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Energy level alignment at semiconductor–water interfaces from atomistic and continuum solvation models†

Cite this: RSC Adv., 2017, 7, 43660

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Juhan Matthias Kahk,^bd Ravishankar Sundararaman,^c
*
Lars Blumenthal,

Paul Tangney^abd and Johannes Lischner^abd

Accurate and efficient methods for predicting the alignment between a semiconductor's electronic energy levels and electrochemical redox potentials are needed to facilitate the computational discovery of photoelectrode materials. In this paper, we present an approach that combines many-body perturbation theory within the GW method with continuum solvation models. Specifically, quasiparticle levels of the bulk photoelectrode are referenced to the outer electric potential of the electrolyte by calculating the change in electric potential across the photoelectrode–electrolyte and the electrolyte–vacuum interfaces using continuum solvation models. We use this method to compute absolute energy levels for the prototypical rutile (TiO₂) photoelectrode in contact with an aqueous electrolyte and find good agreement with predictions from atomistic simulations based on molecular dynamics. Our analysis reveals qualitative and quantitative differences of the description of the interfacial charge density in atomistic and continuum solvation models and highlights the need for a consistent treatment of electrode–electrolyte and electrolyte–vacuum interfaces for the determination of accurate absolute energy levels.

Received 28th July 2017 Accepted 4th September 2017

DOI: 10.1039/c7ra08357b

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1 Introduction

Photocatalysts enable the conversion of solar energy into chemical fuels and there is currently much interest in the search for new materials for water splitting or carbon dioxide reduction.^1–4 An important criterion for photocatalyst materials is the favorable alignment of their electrons' quasi electrochemical potential with the redox potential of the desired chemical reaction.⁵ For example, a photocathode can only drive the hydrogen evolution reaction (HER) if the quasi electrochemical potential of its electrons is above the reduction potential of water, see Fig. 1. Accurate methods to calculate electronic energy levels and align them with redox potentials are therefore needed to enable the computational search for new photocatalysts and their in silico design.
Redox potentials are conventionally reported relative to a reference electrode, such as the standard hydrogen electrode

• Fig.
• 1

• Schematic illustration of
• a
• photoelectrochemical cell

composed of a hydrogen electrode (left) and a photoelectrode (right). Thermodynamically, the HER is only possible if the (quasi) electrochemical potential of an electron (denoted by green lines) is higher in the photoelectrode than in the hydrogen electrode. The electrochemical potential of the majority charge carriers in the semiconductor is mainly determined by the degree of doping and closely related to the semiconductor's conduction band minimum (blue line) or valence band maximum (red line). The difference in electrochemical potential of the electrons between the two electrode compartments, Dm~e, determines the direction of the current and hence whether the photoelectrode can drive the redox reaction of interest.

^aImperial College London, Department of Physics, South Kensington Campus, London SW7 2AZ, UK. E-mail: [email protected] ^bImperial College London, Department of Materials, Royal School of Mines, South Kensington Campus, London SW7 2AZ, UK ^cDepartment of Materials Science and Engineering, Rensselaer Polytechnic Institute, 110 8th St, Troy, New York 12180, USA ^dThomas Young Centre for Theory and Simulation of Materials, UK

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7ra08357b

43660 | RSC Adv., 2017, 7, 43660–43670

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(SHE).^6,7 Trasatti proposed to determine absolute redox poten- cost even further. It has been shown that the free energy of solid– tials by computing the work needed to bring an electron from liquid interfaces can be obtained by means of joint density the SHE to a point in the vacuum above an ideal aqueous functional theory (JDFT) which combines a CDFT treatment of electrolyte.⁸ By performing the same calculation for an electron the liquid with an electronic structure DFT approach to the from the photoelectrode, a direct comparison between the solid.²⁶ JDFT partitions the free energy of the solid–liquid interredox potentials of interest and the photoelectrode's electronic face into three contributions: one contribution depends only on

• energy levels is possible.
• the electron density of the solid, another contribution depends

Many computational studies have used density functional only on the atomic site density of the liquid and the nal theory (DFT) to calculate the Kohn–Sham energy levels of contribution, which captures the interaction between the solid a photoelectrode. Oen, these levels are then referenced to the and the liquid, depends both on the electron density of the solid vacuum outside the photoelectrode and the effect of the elec- and the atomic site density of the liquid. Other successful trolyte is ignored.^9–12 Except for the energy of the highest attempts of directly linking continuum solvation models to DFT occupied state, the Kohn–Sham energies do not have the include those by Anderson et al., Liu et al. and Marzari et al.^27–30 physical meaning of electron addition or removal energies and The former has been employed to calculate reversible potentials

• ´
• Stevanovic and coworkers employed many-body perturbation for redox reactions in solution and at the surface of metal elec-

theory within the GW method to compute accurate quasiparticle trodes.³¹ The latter two have been used to study reaction mechenergy levels.¹³ Aer aligning the GW-corrected energy levels to anisms at metal–water interfaces.^29,32

• the vacuum outside the photoelectrode, they concluded that, on
• Ping and coworkers recently employed a sophisticated PCM

average, an additional shi of 0.5 Æ 0.3 eV towards the vacuum to study band edge positions of several well-known photolevel was needed to bring the computed energy levels into electrodes. By modelling the photoelectrode–electrolyte interagreement with electrochemical measurements at semi- faces, they referenced the photoelectrodes' electronic energy conductor–electrolyte interfaces. However, it has become clear levels to the inner electric potential of the electrolyte. Despite that such a rule of thumb is only of limited value for the search this being a different reference point than the outer electric for new photoelectrodes as the effect of the electrolyte can vary potential of water, which is the one used to calculate the absosignicantly across different semiconductors. For example, lute SHE potential, they reported good agreement of their while the reduction of the ionization energy and the electron results with experimental data.³³ However, so far no explanation affinity by 0.5 eV yields good agreement with electrochemically for this quantitative agreement despite choosing inconsistent

ꢀ

measured band edges at at band conditions for the (1010) reference points has been given. This demonstrates that the surface of ZnO, other materials like TiO₂, CdS, and SnO₂ show determination of absolute electronic energy levels, which is well a shi of 1 eV or more, while the shi for Fe₂O₃ and NiO is only established for atomistic solvation models, is qualitatively

• 0.2 eV.¹³
• different and much less well understood for continuum solva-

To explicitly capture the effect of the electrolyte, several tion approaches.

• groups carried out ab initio molecular dynamics (AIMD) simu-
• In this paper, we rigorously explain the above observation by

lations of the photoelectrode–electrolyte and the electrolyte– directly comparing approaches for calculating absolute energy vacuum interface.^3,14–18 By averaging the electric potential of levels of photoelectrodes based on atomistic and continuum many structural congurations, the difference between the solvation models using both a PCM and CDFT. To reference GW inner electric potential of the photoelectrode and the outer quasiparticle energies of bulk photoelectrodes to the outer electric potential of the electrolyte was determined.⁶ This electric potential of water, we compute the electric potential potential difference was then used to obtain absolute photo- across both the photoelectrode–water interface and the water– electrode energy levels. Unfortunately, the computational vacuum interface. As will be shown, the two approaches yield expense of such simulations is very high as large unit cells and signicantly different results for the change in electric potential long simulation times are needed for precise thermodynamic across the individual interfaces, but their predictions for the averages. As a consequence, such methods cannot easily be absolute positions of the electrode's electronic energy levels used in high-throughput computational searches for photo- agree closely with each other. We explain the observed

• catalyst materials.
• disagreement for the individual interfaces by analyzing how the

Continuum solvation models are computationally inexpensive different approaches describe the interfacial charge distribualternatives to atomistic simulations of liquids at solid–liquid tion. As a test system, we investigate a prototypical rutile (TiO₂) interfaces.^19,20 Within this category exists a wide range of photoelectrode as this material has been extensively studied approaches, from simple polarizable continuum models (PCMs) and there exists a wide range of experimental and theoretical to classical density functional theory (CDFT).^21–24 Many of these reference data.

• approaches are oen based on or at least inspired by the work of
• In the following, we rst outline the methods and compu-

Fattebert and Gygi.²⁵ Continuum solvation models directly yield tational details that were used to align the photoelectrode's averages of properties, such as number densities of atoms or electronic energy levels with the SHE potential. Next, we present electric potentials, and avoid the computationally expensive our results for the electronic structure of bulk rutile, and the sampling of atomic congurations. Furthermore, in many cases structure of the rutile(110)–water and water–vacuum interfaces. they enable the use of much smaller supercells compared to Finally, the main differences between atomistic and continuum atomistic solvation models which reduces the computational solvation models are discussed.

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layer, an electrolyte layer and a vacuum layer, see Fig. 2.^16,17 In practice, rst-principles calculations for such systems are extremely challenging because of the large system size and the need for congurational averages.

2 Methods

2.1 Electronic energy level alignment at photoelectrode– electrolyte interfaces

As the water surface should be independent of the chosen
Aligning electronic energy levels across photoelectrode–electrolyte interfaces requires the choice of a common energy reference point.¹⁶ Redox potentials of chemical reactions are conventionally expressed on a scale whose zero is the potential of the SHE.⁷ By computing the work required to bring electrons from the SHE and a photoelectrode to the same reference point in space, it is possible to compare their electrochemical potentials. Trasatti proposed a reference point in the vacuum located far enough above an ideal aqueous electrolyte so that effects of image charges are negligible.⁸ By analysing the associated Born–Haber cycle and using experimental values for the involved energy differences, he determined the work required to bring an electron from the SHE to this point to be 4.44 Æ 0.02 eV.⁸ The electric potential the electron experiences at this reference point is the outer electric potential of the aqueous

ꢀ^PE

photoelectrode material, the calculation of f À j^{H O} is usually

2

split into two parts. First, the inner electric potential difference

D^P_H^E_Ofhf^PE À f^{H O}

;

(3)

2
2

ꢀ^{H O}

where f

2

denotes the average electric potential in the ideal aqueous electrolyte, is calculated using a unit cell that does not contain any vacuum but only a slab of the photoelectrode whose periodic images are separated by the electrolyte. In the second step, the surface potential of water,

ꢀ^{H O} c^{H O} h f
À j^{H O}

,

(4)

• 2
• 2
• 2

is determined using a unit cell that contains only a liquid water layer and a vacuum layer, but no photoelectrode. As discussed by Kathmann et al., experimental measurements of the water surface potential depend sensitively on the probe used; electron holography experiments suggest a value of around 3.5 V, while electrochemical measurements in which protons, as hydronium ions, sample the potential, yield values of around 0.1 V.³⁷
Finally, the outer electric potential difference is dened as electrolyte, j^{H O}, and the associated potential energy is the zero

2

of Trasatti's absolute scale.
To express the electronic energy levels in the photoelectrode on this absolute scale, we follow a two-step procedure. First, the energy levels of the bulk photoelectrode are calculated using the GW method.^34–36 In this approach, the electron self-energy, which determines the quasiparticle energy levels via the Dyson equation, is expanded in a power series in the screened Coulomb interaction, W, and only the rst term is retained; G denotes the one-electron Green's function. In the second step of the alignment procedure, the quasiparticle energies of the bulk photoelectrode—most importantly, the valence band maximum (VBM) and the conduction band minimum (CBM)—are referenced to the outer electric potential of the aqueous electrolyte,

D^P_H^E_Ojhj^PE À j^{H O} ¼ c^{H O} þ D_H^PE_Of À c^PE

;

(5)

• 2
• 2

• 2
• 2

ꢀ^PE

where c^PE h f À j^PE denotes the surface potential of the photoelectrode. Note that D^P_H^E_Oj is experimentally measurable,

2

for example by two-photon photoemission, and relates electronic energy levels referenced to the vacuum above the photoelectrode, e.g. the electron affinity (EA) and ionization energy

• (IE), to absolute energy levels referenced to j^{H O}
• .
• ³⁸ For example,

2

j^{H O}. For this, we rst reference the quasiparticle energies to

2

the absolute value of the CBM is given by the average electric potential in the bulk of the photoelectrode,

E_CBMðabsÞ ¼ ÀEA À eD^{H O}j:

(6)

2

PE

ꢀ^PE

f

, with the electric potential being here calculated as

À Á ð

n r⁰

X

e

1

3

0

V_i^locðjr_i À rjÞ;

(1)

ꢀꢀ
ꢀꢀ

fðrÞ ¼ À

d r À

0

4p3₀

• e
• r À r

i

where e is the elementary charge, 3₀ is the vacuum permittivity, n(r) is the calculated valence electron density and V^l_i^oc(r) is the local part of the pseudopotential associated with an atom core located at r_i. The latter is not unique and therefore depends on the pseudopotentials used. We dene the average electric

ꢀ

potential, f(z), as

a

2

ð

• À
• Á

1

fðzÞh hfi_xy z þ z⁰ dz⁰;

(2)

a

2

a

À

where a is the lattice parameter of the photoelectrode's surface unit cell in the z-direction which is parallel to the surface normal, and hfi_xy(z) is the average in the (x,y)-plane. Furthermore, it is implied that hfi_xy(z) is averaged over different atomic congurations.

ꢀ^PE

Fig. 2 Qualitative profile of the average electric potential (green) along the z-direction of the vacuum–photoelectrode–electrolyte– vacuum slab structure comprised by the computational supercell. The outer electric potential of water is chosen to be zero.

Aer referencing the quasiparticle energies to f , the

ꢀ^PE

difference between f and j^{H O} is computed. In principle, this

2

requires a unit cell containing a slab with a photoelectrode

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For a detailed discussion, see the work of Cheng et al.¹⁶

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rutile's surface properties with respect to the slab thickness
The computational details of each of these steps are outlined compared to PBE. The PBE exchange–correlation functional was below. We emphasize that we perform GW calculations only for used for the electronic structure calculations for the sake of the bulk photoelectrode, but not for the photoelectrode–water easing comparison with previous work on rutile. or the water–vacuum interfaces. The goal of our work is to align energy levels of bulk states which are unaffected by surface phenomena, such as mutual polarization effects.^3,16 For non-

2.3 Rutile(110)–water interface

ideal photoelectrode–electrolyte interfaces, where surface We calculated the difference between the average electric states are important, GW calculations of the full interface will be potentials in the rutile(110) photoelectrode and the aqueous

TiO₂

• electrolyte, i.e. D^T_H^iO_O f ¼ f
• À f^{H O}, using both JDFT and

2

needed.

2
2

classical molecular dynamics combined with subsequent DFT calculations (MD+DFT).