Home
Search
Collections
Journals
About
Contact us
My IOPscience
Theory of plasmon enhanced interfacial electron transfer
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 134209 (http://iopscience.iop.org/0953-8984/27/13/134209) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.248.155.225 This content was downloaded on 07/05/2015 at 09:42
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 134209 (9pp)
doi:10.1088/0953-8984/27/13/134209
Theory of plasmon enhanced interfacial electron transfer Luxia Wang1 and Volkhard May2 1
Department of Physics, University of Science and Technology Beijing, 100083 Beijing, People’s Republic of China 2 Institute of Physics, Humboldt-University at Berlin, Newtonstraße 15, D-12489 Berlin, Germany E-mail: [email protected] and [email protected] Received 26 June 2014, revised 23 January 2015 Accepted for publication 3 February 2015 Published 13 March 2015 Abstract
A particular attempt to improve the efficiency of a dye sensitized solar cell is it’s decoration with metal nano-particles (MNP). The MNP-plasmon induced enhancement of the local field enlarges the photoexcitation of the dyes and a subsequent improvement of the charge separation efficiency may result. In a recent work (2014 J. Phys. Chem. C 118 2812) we presented a theory of plasmon enhanced interfacial electron transfer for perylene attached to a TiO2 surface and placed in the proximity of a spherical MNP. These earlier studies are generalized here to the coupling of to up to four MNPs and to the use of somewhat altered molecular parameters. If the MNPs are placed close to each other strong hybridization of plasmon excitations appears and a broad resonance to which molecular excitations are coupled is formed. To investigate this situation the whole charge injection dynamics is described in the framework of the density matrix theory. The approach accounts for optical excitation of the dye coupled to the MNPs and considers subsequent electron injection into the rutile TiO2 -cluster. Using a tight-binding model for the TiO2 -system with about 105 atoms the electron motion in the cluster is described. We again consider short optical excitation which causes an intermediate steady state with a time-independent overall probability to have the electron injected into the cluster. This probability is used to introduce an enhancement factor which rates the influence of the MNP. Values larger than 500 are obtained. Keywords: interfacial electron transfer, metal nano-particles, plasmon enhancement, tight-binding model, density matrix theory (Some figures may appear in colour only in the online journal)
solar cell (for a recent overview see [10]). In [11] metal nano-structures combined with thin-film solar cells have been suggested. [12] described photosensitization of bulk TiO2 by embedded MNPs. TiO2 -nanostructures decorated with MNPs were introduced in [13, 14]. In [13] a similar structure was investigated as we will consider in the following, i.e. TiO2 clusters to which dye molecules and MNPs have been attached. In order to demonstrate MNP enhanced heterogeneous electron transfer we described in [9] the TiO2 cluster in a tightbinding model. The dye (perylene) has been accounted for by a two electronic level model as in our earlier work (see, for example, [15–17]). Following our recent studies which treat the dye-MNP system as a uniform quantum system with internal Coulomb coupling [18, 21, 22] this interaction appears
1. Introduction
There is an ongoing interest in theoretical studies on interfacial electron transfer in TiO2 based dye solar cells (see [1–3] and recent work concentrating on electronic structure calculations [4, 5]). The temporal evolution of charge injection was investigated in, for example, [6–8], but concentrating on small TiO2 -clusters. (TiO2 )60 clusters were considered in [6] and (TiO2 )90 clusters in [7, 8]. In line with this work we recently published computations which aim at the enhancement effect of metal nano-particles (MNPs) on interfacial electron transfer [9]. These studies were motivated by the numerous attempts described in literatures to achieve MNP induced efficiency enhancement of a dye-sensitized 0953-8984/15/134209+09$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
V
J Epl Epl Epl
E1
Ee
E2
E3
T
x y
0
z
z
Figure 2. Energy level scheme combining excited states of the dye-MNP complex with charge transfer states formed after electron injection into the TiO2 cluster. The blue area covers the molecular excitation with energy level Ee coupled to the MNP dipole plasmon excitations Epl via the coupling J . It also includes inter MNP coupling. The highlighted stripe around Epl indicates the range of level splitting due to plasmon hybrid state formation. The ruby colored area displays the TiO2 -cluster in z-direction (see figure 1). Charge transfer 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 (V is the inter-atom coupling).
Figure 1. Scheme of a TiO2 -cluster (rutile form) with (1 1 0) surface
decorated with spherical Au MNPs of 20 nm diameter and with a single perylene molecule. (The choice of the coordinate system is indicated. Its origin is defined by the anchor position of the dye to the surface. For all parameters see table 1.)
as an excitation energy exchange coupling between the dye and the MNP. Such a uniform description accounts for local field induced molecular absorption enhancement as well as for molecular excitation energy quenching. The dye-MNP coupling is embedded in a density matrix description of the whole photoinduced charge injection process. Again we compute the MNP induced enhancement of charge injection. Therefore, and as already done in [9], we compare the charge injection process in the presence of MNPs and in their absence. Since the TiO2 -system is considered as a finite cluster it would be not appropriate to focus the computations 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 MNPs and in their absence is used as a quantitative measure of the MNP enhancement effect. The concentration in [9] on a single MNP coupled to the dye requires the molecular excitation to be in near resonance to the MNP dipole plasmon. If the TiO2 surface is decorated with a higher density of MNPs a single dye may couple to more MNPs. Although they might be of identical shape and consequently of identical dipole plasmon energy, hybridization due to MNP–MNP coupling results in an energy splitting. A certain energy band of plasmon resonances is formed and a more efficient absorption enhancement and an increase of injection probability has to be expected. Like the dye-MNP coupling, inter MNP interactions are introduced in the density matrix description. So, it also accounts for MNP hybrid state formation and their coupling to the dye. It is the aim of the subsequent studies to prove this conception. Therefore, systems as shown in figure 1 will be investigated. Again, a single dye molecule, here perylene as an example, attached to the (1 1 0) surface of rutile TiO2 is
considered, but now with up to four Au MNPs in the proximity. We do not focus exclusively on perylene but vary the type of molecule by varying its excitation energy. This allows to present a number of plasmon enhanced injection scenarios (other molecular parameters do not need to be varied). The paper is organized as follows. In the subsequent section we introduce the used model of the dye TiO2 -cluster MNP system (details can be also found in the appendix). Section 3 describes the applied density matrix theory and in section 4 we discuss the results. Some concluding remarks are finally presented in section 5. 2. Model for plasmon enhanced interfacial electron transfer
As already indicated, the description used in the following is based on our earlier studies of [9] but is generalized here to the case of up to four MNPs affecting the charge injection process. It combines coupled excitations among the dye and the MNPs with the charge transfer states formed after electron injection into the TiO2 -lattice. The involved states and their particular couplings are displayed in figure 2. The respective excited states are product states of the dye TiO2 -cluster MNP system and will be labeled by ψα (they correspond to the electronic part of the noninteracting system; vibrational contribution are not considered here). The electronic ground-state energy of the molecule together with the ground-state energy of the different MNPs and the energy of the TiO2 -cluster in the absence of an injected electron form the ground-state energy of coupled system. Conveniently, this energy is set equal to zero, and the excited states energies Eα which ignore the various couplings can be 2
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
simply fixed. The excited state of the dye is denoted as Ee = εe − εg .
(1)
εg and εe are the electronic ground and excited state energy, respectively. The energies of the MNP plasmon excitations are EXI = εpl .
(2)
Since we assume that the inter MNP-surface distance is of some nanometer the restriction to dipole plasmon is justified when considering the formation of hybrid states (see also the reasoning below). X labels the particular MNP, I counts the three degenerated dipole plasmons (I = x, y, z indicates excitations in the three directions of a Cartesian coordinate system), and εpl is the dipole plasmon energy. It’s independence on the MNP index X indicates the restriction to identical MNPs. The MNP arrangements we have in mind are shown in figure 3. The two-MNP case is displayed in the upper panel and the four-MNP case in the lower panel. To generate configurations with different inter-MNP distances we uniformly increase the MNP surface to surface distance. It is (2) labeled by rmnp in the two-MNP case. Here, the two spheres are moved away from each other along the diagonal of the x–y-plane. The actual distance is characterized by the x–yplane projection of the line connecting the molecular center of (2) mass with that of the MNP d = rmnp + rmnp /2. Different configurations for the four-MNP case are also obtained by moving away the MNPs along the diagonals of the x–yplane. Note the different definition of the inter-MNP distance (4) rmnp for the four-MNP case (see figure 3). Obviously, closer molecule-MNP distances can be realized in the two-MNP case. We also introduced the distance z = 0.5 nm between the MNP-surface and that of the TiO2 cluster to avoid charge injection from the MNPs (it may be achieved by appropriate coating of either the MNP or the TiO2 surface; the center of mass of the molecule is also placed z = 0.5 nm upon the surface). It should be noted that a large life-time broadening of the plasmon excitations has to be considered what is carried out here in the framework of a density matrix description (see below). Finally, we have to specify the energies of those states formed after charge injection with the excess electron being localized at Ti-atom m. The energies read Em = ε+ + εm− ,
x y
x y Figure 3. Top view of the TiO2 -cluster of figure 1 decorated with
two MNPs (upper panel) and four MNPs (lower panel). The MNPs are regularly placed on a square around the dye molecule (tiny gray sphere). The MNP radius is always rmnp = 10 nm, and the minimal MNP surface to surface distance rmnp should be 2 nm. The xyz-position √ of the MNPs √ center of mass results in the two-MNP case as (±d/ 2, ±d/ 2, −rmnp − z) (d is the molecule MNP center of mass distance projected into the xy-plane and displayed by the narrow arrow). In the four-MNP case the positions are (4) (4) (±(rmnp + rmnp /2), ±(rmnp + rmnp /2), −rmnp − z) (for further (2) details see text). The MNP surface to surface distances rmnp (4) (two-MNP case) and rmnp (four-MNP case) are also shown.
(3)
where we introduced ε+ = ε+ −εg as the molecular ionization energy (ε+ is the electronic ground-state energy of the cationic dye), and εm− = εm− − εm is the Ti-atom 3d level electron affinity. Noting the restriction to the case of weak optical excitation in photovoltaic systems higher excited state energies of the dye MNP complex are of no importance. Next we introduce the wave functions which belong to the different (zero-order) energies explained beforehand. The overall ground-state of the system takes the form |ψg = |ϕg × |X0 × |φm . (4) X
whole state of the TiO2 -cluster. The states φm describe the neutral state of the Ti-atoms (for more details see the appendix). Molecular excitation is accounted for by |ψe = |ϕe × |X0 × |φm , (5) m
X
where ϕe denotes the first excited electronic state of the dye. The excitation of a single MNP plasmon results in the following state |ψXI = |ϕg × |XI |X 0 × |φm . (6)
m
It includes the electronic ground state ϕg of the dye, the groundstates |X0 of the involved MNPs (labeled by X) and the
X =X
3
m
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
As already indicated it is not necessary to include double or higher excitations of the MNP system as well as a simultaneous excitation of the molecule and of one of the MNPs. When considering the state after charge injection, there is also no need to account for a simultaneous MNP excitation. The charge separated states are denoted as |X0 × |φm− |φn . (7) |ψm = |ϕ+ ×
In principle, the energy exchange type of Coulomb coupling has to be completed by the Coulomb coupling of the injected charge with the molecular cation as well as the mirror charge induced at the MNPs. Both couplings have been considered in our earlier work [9] but found to be of minor importance for the charge injection kinetics. While this has been obtained for the absence of any screening of the Coulomb coupling it’s consideration would further diminish the effect. So we can circumvent here any consideration of related screening effects. One may also emphasize that a screening of the molecule MNP energy exchange coupling would be of some importance (like that in the case of Frenkelexciton formation). Since in the present case this interaction takes place at the surface of a dielectric medium (the TiO2 lattice) and not inside, related screening effects should be only of minor importance. Next we consider the charge transfer coupling which is denoted as Vmn |ψm ψn | . (11) Hct = (T1e |ψ1 ψe | + H.c.) +
n=m
X
The state of the cationic dye is ϕ+ and the state φm− denotes the presence of an injected excess electron at Ti-atom m. According to the energies and states introduced to characterize the ground and the singly excited states of the dye MNP TiO2 system the related Hamiltonian takes the form (a ground-state contributions does not appear because the energy has been assumed to equal zero) H (t) = Ee |ψe ψe | + EXI |ψXI ψXI | + Em |ψm ψm | m
X,I
+ Hmol−mnp + Hmnp−mnp + Hct + HF (t) .
(8)
m,n
Besides the energies and states introduced so far, the Hamiltonian covers a number of different couplings. The molecule-MNP coupling Hmol−mnp and the MNP-MNP coupling Hmnp−mnp are responsible for energy exchange between the molecule and a MNP and among different MNPs, respectively. Charge transfer processes are accounted for by Hct , and the time-dependent part HF (t) shall describe (ultrafast) optical excitation of the dye MNP system (photo transitions in the TiO2 lattice are completely out of resonance). The excitation energy exchange coupling between the dye and the MNP reads JXI,e |ψXI ψe | + H.c. . (9) Hmol−mnp =
It characterizes the charge injection process from the dye to that Ti-atom being in the nearest position to the dye localization at the surface (Ti-atom position m = 1; related transfer coupling is T1e ). Charge motion among the Ti-atoms is accounted for by the second term of Hct with transfer coupling Vmn . To consider optical excitation by an externally applied laser field we have to consider electronic excitation of the dye as well as of the MNPs. The coupling reads dXI |ψXI ψg | HF (t) = − E(t)dmol |ψe ψg | − E(t) X,I
+ H.c. .
X,I
(12)
The molecular transition dipole moment is dmol and that one for the various MNPs is dXI = dpl eI (the eI are the unit vectors of a Cartesian coordinate system; I = x, y, z). The timedependent electric field-strength of a pulsed optical excitation takes the form
It is determined by the dipole–dipole coupling matrix element JXI,e relating the transition dipole moments of MNP X to that of the dye (see the appendix). In a similar form we get the MNP–MNP coupling JXI,X I |ψXI ψX I | + H.c. , (10) Hmnp−mnp =
E(t) = nE E(t) exp(−iω0 t) + c.c. .
X,I X ,I
(13)
It includes the unit vector nE of field polarization, the carrier frequency ω0 and the envelope E(t). The latter shall 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.
where JXI,X I is the dipole–dipole coupling among dipole plasmons of different MNPs. The use of the dipole–dipole coupling for the molecule MNP and inter MNP coupling, i.e. the neglect of higher MNP multipole excitations needs some special attention. With respect to the molecule MNP coupling we checked this in earlier work given in [18, 19] which indicates that for a molecule MNP surface distance larger than 2 nm a dipole– dipole coupling works reasonable (hence, only the later on presented full curve of figure 5 with molecule MNP surface distance of 1 nm might be somewhat questionable). In the case of the MNP–MNP coupling a similar distance argument applies. However, if we also take into consideration higher MNP multipoles the lowest part of the excitation spectrum does not change drastically. There is only a further shift of the levels already produced in a dipole approximation (see also the detailed computations of [20]).
3. Charge injection kinetics
The temporal behavior of the injection process is described in the framework of open system dynamics. Consequently, the following density matrix is introduced ˆ ραβ (t) = ψα |ρ(t)|ψ β .
(14)
It is defined by the states explained in the preceding section, and ρˆ is the density operator reduced to the system formed by these states. The dominating effect due to an environment is 4
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
Table 1. Used parameters (for explanation see text; TiO2 parameters according to [23]; a and c: rutile lattice parameters).
dmol T1e a c Em V (V˜ ) rmnp Epl h ¯ γpl dpl E0 τp (tp )
plasmon dephasing is γpl (see table 1). The molecule is strongly affected by the fast charge injection. Any additional inclusion of excited state life-times and dephasing is not necessary. Now, the photoinduced charge injection dynamics can be characterized by the solution of the equations (15) (for details see also [9]). The whole description of charge injection as based on the equations (15) only uses the external field, equation (13). We never introduced the local field to which the external field has been translated due to the MNP polarization. Such an omission of the local field is possible since our approach accounts for the molecule MNP coupling exactly. The molecule does not feel a local field but is directly coupled to the MNP. This is a more general approach than the one based on the local field (the latter is obtained in a certain type of mean-field theory, see [24]). Accordingly, the molecule MNP system is treated as a unified quantum system. For example, the change of an external to a local field and a resulting changed molecular absorption is considered here via an renormalized (increased) molecular oscillator strength. This is a result of the correct quantum dynamical description of the molecule MNP coupling. To characterize the process of charge injection in the following we take the total probability ρmm (t) (18) Psem (t) =
Figure 4. Charge injection dynamics due to laser pulse excitation with pulse duration τp = 10 fs (see also table 1). Upper panel: absence of the MNP, lower panel: presence of the MNP. Consideration of the case of complete resonance h ¯ ω0 = Ee = Epl . Light propagates parallel to the TiO2 surface (k ⊥ ez and nE = ez ). A single MNP is placed√near the √ dye (dmol nE ). The MNP’s center of mass position is (d/ 2, d/ 2, −rmnp − z) = (7.78, 7.78, −10.5) nm (see upper part of figure 3 with the lower left MNP removed). The cluster extension amounts to 14.2 nm × 14.7 nm × 7.3 nm (see text for details). Solid black curves: Pe (t), red dashed curves: Psem (t), blue chain-dotted curve: Ppl (t) (multiplied by 0.0001).
the rather short plasmon life-time. Therefore, it is sufficient to use the following type of a quantum master equation ∂ i (vαγ (t)ργβ − vγβ (t)ραγ ) ραβ = − iω˜ αβ ραβ − ∂t h ¯ γ − δα,β (kα→γ ραα − kγ →α ργ γ ) . (15)
m
to have an electron in the TiO2 -cluster. Optical excitation of the dye is described by
γ
Pe (t) = ρee (t) ,
The complex transition frequencies are defined as ω˜ αβ = ωαβ − i(1 − δα,β )rαβ
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−1 10 fs (15 fs)
and excitations of dipole plasmons by Ppl (t) = ρXI,XI (t) .
(16)
where h ¯ ωαβ = Eα − Eβ and the rαβ are the dephasing rates. The coupling matrix h ¯ vαβ (t) covers all electron transfer matrix elements T1e and Vmn , the molecule-MNP energy transfer couplings JXI,e , the inter-MNP couplings JXI,X I , and the coupling to the time-dependent external field via −dmol E(t) and −dI E(t). The transition rates kα→β determine (pd) the dephasing rates according to the standard formula (rαβ is the pure-dephasing contribution) 1 (pd) (kα→γ + kβ→γ ) + rαβ . (17) rαβ = 2 γ
(19)
(20)
X,I
The introduced populations are calculated for a cluster of rutile TiO2 with a (1 1 0) surface and a x–y–z-extent of 14.2 nm × 14.7 nm × 7.3 nm what corresponds to more than 50 000 Ti atoms. If we choose ultrafast optical excitation (pulse length τp = 10 fs) we avoid wave function reflection of the injected electron at the TiO2 cluster boundaries in a 200–400 fs time interval. Respective results for the injection kinetics are displayed in figure 4. They correspond to a single MNP (see upper panel of figure 3, but with the lower left MNP removed). The dye is placed at the origin of the coordinate system and the
Here, the rates only cover the contribution due to plasmon decay. Accordingly, we have kI →g = 2γpl , where the dipole 5
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
the mirror charge induced at the MNP is of less importance (see also the appendix A). Therefore, computations similar to those resulting in figure 4 are repeated systematically for larger dye MNP systems. 4. Charge injection enhancement effects
The formation of an intermediate steady state offers the possibility to determine the charge injection enhancement Enh, equation (21), for different MNP dye configurations. We used those introduced in figure 3. The consideration of two or four MNPs means that Enh is the result of a complex interplay of energy exchange coupling between the molecule and the MNPs as well as among the MNPs themselves. The quantity Enh for the two-MNP case is shown in figure 5 and that one for the fourMNP case in figure 7. We assumed complete resonance of the photon energy with the molecular excitation, i.e. h ¯ ω 0 = Ee , and varied the molecular excitation energy in a rather wide range (all other molecular parameters stay constant and are due to table 1). The enhancement effect reaches its maximum if h ¯ ω0 comes into resonance with a (bright) MNP hybrid state. Turning first to the two-MNP case (see figure 5) a nearly three order of magnitude enhancement is reached if the (2) closest MNP surface to surface distance rmnp is considered. According to appendix B the bright MNP hybrid state, formed at optical excitation with polarization perpendicular to the surface, is positioned at Epl + Jpl with Jpl about 0.5 eV (Jpl = dpl2 /X 3 , X is the MNP–MNP center of mass distance). So the maximum enhancement in this case is obtained at h ¯ ω0 = Ee = 3.1 eV. If the MNP surface to surface distance is increased the maximum of Enh is decreased and shifted (2) becomes larger than 6 nm the to lower energies. If rmnp enhancement is marginal. The low energy shift of the Enh maxima is due to the (2) decrease of Jpl with increasing rmnp . The changing height of Enh, however, is mainly determined by the moleculeMNP coupling. It’s dominating 1/R 3 -dependence (R is the molecule-MNP center of mass distance) is modulated by the R-dependence of the geometry factors κI , equations (A3) and (2) (A4). κx,y decreases continuously with an increase of rmnp (increase of d). At the same time κz also decreases to arrive (2) = 8.5 nm. This simultaneous reduction at zero for rmnp explains the behavior shown figure 5. By twisting the molecule and, thus, the transition dipole moment somewhat in the x-direction (see figure 1) we leave the degenerated case of dmol perpendicular to the TiO2 surface where the coupling is restricted to I = z-plasmons. Figure 6 shows the appearance of a second structure located at the energy of a lower hybrid plasmon. Its position at about 1.6 eV indicates that it is from the other type (J-aggregate like type, see appendix B). The non-monotonous change of the peak height with further twist of dmol from its perpendicular position to the TiO2 surface is due to the interplay of the coupling to two types of MNP hybrid plasmons. The charge injection enhancement is less pronounced when changing to the four-MNP case as displayed in figure 7. Enh approaches values which are valid in the two-MNP case for (2) (see figure 5). Moreover, larger MNP–MNP distances rmnp
Figure 5. Enhancement of charge injection for the two-MNP case of
figure 3, upper panel. The factor Enh, equation (21), is drawn versus h ¯ ω0 = Ee (for Epl see table 1, other parameters as in figure 4). (2) = 2 nm, Change of mutual MNP position. Black solid line: rmnp (2) red dashed line: rmnp = 4 nm, blue chain-dotted line: (2) = 6 nm. rmnp
√ √ MNP center of mass is located at (d/ 2, d/ 2, −rmnp − z) = (7.78, 7.78, −10.5) nm. To avoid extra charge transfer from the MNP into the TiO2 cluster we assumed a passivation layer at the MNP of 5 Å thickness. If the MNP is absent (upper panel of figure 4) charge injection follows the resonant optical excitation of the dye rather directly (¯hω0 = Ee ). After Pe approaches zero Psem stays constant on a rather long time interval compared with the duration of the excitation process (the population in the absence of MNPs will be written as (0) in the following). In [9] this period has been named Psem the intermediate steady state. It characterizes the state of the system after the charge injection has been completed and before electron wave function reflection at the TiO2 cluster boundaries take place. The extremely small probabilities are due to the weak excitation (see table 1). It has been chosen in this manner to also stay in the weak excitation regime if the coupling to the MNPs is accounted for. Related populations are drawn in the lower panel of (MNP) ). Due to the energy figure 4 (note that we now write Psem exchange between the MNP and the dye the temporal increase of Pe and Psem is somewhat reduced compared to that one in the absence of the MNP (upper panel of figure 4). Worth noting is the change of the order of magnitude determining the populations and of the multiplication of Ppl by 0.0001. We also emphasize that the huge value of Ppl results in a net excitation energy transfer from the MNP to the dye. (0) (MNP) (tss ) and Psem (tss ) at the steady The populations Psem state at time-region ≈ tss are well suited to judge the charge injection enhancement due to the presences of MNPs. Therefore, we introduce (MNP) (0) Enh = Psem (tss )/Psem (tss ) .
(21)
The quantity amounts to about 190 in the case of figure 4. The intermediate steady state which is essential to introduce the charge injection enhancement is also found for other parameters of the electron transfer coupling between the dye and the TiO2 cluster as well as for longer laser pulse durations. We also demonstrated in [9] that Coulomb coupling between the injected electron and the remaining dye cation as well as 6
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
Figure 6. Enhancement of charge injection for the two-MNP case of
Figure 7. Enhancement of charge injection for the four-MNP case of figure 3, lower panel. The factor Enh, equation (21), is drawn versus h ¯ ω0 = Ee (for Epl see table 1, other parameters as in figure 4). Change of mutual MNP position. Black solid line: (4) (4) rmnp = 2 nm, red dashed line: rmnp = 4 nm, blue chain-dotted (4) (4) = 12 nm, line: rmnp = 6 nm, orange chain-dotted line: rmnp (4) green dotted line: rmnp = 16 nm.
figure 3, upper panel. The factor Enh, equation (21), is drawn versus (2) h ¯ ω0 = Ee (rmnp = 2 nm, for Epl see table 1, other parameters as in figure 4). Change of the molecular dipole orientation according to dx = dmol sin θ, dy = 0, dz = dmol cos θ . Black solid line: θ = 0, red dashed line: θ = π/6, green chain dotted line: θ = π/4, blue dotted line: θ = π/3.
the shift of the Enh maximum compared to Epl amounts to (4) (be aware of the different definition 1.2 eV. Increasing rmnp compared with the two-MNP case) the shift moves to lower energies taking the value of 200 to afterwards decrease. To understand this behavior take into consideration that the used (4) result in larger molecule MNP distances as values of rmnp considered in the two-MNP case. This reduces the maxima of Enh compared to figure 5 (the coupling to four instead of two MNP may compensate this reduction somewhat). Moreover, we note the change of the geometry factors κI (4) entering the molecule-MNP coupling with changing rmnp (4) (see appendix A2). As indicated, κz increases for rmnp >0 to finally arrive at 1 (κx,y go to zero). So, the interplay of an increasing κz with the additional 1/R 3 -dependence of the molecule-MNP coupling (R is the molecule MNP center of mass distance) explains the intermediate larger values of Enh (4) . when increasing rmnp As in the two-MNP case, a multi peak (three peak) structure of the enhancement versus excitation energy is obtained if the molecular transition dipole moment originally perpendicular to TiO2 surface is twisted in the x-direction (see figure 8). In this way larger Enh values are obtained and the resonances are moved to lower energies. So a rather dense decoration of the TiO2 surface seems reasonable to obtain MNP hybrid state formation of four or more interacting MNPs. At the same time the position of the dye relative to the MNPs is not so critical. And, a non-perpendicular positioning of the dye to the TiO2 surface promotes the charge injection enhancement effect. It is also of interest that the short MNP plasmon life time has no dramatic effect on the charge injection process. This is due to the fact that the charge injection time unaffected by MNPs (see figure 4, upper panel) is in the range of the MNP plasmon life-time itself. However, the related single MNP level broadening as well as the resulting hybrid level broadening is large enough to induce an injection enhancement with noticeable values about ±100 meV around the maximum of the respective Enh (see figures 5 and 7).
5. Conclusions
Plasmon enhanced interfacial electron transfer has been discussed theoretically for a dye molecule attached to a TiO2 surface decorated with spherical MNPs. In order to mimic different types of dyes we varied the respective electronic excitation energy. If the MNPs are placed close to each other strong hybridization of their plasmon excitations appears and a broad resonance to which the molecular excitations are coupled is formed. The dye MNP-plasmon interaction enlarges the photoexcitation of the dyes and an improvement of the charge separation efficiency results. In the present study we concentrated on a completely regular arrangement of the dye-MNP systems at the TiO2 surface. In any application one has to take into consideration more or less large structural fluctuations. The MNPs are not placed at the surface with identical distances to each other, and the position of the dye to the MNPs as well as its surface orientation may change. All these disorder effects do not favor a particular dye MNP arrangement but result in a certain average. This is the subject of ongoing work. Of separate interest for the case of dense MNP decoration of the TiO2 surface would be also charge injection via socalled spill out electrons moving directly from the MNP into the semiconductor. Acknowledgments
Financial support by the Deutsche Forschungsgemeinschaft through Sfb 951 and the NSFC, grant no. 1174029 is gratefully acknowledged. Appendix A. Model of the dye TiO2 -cluster MNP system
In the following we quote some details necessary for the composition of the model of section 2. We first describe the 7
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
is dI = dpl eI (the common dipole moment is given by dpl and the eI are unit vectors of a Cartesian coordinate system). Then, the coupling matrix reads κI JI e = dpl dmol 3 (A2) R The magnitude of the molecular transition dipole moment is denoted as dmol , and R is the distance between the molecule and the MNP center of mass. The geometry factor κI = [eI nmol ] − 3[eI n][nnmol ]
(A3)
also includes the unit vector nmol of the molecular transition dipole moment and the unit vector n pointing in the direction of the line connecting the molecule and the MNP center of mass. We specify the geometry factors for the coupling of the molecule to a single MNP (upper right one in the two panels of figure 3). √ √ The MNP center of mass position vector is (d/ 2, d/ 2, −rmnp − z) with d as the distance of the MNP center of mass position projected into the x, y-plane to the coordinate system origin (see figure 3). The respective vector for the molecule reads (0, 0, −z) (since we also set here z = 0.5 nm, the same z appears in both vectors). Introducing ξ = d/rmnp we arrive at
Figure 8. Enhancement of charge injection for the four-MNP case of figure 3, lower panel. The factor Enh, equation (21), is drawn versus (4) h ¯ ω0 = Ee (rmnp = 2 nm, for Epl see table 1, other parameters as in figure 4). Change of the molecular dipole orientation according to dx = dmol sin θ, dy = 0, dz = dmol cos θ . Black solid line: θ = 0, red dashed line: θ = π/6, green chain dotted line: θ = π/4, blue dotted line: θ = π/3, orange chain dotted line: θ = π/2.
dye-MNP complex followed by the some remarks on the dyeMNP coupling. The used TiO2 -cluster is explained in the third section.
3 3 ξ κx,y = √ , κz = 1 − . (A4) 2 1 + ξ2 21+ξ The coupling to dipole plasmons polarized in the x- and y-direction is characterized by a geometry factor becoming maximal at d = rmnp . In contrast, the coupling to dipole plasmons polarized in z-direction changes its sign at d = √ 2rmnp . In the two-MNP case we may write d = rmnp + (2) (2) /2 and note that the sign change appears at rmnp about rmnp 8.3 nm for the chosen type of MNP. Turning to the four-MNP √ (4) case where we have d = 2(rmnp +rmnp /2) we meet the sign (4) change of κz at rmnp = 0 (two adjacent MNPs are in surface contact).
A.1. The dye MNP complex
The types of dye molecules we have in mind (perylene as one example) are characterized by an electronic ground-state ϕg , a first excited state ϕe , and a cationic state ϕ+ which is formed after charge injection. The related energies are εg , εe , and ε+ , respectively. The molecular states have to be understood as total electronic wave functions (a completion by vibrational wave function is of less interest here). Charge injection from the excited molecular state into the adjacent Ti-atom is considered by the transfer integral T1e . According to earlier studies [16, 17] the value of T1e as given in table 1 corresponds to perylene attached via a carboxylic acid anchor group. Optical excitation of the dye is governed by its transition dipole moment dmol . To include the presence of the MNPs we follow our approach already used in the foregoing study of [9]. The molecule-MNP complex is treated as a uniform quantum system with internal Coulomb-couplings between the molecule and the MNPs as well as among them. This coupling appears as an energy transfer coupling between the different excitable units. For the computations we focus on spherical Au nanoparticles with a diameter of 20 nm (see also table 1).
A.3. The TiO2 -cluster
The dye is attached to the (1 1 0) surface of rutile TiO2 . The respective ideal lattice (without any surface relaxation) follows if Ti-atoms are positioned at a mutual distance of 2.96 Å in the x-direction and of 4.59 Å in the y-direction according to the chosen coordinate system shown in figure 1. Oxygen atoms do not participate in the charge injection and are ignored. Ti-atom layers which formed in such a way are repeated in z-direction at a distance of 4.59 Å. The resulting cuboids have an additional Ti-atom at their center. In order to simulate charge injection into a TiO2 -lattice of some thousands of atoms a parameterized tight-binding model is used (parameters are from [23] and are listed in table 1). We consider as in [9] clusters up to an x–y–z-extent of 14.2 nm × 14.7 nm × 7.3 nm. This corresponds to more than 50.000 Ti atoms. The injected electron is mainly localized at the 3d levels of the Ti-atoms [23, 25] (as indicated we ignore 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.
A.2. Molecule MNP coupling
The coupling of the dye to a single MNP takes the form (the MNP index X is suppressed) V = JI e |ϕg |I 0|ϕe | + H.c. . (A1) I
The MNP excitations have been restricted to dipole plasmons since the molecule MNP-surface distance is not too small (larger than 2 nm) [18]. The three degenerated dipole plasmons are labeled by I (= x, y, z) and the transition dipole moment 8
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
After electron injection anionic states of the Ti-atoms are formed which will be denoted by φm− . Related energies are 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 . It comprises nearest neighbor Ti atom coupling V and next nearest neighbor coupling V˜ (see [23]).
References [1] Persson P, Lundqvist M J, Ernstorfer R, Goddard W A and Willig F 2006 J. Chem. Theory Comput. 2 441 [2] Prezhdo O V, Duncan W R and Prezhdo V V 2009 Prog. Surf. Sci. 84 30 [3] Martsinovich N and Troisi A 2011 J. Phys. Chem. C 115 11781 [4] Labat F, Le Bahers T, Ciofini I and Adamo C 2012 Acc. Chem. Res. 45 1268 [5] Pastore M, Fantacci S and De Angelis F 2013 J. Phys. Chem. C 117 3685 [6] Li J, Wang H, Persson P and Thoss M 2012 J. Chem. Phys. 137 22A529 [7] Negre C F A, Fuertes V C, Oviedo M B, Oliva F Y and Sanchez C G 2012 J. Phys. Chem. C 116 14748 [8] Oviedo M B, Zarate X, Negre C F A, Schott E, Arratia-Perez R and Sanchez C G 2012 J. Phys. Chem. Lett. 3 2548 [9] Wang L and May V 2014 J. Phys. Chem. C 118 2812 [10] Hartland G V 2012 J. Phys. Chem. Lett. 3 1421 [11] H¨agglund C and Apell S P 2012 J. Phys. Chem. Lett. 3 1275 [12] Mubeen S, Hernandez-Sosa G, Moses D, Lee J and Moskovits M 2011 Nano Lett. 11 5548 [13] Brown M D, Suteewong T, Kumar R S S, D’Innocenzo V, Petrozza A, Lee M M, Wiesner U and Snaith H J 2011 Nano Lett. 11 438 [14] Kochuveedu S T, Kim D-P and Kim D H 2012 J. Phys. Chem. C 116 2500 [15] Tsivlin D V, Willig F and May V 2008 Phys. Rev. B 77 035319 [16] Wang L, Ernstorfer R, Willig F and May V 2005 J. Phys. Chem. B 109 9589 [17] Wang L, May V, Ernstorfer R, Gundlach J and Willig F 2007 Analysis and Control of Ultrafast Photoinduced Reactions (Springer Series in Chemical Physics vol 87) ed O K¨uhn and L W¨oste (Berlin: Springer) p 437 [18] Zelinskyi I, Zhang Y and May V 2012 J. Phys. Chem. A 116 11330 [19] Zhang Y and May V 2014 Phys. Rev. B 89 245441 [20] Nordlander P, Oubre C, Prodan E, Li K and Stockmann M I 2004 Nano Lett. 4 899 [21] Zelinskyi I and May V 2012 Nano Lett. 12 446 [22] Zhang Y, Zelinskyi I and May V 2012 J. Phys. Chem. C 116 25962 [23] Schelling P K, Yu N and Halley J W 1998 Phys. Rev. B 58 1279 [24] Zelinskyy Y, Zhang Y and May V 2013 J. Chem. Phys. 138 114704 [25] Fox H, Newman K E, Schneider W F and Corcelli S A 2010 J. Chem. Theorey Comput. 6 499
Appendix B. Plasmon hybrid states
To get the inter MNP coupling between the MNP X (with dipole excitation I ) and the identical MNP X (with dipole excitation I ) we take an expression similar to equation (A1) and arrive at dipole–dipole coupling of the form (X and X are suppressed in the following) JI I = dpl2
κI I , R3
(B1)
with the geometry factor κI I = [eI eI ] − 3[eI n][neI ] ,
(B2)
and the MNP–MNP center of mass distance R. Using these relations we shortly comment on the possible hybrid state formation of two (identical) MNPs placed along the x-axis. First we note that only certain elements of JI I remain finite: Jxx = −2Jpl , Jyy = Jzz = Jpl with Jpl = dpl2 /R 3 . It amounts to 502 meV if the MNP surface to surface distance rmnp is 2 nm. Obviously, plasmon excitations of different orientation are decoupled, but those of the same orientation may hybridize. For further use we quote the plasmon hybrid levels of the pair of spherical leads: Ex± = Epl ± 2Jpl and Ey± = Ez± = Epl ± Jpl . Respective optical absorption becomes J-aggregate like (dipole moments point in the direction of the connecting line; the lower hybrid level takes all oscillator strength) if the radiation field is polarized in the x direction. Note, that the treatment described in the main part does not require the introduction of hybrid states (they are accounted for directly by the chosen density matrix approach).
9
Search
Collections
Journals
About
Contact us
My IOPscience
Theory of plasmon enhanced interfacial electron transfer
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 134209 (http://iopscience.iop.org/0953-8984/27/13/134209) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.248.155.225 This content was downloaded on 07/05/2015 at 09:42
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 134209 (9pp)
doi:10.1088/0953-8984/27/13/134209
Theory of plasmon enhanced interfacial electron transfer Luxia Wang1 and Volkhard May2 1
Department of Physics, University of Science and Technology Beijing, 100083 Beijing, People’s Republic of China 2 Institute of Physics, Humboldt-University at Berlin, Newtonstraße 15, D-12489 Berlin, Germany E-mail: [email protected] and [email protected] Received 26 June 2014, revised 23 January 2015 Accepted for publication 3 February 2015 Published 13 March 2015 Abstract
A particular attempt to improve the efficiency of a dye sensitized solar cell is it’s decoration with metal nano-particles (MNP). The MNP-plasmon induced enhancement of the local field enlarges the photoexcitation of the dyes and a subsequent improvement of the charge separation efficiency may result. In a recent work (2014 J. Phys. Chem. C 118 2812) we presented a theory of plasmon enhanced interfacial electron transfer for perylene attached to a TiO2 surface and placed in the proximity of a spherical MNP. These earlier studies are generalized here to the coupling of to up to four MNPs and to the use of somewhat altered molecular parameters. If the MNPs are placed close to each other strong hybridization of plasmon excitations appears and a broad resonance to which molecular excitations are coupled is formed. To investigate this situation the whole charge injection dynamics is described in the framework of the density matrix theory. The approach accounts for optical excitation of the dye coupled to the MNPs and considers subsequent electron injection into the rutile TiO2 -cluster. Using a tight-binding model for the TiO2 -system with about 105 atoms the electron motion in the cluster is described. We again consider short optical excitation which causes an intermediate steady state with a time-independent overall probability to have the electron injected into the cluster. This probability is used to introduce an enhancement factor which rates the influence of the MNP. Values larger than 500 are obtained. Keywords: interfacial electron transfer, metal nano-particles, plasmon enhancement, tight-binding model, density matrix theory (Some figures may appear in colour only in the online journal)
solar cell (for a recent overview see [10]). In [11] metal nano-structures combined with thin-film solar cells have been suggested. [12] described photosensitization of bulk TiO2 by embedded MNPs. TiO2 -nanostructures decorated with MNPs were introduced in [13, 14]. In [13] a similar structure was investigated as we will consider in the following, i.e. TiO2 clusters to which dye molecules and MNPs have been attached. In order to demonstrate MNP enhanced heterogeneous electron transfer we described in [9] the TiO2 cluster in a tightbinding model. The dye (perylene) has been accounted for by a two electronic level model as in our earlier work (see, for example, [15–17]). Following our recent studies which treat the dye-MNP system as a uniform quantum system with internal Coulomb coupling [18, 21, 22] this interaction appears
1. Introduction
There is an ongoing interest in theoretical studies on interfacial electron transfer in TiO2 based dye solar cells (see [1–3] and recent work concentrating on electronic structure calculations [4, 5]). The temporal evolution of charge injection was investigated in, for example, [6–8], but concentrating on small TiO2 -clusters. (TiO2 )60 clusters were considered in [6] and (TiO2 )90 clusters in [7, 8]. In line with this work we recently published computations which aim at the enhancement effect of metal nano-particles (MNPs) on interfacial electron transfer [9]. These studies were motivated by the numerous attempts described in literatures to achieve MNP induced efficiency enhancement of a dye-sensitized 0953-8984/15/134209+09$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
V
J Epl Epl Epl
E1
Ee
E2
E3
T
x y
0
z
z
Figure 2. Energy level scheme combining excited states of the dye-MNP complex with charge transfer states formed after electron injection into the TiO2 cluster. The blue area covers the molecular excitation with energy level Ee coupled to the MNP dipole plasmon excitations Epl via the coupling J . It also includes inter MNP coupling. The highlighted stripe around Epl indicates the range of level splitting due to plasmon hybrid state formation. The ruby colored area displays the TiO2 -cluster in z-direction (see figure 1). Charge transfer 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 (V is the inter-atom coupling).
Figure 1. Scheme of a TiO2 -cluster (rutile form) with (1 1 0) surface
decorated with spherical Au MNPs of 20 nm diameter and with a single perylene molecule. (The choice of the coordinate system is indicated. Its origin is defined by the anchor position of the dye to the surface. For all parameters see table 1.)
as an excitation energy exchange coupling between the dye and the MNP. Such a uniform description accounts for local field induced molecular absorption enhancement as well as for molecular excitation energy quenching. The dye-MNP coupling is embedded in a density matrix description of the whole photoinduced charge injection process. Again we compute the MNP induced enhancement of charge injection. Therefore, and as already done in [9], we compare the charge injection process in the presence of MNPs and in their absence. Since the TiO2 -system is considered as a finite cluster it would be not appropriate to focus the computations 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 MNPs and in their absence is used as a quantitative measure of the MNP enhancement effect. The concentration in [9] on a single MNP coupled to the dye requires the molecular excitation to be in near resonance to the MNP dipole plasmon. If the TiO2 surface is decorated with a higher density of MNPs a single dye may couple to more MNPs. Although they might be of identical shape and consequently of identical dipole plasmon energy, hybridization due to MNP–MNP coupling results in an energy splitting. A certain energy band of plasmon resonances is formed and a more efficient absorption enhancement and an increase of injection probability has to be expected. Like the dye-MNP coupling, inter MNP interactions are introduced in the density matrix description. So, it also accounts for MNP hybrid state formation and their coupling to the dye. It is the aim of the subsequent studies to prove this conception. Therefore, systems as shown in figure 1 will be investigated. Again, a single dye molecule, here perylene as an example, attached to the (1 1 0) surface of rutile TiO2 is
considered, but now with up to four Au MNPs in the proximity. We do not focus exclusively on perylene but vary the type of molecule by varying its excitation energy. This allows to present a number of plasmon enhanced injection scenarios (other molecular parameters do not need to be varied). The paper is organized as follows. In the subsequent section we introduce the used model of the dye TiO2 -cluster MNP system (details can be also found in the appendix). Section 3 describes the applied density matrix theory and in section 4 we discuss the results. Some concluding remarks are finally presented in section 5. 2. Model for plasmon enhanced interfacial electron transfer
As already indicated, the description used in the following is based on our earlier studies of [9] but is generalized here to the case of up to four MNPs affecting the charge injection process. It combines coupled excitations among the dye and the MNPs with the charge transfer states formed after electron injection into the TiO2 -lattice. The involved states and their particular couplings are displayed in figure 2. The respective excited states are product states of the dye TiO2 -cluster MNP system and will be labeled by ψα (they correspond to the electronic part of the noninteracting system; vibrational contribution are not considered here). The electronic ground-state energy of the molecule together with the ground-state energy of the different MNPs and the energy of the TiO2 -cluster in the absence of an injected electron form the ground-state energy of coupled system. Conveniently, this energy is set equal to zero, and the excited states energies Eα which ignore the various couplings can be 2
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
simply fixed. The excited state of the dye is denoted as Ee = εe − εg .
(1)
εg and εe are the electronic ground and excited state energy, respectively. The energies of the MNP plasmon excitations are EXI = εpl .
(2)
Since we assume that the inter MNP-surface distance is of some nanometer the restriction to dipole plasmon is justified when considering the formation of hybrid states (see also the reasoning below). X labels the particular MNP, I counts the three degenerated dipole plasmons (I = x, y, z indicates excitations in the three directions of a Cartesian coordinate system), and εpl is the dipole plasmon energy. It’s independence on the MNP index X indicates the restriction to identical MNPs. The MNP arrangements we have in mind are shown in figure 3. The two-MNP case is displayed in the upper panel and the four-MNP case in the lower panel. To generate configurations with different inter-MNP distances we uniformly increase the MNP surface to surface distance. It is (2) labeled by rmnp in the two-MNP case. Here, the two spheres are moved away from each other along the diagonal of the x–y-plane. The actual distance is characterized by the x–yplane projection of the line connecting the molecular center of (2) mass with that of the MNP d = rmnp + rmnp /2. Different configurations for the four-MNP case are also obtained by moving away the MNPs along the diagonals of the x–yplane. Note the different definition of the inter-MNP distance (4) rmnp for the four-MNP case (see figure 3). Obviously, closer molecule-MNP distances can be realized in the two-MNP case. We also introduced the distance z = 0.5 nm between the MNP-surface and that of the TiO2 cluster to avoid charge injection from the MNPs (it may be achieved by appropriate coating of either the MNP or the TiO2 surface; the center of mass of the molecule is also placed z = 0.5 nm upon the surface). It should be noted that a large life-time broadening of the plasmon excitations has to be considered what is carried out here in the framework of a density matrix description (see below). Finally, we have to specify the energies of those states formed after charge injection with the excess electron being localized at Ti-atom m. The energies read Em = ε+ + εm− ,
x y
x y Figure 3. Top view of the TiO2 -cluster of figure 1 decorated with
two MNPs (upper panel) and four MNPs (lower panel). The MNPs are regularly placed on a square around the dye molecule (tiny gray sphere). The MNP radius is always rmnp = 10 nm, and the minimal MNP surface to surface distance rmnp should be 2 nm. The xyz-position √ of the MNPs √ center of mass results in the two-MNP case as (±d/ 2, ±d/ 2, −rmnp − z) (d is the molecule MNP center of mass distance projected into the xy-plane and displayed by the narrow arrow). In the four-MNP case the positions are (4) (4) (±(rmnp + rmnp /2), ±(rmnp + rmnp /2), −rmnp − z) (for further (2) details see text). The MNP surface to surface distances rmnp (4) (two-MNP case) and rmnp (four-MNP case) are also shown.
(3)
where we introduced ε+ = ε+ −εg as the molecular ionization energy (ε+ is the electronic ground-state energy of the cationic dye), and εm− = εm− − εm is the Ti-atom 3d level electron affinity. Noting the restriction to the case of weak optical excitation in photovoltaic systems higher excited state energies of the dye MNP complex are of no importance. Next we introduce the wave functions which belong to the different (zero-order) energies explained beforehand. The overall ground-state of the system takes the form |ψg = |ϕg × |X0 × |φm . (4) X
whole state of the TiO2 -cluster. The states φm describe the neutral state of the Ti-atoms (for more details see the appendix). Molecular excitation is accounted for by |ψe = |ϕe × |X0 × |φm , (5) m
X
where ϕe denotes the first excited electronic state of the dye. The excitation of a single MNP plasmon results in the following state |ψXI = |ϕg × |XI |X 0 × |φm . (6)
m
It includes the electronic ground state ϕg of the dye, the groundstates |X0 of the involved MNPs (labeled by X) and the
X =X
3
m
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
As already indicated it is not necessary to include double or higher excitations of the MNP system as well as a simultaneous excitation of the molecule and of one of the MNPs. When considering the state after charge injection, there is also no need to account for a simultaneous MNP excitation. The charge separated states are denoted as |X0 × |φm− |φn . (7) |ψm = |ϕ+ ×
In principle, the energy exchange type of Coulomb coupling has to be completed by the Coulomb coupling of the injected charge with the molecular cation as well as the mirror charge induced at the MNPs. Both couplings have been considered in our earlier work [9] but found to be of minor importance for the charge injection kinetics. While this has been obtained for the absence of any screening of the Coulomb coupling it’s consideration would further diminish the effect. So we can circumvent here any consideration of related screening effects. One may also emphasize that a screening of the molecule MNP energy exchange coupling would be of some importance (like that in the case of Frenkelexciton formation). Since in the present case this interaction takes place at the surface of a dielectric medium (the TiO2 lattice) and not inside, related screening effects should be only of minor importance. Next we consider the charge transfer coupling which is denoted as Vmn |ψm ψn | . (11) Hct = (T1e |ψ1 ψe | + H.c.) +
n=m
X
The state of the cationic dye is ϕ+ and the state φm− denotes the presence of an injected excess electron at Ti-atom m. According to the energies and states introduced to characterize the ground and the singly excited states of the dye MNP TiO2 system the related Hamiltonian takes the form (a ground-state contributions does not appear because the energy has been assumed to equal zero) H (t) = Ee |ψe ψe | + EXI |ψXI ψXI | + Em |ψm ψm | m
X,I
+ Hmol−mnp + Hmnp−mnp + Hct + HF (t) .
(8)
m,n
Besides the energies and states introduced so far, the Hamiltonian covers a number of different couplings. The molecule-MNP coupling Hmol−mnp and the MNP-MNP coupling Hmnp−mnp are responsible for energy exchange between the molecule and a MNP and among different MNPs, respectively. Charge transfer processes are accounted for by Hct , and the time-dependent part HF (t) shall describe (ultrafast) optical excitation of the dye MNP system (photo transitions in the TiO2 lattice are completely out of resonance). The excitation energy exchange coupling between the dye and the MNP reads JXI,e |ψXI ψe | + H.c. . (9) Hmol−mnp =
It characterizes the charge injection process from the dye to that Ti-atom being in the nearest position to the dye localization at the surface (Ti-atom position m = 1; related transfer coupling is T1e ). Charge motion among the Ti-atoms is accounted for by the second term of Hct with transfer coupling Vmn . To consider optical excitation by an externally applied laser field we have to consider electronic excitation of the dye as well as of the MNPs. The coupling reads dXI |ψXI ψg | HF (t) = − E(t)dmol |ψe ψg | − E(t) X,I
+ H.c. .
X,I
(12)
The molecular transition dipole moment is dmol and that one for the various MNPs is dXI = dpl eI (the eI are the unit vectors of a Cartesian coordinate system; I = x, y, z). The timedependent electric field-strength of a pulsed optical excitation takes the form
It is determined by the dipole–dipole coupling matrix element JXI,e relating the transition dipole moments of MNP X to that of the dye (see the appendix). In a similar form we get the MNP–MNP coupling JXI,X I |ψXI ψX I | + H.c. , (10) Hmnp−mnp =
E(t) = nE E(t) exp(−iω0 t) + c.c. .
X,I X ,I
(13)
It includes the unit vector nE of field polarization, the carrier frequency ω0 and the envelope E(t). The latter shall 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.
where JXI,X I is the dipole–dipole coupling among dipole plasmons of different MNPs. The use of the dipole–dipole coupling for the molecule MNP and inter MNP coupling, i.e. the neglect of higher MNP multipole excitations needs some special attention. With respect to the molecule MNP coupling we checked this in earlier work given in [18, 19] which indicates that for a molecule MNP surface distance larger than 2 nm a dipole– dipole coupling works reasonable (hence, only the later on presented full curve of figure 5 with molecule MNP surface distance of 1 nm might be somewhat questionable). In the case of the MNP–MNP coupling a similar distance argument applies. However, if we also take into consideration higher MNP multipoles the lowest part of the excitation spectrum does not change drastically. There is only a further shift of the levels already produced in a dipole approximation (see also the detailed computations of [20]).
3. Charge injection kinetics
The temporal behavior of the injection process is described in the framework of open system dynamics. Consequently, the following density matrix is introduced ˆ ραβ (t) = ψα |ρ(t)|ψ β .
(14)
It is defined by the states explained in the preceding section, and ρˆ is the density operator reduced to the system formed by these states. The dominating effect due to an environment is 4
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
Table 1. Used parameters (for explanation see text; TiO2 parameters according to [23]; a and c: rutile lattice parameters).
dmol T1e a c Em V (V˜ ) rmnp Epl h ¯ γpl dpl E0 τp (tp )
plasmon dephasing is γpl (see table 1). The molecule is strongly affected by the fast charge injection. Any additional inclusion of excited state life-times and dephasing is not necessary. Now, the photoinduced charge injection dynamics can be characterized by the solution of the equations (15) (for details see also [9]). The whole description of charge injection as based on the equations (15) only uses the external field, equation (13). We never introduced the local field to which the external field has been translated due to the MNP polarization. Such an omission of the local field is possible since our approach accounts for the molecule MNP coupling exactly. The molecule does not feel a local field but is directly coupled to the MNP. This is a more general approach than the one based on the local field (the latter is obtained in a certain type of mean-field theory, see [24]). Accordingly, the molecule MNP system is treated as a unified quantum system. For example, the change of an external to a local field and a resulting changed molecular absorption is considered here via an renormalized (increased) molecular oscillator strength. This is a result of the correct quantum dynamical description of the molecule MNP coupling. To characterize the process of charge injection in the following we take the total probability ρmm (t) (18) Psem (t) =
Figure 4. Charge injection dynamics due to laser pulse excitation with pulse duration τp = 10 fs (see also table 1). Upper panel: absence of the MNP, lower panel: presence of the MNP. Consideration of the case of complete resonance h ¯ ω0 = Ee = Epl . Light propagates parallel to the TiO2 surface (k ⊥ ez and nE = ez ). A single MNP is placed√near the √ dye (dmol nE ). The MNP’s center of mass position is (d/ 2, d/ 2, −rmnp − z) = (7.78, 7.78, −10.5) nm (see upper part of figure 3 with the lower left MNP removed). The cluster extension amounts to 14.2 nm × 14.7 nm × 7.3 nm (see text for details). Solid black curves: Pe (t), red dashed curves: Psem (t), blue chain-dotted curve: Ppl (t) (multiplied by 0.0001).
the rather short plasmon life-time. Therefore, it is sufficient to use the following type of a quantum master equation ∂ i (vαγ (t)ργβ − vγβ (t)ραγ ) ραβ = − iω˜ αβ ραβ − ∂t h ¯ γ − δα,β (kα→γ ραα − kγ →α ργ γ ) . (15)
m
to have an electron in the TiO2 -cluster. Optical excitation of the dye is described by
γ
Pe (t) = ρee (t) ,
The complex transition frequencies are defined as ω˜ αβ = ωαβ − i(1 − δα,β )rαβ
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−1 10 fs (15 fs)
and excitations of dipole plasmons by Ppl (t) = ρXI,XI (t) .
(16)
where h ¯ ωαβ = Eα − Eβ and the rαβ are the dephasing rates. The coupling matrix h ¯ vαβ (t) covers all electron transfer matrix elements T1e and Vmn , the molecule-MNP energy transfer couplings JXI,e , the inter-MNP couplings JXI,X I , and the coupling to the time-dependent external field via −dmol E(t) and −dI E(t). The transition rates kα→β determine (pd) the dephasing rates according to the standard formula (rαβ is the pure-dephasing contribution) 1 (pd) (kα→γ + kβ→γ ) + rαβ . (17) rαβ = 2 γ
(19)
(20)
X,I
The introduced populations are calculated for a cluster of rutile TiO2 with a (1 1 0) surface and a x–y–z-extent of 14.2 nm × 14.7 nm × 7.3 nm what corresponds to more than 50 000 Ti atoms. If we choose ultrafast optical excitation (pulse length τp = 10 fs) we avoid wave function reflection of the injected electron at the TiO2 cluster boundaries in a 200–400 fs time interval. Respective results for the injection kinetics are displayed in figure 4. They correspond to a single MNP (see upper panel of figure 3, but with the lower left MNP removed). The dye is placed at the origin of the coordinate system and the
Here, the rates only cover the contribution due to plasmon decay. Accordingly, we have kI →g = 2γpl , where the dipole 5
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
the mirror charge induced at the MNP is of less importance (see also the appendix A). Therefore, computations similar to those resulting in figure 4 are repeated systematically for larger dye MNP systems. 4. Charge injection enhancement effects
The formation of an intermediate steady state offers the possibility to determine the charge injection enhancement Enh, equation (21), for different MNP dye configurations. We used those introduced in figure 3. The consideration of two or four MNPs means that Enh is the result of a complex interplay of energy exchange coupling between the molecule and the MNPs as well as among the MNPs themselves. The quantity Enh for the two-MNP case is shown in figure 5 and that one for the fourMNP case in figure 7. We assumed complete resonance of the photon energy with the molecular excitation, i.e. h ¯ ω 0 = Ee , and varied the molecular excitation energy in a rather wide range (all other molecular parameters stay constant and are due to table 1). The enhancement effect reaches its maximum if h ¯ ω0 comes into resonance with a (bright) MNP hybrid state. Turning first to the two-MNP case (see figure 5) a nearly three order of magnitude enhancement is reached if the (2) closest MNP surface to surface distance rmnp is considered. According to appendix B the bright MNP hybrid state, formed at optical excitation with polarization perpendicular to the surface, is positioned at Epl + Jpl with Jpl about 0.5 eV (Jpl = dpl2 /X 3 , X is the MNP–MNP center of mass distance). So the maximum enhancement in this case is obtained at h ¯ ω0 = Ee = 3.1 eV. If the MNP surface to surface distance is increased the maximum of Enh is decreased and shifted (2) becomes larger than 6 nm the to lower energies. If rmnp enhancement is marginal. The low energy shift of the Enh maxima is due to the (2) decrease of Jpl with increasing rmnp . The changing height of Enh, however, is mainly determined by the moleculeMNP coupling. It’s dominating 1/R 3 -dependence (R is the molecule-MNP center of mass distance) is modulated by the R-dependence of the geometry factors κI , equations (A3) and (2) (A4). κx,y decreases continuously with an increase of rmnp (increase of d). At the same time κz also decreases to arrive (2) = 8.5 nm. This simultaneous reduction at zero for rmnp explains the behavior shown figure 5. By twisting the molecule and, thus, the transition dipole moment somewhat in the x-direction (see figure 1) we leave the degenerated case of dmol perpendicular to the TiO2 surface where the coupling is restricted to I = z-plasmons. Figure 6 shows the appearance of a second structure located at the energy of a lower hybrid plasmon. Its position at about 1.6 eV indicates that it is from the other type (J-aggregate like type, see appendix B). The non-monotonous change of the peak height with further twist of dmol from its perpendicular position to the TiO2 surface is due to the interplay of the coupling to two types of MNP hybrid plasmons. The charge injection enhancement is less pronounced when changing to the four-MNP case as displayed in figure 7. Enh approaches values which are valid in the two-MNP case for (2) (see figure 5). Moreover, larger MNP–MNP distances rmnp
Figure 5. Enhancement of charge injection for the two-MNP case of
figure 3, upper panel. The factor Enh, equation (21), is drawn versus h ¯ ω0 = Ee (for Epl see table 1, other parameters as in figure 4). (2) = 2 nm, Change of mutual MNP position. Black solid line: rmnp (2) red dashed line: rmnp = 4 nm, blue chain-dotted line: (2) = 6 nm. rmnp
√ √ MNP center of mass is located at (d/ 2, d/ 2, −rmnp − z) = (7.78, 7.78, −10.5) nm. To avoid extra charge transfer from the MNP into the TiO2 cluster we assumed a passivation layer at the MNP of 5 Å thickness. If the MNP is absent (upper panel of figure 4) charge injection follows the resonant optical excitation of the dye rather directly (¯hω0 = Ee ). After Pe approaches zero Psem stays constant on a rather long time interval compared with the duration of the excitation process (the population in the absence of MNPs will be written as (0) in the following). In [9] this period has been named Psem the intermediate steady state. It characterizes the state of the system after the charge injection has been completed and before electron wave function reflection at the TiO2 cluster boundaries take place. The extremely small probabilities are due to the weak excitation (see table 1). It has been chosen in this manner to also stay in the weak excitation regime if the coupling to the MNPs is accounted for. Related populations are drawn in the lower panel of (MNP) ). Due to the energy figure 4 (note that we now write Psem exchange between the MNP and the dye the temporal increase of Pe and Psem is somewhat reduced compared to that one in the absence of the MNP (upper panel of figure 4). Worth noting is the change of the order of magnitude determining the populations and of the multiplication of Ppl by 0.0001. We also emphasize that the huge value of Ppl results in a net excitation energy transfer from the MNP to the dye. (0) (MNP) (tss ) and Psem (tss ) at the steady The populations Psem state at time-region ≈ tss are well suited to judge the charge injection enhancement due to the presences of MNPs. Therefore, we introduce (MNP) (0) Enh = Psem (tss )/Psem (tss ) .
(21)
The quantity amounts to about 190 in the case of figure 4. The intermediate steady state which is essential to introduce the charge injection enhancement is also found for other parameters of the electron transfer coupling between the dye and the TiO2 cluster as well as for longer laser pulse durations. We also demonstrated in [9] that Coulomb coupling between the injected electron and the remaining dye cation as well as 6
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
Figure 6. Enhancement of charge injection for the two-MNP case of
Figure 7. Enhancement of charge injection for the four-MNP case of figure 3, lower panel. The factor Enh, equation (21), is drawn versus h ¯ ω0 = Ee (for Epl see table 1, other parameters as in figure 4). Change of mutual MNP position. Black solid line: (4) (4) rmnp = 2 nm, red dashed line: rmnp = 4 nm, blue chain-dotted (4) (4) = 12 nm, line: rmnp = 6 nm, orange chain-dotted line: rmnp (4) green dotted line: rmnp = 16 nm.
figure 3, upper panel. The factor Enh, equation (21), is drawn versus (2) h ¯ ω0 = Ee (rmnp = 2 nm, for Epl see table 1, other parameters as in figure 4). Change of the molecular dipole orientation according to dx = dmol sin θ, dy = 0, dz = dmol cos θ . Black solid line: θ = 0, red dashed line: θ = π/6, green chain dotted line: θ = π/4, blue dotted line: θ = π/3.
the shift of the Enh maximum compared to Epl amounts to (4) (be aware of the different definition 1.2 eV. Increasing rmnp compared with the two-MNP case) the shift moves to lower energies taking the value of 200 to afterwards decrease. To understand this behavior take into consideration that the used (4) result in larger molecule MNP distances as values of rmnp considered in the two-MNP case. This reduces the maxima of Enh compared to figure 5 (the coupling to four instead of two MNP may compensate this reduction somewhat). Moreover, we note the change of the geometry factors κI (4) entering the molecule-MNP coupling with changing rmnp (4) (see appendix A2). As indicated, κz increases for rmnp >0 to finally arrive at 1 (κx,y go to zero). So, the interplay of an increasing κz with the additional 1/R 3 -dependence of the molecule-MNP coupling (R is the molecule MNP center of mass distance) explains the intermediate larger values of Enh (4) . when increasing rmnp As in the two-MNP case, a multi peak (three peak) structure of the enhancement versus excitation energy is obtained if the molecular transition dipole moment originally perpendicular to TiO2 surface is twisted in the x-direction (see figure 8). In this way larger Enh values are obtained and the resonances are moved to lower energies. So a rather dense decoration of the TiO2 surface seems reasonable to obtain MNP hybrid state formation of four or more interacting MNPs. At the same time the position of the dye relative to the MNPs is not so critical. And, a non-perpendicular positioning of the dye to the TiO2 surface promotes the charge injection enhancement effect. It is also of interest that the short MNP plasmon life time has no dramatic effect on the charge injection process. This is due to the fact that the charge injection time unaffected by MNPs (see figure 4, upper panel) is in the range of the MNP plasmon life-time itself. However, the related single MNP level broadening as well as the resulting hybrid level broadening is large enough to induce an injection enhancement with noticeable values about ±100 meV around the maximum of the respective Enh (see figures 5 and 7).
5. Conclusions
Plasmon enhanced interfacial electron transfer has been discussed theoretically for a dye molecule attached to a TiO2 surface decorated with spherical MNPs. In order to mimic different types of dyes we varied the respective electronic excitation energy. If the MNPs are placed close to each other strong hybridization of their plasmon excitations appears and a broad resonance to which the molecular excitations are coupled is formed. The dye MNP-plasmon interaction enlarges the photoexcitation of the dyes and an improvement of the charge separation efficiency results. In the present study we concentrated on a completely regular arrangement of the dye-MNP systems at the TiO2 surface. In any application one has to take into consideration more or less large structural fluctuations. The MNPs are not placed at the surface with identical distances to each other, and the position of the dye to the MNPs as well as its surface orientation may change. All these disorder effects do not favor a particular dye MNP arrangement but result in a certain average. This is the subject of ongoing work. Of separate interest for the case of dense MNP decoration of the TiO2 surface would be also charge injection via socalled spill out electrons moving directly from the MNP into the semiconductor. Acknowledgments
Financial support by the Deutsche Forschungsgemeinschaft through Sfb 951 and the NSFC, grant no. 1174029 is gratefully acknowledged. Appendix A. Model of the dye TiO2 -cluster MNP system
In the following we quote some details necessary for the composition of the model of section 2. We first describe the 7
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
is dI = dpl eI (the common dipole moment is given by dpl and the eI are unit vectors of a Cartesian coordinate system). Then, the coupling matrix reads κI JI e = dpl dmol 3 (A2) R The magnitude of the molecular transition dipole moment is denoted as dmol , and R is the distance between the molecule and the MNP center of mass. The geometry factor κI = [eI nmol ] − 3[eI n][nnmol ]
(A3)
also includes the unit vector nmol of the molecular transition dipole moment and the unit vector n pointing in the direction of the line connecting the molecule and the MNP center of mass. We specify the geometry factors for the coupling of the molecule to a single MNP (upper right one in the two panels of figure 3). √ √ The MNP center of mass position vector is (d/ 2, d/ 2, −rmnp − z) with d as the distance of the MNP center of mass position projected into the x, y-plane to the coordinate system origin (see figure 3). The respective vector for the molecule reads (0, 0, −z) (since we also set here z = 0.5 nm, the same z appears in both vectors). Introducing ξ = d/rmnp we arrive at
Figure 8. Enhancement of charge injection for the four-MNP case of figure 3, lower panel. The factor Enh, equation (21), is drawn versus (4) h ¯ ω0 = Ee (rmnp = 2 nm, for Epl see table 1, other parameters as in figure 4). Change of the molecular dipole orientation according to dx = dmol sin θ, dy = 0, dz = dmol cos θ . Black solid line: θ = 0, red dashed line: θ = π/6, green chain dotted line: θ = π/4, blue dotted line: θ = π/3, orange chain dotted line: θ = π/2.
dye-MNP complex followed by the some remarks on the dyeMNP coupling. The used TiO2 -cluster is explained in the third section.
3 3 ξ κx,y = √ , κz = 1 − . (A4) 2 1 + ξ2 21+ξ The coupling to dipole plasmons polarized in the x- and y-direction is characterized by a geometry factor becoming maximal at d = rmnp . In contrast, the coupling to dipole plasmons polarized in z-direction changes its sign at d = √ 2rmnp . In the two-MNP case we may write d = rmnp + (2) (2) /2 and note that the sign change appears at rmnp about rmnp 8.3 nm for the chosen type of MNP. Turning to the four-MNP √ (4) case where we have d = 2(rmnp +rmnp /2) we meet the sign (4) change of κz at rmnp = 0 (two adjacent MNPs are in surface contact).
A.1. The dye MNP complex
The types of dye molecules we have in mind (perylene as one example) are characterized by an electronic ground-state ϕg , a first excited state ϕe , and a cationic state ϕ+ which is formed after charge injection. The related energies are εg , εe , and ε+ , respectively. The molecular states have to be understood as total electronic wave functions (a completion by vibrational wave function is of less interest here). Charge injection from the excited molecular state into the adjacent Ti-atom is considered by the transfer integral T1e . According to earlier studies [16, 17] the value of T1e as given in table 1 corresponds to perylene attached via a carboxylic acid anchor group. Optical excitation of the dye is governed by its transition dipole moment dmol . To include the presence of the MNPs we follow our approach already used in the foregoing study of [9]. The molecule-MNP complex is treated as a uniform quantum system with internal Coulomb-couplings between the molecule and the MNPs as well as among them. This coupling appears as an energy transfer coupling between the different excitable units. For the computations we focus on spherical Au nanoparticles with a diameter of 20 nm (see also table 1).
A.3. The TiO2 -cluster
The dye is attached to the (1 1 0) surface of rutile TiO2 . The respective ideal lattice (without any surface relaxation) follows if Ti-atoms are positioned at a mutual distance of 2.96 Å in the x-direction and of 4.59 Å in the y-direction according to the chosen coordinate system shown in figure 1. Oxygen atoms do not participate in the charge injection and are ignored. Ti-atom layers which formed in such a way are repeated in z-direction at a distance of 4.59 Å. The resulting cuboids have an additional Ti-atom at their center. In order to simulate charge injection into a TiO2 -lattice of some thousands of atoms a parameterized tight-binding model is used (parameters are from [23] and are listed in table 1). We consider as in [9] clusters up to an x–y–z-extent of 14.2 nm × 14.7 nm × 7.3 nm. This corresponds to more than 50.000 Ti atoms. The injected electron is mainly localized at the 3d levels of the Ti-atoms [23, 25] (as indicated we ignore 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.
A.2. Molecule MNP coupling
The coupling of the dye to a single MNP takes the form (the MNP index X is suppressed) V = JI e |ϕg |I 0|ϕe | + H.c. . (A1) I
The MNP excitations have been restricted to dipole plasmons since the molecule MNP-surface distance is not too small (larger than 2 nm) [18]. The three degenerated dipole plasmons are labeled by I (= x, y, z) and the transition dipole moment 8
J. Phys.: Condens. Matter 27 (2015) 134209
L Wang and V May
After electron injection anionic states of the Ti-atoms are formed which will be denoted by φm− . Related energies are 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 . It comprises nearest neighbor Ti atom coupling V and next nearest neighbor coupling V˜ (see [23]).
References [1] Persson P, Lundqvist M J, Ernstorfer R, Goddard W A and Willig F 2006 J. Chem. Theory Comput. 2 441 [2] Prezhdo O V, Duncan W R and Prezhdo V V 2009 Prog. Surf. Sci. 84 30 [3] Martsinovich N and Troisi A 2011 J. Phys. Chem. C 115 11781 [4] Labat F, Le Bahers T, Ciofini I and Adamo C 2012 Acc. Chem. Res. 45 1268 [5] Pastore M, Fantacci S and De Angelis F 2013 J. Phys. Chem. C 117 3685 [6] Li J, Wang H, Persson P and Thoss M 2012 J. Chem. Phys. 137 22A529 [7] Negre C F A, Fuertes V C, Oviedo M B, Oliva F Y and Sanchez C G 2012 J. Phys. Chem. C 116 14748 [8] Oviedo M B, Zarate X, Negre C F A, Schott E, Arratia-Perez R and Sanchez C G 2012 J. Phys. Chem. Lett. 3 2548 [9] Wang L and May V 2014 J. Phys. Chem. C 118 2812 [10] Hartland G V 2012 J. Phys. Chem. Lett. 3 1421 [11] H¨agglund C and Apell S P 2012 J. Phys. Chem. Lett. 3 1275 [12] Mubeen S, Hernandez-Sosa G, Moses D, Lee J and Moskovits M 2011 Nano Lett. 11 5548 [13] Brown M D, Suteewong T, Kumar R S S, D’Innocenzo V, Petrozza A, Lee M M, Wiesner U and Snaith H J 2011 Nano Lett. 11 438 [14] Kochuveedu S T, Kim D-P and Kim D H 2012 J. Phys. Chem. C 116 2500 [15] Tsivlin D V, Willig F and May V 2008 Phys. Rev. B 77 035319 [16] Wang L, Ernstorfer R, Willig F and May V 2005 J. Phys. Chem. B 109 9589 [17] Wang L, May V, Ernstorfer R, Gundlach J and Willig F 2007 Analysis and Control of Ultrafast Photoinduced Reactions (Springer Series in Chemical Physics vol 87) ed O K¨uhn and L W¨oste (Berlin: Springer) p 437 [18] Zelinskyi I, Zhang Y and May V 2012 J. Phys. Chem. A 116 11330 [19] Zhang Y and May V 2014 Phys. Rev. B 89 245441 [20] Nordlander P, Oubre C, Prodan E, Li K and Stockmann M I 2004 Nano Lett. 4 899 [21] Zelinskyi I and May V 2012 Nano Lett. 12 446 [22] Zhang Y, Zelinskyi I and May V 2012 J. Phys. Chem. C 116 25962 [23] Schelling P K, Yu N and Halley J W 1998 Phys. Rev. B 58 1279 [24] Zelinskyy Y, Zhang Y and May V 2013 J. Chem. Phys. 138 114704 [25] Fox H, Newman K E, Schneider W F and Corcelli S A 2010 J. Chem. Theorey Comput. 6 499
Appendix B. Plasmon hybrid states
To get the inter MNP coupling between the MNP X (with dipole excitation I ) and the identical MNP X (with dipole excitation I ) we take an expression similar to equation (A1) and arrive at dipole–dipole coupling of the form (X and X are suppressed in the following) JI I = dpl2
κI I , R3
(B1)
with the geometry factor κI I = [eI eI ] − 3[eI n][neI ] ,
(B2)
and the MNP–MNP center of mass distance R. Using these relations we shortly comment on the possible hybrid state formation of two (identical) MNPs placed along the x-axis. First we note that only certain elements of JI I remain finite: Jxx = −2Jpl , Jyy = Jzz = Jpl with Jpl = dpl2 /R 3 . It amounts to 502 meV if the MNP surface to surface distance rmnp is 2 nm. Obviously, plasmon excitations of different orientation are decoupled, but those of the same orientation may hybridize. For further use we quote the plasmon hybrid levels of the pair of spherical leads: Ex± = Epl ± 2Jpl and Ey± = Ez± = Epl ± Jpl . Respective optical absorption becomes J-aggregate like (dipole moments point in the direction of the connecting line; the lower hybrid level takes all oscillator strength) if the radiation field is polarized in the x direction. Note, that the treatment described in the main part does not require the introduction of hybrid states (they are accounted for directly by the chosen density matrix approach).
9
