proved by Mete et al.68 who calculated the HOMO-LUMO gap of ruthenium bipyridine dyeusing the conventional PBE functional in DFT as well as the hybrid B3LYP in TD-DFT. DFT

calculations led to a bandgap of 1.85 eV whereas TD-DFT calculations gave a result of 2.64 eV

which is much closer to the experimental value of 2.7 eV.68 Following this conclusion, Mete

et al. adopted TD-DFT to study the effect of attaching different ligands and halogen atoms onthe electronic properties of perylene diimide (PDI) dye molecules. They pointed out that the

energy gap depends on the size of the halogen atom as well as the geometric structure of the

molecule. As the size of the halogen increases, the bandgap decreases and the bond length

increases due to increasing the delocalization of the density of states68 (Table II). On the other

hand, the introduction of more carboxylic groups from ligands does not affect the energy gap

since they introduce energy levels below the HOMO (shown in Figure 3).68 However, introduc-

ing more of these levels by adding more carboxylic groups increases photon harvesting, which

widens the absorption spectrum of the dye. Also, increasing the number of carboxylic groups

improves the electron transfer from the dye to the semiconductor due to better anchoring.

Although B3LYP tends to be in close agreement with experimental values, the calculated

values of excitation energy are underestimated (2.64 eV versus 2.7 eV as reported in Ref. 68).

This underestimation was confirmed by Hagberg et al.52 who found that the extent of underesti-mation increases by increasing the distance between the donor and acceptor moieties, i.e., size

of the dye. This is a severe drawback of B3LYP since the use of long-chain dyes is inevitable

due to their high extinction coefficient.52 Miao et al.69 tried to find a better agreement withexperiments by trying other hybrid functionals. They compared the accuracy of B3LYP and

PBE070 in calculating the excitation energy of 39N-substituted 1,8-naphthalimides dyes and

found that PBE0 provided a slightly better accuracy with a mean absolute error of 0.21 eV

FIG. 3. Calculated TDDFT molecular orbital levels of PDI based dye molecules. Adapted from Ref. 68.

TABLE II. The effect of halogen on the bandgap and carboxylic group-PDI bond length. Adapted from Ref. 68.

Chromophore Bandgap (eV) Bond length (A)

F-PDI 2.50 1.38

Cl-PDI 2.44 1.74

Br-PDI 2.39 1.88

I-PDI 2.31 2.12



compared to 0.28 eV obtained from B3LYP with the error in the excitation energy depending

on the solvent.69 This has been asserted by Zhang et al.71 who compared B3LYP, PBE0, andMPW1K72 in calculating the electronic absorption of TA-St-CA dye sensitizer and found that

PBE0 and MPW1K are somewhat better than B3LYP with maximum absorption occurring at

378 nm for PBE0 and MPW1K and at 395 nm for B3LYP as shown in Figure 4.71 This is com-

parable to the experimental result of 386 nm.73

Another interesting observation from Figure 4 is the red shift of the dye absorption spec-

trum when being in a solvent compared to that in vacuum. The same phenomenon was

observed by Kurashige et al.74 as well as by Pastore et al.75 and was attributed to the polariza-tion of the solvent, which led to electrostatic interaction with the charge-separated excited state

leading to its stabilization.74,75 To model this interaction, a Polarizable Continuum Model

(PCM) is introduced while doing the calculations. When ignoring this phenomenon in calcula-

tions, Pastore et al. had an error of about 0.2 to 0.3 eV in the excitation energy compared toabout 0.1 eV when PCM is included.75 However, Rocca et al.76 ignored this phenomenon dur-ing their study on squaraine dyes and their calculations were within 0.14 eV of the experimental

results, which suggests that this phenomenon is very system-dependent and that the strength of

electrostatic interaction between the dye and the solvent should be considered first before decid-

ing to include or exclude PCM in the calculations.

The adsorption of the dye on the semiconductor surface represents one more parameter that

affects its absorption spectrum. It was shown experimentally7780 that the adsorption of dyes on

TiO2 leads to a blue shift of the dyes absorption spectrum due to the deprotonation of the dye

where the proton is replaced by the metal ion leading to an upward shifting of the LUMOs

energy.80 Two TD-DFT recent works by Agrawal et al.81 and Sanchez-de-Armas et al.82 wereconducted in parallel on coumarin derivatives and both studies asserted the upward shifting of

the LUMO, which led to blue shifting of the dyes absorption spectrum. Deprotonation of these

dyes when adsorbed on TiO2 was confirmed by FTIR indicating the presence of carboxylate

ion after adsorption.83 In their analysis, Sanchez-de-Armas et al. ignored long range interactionsassuming the considered systems are not very large and hence having negligible long range

interaction.82 Although this is true for the smallest dyes considered, the error becomes

FIG. 4. Calculated electronic absorption spectra of TA-St-CA. Reprinted with permission from Zhang et al., Curr. Appl.Phys. 10(1), 77-83 (2010). Copyright 2010 Elsevier.



sufficiently large for larger dyes. The error increases from 0.15 eV for the smallest dye to

0.47 eV for the largest dye considered.82 This is in agreement with the results of Agrawal et al.,where the error was 0.01 eV for the smaller dye and 0.37 eV for the larger one.81 They attrib-

uted this error to using B3LYP functional.

Almost all of the previously mentioned works depend on utilizing hybrid functionals in cal-

culating the electronic properties of the system. These functionals are not very accurate as they

ignore long range interaction, which has a pronounced effect in large systems. In other words,

these functionals provide a very fast decay of the potential and hence fail to keep the 1/r de-

pendence of the potential especially at large distances. This limits these methods to qualitative

descriptions and trends of the system only. Unfortunately, quantitative description is still

required in order to design an optimized DSSC. Several efforts have been made to find more

accurate methods. Kurashige et al.74 resorted to wavefunction-based methods, namely configu-ration interaction single (CIS) method and approximate coupled cluster singles and doubles

(CC2) method, to investigate the excited states of coumarin dyes. The main advantage of

wavefunction-based methods is that they do not suffer from self-interaction error, which is con-

sidered the main drawback of DFT. They compared these methods with TD-DFT using the pop-

ular B3LYP functional. As expected, they found that B3LYP provides good results for small

dyes but introduces relatively large errors as the size of the dye increases. Also, CIS is likely to

overestimate the excitation energy and oscillation strengths due to the lack of electronic correla-

tion. CC2 seems to be the best candidate especially for large dyes. For example, for C343 dye,

the experimental value for the excitation energy is 2.81 eV. The calculated value from B3LYP

is 3.09 eV leading to an error of 0.28 eV whereas CC2 gives 3.19 eV leading to an error of

0.38 eV. CIS overestimates these values by giving 4.15 eV leading to an error of 1.34 eV. On

the other hand, B3LYP does not seem to be a good choice for large NKX-2586 dye underesti-

mating the excitation energy value by 0.24 eV. CC2 overestimates the experimental value by

only 0.07 eV and again CIS overestimates the experimental value by 0.79 eV.74 Table III lists

the experimental together with the calculated excitation energies for different coumarin dyes

using different functionals.74

Although CC2 seems appealing from the accuracy point of view, being a wavefunction-

based method renders it computationally expensive. To solve this problem, Wong and

Cordaro50 used a LC functional, namely LC-BLYP, which was proposed by Hirao et al.49,84,85

in TD-DFT calculations and compared the results with that of Kurashige and co-workers.74 The

main idea of LC-TDDFT functional, which was initially proposed by Savin,86 is to provide a

higher percentage of the exact Hartree-Fock exchange energy as the distance increases to

recover the 1/r dependence of the potential. Wong and Cordaro studied the same dyes of

Kurashige et al. and compared the results obtained by LC-BLYP with that of CC2. Table IVcompares the values of the calculated excitation energy of the dyes in the gas phase by LC-

BLYP method with those of CC2, CIS, and B3LYP obtained from Kurashige et al.50,74 fromwhich we can observe the exceptional agreement between LC-BLYP and CC2.

This agreement between LC-BLYP and CC2 is considered a remarkable achievement

owing to the relatively modest computational cost of electron density-based LC-BLYP com-

pared to wavefunction-based CC2. However, this achievement is questionable since it was

shown that sometimes CC2 gives underestimated values as pointed out by Schreiber et al.87

This was confirmed by Pastore et al.75 who compared several TD-DFT functionals with

TABLE III. Vertical excitation energy for different coumarin dyes in methanol. Adapted from Ref. 74.

CC2 CIS B3LYP Experimental

C343 3.19 4.15 3.09 2.81

NKX-2388 2.77 3.72 2.70 2.51

NKX-2311 2.63 3.47 2.43 2.46

NKX-2586 2.52 3.24 2.21 2.45



wavefunction methods on a number of organic dyes. The long-range corrected (LC) functional,

CAM-B3LYP,88 was specifically attractive as compared to B3LYP and MPW1K hybrid func-

tionals as it always gave values higher than that of CC2 and hence compensating the underesti-

mating nature of CC2.75 Also, CAM-B3LYP was preferred to other wavefunction methods dis-

cussed as these methods tend to be computationally expensive and unpractical for large

systems. However, the performance of this functional was not compared to experiment.

From the above analysis, it seems that LC functionals are promising methods to quantita-

tively describe the electronic behavior of the system. However, more experiments should be

carried out to assess the performance of these functionals. Also, LC functionals suffer from a

practical problem which is the choice of the range-separation parameter, l, which was found tobe system and property dependent89,90 and therefore there should be a systematic way to obtain

the optimum value of l.

IV. DESIGNING EFFICIENT PHOTOANODES FOR SOLAR HYDROGEN PRODUCTION

Hydrogen is considered as one of the most promising energy carriers to replace fossil fuels

mainly because it does not produce any environmental unfriendly emissions91 besides being the

richest fuel in energy per unit mass.7 Solar hydrogen production is the most appealing method

to produce hydrogen92,93 as it depends on clean and abundant sources of energy, that is, the so-

lar energy.

To produce hydrogen efficiently, certain criteria should be met in the photocatalyst,

namely, (i) the conduction band minimum (CBM) should be located above the hydrogen evolu-

tion potential (0 eV vs. NHE) and the valence band maximum (VBM) should be located below

the oxygen evolution potential (1.23 eV vs. NHE), therefore, the bandgap should be greater

than 1.23 eV, (ii) The bandgap should be low enough to absorb light efficiently, (iii) the charge

carriers should have low effective mass to allow for good charge separation and decrease the

probability of recombination, and (iv) the photocatalyst should be stable in aqueous solutions.

In 1972, Fujishima and Honda94 used TiO2 as a photocatalyst to produce hydrogen using

solar energy. TiO2 is well-known for its high catalytic activity and stability in aqueous solution.

However, its bandgap is about 3.2 eV,95 which limits its absorption capabilities to the ultraviolet

region only. Also, the conduction band edge should be raised to allow for the hydrogen evolu-

tion.96 A lot of efforts have been made to reduce the bandgap of TiO2 by doping with nonme-

tallic elements, such as N and C, or metallic elements, such as Cr and V.97104 Adding small

amounts of these elements leads to marginal improvements. Although it reduces the bandgap, it

causes other detrimental effects due to introduction of recombination centers. Passivated co-

doping with isovalent donor-acceptor elements can provide better engineering of the bandgap

and at the same time suppress the recombination centers.96,105 It can also provide better solubil-

ity for the alloying elements, which permits higher alloying concentrations.106109

In this regard, DFT is considered a powerful tool that can give a deep understanding about

the changes in the electronic and optical characteristics of the material upon incorporating the

TABLE IV. Cap Comparison between the excitation energy calculated by different methods for coumarin dyes in the gas

state. Adapted from Refs. 50 and 74.

Dye LC-BLYP (eV) CC2 (eV) CIS (eV) B3LYP (eV)

C343 3.36 3.44 4.40 3.32

NKX-2388 (s-trans) 3.01 2.99 3.94 2.90

NKX-2388 (s-cis) 2.85 2.80 3.75 2.78

NKX-2311 (s-trans) 2.91 2.89 3.73 2.70

NKX-2311 (s-cis) 2.73 2.71 3.50 2.56

NKX-2586 (s-trans) 2.81 2.81 3.53 2.50

NKX-2586 (s-cis) 2.66 2.66 3.34 2.40

NKX-2677 2.67 2.71 3.12 2.23



co-dopants and thus help in selecting the best dopant candidates. However, the accuracy by

which the bandgap is determined is very sensitive to the functional used. A recent study on

Ta2O5 showed that the error in calculating the bandgap can be reduced from 95% to only 5%

when using PEB0 hybrid functional.110 In this study, Allam et al.110 compared between GGA-PBE and three hybrid functionals, namely HSE06, B3LYP, and PBE0. The hybrid functionals

show more accurate results because they partially incorporate the exact Hartree-Fock exchange.

PBE0 gave the most accurate result since it incorporates the highest Hartree-Fock percentage.

PBE0 and HSE06 are essentially similar. The main difference between them is that HSE

divides the exchange energy into short range and long range with the short range only incorpo-

rating Hartree-Fock, whereas the long range uses pure PBE functional. In PBE0, this split does

not occur which makes it more accurate especially in systems with large atoms where there is

long range interaction. For systems with small atoms, the two methods would give very similar

results since the long range interaction can be neglected. Although this paper does not study

the effect of doping on the band structure of metal oxides, it gives a good insight about how

one can choose a suitable functional for the system under test. The effect of doping on the

band structure of metal oxides was studied by Yin et al.96 who studied the codoping of TiO2 toremedy its limitations. The idea was based on replacing Ti with 4d and 5d cations, which have

higher energy than the 3d of Ti and hence the CBM will be raised above the hydrogen evolu-

tion potential. Also, O was replaced by 2p and 3p anions to raise the VBM and reduce the

bandgap. Figure 5 shows the resulting bandgap for high and low codoping concentration.96 For

low concentration, the symbol (X, Y) denotes that 3.1% of Ti was replaced by X and 3.1% of

O was replaced by Y. (X, 2Y) denotes that 3.1% of Ti was replaced by X and 6.25% of O was

replaced by Y and so on. For high concentration, the 3.1% is replaced by 12.5% and 6.25% is

replaced by 25%. At low concentration, (Ta, P) and (Nb, P) tend to give the lowest energy gap

as well as a slight upshift of the CBM. However, there is a large mismatch between the atomic

size of P and that of O which would affect the charge transfer negatively. (Mo, C) and (W, C)

have good bandgap but the CBM is downshifted which makes them unappealing. (2Nb, C),

(2Ta, C), (Mo, 2N), and (W, 2N) have suitable bandgaps as well as upshift of the CBM.

However, (Mo, 2N) and (W, 2N) provide better carrier mobility due to more dispersive bands

and hence they represent the best candidates for low doping concentration. For high concentra-

tion, (Mo, 2N) and (W, 2N) suffer from CBM lowering and hence they are discarded. (2Ta, C)

and (2Nb, C) have suitable bandgaps but the VBM is shifted above the oxygen evolution poten-

tial and so they are discarded too. The best candidates for high doping concentration are

(Nb, N) and (Ta, N). These calculations were based on General Gradient Approximation

FIG. 5. Calculated GGA band offsets (at the C point) for TiO2 and TiO2 alloyed with various passivated donor-acceptorcombinations in the (a) low-concentration regime, (b) high-concentration regime. The CBM of pure TiO2 is set to zero as

the reference and the band gap is corrected using a scissor operator. Reprinted with permission from Yin et al., Phys.Rev. B 82, 045106 (2010). Copyright 2010 American Physical Society.



(GGA) functional of the DFT which is well-known for its underestimation of bandgaps. To test

the accuracy of these calculations, the authors performed the calculations using the hybrid func-

tional HSE111 with 22% of the exact Hartree-Fock exchange and found that the position of the

CBM agrees well whereas the VBM is underestimated by 0.3 eV in the GGA calculations

developed by Perdew and Wang,112 which means that the bandgap reduction shown in Figure 5

is underestimated by 0.3 eV. To study the effect of doping concentration on the carrier mobility,

Yin et al.96 studied the effect of N concentration in (Ta, N) codopant system of TiO2 and foundthat increasing the concentration of N leads to higher hole mobility while leaving the electron

mobility nearly intact. This is illustrated from the density of states given in Figure 6 which

shows higher curvature at the VBM as the N concentration increases.96

Zhang et al.113 studied the effect of Ag-La codoping on the electronic and optical proper-ties of CaTiO3, known as perovskite. Although it has a wide bandgap of 3.5 eV, perovskite is

an attractive material due to its low cost, ease of synthesis, and high stability.113 Zhang et al.based their DFT calculations on GGA functional of Perdew et al.40 The band structures ofundoped CaTiO3, Ag-doped CaTiO3, and Ag-La codoped CaTiO3 are shown in Figure 7.

113

The calculated bandgap of undoped CaTiO3 is 2.45 eV, which is about 1 eV lower than the ex-

perimental value. This is due to the use of the inaccurate GGA functional. However, qualitative

conclusions can still be drawn from these values. Adding Ag to CaTiO3 shifts the VBM by

0.23 eV while keeping the CBM nearly unchanged leading to a reduction in the bandgap by

0.23 eV. Further codoping with Ag and La further decreases the bandgap by 0.55 eV. This cor-

responds to about 23% reduction in the bandgap, which is considered a significant bandgap nar-

rowing. However, the VBM is nearly flat indicating a small hole mobility which will in-turn

decrease the rate of water oxidation and increase the probability of carrier recombination.

As the reduction in bandgap comes from the upward shifting of the VBM, it is suggested

that the dopant materials introduce energy states around the valence band. This is illustrated in

Figure 8, which shows the DOS for each element in the undoped as well as the doped

CaTiO3.113 It is clear that for both undoped and doped CaTiO3, the conduction band is domi-

nated by Ti-3d orbital, which is the reason why the position of the CBM was not changed in

FIG. 6. Calculated GGA band structures for (a) pure TiO2;(b)(d)] TiO2 coincorporated with (Ta, N) with different concen-

trations in which 3.1% O, 12.5% O, and 25% O were replaced by N, respectively. Band offsets are taken into account in

these plots. Reprinted with permission from Yin et al., Phys. Rev. B 82, 045106 (2010). Copyright 2010 AmericanPhysical Society.

FIG. 7. Band structures of (a) CaTiO3, (b) CaTiO3 doped with Ag, and (c) CaTiO3 codoped with AgLa. Reprinted with

permission from Zhang et al., J. Alloys Compd. 516, 9195 (2012). Copyright 2012 Elsevier.



the different plots of Figure 7. On the other hand, for Ag-doped CaTiO3, Ag-5s extended above

the valence band and mix with O-2p orbital leading to a shift in the VBM. The same argument

applies for Ag-La codoped CaTiO3 in which La-4s extends the valence band.

Cu delafossites, CuMO2 (M group III-A and III-B elements) represent another promisingclass of photocatalysts for hydrogen production due to their excellent hole mobility and high

stability in aqueous solutions.114120 Nie et al.121 studied the electronic structure of CuMO2with M from group III-A using (LDA) of DFT. All studied compounds had indirect bandgap of

1.97 eV, 0.95 eV, and 0.41 eV for CuAlO2, CuGaO2, and CuInO2, respectively. The direct

bandgap at C was found to decrease from 2.93 eV for CuAlO2, to 1.63 eV for CuGaO2, and to0.73 eV for CuInO2 as shown in Figure 9.

121 The LDA calculations are expected to introduce

an underestimation of 0.8 eV.121 However, even with the compensation for this error, the trend

in the calculated bandgap does not follow the experimental results, which show an increase in

the bandgap from 3.5 eV in CuAlO2, to 3.6 eV in CuGaO2, and to 3.9 eV in CuInO2. The reason

for this discrepancy is that the absorption near the fundamental gap at C is negligible due tosame (even) parity of the valence and conduction band states. This low absorption is depicted

in Figures 9(d)9(f).121

FIG. 8. DOS of (a) CaTiO3, (b) CaTiO3 doped with Ag, and (c) CaTiO3 codoped with AgLa. Reprinted with permissions

from Zhang et al., J. Alloys Compd. 516, 9195 (2012). Copyright 2012 Elsevier.

FIG. 9. (a) to (c) The calculated LDA band structures for CuAlO2, CuGaO2, and CuInO2, respectively. Energy zero is at

the highest valence band at F. The VBMs appeared off F are marked by the black circles. (d) to (f) The corresponding tran-

sition matrix elements between the band edge states. Reprinted with permissions from Nie et al., Phys. Rev. Lett. 88,066405 (2002). Copyright 2002 American Physical Society.



A similar observation was reported by Huda et al.105 for group III-B delafossites in whichthe transition probability at the C point for CuLaO2 is nearly zero (Figure 10(a)) due to thesame parity violation observed for group III-A delafossites. The reason for this parity violation

stems from the inversion symmetry of their crystal structure.120 To break this symmetry, Huda

and coworkers coincorporated group III-A and group III-B delafossites105 Figure 10(a) shows

that the probability of absorption at the C point is no longer zero when La is mixed with Gaand Figure 10(b) illustrates the enhancement in the optical absorption which is around 1 eV.104

The calculations were carried out with GGA of the DFT and hence might suffer from underesti-

mation of errors. However, it gives a qualitative insight about the effect of mixing group III-A

with group III-B delafossites. Furthermore, Huda et al. tried substitutional alloying of Mn andCr with Y in CuYO2; however, the material suffered from a large electron effective mass as

well as unwanted defect levels due to the multiple ionizations of these transition metals.105

Although mixing group III-A with group III-B delafossites provided local symmetry break-

ing, it failed to provide global symmetry breaking.122 To effectively break the symmetry, an

element with large size as well as chemical potential mismatch should be incorporated.122 Huda

et al.122 proposed Bi to be incorporated in group III-B delafossites. Group III-B delafossites arepreferred to group III-A delafossites because they are direct bandgap materials113 and hence

will provide better absorption. The rationale behind choosing Bi was based on: (i) Bi is isova-

lent to group III elements and so it will not create any unwanted recombination centers, (ii) Bi

has large size and potential mismatch with group III elements, and (iii) Bi is expected to reduce

the hole effective mass by introducing delocalized states at the valence band maxima due to Bi

6 s lone pair electron.123125 Figure 11 (Ref. 122) shows the partial density of states for differ-

ent Bi-alloyed group III-B delafossites, namely, Cu(Sc, Bi)O2, Cu(Y, Bi)O2, and Cu(La, Bi)O2.

It is clear that the conduction band minimum is dominated by Bi-p states, which is the reason

for reducing the bandgap. For CuScO2, the fundamental bandgap is reduced from 3.05 eV to

FIG. 10. (a) Transition probability for CuLaO2 (blue) and Cu(La,Ga)O2 (red) at different symmetry points to show the

effect of mixing group III-A and group III-B on the transition probability of delafossites. (b) Calculated optical absorption

coefficients for Cu(La,Ga)O2. The arrow in the x-axis shows the band gap for pristine CuLaO2. The reduction of optical

band gap due to this isovalent alloying is clear from the figure. Reprinted with permissions from Huda et al., Sol. HydrogenNanotechnol. V, 77700F (2010). Copyright 2010 SPIE.

FIG. 11. Partial density of state (p-DOS) of Bi alloyed (a) CuScO2, (b) CuYO2, and (c) CuLaO2. Reprinted with permission

from Huda et al., J. Appl. Phys. 109, 113710 (2011). Copyright 2011 American Institute of Physics.



1.44 eV after doping with Bi whereas for CuYO2, the bandgap is reduced from 2.95 to

1.24 eV.122 These results would have been a real breakthrough if the material could sustain its

direct bandgap property. However, incorporation of Bi changes group III-B delafossites to indi-

rect bandgap materials.122 The minimum calculated direct bandgap is 2.543 eV for Cu(Sc,

Bi)O2 and 2.348 eV for Cu(Y, Bi)O2 which is still large for efficient visible light absorption.122

The band structures of pristine delafossites as well as the alloyed ones are shown in Figure

12.121 Another observation that can be drawn from the figure, besides the reduction in the

bandgap, is the carriers effective masses. It is clear that the electrons as well as the holes

effective masses are reduced, especially for Cu(Y, Bi)O2. The data given in Figures 11 and 12

represent high Bi concentration in which the ratio between Bi and group III-B element is 1:1.

To study the effect of concentration, low Bi concentration is considered for Cu(Y, Bi)O2 in

which Bi:Y is 11:1. For this low concentration, the reduction in the bandgap was negligible

because the width of Bi-p, which dominates the CBM, is very narrow compared to the high

concentration case.122 To decrease the bandgap at this low Bi concentration, Ga has been co-

incorporated with Bi since Ga was found to decrease the bandgap of CuYO2.104 However, the

bandgap remained indirect with the minimum direct bandgap being around 2.73 eV.122 The

reduction in the bandgap was not significant since the Ga-s states are at higher energy than that

of Bi-p.122 The calculations were done using GGA functional with U-correction125,126 to correct

for the underestimation in the calculated energy.

MCu2O2, where M is a group II-A element, is another Cu-based promising candidate for

solar water splitting. Although they have been used as transparent metal oxides, they can also

be used as photocatalysts owing to their direct bandgap. Nie et al.127 studied these materialsand found that the bandgap increases with increasing the size of incorporated group II-A ele-

ment, which suggests using small-size elements, namely Mg or Ca. The calculated bandgap for

CaCu2O2 and MgCu2O2 are 3.01 eV and 2.45 eV,128 respectively, that is comparable to the fun-

damental direct bandgap of Bi-doped group III-B delafossites.122 Figure 13 shows the bandgap

structure for the different MCu2O2 materials.128 The calculations were based on LDA and as a

result the calculated conduction bands were shifted by 1.5 eV to account for the underestimation

of LDA. Nie and coworkers showed that the contribution to the CBM comes mainly from the

group II-A element as seen in Figure 14.128 Incorporation of Bi might further reduce the

bandgap as it was the case for group III-B delafossites.122 However, the position of Bi-p states

relative to the states of group II-A element should be calculated to determine whether Bi would

introduce new states near the CBM. Also, the effect of Bi on changing the material into an

indirect bandgap still needs to be investigated.

The strong effect that Bi has on the host materials has intrigued researchers to think of Bi-

based compounds as efficient photocatalysts. BiVO4 is one of the recent compounds that were

studied intensively as a photocatalyst129133 owing to its direct bandgap of 2.4-2.5 eV (Refs.

125, 134, and 135) good optical properties, its suitable bandgap alignment for H2 and O2

FIG. 12. Band structures of (a) CuScO2, (b) Cu(Sc,Bi)O2, (c) CuYO2, and (d) Cu(Y,Bi)O2. Reprinted with permission from

Huda et al., J. Appl. Phys. 109, 113710 (2011). Copyright 2011 American Institute of Physics.



evolution131,134 and its capability to be doped with a wide range of dopants with 10 atomic per-

cent or more,136,137 which allows for its bandgap engineering. Liu et al.138 studied the photoca-talytic properties of Bi0.5M0.5VO4 (MLa, Eu, Sm, and Y). The bandgap of BiVO4 wasreduced from 2.82 eV in the undoped case to 2.58 eV in Bi0.5Y0.5VO4 (see Figure 15).

138

However, co-doping of Bi and Y resulted in an indirect bandgap. The fundamental direct

bandgap for Bi0.5Y0.5VO4 was 2.90 eV.138 These results were higher than the experimental val-

ues of 2.42.5 eV for pristine BiVO4 because the studied crystal is of zircon type whereas that

discussed in experiments has a monoclinic crystal structure. The presence of Y does not affect

the bandgap significantly because the VBM is dominated by O-2p whereas CBM is dominated

by V-3d leaving Bi and Y with little contribution as shown in Figure 15. To determine the ori-

gin of photocatalytic activity, Liu and co-workers considered different VO4 compounds,

namely, Bi0.5Y0.5VO4, La0.5Y0.5VO4, and Ce0.5Y0.5VO4 and found that only Bi0.5Y0.5VO4showed photocatalytic activity concluding that Bi ion and not VO4 was responsible for enhanc-

ing the photocatalytic response.138 This is mainly due to reducing the carriers effective masses

as reported by Huda et al. in their study of Bi-doped group III-B delafossites.121

Yin et al.139 studied monoclinic BiVO4 (which is more active and hence a better photocata-lyst than zircon BiVO4

128) and calculated the bandgap to be ca. 2.06 eV which is underesti-

mated by about 0.44 eV owing to the use of GGA functional. Figure 16(a) illustrates the calcu-

lated band structure whereas Figure 16(b) shows the density of states.139 In agreement with Liu

et al.,138 the VBM is dominated by O-2p states whereas V-3d dominates the CBM.To optimize the electron and hole conductivities, Yin et al. considered doping BiVO4 with

different group I-A, group II-A, group II-B, group IV-B, and group VI-B elements.139 They

also considered substituting O with C or N or F. Table V lists the electron ionization energies

of different donors, whereas the transition energy levels of the different acceptors are listed in

Table VI.139 (Xvac) denotes an intrinsic defect occurring due to a vacancy position of X, (Xy)

denotes a substitutional defect due to replacing X with Y, and (Xint) is an interstitial defect of

element X. Tables IV and V show that intrinsic defects (Ovac, VBi, Vint, Bivac, Vvac, BiV, Oint)

FIG. 13. Calculated semi relativistic electronic band structure for (a) MgCu2O2, (b) CaCu2O2, (c) SrCu2O2, and (d)

BaCu2O2. Energy zero is at VBM. The energy of the conduction bands are shifted upwards by 1.5 eV to correct the LDA

band gap error. Reprinted with permission from Nie et al., Phys. Rev. B 65, 075111 (2002). Copyright 2002 AmericanPhysical Society.



can only provide moderate n- and p- conductivity, respectively, if compared to the extrinsic

defects.139

Interstitial Li, Na, and K have very low ionization energy. However, due to the compensa-

tion between intrinsic and extrinsic defects, the Fermi level is pinned at 0.6, 1.2, and 0.9 eV

below the CBM for Li, Na, and K, respectively, which makes them poor electron conductors.

The same effect is noticed for group II-A and group II-B elements. For group IV-B, it is

FIG. 14. Calculated total and local density of states (DOS) for SrCu2O2. Reprinted with permission from Nie et al., Phys.Rev. B 65, 075111 (2002). Copyright 2002 American Physical Society.

FIG. 15. DOS plots of zircon BiVO, YVO, and BYV solid solution. Reprinted with permission from J. Solid State Chem.

186, 7075 (2012), Copyright 2012 Elsevier.



relatively small with only 0.2 eV pinning of Fermi level below CBM for Zr and Hf. Group

VI-B represents the best n-type conductors, especially MoV whose Fermi level is limited by its

ionization energy and does not suffer donor-acceptor compensation.139

For p-type conductivity, MgBi and ZnBi have the lowest transition energy of 0.02 eV.

However, they have high formation energy (1 eV), which makes CaBi and SrBi more attractive

as they have lower formation energy and their Fermi level position is determined by the

FIG. 16. (a) Band structure of monoclinic BiVO4, (b) Partial density of states of bulk monoclinic BiVO4. Reprinted with

permissions from Yin et al., Phys. Rev. B 83, (2011), 155102. Copyright 2011 American Physical Society.

TABLE V. Electron ionization energies of donors in intrinsic and different dopings of BiVO4. Reprinted with permission

from Yin et al., Phys. Rev. B 83, 155102 (2011). Copyright 2011 American Physical Society.

Ei(0/1) Ei(0/2) Ei(0/3) Ei(0/4) Ei(0/5) Ei(0/6)

Ovac 0.22 0.17VBi 0.52 0.68

Vint 0.31 0.46 0.56 0.76 0.94

Liint 0.02

Naint 0.03

Kint 0.04

Mgint 0.06 0.12

Caint 0.04 0.11

Srint 0.04 0.10

Znint 0.08 0.15

TiBi 0.20

Tiint 0.20 0.27 0.39 0.50

ZrBi 0.04

Hfint 0.05 0.18 0.30 0.40

CrBi 1.08 0.85 1.16

CrV 0.41

Crint 0.70 0.77 1.03 1.08 1.30 1.40

MoBi 0.21 0.42 0.57

MoV 0.04

Moint 0.67 0.74 0.90 0.95 1.14 1.22

WBi 0.09 0.16 0.23

WV 0.01

Wint 0.50 0.55 0.68 0.72 0.88 0.93



transition energy (0.1 eV above VBM). Also, NaBi and KBi show similar values of 0.1 and0.2 eV above the VBM, respectively. For substitutional alloying of O, the results did not give

good n- or p-conductivity due to strong compensation from intrinsic defects.139

Although Table VI suggests that WV would provide shallower donor level than MoV and

hence provide better electron conduction, Yin et al. suggested that incorporation of Mo wouldgive the most efficient electron conduction due to donor-acceptor compensation, which pins the

Fermi level at 0.2 eV below the CBM.138 However, Park et al.140 proved experimentally thatincorporation of W gave higher photocurrent than Mo. Incorporation of W increases the photo-

current by 6 times than that of pristine BiVO4 whereas incorporation of Mo increases the cur-

rent by only 3.5 times.139 This is because the difference in ionization energy between W and

Mo is about 0.02 eV,140 which is in close agreement with the 0.03 eV shown in Table V.140

Consequently, W can give electrons more efficiently to the host material than Mo. There seems

to be a controversy between the experimental results of Park et al. and the theoretical calcula-tions of Yin et al., which requires more work to resolve. Coincorporation of W and Mo leadsto a better improvement compared to incorporation of W alone as shown in Figure 17(a).139

This is due to increasing the carrier density which in turn increases the electric field in the

depletion region and hence provides better separation of charge carriers.140 The charge density

for W/Mo-doped BiVO4 is nearly twice that of W-doped BiVO4 as depicted from the slope of

the Mott-Schottky curve in Figures 17(b) and 17(c).140 DFT calculations show that the shape of

the band structure as well as the bandgap is nearly equal for pristine BiVO4, W-doped BiVO4,

and Mo-doped BiVO4 asserting that the improvement of the photocatalytic activity is due to

better charge carrier separation (see Figure 18).140

V. CONCLUSION

DFT represents a very effective means that can provide a systematic approach towards the

design of efficient solar energy conversion systems such as sensitizers for dye-sensitized solar

TABLE VI. Transition energy levels of acceptors in intrinsic and different dopings of BiVO4. Reprinted with permission

from Yin et al., Phys. Rev. B 83, 155102 (2011) Copyright 2011 American Physical Society.

ei(0/-1) ei(0/-2) ei(0/-3) ei(0/-4) ei(0/-5)

Bivac 0.14 0.16 0.18

Vvac 0.23 0.25 0.27 0.30 0.26

BiV 1.23 1.15

Oint 0.74 0.49

LiBi 0.09 0.11

LiV 0.22 0.24 0.27 0.29

NaBi 0.08 0.11

NaV 0.23 0.25 0.28 0.30

KBi 0.16 0.18

KV 0.22 0.24 0.27 0.28

MgBi 0.02

MgV 0.20 0.22 0.25

CaBi 0.08

CaV 0.21 0.24 0.25

SrBi 0.09

SrV 0.21 0.25 0.26

ZnBi 0.02

ZnV 0.21 0.24 0.26

TiV 0.15

ZrV 0.17

HfV 0.16



cells and photoanode materials for water splitting. DFT is preferred over ab-initio methods dueto its considerably reduced computational cost as it is based on calculating the electron density

instead of the wavefunction. Different functionals have been reported to investigate the best

candidate dyes for use in dye-sensitized solar cells. LDA- and GGA-based functionals tend to

underestimate the excitation energy. Implementing hybrid functionals in TD-DFT give good

results for small dyes but tend to underestimate the results for large ones. LC-based functionals

incorporated in TD-DFT are the best candidates for larger dyes allowing one to reproduce the

electronic absorption spectra of organic dyes. Using TD-DFT, the LC formalism can be used to

correctly predict any increase in the excited-state electric dipole moment of the dyes. As a

result, the LC/TD-DFT formalism will help provide the insight needed to guide the design of

efficient solar cell dyes. In particular, one promising approach will be to investigate light-

harvesting organic sensitizers, which selectively absorb photons with specific wavelengths. To

this end, the understanding of critical features such as excitation energies and charge-transfer

states using the LC/TD-DFT technique can provide a step towards this goal

For solar-driven hydrogen production, metal oxides are the most promising candidates for

robust photoanodes. However, metal oxides should be modified to satisfy the requirements for

efficient photoanodes. DFT is a very effective means to screen materials and indentify the most

efficient mixed metal oxides. Hybrid functionals seem to give very accurate results for such

systems, reducing the discrepancy between theoretical calculations and experiments to only 5%,

provided the right functional is chosen. In order to be able to select the right functional for the

system, one should look closely at the expression of the exchange-correlation energy of the

functional and have a rough understanding about the correlation between the electrons in the

system under study. For example, for systems like Ta2O5, functionals that implement the exact

FIG. 17. (a) Linear sweep voltammograms of undoped BiVO4 (blue), W-doped BiVO4 (red), and W/Mo-doped BiVO4

(black) with chopped light under visible irradiation in the 0.1M Na2SO4 aqueous solution (pH 7, 0.2M sodium phosphatebuffered). Beam intensity was about 120 mW cm2 from a full xenon lamp, and the scan rate was 20mV s1; (b) Mott-Schottky plots of W-doped BiVO4, (c) Mott-Schottky plots of W/Mo-doped BiVO4. AC amplitude of 10mV was applied

for each potential, and three different AC frequencies were used for the measurements: 1000Hz (blue), 500Hz (red), and

200Hz (black). Tangent lines of the M-S plots are drawn to obtain the flat band potential. Adapted from Ref. 140.



Hartree-Fock on longer range are more desirable since the Ta atoms are quite large and so they

are expected to exert a long range force on the nearby nuclei and electrons. For this reason,

PBE0 gives a better approximation to the bandgap of Ta2O5 than HSE06. In fact, this is the

main source of discrepancy that is usually found between theoretical and experimental data.

This opens the vista for further work to resolve these controversies. In this regard, DFT can be

considered as a fast computational screening method, with respect to stability and bandgap, to

discover new light harvesting materials for water splitting.

FIG. 18. Density of states projected onto the Bi 6s (red), Bi 6p (pink), O 2p (blue), and V 3d (green) states for: (a) pristine

BiVO4, (b) W-doped BiVO4, (c) Mo-doped BiVO4. Adapted from Ref. 140.



ACKNOWLEDGMENTS

This research was partially funded by The American University in Cairo, Zewail City of

Science and Technology, the STDF, Intel, Mentor Graphics, andMCIT.

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