Semiconductors Used in Photovoltaic and Photocatalytic Devices

Article pubs.acs.org/JPCC

Semiconductors Used in Photovoltaic and Photocatalytic Devices: Assessing Fundamental Properties from DFT Tangui Le Bahers,*,† Michel Rérat,‡ and Philippe Sautet† †

Université de Lyon, Université Claude Bernard Lyon1, ENS Lyon, Centre Nationale de Recherche Scientifique, 46 allée d’Italie, 69007 Lyon Cedex 07, France ‡ Equipe de Chimie-Physique, IPREM UMR5254, Université de Pau et des Pays de l’Adour, Hélioparc, 2 avenue du Président P. Angot, 64053 Cedex, France S Supporting Information *

ABSTRACT: The photovoltaic and photocatalytic systems generally use at least one semiconductor in their architecture which role is to absorb the light or to transport the charge carriers. Despite the large variety of working principles encountered in these systems, they share some fundamental steps such as light absorption, exciton dissociation, and charge carrier diffusion. These phenomena are governed by fundamental properties of the semiconductor like the bandgap, the dielectric constant, the charge carrier effective masses, and the exciton binding energy. The ability of density functional theory to compute all of these properties is evaluated. From the particularly good results obtained with the HSE06 functional, it can be concluded that DFT is a reliable tool for the evaluation and prediction of these key properties which open nice perpectives for in silico design of improved semiconductors for solar energy application. In the light of these calculations, some experimental observations on the difference of efficiencies between semiconductors like TiO2 anatase and rutile or ZnO are interpreted.

1. INTRODUCTION

Although the working principle of this type of devices can be complicated and different from one system to another, some fundamental steps are common to all of them. These steps are summed up in Figure 1, each step inducing a specific requirement on the SC properties. Step 1: The light absorption. This is the first step for devices where a SC has to absorb the light. It promotes an electron (e−) from the valence band to the conduction band of the SC, leaving an electronic vacancy in the valence band, called a hole (h+). The electron and the hole interact through a Coulomb attraction, giving an entity called exciton. In the case of sunlight absorption and for photovoltaic application, an optimum gap between the valence and conduction bands of the SC exists. As the curve of the maximum efficiency as a function of the bandgap, Eg, is pretty flat around the optimum,26 one can defined a region of optimum, between 1.1 and 1.4 eV,26 more than a unique optimum. This zone of Eg optimum represents a compromise between a high photocurrent in the solar cells (obtained by diminishing the gap) and a high photovoltage (obtained by increasing the gap). This requirement on the gap can be different if the light does not belong to the sunlight spectrum (indoor application for instance) or if the objective is not electricity production but chemical reactions (water splitting for instance). In the case of chemical reaction, a compromise needs

Harvesting energy from sunlight is a major challenge for sustainable development. Devices to convert the energy of light into other forms of energy such as electricity (photovoltaic systems) or chemical compound (photocatalysis, water splitting, CO2 photoreduction, etc.) play a key role for that objective since they represent a real opportunity to produce electricity, chemicals, hydrogen fuels, etc. at low environmental and economical costs.1−5 Generally these systems involve a semiconductor (SC) for the light absorption or for the conduction of the photogenerated charge carriers or both. Table 1 presents some examples of applications where a SC is used for the light conversion process. Table 1. Example of applications of the light conversion along with SCs used for this application. The stars indicate SCs that are used for the light absorption. example of semiconductors found for these application

applications inorganic photovoltaic̈ dye-sensitized solar cells and quantum dots-sensitized solar cells CO2 photoreduction water splitting waste photodegradation

Si*,6 Ge*,7 GaAs*,6,8 CdTe*,6,9 CuInS2*,10 CdS11 3 TiO2, ZnO,12 SnO2,13 NiO,14 CdSe*,15 CdTe*15 16 TiO2, TiO2*,17,18 In(OH)3*19 Fe2O3*,20 CdS*,21 NiO,22 Bi2S3*,2 Ta3N5*23 BiVO4*,24 BiWO6*,2 TiO2*25

© 2014 American Chemical Society

Received: September 30, 2013 Revised: February 25, 2014 Published: February 28, 2014 5997

dx.doi.org/10.1021/jp409724c | J. Phys. Chem. C 2014, 118, 5997−6008

The Journal of Physical Chemistry C

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Figure 1. Schematic representation of the three fundamental steps encountered in photovoltaic and photochemical systems involving a semiconductor. The quantitative values of the requirements are determined for photovoltaica and water splittingb applications.

Table 2. Properties of the Selected Semiconductorsa compound

structure

εr

Eg

C

diamond

5.50 (i)34

5.735

Si

diamond

1.17 (i)34

12.134

Ge

diamond

0.74 (i)34

16−16.534

CdO CdS

rocksalt wurtzite

1.10 (i)34 2.48 (d)34

CdSe

wurtzite

1.73 (d)34

CdTe

zincblende

1.47 (d)34

18.938 8.3 (⊥)34 8.7 (∥)34 9.1 (⊥)34 9.3 (∥)34 10.434

GaAs

zincblende

1.52 (d)42

12.943

TiO2-A

anatase

3.2 (i)46

TiO2-R

rutile

3.0 (d)46

ZnO

wurtzite

3.4 (d)34

45.1 (⊥)47 22.7 (∥)47 110 (⊥)48 260 (∥)48 7.8 (⊥)34 8.7 (∥)34

me*

mh*

1.4 (⊥)34 0.36 (∥)34 0.19 (⊥)34 0.92 (∥)34 0.08 (⊥)34 1.57 (∥)34 0.2139 0.2534 0.1134 0.0934 0.0742

0.7534

Eb 80−19034,36

(h)34 (l)34 (⊥)34 (∥)34

1537

0.7 (⊥)34 5 (∥)34 0.45 (⊥)34 >1 (∥)34 0.12 (l)34 0.81 (h)34 0.55 (h)44 0.08 (l)44

2740

0.54 0.15 0.04 0.28

437

1537 1041 545

10−2049 0.2834

0.5934

6050

Eg is the bandgap (in eV), (d) and (i) mean direct and indirect bandgap respectively. εr is the total dielectric constant. me* and mh* are the electron and hole effective masses respectively (given in electron mass). The symbols ⊥ and ∥ mean transverse and longitudinal directions, respectively, while (h) and (l) for the hole mean heavy and light. Eb is the exciton binding energy (in meV). The empty cells mean that no data or reliable data were found. a

also to be achieved: a low band gap insures the absorption of a large fraction of the sunlight but the energy of the electron/ hole should be high enough to perform the given reaction. For water splitting, the optimum gap is around 2 eV. Generally whatever the application, an optimum gap can be defined. The nature of the bandgap (i.e., direct or indirect) plays also an important role since it modifies the absorption coefficient of the SC. An indirect bandgap gives rise to a low absorption coefficient that must be compensated by an important thickness of the SC in the final device. Consequently, the diffusion length of charge carriers must be high to allow them to diffuse toward the electrodes. The consequence is that in the case of an indirect bandgap SC, the diffusion properties play a more important role than in direct bandgap SC. A discussion on the diffusion properties of charge carriers is presented in step 3.

Step 2: The exciton dissociation. The electron and the hole have to be dissociated to obtain free charge carriers, which will be used in the working of the device. This is called the exciton dissociation. The exciton is characterized by its binding energy, Eb. Eb has to be as low as possible to facilitate the dissociation. If we assume that the exciton will be dissociated by the thermal energy, Eb should be lower than kBT (around 25 meV at room temperature). In a dielectric environment, the electrostatic force between the electron and the hole can be approximated by the eq 1.

F=

e2 4πε0εr R2

(1)

We can see that the strength of the binding (and consequently Eb) is reduced if the dielectric constant of the 5998



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SC, εr, is high. If we look at dielectric constants frequently encountered in SCs used in photovoltaic (table 2), a value of 10 or more is good enough to obtain a Eb value lower than 25 meV (see the Results and Discussion section for more discussion on this). As a counter example, polymers studied for organic photovoltaic have low εr, generally less than 4.27 The consequence is that the exciton cannot be dissociated by the thermal energy and the entire solar cell architecture had to be rethought to permit the exciton dissociation giving a new architecture called the “bulk heterojunction”.27 Step 3: The diffusion of free charge carriers. In this last step, the electron and the hole are free to move separately. They have to diffuse to the active sites where they will be used (the back electrodes in a photovoltaic system or the catalytic site in a photocatalytic device) before their recombination. The diffusion coefficient, D, represents the facility of diffusion of a charge carrier in a SC. D is linked to the mobility, μ, of the charge carrier through the Einstein relation (eq 2) which in turn is linked to effective mass (m*) and the collision time (τ) of the charge carrier (eq 3). μ and D are increased if the effective mass is low, which means a high delocalization of the charge carrier, and if the collision time is high which is obtained in a SC without defects. D=

kBT μ e

(2)

τ m*

(3)

μ=e

comparison purposes among isoelectronic systems (C, Si, and Ge) having the same structure (diamond). GaAs and CdTe were selected for their important use in photovoltaic (inorganic photovoltaic and quantum dots-sensitized solar cells).6,15 CdS and CdSe are semiconductors frequently found in photochemical (water spitting21) or photovoltaic11,15 systems where they can play the role of light absorbers or charge carrier conductors. The CdO was selected to see the influence of the electronegative oxygen on the electronic properties of II−VI semiconductors containing Cd atoms. Finally, the oxides TiO2 (rutile and anatase) and ZnO were selected since they are frequently encountered in dye-sensitized solar cells (DSSCs)3,12 and in photocatalytic systems (waste degradation, water splitting, etc.).25,33 From the DFT point of view, the idea is not to perform an exhaustive benchmark with all possible functionals but to only use a selected set of functionals chosen for their importance in the DFT community covering the different families of functionals. The selected functionals are PBE (GGA), B3LYP (global hybrid, 20% of Hartree−Fock exchange), PBE0 (global hybrid, 25% of Hartree−Fock exchange), and HSE06 (rangeseparated hybrid). The manuscript is organized as follows. In section 2 , the methodology of calculation of the selected properties is presented. Section 3 is dedicated to the computational details. Then the results and discussion appear in section 4. To facilitate reading of the results, the computed and experimental properties are gathered in graphics, but numerical values are fully available in the Supporting Information.

In Table 2, the effective mass of electrons and holes of common SCs found in photovoltaic devices (Si, CdTe, GaAs) is generally lower than 0.5 me (me being the electron mass), at least in one direction. This value can be seen as a requirement for the effective mass of the charge carrier; that is, the effective mass should be lower than 0.5 m e , at least in one crystallographic direction, in order to have a good mobility. Nowadays, most of DFT codes, using periodic boundary conditions, are able to compute some properties presented in the previous paragraph. Consequently, DFT can help scientists to design new semiconductors for photovoltaic or photochemistry applications by computing these properties since some of them, such as the effective mass, can be quite difficult to obtain experimentally. If several studies addressed the reliability of DFT to reproduce a specific property such as the band gap,28−30 the dielectric constant31 or the effective masses,32 to the best of our knowledge, no study were performed to determine the ability of DFT to reproduce the exciton binding energy (Eb) and simultaneously all the fundamental properties (Eg, εr, m*, Eb) determining the ability of a SC for its use in photophysical or photochemical devices. The final objective of the article is to develop a theoretical protocol for computing and predicting all these properties. Thus DFT calculations will support experimenters for the choice of the SC to study before starting a time and resourceconsuming synthesis and device characterizations. The accuracy of DFT calculations is directly linked with quality of the functional used to describe electronic exchange and correlation. In this work, several popular functionals are benchmarked for the computation of the properties presented above on a set of SCs. The SCs chosen for this study are gathered in the table 2 along their experimental properties. The Si and Ge were chosen as good representatives of SCs used in photovoltaic devices.6 The diamond (C) was selected for

2. METHODOLOGY In this section, the way of computing the different properties is presented. I. Bandgap Calculation. The bandgap, Eg, is computed as the difference between the minimum energy of the conduction band and the maximum energy of the valence band. These energies are obtained by solving self-consistently the monoelectronic Khon−Sham equations. The bandgap here obtained is called the monoelectronic bandgap. Post-DFT calculations, such as GW along with the Bethe− Salpeter equations, can compute an excitonic bandgap by taking into account the exciton formation during the light absorption process. A postanalysis of TD-DFT calculations can also be done to estimate the excitonic bandgap.51 Nevertheless, the comparison between the monoelectronic bandgap and the excitonic one is not in the scope of this manuscript. II. Dielectric Constants. When an electric field (E⃗ ) is applied on a dielectric material, an electric displacement field (D⃗ ) appears within it. Both electric fields are linked by the dielectric constant εr. D⃗ = ε0εrE ⃗

(4)

This dielectric constant represents the ability of a dielectric material to screen the external electric field by the apparition of a polarization. This polarization comes from the reorganization of the electronic density or from the motion the ions constituting the material. The dielectric constant induced by the electronic density is noted ε∞ and the contribution of the dielectric constant involving the ionic motions is named εvib. Finally we have

εr = ε∞ + εvib 5999

(5)



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1 1 1 = + * μ me mh*

The electronic contribution to the static dielectric tensor is evaluated through a coupled-perturbed Hartree−Fock/Kohn− Sham (CPKS) scheme adapted to periodic systems. This is a pertubative, self-consistent method that focuses on the description of the relaxation of the crystalline orbitals under the effect of an external electric field. The perturbed wave function is then used to calculate the dielectric properties as energy derivatives. Further details about the method and its implementation in the CRYSTAL and VASP programs can be found elsewhere as well as some recent examples of its application.31,52,53 Here, let us simply remind that, the calculation of the ε∞ involves the gap between occupied and empty orbitals. A poor description of the SC gap generally gives rise to a bad description of ε∞. The vibrational contribution is computed from the phonon spectrum, following the formula 6.54 νp is the phonon frequency of mode p, V is the unit cell volume, and Zp is the mass-weighted mode effective Born vector. The intensity Ip of IR absorbance for a given mode p is proportional to |Zp|.2 εr = ε∞ + εvib = ε∞ +

4π V

∑ p

For the cubic structures, the effective masses of the electron and the hole are taken as the average of the effective masses computed in the different crystallographic directions. For the noncubic systems, the effective masses of the electron and the hole and the dielectric constant are computed using the eqs 10−12.55 εr, ⊥ 1 2 = + * * me 3me, ⊥ 3εr, me,* (10)

(6)

III. Effective Mass. The effective mass is related to the curvature of the band at the top of the valence band or at the bottom of the conduction band. It can be obtained by fitting these band extrema by the formula:34 ℏ2 ℏ2 E(k) = E(k 0) ± (k 0 − k )2 ± (k 0 − k⊥)2 m* m⊥*

μ εr 2

(11)

εr =

(12)

εr, ⊥εr,

3. COMPUTATIONAL DETAILS Calculations using the PBE,56 B3LYP,57 and PBE058 functionals were performed with the ab initio CRYSTAL09 code,59 making use of localized (Gaussian) basis sets and solving selfconsistently the Hartree−Fock and Kohn−Sham equations thus allowing the efficient use of hybrid functionals. For the S, Se, Ge, Ga, and As atoms, the all-electron triple-ζ valence basis set containing polarization functions developed by Peintinger et al. was used.60 For the O, Si, Ti, and Zn atoms, the 62111G(d),61 86-311G(d),62 86-411G(2d),63 and 86-411G(2d)64 basis sets were used, respectively. For the Cd and Te atoms, the relativistic pseudopotentials developed in the Stuttgart university along with the double-ζ valence basis set (cc-pVDZ) were used.65,66 For the cc-pVDZ basis sets, the most diffuse functions (i.e., α < 0.06) were removed to facilitate the SCF cycles convergence. Reciprocal space is sampled according to a sublattice with a 8 × 8 × 8 k-points mesh for geometry optimization, CPKS, and vibrations calculations. A 12 × 12 × 12 k-points sampling was used for gap and band structure calculation. The convergence criterion for the SCF cycle was fixed at 10−10 Ha per unit cell for geometry optimization, CPKS, and frequencies calculations. The calculation of vibration frequencies has been performed within the harmonic approximation to the lattice potential and infrared intensities are obtained through the Berry phase method.67 The calculations performed with the HSE0668,69 functional were done with the Vienna Ab-Initio Software Package (VASP) along with the 5.3 version. For this functional, the Coulomb screen parameter, ω, was fixed to 0.11 a0−1 according to the results presented in the literature.70 The convergence criterion for the SCF cycle was fixed at 10−6 eV per unit cell. The Encut value, defining the basis set size, was chosen as Enmax * 1.33, where Enmax is the recommended cutoff value for the PAW method. The k-points sampling was done with 7 × 7 × 7MonkhorstPack grid for the geometry optimization and a 15 × 15 × 15 Monkhorst-Pack grid for the infinite dielectric constant

(7)

The ± is + for the conduction band and − for the valence band. k0 is the vector of the reciprocal space where the band is at its extremum. The symbols ∥ and ⊥ refer to the longitudinal and perpendicular axes, which are the {001} and {110} directions, respectively, of the noncubic structures. For the cubic structures (diamond and zincblende), the longitudinal or transverse directions can change from one compound to another. The effective masses were obtained by fitting the bottom of the conduction band (for the electron effective mass) and the top of the valence band (for the hole effective mass) with a parabolic function. This was done in the symmetric directions {100}, {110}, and {111} for the cubic structures and for the {100}, {001}, {110}, {011}, and {111} directions for the noncubic structures. In the case of cubic and wurtzite structures, the top of valence band is degenerated, a degeneracy which is lost when k ≠ k0, implying the existence of heavy and light holes. The effective masses of both holes were computed by fitting the bands corresponding to the light and the heavy holes separately. IV. Binding Energy of the Exciton. If the exciton is delocalized on several unit cells, it can be treated as a hydrogen atom (the hole being the proton). This model is called the Wannier exciton.37 In that model, the energy of the 1S state of the exciton is the binding energy of the exciton, Eb, and it can be computed with the formula:55

E b = EH

εr, ⊥ 1 2 = + * * mh 3mh, ⊥ 3εr, mh,*

If the exciton has a localized behavior, like in the Frenkel’s exciton or in excitons of molecular crystals, the Wannier’s model is no longer relevant. Nevertheless, it can be used as a first step, easy to implement, to estimate the order of magnitude of the exciton binding energy. In the solids considered in this study, the exciton is delocalized on at least three uni cells (see Table S1 in Supporting Information), so the Wannier’s exciton model is always valid.

Zp 2 vp 2

(9)

(8)

EH is the energy of the 1s orbital of hydrogen (−13.6 eV) and μ is the reduced mass of the exciton. 6000



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B3LYP show a larger discrepancy for the cell parameter calculations while it is widely used for studying SCs.28,71−74 II. Bandgaps. As it was shown in the Introduction, the bandgap is the very fundamental property of a SC for its application in a photovoltaic or a photochemical device. Remember the optimum band gap of a photovoltaic device is between 1.1 and 1.4 eV (for a simple p−n junction), and for the water splitting application the bandgap should ideally be between 1.8 and 2.2 eV.23 Nevertheless, even if a SC does not play the role of light absorber in a device, a requirement on the gap can be defined. Indeed, in that case, the SC should not compete with the other light absorbing materials. So the bandgap of this SC has to be higher than the photon energy of the light source. In the case of sunlight-based devices, the gap of the SC should be higher than 3.1 eV (corresponding to 400 nm) since few photons have energy higher than this value in the solar spectrum. A good example is TiO2-anatase with a bandgap of 3.2 eV, which is used in DSSCs but only for electron conduction and not for light absorption.3 A counter-example is the use of CdS (Eg = 2.48 eV) for the electron conduction in CIGS solar cells, which absorbs a part of the sunlight normally dedicated to CIGS compound reducing the final photoconversion efficiency.11 In this study, the set of SCs covers a broad range of Eg, from 0.66 eV (Ge) to 5.50 eV (C). General trends can be extracted from the computed values, which are depicted in Figure 3. First, as it is well-known, the pure GGA functional (PBE) always gives a lower bandgap than the experimental one while global hybrid functionals (B3LYP and PBE0) compute always bandgaps higher than the experimental references. As it was noticed in the literature, a higher amount of Hartree−Fock exchange in the global hybrid functionals increases bandgap values.28,75 Unfortunately, some previous works have shown that the best fraction of Hartree− Fock for the Eg reproduction does not correspond to the best fraction of Hartree−Fock exchange for the lattice parameters simulation.76 This is what we obtain in the present study since PBE0 (25% of exact exchange) gives better cell parameters and worse Eg than B3LYP (20% of exact exchange). So a compromise has to be done for the use of global hybrids. HSE06 functional gives the best agreement with the experiments with a MAE of only 10.6% of discrepancy. The values of HSE06 are generally lower than the experimental ones. On the influence of the amount of exact HF exchange, Marques et al.77 and Conesa78 did an interesting observation. If this amount is close to 1/ε∞ (the inverse of the optical dielectric constant), the bandgap is generally well reproduced. For PBE0, there is 25% of HF exchange. So this functional should be adapted to compounds having ε∞ around 4 (see Figure 4 and Table S5). This is the case of ZnO, for which Eg is well reproduced by PBE0. As the exact exchange amount is lower in B3LYP, this functional should be more suited than PBE0 for SCs having higher ε∞ than in ZnO. This corresponds to what we observe qualitatively on our results. As a mater of fact, this method was tested on Ge and GaAs (which have high ε∞). The methodology and the results are presented in the Supporting Information. The major conclusion is that results in good agreement with experiment can be obtained with this way to optimize a functional to a particular SC. Consequently, since the right amount of exact HF exchange is system dependent, to choose a value corresponding to 1/ε∞ gives a good starting point to built a functional suited for a compound.

calculation. The only exception is for the calculation of the infinite dielectric constant of anatase which was done with a 12 × 12 × 12 Monkhorst-Pack grid for the k-points. The core electrons were described with the projector-augmented plane wave (PAW) approach. The characteristics of the pseudopotentials used are presented in Supporting Information (Table S2). For the calculation of the mean absolute error (MAE), when experimentally only a data range is available, the average value of the range is used as experimental value.

4. RESULTS AND DISCUSSIONS I. Cell Parameters. The good reproduction of the cell parameters by DFT calculations is essential for the study of SC systems. Indeed, a major shift between theory and experiment will signify that the interaction between the atoms of the compounds are poorly described and probably also the electronic structure. It will have a negative consequence on the calculation of the electronic properties presented here. Moreover, in the particular case of the computation of the vibration contribution to the dielectric constant, the good reproduction of cell parameters is essential for an adequate modeling of the phonon spectrum. Since for this property the difference between the functionals is very small, only the mean absolute error is presented in Figure 2. We can observe that there is a fairly good agreement

Figure 2. Mean absolute error (in %) on the computed cell parameters. The colors blue, orange, red, and green correspond to PBE, B3LYP, PBE0, and HSE06, respectively.

between theory and experiment for the lattice parameters determination. The error is generally less than 1% for all functionals, except for CdSe and CdTe (see the Supporting Information, Table S3). For these latter compounds, the relativistic effects are probably too important to be well enough taken into account only by a scalar-relativistic pseudo potential on Cd, Se, and Te atoms. Probably, the addition of the spin− orbit coupling to the Hamiltonian may improve the good reproduction of cell parameters. Nevertheless, the error is reasonably low since the maximum error is below 3% (obtained by B3LYP for CdTe). From the functionals point of view, the best results are obtained with PBE0 with a mean absolute error (MAE) of 0.42%, while HSE06 and PBE give also results in quantitative agreements with the experiments (see Figure 2). Surprisingly, 6001



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Figure 3. Computed gaps (in eV) along with the MAE (in %). The colors blue, orange, red, green, and black correspond to PBE, B3LYP, PBE0, HSE06, and experiment, respectively.

III. Dielectric Constants. The dielectric constant plays an important role in the exciton dissociation, since the electrostatic force bonding the electron and the hole is reduced by the increase of εr (eq 1). In the Methodology section, it was recalled that the dielectric constant represents the ability of charges constituting the SC to reorganize when an electric field is applied (caused by the exciton), which screens the electron and the hole. The charges constituting a crystal are the electronic density and the ions of the lattice. So two contributions are present in the dielectric constant, the electronic one, which is linked to the polarizability of the electron density, and the vibrational one, which is linked to the ionic degrees of freedom. Figure 4 presents ε∞, which takes into account only the electronic contribution, and Figure 5 gathers the total εr, taking into account the electronic and vibrational contributions to the dielectric constant. Vibrational contributions were not computed with HSE06 functional since the needed frequencies calculations request too much calculation resources. From the functionals point of view, all of them give values of ε∞ in fairly good agreement with the experiment. Generally, PBE gives higher ε∞ than the other functionals because the gap computed by PBE is lower. As for Eg, HSE06 clearly outperforms the other functionals for the calculation of this property, with a MAE of only 2.7%. The agreement is good also for ZnO, for which Eg was not so well reproduced by HSE06. This confirms, along with the good results obtained for lattice parameters and Eg calculations, that HSE06 reproduces well the electronic structures of SCs. For the global dielectric constant εr (containing both electronic and vibrational contributions), the agreement with experiments is generally good as well for all functionals (MAE between 8 and 10%). More precisely, for SCs having an important vibrational contribution to εr such as TiO2 (anatase and rutile) and CdO, the different functionals reproduce this large amount of vibrational contribution. However, PBE, and to a smaller extent B3LYP, largely overestimates this strong vibrational contributions for TiO2 rutile. This is a consequence of the bad reproduction of the phonon spectrum, already noticed in the literature, which causes this discrepancy.79 We have to keep in mind that only the softest modes (below 100 cm−1) contribute to the dielectric constant (eq 6). A small error of 10 cm−1 induces a large error on the vibrational contribution to the dielectric constant. As a

We can notice that the discrepancy of the computed gap with global hybrid functionals increases in the series CdS, CdSe, and CdTe, while the agreement improves for the GGA functional PBE. As for the cell parameter calculation, this is certainly a consequence of relativistic effects. To support this hypothesis, a single-point relativistic calculation on CdTe (which contains the heaviest elements in the list of SCs we selected) was performed with VASP by taking into account spin−orbit coupling along with the noncollinear formalism and the HSE06 functional (see the Supporting Information). The computed Eg is 1.23 eV, so 0.3 eV lower than the result obtained when scalarrelativistic effects are considered through the pseudopotential. Consequently, PBE artificially reproduces this effect by underestimating the bandgap. In other words, the apparent fairly good description of Eg by PBE does not mean that the electronic structure of the system is well reproduced by PBE. From a photochemical point of view, we understand better why Si, CdTe, and GaAs are used as sunlight absorber in photovoltaic devices since they have a Eg almost in the optimum range, i.e., between 1.1 and 1.4 eV. The bandgap of Ge is too small for a use in a simple p-n junction, but it is employed in multijunction solar cells where the requirement on the bandgap is different. Despite the good position of the gap of CdO for a photovoltaic application, this SC is not used in this field. Actually, the gap of CdO is indirect which means that the light absorption is not efficient. The first direct bandgap in CdO is around 2.21 eV (computed with HSE06 at the Γ point), which is too high for a photovoltaic application. For the water splitting application, the CdS compound is reasonably well placed with a gap not far from the optimum area (being between 1.8 and 2.2 eV).23 This explains why this compound is intensively studied for this application.2,4,21 The oxides TiO2 (rutile and anatase) and ZnO have a very high Eg, which means that they almost do not absorb the sunlight except the UV fraction. That is the reason why they are used in DSSCs or in Quantum-Dots-Sensitized Solar Cells (QDSSCs), since these oxides have to conduct the photogenerated electrons without absorbing the sunlight, which is done by dyes or quantum dots. However, for some photochemical reactions, there is a minimum photovoltage below which the photoreaction does not occur, for example in water splitting devices or for the photodegradation of pollutant. For these applications, oxides like TiO2 or ZnO are efficient since they can achieve high photovoltage because of their high Eg. 6002



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Figure 4. Computed ε∞ along with the MAE (in %). The full-colored and dashed columns correspond to the values along the perpendicular and parallel axis respectively. The colors blue, orange, red, green, and black correspond to PBE, B3LYP, PBE0, HSE06, and experiment, respectively.

Figure 5. Computed εr along with the MAE (in %). The full-colored and dashed columns correspond to the values along the perpendicular and parallel axis respectively. The colors blue, orange, red, and black correspond to PBE, B3LYP, PBE0, and experiment, respectively. For that property only, the MAE does not include the values computed for TiO2 rutile.

with elements having a high polarizability, so generally in the bottom of the periodic table. The second contribution is significantly strong in ionic materials with soft vibrational modes, so involving elements having high electronegativity, localized at the top of the periodic table. IV. Effective Mass. The effective masses of electrons or holes are the masses they seem to carry for transport properties. It is related to the curvature of the conduction band (for the electrons) or of the valence band (for the holes) around the extremum of the band, as it was presented in the Methodology section. The determination of the effective mass of charge carriers by fitting the valence and conduction bands for different crystallographic direction gives access to different information such as the anisotropy of charge carrier conduction or the potentially high mobility of charge carriers (assumed to occur for m* < 0.5). Moreover, m* also plays a role in the binding energy of the exciton since the lower m*, the lower Eb (eq 8). Figures 6 and 7 present the smallest computed and experimental values of m* for electrons and holes. All the values computed for the different crystallographic directions are reported in the Supporting Information. The experimental data for the effective masses suffer from an important uncertainty and the values can change from one data series to another. For example, for the light hole effective mass in CdTe, values can go from 0.1234 to 0.35.80 This discrepancy

consequence, only PBE0 provides a reasonable value for the ionic contribution of the dielectric constant for TiO2 rutile. We can now focus on the chemical interpretation of the dielectric constants. In the case of the C, Si, and Ge series, εr is rigorously equal to ε∞ since these materials are totally nonionic so have no vibrational contribution to εr. For this series, all functionals reproduce the increase of εr with the atomic number of the elements constituting the SC. The dielectric constant increases in this order since the polarizability of the elements increases when we go down in a column of the periodic table. For the O-free SCs, the vibrational contribution is much lower than for the oxides since these SCs are less ionic than oxides. As a result, for example, CdS keeps a low value of εr (∼8) which compromises its efficiency as a photocatalyst even if the bandgap is in the favorable range as discussed earlier. Since the oxides are generally very ionic, from the high electronegativity of oxygen, they yield important Born charges. This can induce a large vibrational contribution to εr, at the condition that the SC possesses soft vibrational modes. For example, TiO2 (anatase or rutile) possesses soft vibrational modes, while it is not the case for ZnO. The consequence is that εr is much higher for TiO2 than for ZnO. In conclusion, in order to have a high dielectric constant εr, a high ε∞ (electronic contribution) and/or a high εvib (ionic contribution) are needed. The first contribution is obtained 6003



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Figure 6. Smallest value of the computed electron effective masses, me*, along with the MAE (in %). The masses in all the crystallographic directions are in the Supporting Information. The colors blue, orange, red, green, and black correspond to PBE, B3LYP, PBE0, HSE06, and experiment, respectively. No MAE are computed for this property; see the text for explanations.

Figure 7. Smallest value of the computed hole effective masses, mh*, along with the MAE (in %). The masses in all the crystallographic directions are in the Supporting Information. The colors blue, orange, red, green, and black correspond to PBE, B3LYP, PBE0, HSE06, and experiment, respectively. No MAE are computed for this property, see the text for explanations.

B3LYP respectively while it is 1.73 and 2.10 for PBE0 and HSE06, respectively. Compared to experimental values, the computed m* have the same order of magnitude than the experimental one. The functionnals PBE, PBE0 and B3LYP give generally very similar results while m* computed with HSE06 is generally higher and in better agreement with the experiments than those obtained with the other functionals. For example, for CdTe, the smallest value of me* with HSE06 is 0.09 which corresponds to the experimental one while PBE, B3LYP and PBE0 compute a much lower value (0.03, 0.04, and 0.05 respectively). Another * with HSE06 is 0.57 example, for GaAs, the smallest value of mhh while the experimental reference is 0.50 and the values obtained with PBE, B3LYP, and PBE0 are 0.19, 0.19, and 0.17, respectively. Once again, HSE06 is much better than the pure GGA or the global hybrid functionals. The DFT results also reproduce the anisotropies observed experimentally. For CdS, the light hole effective mass is very high along the {0001} direction (1.56 obtained with B3LYP) while, it is much lower in the {11−20} direction (0.13 with B3LYP), in agreement with the experiments.

is related with the difficulty to obtain experimentally the effective masses. For that reason, the mean absolute error (MAE) between theory and experiment are not calculated for this property. The discussion will focus only on the ability to reproduce by DFT the order of magnitude of m* and the anisotropy of the m* tensor. From a general point of view, the different functionals give a good description of the evolution of m* from one compound to another. For example, in the series C, Si and Ge, the m*e,{100} decreases by almost the same intensity from C to Si as from Si to Ge for all functionnals while m*hh,{100} is computed slightly smaller (around 5% smaller) for Si than for C and significantly smaller (around 25% smaller) for Ge than for Si with all functionals. Moreover, for a given SC, all functionals reproduce the same evolution of m* from one crystallographic direction to another. For example with TiO2-anatase, all functionnals obtain the same variation of m*e , m*e,{001} < m*e,{100} ≈ m*e,{110}. The major deviation between functionals is obtained for wurtzite structures. While the electron and heavy hole effective masses are similar for the different functionals, the light hole effective masses can change considerably from one functional to another. * is 0.21 and 0.25 for PBE and For instance for ZnO, mlh,{110} 6004



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Figure 8. Exciton binding energy, Eb (in meV), along with the MAE (in %). The colors blue, orange, red, green, and black correspond to PBE, B3LYP, PBE0, HSE06, and experiment, respectively.

parameters such as the temperature and the presence of impurities. This is one of the reasons of the difficulty to have access to Eb experimentally. As shown in the methodology section, Eb can be evaluated with a hydrogenoid model of the exciton that involves only the knowledge of the dielectric constant of the SC and the effective masses of the charge carriers. The relaxed and unrelaxed exciton binding energies can be computed using εr and ε∞ respectively. Only the Eb values corresponding to the relaxed exciton are presented in Figure 8, but both the relaxed and unrelaxed Eb are in the Supporting Information. The Eb values obtained with m* and εr computed at the DFT level are gathered in Figure 8 along with experimental values. Eb can be computed by HSE06 only for the C, Si, and Ge series since for that compound εr = ε∞. For the other compounds, for the calculation of Eb with HSE06, we used the experimental value of εr. This is certainly a good approximation, looking at the good agreement between HSE06 and the experimental data already observed for the other properties. For the exciton binding energy, a clear separation between the different functional families can be seen. The MAE is higher for PBE (64%) than for global hybrid functionals (33% for PBE0 and 47% for B3LYP). The range separated functional, HSE06, is still the best functional with a MAE of 15%. PBE gives values generally much smaller (by a factor 2) than the experimental ones. This is a consequence of the overestimation of εr given by PBE. As εr and m* are fairly well reproduced by the hybrid functionals, it is not a surprise that PBE0, B3LYP and HSE06 gives good results for the Eb calculation. It is interesting to see that in the C, Si and Ge series, the Eb of C is the only one higher than 25 meV. It means that the exciton is strongly bound in diamond and special device architectures should be conceived in order to achieve a good charge extraction in a photochemical device involving diamond. ZnO has also a much higher Eb than TiO2 (anatase and rutile), which is a consequence of the low dielectric constant εr of ZnO. Finally, despite the very high dielectric constant of the rutile phase of TiO2, Eb is not as low as we could expect because the influence of εr is completely compensated by the high value of m*. As a result, Eb of rutile is slightly higher than Eb of anatase. Fortunately, both are lower than 25 meV, which means that excitons can be dissociated at room temperature. Putting together all properties computed on TiO2 anatase and rutile help us to understand much better the different

From a chemical point of view now, it is interesting to see that for a given crystal structure (for example the C, Si and Ge series or for the ZnO, CdS and CdSe series), m* decreases when the atomic number of the elements constituting the SC increases. This can be a consequence of the higher covalency of bonds involving atoms with higher atomic numbers because of their very diffuse orbitals. Indeed, in the tight binding theory of solids, the effective mass of charge carrier is related to the resonance overlap β with the formula 13 (where a is the cell parameter of the compound). β is related to the covalency of the bonds between neighbor atoms. A larger β, i.e., a good covalency, decreases the effective mass. m* =

ℏ2 2βa 2

(13)

Moreover, the effective mass can also be strongly affected by the crystal structure. TiO2 in the rutile or in the anatase phases does not have the same electron and hole effective mass. Clearly, the anatase phase is much more conductive for the electrons than the rutile phase. This can explain why the anatase gives better efficiency than rutile for applications where TiO2 must be a good electron-conductor like in DSSCs.3 To our knowledge, no experimental data has been proposed for TiO2-anatase so that DFT calculations, after the validation step carried here, can bring an important new insight on the comparison between these two solids. V. Exciton Binding Energy. As presented in the Introduction, the binding energy of the exciton should be lower than the thermal energy (25 meV at room temperature) in order to achieve an efficient dissociation of this photogenerated exciton. Experimentally, the exciton generation is faster than atomic motions in the crystal. This is the vertical transition principle. It means that, the charge- screening felt by the exciton when it is generated only comes from ε∞ since only the electronic density can reorganize at this time scale. This is the unrelaxed exciton. Then the atoms move to adapt to the exciton. At that time, the charge screening will be governed by εr, giving the relaxed exciton. In the present study, it is assumed that for photovoltaic or photochemical devices, the time scale of the exciton dissociation is higher than the atomic motions and, consequently, Eb corresponds to the binding energy of the relaxed exciton. The values given in Table 2 correspond to the relaxed exciton. To conclude about the experimental determination of Eb, these values can be affected by external 6005



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desired properties (especially the vibrational contribution to the dielectric constant). Global hybrid functionals offer a satisfactory alternative since they obtained very good results in the present study. For this comparison, we decided to study the most wellknown semiconductors (Si, GaAs, TiO2, etc.) used in the most familiar applications (photovoltaic, water splitting, etc.). Nevertheless, these calculations can be used on much more complex materials including complex oxides or doped semiconductors for numerous applications involving the electronic structures of these materials such as the transparent conducting oxides or the high-k dielectrics, for instance. DFT is hence potentially able to play an important role in the design of new semiconductor materials for very complex devices.

efficiencies experimentally observed for these two phases, anatase being more efficient than rutile.17,81,82 Their bandgaps, dielectric constants and exciton binding energies are very similar and cannot be the reason of the observed differences. Only their effective masses are significantly different. For m*e , the smallest values are 0.06 (HSE06, Figure 6) for anatase and 1.01 (HSE06, Figure 6) for rutile. The electron is almost 17 times lighter in anatase than in rutile. Looking at these results, anatase is more efficient than rutile because of the easier diffusion of charge carrier in anatase than in rutile. These results help us to understand better the differences observed between TiO2-Anatase and ZnO for dye-sensitized solar cells applications. The similar bandgap value between ZnO and TiO2 makes the community of DSSCs think that ZnO can give as good photoconversion efficiency as TiO2Anatase.83,84 However, DSSCs based on ZnO have never exceeded the 5.6% value for photoconversion efficiency obtained by Yoshida et al.85 while TiO2 based DSSCs are now reaching 12% of photoconversion efficiency.86 From the results obtained in the present work, some interpretations of the DSSCs efficiencies can be done. The major differences between ZnO and TiO2 anatase come from εr (5 time lower for ZnO than for anatase), me* (6 time larger for ZnO than for anatase) and consequently on Eb (19 larger for ZnO than for anatase). This clearly makes a difference between ZnO and TiO2. The low value of εr of ZnO signifies that, from a DSSC working principle point of view, the injected electron in ZnO is not screened by the solid and stays near the surface, interacting with the electrolyte which favors the recombination and lowers the final efficiency of the cell. Furthermore, due to the high m*e , it appears that ZnO is not as good an electronic conductor as TiO2. All together, these are reasons explaining why ZnO is not as good as TiO2 for DSSCs applications.

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ASSOCIATED CONTENT

S Supporting Information *

Full listing of the numerical values and characteristics of the pseudopotentials used. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +334 72 72 88 46. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Shah, A.; Torres, P.; Tscharner, R.; Wyrsch, N.; Keppner, H. Photovoltaic Technology: The Case for Thin-Film Solar Cells. Science 1999, 285, 692−698. (2) Osterloh, F. E. Reviews Inorganic Materials as Catalysts for Photochemical Splitting of Water. Chem. Mater. 2008, 20, 35−54. (3) Hagfeldt, A.; Boschloo, G.; Sun, L.; Kloo, L.; Pettersson, H. DyeSensitized Solar Cells. Chem. Rev. 2010, 110, 6595−6663. (4) Hernández-Alonso, M. D.; Fresno, F.; Suárez, S.; Coronado, J. M. Development of Alternative Photocatalysts to TiO2: Challenges and Opportunities. Energ. Environ. Sci. 2009, 2, 1231−1257. (5) Nowotny, J.; Sorrell, C.; Sheppard, L.; Bak, T. Solar-Hydrogen: Environmentally Safe Fuel for the Future. Int. J. Hydrog. Energ. 2005, 30, 521−544. (6) Green, M. A.; Emery, K.; Hishikawa, Y.; Warta, W.; Dunlop, E. D. Solar Cell Efficiency Tables (Version 42). Prog. Photovolt Res. Appl. 2013, 21, 827−837. (7) King, R. R.; Law, D. C.; Edmondson, K. M.; Fetzer, C. M.; Kinsey, G. S.; Yoon, H.; Sherif, R. a.; Karam, N. H. 40% Efficient Metamorphic GaInP/GaInAs/Ge Multijunction Solar Cells. App. Phys. Lett. 2007, 90, 183516. (8) Bosi, M.; Pelosi, C. The Potential of III-V Semiconductors as Terrestrial Photovoltaic Devices. Prog. Photovolt Res. Appl. 2007, 15, 51−68. (9) Wu, X. High-Efficiency Polycrystalline CdTe Thin-Film Solar Cells. Sol. Energy 2004, 77, 803−814. (10) Jackson, P.; Hariskos, D.; Lotter, E.; Paetel, S.; Wuerz, R.; Menner, R.; Wischmann, W.; Powalla, M. New World Record Efficiency for Cu(In,Ga)Se2 Thin-Film Solar Cells beyond 20%. Prog. Photovolt Res. Appl. 2011, 19, 894−897. (11) Singh, U. P.; Patra, S. P. Progress in Polycrystalline Thin-Film Cu(In,Ga)Se2 Solar Cells. Int. J. Photoenergy 2010, 2010, 1−19. (12) Guerin, V. M.; Magne, C.; Pauporté, T.; Le Bahers, T.; Rathousky, J. Electrodeposited Nanoporous versus Nanoparticulate ZnO Films of Similar Roughness for Dye-Sensitized Solar Cell Applications. ACS Appl. Mater. Inter. 2010, 2, 3677−3685.

5. CONCLUSION In the work presented here, we focused on the important fundamental properties (bandgap Eg, dielectric constant εr, effective mass m* and exciton binding energy Eb) required from a semiconductor to make it usable in a photovoltaic or a photochemical device. The question to answer was: Is DFT able to calculate all of these fundamental properties in order to use this computational method as a predictive tool? To answer this question, a set of some of the most important semiconductors was considered and the DFT capabilities were tested along with four different functionals. Remember that the aim of the work here presented is not to do an extensive benchmark but more to determine if there are major differences between the different functional families. We observed that all properties do not have the same sensibility to the choice of the functional. For instance, the cell parameters are reasonably well reproduced by all functionals while bandgap, dielectric constants, and exciton binding energies are much more functional dependent. Finally, except for the cell parameters, the HSE06 functional always obtained the best agreement compared to experiments. The reason of the good reliability of this functional comes from the suitable choice of the coulomb screen parameter (ω = 0.11 a0−1) giving the good amount of nonlocal exchange to reproduce the electronic properties of small gap semiconductors.87 Consequently, this parameter should be reoptimized to study other systems such as insulators.87 Unfortunately, due to implementation reasons (high CPU time cost), this range separated functional cannot be used efficiently to compute all of the 6006



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(35) Bass, M. Handbook of Optics, Vol. IV, Optical Properties of Materials, Non Linear Optics, Quantum Optics; 3rd ed.; Baas, M., Ed.; McGraw Hill: New York, 2010. (36) Morar, J. F.; Himpsel, F. J.; Hollinger, G.; Hughes, G.; Jordan, J. L. Oervation of a C-1s Core Exciton in Diamond. Phys. Rev. B 1985, 54, 1960−1963. (37) Pelant, I.; Valenta, J. Luminescence of Excitons. In Luminescence Spectroscopy of Semiconductors; Oxford University Press: Oxford, U.K., 2012. (38) Rieder, K. H.; Ishigame, M.; Genzel, L. Infrared Absorption by Coupled Surface-Phonon-Surafce-Plasmon Modes in Microcrystals of CdO. Phys. Rev. B 1972, 6, 3804−3810. (39) Jefferson, P. H.; Hatfield, S. a.; Veal, T. D.; King, P. D. C.; McConville, C. F.; Zúñiga−Pérez, J.; Muñoz−Sanjosé, V. Bandgap and Effective Mass of Epitaxial Cadmium Oxide. App. Phys. Lett. 2008, 92, 022101. (40) Zhang, J.; Ruf, T.; Lauck, R.; Cardona, M. Isotope Effects on Exciton Energies in CdS. Phys. Rev. B 1998, 57, 9716−9722. (41) Taguchi, T.; Shirafuji, J.; Inuishi, Y. Excitonic Emission in Cadmium Telluride. Phys. Status Solidi b 1975, 68, 727−738. (42) Vurgaftman, I.; Meyer, J. R.; Ram-Mohan, L. R. Band Parameters for III−V Compound Semiconductors and Their Alloys. J. Appl. Phys. 2001, 89, 5815−5875. (43) Young, K. F.; Frederikse, H. P. R. Compilation of Static Dielectric Constant of Inorganics Solids. J. Phys. Chem. Ref. Data 1973, 2, 313−409. (44) Adachi, S. GaAs and Related Materials; World Scientific Publishing Co. Pte. Ltd.: Singapore, 1994. (45) Gilleo, M. A.; Bailey, P. T.; Hill, D. E. Free-Carrier and Exciton Recombination Radiation in GaAs. Phys. Rev. 1968, 174, 898−905. (46) Kavan, L.; Gra, M.; Gilbert, S. E.; Klemenz, C.; Scheel, H. J. Electrochemical and Photoelectrochemical Investigation of SingleCrystal Anatase. J. Am. Chem. Soc. 1996, 118, 6716−6723. (47) Gonzalez, R. J.; Zallen, R.; Berger, H. Infrared Reflectivity and Lattice Fundamentals in Anatase TiO2. Phys. Rev. B 1997, 55, 7014− 7017. (48) Parker, R. A. Static Dielectric Constant of Rutile. Phys. Rev. 1961, 124, 1719−1722. (49) Amtout, A.; Leonelli, R. Time-Resolved Photoluminescence from Exciton TiO2. Solid State Commun. 1992, 84, 349−352. (50) Teke, A.; Ö zgür, Ü .; Doğan, S.; Gu, X.; Morkoç, H.; Nemeth, B.; Nause, J.; Everitt, H. Excitonic Fine Structure and Recombination Dynamics in Single-Crystalline ZnO. Phys. Rev. B 2004, 70, 195207. (51) Yang, Z.; Ullrich, C. A. Direct Calculation of Exciton Binding Energies with Time-Dependent Density-Functional Theory. Phys. Rev. B 2013, 87, 195204. (52) Ferrero, M.; Rérat, M.; Orlando, R.; Dovesi, R. The Calculation of Static Polarizabilities of 1−3D Periodic Compounds . The Implementation in the CRYSTAL Code. J. Comput. Chem. 2008, 29, 1450−1459. (53) Ferrero, M.; Rérat, M.; Orlando, R.; Dovesi, R. Coupled Perturbed Hartree-Fock for Periodic Systems: The Role of Symmetry and Related Computational Aspects. J. Chem. Phys. 2008, 128, 014110. (54) Gonze, X.; Lee, C. Dynamical Matrices, Born Effective Charges, Dielectric Permittivity Tensors, and Interatomic Force Constants from Density-Functional Perturbation Theory. Phys. Rev. B 1997, 55, 355− 368. (55) Rodina, A.; Dietrich, M.; Göldner, A.; Eckey, L.; Hoffmann, A.; Efros, A.; Rosen, M.; Meyer, B. Free Excitons in Wurtzite GaN. Phys. Rev. B 2001, 64, 115204. (56) Perdew, J.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (57) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (58) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0Model. J. Chem. Phys. 1999, 110, 6158−6170. (59) Dovesi, R.; et al. Crystal09 1.0.0. Crystal09 User’s Manual; Universita di Torino: Italy, 2009.

(13) Ramasamy, E.; Lee, J. Ordered Mesoporous SnO2-Based Photoanodes for High-Performance Dye-Sensitized Solar Cells. J. Phys. Chem. C 2010, 114, 22032−22037. (14) Odobel, F.; Le Pleux, L.; Pellegrin, Y.; Blart, E. New Photovoltaic Devices Based on the Sensitization of p-Type Semiconductors. Acc. Chem. Res. 2010, 43, 1063−1071. (15) Kamat, P. V. Quantum Dot Solar Cells. Semiconductor Nanocrystals as Light Harvesters. J. Phys. Chem. C 2008, 112, 18737−18753. (16) Woolerton, T. W.; Sheard, S.; Reisner, E.; Pierce, E.; Ragsdale, S. W.; Armstrong, F. A. Efficient and Clean Photoreduction of CO2 to CO by Enzyme-Modified TiO2 Nanoparticles Using Visible Light. J. Am. Chem. Soc. 2010, 132, 2132−2133. (17) Liu, L.; Zhao, H.; Andino, J. M.; Li, Y. Photocatalytic CO2 Reduction with H2O on TiO 2 Nanocrystals: Comparison of Anatase, Rutile, and Brookite Polymorphs and Exploration of Surface Chemistry. ACS Catal. 2012, 2, 1817−1828. (18) Liu, L.; Gao, F.; Zhao, H.; Li, Y. Tailoring Cu Valence and Oxygen Vacancy in Cu/TiO2 Catalysts for Enhanced CO2 Photoreduction Efficiency. Appl. Catal. B-Environ. 2013, 134−135, 349−358. (19) Guo, J.; Ouyang, S.; Kako, T.; Ye, J. Mesoporous In(OH)3 for Photoreduction of CO2 into Renewable Hydrocarbon Fuels. App. Surf. Sci 2013, 280, 418−423. (20) Sivula, K.; Le Formal, F.; Grätzel, M. Solar Water Splitting: Progress Using Hematite (α-Fe2O3) Photoelectrodes. ChemSusChem 2011, 4, 432−449. (21) Li, Q.; Guo, B.; Yu, J.; Ran, J.; Zhang, B.; Yan, H.; Gong, J. R. Highly Efficient Visible-Light-Driven Photocatalytic Hydrogen Production of CdS-Cluster-Decorated Graphene Nanosheets. J. Am. Chem. Soc. 2011, 133, 10878−10884. (22) Ji, Z.; He, M.; Huang, Z.; Ozkan, U.; Wu, Y. Photostable P-Type Dye-Sensitized Photoelectrochemical Cells for Water Reduction. J. Am. Chem. Soc. 2013, 135, 11696. (23) Moriya, Y.; Takata, T.; Domen, K. Recent Progress in the Development of (oxy)nitride Photocatalysts for Water Splitting under Visible-Light Irradiation. Coord. Chem. Rev. 2013, 257, 1957−1969. (24) Shang, M.; Wang, W.; Sun, S.; Ren, J.; Zhou, L.; Zhang, L. Efficient Visible Light-Induced Photocatalytic Degradation of Contaminant by Spindle-like PANI/BiVO4. J. Phys. Chem. C 2009, 113, 20228−20233. (25) Konstantinou, I. K.; Albanis, T. a TiO2-Assisted Photocatalytic Degradation of Azo Dyes in Aqueous Solution: Kinetic and Mechanistic Investigations. Appl. Catal. B-Environ. 2004, 49, 1−14. (26) Lunt, R. R.; Osedach, T. P.; Brown, P. R.; Rowehl, J. a.; Bulović, V. Practical Roadmap and Limits to Nanostructured Photovoltaics. Adv. Mater. 2011, 23, 5712−5727. (27) Clarke, T. M.; Durrant, J. R. Charge Photogeneration in Organic Solar Cells. Chem. Rev. 2010, 110, 6736−6767. (28) Xiao, H.; Tahir-Kheli, J.; Goddard, W. a. Accurate Band Gaps for Semiconductors from Density Functional Theory. J. Phys. Chem. Lett. 2011, 2, 212−217. (29) Marsman, M.; Paier, J.; Stroppa, A.; Kresse, G. Hybrid Functionals Applied to Extended Systems. J. Phys.: Condens. Matter 2008, 20, 064201. (30) Paier, J.; Marsman, M.; Hummer, K.; Kresse, G.; Gerber, I. C.; Angyán, J. G. Screened Hybrid Density Functionals Applied to Solids. J. Chem. Phys. 2006, 124, 154709. (31) Gajdoš, M.; Hummer, K.; Kresse, G.; Furthmüller, J.; Bechstedt, F. Linear Optical Properties in the Projector-Augmented Wave Methodology. Phys. Rev. B 2006, 73, 045112. (32) Kim, Y.-S.; Hummer, K.; Kresse, G. Accurate Band Structures and Effective Masses for InP, InAs, and InSb Using Hybrid Functionals. Phys. Rev. B 2009, 80, 035203. (33) Walter, M. G.; Warren, E. L.; McKone, J. R.; Boettcher, S. W.; Mi, Q.; Santori, E. A.; Lewis, N. S. Solar Water Splitting Cells. Chem. Rev. 2010, 110, 6446−6473. (34) Madelung, O. Semiconductors: Data Handbook, 3rd ed.; Springer: New York, 2004. 6007



Article

(80) Kissling, G. P.; Fermín, D. J. Electrochemical Hole Injection into the Valence Band of Thiol Stabilised CdTe Quantum Dots. Phys. Chem. Chem. Phys. 2009, 11, 10080−10086. (81) Magne, C.; Dufour, F.; Labat, F.; Lancel, G.; Durupthy, O.; Cassaignon, S.; Pauporté, T. Effects of TiO2 Nanoparticle Polymorphism on Dye-Sensitized Solar Cell Photovoltaic Properties. J. Photochem. Photobiol. A 2012, 232, 22−31. (82) Sclafani, A.; Hermann, J. M. Comparison of the Photoelectronic and Photocatalytic Activities of Various Anatase and Rutile Forms of Titania in Pure Liquid Organic Phases and in Aqueous Solutions. J. Phys. Chem. 1996, 3654, 13655−13661. (83) Anta, J. A.; Guillen, E.; Tena-Zaera, R. ZnO-Based DyeSensitized Solar Cells. J. Phys. Chem. C 2012, 116, 11413−11425. (84) Guérin, V.-M.; Elias, J.; Nguyen, T. T.; Philippe, L.; Pauporté, T. Ordered Networks of ZnO-Nanowire Hierarchical Urchin-like Structures for Improved Dye-Sensitized Solar Cells. Phys. Chem. Chem. Phys. 2012, 14, 12948−12955. (85) Yoshida, T.; Zhang, J.; Komatsu, D.; Sawatani, S.; Minoura, H.; Pauporté, T.; Lincot, D.; Oekermann, T.; Schlettwein, D.; Tada, H.; et al. Electrodeposition of Inorganic/Organic Hybrid Thin Films. Adv. Funct. Mater. 2009, 19, 17−43. (86) Yella, A.; Lee, H.-W.; Tsao, H. N.; Yi, C.; Chandiran, A. K.; Nazeeruddin, M. K.; Diau, E. W.-G.; Yeh, C.-Y.; Zakeeruddin, S. M.; Grätzel, M. Porphyrin-Sensitized Solar Cells with Cobalt (II/III)Based Redox Electrolyte Exceed 12% Efficiency. Science 2011, 334, 629−634. (87) Paier, J.; Marsman, M.; Hummer, K.; Kresse, G.; Gerber, I. C.; Angyán, J. G. Screened Hybrid Density Functionals Applied to Solids. J. Chem. Phys. 2006, 124, 154709.

(60) Peintinger, M. F.; Oliveira, D. V.; Bredow, T. Consistent Gaussian Basis Sets of Triple-Zeta Valence with Polarization Quality for Solid-State Calculations. J. Comput. Chem. 2013, 34, 451−459. (61) Baima, J.; Erba, A.; Rérat, M.; Orlando, R.; Dovesi, R. Beryllium Oxide Nanotubes and Their Connection to the Flat Monolayer. J. Phys. Chem. C 2013, 117, 12864−12872. (62) Pascale, F.; Zicovich-wilson, C. M.; Orlando, R.; Roetti, C.; Ugliengo, P.; Dovesi, R.; Orientale, P.; G, V. B.; Giuria, V. P. Vibration Frequencies of Mg3Al2Si3O12 Pyrope. An Ab Initio Study with CRYSTAL Code. J. Phys. Chem. B 2005, 109, 6146−6152. (63) Bredow, T.; Heitjans, P.; Wilkening, M. Electric Field Gradient Calculations for LixTiS2 and Comparison with 7Li NMR Results. Phys. Rev. B 2004, 70, 115111. (64) Jaffe, J. E.; Hess, A. C. Hartree-Fock Study of Phase Changes in ZnO at High Pressure. Phys. Rev. B 1993, 48, 7903−7909. (65) Peterson, K. A.; Figgen, D.; Goll, E.; Stoll, H.; Dolg, M. Systematically Convergent Basis Sets with Relativistic Pseudopotentials. II. Small-Core Pseudopotentials and Correlation Consistent Basis Sets for the Post-D Group 16−18 Elements. J. Chem. Phys. 2003, 119, 11113−11123. (66) Figgen, D.; Rauhut, G.; Dolg, M.; Stoll, H. Energy-Consistent Pseudopotentials for Group 11 and 12 Atoms: Adjustment to MultiConfiguration Dirac−Hartree−Fock Data. Chem. Phys. 2005, 311, 227−244. (67) Noel, Y.; Zicovich-Wilson, C.; Civalleri, B.; D’Arco, P.; Dovesi, R. Polarization Properties of ZnO and BeO: An Ab Initio Study through the Berry Phase and Wannier Functions Approaches. Phys. Rev. B 2001, 65, 014111. (68) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Erratum: “Hybrid Functionals Based on a Screened Coulomb Potential” [J. Chem. Phys. 118, 8207 (2003)]. J. Chem. Phys. 2006, 124, 219906. (69) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid Functionals Based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207−8215. (70) Krukau, A. V; Vydrov, O. A.; Izmaylov, A. F.; Scuseria, G. E. Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals. J. Chem. Phys. 2006, 125, 224106. (71) Graciani, J.; Marquez, A. M.; Plata, J. J.; Ortega, Y.; Hernandez, N. C.; Meyer, A.; Zicovich-Wilson, C. M.; Sanz, J. F. Comparative Study on the Performance of Hybrid DFT Functionals in Highly Correlated Oxides: The Case of CeO2 and Ce2O3. J. Chem. Theor. Comput. 2011, 7, 56−65. (72) Zicovich-Wilson, C. M.; San-Román, M. L.; Camblor, M. A.; Pascale, F.; Durand-Niconoff, J. S. Structure, Vibrational Analysis, and Insights into Host-Guest Interactions in as-Synthesized Pure Silica ITQ-12 Zeolite by Periodic B3LYP Calculations. J. Am. Chem. Soc. 2007, 129, 11512−11523. (73) Martsinovich, N.; Ambrosio, F.; Troisi, A. Adsorption and Electron Injection of the N3Metal-Organic Dye on the TiO2 Rutile (110) Surface. Phys. Chem. Chem. Phys. 2012, 14, 16668−16675. (74) Nilsing, M.; Persson, P.; Lunell, S.; Ojamae, L. Dye-Sensitization of the TiO2 Rutile (110) Surface by Perylene Dyes: QuantumChemical Periodic B3LYP Computations. J. Phys. Chem. C 2007, 111, 12116−12123. (75) Labat, F.; Ciofini, I.; Adamo, C. Modeling ZnO Phases Using a Periodic Approach: From Bulk to Surface and Beyond. J. Chem. Phys. 2009, 131, 044708. (76) Dahan, N.; Jehl, Z.; Hildebrandt, T.; Greffet, J.-J.; Guillemoles, J.-F.; Lincot, D.; Naghavi, N. Optical Approaches to Improve the Photocurrent Generation in Cu(In,Ga)Se2 Solar Cells with Absorber Thicknesses down to 0.5 Mm. J. Appl. Phys. 2012, 112, 094902. (77) Marques, M. A. L.; Vidal, J.; Oliveira, M. J. T.; Reining, L.; Botti, S. Density-Based Mixing Parameter for Hybrid Functionals. Phys. Rev. B 2011, 83, 035119. (78) Conesa, J. C. Band Structures and Nitrogen Doping Effects in Zinc Titanate Photocatalysts. Catal. Today 2013, 208, 11−18. (79) Montanari, B.; Harrison, N. M. Lattice Dynamics of TiO2 Rutile: Influence of Gradient Corrections in Density Functional Calculations. Chem. Phys. Lett. 2002, 364, 528−534. 6008


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