Quantum Dynamics of Light-Induced Charge ... - ACS Publications

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Quantum Dynamics of Light-Induced Charge Injection in a Model Dye−Nanoparticle Complex Christian F. A. Negre,† Valeria C. Fuertes,† M. Belén Oviedo,† Fabiana Y. Oliva,‡ and Cristián G. Sánchez*,† †

Departamento de Matemática y Física, Facultad de Ciencias Químicas, INFIQC, Universidad Nacional de Córdoba, Ciudad Universitaria, X5000HUA, Córdoba, Argentina ‡ Departamento de Fisicoquímica, Facultad de Ciencias Químicas, INFIQC, Universidad Nacional de Córdoba, Ciudad Universitaria, X5000HUA, Córdoba, Argentina ABSTRACT: We present a detailed description of the direct charge injection mechanism in a coupled dye−TiO2 nanoparticle (NP) using a full quantum dynamical simulation framework. The method employed here is based on a timedependent tight-binding model to describe the system under nonequilibrium conditions. By using this tool, we performed a simulation showing a full timedependent picture of the photoabsorption process in type-II dye-sensitized solar cells (direct charge injection mechanism from the dye to the TiO2 NP). This task is accomplished by tuning the frequency of an applied sinusoidal time-dependent electric field with the frequency of the dye−TiO2 NP’s main absorption peak. We find that during the field irradiation there is a net charge transfer from the dye to the NP superposed with a typical charge oscillation due to absorption of radiant energy.

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INTRODUCTION From the first paper of Grätzel and co-workers,1 dye-sensitized solar cells (DSSCs) are considered promising devices for solar energy conversion. This is due to their low cost of production, robustness, and relatively high conversion efficiency, which is slowly reaching that of traditional silicon based cells.2 DSSCs are composed of a metal complex or organic dye attached to the surface of a meso- or nanostructured semiconductor. TiO2 and ZnO semiconductors are commonly used due to their appropriate band gap structures.3 The understanding of the DSSC operation mechanism is of major importance to find ways of enhancing their efficiency and lifetime.4 The main step of their operating principle is the photoinjection process in which an electron in the ground state (GS) of the dye is injected into the conduction band of the semiconductor during or following the absorption of a photon.5 Although the photoinjection process has been exhaustively studied during the last 20 years, many of its aspects remain unclear. Issues such as whether the excited state is localized at the dye or within the semiconductor phase, if the electron transfer (ET) mechanism is adiabatic or nonadiabatic, as well as the role of temperature are currently under discussion.5,6 The injection mechanism depends on both the semiconductor and the dye as well as their distance of separation. Even the cell electrolyte can indirectly influence the electron injection rate.2 DSSC devices can be classified into two categories depending on whether the electron is injected during or after the absorption of the photon. The Type I mechanism (indirect injection) proceeds in two steps: chromophore excitation caused by photon absorption and subsequent electron injection © 2012 American Chemical Society

(see Figure 1). On the other hand, the Type II mechanism occurs via direct promotion of an electron from the GS of the dye to the first unoccupied level (the lowest unoccupied molecular orbital (LUMO)) of the conduction band of the semiconductor upon photoabsorption.5,7 The direct mechanism is evidenced by a new band in the absorption spectrum that arises when the dye is attached to the semiconductor surface.8 Catechol and alizarin are the benchmark dyes used for studying Type II and Type I mechanisms, respectively. For catechol, a new absorption band centered at 3.0 eV appears when the dye is attached to the TiO2 semiconductor.8−11 Results in this paper are related to this dye in combination with an anatase NP as a model system. Although expensive, Ru complexes dyes are still the highest in performance for DSSC purposes. On the other hand, simple organic molecules having delocalized π−π* orbital excitations have larger cross sections and are conceived from cheaper synthesis. Understanding the details of the photoinjection process would provide information that can guide the synthesis of new dyes for DSSCs. There has been significant theoretical research focused on the details of the injection mechanism under many different considerations:7,12−18 from the time-dependent extended Hückel simulations of the ET in the Type I mechanism of Rego et al.12 to the more sophisticated time-dependent density functional theory (TDDFT) calculations in the frequency Received: October 25, 2011 Revised: June 19, 2012 Published: June 22, 2012 14748

dx.doi.org/10.1021/jp210248k | J. Phys. Chem. C 2012, 116, 14748−14753

The Journal of Physical Chemistry C

Article

in refs 22 and 23, and is based on the time propagation of the one electron density matrix under the influence of external time-varying electromagnetic fields in order to obtain optical information about the system. For further details on the method and its validation for the description of absorption spectra of molecular systems, refer to the work of Sánchez and collaborators.22 In order to obtain optical absorption spectra, we introduce an initial perturbation in the shape of a Dirac delta pulse (Ĥ = Ĥ 0 + E0δ(t − t0)μ̂ ) to the initial GS density matrix.24 After pulse application, the density matrix evolves in time, and its evolution can be calculated by time integration of the Liouville−von Newmann equation of motion in the nonorthogonal basis: ∂ρ ̂ 1 = (S −1Ĥ [ρ ]̂ ρ ̂ − ρ ̂Ĥ [ρ ]̂ S −1) ∂t iℏ

(1)

where S−1 is the inverse of the overlap matrix. Note that Ĥ depends on ρ̂ through the onsite charges, introducing nonlinear effects in the dynamics and renormalizing the excitations of the non-self-consistent DFTB Hamiltonian.25 In the linear response regime, when the applied electric field pulse is small, the dipole moment is

Figure 1. Energy diagram representation of the different processes involved in the operation principle of a DSSC.

∞

μ(t ) =

domain to obtain absorption spectra in order to reveal the prevalent mechanisms.7,13 However, to the best of our knowledge, no simulations have been shown involving a complete picture from photon striking to electron injection. These two processes are usually treated separately, and a concerted picture is lacking. Traditional calculations of the injection time are performed by simulating the adiabatic ET from the LUMO of the dye molecule to the conduction band of the semiconductor in Type I dyes.12 This simulated injections times are typically of the order of tens of femtoseconds, reflecting the ultrafast coherent quantum transport between states at the same energy. The Type II (direct injection) mechanism requires the photoabsorption to be coupled with the ET as, in this case, an electron is promoted from the dye highest occupied molecular orbital (HOMO) to states in the conduction band of the semiconductor that are higher in energy. Tools to be developed in order to simulate this direct mechanism need to be able to handle, at the same time, the photon absorption together with the interfacial ET. The aim of the present work is to understand the direct process of charge injection between the dye and the TiO2 nanoparticle (NP), providing a complete time-dependent picture of it, including the nonadiabatic interaction with the electromagnetic fields.

∫−∞ α(t − τ)E(τ) dτ

(2)

where α(t − τ) is the polarizability along the axis over which the external field E(t) is applied. The absorption spectrum is proportional to the imaginary part of the frequency dependent polarizability, obtained from the Fourier transform of the timedependent dipole moment, after deconvolution of the applied electric field (E(t) = E0δ(t0 − t), with E0 = 0.001V/Å): α(E ) =

μ(E) E0

(3)

In order to make the dipole signal die out within the simulated time window, an exponential damping with a time constant of 10 fs is applied before transforming to frequency space, uniformly broadening all of the spectral lines. This method determines the frequency-dependent polarizability along the direction of the initially applied field. The average of the polarizability over the three Cartesian axes is taken as the absorption spectra of the system. We have checked that spectra are independent of the magnitude of the applied field, indicating that the simulations are done within the linear response regime under which the frequency-dependent polarizability can be extracted from the response using equation eq 3. We have found that an optimal time step integration for performing electronic dynamics is 9.675 × 10−4 fs, providing adequate resolution of low energy modes and ensuring numerical stability. There is no restriction on the function E(t) to be used as a perturbation, and this will depend on the information to be extracted from the dynamics. In order to model the overall picture of an electromagnetic wave striking on a dye-cluster system, we used the method described in refs 26 and 22, in which a sinusoidal shape perturbation E(t) = E0 sin(ωt) is employed. Application of this type of perturbation in tune with a particular excitation can describe the electron dynamics that occurs during light absorption. This reveals the underlying dynamics that characterizes the excited electron motion at frequency ω.

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COMPUTATIONAL METHOD We have used a self-consistent density functional tight-binding method (SCC-DFTB) to model the electronic structure of the cluster-molecule system. This method has been successfully applied before to describe the electronic structure of large molecular systems.19 SCC-DFTB is based on the second-order expansion of the Kohn−Sham energy functional around a reference density of neutral atomic species. The DFTB+ code20 is a sparse matrix-based implementation of the SCC-DFTB method. We have used this code in order to compute the Hamiltonian, overlap matrix elements, and the initial GS single electron density matrix. [For the calculations shown here, we used the tiorg-0-1 and mio-0-1 DFTB parameter sets.19,21] The methodology applied in this work has been recently described 14749



Article

Once we have the evolution of the atomic charges produced by the application of a laser type perturbation, in order to characterize the excitation, we evaluate the following function: ς(r ) =

∑ i

⟨qi2⟩ − ⟨qi⟩2 (r − ri)2

(4)

where ri and qi are the atomic positions and charges, respectively. ⟨qi⟩ = (1/Nt)∑kqi(tk) and ⟨q2i ⟩ = (1/Nt)∑kqi(tk), where Nt is the total number of time steps and tk stands for the time at step k. Equation 4 is a way of visualizing the electrostatic potential of hot electrons involved in the dynamics and serves the purpose of depicting the spatial distribution of electrons involved in a particular optical excitation.

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Figure 2. Ti−Ti radial distribution function for a 570 atoms anatase NP compared with bulk.

RESULTS AND DISCUSSION For all calculations performed in this work, we used a 270 atom (90 TiO2 units) anatase nanocluster, and catechol and cresol as sensitizer dyes. Despite the fact that cresol is not commonly used in DSSCs, probably because of photochemical degradation,27 we have considered it here because it has a chemical structure similar to that of catechol. On the other hand, it has a different direction of the transition dipole moment with respect to the anchoring group (see below). TiO2 nanoclusters are generated by performing molecular dynamic simulations of a spherical piece cut out from bulk anatase having the desired radius. Before starting the dynamics, O surface atoms are reordered to obtain an homogeneous distribution that satisfies local charge neutrality. The LAMMPS molecular dynamics code has been used for all MD simulations.28 We have used the Matsui−Akaogi potential detailed in ref 29 in order to describe TiO2. This potential has been used before for several structural nanoscale TiO2 studies.30−36 MD simulations were carried out in the canonical ensemble (NVT) at 300 K for times of 3.5 ns production runs following equilibration using a time step of 3.5 fs. We have taken the last frame of the production run for performing all optical calculations. A full study of the dependence of the optical properties with cluster size for the three TiO 2 polymorphs (anatase, rutile, and brookite) will be the subject of an incoming work.37 We have chosen a 90 TiO2 unit anatase NP (270 atoms) for performing the electron dynamics calculations shown below. Although a further geometry optimization of the whole cluster with the DFTB method is desirable, it is unfordable because of the computational cost. As soon as the size of these clusters becomes larger, they reveal an internal crystal structure that is similar to its bulk counterpart, as depicted by the Ti−Ti radial distribution function shown in Figure 2. The selected cluster is in turn small enough to allow quantum dynamical simulations of the photoinjection process. Ti−Ti radial distributions are very similar to those obtained in ref 34. Catechol is found to adsorb onto the TiO2 surface in three different modes: molecular, dissociative monodentate, and dissociative bidentate.7,14,38−40 The molecular adsorbing mode is not considered here because it is less energetically favorable. Although the dissociative bidentate mode is energetically preferred, we found that for this TB model, this form is unstable. For this reason, we have considered only the monodentate form. Moreover, cresol adsorbs in the same way as the catechol monodentate adsorbing mode. A further geometry optimization of the dye onto the cluster surface was performed using the DFTB method. In order to find the

geometry of equilibrium, only coordinates from the dye and of the dye-coupled TiO2 units were allowed to change during geometry optimization. This allows one to obtain dye-relaxed geometry onto the NP surface within an affordable computational cost. For catechol and cresol, the calculated adsorption energies within the DFTB model are −1.34 and −1.86 eV, respectively. Upon adsorption of the dye onto the NP, there is a localized state that appears within the band gap and has projection over the atoms of the dye (see Figure 3). This state is responsible for

Figure 3. Band gap structure of catechol−TiO2 anatase NP. Total density of states (black line) and partial density of states projected over the catechol molecule (red dashed line). The inset shows the molecular orbital responsible for the new state that appears within the gap region upon adsorption.

enhancing the absorption cross section at the visible region of the spectrum, creating a new band at longer wavelengths than the usual pure TiO2 UV band (see Figure 5). Figure 4 shows the optical spectrum of catechol (a) and cresol (b) molecules. Both energy position and cross section are very similar for the lowest energy absorption band. For catechol and cresol, these bands appear at 4.72 and 4.75 eV, respectively. The figure also shows the calculated transition dipole moment vector corresponding to the lowest electronic transition energy. The magnitude of the corresponding vector for catechol and cresol molecules are 1.73 and 1.64 D, respectively. As can be observed, the two magnitudes are similar, but the directions with respect to the anchoring group are different. The magnitude of this vector is calculated by the linear response of the dipole moment, when the system is 14750



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Figure 4. Absorption spectra for catechol (a) and cresol (b) molecules. The inset shows the direction of the transition dipole moment vector corresponding to the lowest energy transition (red line).

perturbed by a monochromatic electric field polarized in the direction of the electronic transition. By fitting the slope of the oscillation amplitude of the dipole moment, the oscillator strength is obtained.23 We have calculated absorption spectra for both catechol and cresol NP systems (see Figure 5). Main peak positions of

(Figure 6). We have calculated the Mulliken charge change with respect to the GS value in both the dye and the NP as a function of time. A net charge transfer from the dye to the NP is observed as time evolves during the simulation. We have seen that the net negative charge that is transferred from the dye is finally homogeneously distributed over the cluster Ti atoms. Figure 6 represents, as far as we know, a new way to describe the Type II photoabsorption process. Figure 7 shows that this process is of second order with respect to the applied field: ∂q/ ∂t = kiE2. In other words, the slopes of the time-dependent averages of the Mulliken charges shown in Figure 6, ∂q/∂t, increase linearly as E2 increases with kt as the proportionality constant. The theoretical maximum power per unit area for this cell would be Pmax = (ktE2V)/s, where V is the voltage produced by the cell, and s is the transverse section of the cluster. On the other hand, the light intensity that arrives at the nanosystem is related to the field as Pin = (1/2)ε0cE2, where ε0 is the vacuum permittivity, and c is the speed of light. Provided that the power produced by the theoretical cell cannot be larger than the power arriving from the sun, we have that the theoretical upper bound for the efficiency is

Figure 5. Absorption spectra for the catechol/cresol−anatase NP system.

η = 100 ×

spectra calculated with this method are close to the experimental for the catechol−NP system at 2.91 eV (425 nm).8 Inhomogeneous effects cause experimental spectra of catechol−anatase NP systems to exhibit a shoulder in the absorption spectrum rather than a well-formed peak, as shown here for a single realization of the system geometry.9−11 Although here catechol is attached to an anatase NP, the conclusion drawn from the calculated absorption spectra and DOS are in good agreement with results from Persson et al.41 for catechol attached to an anatase cluster surface using the INDO/S−CI method. In order to explore the electron dynamics for the Type II mechanism, a sinusoidal time-dependent electric field tuned with the absorption maximum of each spectra was applied for both dye−NP systems. The illuminated frequency corresponds to the new peaks appearing upon adsorption just below the bare particle absorption edge (Figure 5). The polarization direction of incoming radiation was chosen so that it matches the transition dipole moment of the entire system. By plotting the ς(r) function, we can see that the excitation is mainly localized in the dye and the Ti4+ atom directly bonded to the catechol O, as well as some surface Ti atoms of the cluster

2V

ktE 2 s 2

ε0cE

= 100 ×

2kt V sε0c

(5)

Considering the band gap energy as an upper bound for the cell voltage (3.06 V), we conclude that this cell has a maximum efficiency of 0.69%, coincident with other authors findings.42 An upper bound for the photocurrent value can be calculated as follows: I = (ktE2)/s. The field E was calculated considering that all the power (1000 W/m2) of AM1.5 G solar conditions is adsorbed by the system. Transverse section s of the 90 TiO2 unit anatase NP was estimated to be approximately 3.14 nm2. The photocurrent upper bound calculated in this way is approximately 0.23 mA/cm2. This theoretical current limit is the current that could be measured if all the electron back transfer processes were neglected. The calculated photocurrent value is 23 times larger than the one measured by Oliva et al.43 for bare nanostructured TiO2, and 53 times smaller than the value measured in ref 1 for a Type I cell. The time taken to transfer a whole electron from the dye to the semiconductor is, in this case, 23 ms for both dye−NP systems. This time is much longer than adiabatic injection times for Type I mechanisms, which are on the order of tens of femtoseconds, reaffirming the fact that theses are Type II dyes. Although injection times are 14751



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Figure 6. Changes in Mulliken charges with respect to their GS values as a function of time for both dye and NP together with the plots of ς(r) functions.

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CONCLUSIONS Optical absorption spectra for both catechol and cresol dyes attached to TiO2 NPs were calculated within a full quantum dynamic framework. Calculated optical spectra for both attached and free dyes are in good agreement with experimental results. This implies that this tool can serve to determine the mechanism (Type I or Type II) by which the photoinjection process would proceed for a particular dye. By using the ς(r) function, we could reveal the excitation projected over real space, and conclude that this excitation is located over the dye and surface Ti atoms. We have evidenced a description of photoinjection that provides a real time-dependent picture of the Type II process. This mechanism can be viewed as a photoassisted quantum electron pumping from the dye to the TiO2 cluster. The mechanism is of second order with respect to the applied field, and this could be the reason for the low efficiency measured in Type II DSSCs. Similar molecules (catechol and cresol) give similar results, even for different transition dipole moment orientations. Estimated photocurrents are within reasonable values.

Figure 7. Slope of the time-dependent averages of the Mulliken charges as a function of E2. Straight lines are linear regressions to the data. Slopes (kt) and correlation coefficients (R) are also shown in the figure.

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very different from experimental values informed for catechol,44 probably because of differences in the field of the applied perturbation, the underlying electron hole interaction could be retarding the whole injection process. This electron hole interaction is taken into account by using the density matrix formalism together with the SCF-DFTB Hamiltonian. Simulations of photoinjection from Cresol dyes show the same general features. From this, we can conclude that the direction of the molecular excitation transition dipole moment with respect to the anchoring group does not determine the efficiency of the Type II DSSC photoinjection process.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support by Consejo Nacional de Inveś tigaciones Cientificas y Técnicas (CONICET) through Grant PIP 112-200801-000983 and ANPCYT through grant Program 14752



Article

(36) Koparde, V. N.; Cummings, P. T. J. Phys. Chem. C 2007, 111, 6920−6926. (37) Fuertes, V. C.; Negre, C. F. A.; Oliva, F. Y.; Sánchez, C. G. Manuscript in preparation, 2011. (38) Redfern, P. C.; Zapol, P.; Curtiss, L. A.; Rajh, T.; Thurnauer, M. C. J. Phys. Chem. B 2003, 107, 11419−11427. (39) Liu, L.-M.; Li, S.-C.; Cheng, H.; Diebold, U.; Selloni, A. J. Am. Chem. Soc. 2011, 133, 7816−7823. (40) Köppen, S.; Langel, W. Phys. Chem. Chem. Phys. 2008, 10, 1907−1915. (41) Persson, P.; Bergström, R.; Lunell, S. J. Phys. Chem. B 2000, 104, 10348−10351. (42) Tae, E. L.; Lee, S. H.; Lee, J. K.; Yoo, S. S.; Kang, E. J.; Yoon, K. B. J. Phys. Chem. B 2005, 109, 22513−22522. (43) Oliva, F. Y.; Avalle, L. B.; Santos, E.; Cámara, O. R. J. Photochem. Photobiol., A 2002, 146, 175−188. (44) Hao, E.; Anderson, N. A.; Asbury, J. B.; Lian, T. J. Phys. Chem. B 2002, 106, 10191−10198.

BID 1728/OC-AR PICT No. 629 and PME-2006-01581. V.C.F., C.F.A.N., and M.B.O. are grateful for studentships from CONICET.

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Quantum Dynamics of Light-Induced Charge ... - ACS Publications

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