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Control of Intermolecular Electronic Excitation Energy Transfer: Application of Metal Nano-Particle Plasmons Luxia Wang, and Volkhard May J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 02 Jun 2017 Downloaded from http://pubs.acs.org on June 4, 2017

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The Journal of Physical Chemistry

Control of Intermolecular Electronic Excitation Energy Transfer: Application of Metal Nano-Particle Plasmons Luxia Wang∗ Department of Physics, University of Science and Technology Beijing, 100083 Beijing, China Volkhard May† Institute of Physics, Humboldt-University at Berlin, Newtonstraße 15, D-12489 Berlin, Germany (Dated: May 31, 2017)

Abstract Control of intermolecular electronic excitation energy transfer via metal nano-particles is analyzed. Different control scenarios are presented which all utilize the effect of field-enhancement close to an optically excited metal nano-particle. Due to this field-enhancement that part of a molecular chain or a molecular cluster gets excited which is in close proximity to the nano-particle. Various simulation results are discussed related to the energy transfer kinetics in a uniform chain of 50 molecules in the vicinity of a single or two spherical gold nano-particles. Interesting changes of the energy transfer pathways are found if the spatial and energetic arrangement between the molecular chain and the nano-particles is altered.

∗

Electronic address: [email protected]

†

Electronic address: [email protected]

1

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INTRODUCTION It represents one topic of actual research to achieve control of dynamical phenomena in quantum systems1,2 . For example, such a control plays a vital role when searching for new functional principles in nano-systems. Focusing on electron transfer or electronic excitation energy transfer (EET)3 control of the transfer dynamics would allow to explore structureproperty relations, in particular, in novel types of hybrid systems4 . In supramolecular complexes or dye aggregates it would be of interest to realize a particular type of control. This type aims at an electronic excitation energy localization (EEL) on a single molecule at a particular time. If EEL is possible distinct parts of the complex can be excited and, subsequently, different transfer pathways within the system can be tested5 . In somewhat earlier studies we suggested the use of shaped laser pulses to achieve this6–8 . When trying to realize spatio-temporal EEL in such a way one has to notice that only the exciton states φα representing the eigenstates of the molecular system are optically addressable. However, applying ultrashort pulses with a spectral broadening which is comparable or P larger than the width of the exciton band, a superposition of exciton states α Aα (t)φα i.e. an excitonic wave packet is formed. If this formation has been done in using a particularly shaped laser pulse, EEL shall be realized. Such a methodology, however, is not applicable to chain-like molecular systems since oscillator strength is concentrated to a few exciton levels, only. For example, in so–called H–aggregates (molecular transition dipole moments point perpendicular to the chain direction) the highest lying exciton level dominates the absorption and the formation of a proper wave packet is prevented. An alternative to realize EEL also in linear systems represents the use of metal nanoparticles (MNPs). If a MNP is resonantly excited efficient local-field formation with a strongly enhanced amplitude appears. For a MNP of small or medium size compared to a molecular chain or molecular cluster strong and localized excitation of that part of the molecular system may take place which is in close proximity to the MNP. This approach to EEL has already been suggested some years ago5,9,10 . But recent experimental studies continue this work. These studies report on the demonstration of strong-coupling effects, enhancement phenomena, and EEL (Refs.11–13 concentrated on MNP semiconductor nanosystems and Refs.14–16 on MNP molecule complexes). Here, we will theoretically study EEL in a MNP-molecule hybrid system similar to the one presented in Ref.16 (cf. also Fig. 1). Some preliminary work on EEL could be already 2


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offered in17 but for the presence of a single MNP only and with the restriction to weak optical excitation. However, such a restriction to single excitations in the molecular system is questionable. Although the overall optical excitation may be weak, field enhancement due to the presence of MNPs results in strong molecular excitation accompanied by multiple plasmon formation. Accordingly, a theory has to be applied which allows for multiple excitations in the MNP and in the molecular system (cf.18–20 ). A measure for the degree of P excitation would be the total molecular excitation Pexc (t) = m Pm (t), where the quantity Pm ≡ Pme represents the excited state population of molecule m. The temporal evolution of the overall MNP-plasmon excitation can be characterized by Ppl (t). In order to describe photoinduced EET and EEL in a molecule-MNP system we utilize a uniform quantum theory on MNPs strongly interacting with molecules and other MNPs. This approach avoids the introduction of MNP dielectric functions and local fields. MNP-molecule coupling is due to a pure Coulomb-interaction of the energy exchange type. The involved neglect of retardation effects restricts the description to molecule-MNP complexes with some 100 nm extension. Mirror charge effects originated by an unbalanced charge distribution in the molecules are assumed to be small and are also neglected. The photoinduced kinetics are formulated in the framework of a density matrix theory which, in particular, allows for a consideration of the short MNP plasmon life time21–23 . Below we offer some further details on the used theory (a complete description can be found in the Supporting Information). Then, photoinduced EET and EEL are discussed in considering different molecule-MNP arrangements.

THEORETICAL METHODS

The Molecule MNP Model. Photoinduced EET will be investigated for a molecular chain of Nmol entities coupled to different MNPs. The molecular system is represented by a model which restricts the description to the ground-state of molecule m and its first excited state (with excitation energy Em ). To allow for considerable population of the first excited molecular state and, thus, intensive generation of excitons we chose molecules with 2Em < Em (twice of the ground-state first excited state transition energy Em is smaller than the transition energy Em from the ground-state to the next higher excited state). In such a case strong population of the first excited state is not quenched by the process known as 3



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exciton-exciton annihilation (see the detailed discussion in Ref.18 ). The related Hamiltonian which accounts for multiple molecular excitation takes the form18 Hmol =

X

+ Em Bm Bm +

m

X

+ Jmn Bm Bn .

(1)

m,n

+ The transition operator Bm moves molecule m from it’s ground-state to it’s first excited

state. The energy transfer (excitonic) coupling is denoted by Jmn and is used in dipole approximation. The usefulness of this approximation is justified by the considered values ∆mol of inter-molecular distances (see below). Moreover, any type of disorder is neglected by assuming a regular molecular chain with identical molecules and a common ∆mol . The MNP plasmon Hamiltonian is introduced in similarity to Hmol 17,19–23 , but with adopted energies EN and inter-MNP couplings JM N . The M and N label a special plasmon mode of a particular MNP. Respective plasmon excitations are achieved via harmonic oscil+ and CN . As in the case of the molecular chain the intern-MNP couplings lator operators CM

are responsible for hybridization and energy level splitting. The ultrashort plasmon life time will be accounted for in the utilized density matrix approach. The molecule MNP coupling P + Hamiltonian is of the form m,N VmN Bm CN (plus conjugated complex term). The matrix elements VmN are the energy transfer part of the general Coulomb coupling matrix element between the m’th molecule and the particular type of MNP plasmon excitation. For the concrete computations we consider dipole plasmons of spherical MNPs. The dipole moment dN is given by dpl eXI where the eXI are the unit vectors of a Cartesian coordinate system (I = x, y, z) and X counts the various MNPs. Molecule-MNP and MNP-MNP configurations are chosen which all allow a restriction to these dipole plasmons and to a dipole-dipole type of coupling (see Ref.19 and the Supporting Information for justification). The coupling of the molecules and the MNPs to the radiation field is taken in the dipole form. Optical excitation is realized via a single laser pulse with carrier frequency ω0 , field amplitude E0 and pulse duration τp . For the latter two values are used: 1 ps and 10 fs. The field-amplitude E0 is chosen in such a way to get weak molecular excitation in the absence of any MNP. We set E0 = 105 V/m for τp = 1 ps. The same measure for the pulse area ap = τp E0 is obtained if we take E0 = 107 V/m for τp = 10 fs. Photon energy ~ω0 always coincides with the molecular excitation energy and in any described example laser pulse polarization points in the direction of the uniformly oriented molecular transition dipole moments. 4


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Kinetic Equations.

Kinetic equations for plasmon assisted EET are derived in

+ considering expectation values of different arrangements of the basic operators Bm , Bn , + CM , and CN 18–20,24 . These expectation values are defined by means of the reduced density

operator which obeys a specifically defined quantum master equation (see the Supporting Information for details). The dissipative part of the latter accounts for molecular excited state and plasmon decay. However, the main obstacle of this approach is the appearance of higher expectation values with three and four operators, what indicates that the derived set of equations does not represent a closed one. To get a closed set we follow a simple approximation (decoupling) scheme which has been recently used in Refs.18–20,24 . It ignores two-exciton and higher types of correlations and replaces operator expressions of type + + 1 − 2Bm Bm by their expectation values < 1 − 2Bm Bm >= 1 − 2Pm . It is just the population

inversion Pmg − Pme between the ground and the excited state (be aware of Pmg = 1 − Pm and Pme = Pm ). These expressions affect the EET among different molecules and let become the whole kinetics nonlinear.

RESULTS AND DISCUSSION While different examples of interacting molecule-MNP systems have been already investigated14–16 a molecular chain decorated by two MNPs as discussed here (cf. Fig. 1) was not realized in the experiment so far. But different examples for suitable chains can be found in literature. Thin films of the quasi-one-dimensional organic semiconductor 3,4,9,10-perylene tetracarboxylic dianhydride (PTCDA) have been presented in25 . Individual perylene bisimide J-aggregates were investigated in26 and single perylene-based H-aggregates could be described in27 . The quasi-one-dimensional π-conjugated organic rigid-rod quantum nanowire of methyl-substituted ladder-type poly(para-phenylenes) was used in28 , and Ref.29 suggests cyclic structures of π-conjugated materials covering several monomers.

All these different types of one-dimensional molecular systems represent

potential candidates for the molecule-MNP configurations we have in mind. Therefore, we do not focus on a particular molecular chain but introduce a fictitious linear arrangement of Nmol = 50 identical molecules. The uniform excitation energies Em = Emol are assumed to be in resonance to the MNP plasmon excitations. The molecules form a H-aggregate and the transition dipole moment dm amounts to 8 D. To give a precise description of the molecule-MNP position we place all molecules along 5



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the x-axis of a Cartesian coordinate system. The uniform intermolecular distance ∆mol is assumed to range from 1.2 to 2.5 nm. The first molecule is placed at x = 0 and the last one at x = (Nmol −1)∆mol . The chain length ℓ varies between 58.8 nm (∆mol = 1.2 nm) and 122.5 nm (∆mol = 2.5 nm). In any case ℓ is much larger than the used MNPs of 20 nm diameter (see below). According to the chosen H-aggregate configuration (dm ||ez ) we obtain the nearest neighbor excitonic coupling (in dipole approximation) between J(∆mol = 1.2 nm) ≈ 23 meV and J(∆mol = 2.5 nm) ≈ 2.6 meV. The exciton energies cover the range between ≈ Emol −2J and ≈ Emol + 2J. With the introduction of different values of ∆mol we have a flexible model at hand to change the extension of the exciton spectrum. For the MNPs we take identical spherical Au particles (with radius of 10 nm, dipole plasmon energy of 2.6 eV, dipole moment of 2925 D, and plasmon decay rate ~γpl = 57 meV). As an initial configuration we assume a placement with the center of mass in the xy-plane: Rmnp = (x, y, z = 0). In any case the closest distance ∆mol−mnp between the MNP-surface and a molecule equals 2.5 nm. The MNPs may be arranged at the center of the chain with Rmnp = ((Nmol −1)∆mol /2, ±(∆mol−mnp +rmnp ), 0) or in front of the respective molecule positioned at the left or right end of the chain. Absence of MNPs. To have a reference case at hand we start the discussion in considering a molecular chain being isolated from any MNP. Fig. 2 displays the populations Pm versus time using a contour plot (an interpolation has been carried out along the abscissa). Due to the chosen strong excitonic coupling (∆mol = 1.2 nm) the exciton band width amounts more than 90 meV. This width overcomes the frequency broadening of the 1 ps long pulse considerably. Only some selected exciton levels are excited what results in the oscillatory behavior of the Pm versus m (note the somewhat larger values of Pm at the chain ends). Due to efficient EET the oscillations are smeared out if the laser pulse is over. Some selected Pm approach values not larger than 10−6 and Pexc ≈ 0.03 what indeed underlines the weak character of excitation (it is due to the chosen laser pulse intensity and the excitation in the center of the exciton band with small values of oscillator strength). If one turns to the case of weak excitonic coupling (larger inter-molecular distance) a much larger number of exciton levels can be addressed by the 1 ps laser pulse and the Pm versus m appear nearly structureless (not shown). But the degree of excitation Pexc does not changes much compared to the foregoing case. Presence of a Single MNP. The degree of excitation increases dramatically if we turn 6


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to the case where a single MNP is placed at the right end of the chain (cf. Fig. 3). To avoid an overlay of EEL by effects of strong excitonic coupling we took ∆mol = 2.5 nm. Since the MNP center of mass is placed in the xy-plane the molecules couple to the I = z-plasmons only. Now, we notice excited state populations of nearly 0.2 (Pexc ≈ 0.5) but close to the chain end (see upper left panel of Fig. 3). So, the uniform values of Pm realized if the MNP is absent have been changed to an EEL with more than five orders of magnitude larger molecular excitation in that part of the chain close to the MNP. In the present scheme of description this is due to the strong molecule MNP-plasmon coupling Vm,I=z . Its influence can be either understood as the effect of molecular oscillator strength renormalization or as the result of fast EET from the MNP to the molecule. Taking the point of view of an increase of the molecular transition dipole moment a considerable increase of molecular photoexcitation is obvious. Choosing the perspective of strong MNP-molecule EET the large MNP excitation results in a large molecular excitation. In the course of the comparable long 1 ps laser pulse excitation, EET proceeds and moves molecular excitation somewhat away from the chain end. Afterwards, quenching of molecular excitation due to EET into the MNP and subsequent plasmon decay dominates the kinetics. Real plasmon population is only present during laser pulse excitation. The remaining panels of Fig. 3 correspond to situations where the MNP center of mass is moved upon the xy-plane. This introduces a coupling of molecules also to the I = xplasmon. Moreover, the excitation region in the chain can somewhat moved into the chain and even can be split (lower left panel). Such a spatial separation of EEL results from the specific form of the molecule-MNP coupling VmI . When moving the MNP out of the plane where the molecular chain is placed the behavior of the various VmI is determined by the interplay of the molecule-MNP distance factor 1/R3 and the geometry factor (cf. the Supporting Information). The overall strength of excitation, of course, decreases with this displacement. Presence of Two MNPs. Next, we consider the case of two MNPs. Every molecule is affected by the presence of these two spheres and, possibly, if the inter-MNP coupling is strong enough plasmon hybridization appears. Fig. 4 displays EET dynamics if one MNP is placed at the left and the other at the right end of the chain. For the considered case of strong excitonic coupling the inter-MNP distance (center of mass to center of mass) amounts ℓ + 2∆mnp−mol + 2rmnp ≈ 84 nm. According to this large value any effect of MNP plasmon 7



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hybridization is of less importance (the coupling just attains 9 meV). The chosen value for ~ω0 = Emol stays roughly in resonance with the dipole plasmons. Due to the short pulse length of 10 fs and a higher field strength the molecules at the chain end get rather strongly excited (Pm ≈ 0.9). This is accompanied by fast EET to the chain center with interesting interference patterns. The strong coupling of the terminal molecules to the adjacent MNP (and the short excitation interval) induces energy oscillations between the MNP and the molecule. This phenomenon produces the double structure of moving electronic excitation energy (green area in the upper and lower part of Fig. 4 separated by the blue stripe). The case where the middle part of the chain is sandwiched by the two MNPs is shown in Fig. 5. Here, MNP plasmon hybridization is of huge importance. EEL due to four different MNP configurations is considered by placing the center of mass of the MNPs in the xy-plane but move them step by step away from each other. To avoid a detuning between the hybrid plasmon level and the molecular exciton band, Emol as well as the photon energy are set equal to the upper hybrid level corresponding to the chosen MNP-MNP distance. Because of the weak excitonic coupling EET (following laser pulse excitation) is less visible. We notice an increase of the molecular excitation in the chain center up to an MNP molecular chain distance of 17.5 nm followed by a decrease. This behavior is identical with the change of the maximum of Pexc (cf. Fig. 6). At the same time the maximum value of Ppl increases from a value of about 0.01 to a value of 0.1 (see also Fig. 6). The smaller value at short distances indicates that a strong MNP-molecule coupling somewhat decreases the plasmon excitation (change of oscillator strength and life-time). At larger MNP chain distances (weaker MNP-molecule coupling), however, plasmon excitation is similar to that of an isolated MNP pair. Less intensive plasmon excitation at short distances and increasing molecular population with increasing MNP chain distances can also be understood as a signature of energy oscillations, i.e. as EET beyond rate determined transfer. While for 10 fs laser pulse excitation these energy oscillations become noticeable (cf. Fig. 4) they are hidden in the present 1 ps case. Finally, Fig. 7 starts with the same two MNP molecular chain configuration as in the foregoing figure. But now, as in Fig. 3, the MNPs center of mass are moved upon the xy-plane (cf. Fig. 1 where both MNPs should be moved simultaneously upwards). And, we consider a pulse length of 10 fs. If the MNP centers of mass stay in the xy-plane (upper left panel) energy oscillations between the molecular chain and the MNPs appear (again, note 8


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that Emol = ~ω0 coincide with the upper plasmon hybrid level). Since the molecules couple to two MNPs the energy oscillations are more pronounced as in the case of a Fig. 4 (there, the two MNPs are placed separately at the chain ends). EET away from the center is less obvious because of the chosen weak excitonic coupling (Pexc amounts to values above 10). Moving the MNPs away from the xy-plane the range of excited molecules now covers the range from molecule 20 to molecule 30. Later on, a threefold EEL at molecule 17, 25 and 33 is obtained. It again results from the interplay of the distance factor 1/R3 and the geometry factor in the molecule-MNP dipole-dipole coupling. If the pair of MNPs is completely placed upon the xy-plane (lower right panel of Fig. 7) molecular excitation is concentrated around the center of the chain (molecule 24 to 26). Here, Pexc amounts to values around 4.

CONCLUSIONS Let us summarize our findings on MNP mediated control of photoinduced EET in a molecular chain. The simulations concentrated on 1 ps and 10 fs laser pulses working in the low intensity regime. Then, an increase of molecular population by five orders of magnitude can be achieved if a single or two MNPs are placed a few manometer close to the chain. Moving the MNP center of mass out of the plane defined by the molecular chain EEL can be split up into two spatially separated excitation spots in the chain. In the case of two MNPs which sandwich the chain in its middle part such a MNP replacement even results in a threefold spatially separated excitation. Interesting EET interference patterns appear if the two MNPs are placed each at the end of the molecular chain. All these finding do not change principally when turning to another 1d arrangement of molecules, for example, a J-aggregate. While the phenomenon of EEL does not require a rather ordered molecular system, subsequent EET only appears if inhomogeneities are of less influence. In contrast, a precise positioning of the MNPs is essential. Also in the examples with 10 fs laser pulses where the excitation interval coincides with the plasmon life time respective quenching of molecular excitation only appears on a time scale of several hundreds of fs. The subtle effect of the coupling of the molecules to different MNPs ask for the consideration of 2d and 3d molecular complexes and a further elaborated placement of MNPs. Respective simulations will be the subject of future work. 9



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ASSOCIATED CONTENT Supporting Information A description of the theoretical background is available. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b11332

AUTHOR INFORMATION Corresponding Author Volkhard May E-Mail: [email protected] ORCID Volkhard May: 0000-0003-4094-181X Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work has been supported by the National Natural Science Foundation of China (Grant No. 11174029) (L. Wang) and the Deutsche Forschungsgemeinschaft through Sfb 951 (V. May).

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[29] Thiessen, A.; W¨ ursch, D.; Jester, S.-S.; Vikas Aggarwal, A.; Idelson, A.; Bange, S.; Vogelsang, J.; H¨ oger, S.; Lupton, J. M. Exciton Localization in Extended π-Electron Systems: Comparison of Linear and Cyclic Structures. J. Phys. Chem. B 2015, 119, 9949-9958.

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FIG. 1: Arrangement of two MNPs (blue spheres) close to a molecular chain (ruby ellipsoids).

FIG. 2: Molecular excited state populations Pm (m = 1, ..., 50) versus time and in the absence of a MNP (∆mol = 1.2 nm, Emol = ~ω0 = 2.6 eV, τp = 1 ps, E0 = 105 V/m).

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FIG. 3: Molecular excited state populations Pm (m = 1, ..., 50) versus time (∆mol = 2.5 nm, Emol = ~ω0 = 2.6 eV, τp = 1 ps, E0 = 105 V/m). A single MNP is placed at the right end of the chain and is moved upwards with Rmnp = (135, 0, a) nm. Upper left: a = 0, upper right: a = 5, lower left: a = 10, lower right: a = 15 (colour scale of upper left panel: 0-0.1715, of upper right panel: 0-0.05550, of lower left panel: 0-0.00686, and of lower right panel:0-0.0343).

FIG. 4: Molecular excited state populations Pm (m = 1, ..., 50) versus time (∆mol = 1.2 nm, Emol = ~ω0 = 2.6 eV, τp = 10 fs, E0 = 107 V/m). One MNP is placed at the left and the other at (1)

(2)

the right end of the chain, Rmnp = (−12.5, 0, 0) nm, Rmnp = (71.3, 0, 0) nm.

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FIG. 5: Molecular excited state populations Pm (m = 1, ..., 50) versus time (∆mol = 2.5 nm, τp = 1 ps, E0 = 105 V/m). The middle part of the chain is sandwiched by two MNPs which are shifted away from the chain step by step. Upper left: Rmnp = (61.25, ±12.5, 0) nm, Emol = ~ω0 = 2.942 eV, upper right: Rmnp = (61.25, ±15, 0) nm, Emol = ~ω0 = 2.798 eV, lower left: Rmnp = (61.25, ±17.5, 0) nm, Emol = ~ω0 = 2.725 eV, lower right: Rmnp = (61.25, ±20, 0) nm, Emol = ~ω0 = 2.684 eV.

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FIG. 6: Total molecular excitation Pexc (upper panel) and overall MNP-plasmon excitation Ppl (lower panel) versus time and corresponding to the situation shown in Fig. 5. Black solid line: Rmnp = (61.25, ±12.5, 0) nm, Emol = ~ω0 = 2.942 eV, red dashed line: Rmnp = (61.25, ±15, 0) nm, Emol = ~ω0 = 2.798 eV, green chain dotted line: Rmnp = (61.25, ±17.5, 0) nm, Emol = ~ω0 = 2.725 eV, blue dotted line: Rmnp = (61.25, ±20, 0) nm, Emol = ~ω0 = 2.684 eV.

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FIG. 7: Molecular excited state populations Pm (m = 1, ..., 50) versus time (∆mol = 2.5 nm, Emol = ~ω0 = 2.942 eV, τp = 10 fs, E0 = 107 V/m). The middle part of the chain is sandwiched by two MNPs which are shifted away perpendicular to the xy-plane, Rmnp = (61.25, ±12.5, a) nm. Upper left: a = 0, upper right: a = 5, lower left: a = 10, lower right: a = 15 (colour scale of upper left panel: 0 - 0.956, of upper right panel: 0 - 0.956, of lower left panel: 0 - 0.239, and of lower right panel: 0 - 0.9560).

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Control of Intermolecular Electronic Excitation ... - ACS Publications

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