Shell Quantum Dot Sensitized Solar

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First-Principles Modeling of Core/Shell Quantum Dot Sensitized Solar Cells Jon Mikel Azpiroz, Ivan Infante, and Filippo De Angelis J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 07 May 2015 Downloaded from http://pubs.acs.org on May 7, 2015

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The Journal of Physical Chemistry

First-Principles Modeling of Core/Shell Quantum Dot Sensitized Solar Cells Jon M. Azpiroz*,a,b Ivan Infante,c and Filippo De Angelisa a

Computational Laboratory for Hybrid/Organic Photovoltaics (CLHYO), Istituto CNR di Scienze e Tecnologie Molecolari (ISTM-CNR), Via Elce di Sotto 8, 06123, Perugia, Italy.

b

Kimika Fakultatea, Euskal Herriko Unibertsitatea (UPV/EHU), and Donostia International Physics Center (DIPC), P. K. 1072, 20080 Donostia, Euskadi, Spain. c

Department of Theoretical Chemistry, Faculty of Sciences, Vrije Universiteit Amsterdam, De Bolelaan 1081, 1083 HV, Amsterdam, The Netherlands.

Abstract We report on the Density Functional Theory (DFT) modeling of core/shell Quantum Dot (QD) Sensitized Solar Cells (QDSSCs), a device architecture that holds great potential in photovoltaics, but has not been fully exploited so far. To understand the working mechanisms of this kind of solar cells, we have investigated ZnSe- and ZnSe/CdS-sensitized TiO2 models. Both the core-only and the core/shell QDs are predicted to strongly adsorb on the oxide surface, driven by the electrostatic interaction between the metal atoms on the QD surface and the O atoms exposed by the oxide substrate. Accordingly, the QD conduction states are strongly mixed with the TiO2 acceptor states, giving rise to bridge states that should funnel the interfacial electron transfer. Accordingly, quite fast electron injection processes are predicted, with computed rates of 135 and 163 fs. The backelectron transfer is much slower for ZnSe/CdS, due to the weak coupling between the newly injected charge and the holes trapped in the sensitizer core. Therefore, the core/shell QDs deliver much better efficiencies. Moreover, the interfacial dipole established between the TiO2-injected electrons and the holes confined in the QD are found to shift the conduction band edge of the oxide, which further improves the performance of the device in terms of the open circuit voltage (VOC). We believe that this work sets the ground for future computational works in the field, which could in turn guide the fabrication of new device architectures with improved efficiencies.

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1.Introduction

Over the last decades the increasing world energy demand has triggered the development of new energy sources. Among them, solar energy has gained momentum as a clean, efficient, costeffective, renewable, and sustainable alternative to solid fuels. Dye sensitized solar cells (DSSCs) firstly developed by O’Regan and Grätzel represented a significant breakthrough in the field.1 They consist of a nanostructured wide gap semiconductor layer, usually TiO2 or ZnO, sensitized with a dye monolayer, which absorbs solar radiation and injects the photogenerated electrons into the conduction band (CB) of the oxide.2 The concomitant hole is then trapped by an electrolyte, closing the circuit. More recently, semiconductor quantum dots (QDs) have gained ground as sensitizers against the classical organometallic dyes.3-5 The high extinction coefficients and large dipole moments typical in QDs, along with their photostability, make them amenable for light harvesting.6 Moreover, the on purpose modification of the band edges with the size and the shape of the QD enables a better overlap with the solar spectrum. It could also promote the electron injection process, due to an improved alignment of electronic levels. Besides, the Multiple Exciton Generation (MEG) inherent in some semiconductor QDs provides a mean to increase solar cell voltage and current,7, 8 making QD sensitized solar cells (QDSSCs) potentially able to overcome the Shockley-Queisser limit of 33%.9 However, the potential of QDSSCs has not been fully exploited to date, with record efficiencies of 7 %,10 far below their dye- (13%)11 and perovskite- (19%)12,

13

sensitized

counterparts. Many hypothesis have been put forward to explain such a poor performance, including stability problems due to the electrolyte.14 In many cases, the fast recombination processes ocurring in the QD (including exciton and Auger recombination),15 which compete with the interfacial electron injection, are probably the main reason behind the low efficiency of current QDSSCs. QDs often develop localized states on their surface that might act as trap states for the photogenerated carriers.16, 17 Moreover, the confinement in QDs implies strong Coulomb interaction between charge carriers, which in turn favours electron-hole recombination.18 The spatial separation of the photogenerated carriers is known to prevent the exciton recombination. In this sense, type-II core/shell QDs, composed of a core localizing the hole and a shell localizing the electron, hold potential as light harvesters, see Scheme 1.19 As a drawback, the localization of the hole in the core hinders its transport to the electrolyte, which in turn could lower the performance of the device. Besides, the staggered alignment of electronic levels in type-II core/shell QDs results in a significant redshift of the absorption features relative to the semiconductor constituents, leading to


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improved absorption characteristics. With regard to the single-material sensitizers, narrow bandgap QDs such as PbS and InP are able to cover an important portion of the solar spectrum but they might deliver poor efficiencies,20 due to their low CB edge,21 which often could lie below the acceptor conduction states.8, 22-24 Moreover, the inherently high valence band (VB) edge of this kind of materials, makes them unstable and prone to oxidation. CdS and CdSe, with their higher CB edge, exhibit prominent photon-to-current conversion efficiencies, at the expense of a narrow optical range.25-33 Due to the spatially indirect bandgap, type-II core/shell QDs combine broad absorption spectra and efficient electron injection. Besides, embedding the sensitizer core in an inorganic shell is known to passivate the dangling bonds, prevent the development of localized states, and mitigate trap recombination, which would otherwise lower the performance of the QDbased device.34 ZnSe/CdS heterostructures represented the first example of core/shell sensitizers in the field of QDSSCs. In their seminal work, Ågren et al. reported a discouraging overall conversion efficiencies of 0.27%.35 Since then, a great deal of experimental effort has been devoted to the development of this class of devices,19, 36-44 although their performance is still poor. Here we present a theoretical investigation on core/shell sensitized TiO2 solar cells, aimed at understanding the atomistic details that govern the functioning of core/shell QDSSCs. To the best of our knowledge, this is the first computational study on this kind of nanocomposites. Therefore, we believe that it will pave the way for future works in the field, which could in turn guide the rational fabrication of solar cell architectures with improved performances.



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Scheme 1. Schematic representation of photoexcitation (blue arrow), injection (green arrow), and recombination (red arrow) processes in core/shell QDSSCs. The valence and conduction bands of the QD sensitizer, localized in the core (red blocks) and the shell (blue blocks) respectively, are highlighted, along with the tail of unoccupied states of the oxide substrate (red line). Black and white dots stand for electrons and holes, respectively.

2.Model and Computational Details

The models studied throughout this work are depicted in Figure 1. We opted for the cluster models to simulate the the TiO2 nanoparticles, because it allows to perform excited state calculations with hybrid functionals and to include solvation effects, which are mandatory for a proper description of the electronic structure of the metal oxide. In particular, we considered the (101) surfaces often exposed in TiO2 nanocrystals, by appropriately cutting a (TiO2)82 from the underlying anatase structure.45 To reproduce the core/shell QD, a (ZnSe)12/(CdS)48 cluster has been built up (1 hereafter). This onion-like model mimics the (ZnS)12/(ZnS)48 geometry firstly reported by Hamad et al. as the lowest-lying isomer for the (ZnS)60 stoichiometry.46 For comparison, the core-only (ZnSe)12/(ZnSe)48 (2 hereafter) has also been considered.

Figure 1. Optimized structures of the standalone TiO2 slab, the ZnSe/CdS QD (1), the ZnSe QD (2), and the interacting 1@TiO2 and 2@TiO2 complexes. Pink = Ti, red = O, blue = Zn, orange = Se, grey = Cd, and green = S atoms.


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Experimentally, there are two main methodologies to couple the QD sensitizer and the TiO2 substrate.4 The in situ approaches, which include the chemical bath deposition (CBD) and the succesive ionic layer adsorption and reaction (SILAR), imply the direct growth of the QD on the oxide surface. These techniques provide high surface coverage and direct connection between the cell constituents, but suffer from poor control over the QD size, shape, and surface passivation. To overcome this drawback, presynthesized colloidal QDs are often adsorbed on the oxide surface via bifunctional linkers. For simplicity, we opted by the direct adsorption of the QDs on the oxide surface, as in previous theoretical works.

47

Although ligand-capped QDs would probably better

reproduce the experiment, this work aims at unraveling the role of the core/shell architecture on the improved performance of sensitized solar cells. Therefore, our bare models will probably capture in any case the experimental trends, even if they lack of the surface molecules. 1 and 2 expose flat facets that could strongly interact with the TiO2 slab. To maximize the contact with the oxide surface, we decided to deposit the QD facing these planes the oxide surface. All the models have been fully optimized, with no symmetry constraint, by means of the dispersion corrected (D3)48 Perdew-Burke-Ernzerhof (PBE)49 xc functional, in combination with a DZ set of Slater-type orbitals, as implemented in the ADF 2012.01 software package.50 Atomic orbital lying below the 1s, 2p, 3p, 3p, 3d, and 4p have been treated with the frozen core approximation for O, S, Ti, Zn, Se, and Cd respectively. Relativistic effects have been considered by means of the Zero-Order Regular Approximation (ZORA) of the Dirac equation.51 For simplicity, optimizations have been conducted in vacuum. Isolated QDs and TiO2 have been relaxed until the maximum norm of the Cartesian gradients was smaller than 1·10-3 Hartree/Angstrom. For the QD@TiO2 nanocomposites, a looser convergence criteria of 5·10-3 Hartree/Angstrom has been set. Although reliable for ground state geometries and normal modes, GGA functionals often fail to describe the electronic and optical properties of semiconductor nanostructures.52 To overcome this drawback, the hybrid B3LYP53 functional has been employed, in conjunction with the 3-21G* (O and Ti atoms) and the LANL2DZ (S, Zn, Se, and Cd atoms) basis sets, to perform ground state and excited state calculations on top of the PBE geometries. For the excited states, we have taken advantage of the simplified Tamm-Dancoff-Approach (sTDA) recently developed by Grimme et al.,54 which allows the calculation of thousands of electronic excitations in systems containing hundreds of atoms. For computational performance, calculations involving B3LYP have been conducted by means of the Gaussian09 package.55 To include the effect of the water solvation, the Conductor-like Polarizable Continuum Model (CPCM) has been used.56



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Radial Distribution Functions (RDFs) have been computed employing the Virtual Molecular Dynamics (VMD) software.57 Charge transfer integrals have been calculated with a locally developed program.

3.Results and Discussion

3.1. Isolated Quantum Dots

To understand the functioning of core/shell QDSSCs, it is worth to have a look to the electronic structure of the freestanding QD. In Figure 2 the electronic levels of 1 are depicted, along with their projection on the core and the shell regions. For comparison, the orbitals of 2 are shown. Irrespective of the QD, the LUMO is spread over the shell, as the rest of the conduction band states. We found a completely different picture in our recent calculations on PbS QDs, for which the LUMOs locate in the core.47 However, the delocalization of the virtual orbitals in the shell is an interesting feature for the functioning of the solar cell, because it favors electron injection into the TiO2, due to the effective coupling between the QD donor and the oxide acceptor states. Moreover, since 1 and 2 share roughly the same conduction band edge, a similar electron injection process might be anticipated (see below). Regarding the HOMO, for 2 it is evenly distributed over the core and shell regions. This is in sharp contrast with the HOMO of 1, which is located in the core region, in such a way that the coupling with the TiO2 conduction states is small and the back transfer of the newly injected electron slowed down.


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Figure 2. Electronic levels of the freestanding 1 and 2 QDs (grey sticks), along with the contribution due to the core (red sticks). The isodensity surfaces of the HOMO and the LUMO orbitals are also shown. The retarded recombination process is not the only advantage of the core/shell QDs. As one might notice from Figure 2, 2 displays a quite large HOMO-LUMO gap of 3.53 eV, providing a poor overlap with the sunlight spectrum. 1 exhibits a smaller bandgap of 3.13 eV, which is better suited for light harvesting. Interestingly, the shrinkage of the gap when passing from 2 to 1 related to the destabilization of the HOMO. The LUMO is still high enough (1.81 eV) with respect to the conduction band edge of the TiO2 to drive the electron injection. The redshift of the absorption features of 1 relative to 2 are evident from Figure 3, where the simulated absorption spectra of both QD models are depicted. The first excitonic features locate at 2.69 and 3.08 eV for 1 and 2, respectively. The shift between them amounts to 0.4 eV, in fair agreement with the experiments by Ågren et al., who reported a difference of 0.5 eV. Note, however, that the calculated lowest-lying peaks are blueshifted with respect to the experiment by ca. 0.5 eV due to the reduced size of our QD models, which experience strong quantum confinement effects.



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Figure 3. Simulated absorption spectrum of the standalone 1 (black line) and 2 (red line) sensitizers, obtained by a Gaussian convolution of σ = 0.15 eV of the individual electronic transitions (vertical sticks). The height of the sticks represents the oscillator strength of the transition.

At this point, we decided to investigate the nature of the main optical features of 1 and 2. In particular, we break down the absorption spectra into their several contributions, depending on the conduction state populated upon photoexcitation, see Supporting Information, Figure S1. Irrespective of the QD models, the first optical peak comprises transitions from the valence band edge states to the LUMO exclusively. In fact, in spite of being similar to higher-lying unoccupied orbitals, the LUMO is quite separated in energy from the rest of the conduction band states. This is an interesting result, pointing out the LUMO as the main orbital from which injection will take place. From our calculations, hot electron injection will only be possible at higher excitation energies (assuming that electron cooling to the LUMO of the QD is slow enough). For 1, we studied a second partition of the absorption spectra, which depends on the spatial localization of the orbitals involved in the electronic transition. From our results, the lowest-lying excitations imply transition from occupied orbitals confined in the ZnSe core to states spread over the CdS carapace, see Supporting Information, Figure S2. Shell-to-shell transitions dominate the spectrum at higher excitation energies.

3.2. Interacting Complexes


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3.1. Geometries and Energetics

The adsorption of the QDs on the wide gap semicondutor plays a crucial role on QDSSCs, because the strength of the QD-substrate interaction determines the stability of the nanocomposite lying at the heart of the solar cell. Besides, the electronic levels of both the sensitizer and the semiconductor are modified in the interacting system. Therefore, a correct prediction of the adsorption geometry is essential to properly reproduce the complex electronic structure of the QD@TiO2 compound. At first glance to Figure 1, the QDs seem to be tightly bound to the TiO2 slab. To better characterize the structure of the interacting complexes, the M(QD) – O(TiO2) (M = Cd, Zn) and the X(QD) – Ti(TiO2) (X = S, Se) Radial Distribution Functions (RDFs) have been calculated, see Supporting Information, Figure S3. Both 1@TiO2 and 2@TiO2 display a broad distribution of M – O and X – Ti pairs. However, the calculated distances are short enough to foresee chemical bonds between the M (X) atoms on the QD and O (Ti) on the TiO2 surface. From the RDFs, the interactions between the M atoms on the QD and the 2-fold coordinated O atoms on the TiO2 slab appear to be particularly important, as shown by the short M – O bonds at R < 2.5 Å and the formation of a large number of M – O pairs (note the steep increment of the integrated RDF with the increasing R). Irrespective of the model, the X – Ti contacts are longer and scarcer. The M – O bonds are particularly short for the 2@TiO2 system, as shown by the prominent feature at R = 2.02.2 Å. However, the integrated M – O and X – Ti RDFs reveal a higher coordination between 1 and the TiO2 slab, suggesting that the core/shell model better accommodates on the oxide surface. To characterize the interaction between the QDs and the TiO2 slab, we have taken advantage of the bond energy decomposition method developed by Ziegler and Rauk,58-60 which partitions the overall bond energy between two interacting fragments (i.e. the QD and the TiO2 slab) into two main contributions:

∆E = ∆E prep + ∆Eint

(1)

The preparation energy ∆Eprep refers to the energy required to distort the separated fragments from their equilibrium structure to their geometry in the complex. The interaction energy

∆Eint accounts for the instantaneous interaction between the prepared fragments. This term is further decomposed into four contributions, namely the Pauli repulsion ∆EPauli, the electrostatic interaction

∆Velst, the orbital interaction ∆Eoi, and the dispersion interaction ∆Edisp:

∆Eint = ∆EPauli + ∆Velst + ∆Eoi + ∆Edisp

(2)



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Within this energy decomposition scheme the attractive and repulsive terms are negative and positive, respectively. Therefore, the more negative the energy term is, the more attractive is the corresponding interaction. The energy decomposition analyses of 1@TiO2 and 2@TiO2 are summarized in Table 1. From our simulations, 1 interacts quite strongly with the oxide surface, with a calculated ∆Eint of 334.84 kcal/mol. However, the preparation energy implies a large energy penalty of 159.54 kcal/mol, which leaves the overall bond energy ∆E in -175.30 kcal/mol. The sizable preparation energy is mainly related to the geometrical rearrangement experienced by the oxide slab upon interaction, see Table 1. Both the instantaneous interaction energy and the preparation term are smaller (in absolute value) for 2@TiO2, which summed up lead to a bond energy ∆E of -167.25 kcal/mol, slightly weaker than that calculated for 1@TiO2. Irrespective of the model, the interaction is electrostatically driven, covering ca. 60% of the attractive interactions. The orbital terms are also significant (ca. 30%), suggesting that polarization effects are relevant at the QD/TiO2 interface. Dispersion forces are meaningful as well, since they comprise ca. 10% of the attraction. Therefore, considering long-range interactions is important to properly describe the QD/TiO2 interface.

Table 1. Energy decomposition analysis of the 1@TiO2 and 2@TiO2 complexes. Energies written in kcal/mol. Values in parentheses give the percentage of each attractive term with respect to the sum of the attractive terms.

∆Ebond ∆Eprep QD TiO2 ∆Eint ∆EPauli ∆Eelst ∆Eoi ∆Edisp

1@TiO2 -175.3 159.5 51.1 108.4 -334.8 585.1 -526.1 (57%) -283.1 (31%) -110.7 (12%)

2@TiO2 -167.2 96.9 33.1 63.8 -264.2 556.8 -463.2 (56%) -255.8 (32%) -102.0 (12%)

3.2. Electronic Structure and Optical Properties


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In Figure 4 the Density of States (DOS) of the 1@TiO2 and 2@TiO2 complexes are depicted, along with their projection into the QD and the TiO2 moieties (PDOS). For both complexes, the conduction band edge of TiO2 emerges at ca. -3.5 eV, in nice agreement with the experiment.61 The valence band, instead, locates at -7.0 eV. Irrespective of the model, the QD occupied states intrude into the bandgap of TiO2, with the unoccupied orbitals immersed in the manifold of conduction band states of the oxide. Such an alignment of electronic levels is a crucial prerequisite for the electron injection from the sensitizer into the TiO2. The PDOS of 1 and 2 essentially coincide with the DOS of the freestanding QDs, as it does the PDOS of TiO2 in the 2@TiO2 complex, see Supporting Information, Figure S4. On the contrary, the CB edge of the oxide in the 1@TiO2 complex rises sizably with respect to the isolated TiO2 slab, due probably to the electrostatic effect induced by 1 (see below). This upward shift increases the open circuit voltage (VOC), which is in turn desirable for the functioning of the solar cell.

Figure 4. Top panels: Density of States (DOS) of 1@TiO2 (a) and 2@TiO2 (b), along with the projection into the sensitizer (blue) and the oxide slab (red), obtained by a Gaussian convolution of

σ = 0.20 eV of the individual molecular orbitals. Bottom panels: Molecular orbitals of the interacting complexes, along with the contributions from the QD (blue) and the TiO2 (red) fragments. The height of the stick represents the % of the orbital localized in each of the corresponding fragments.

At this point, it is worth to identify the QD virtual states in the interacting complexes, from which electron injection will presumably take place. From the bottom panels of Figure 4, where the fragment contributions to the complex states are sketched, sensitizer orbitals appear ca. 2 eV above



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the conduction band edge of the oxide. Importantly, they are strongly hybridized with the TiO2 orbitals. This kind of mixed states should funnel adiabatic electron injection from the QD to the oxide. These findings are in sharp contrast with our recent calculations on PbS QD@TiO2 solar cells, for which the sensitizer orbitals appear closer to the oxide conduction band edge, and weakly coupled with the acceptor states.47 The wider gap of the II-VI QDs and their stronger interaction with the TiO2 slab should explain the alignment of the electronic levels and their mixing with the oxide orbitals, respectively. As shown later, these differences will play an important role on the electron injection process. To simulate the optical response of the QD@TiO2 nanocomposites, we have calculated their excited states. In Figure 5 the absorption spectra of the interacting complexes is shown, along with the absorption features of the freestanding sensitizers. The electronic excitations in the QD@TiO2 are in general weak, due probably to the small spatial overlap between the VB and the CB states involved in the transitions, which are mostly localized in the QD and the TiO2 slab, respectively. 1@TiO2 and 2@TiO2 display a similar absorption onset, which emerges at 1.75 eV, in excellent agreement with the experiments (ca. 1.7 eV).40 The first optical feature of 1@TiO2, instead, is redshifted by 0.42 eV with respect to that of 2@TiO2. This difference essentially coincides with that calculated for the standalone 1 and 2. However, we observe a consistent blueshift of the lowestlying excitonic feature of the sensitizers upon complexation with the TiO2 surface, which amounts to ca. 0.1 eV, along with a sizable broadening of the first peak. This is expected as a consequence of the strong interaction with TiO2.


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Figure 5. Simulated absorption spectrum of the interacting systems 1@TiO2 (solid black line) and 2 TiO2 (solid red line), obtained by a Gaussian convolution of σ = 0.15 eV of the individual electronic transitions (vertical sticks). The height of the sticks represents the oscillator strength of the transition. For comparison, the spectra of the standalone 1 (dashed black line) and 2 (dashed red line) are also shown.

3.3. Injection and Recombination Processes

The electron injection and recombination processes are crucial for the functioning of QDSSCs. To simulate them, we have taken advantage of the Fermi’s golden rule,62, 63 as in our previous work on PbS QDSSCs.47 In brief, the rate constant of electron injection, kinj, from a single donor state, QD, to a manifold of acceptor states, TiO2, can be expressed according to Eq. 3:

∑ , (3)

where ħ is the reduced Planck constant, ρ(ε) is the DOS of the oxide substrate evaluated at the energy of the donor state ε, and VQD,TiO2 is the electronic coupling between the donor and the acceptor states. The VQD,TiO2 terms are nothing but the off-diagonal elements of the Hamiltonian of the interacting QD@TiO2 complex, written on the basis of the QD (φ) and the TiO2 states (φTiO2): (4) Alternatively, the injection rate can be expressed as:



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(5)

where Γinj stands for the injection function, which provides a visual interpretation of the coupling between the QD and the TiO2 states. In Figure 6 we depict the electronic coupling between the QD LUMO and the manifold of oxide CB states, along with the corresponding injection function. At a first glance to Figure 6, the LUMO of 1 and 2 couple quite similarly with the oxide states. Accordingly, evaluation of the Γinj at the LUMO energy delivers almost coincident injection rates of 135 and 163 fs for 1 and 2, respectively.

Figure 6. Scheme of the injection and recombination processes in 1@TiO2 (left) and 2@TiO2 (right). For each model, the DOS of the TiO2 states (blue, left) and the injection function Γinj(ε) (red curve, right) are shown, along with the electronic couplings between the QD LUMO and the manifold of the TiO2 k acceptor states (black sticks, right). For sake of clarity, the coupling elements have been reduced by a factor of 5. ∆G, calculated as the energy difference between the TiO2 LUMO and the QD HOMO, represents the driving force for the recombination process. The excited state calculations in the freestanding QDs revealed us that the LUMO is the only virtual orbital involved in the lowest-lying excitations. In addition, Tisdale et al. recently showed that, at room temperature, hot electrons populating higher-lying CB states cool down


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rapidly to the CB edge.64, 65 Therefore, hot injection in QDSSCs only would happen under particular circumstances. However, to simulate such a process, we also calculated the interfacial electron transfer for higher-lying unoccupied states. As found for the LUMO, no clear differences are observed between 1@TiO2 and 2@TiO2. Interestingly, in both cases injection seems to be favored with the increasing energy of the donor CB state, see Supporting Information, Figure S5. Coming back to the interfacial electron injection from the sensitizer LUMO, to properly simulate the functioning QDSSCs other physical processes lying at the heart of the device have to be considered, including the radiative decay of the photoexcited QD and the recombination of the newly injected electron and the hole residing in the QD. Therefore, the injection efficiency of the device should be expressed as:

!"## $

$

%$&'( %$&)*

(6)

where kinj, krec, and krad are the rate constants for the electron injection, electron-hole recombination, and the radiative decay of the QD excited state, respectively. The latter occurs in the ns time scale, considerably slower than the interfacial electron injection and recombination processes, so it can be neglected in Eq. 6. According to the Marcus theory, the back-electron transfer from the TiO2 LUMO to the QD HOMO can be computed as follows:

+",

|.*/ |

012$3

456 7

89:;&'( %2

Shell Quantum Dot Sensitized Solar

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