Plasmon Enhanced Heterogeneous Electron Transfer: A Model Study

Article pubs.acs.org/JPCC

Plasmon Enhanced Heterogeneous Electron Transfer: A Model Study Luxia Wang†,* and Volkhard May‡,* †

Department of Physics, University of Science and Technology Beijing, 100083 Beijing, China Institute of Physics, Humboldt University at Berlin, Newtonstraße 15, D-12489 Berlin, Germany

‡

ABSTRACT: Plasmon enhancement of photoinduced charge injection from a perylene dye into a large TiO2 cluster is studied theoretically. A system is investigated where a spherical metal nanoparticle (MNP) is placed near the dye at the cluster surface. The simulations account for optical excitation of the dye coupled to the MNP and subsequent electron injection into the rutile TiO2 cluster with (110) surface. The electron motion in the cluster is described in a tight-binding model and focuses on excess electron localization at the Ti atoms and inter Ti charge transfer. Clusters with about 105 atoms can be treated. Charge injection dynamics is described in the framework of the density matrix theory which, however, ignores in this first attempt molecular vibrations. Considering short optical excitations, the overall probability to have the electron injected into the cluster reaches an intermediate steady state. This probability is used to introduce an enhancement factor which rates the influence of the MNP. Values larger than 1000 are obtained mainly caused by MNP induced photoabsorption enhancement. The also considered Coulomb coupling of the injected electron with the molecular cation and with the image charge induced at the MNP is of minor importance.

I. INTRODUCTION Numerous attempts to achieve metal nanoparticle (MNP) induced efficiency enhancement of a dye-sensitized solar cell have been described in literature (see ref 1 for a recent overview). Metal nanostructures combined with thin-film solar cells have been suggested in ref 2. Photosensitization of bulk TiO2 by embedded MNPs were described in ref 3, and refs 4 and 5 reported on TiO2 nanostructures decorated with MNPs. In particular, ref 4 considered a structure similar to what we have in mind: TiO2 clusters to which the dye molecules (Z907 ruthenium complexes) and the MNPs have been attached (cf. also Figure 1). It is argued in ref 4 that charge injection into the TiO2 cluster may become more efficient due to the MNP induced local field enhancement of photoabsorption. At the

same time it might be possible to observe a decrease of charge injection efficiency. Now, the excited dye state does not initiate charge injection but plasmon excitation in the MNP with subsequent energy dissipation due to plasmon decay. Although MNP induced dye absorption enhancement and the resulting photocurrent increase has been already addressed in the theoretical studies of ref 6, the essential dye excitation quenching was not considered so far. Concerning theoretical studies on TiO2 based dye solar cells without MNPs, a number of publications exist (see, for example, refs 7−9). Recent work with emphasis on electronic structure calculations has been reviewed in refs 10 and 11. The temporal evolution of charge injection was simulated in ref 12 by including nuclear vibration of the dye molecule or in refs 13 and 14 by neglecting them. In any case small TiO2 clusters have been considered ((TiO2)60 clusters in ref 12 and (TiO2)90 clusters in refs 13 and 14). In the latter work a density matrix description was applied similar to what has been done by us in ref 15 and what will be utilized here. Both studies in refs 13 and 14, however, employ a self-consistent density functional tightbinding method, an approach which restricts the size of the TiO2 cluster. To treat much larger clusters and to extend the dynamical simulations behind the 70 fs time limitation in refs 13 and 14, we move back to an empirical tight-binding model. We also describe the dye (here perylene) in a simple two electronic level model as in our earlier work (see, for example, refs 15−17). However, this somewhat reduced description is

Figure 1. Scheme of the dye (perylene) placed in the vicinity of a spherical MNP and attached to a huge TiO2 cluster (rutile form, (110) surface). The orange area around the dye indicates the part of the TiO2 system which has been explicitly considered in the simulations. (The choice of the coordinate system is indicated. Its origin defines the anchor position of the dye to the surface.) © 2014 American Chemical Society

Received: October 28, 2013 Revised: January 15, 2014 Published: January 15, 2014 2812

dx.doi.org/10.1021/jp410638r | J. Phys. Chem. C 2014, 118, 2812−2819

The Journal of Physical Chemistry C

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charge injection into a TiO2 lattice of some thousands of atoms (necessary parameters are taken from ref 22 and can be found in Table 1). Therefore, clusters are considered up to an x−y−z extent of 14.2 nm × 14.7 nm × 7.3 nm (orange area in Figure 1). This corresponds to more than 50 000 Ti atoms. Since the injected electron is mainly localized at the 3d levels of the Ti atoms22,24 we do not need to consider the oxygen atoms. The Ti atoms are counted by m and we use the convention to label the Ti atom closest to the anchor position of the dye with m = 1. The anionic states of the Ti atoms formed after electron injection are denoted by ϕm−. Since the excess electron states in the TiO2 lattice are the outcome of a tight-binding description they represent single (excess) electron states. The energies are denoted as εm− (if the electron is absent the electronic energy is εm and the respective state is denoted by ϕm). Excess electron motion in the TiO2 lattice is due to the transfer coupling Vmn (nearest neighbor Ti atom coupling: V, next nearest neighbor coupling: Ṽ , cf. ref 22). Moreover, we consider the ground-state φg of the perylene dye, the first excited state φe, and the cationic state φ+ which remains after charge injection. The related energies of the dye are εg, εe, and ε+, respectively. The molecular states have to be understood as total electronic wave functions which can be completed by a vibrational part χ. Charge injection from the excited molecular state into the adjacent Ti atom is considered by the transfer integral T1e. According to earlier studies16,17 the larger value of T1e as given in Table 1 corresponds to perylene attached via a carboxylic acid anchor group and the smaller one is due to a propionic acid anchor group.25 Optical excitation of the dye is governed by its transition dipole moment deg (we assume strict orientation in z direction, i.e., deg = ezdeg). To account for the presence of the MNP different ways have been suggested in literature (see the recent overviews in refs 26 and 27) We follow our own approach which treats the molecule−MNP complex as a uniform quantum system with internal Coulomb coupling.18 The latter appears as an energy transfer coupling between the molecular excitation and the collective plasmon excitations of the MNP (this instantaneous interaction requires an extension of the molecule−MNP complex smaller than typical wavelengths of light). The description includes the simultaneous optical excitation of the molecule and the MNP. It results in absorption enhancement of the molecule due to the MNP coupling induced oscillators strength renormalization (the short plasmon life-time is considered below). For the subsequent computations we have in mind a spherical Au nanoparticle with a diameter of 20 nm (see also Table 1). Accordingly, the complete molecule−MNP coupling Hamiltonian takes the form ΣIJIe|φg⟩|I⟩⟨0|⟨φe| + H.c. (the Hermitian conjugated part is abbreviated by H.c.). If the molecule MNP− surface distance overcomes 2 nm the coupling matrix element JIe can be approximated as an interaction between molecular and MNP transition dipole moments.18 The three degenerated dipole plasmons are labeled by I (= x,y,z), and the transition dipole moment is dI = dpleI (the common dipole moment is given by dpl and the eI are unit vectors of a Cartesian coordinate system). Then, the coupling matrix reads JIe = dpldegκI/R3. The magnitude of the molecular transition dipole moment is denoted as deg, and R is the distance between the molecule and MNP center of mass. The geometry factor κI = ([eIneg] − 3[eIn][nneg] also includes the unit vector neg (ez) of the molecular transition dipole moment and the unit vector n

combined with the excitation energy exchange coupling between the dye and the MNP. We follow our recent studies which treat the dye−MNP system as a uniform quantum system with internal Coulomb coupling.18−20 This uniform description accounts for local field induced molecular absorption enhancement as well as for molecular excitation energy quenching (see ref 21 for the a discussion of the shortcomings of a local field description). In order to evaluate the MNP enhancement effect photoinduced charge injection in the presence of the MNP and without the MNP have to be compared. Since the TiO2 system is considered as a finite cluster it would be not appropriate to focus the computation on steady-state quantities. Instead, the system is excited with a radiation field of finite duration. If the TiO2 cluster is large enough an intermediate steady state is formed where the overall excess electron population of the TiO2 lattice stays constant (at the same time the probability to have the molecule in its excited electronic state becomes zero). The ratio of this population computed in the presence of the MNP and in its absence is used as a quantitative measure of the MNP enhancement effect. The paper is organized as follows. In the subsequent section, we introduce the used model of the dye TiO2 cluster MNP system. Section III describes the applied density matrix theory, and in section IV we discuss the results. Some concluding remarks are finally presented in section VI.

II. MODEL OF THE DYE TIO2 CLUSTER MMP SYSTEM We consider a single dye molecule (perylene) attached to the (110) surface of rutile TiO2 with a MNP placed nearby (cf. Figure 1, for system parameters see Table 1). According to the Table 1. Parameters Useda Ee deg T1e a c Em V(Ṽ ) rmnp Epl ℏγpl dpl E0 τp(tp)

2.79 (2.6)eV 3D 0.26 eV 0.459 nm 0.296 nm 3.23 eV −0.72 (−0.19) eV 10 nm 2.6 eV 28.6 meV 2925 D 5 × 105 V/m 10 fs (15 fs)

a

For explanation see text; TiO2 parameters according to ref 22; a and c: rutile lattice parameters.

chosen coordinate system as displayed in Figure 1, Ti atoms are positioned at a distance of 2.96 Å in x direction and of 4.59 Å in y direction (oxygen atoms do not participate in the charge injection and are ignored). Such layers are repeated in z direction at a distance of 4.59 Å. The resulting cuboids have an additional Ti atom at their center. The given distances are those formed inside a bulk TiO2 system and, obviously, there is surface relaxation (see, for example, refs 22−24). Since we will evaluate the MNP injection enhancement via the total excess electron population in the TiO2 lattice these details are of less interest and will be ignored.In contrast to more involved descriptions utilized when small TiO2 clusters are concerned (less than 100 atoms) we use a parametrized model to simulate 2813



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pointing in the direction of the line connecting the molecule and the MNP center of mass. Dipole plasmon states are written as |I⟩, and the MNP ground-state is |0⟩. ℏΩ0 is the plasmon ground-state energy and Epl the excitation energy of a dipole plasmon. While the described part of the Coulomb coupling relates excited states to each other, two further interactions have to be mentioned. There is the Coulomb attraction between the injected electron and the cationic state of the dye. At the same time the injected electron induces an image charge at the MNP. Again an attractive interaction appears. The attractive interaction results in a lowering of the tight-binding energies εm− if the Ti atoms are closer to the surface (when injected, the electron has to climb an energy ramp). A. Singly Excited State Hamiltonian. For a proper formulation of the injection dynamics in the framework of density matrix theory the dye TiO2 cluster MNP system has to be represented by common product states. Those will be labeled by ψα and correspond to the electronic part of the noninteracting system. (Vibrational contribution are of less importance when studying the MNP enhancement effect and will be considered elsewhere.) The overall ground-state is |ψg⟩ = |φg⟩|0⟩Πm|ϕm⟩. To consider the presence of plasmon excitations we introduce |ψI⟩ = |φg⟩|I⟩Πm|ϕm⟩. According to the weak optical excitation, higher excited MNP states are of no interest, and we also neglect simultaneous molecule and MNP excitations. Then, the other excited state of the system belongs to the molecule |ψe⟩ = |φe⟩|0⟩Πm|ϕm⟩. From this state the charge separated states |ψm⟩ = |φ+⟩|0⟩|ϕm−⟩ Πn≠m |ϕn⟩ with the injected electron moving in the TiO2 lattice is formed. The electronic ground-state energy of the molecule together with that of the MNP and the energy of the TiO2 lattice without the injected electron form the reference energy which is set equal to zero. Then, the electronic energies of the excited states ψe and ψI can be written as Ee = εe − εg and as EI = Epl, respectively. Finally, we have the energies of the states formed after charge injection (with the electron at Ti atom m) as Em = Δε+ + Δεm−. Here, Δε+ = ε+ − εg can be understood as the molecular ionization energy, and Δεm− =εm− − εm gives the Ti 3d level electron affinity. The resulting singly excited state Hamiltonian takes the form (Figure 2 offers a graphical scheme): H1(t ) = Ee|Ψ⟩⟨Ψ| e e +

Figure 2. Energy level arrangement displaying the energies which constitute the Hamiltonian, eq 1 (the blue area indicates the TiO2 cluster in z direction, cf. Figure 1). The molecular excitation with energy level Ee is coupled to MNP dipole plasmon excitations Epl via the coupling J. Charge injection to the Ti atom m = 1 is realized by the transfer coupling T. The excess electron energy levels Em=1,2,3 of the TiO2 cluster are placed at identical energetic positions (the highlighted stripe around the Em symbolizes the TiO2 conduction band region).

III. DENSITY MATRIX DESCRIPTION OF CHARGE INJECTION Concerning MNP plasmon excitations it is essential to account for their fast nonradiative decay. Consequently, the description of charge injection kinetics is put into the frame of open system dynamics. It is based on the reduced density operator ρ̂. Since dissipation is dominated by plasmon decay an application of the standard quantum master equation is appropriate. The related density matrix reads ραβ(t) = ⟨ψα|ρ̂(t)|ψβ⟩. For the present purposes it is sufficient to chose a variant of the quantum master equation where dissipation does not couple diagonal and off-diagonal density matrix elements. (Possible small corrections due to a plasmon decay on a 10 fs time scale are of no interest.) Therefore, we take the following equations of motion i ∂ ρ = −iωαβ ̃ ραβ − ∑ (υαγ (t )ργβ − υγβ (t )ραγ ) ∂t αβ ℏ γ − δα , β ∑ (kα → γραα − k γ → αργγ ) γ

The complex transition frequencies are defined as ω̃ αβ = ωαβ − i(1 − δα,β)rαβ where ℏωαβ = Eα − Eβ and the rαβ are the dephasing rates. The coupling matrix ℏvαβ(t) covers all electron transfer matrix elements T1e, Vmn, the molecule−MNP energy transfer coupling JIe, and the coupling to the time-dependent external field via −degE(t) and −dIE(t). The transition rates kα→β determine the dephasing rates rαβ according to the standard formula 1/2 ∑γ(kα→γ + kβ→γ). Here, the rates only cover the contribution due to plasmon decay. Accordingly, we have kI→g = 2γpl, where the dipole plasmon dephasing is γpl (cf. Table 1). The molecule is strongly affected by the fast charge injection (further inclusion of excited state life times and dephasing are without effect). TiO2 lattice vibrations may slightly alter details of the charge transport but also have a minor influence on the overall charge injection efficiency. The described approach is ready to simulate optical excitation of the dye, the plasmon enhancement and the temporal evolution of charge injection.28 The latter will be characterized by the total probability Psem(t) = ∑mρmm(t) to have an electron in the TiO2 cluster. Optical excitation of the dye is characterized by Pe(t) = ρee(t) and of dipole plasmons by Pp1(t) = ∑IρII(t).

∑ Em|Ψm⟩⟨Ψm| + ∑ EI |Ψ⟩⟨Ψ| I I m

+ (T1e|Ψ⟩⟨Ψ| 1 e + H.c.) +

I

∑ Vmn|Ψm⟩⟨Ψn| + Hmol − mnp mn

+ HF(t )

(2)

(1)

The excitation energy exchange coupling between the dye and the MNP reads Hmol−mnp = ∑I JIe|ψI⟩⟨ψe| + H.c.. Optical excitation by an externally applied laser field is considered by HF(t) = −E(t)deg |ψe⟩ ⟨ψg| − E(t) ∑IdI |ψI⟩⟨ψg| + H.c. The time-dependent electric field strength of a pulsed optical excitation takes the form E(t) = nEE(t) exp(−iω0t) + c.c. It includes the unit vector nE of field polarization, the carrier frequency ω0, and the envelope E(t). The latter is assumed to have a Gaussian shape E(t) = E0 exp(2(t − tp)2/τp2) with amplitude E0, pulse center at tp, and duration τp. All necessary parameters are collected in Table 1. 2814



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IV. KINETICS OF CHARGE INJECTION Since a finite TiO2 cluster is considered (rutile form, (110) surface, some thousands of Ti atoms) the electron wave function expanding in the lattice will reach the boundaries after a certain time interval and will undergo reflection. This reflection can be suppressed by introducing absorbing boundary conditions. In such a case the steady state of the system due to continuous optical excitation can be realized. However, it requires separate studies to construct appropriate absorbing boundaries. To avoid this we proceed in a different way and focus on the transient state formed after sufficiently short optical excitation and before substantial wave function reflection at the TiO2 cluster boundaries. It is this state which will be used later to evaluate the MNP induced enhancement. The characteristic time at which the transient state is formed shall be introduced as τss (time of transient ”steady state”). The upper part of Figure 3 displays Pe(t) and Psem(t) for different TiO2 cluster sizes (in the absence of the MNP) and

reached for the largest cluster size indicating that this system is well suited for an estimate of the MNP enhancement. The curves related to smaller clusters show oscillations and even repopulation of the excited molecular level. Nevertheless the value of Psem(t) is rather identical to the one obtained by a larger cluster. The choice of the transfer coupling T1e merge optical excitation and charge injection to a single process. The rise of Psem(t) follows rather directly that one of Pe(t). The effect of the MNP on the injection dynamics is shown in the lower panel of Figure 3 for the same cluster sizes as in the upper part (note ℏω0 = Ee = Epl; deviations from this completely degenerated case are considered in the following section). An increase of Psem by nearly 3 orders of magnitude becomes visible. This means that the probability to have an electron injected due to the action of a laser pulse (the charge injection efficiency) has been considerably enhanced. At the same time the maximum value of the dipole plasmon population Ppl reaches about 0.14 (note the necessary enlargement by the factor 100). This indicates the suitable choice of the external field strength. The rise of Ppl directly follows optical excitation, but the population decays fast due to the short plasmon lifetime (here about 11 fs). In contrast, the maximum of Pe is shifted compared to the case without the MNP. This is due to excitation energy transfer between the dye and the MNP which simultaneously to optical excitation takes place. Since Ppl ≫ Pe during optical excitation net energy transfer appears from the MNP to the dye. The situation is reverse when Ppl ≪ Pe. As a result, the temporal profile of Pe compared to the absence of the MNP is shifted to later times. Any counteraction of fast plasmon decay is absent. The effect of the MNP on charge injection dynamics is demonstrated again in Figure 4, but for a 50 fs laser pulse

Figure 3. Laser pulse induced charge injection dynamics (τp = 10 and tp = 15 fs). Upper panel: absence of the MNP, lower panel: presence of the MNP (center of mass at x = y = 5 nm, z = 10 nm, cf. Figure 1; the dye is placed at the coordinate system origin). Shown are the molecular excited state population Pe(t) (black curves), the total probability Psem(t) of charge injection (red curves), and the total dipole plasmon population Ppl(t) (blue dotted curve of lower panel, multiplied by 0.01) for different TiO2 cluster sizes (nE∥deg, case of resonant excitation: ℏω0 = Ee = Epl = 2.6 eV). Full curves: 14.2 nm × 14.7 nm × 7.3 nm cluster; dashed curves: 11.2 nm × 11.0 nm × 5.5 nm cluster; chain-dotted curves: 8.3 nm × 8.3 nm × 4.1 nm cluster. Upper panel green and blue curves: comparison of the full density matrix treatment (green) and the approximate description (blue, and solid curves: Pe, dashed curves: Psem) for 7.1 nm × 7.3 nm × 3.7 nm cluster (see also.28). Figure 4. Laser pulse induced charge injection dynamics as in Figure 3 but with a 50 fs long laser pulse excitation (τp = 50 and tp = 80 fs, amplitude reduced to E0/3).

for an ultrashort optical excitation of 10 fs. The dye excited state population Pe(t) succeeds rather directly the laser pulse, and charge injection follows. (It is this behavior which has been reproduced to meet earlier computations and which result in the concrete value of the perylene Ti transfer coupling T1e.25) To be consistent with the general restriction to a single excitation in the system (prevention of nonlinear effects when introducing the coupling to the MNP) the electric field strength was chosen rather small resulting in the small value of Psem(t). A constant value of Psem(t) in the 200 fs interval is only

excitation (the pulse amplitude E0 has been properly reduced to achieve the same value of Psem). As in the foregoing study the largest cluster type is used to avoid recurrence phenomena (repopulation of the excited dye state). Now, Psem stays constant up to 400 fs. While the longer pulse increases the time interval of intermediate population of the excited dye state the enhancement effect of the MNP is not affected. To find out if 2815



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enhancement of Psem remains at a value somewhat below 1000. This observation is in contrast to what has been stated in ref 4 and justifies the subsequent considerations which exclusively focus on 10 fs laser pulse excitations and on T1e = 0.26 eV.

this is also the case if the injection time is increased. Figures 5 and 6 display the injection dynamics also for smaller transfer

V. PLASMON ENHANCEMENT FACTOR To valuate the MNP enhancement effect, we take the intermediate steady-state probability Psem(τss) and compute this quantity in the absence of the MNP (P(0)) and in its presence (P(MNP)). Then, an enhancement factor can be introduced as (MNP) (0) Enh = Psem /Psem

(3)

If we strictly work in the low excitation regime (as dictated by the chosen model) the enhancement factor should not depend on the field intensity and should only slightly depend on the pulse duration. Of interest would be, however, the dependence of Enh on the dye position relative to the MNP. Before discussing this we present Enh in dependence on the molecular excitation energy Ee (we allow for small deviations from the value typical for perylene) and in dependence on the exciting photon energy ℏω0. Figure 7 shows respective results.

Figure 5. Laser pulse induced charge injection dynamics as in Figure 3 (τp = 10 and tp = 15 fs) but for different transfer couplings Tle (choice of the largest cluster size; lower panel: presence of the MNP). Black curves: Pe(t), red curves: Psem(t), blue dotted curve of lower panel: Ppl(t) (multiplied by 0.01). Solid curves: Tle = 0.26 meV, dashed curves: Tle = 0.13 eV, chain-dotted curves: Tle = 0.05 eV.

Figure 7. Energy dependence of the enhancement factor, eq 3 (MNP center of mass position at x = y = 7 nm, and z = 10 nm, Epl = 2.6 eV, nE∥ez). Black solid curve: fixed Ee = 2.79 eV but variation of ℏω0 between 2.4 and 2.9 eV, blue dotted curve: as black curve but with Ee = 2.6 eV, green chain-dotted curve: fixed ℏω0= 2.6 eV but variation of Ee between 2.4 and 2.9 eV, red dashed curve: as green curve but with ℏω0= 2.7 eV.

As it has to be expected Enh reaches its maximum in the complete resonant case. However, comparable large deviations of the exciting photon energy ℏω0 from Ee = Epl are still possible to just result in an enhancement factor larger than 100. These values are also ensured by a change of the molecular excitation energy Ee. (The black curve has a resonance below Epl due to the strong decrease of P(0) sem when moving away from Ee.) A. Additional Electrostatic Couplings. If charge injection took place a single electron moves within the TiO2 lattice. It is motion is disturbed by the Coulomb interaction with the molecular cation and by the coupling to its image charge induced at the MNP. The simplest expression for the coupling between the injected electron and the remaining molecular cation is Vm = −e2/Xm where Xm denotes the distance between the cation and the injected electron localized at Ti atom m. This coupling energy alters the Em and has its largest negative value for m = 1 (TiO2 lattice position to which the molecule is attached). Here, the concrete spatial distribution of the positive charge of the molecular cation may contribute. V1 according to the above given expression overestimates the coupling and a changed quantity Ṽ 1 has to be introduced. Since Ṽ 1 has to be calculated separately we proceed differently. Therefore, V1 is

Figure 6. Laser pulse induced charge injection dynamics as in Figure 5 but with a 50 fs long laser pulse excitation (τp = 50 and tp = 80 fs, amplitude reduced to E0/3).

couplings T1e. The smallest value of T1e results in a separation of excited dye state population and charge injection. However, the presence of the MNP increases Psem also in this case by somewhat less than 3 orders of magnitude. Excitation energy transfer from the excited dye to the MNP only slightly decreases the slope of Pe and Psem. In addition, the reverse process extends finite values of Pe to a somewhat larger time region. We may conclude that also in the case where the duration of the optical excitation and charge injection strongly overcomes the time of plasmon decay of about 11 fs, the MNP induced 2816



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removed from the description by introducing the Coulombcorrected energy Ẽ 1 = Δε+ + Δε1− + Ṽ 1 as a parameter. For m > 1 we write: Ẽ m = Ẽ 1 + (Δεm + Δε1−) + (Vm − Ṽ 1). Next, we assume Δεm− − Δεn− = 0, and the Coulomb effect is only accounted for by the difference Vm − Ṽ 1 what is finally estimated by −e2/Xm + e2/X1. We further comment on the treatment of the Coulomb coupling below. Turning to the coupling of the injected electron to its image charge induced at the MNP we use the relation of classical electrodynamics Wm = −e2rmnp3/(2Ym2(Ym2 − rmnp2)).29 rmnp is the MNP radius and Ym the distance between the injected electron at Ti atom m and the MNP center of mass. If the charge is positioned at the MNP surface the energy gets divergent (this indicates the breakdown of the image charge approximation). Assuming a molecular coating of the MNP of 0.5−1 nm there will be no direct Ti atom MNP surface contact (such a coating has been considered in the concrete computations). After charge injection the molecular cation also interacts with the MNP. The respective energy can be also computed with the above given formula and is denoted by Wmol. Accordingly, the total interaction energy with the MNP follows as Wm + Wmol. This energy has to be confronted with the interaction energy before charge injection (the electron to be injected, is just at the molecule). The most simple expression would be 2Wmol. The electron realizes the same image charge energy as the positive charge related to the molecular cation (there is no local charge compensation). We further assume that this energy is valid for the dye ground state as well as it is excited state. Accordingly, the new overall ground-state energy is Ẽ g = Eg + 2Wmol (we temporarily ignore the former assumption Eg = 0). The molecular excitation energy Ẽ 1 is obtained as Ẽ e + Ẽ g. The energy Em turns into the form Ẽ m = Δε+ + Δεm− + Wm + Eg + Wmol or Ẽ m = Δε+ + Δεm− + Wm + Wmol + Ẽ g. Finally, we set Ẽ g = 0 and get a correction of Em by Wm − Wmol. Since both additionally interactions discussed beforehand are attractive the electron moves energetically uphill when it departs from the anchor position of the dye at the surface or from the MNP position. However, the energetic shift of the excess electron energies Em at the Ti atoms is much smaller than the transfer couplings V, and consequently, the effect remains small. Therefore, we do not present numbers for the enhancement factor without and with these additional electrostatic couplings but show their temporal behavior. Figure 8 displays the influence of the coupling of the injected electron with its image charge induced in the MNP, and Figure 9 shows the effect of the electrostatic interaction of the moving electron with the molecular cation. Both types of coupling are of minor influence and can be neglected in the following. B. Change of MNP Position and Exciting Field Polarization Averaging. Next, we study the MNP enhancement in dependence of the MNP position relative to the dye at the surface. In a first step we assume resonant excitation of the dye and external field polarization parallel to deg (in z direction). Figure 10 (red dashed line) indicates the fast decrease of Enh when moving the MNP away from the dye anchor position. To consider charge injection initiated by external fields with random polarization we introduce the polarization averaged enhancement factor (MNP) (0) Enh = Psem / Psem

Figure 8. Enhancement factor, eq 3, versus time with the coupling of the injected electron to its image charge (solid curves) and without this coupling (dashed curves; charge of molecular cation localized at x = y = 0, and z = −0.5 nm; MNP center of mass position at x = y = 7 nm and with variable zmnp, resonant case: ℏω0 = Ee = Epl = 2.6 eV). Blue dashed curve: absence of image charge coupling, zmnp = −10 nm. Red curves: zmnp = −10.5 nm (0.5 nm gap between MNP and semiconductor surface), black curves: zmnp = −11.

Figure 9. Enhancement factor, eq 3, versus time with the Coulomb coupling of the injected electron to the molecular cation (solid curves) and without this coupling (dotted curves; MNP center of mass position at x = y = 7 nm, and z = 10 nm; resonant case: ℏω0 = Ee = Epl = 2.6 eV). Black curves: charge of the molecular cation localized at x = y = 0, and z = −0.5 nm, red curves: z = −1 nm.

Figure 10. Enhancement factor versus MNP position relative to the dye. The MNP center of mass position xmnp = ymnp is varied (zmnp = 10 nm, other parameters according to Table 1). Black solid line: averaged enhancement factor, eq 4, red dashed line: enhancement factor, eq 3, with field polarization nE = ez, blue chain-dotted line: as red line but with nE = (ez + ey)/√2, green dotted line: as red line but with nE = (ez − ey)/√2.

It relates the averaged steady-state probability of charge injection in the presence of the MNP to that in the absence of the MNP. To carry out the averaging analytically we take into consideration that we work in the linear excitation regime where P(MNP) and P(0) sem sem depend quadratically on the field strength. Noting, further, the coupling terms −degE(t) and −dIE(t) we expect the following expansions to be valid: P(MNP) sem = A[nEdeg]2 + ΣIBI[nEdI][ndeg] + ΣICI[nEdI]2 and P(0) sem = A(0)[nEdeg]2 (nE is the unit vector of field polarization). Now, the expressions for the probabilities are ready for averaging. The coefficients A, BI, CI, and A(0) can be determined by a numerical computation of P(MNP) and P(0) sem sem for different nE (see below).

(4) 2817



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coupled to the MNP. The overall probability to have the electron injected into the cluster reaches an intermediate steady state which is used to rate the influence of the MNP. Introducing an enhancement factor as the ratio of this probability in the presence of the MNP and in its absence values larger than 1000 are obtained. MNP enhanced charge injection is dominated by absorption enhancement of the dye. Therefore, the injection efficiency is largest if the MNP dipole plasmon excitation comes into resonance to the molecular excitation. Any change of the energetic position of the injected electron as caused by the Coulomb coupling with its image charge (induced at the MNP) or with the molecular cation is of minor importance. This is mainly due to the “mid-band” injection position of perylene. If this continuum of states is slightly reorganized by additional interaction mechanisms (change of the density of states) there is no effect on the injection efficiency. To judge the influence of the ultrashort MNP plasmon lifetime of about 11 fs the duration of the exciting laser pulse and of the charge injection process has been varied. Also in the case where these two processes are much slower than plasmon decay the resulting probability of charge injection does not change essentially. The chosen weak optical excitation results in an intermediate plasmon population which is nearly 3 orders of magnitude larger than the molecular excited state population. So, an efficient energy transfer from the MNP to the molecule occurs avoiding any charge injection suppression due to a quenching of the molecular excitation. Imagine a decoration of the TiO2 surface with MNPs forming dimers, trimers, etc. strong inter-MNP interaction may appear. Depending on the inter-MNP distance the energetic position of the interaction induced plasmon hybrid states changes. Such a system would offer a higher flexibility to reach the resonance case with the molecular excitation. Related studies are in progress.

The averaging of the incoming light polarization at the surface can be carried out as follows. First, we note that the propagation direction defined by the wave vector k has to be averaged in the half space defined by all positive z components [kez]≥0. This is combined with an averaging of all transversal polarization directions nE⊥k (introducing spherical coordinates the two combined averaging procedures can be easily carried 2 (0) (MNP) 2 out). We get ⟨P(0) sem⟩ = 4deg A /15 and ⟨Psem ⟩ = 4deg A/15 + 4degdplBz/15 + dpl2(Cx + Cy + 4Cz/3)/5. Then, the averaged enhancement factor follows as (a = A/A(0), b = B/A(0), etc., and r = dpl/deg) ⟨Enh⟩ = a + rbz + 3r2(cx + cy)/4 + r2cz. If we finally compute the nonaveraged enhancement factor, eq 3, in using the introduced expansion of the probabilities and for the different polarizations ne = ez, (ez ± ex)/21/2, and (ez ± ey)/21/2 we arrive at Enh

=

3 (Enhz + x + Enhz − x + Enhz + y + Enhz − y) 8

1 Enhz (5) 2 The indices at the enhancement factors indicate the chosen polarization. Calculating these enhancement factors via a solution of the density matrix equations they can be combined to the averaged enhancement factor. Figure 11 displays the −

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Figure 11. Averaged enhancement factor, eq 5, versus MNP center of mass position xmnp and ymnp (zmnp = 10 nm, other parameters according to Table 1).

ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaft through Sfb 951 and the NSFC, Grant No. 1174029 is gratefully acknowledged.

results for various positions of the MNP relative to the dye. This has to be compared with the results of Figure 10 which indicates a doubling of ⟨Enh⟩ compared to Enh. To meet realistic conditions one may also introduce an averaging of the enhancement factor with respect to the dye MNP position and the photon energy of the incoming light. This will be the subject of future studies. The respective figures indicate that Enh remains in a range larger than 100.

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REFERENCES

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VI. CONCLUSIONS Charge injection was described from a perylene dye into a large TiO2 cluster in the presence of a nearby placed MNP. To simulate electron motion in the rutile TiO2 cluster with (110) surface, a tight-binding model for excess electron localization at the Ti atoms and inter-Ti charge transfer has been utilized. Charge injection is initiated by optical excitation of the dye 2818



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(25) In our earlier studies in refs 16 and 17 (see also further references therein), we used continuous band states of bulk TiO2 and found averaged transfer couplings of about 0.1 and 0.06 eV (carboxylic acid anchor group and propionic acid anchor group, respectively). Those values have been translated to the local couplings T1e as given in Table 1. Therefore, we searched for ultrafast laser pulse induced injection dynamics described in the present tight-binding model which shows the same temporal behavior. (26) Odom, T. W.; Schatz, G. C. Introduction to plasmonics. Chem. Rev. 2011, 11, 3667−3668. (27) Halas, N. J. Plasmonics: an emerging field fostered by nano letters. Nano Lett. 2010, 10, 3816−3822. (28) To solve the density matrix eq (2) we first apply the rotating wave approximation (RWA) by separating ρeg and ρIg into a fast oscillating term exp(−iω0t) and a remaining contribution reg and rIg, respectively, which is free of this oscillation (terms ∼exp(±2iω0t) are neglected). Second, we simplify the determination of the density matrix elements ρmn If we temporarily ignore the coupling of the TiO2 cluster to the dye and notice the neglect of lattice vibrations the motion of the extra electron can be described by the time-dependent Schrödinger equation. It determines the expansion coefficients cm(t), and the density matrix ρmn(t) can be identified by cm(t)cn*(t). For large clusters the numerical computation of the cm(t) is extremely advantageously compared to the determination of the related density matrix. We translate the identification in an approximate way to the case of possible charge injection. Now, the expansion coefficient ce(t) related to the dye in its excited state enters the equations of motion for the cm(t). They can be solved if the inhomogeneity ce(t) is approximated by (ρee(t))1/2. Moreover, density matrix elements of the type ρ1g, ρ1e, and ρ1I (and complex conjugated expressions) appear. Those are replaced by c1(t)(ρgg(t))1/2, c1(t)(ρee(t))1/2, and c1(t) (ρII(t))1/2, respectively (if the RWA is considered the oscillating term has to split). A closed set of coupled equations is realized. The resulting charge injection dynamics only slightly deviates from the one obtained from a solution of the complete density matrix equations (see Figure 3). (29) Landau, L. D.; Lifschitz, E. M. Lehrbuch der Theoretischen Physik, Band VIII; Akademie−Verlag: Berlin, 1974; p 13.

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Plasmon Enhanced Heterogeneous Electron Transfer: A Model Study

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