Abstract
Organic molecules store the energy of absorbed light in the form of charge-neutral molecular excitations – Frenkel excitons. Usually, in amorphous organic materials, excitons are viewed as quasiparticles, localized on single molecules, which diffuse randomly through the structure. However, the picture of incoherent hopping is not applicable to some classes of molecular aggregates – assemblies of molecules that have strong near-field interaction between electronic excitations in the individual subunits. Molecular aggregates can be found in nature, in photosynthetic complexes of plants and bacteria, and they can also be produced artificially in various forms including quasi-one dimensional chains, two-dimensional films, tubes, etc. In these structures light is absorbed collectively by many molecules and the following dynamics of molecular excitation possesses coherent properties. This energy transfer mechanism, mediated by the coherent exciton dynamics, resembles the propagation of electromagnetic waves through a structured medium on the nanometer scale. The absorbed energy can be transferred resonantly over distances of hundreds of nanometers before exciton relaxation occurs. Furthermore, the spatial and energetic landscape of molecular aggregates can enable the funneling of the exciton energy to a small number of molecules either within or outside the aggregate. In this review we establish a bridge between the fields of photonics and excitonics by describing the present understanding of exciton dynamics in molecular aggregates.
Edited by Volker Sorger
1 Introduction
Advances in nanotechnology supported by our understanding of material properties on the microscopic level persistently drive the field of photonics to the nanometer scale. With the development of photonic crystals [1, 2] and semiconductor optical cavities [3, 4] the size of optical devices, usually limited by light diffraction in free space, can been scaled down to hundreds of nanometers just by exploiting the material’s dielectric constant. Advances in the design of plasmonic metamaterials [5–7] permitted this limit to be pushed even further down to the order of tens of nanometers. The natural question which arises is: what will be the next limit and how can it be approached?
In this review we consider one of possible pathways towards reaching the goal of optical devices on the nanometer scale: using molecular aggregates as photon processing elements on the true nanometer scale, see Figure 1. Molecular aggregates are assemblies of molecules held in place by non-covalent interactions. These single molecules mostly keep their nuclear and electronic structure. Examples are molecular crystals as well as nanoscale self-assembled structures, molecular films and light harvesting systems in photosynthesis. If the lowest allowed electronic transitions in the composing molecules are within the visible part of the optical spectrum and if the molecules have large absorption and fluorescence cross-sections, the interaction between molecular electronic transitions is strong enough to transfer the light absorbed from one molecule to the others, via a resonant de-excitation/excitation process.
Schematic illustration of the length scales characterizing the different physical phenomena present the different fields discussed in this review.
This mechanism is closely related to the near-field energy transfer between classical dipoles (antennas) [8–13]. In this sense, the exciton dynamics in molecular aggregates resembles the propagation of light in a metamaterial where dye molecules play the role of functional elements. In contrast to excitons in inorganic semiconductors, where free charge carrier mobility determines the exciton transport, in aggregates, electrons remain localized on molecules while the excitation is transferred.
A canonical example of molecular aggregates that possess coherent exciton properties are J-aggregates. These are aggregates of fluorescent molecules, discovered about 80 years ago independently by Scheibe [14, 15] and Jelley [16]. They observed that the formation of aggregates is accompanied by drastic changes in the optical properties. The broad absorption line corresponding to the molecular excitation is shifted to the red side of the spectrum and becomes much narrower. Later, these types of molecular aggregates were named J-aggregates after Jelley. J-aggregates represent only one class of molecular aggregates. For instance, there exist aggregates where the absorption line is blue-shifted, so called H-aggregates. Some aggregates exhibit several J-bands [17] and some have both J-and H-bands [18]. In aggregates that consist of non-equivalent monomers even more complicated absorption structures can be found [19].
Molecular aggregates also appear as functional units in nature. For instance, they form the absorbing and energy transferring parts of light-harvesting complexes in plants and some types of bacteria and algae [19]. In these complexes the exciton absorbed by the antenna aggregate has to be funneled to the reaction center – the part where the exciton energy is used to create free charges to be employed in a chemical reaction. The efficiency and robustness of light absorption and exciton transfer in light harvesting complexes may be crucial for the survival of photosynthetic organisms under evolutionary pressure.
Shortly after the discovery of the self-assembled organic dye aggregates the close connection of these artificial structures to the natural light harvesting systems was recognized [20, 21] and the exciton model of Frenkel [9, 22] has been used to explain the observed changes in optical properties and the transfer of energy along the aggregate [21].
In Frenkel’s exciton theory, which is based on the classical resonance interaction theory of Holtsmark [8], the electronic excitation in the aggregate is not confined to a single monomer, but it is coherently delocalized over many monomers in the form of “excitation waves”. From superpositions of these excitation waves “excitation packets” can be formed, which describe the coherent motion of (localized) excitations [22]. Already in these early works it was established that coupling to internal and external vibrational modes, imperfections of the aggregates and disorder induced by the environment strongly influences the “coherence size” of the exciton waves and modifies the optical and transport properties.
When the interaction with the environment (internal vibrations, solvent, etc.) is much stronger than the resonant excitation transfer interaction, the excitation becomes more or less localized on one monomer and the transfer is no longer described by coherent exciton motion but it becomes an incoherent hopping process. Förster derived an elegant formula for the rate constants for transport of excitation from one monomer to the other [23, 24]. This rate is proportional to the overlap of the donor emission spectrum and the absorption spectrum of the acceptor molecule and depends on the inverse sixth power of the distance between donor and acceptor. Typically, in molecular aggregates, one is in a regime in-between these two extreme cases, i.e., the transport is neither fully coherent nor incoherent. This complicates the theoretical modeling and the interpretation of experiments.
The remarkable optical and transport properties of molecular aggregates have led to a variety of applications. Right from the beginning, molecular aggregates were employed as wavelength selective sensitizers in photography [25, 26]. Recent applications include the measurement of membrane potentials [27] or the design of colorants [28]. Some molecular aggregates form self-assembled supramolecular flexible fluorescent fibers [29] which may have applications in thin-film optical and optoelectronic devices, for example by employing optical bistability of J-aggregates [30]. Molecular aggregates can be also utilized for sensing applications. For instance, it has been demonstrated that a large absorption cross section combined with fast exciton diffusion may be used to enhance fluorescence from a small number of dye molecules adsorbed or embedded in an aggregate film [31, 32]. Due to fast exciton diffusion within the aggregate the excitons explore the aggregate. Once they find a molecule with an electronic transition close to the J-band the exciton can be transferred inelastically to the adsorbed molecule. Moreover, dye aggregates might play an important role in the development of efficient, low-cost artificial light harvesting units (like organic solar cells) [33].
This review gives a brief, yet by no means complete, overview of our experimental and theoretical understanding of excitons in molecular aggregates. Most of the aspects in this work are covered at the advanced introductory level, and for a deeper understanding we refer to more detailed studies such as [19, 34–38]. The review is structured as follows: In Section 2, we describe basic excitonic properties of molecular aggregates starting with single molecules and molecular dimers and then introducing several examples of artificial and natural molecular aggregates. In Section 3, we overview the main experimental techniques that allow for probing the structural properties of molecular aggregates and excitation dynamics. Section 4 introduces the main theoretical approaches utilized for modeling of excitons in aggregates. Section 5 shows how molecular aggregates can be combined with photonic and plasmonic structures. Finally, we conclude the review in Section 6.
2 Basic properties of molecular aggregates
2.1 Properties of monomers
Molecules in the aggregates largely retain their electronic and nuclear structure. Thus, it is natural to start our discussion with individual molecules interacting with light. The interaction of a single molecule with light is schematically illustrated in Figure 2. If the molecule has an optically allowed electronic transitions within the photon spectrum it can be excited by absorbing light at the corresponding frequency. The transition time between the ground and excited state of the molecule is of the order of tens of attoseconds – the time that a single photon interacts with a molecule. During this short time interval the positions of the nuclei in the molecule are not changed. Initially the molecule is in its ground state geometry, i.e., in a minimum of the electronic ground state potential |g〉 indicated by the blue wave-packet in Figure 2. The equilibrium positions of atoms for the excited states, usually, are different from those of the ground state. Therefore, after the transition to the state |e〉 the molecule is in a transient non-equilibrium state where molecular vibrational modes are also excited (green wave-packet in Figure 2). Then, due to the interaction of the molecule with its environment the molecule relaxes towards the energetic minimum of the excited state, i.e., the excited vibrational modes are relaxed (wavy arrow and orange wavepacket in Figure 2). This relaxation of the molecular geometry can also be accompanied by a rearrangement of the environmental molecules to a configuration with lower total energy. At ambient conditions this process occurs over a timescale of hundreds of femtoseconds to several picoseconds. Finally, on timescales of tens of picoseconds to nanoseconds the molecule relaxes to its electronic ground state either emitting a photon (fluorescence) or transferring energy to other degrees of freedom, for instance vibrations (non-radiative relaxation [39]). Usually, the absorption and emission spectra exhibit a progression of peaks stemming from the coupling to vibrational modes with high energy (~150 meV). These peaks are broadened by the same order of magnitude, due to coupling to a multitude of low energy modes of the molecule and the surroundings. Emission takes place from the thermally relaxed excited state, which is typically located in the low energy wing of the absorption spectrum. The relaxation energy is related to the energy difference between the maxima of the absorption and emission spectra, the so-called Stokes shift.
Excitation/de-excitation processes in a single molecule illustrated on a two-level molecular energy diagram. The parabolic surfaces correspond to the ground state |g〉 and the first excited |e〉 electronic states of the molecule. The horizontal axis indicates the displacements of nuclei from their equilibrium positions. The molecule, initially in the ground electronic and vibrational states, is excited by absorbing light – blue arrow. During the absorption process the positions of the nuclei are not changed. The light absorption also induces molecular vibrations. Due to the interaction with the environment the molecular vibrations are equilibrated and the molecule relaxes to the bottom of the excited electronic state – orange wavy arrow. Finally, the molecule relaxes down to the ground electronic state by emitting a photon – red arrow (fluorescence), or without photon emission – green dashed arrow (non-radiative process). In the ground state the induced vibrations are also equilibrated due to interaction with the environment.
The absorption efficiency of a particular electronic transition can often be characterized by the corresponding transition dipole – a matrix element of a dipole operator between the ground and the excited molecular states.
Here and in the following we use an arrow symbol to indicate three-dimensional vectors in real space. The dipole operator is characterized by its charge q and the position operator
Structures and optical spectra of representative molecules are shown in Figures 3 and 4. In Figure 3 we show a TDBC molecule, which can be used as an illustrative example for a broader class of cyanine dyes. Beside the monomeric absorption and fluorescence spectra (in solution), spectra of aggregates formed on a glass substrate are also shown. The lowest electronic transition in TDBC is characterized by a large transition dipole of the order of
TDBC fluorescent dye that forms 1D and 2D molecular aggregates. (A) Molecular structure, gray and white spheres represent carbons and hydrogens respectively. (B) Normalized absorption and fluorescence spectra of TDBC molecules taken in a solution (solid lines) and in a 2D aggregated form (dashed lines).
(A) Bacteriochlorophyll (Bchl) molecule – light absorbing pigment that is a basic structural element of light-harvesting complexes in phototrophic bacteria. Its structure is similar to chlorophyll – the photosynthetic pigment in plants.; (B) Absorption and fluorescent spectra of Bchl molecules.
Another representative example of molecules forming aggregates is the bacteriochlorophyll (Bchl) – a pigment molecule, which is a functional element of photosynthetic systems in phototrophic bacteria, see Figure 4(A). The structure of Bchl is similar to that of chlorophyll – the photosynthetic pigment in plants. Both are derivatives of porphyrin, compelexed with Mg2+. The lowest electronic transition, which is involved in the energy transfer, is the so called Qy-band. The corresponding transition dipole lies in the plane of the porphyrin ring and is about
2.2 The aggregate
Organic dye molecules can self-aggregate in different types of structures. Sometimes the aggregation is driven by electrostatic forces pushing molecules to adsorb on a surface; sometimes hydrophobic parts of the molecules repel the water and tend to collect together such as in synthetic tube aggregates [17].
A specific property of molecular aggregates with strong inter-monomer couplings is that the absorption and fluorescent spectra substantially differ from the spectra of the molecules which form the aggregates while the electronic structure of the molecules is not modified. The intermolecular distance in an aggregate is large enough that the electron tunneling between different molecules can be neglected. Therefore, only the Coulomb interaction between the electronic transitions in monomers is responsible for the spectral modifications and the intermolecular energy transfer. This interaction is similar to the non-radiative near-field coupling between plasmonic structures, which is mediated by virtual photons. It turns out that the major contribution of the Coulomb interaction in the spectra of aggregates is from the excitation transfer via resonant transitions. We will refer to this main interaction term between the monomers as Förster coupling. The Förster coupling, to a good approximation, can be described by restricting to dipole transitions only (see below Eq. 2). The interaction is visualized in Figure 5 for a situation in which initially monomer 1 is electronically excited and monomer 2 is in the electronic ground state. Then, in a resonant process molecule 1 is deexcited and simultaneously the second monomer is excited. While this time-ordered discrete picture is easy for visualization in reality the exciton energy is transferred coherently. For large distances between the monomers the interaction between monomers 1 and 2 can be approximated by the dipole-dipole term
Illustration of the excitation transfer due to resonant near-field interaction between the state |e〉|g〉 and |g〉|e〉.
where R12 denotes the distance between the monomers,
If the intermolecular distance is comparable with the size of the molecules, then the interaction between the two monomers can no longer be described adequately using the point dipole-dipole interaction of Eq. (2). Often, it is then sufficient to replace the point-dipoles in Eq. (2) by extended dipoles [41]. For even higher accuracy more elaborate schemes have been developed (see e.g., [42]).
Before discussing general molecular aggregates, let us describe some basic results for the case of two coupled identical monomers, ignoring nuclear degrees of freedom for the moment. The eigenstates of the uncoupled monomers can be taken as a basis. These states are |g〉|g〉, |e〉|g〉, |g〉|e〉, |e〉|e〉. While the states |g〉|g〉 and |e〉|e〉 are still suitable to describe the ground state and the doubly excited state of the dimer, respectively, the two degenerate single exciton states |e〉|g〉 and |g〉|e〉 are no longer eigenstates of the coupled system, because of the resonant transfer interaction. The corresponding Hamiltonian is written as H = ε1 |1〉 〈1| +ε2|2〉 〈2|+V12(|1〉 〈2|+|2〉 〈1|) (we choose the energy of the monomer ground state as zero), where we have introduced the shorthand notation |1〉 ≡ |e〉|g〉 and |2〉 ≡ |g〉|e〉 or equivalently in a matrix representation
For identical energies of excited states ε19 = 9ε2 ≡ ε the eigenstates of the dimer are superpositions
Interaction between electronic transitions dipoles (blue arrows) of two molecules. The directions of the arrows reflect the relative phase of the transition dipoles. For the considered molecular alignments only one of the optical transitions is allowed for the dimer (green arrows). (A) The optically allowed transition of the dimer is shifted to the blue part of the spectrum, which is similar to H-aggregation; (B) the optically allowed transition is shifted to the red, which correspond to J-aggregation.
Let us briefly mention that the dimer often appears as a first step of aggregation [43] and thus has been investigated intensively (see e.g., [44–49]). This interest in the dimer is also due to the fact that many of the more realistic models for aggregates where the environment and the internal vibrations are included can only be solved efficiently for the dimer.
The theoretical description of the dimer can easily be extended to arbitrarily arranged monomers. As in the case of the dimer the total ground state is taken as a product of states where all monomers are in the ground state
in which monomer n is electronically excited and all other monomers are in their electronic ground state. In this one-exciton manifold approximation the Hamiltonian can be written as
where Vnm is the Coulomb dipole-dipole interaction, see Eq. 2, which transfers excitation from monomer n to m.
Examples of commonly discussed 1D and 2D planar structures are shown in Figure 7. While 1D models are well studied theoretically (see e.g., Refs. [50–58]) and are frequently used to characterize excitons in self-assembled molecular aggregates, it is not clear whether ideal 1D molecular chains are formed in experiments [34]. Usually, they are subsystems of higher dimensional structures such as films or crystals.
Illustrations of 1D and 2D planar molecular aggregates. Each block corresponds to a monomer forming the aggregate. (A) staircase and (B) ladder models for 1D packing; (C) brickstone and (D) herringbone 2D packing models.
The 1D model of molecular aggregates is convenient because many results can be obtained analytically [19, 59]. For a perfect very long chain with N molecules the eigenstates are well described by “exciton waves”
with the corresponding eigenenergies
In the last equation, for the sake of simplicity, we have considered the interactions between nearest neighbors only, which are denoted by V. A discussion of finite one-dimensional aggregates can be found e.g., in Refs. [19, 59].
For the perfect linear chain with parallel monomers, the largest transition dipole correspond to the excitation of the state with a minimal number of nodes. In the case of J-aggregation, Figure 7A, this state is at the bottom of the exciton spectrum shifted by about several hundreds of meV from the monomer transition while for H-aggregates, Figure 7B, this shift is to the blue part of the spectrum. In reality, to compare exciton transitions in aggregates with excitations in single molecules one also has to include Van der Waals interaction with off-resonant excitations [19, 47] and the vibronic structure of molecular excitation [19, 47, 60]. However, the simplified model presented above qualitatively describes the exciton states in aggregates.
2D molecular aggregates can be formed, for instance, as Langmuir-Blodgett films [61], deposited molecule-by-molecule on a surface [40, 62], or by spin coating [63]. The structure of the aggregate is determined by the molecular properties such as their shape, charge, etc., and the assembly method. For instance, for cyanine dyes [64] (TDBC is one example of the class of molecules) some commonly assumed aggregation structures are brickstone, Figure 7C, and herringbone, Figure 7D.
3D molecular aggregates with translational symmetry are usually known as molecular crystals. Frequently, in these structures is possible to identify lower dimensional subsystems with a preferred interaction between molecular transition dipoles [65]. These directions would determine anisotropy of exciton transport. While on average the exciton transport in a crystal can be incoherent one may expect specific directions with large coherent exciton delocalization.
The above-mentioned structures are only very simple examples of molecular aggregates. For instance, when driven by hydrophobic and hydrophilic forces molecules can aggregate in two-layer tubes of about 10 nanometers in diameter [17, 66] or one can use templates like polypeptides or DNA to induce helical structures [67, 68].
Because of the huge variety of organic dyes that can aggregate it is possible to create narrow J-band absorption within an arbitrary part of visible and near IR spectrum, see Figure 8.
![Figure 8 Absorption spectra of four different cyanine dyes in monomeric and aggregated forms illustrating the variability in the aggregate optical properties. Adapted with permission from Ref. [69]. Copyright 2011 American Chemical Society.](/document/doi/10.1515/nanoph-2012-0025/asset/graphic/nanoph-2012-0025_fig8.jpg)
Absorption spectra of four different cyanine dyes in monomeric and aggregated forms illustrating the variability in the aggregate optical properties. Adapted with permission from Ref. [69]. Copyright 2011 American Chemical Society.
2.3 Natural aggregates
In nature, photosynthetic complexes of plants, phototrophic bacteria and algae are composed of molecular aggregates that efficiently transfer excitons. As an example, in Figure 9 we sketch the energy transfer along the light harvesting system of green sulfur bacteria (GSB). These GSB can be found e.g., in deep sea. By developing a sophisticated light-harvesting antennae structure these organisms adapted to survive at very low light intensities [70, 71]. Energy absorption and transfer in GSB goes through a network of Bchl molecules, Figure 4 aggregated in several types of functional structures – chlorosome antenna, baseplate, Fenna-Matthews-Olson protein complex (FMO), and the reaction center.
Schematic structure of light-harvesting complex in green sulfur bacteria. The solar light is absorbed by chlorosome antenna aggregates of Bchl molecules. Then, the energy in the form of excitons is transferred through the baseplate and Fenna-Matthews-Olson (FMO) protein complexes to the reaction center, where the charge separation occurs.
The photons are absorbed by the chlorosome – an organelle which is bound to the bacterial membrane and has a size of several hundreds of nanometers. The chlorosome contains a disordered array of cylindrical or ellipsoidal molecular aggregates – antennas. While different types of geometrical and structural disorders complicate the molecular structure characterization of natural aggregates, recent NMR analysis of mutant chlorosome antennas [72] suggested that Bchl molecules in them are arranged in an array of concentric helical structures illustrated in Figure 10A. The Förster coupling between nearest-neighbor Bchl molecules in the chlorosome antenna complexes is of the order of 100 meV and the proposed molecular arrangement of the mutant should result in the formation of a J-band. Both experimental [74] and theoretical studies [75], show that the exciton spreads over a single antennae on the timescale of hundreds of femtoseconds.
![Figure 10 Examples of molecular aggregates in the light-harvesting complex of green sulfur bacteria. (A) Chlorosome antenna complex – a cylindrical or ellipsoidal aggregate of Bchl molecules, several concentric aggregates are enclosed; (B) A Bchl dimer – a unit block of the antenna complex [72]; (C) Fenna-Matthews-Olson (FMO) protein complex – an excitonic “wire”; (D) organization of Bchls in a monomer unit of FMO complex [73].](/document/doi/10.1515/nanoph-2012-0025/asset/graphic/nanoph-2012-0025_fig10.jpg)
Examples of molecular aggregates in the light-harvesting complex of green sulfur bacteria. (A) Chlorosome antenna complex – a cylindrical or ellipsoidal aggregate of Bchl molecules, several concentric aggregates are enclosed; (B) A Bchl dimer – a unit block of the antenna complex [72]; (C) Fenna-Matthews-Olson (FMO) protein complex – an excitonic “wire”; (D) organization of Bchls in a monomer unit of FMO complex [73].
Another functional element of GSB light-harvesting structure, is the Fenna-Matthews-Olson (FMO) complex, a protein which is depicted in Figure 10C. FMO plays the role of a molecular wire transferring energy from the chlorosome to the reaction center. It is a trimer containing 8 Bchl molecules in each subunit. Unlike in the chlorosome antenna, Bchl molecules in FMO are held together by a protein cage, and also the Förster coupling between the molecules is several times weaker. However, from the multiple experimental and theoretical studies it is suggested that the exciton states in FMO are partly delocalized [76–78].
3 Experimental characterization of excitonic systems
Much effort is devoted to obtain the microscopic structure of the various aggregates (which in turn are needed for theoretical models), including the arrangement of the monomers, their spacial and energetic disorder, etc. Beside these conformational aspects there is also large interest in the dynamic properties of excitons such as exciton relaxation and dephasing rates, diffusion coefficients, etc.
X-ray crystallography can be used to obtain lattice properties of molecular aggregates, if the latter either possess an intrinsically long-range order (like molecular crystals or two-dimensional monolayers on substrate [79]) or can be crystallized. For instance, crystallization of photosynthetic light-harvesting complexes [80] had a large impact on the understanding of natural photosynthesis [19]. However, most molecular aggregates cannot be crystallized. In recent years Cryo-Transmission electron microscopy has provided valuable information on the geometrical structure of many aggregates [66, 81–83].
The classical way to obtain information on the geometry of the aggregate is by optical spectroscopy (in particular absorption, linear dichroism and circular dichroism) combined with theoretical modeling. For instance, the positions, intensities, polarizations and shapes of the aggregate absorption bands provide information about the strength of intermolecular coupling as well as relative orientations of the monomers [19, 47, 82]. Circular dichroism spectra can be used to analyze chirality of the structures, for instance, in studies of J-aggregate helices and tubes [18, 84–87]. Similarly, the emission spectra can be used to obtain information on the aggregate structure. These optical experiments are often performed under varying environmental conditions like solvent, temperature, concentration of monomers, alignment of the aggregates or pressure [88–90]. More detailed information including separation of homogeneous and inhomogeneous line widths and femtosecond exciton dynamics can be obtained from non-linear 2D spectra [74, 91]. Beside optical spectroscopy of electronic excitations a multitude of various experimental techniques is used to characterize aggregates, e.g., electroabsorption [92], Fourier transform infrared (FTIR) spectroscopy [93], nuclear magnetic resonance NMR [94] and Raman spectroscopy [95].
With optical spectroscopy one probes the structure of energetic levels in the aggregate and phase relations between corresponding electronic transitions, from which one can infer the excitation dynamics in the aggregate. This might be a viable way for small systems, where the exciton is delocalized along the aggregate. However, one would like to follow the dynamics of the exciton also in real space, that means to measure the time-dependent probability to find excitation on a certain monomer. Here in particular, diffusion constants are of interest, which characterize the spreading of the exciton after the initial coherences have died out. Exciton diffusion coefficients cannot be measured directly using existing experimental techniques. The indirect methods include quenching of photoluminescence [96, 97], photocurrent response [98], transient grating [99] and exciton-exciton annihilation [100–103]. Among the state-of-the-art techniques one could mention the recently developed coherent nanoscopy [104], which would allow for spatial resolution comparable with the exciton diffusion length.
4 Models of exciton dynamics
While the simple electronic model from Section 2 [see in particular Eq. (5)] already allows us to understand basic properties of molecular aggregates (for instance, the position of the J-band) it is not sufficient to describe important features such as the narrowing of the J-band (in particular in contrast to the broad H-band with vibronic structure). Also, many transport properties, e.g., dependence on temperature [105], cannot be explained. To this end it is necessary to include vibrational modes and the influence of the environment into the description. This can be done in various ways and there exists a multitude of theoretical models in the literature where each is best suited for a particular question and/or situation. For example, the Förster rate theory [24] works well when the timescale of the transfer is very long compared to decoherence and vibrational relaxation of the excitation. In this Section we review the most common models.
4.1 Static disorder
Since usually the experiments are performed on ensembles of aggregates where each aggregate experiences a slightly different environment, it is common to average over different “configurations” of the aggregate. Most often the influence of the environment is treated simply as inducing random shifts of the transition frequencies of the monomers. For early works see e.g., [54, 106, 107]. Often, it is assumed that each monomer sees “its own” local environment and the shifts induced by this local environment are uncorrelated from the shifts of the neighboring monomers and disorder of the coupling is ignored. This assumption could be called “uncorrelated diagonal disorder”. Sometimes also the positions or couplings of the monomers are treated as random variables (off-diagonal disorder) [107, 108]. For each disorder realization the exciton states are typically no longer delocalized over the full aggregate as in a perfect chain. This localization becomes stronger for larger values of disorder [54, 107, 109]. In the literature, the influence of various forms of correlations has been discussed, see e.g., [54, 110, 111]. To account for temperature a variant of the open system model described below is adapted [112–114], where the environment of the disordered chain can lead to scattering and relaxation between the localized exciton states. Such a model has been successfully used to describe optical and transfer properties of some aggregates [113, 115, 116]. In particular at low temperatures the increase of mobility with increasing temperature could be explained.
In the above formulation the influence of internal vibrations of the monomers is neglected. This seems to be a good approximation for the J-band where one can show that the electronic excitation is to a large extend decoupled from vibrational modes [55, 111, 117].
4.2 Dynamic disorder, Haken-Strobel-Reineker model
Fluctuations of the monomer transition frequencies or the intermolecular couplings that are fast compared to the exciton transfer time-scale are usually referred to as dynamic disorder. Dynamical fluctuations play a dual role in exciton transport [118–120]. In structures with large static disorder excitons are localized. Dynamic fluctuations in this case remove localization. However, if the fluctuations are strong this results in the dynamical localization of the exciton. A seminal model to treat these fast fluctuations is due to Haken, Strobl and Reineker [121–123], where the dynamical fluctuations are described by real stochastic processes, εn(t) and Vnm(t) and modeled using the density matrix [121] or stochastic Schrödinger [124] equations. In the original work the stochastic processes have been chosen as white noise, i.e., delta-correlated in time. This model has been shown to capture coherent and incoherent excitation transfer on the same footing. Moreover, one should notice that the equation describing Haken-Strobl-Reineker (HRS) model for 1D systems is equivalent to telegrapher’s equation, which further reflects the similarity between the exciton propagation in molecular aggregates and a wave propagation in a medium.
The Haken-Strobl-Reineker model, with several extensions, has been utilized to study exciton dynamics in natural and artificial molecular aggregates [120, 123–127]. The drawback of this model is that the exciton population does not relax to thermal equilibrium. There have been many investigations and extensions which have included colored noise and tried to cure the thermalization issue, e.g., [128, 129]. The original model also does not include the effect of strongly coupled high frequency modes of the monomers which is essential to describe the vibronic structure present in many H-aggregates. The recent development of so-called quantum state diffusion methods [130, 131] can be considered as an extension of the HSR model which cures these deficiencies at the cost of colored complex noise.
Figure 11A illustrates how the HRS model can be applied to exciton propagation on a brickstone lattice of TDBC dyes at ambient conditions [124]. The second moment of the exciton wavefunction, Figure 11B shows that the initial propagation on the time scale of 20–30 fs is ballistic or wave-like, which is associated with the quadratic dependence of the second moment on time. For the longer timescale the time dependence is linear, which characterizes a diffusive motion.
![Figure 11 (A) Time dynamics of exciton wave function on a 2D lattice of TDBC molecules subject to static disorder (linewidth σ = 70 meV) and dynamic fluctuations (linewidth Γ = 30 meV) of molecular electronic transitions. At time zero the exciton is localized in the center of the lattice. The contour plots show the population of lattice sites at four different times. (B) Second moments of the wave function corresponding to the exciton dynamics shown in (A). Two different transport regimes can be observed: the ballistic or a wave-like propagation on timescales of 20–30 fs (M(2) ~ t2) and the diffusive motion (M(2) ~ t) at longer times. Reprinted with permission from Ref. [124]. Copyright 2012, American Institute of Physics.](/document/doi/10.1515/nanoph-2012-0025/asset/graphic/nanoph-2012-0025_fig11.jpg)
(A) Time dynamics of exciton wave function on a 2D lattice of TDBC molecules subject to static disorder (linewidth σ = 70 meV) and dynamic fluctuations (linewidth Γ = 30 meV) of molecular electronic transitions. At time zero the exciton is localized in the center of the lattice. The contour plots show the population of lattice sites at four different times. (B) Second moments of the wave function corresponding to the exciton dynamics shown in (A). Two different transport regimes can be observed: the ballistic or a wave-like propagation on timescales of 20–30 fs (M(2) ~ t2) and the diffusive motion (M(2) ~ t) at longer times. Reprinted with permission from Ref. [124]. Copyright 2012, American Institute of Physics.
4.3 The vibronic model
Let us first consider a single monomer, where only the two lowest electronic states are taken into account, see Figure 2. In the Born-Oppenheimer approximation [132] we write the nuclear Hamiltonian of the electronic ground and excited states as
where Qnj are the (mass-scaled) nuclear coordinates of monomer n and Pnj are the corresponding momenta. Ug/e(Qnj) is the Born-Oppenheimer potential in the electronic ground/excited state (see Figure 2).
Then, for a molecular aggregate the Hamiltonian in the electronic ground state is given by
with the “collective” Born-Oppenheimer surfaces
The coupling to vibrations usually slows down the propagation of the exciton, however, it can help to overcome energetic barriers caused by differences in the electronic transition energies. In this way efficient directed transport along a biased chain can be achieved.
Often it is justified to approximate the Born-Oppenheimer potentials, Eq. (8), as harmonic potentials where the surface of the electronically excited state is just shifted relative to that of the ground electronic state, i.e., we have
4.4 Open system approaches
Open system approaches are closely related to the vibronic model discussed in the previous subsection. In these approaches one typically considers an infinite set of harmonic oscillators forming the bath. It is easy to see that the Hamiltonian (9) can be written as He = Hsys+ Hsys–bath+Hbath with Hsys given by the purely electronic part (5) with transition energies εn that already include static overall shifts induced by the environment. The bath is taken as
4.5 Mixed QM/MM
With the increase of computer capabilities, in recent years it has become possible to simulate aggregates starting from a microscopic description [142–145]. Since a full quantum mechanical treatment is still out of reach, one uses mixed quantum classical approaches. In these approaches usually the nuclei are propagated classically in the electronic ground state using molecular dynamics. Along these nuclear trajectories one calculates the (time-dependent) transition energies using quantum chemistry methods. These time-dependent transition energies can then be used to obtain input parameters for open system models or to use them directly as the energy fluctuations in HSR model. However, both approaches have their problems, since one is restricted to a classical propagation in the electronic ground state.
4.6 Semi-empirical approach based on polarizabilities
While the approaches discussed above start from a microscopic description of the exciton-phonon coupling and the environment, in the semi-empirical approaches one can take the standpoint that all the relevant information of the coupling of an electronic transition to other degrees of freedom is already contained in the shape of the absorption spectrum. The absorption spectrum in turn is linked to the polarizabilities of the monomers.
The induced electric moment of a monomer depends on the local field at that monomer, which is produced by the electric moments of all the other monomers and the external field. This leads to a set of coupled equations from which one can extract, for a given arrangement of the monomers, the polarizability of the aggregate and thus the optical properties. Such equations have been derived using classical treatments [10] or quantum approaches [117, 146]. This method implicitly assumes that the frequency dependent polarizability does not change (beside an overall shift in energy) when going from the monomer to the aggregate. Recently, it has been demonstrated that using measured monomer spectra as input one can accurately describe the band-shape of J- and H-aggregates [60, 147]. This approach also gives an intuitive explanation of the vibrational structure and the broadenings of the absorption bands of the aggregate.
4.7 Beyond the single-exciton approximation
Photonic device applications may require structures operating in a nonlinear response regime, where interactions between excitons cannot be neglected. For example, thin J-aggregate films switches based on optical bistability have been suggested in [30]. At higher intensities, however, an additional loss-channel – exciton-exciton annihilation – appears. The underlying physical process is similar to the Auger effect, followed by a fast non-radiative energy conversion. Within an approximate picture this can be viewed as a process by which two excitons localized on neighboring molecules are combined into a higher-energy electronic excitation of a single molecule. Then, the resulting excitation quickly decays to the lowest excited state or ground state dissipating the exceeding energy through vibrations. A detailed kinematic model for the exciton-exciton annihilation in molecular crystals of different dimensions has been introduced by Suna in [148], and a similar master equation approach for exciton annihilation dynamics in natural photosynthetic complexes has been suggested in [149]. While the microscopic approaches discussed above are still valid, the molecular Hamiltonian (5) should be extended to account for at least two excited states per each monomer [38, 150].
5 Hybrid excitonic structures
In this section we specifically focus on J-aggregates. Their large absorption and fluorescence cross-sections combined with a narrow line width and a small Stokes shift allow for coherent coupling between excitons and photons in optical cavities or excitons and plasmon modes in metal structures. This property can be utilized as an interface between photonic, plasmonic and excitonic circuits.
4.1 Exciton-polariton structures
Most of the studies of strong coupling between excitons in J-aggreagtes and photons have been done using organic microcavities [151–154]. A schematic illustration of an organic microcavity – a planar λ/2 cavity with a J-aggregate layer embedded in it – is shown in Figure 12A. Strong coupling of cavity photons with excitons in a J-aggregate result in formation of exciton-polariton modes. This is associated with the splitting of a cavity mode – vacuum Rabi splitting, see Figure 12B.
![Figure 12 (A) Structure of a planar λ/2 organic microcavity with a thin layer of J-aggregates embedded. (B) An example of a cavity photon and a J-aggregate exciton energy level anticrossing. Reprinted with permission from Ref. [102]. Copyright 2010 by the American Physical Society.](/document/doi/10.1515/nanoph-2012-0025/asset/graphic/nanoph-2012-0025_fig12.jpg)
(A) Structure of a planar λ/2 organic microcavity with a thin layer of J-aggregates embedded. (B) An example of a cavity photon and a J-aggregate exciton energy level anticrossing. Reprinted with permission from Ref. [102]. Copyright 2010 by the American Physical Society.
Within a simple model assuming only one exciton state and a single photon mode the value of the Rabi splitting, ΩR, is just
where
5.2 Plexciton structures
A strong coupling of excitons in molecular aggregates with plasmons in noble metals has been demonstrated for both propagating plasmon modes in films [164] as well as localized modes in various types of nanostructures [165, 166] and nanostructure arrays [167]. The hybrid plasmon-exciton modes were also named plexcitons [166]. The observed values of the splitting between the plexciton modes are of the order of several hundreds of meV. The linewidth of plasmon modes is usually sufficiently larger than the linewidth of excitons in molecular aggregates (plasmon lifetime is about tens of femtoseconds as compared to picosecond exciton lifetime in J-aggregates). Therefore, the interaction between the exciton and the plasmon modes can frequently be considered as the coupling of a single mode (exciton) to a broader continuum (plasmon). This results in the formation of a Fano resonance [168, 169]. It is interesting to notice that in the case of a strong optical pumping of plexcitonic structures non-linear Fano effects can be observed [170].
6 Conclusions
Aggregates of organic molecules – supramolecular assemblies with strong resonant near-field interactions between electronic transitions – could be exploited in the design of nanophotonic devices at the true nanometer scale. The molecules, forming the aggregates, interact collectively with optical fields, and the absorbed energy is transferred in the form of excitons on a sub-micron scale. The exciton transport within the aggregates possesses coherent properties even at room temperatures, similar to the electromagnetic wave propagation through a medium. Moreover, molecular aggregates can be coupled coherently to photonic and plasmonic structures. While at the present stage there is a sufficient gap between the research communities studying excitonics and photonics, this review calls for merging the knowledge from the two fields.
The authors thank Gleb Akselrod and Brian Walker for providing experimental spectra. We also appreciate comments from Alex Govorov on hybrid structures. The Harvard contribution of this work was supported by the Defense Threat Reduction Agency under Contract No HDTRA1-10-1-0046. S. V. acknowledges support from the Center for Excitonics, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science and Office of Basic Energy Sciences under Award Number DE-SC0001088 as well as support from the Defense Advanced Research Projects Agency under award number N66001-10-1-4060.
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