Near-Field Spectroscopy of Nanoscale Molecular Aggregates - The

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Spectroscopy and Photochemistry; General Theory

Near-Field Spectroscopy of Nanoscale Molecular Aggregates Xing Gao, and Alexander Eisfeld J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02482 • Publication Date (Web): 25 Sep 2018 Downloaded from http://pubs.acs.org on September 27, 2018

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The Journal of Physical Chemistry Letters

Near-field Spectroscopy of Nanoscale Molecular Aggregates Xing Gao∗ and Alexander Eisfeld∗ Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden, Germany E-mail: [email protected]; [email protected]

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Abstract When molecules are assembled into an aggregate, their mutual dipole-dipole interaction leads to electronic eigenstates that are coherently delocalized over many molecules. Knowledge about these states is important to understand the optical and transfer properties of the aggregates. Optical spectroscopy, in principle, allows one to infer information of these eigenstates and about the interactions between the molecules. However, traditional optical techniques using an electromagnetic field which is uniform over the relevant size of the aggregate cannot access most of the excited states because of selection rules. We demonstrate that by using localized fields one can obtain information about these otherwise inaccessible states. As an example, we discuss in detail the case of local excitation via radiation from the apex of a metallic tip, which allows also scanning across the aggregate. The resulting spatially resolved spectra provide extensive information on the eigenenergies and wavefunctions. Finally we show that the technique will elucidate the anomalous temperature dependence of superradiance found recently for two-dimensional aggregates of the semiconductor PTCDA formed on a KCl surface.

Graphical TOC Entry

A metallic nano-tip is used to create an optical near field which interacts with the delocalized states of an PTCDA aggregate on a dielectric surface. Right: Absorption spectrum as a function of tipposition along a 1D aggregate.

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Molecular aggregates are assemblies of molecules where the individual molecules interact via a transition dipole-dipole interaction. 1,2 This leads to coherently delocalized electronic excitations which are responsible for the remarkable optical and transfer properties of such aggregates. Because of their properties molecular aggregates have found application as building blocks of organic semiconductors 3 and photovoltaic systems. 4 In particular, the strong interaction with light caused by the coherently delocalized states has been extensively studied both theoretically and experimentally (see e.g. Ref. 5–14 and references therein). To understand and exploit the optical and transfer properties of molecular aggregates detailed knowledge of the collective excited states is crucial. To this end, optical far-field spectroscopy is typically employed, which in principle can provide information about the energies of the excited eigenstates, their coherence lengths, and about relaxation and decoherence dynamics. 15–17 A major obstacle of these far-field techniques with a spatially uniform electric field over the aggregate is, that in many relevant cases (for example J-aggregates 1 ) most of the excited states are optically inaccessible because of selection rules. 1,11,18 In the present work we show that one can gain access to these inaccessible states by using fields that are spatially inhomogeneous on the length-scale of a few molecules. Our theoretical calculations show that in this way one can gain insight into the eigenstate structure of the molecular aggregates. One way of obtaining a strongly inhomogeneous field is from near field radiation. Nearfield light-matter interaction has been studied theoretically and experimentally for small molecules, 19–22 inorganic semiconductor nanorods, 23,24 graphene 25 and carbon nanotubes. 26–28 Recent experimental progress has made it possible to obtain fields that are localized on a size about 10 nm 29–32 reaching even the sub-5 nm regime. 33–35 However, for typical molecular sizes of the order of a nanometer it is still challenging to clearly identify the effects of field-variation across the molecule, in particular because of the small variation of the fields over the distance of a molecule. Molecular aggregates, where the electronic excitation can be coherently delocalized over tens of molecules with center-to-center distances in the or-

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Figure 1: (A): Basic setup. An emitting (point-)dipole with dipole moment d~ is placed at ~ dip = (xdip , ydip , zdip ). The emitted field interacts with the molecular transition position R ~ m (see Eq. (5)). In the figure we dipoles µ ~ m which are located at the molecular positions R sketch the specific arrangement that we have used for the calculations presented in Figs. 2 and 3. Panel B shows cuts through the field distribution of the dipole field. Here d~ is chosen perpendicular to the x-y plane. The distance zdip is chosen as unit of distance. der of a nanometer offer a promising platform to investigate basic effects of the interaction of near-fields with matter. Previously, near-field spectroscopy with less confined fields has been successfully applied to obtain information about the molecular arrangement of certain aggregates. 36–38 However, in these works, the field does not change fast enough between the molecules to appreciably alter the far-field selection rules. As a concrete example of a setup with a sufficiently strong field gradient we discuss a near-field generated by a localized plasmon of a metallic tip, as in the experiments of Ref. 29 The tip is positioned above a transparent dielectric surface on which the molecular aggregate is formed. 11,39–45 A particular feature of metallic tips as the excitation source is the possibility to move the tip parallel to the surface. This allows one to record spectra for different tip position. Such a scan allows one to extract information about the extent of the electronic wavefunctions of the excited states of the aggregate. This setup is motivated by the increasing number of experiments where aggregates are created on dielectric surfaces. 11,39–45 In contrast to metallic surfaces a dielectric surface does not quench the molecular excitation, thus enabling the use of spectroscopic techniques that 4


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rely on the detection of molecular fluorescence. In the present context, where we are interested in electronic absorption one such technique is laser induced fluorescence (LIF). 46 Here, one excites the molecules with a narrow laser and records the dependence of the fluorescence intensity on the excitation frequency. If there are no non-radiative decay channels the observed LIF spectrum is proportional to the absorption spectrum. In Ref. 47 such LIF spectra were recorded with high resolution for two-dimensional aggregates of the organic dye PTCDA on a potassium chloride (KCl) surface using far-field radiation. These spectra show clear indications of excited state wavefunctions that are coherently delocalized over tens of molecules. For this system the temperature dependence of the superradiance shows an anomalous behavior at low temperatures, which was theoretically traced back to optically dark states at the bottom of the excited state manifold. 11 However, a direct confirmation of this theoretical prediction is missing. As we will show below, with near field excitation one would also have access to this relevant state.

Basic Hamiltonian of the aggregate. In our theoretical modeling we use a widely adopted description of the aggregate. 48 For each monomer in the aggregate we take two electronic states into account: the ground state |gin and the first excited state |ein , where the index n labels the monomers. The transition dipole between these two states is denoted by µ ~ n . Initially the aggregate, which consists of N molecules (monomers), is in the total ground state |gagg i = |gi1 · · · |giN in which all monomers are in their ground state. For linear absorption we are interested in states with one excitation. Using as a basis state Q |mi = |eim N n6=m |gin the excited state Hamiltonian for the system is written as

Hex =

X

εm |mihm| +

m

X

Vmn |mihn|.

(1)

m6=n

Here εm is excitation energy for the monomer m and Vmn is the transition dipole-dipole ~ mn ~ mn R ~m ·µ ~ n − 3(~µm · R )(~ µ · ) interaction, which for our calculations we take as Vmn = R31 µ n Rmn Rmn mn

~ mn the distance vector from monomer m to n and Rmn = |R ~ mn |. This form of the with R 5



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interaction is sufficiently accurate for the purpose of the present paper. In the Supplemental Information we present for the case of PTCDA also results stemming from more advanced approaches to calculate the interaction. From solving the time-independent Schrödinger equation Hex |φ` i = E` |φ` i,

(2)

one obtains the N eigenenergies E` with corresponding eigenstates

|φ` i =

N X

cm` |mi.

(3)

m=1

The coefficients cm` depend on the arrangement of the molecules in the aggregate. Absorption spectrum for a spatially varying electromagnetic field: The space (~r) and time (t) dependence of the electric field of a monochromatic light source with frequency ω is given by n o ~ r, t) = Re E(~ ~ r)eiωt . E(~

(4)

Note the explicit dependence of the electric field on the position ~r. Absorption spectra for fields that vary over the extent of the aggregate (but have only a small variation over the extent of a single molecule) can be approximately written as P σ(ω) = ` A` δ(ω − E` ) with the absorption strength for a transition to the state ` given by 2 N X ~ R ~ m ) . A` = cm` µ ~ m · E( m=1

(5)

Here, crucially we have made use of the assumption that there is no overlap between the electronic wavefunctions of the monomers, as is justified for typical molecular aggregates. Details of the derivation can be found in the Supplemental Information sections I.1 and I.2. In the Supplemental Information we also discuss corrections to equation (5) for the case of 6


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field distributions that vary considerably over the size of a single molecule (Supplemental Information sections I.3 and I.4). Let us briefly relate to typical far-field spectra where the electric field variation is so small over the extent of the aggregate (or more precisely over the extent of the coherent size of the eigenstates of the aggregate) that one can take the field equal for all molecules, ~ R ~ m) = E ~ for all m. This use of the term far-field is somewhat sloppy, because there i.e. E( are certain far field techniques (such as circular dichroism) where the phase-variation of the electric field over the aggregate matters. 49,50

Dipolar light field: As a concrete example for an electric field distribution we use the field from a Hertzian dipole. 51 Such a field could, for example, stem from a tapered metallic tip. In recent years it has been shown that such setups can be used for spectroscopy. 52,53 In ~ located at Fig. 1 our basic setup is discussed. Here a Hertzian dipole with dipole moment d, ~ dip creates an electromagnetic field. In the near field zone this field can be written as R

~ − d~ r(ˆ r · d) ~ R ~ dip + ~r) = 3ˆ . E( r3

(6)

~ dip and rˆ and r are the corresponding Here with ~r we denote the spatial position relative to R direction and magnitude, respectively. This near-field formula is appropriate for our situation since the wavelength (> 400 nm ) is much larger than the distance between tip and aggregate (r . 10 nm). For aggregate molecules that are far away from the tip the electric field strength is quite small, so that deviations from the ideal dipole field do not matter.

Ideal spectra: As a first instructive example we consider the case of a linear chain of identical molecules (i.e. m and µm are the same for all molecules). All molecules are arranged in a line (which we choose as the x-axis) and are aligned parallel with the same distance a between one another. We use a as the unit for distance; thus the monomeric positions

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Figure 2: Spectra resulting from a Hertzian-dipole aligned along the z-direction, which is located at different positions xdip with a vertical distance zdip = 3a above a chain of N = 25 ~ n = (na, 0, 0). All monomeric transition dipoles are oriented along the molecules located at R x-direction All energies are expressed in units of the absolute value of the nearest neighbor 2 interaction strength 2V0 with V0 = µa3 . Left: (a)-(c) Examples of three tip positions xdip . For comparison, (d) is the spectrum corresponding to the same molecular aggregate but with uniform distributed electric field. Right: Continuous scan of xdip . The horizontal dashed green lines correspond to the spectra of panels (a)-(c). Note that xdip = a corresponds to the dipole located exactly above the first molecule of the chain. ~ m = (ma, 0, 0). This situation is also sketched in Fig. 1. It is well known that are R for this arrangement only states at the edge of the ’exciton’ band absorb in the far-field limit 49 (for the present situation the absorption strength in the far-field limit reduces to P 2 Afar−field = const · | ` m cm` | ). As a concrete example, in Fig. 2 we discuss a linear chain of N = 25 molecules where the dipoles of the molecules are aligned parallel to the aggregate axis. This arrangement 2

1 The first three rows (Fig. 2(a)-(c)) results in negative interactions Vnm = −2 µa3 |n−m| 3.

~ dip = (xdip , 0, 3a) of the tip for a fixed show spectra stemming from different x-positions R distance zdip = 3a (recall that the a is the distance between the monomers). The last row (d) shows the corresponding far field spectrum, which consists of one dominant peak at ωex ≈ −2.4 · (2µ2 /a3 ). One can clearly see that for the case of a dipolar light field much more information about the eigenstates is available than in the far field spectra Fig. 2(d). In particular many more spectral lines can be found at high energies. The energies ωex at which these lines appear correspond to the eigenstates of the aggregate. For the present example most of the eigenstates obtain an appreciable absorption strength. Only for the energetically

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high lying states the increase of absorption strength is negligible. From Fig. 2 (a)-(c), one sees that the intensity of the absorption lines depends sensitively on the position xdip of the tip. In Fig. 2 (right) the dependence of the spectra on xdip is shown in more detail. The horizontal dashed green lines indicate the values of xdip for which the spectra on the left hand side are shown. One sees that when scanning the tip position xdip for a fixed frequency ωex there are pronounced changes in absorption strength, with regions of zero absorption appearing in a regular oscillatory manner. The higher in energy an eigenstate is the more minima and maxima occur. This provides useful information for identifying the eigenstates. The appearance of minima and maxima can be easily understood q P πn` by noting that the eigenstates of a linear chain have the form |ψ` i = N2+1 N n=1 sin N +1 |ni. That means in particular that they are alternatively symmetric and antisymmetric and the number of nodes is given by ` − 1. Note also that the field-component of the Hertzian dipole that is parallel to the molecular transition dipole is anti-symmetric with respect to the x-position of the dipole.

Interaction with the plasmon of the tip. So far we have discussed an ideal situation where there tip only creates an electro-magnetic field. However, the transition dipoles of the molecules can also interact with the metallic tip. To estimate the influence of the tip we use local field theory which has been used before to study the optical response of metallic nanoparticles arrays 56,57 or molecular aggregates. 58,59 It is also closely related to the Table 1: Parameters used in the modeling of the polarizabilities of the molecules (top row) and the tip (bottom row). The respective formulas are given in the main text after Eq. (10). The molecular parameters correspond roughly to PTCDA on KCl at low temperatures. 54 The parameters of the tip correspond to gold nano-spheres. 55 ωm [cm−1 ] 2 × 104 ar [nm] b 2.52 9

env 1

γm [cm−1 ] µm [Debye] 1 7.4 ωp [cm−1 ] 7.26 × 104

9

γp [cm−1 ] 400


vF [cm s−1 ] 1.39 × 108


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Green-operator based CES method, 60 which has been shown to be well suited to describe experimental spectra of molecular aggregates, even in situations when coupling to vibrations becomes relevant. 50,61 In the local field theory approach, both the tip and the molecules are treated on the same footing: they are characterized by their frequency dependent po↔

larizabilities α m (ω). The double-arrow indicates that the polarizabilities are 3 × 3 tensors which connect the polarization vector with the electric-field. As before, we use the labels m = 1, . . . , N for the molecules of the aggregate. We use m = 0 to refer to quantities belonging to the tip. The fundamental equation of local field theory relates the induced dipole moment P~m (ω) ~ ext (ω) = of particle m to the total field at its position which consists of the external field E m ~ R ~ m ) and the ’internal’ fields E ~ int produced by all other particles n: E( mn N X ↔ ext int ~ ~ ~ Pm (ω) = α m (ω) Em (ω) + Emn (ω)

(7)

n6=m

~ int originates from a dipole moment at particle n and is given by The field contribution E mn ↔

int ~ mn E (ω) = − T mn P~n (ω)

(8)

↔

in which the transfer tensor T mn between m and n is ↔

T mn

~ mn ⊗ R ~ mn R 1 ↔ I −3 , = 3 2 Rmn Rmn

(9)

~ mn is the separation vector between m and where ⊗ denotes the outer product. As before R ↔

n; with I we denote the identity tensor. The use of Eq. (9) for the interaction between the particles is equivalent to using the point dipole-dipole interaction for Vnm as given below Eq. (1). We use the same form for the interaction between molecules and tip. 62 Solving the coupled system of equations (7) and (8) for given external fields one can obtain the induced

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dipole moments P~m of all particles. Finally the linear absorption spectrum is obtained from

σ(ω) = −Im

N X

~ ext (ω) P~m (ω) · E m

(10)

m=0

~ 0ext = 0, since the In our setup the external field stems solely from the nano-tip and thus E index 0 stands for the tip. Before we present calculations using Eq. (10) we will briefly comment on the relation between Eq. (10) and the description using Eq. (5). To establish the connection all terms belonging to the tip have to be omitted in Eq. (10), since in Eq. (5) there is no explicit appearance of the tip. The external fields are the fields entering Eq. (5). In addition the molecular polarizability has to be chosen as an infinitely sharp resonance. For more extended discussions see for example Refs. 58,61 We will now use Eq. (10) to investigate the influence of the tip on the spectra. For the following calculations we have used particularly simple models for the polarizability of the tip and the molecules, since we are interested in the basic effects. For the nanotip we use ↔

↔

↔

(ω)−env I, the polarizability of a sphere with radius ar which is 56 α 0 (ω) = α tip (ω) = a3r (ω)+2 env

where env is the dielectric constant of the surrounding medium. We take the polarizability of the tip to be isotropic. The complex dielectric function, (ω) is evaluated according to a generalized Drude model with finite-size effects correction for particles with small radius ω2

ω2

p p − ω(ω−iγp −iv . Here b is an adjustable constant, and (below ∼ 5 nm), (ω) = b + ω(ω−iωγ p) F /ar )

ωp , γp and vF are the plasma frequency, Ohmic damping constant and Fermi velocity in the bulk material of the tip, respectively. The polarizability tensor for the m-th monomer 61 is ↔

taken as α m (ω) =

µ ~ m ⊗~ µm , ω−ωm +iγm

where decoherence is taken into account via the parameter γ.

As before µ ~ m and ωm are the transition dipole and the transition frequency of molecule m. Note that because of the dyadic product µ ~m ⊗ µ ~ m the polarizability tensor of the molecules is highly anisotropic. In Fig. 3 results are shown using this model. The arrangement of the molecules is sketched

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in Fig. 3A (it is the same as for the calculations of Fig. 2). The exciting dipole field is taken to be centered in the sphere that we use to model the tip apex. The polarizabilities of the tip and the molecules are shown in panel B; the corresponding parameters are provided in Table 1. The polarizability of the molecules is chosen as a single very narrow resonance. This choice is motivated by the experiments of PTCDA molecules on a KCl surface at low temperatures. 11,47,63 The parameters of the tip correspond to those used to describe gold nano-particles. The radius of the apex of the tip which enters the formula for the tippolarizability is chosen as ar = 2.5 nm. As the external field stemming from the tip we used a Hertzian dipole oriented along z direction. Calculated spectra using the above parameters are shown in panels C and E of Fig. 3. In contrast to the calculations of Fig. 2 we now have interactions with the tip and broadening of the lines because of the finite γm as discussed above. Neverthess, the spectra are essentially identical to the ’stick’ spectra of Fig. 2 showing that the presence of the tip does not destroy the coherence properties of the aggregate eigenstates. It is now interesting to investigate how stronger couplings to the tip would influence the observed spectra. To this end we have artificially scaled the tip-polarizability by a factor s = 100, which results in the spectra presented in panels D and F. One clearly sees that the weight of absorption is now shifted to higher energies. Because of the strong coupling new ’hybrid’ eigenfunctions are formed which are localized around the tip. How far these spectra in the ’strong exciton plasmon coupling regime’ can be used to infer properties of the undisturbed molecular aggregate is not yet clear. However we believe that the interaction with the tip might even be advantageous to study high lying excitonic states.

2D aggregates of PTCDA molecules: Let us now look at two dimensional aggregates of PTCDA molecules on a KCl surface. In Refs. 11,47,63,64 it has been found that the PTCDA molecules arrange on the KCl surface into a rectangular lattice where the molecules are arranged parallel to each other forming an angle of 45◦ with respect to the lattice axis. From

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Figure 3: Spectra for the same linear chain as Fig. 2 but now including interaction with the tip and a frequency dependent polarizablity of the molecules. Panel A shows the setup where the tip is modeled by a gold nano-sphere for calculating the polarizability. Calculations are performed using Eq. (7) - (10). In panel B the used polarizabilities for the tip (αtip ) and the molecules (αmol ) are shown (note that the range of the frequency axis of the tip is three orders of magnitude larger than that of the molecule ). The corresponding parameters, can be found in Table 1. The spectra in the top row (panels D and F) are calculated with these parameters. For the spectra shown in the bottom row (panel D and F) the polarizability has been multiplied by a factor 100. Panel G shows the corresponding far field spectrum. The zero of energy is at the excitation frequency of the monomer.

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Figure 4: Two-dimensional aggregate of PTCDA molecules. (A): Sketch of the aggregate and gold nanosphere used to model the tip. The molecules are represented by blue arrows which indicate the direction of the transition dipole of the molecue. Two PTCDA molecules are shown in the inset. The transition dipoles are aligned along the long axis. (B) Top view of the arrangement of the PTCDA molecules (10 × 30 molecules). The distance between the molecules (see inset of panel A) is a = 1.26 nm. (E) and (F): subpanels (a)-(c) show near-field spectra for different tip positions (indicated by the purple circles in panel B). subpanel (d) shows the far-field spectrum. The zero of energy is at the monomeric transition frequency ωm . Panel D shows a zoom into the low energy region. The vertical dashed lines indicate energies where the Hamiltonian has eigenstates. For comparison, in panel C we also show stick spectra using Eq. (5). The calculations are performed using µn = 3 Debye which is the magnitude of the relevant effective dipole; 11 the direction of the transition dipoles is µ ˆn = √12 (1, 1, 0). For the polarizabilities the same parameters as for the calculation in Fig. 3 have been used.

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evaluating low energy electron diffraction (LEED) and far field absorption and emission, there is evidence that finite size domains are formed. Examples of such a domain are shown in panels A and B of Fig. 4. An unexpected behavior of the temperature dependence of the superradiance (i.e. the enhancement of the collective radiative decay rate) was observed, 11 which was explained by the appearance of domain shapes that exhibit a dark states (for far field radiation) as the lowest excited state for certain domain sizes. So far there is no direct access to this state to confirm this prediction. In the following we will show that using the near-field radiation from a tip this will be possible. To this end we consider as example the particular finite domain shown in Fig. 4 B, which exhibits a dark state at the bottom of the exciton band. Let us first take a look at the far-field spectrum which is shown in subpanel (d) of Panels E and F of Fig. 4; it consists of a few peaks at energies around −135 cm−1 with respect to the monomeric transition energy. We do not explicitly consider an overall energy shift caused by off-resonant interactions. In panel D a zoom into the low frequency region is shown. That indeed the lowest eigenstate is a dark state can be seen better when ignoring line broadening as done in panel C where all eigenstates with their transition strength in this low energy region are shown. Let us now look at the near field spectra. Spectra for several positions of the tip are shown in panels E and F of Fig. 4. The respective positions are indicated by purple circles in panel B. As for the far field spectra panel D shows a zoom into the low frequency region of panel E and in panel C the corresponding ideal ’stick-spectra’ (i.e. eigenenergies and transition strengths) are shown. For the calculations we have used the same parameters for the polarizabilities and for modeling the tip as in the 1D calculations above (see Table 1. We have verified by density functional tight binding calculations that for PTCDA and our choice of parameters the variation of the field over a single molecule is so small that the point dipole approximation in the coupling of the external field to the molecule is still a good description for a single molecule; that means that Eq. (5) and Eq. (10) are applicable. Our

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calculations indicate that the error resulting from the point dipole approximation for the interactions Vnm is much more significant than the error in the interaction with the external field. Details can be found in the Supplemental Information (Section II). As for the 1D case one sees in Fig. 4 from the comparison of the near field spectra with the far field spectrum that many states at high energies become accessible (for our example there are 300 eigenstates in total). One also sees that there is a strong dependence of the spectra on the position on the tip. This dependence is now more complicated as in the 1D case. Panel D which shows a zoom into the low energy region where the particular interesting dark states are located. The horizontal dashed lines indicate energies where the Hamiltonian has eigenstates (cf. panel C). One sees that all dark states can become accessible for suitable positions of the tip. One also sees that the inclusion of tip and broadening makes it difficult to distinguish the nearly degenerate peaks at −136 and −134 cm−1 . Most importantly, however, is the fact that the lowest energy state which is dark in the far field now becomes clearly visible.

Conclusions: In the present Letter we have theoretically investigated the advantages of spatially localized electromagnetic field for the optical spectroscopy of molecular aggregates. We have shown that one obtains new information on the eigenfunctions. In particular, one gains access to states that are optically forbidden by far-field selection rules. A particular focus was on aggregates on transparent dielectric surfaces and an excitation field stemming from a metallic nano-tip, which also allows one to record spectra at different tip positions. In experiment such localized spectroscopy can be performed analogously to far-field spectroscopy of molecules on surfaces using the principles of light induced fluorescence. 47 By varying the distance between tip and aggregate one has an additional degree of freedom (not discussed in the present Letter) to monitor the spectra. This might allow one to obtain additional information on the excitonic eigenfunctions. We have explicitly considered PTCDA molecules in an arrangement found on KCl sur-

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faces. However, our results should remain valid also for other molecules and other transparent dielectric surfaces (where different arrangements and thus different eigenstates are present). The novel near-field selection rules will also be of interest for non-linear spectroscopic techniques such as multidimensional femtosecond spectroscopy. It has already been shown experimentally that metallic tips can be used to provide localized femtosecond pulses. 29,65–68 Thus our proposal opens new ways to investigate the optical and transport properties of molecular aggregates, which will be helpful also for the design of aggregates with specific properties.

Supporting Information Available Derivation of the absorption spectrum (Eq 5). Ab initio calculation of the relevant parameters using quantum chemical methods. Comparison of different approximations.

Acknowledgement We thank Semion Saikin for inspiring discussions.

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