Reflection and extinction of light by self-assembled monolayers of a quinque-thiophene derivative: A coherent scattering approach

Our current understanding of the propagation of light through condensed matter and of the absorption and emission of radiation inside bulk material is still problematic. Well-known is the Minkowski-Abraham controversy on the momentum of electromagnetic waves in a solid.^1,2 A second open issue concerns the polariton, a hybrid exciton-photon quasi-particle, that represents the quantum of light inside a solid. Following the pioneering work of Hopfield and Pekar, coupling between excitons and photons with frequency close to an allowed electronic transition inside a crystalline solid leads to the formation of polaritons.^3–5 Although the concept of a polariton is well supported by experimental data, a major controversy persists until the present day concerning the boundary conditions that should be applied to the polariton. The traditional boundary conditions for purely electromagnetic waves at the interface between different media, from which, e.g., the Fresnel equations for reflection and refraction can be derived, are not sufficient for the polariton. Additional boundary conditions (ABCs) are required but the exact nature and formulation of these ABC’s is still under debate.^6–13 Without certainty with regards the boundary conditions, the process of calculating reflection and transmission spectra becomes difficult and involuntarily invokes an element of arbitrariness. Here we use the K-matrix formalism, developed originally to describe neutron scattering spectra in nuclear physics to calculate reflection transmission from coherent scattering. This avoids the need for introducing additional boundary conditions for the polariton.

A systematic approach to these problems involves the study of systems of reduced dimensionality. In a zero-dimensional system consisting of an isolated molecule in a radiation field, the interaction between light and the molecular quantum system is well understood. In higher dimensions, the emergence of quasi-particles involving coupled excitation of the electromagnetic field and the molecules is a main source of complication. Recent advances in the synthesis and preparation of self-assembled monolayers (SAMs) of chromophoric molecules offer new experimental possibilities to study the optical response of two-dimensional (2D) systems. Closely packed, highly ordered two-dimensional self-assembled monolayers (SAMs) of a quinquethiophene (5T) derivative (1) have become available on different substrates. Such layers are ideal model systems for studying the quantum electrodynamical behavior of optical excitations in a 2D molecular arrangement. In contrast to earlier studies involving monolayers of thiol functionalized thiophene derivatives on metallic substrates,^14–16 SAMs of (1) form on optically silent substrates, e.g., SiO₂. The layers are characterized by a high degree of order and are electronically active in transistor geometry. Self-assembled monolayer field-effect transistors made using the 5T derivative exhibit a bulk-like carrier mobility.^17,18

In this contribution, we study optical properties of a monolayer of quinquethiophene (5T) based molecules (see Fig. 1) on a solid support, both experimentally and theoretically. Oligothiophene based molecular materials are well known organic semiconductors. The 5T moiety has an allowed optical transition between the ground state (S₀) and the first excited singlet state (S₁). The transition dipole moment μ for the S₀-S₁ transition as determined from the absorption cross section has a magnitude of 8.1 D.¹⁹ The transition dipole moment is oriented along the long axis of the molecule.²⁰ The optical response of oligothiophenes is characterized by a strong coupling between electronic and nuclear motion, resulting from a change in bond-length alternation in the π-system. In the excited state, the π-system has more quinoidal character, with more pronounced double bond character for the carbon-carbon bonds between the thiophene rings. As a result, upon excitation to the lowest excited singlet state (S₁), the bonds joining the thiophene rings tend to get shorter, while the double bonds within the rings become longer. The change in geometry gives rise to vibronic progressions in fluorescence emission and excitation spectra. From the Huang-Rhys factor describing the relative intensity of the various vibronic peaks we calculate a coupling constant, γ, for nuclear and electronic motion equal to 0.17 eV.

The optical properties of an extended monolayer of oriented oligothiophene molecules are essentially governed by three different types of couplings. First, the molecular electronic excitations couple with vibrations, forming so-called vibronic excitations. Secondly, the electronic motion on neighboring molecules is coupled via, e.g., transition dipole–transition dipole interactions, leading to the formation of excitons. Finally, the optical excitations also couple with the electromagnetic modes of radiation. Treating these couplings simultaneously and on an equal footing, poses a theoretical challenge.^21,22 Here, we demonstrate that scattering matrix theory from nuclear physics provides a natural solution to this problem.

In molecular exciton theory, electronic excitations in orga nic semiconductors are modeled as tightly bound electron-hole pairs or Frenkel excitons. Molecular exciton theory can account for the shape of the absorption band of small aggregates of chromophoric molecules.^5,23 The exciton states are calculated by diagonalizing the Hamiltonian describing the excited state interchromophoric interaction. Recently, efficient computational strategies have been developed to include also the coupling of excitations with the molecular vibrations.^23–27 Depending on the mutual orientation of the molecules in the aggregate, they can display J-type or H-type behavior. J aggregates are characterized by a narrow absorption band that is red shifted in comparison to the absorption band of the isolated molecules. The narrowness of the absorption band in the J-aggregate is attributed to quantum mechanical delocalization of the excitation over a significant number of molecules, mitigating the effects of local disorder. In contrast, H aggregates show a blue shifted absorption band, which is usually quite broad. No satisfactory explanation for the large width if the allowed H-aggregate band is yet available. Oligothiophene molecules comparable in structure to the ones under study here, usually form H-type aggregates. Here we extract a coherence length for the Frenkel exciton in the H-type aggregate by comparing calculated and experimental spectra. The width of the exciton absorption band will be discussed.

Studies on oligothiophene crystals have indicated that the broad, blue shifted absorption maximum, associated with H-type aggregates, shows signatures characteristic of polaritons.^28–30 Due to the complexity of the problem, no comprehensive description of the optical properties of oligothiophene crystals is yet available. Already in 1975, it was suggested by Philpott and Sherman³¹ that monolayers of quinque-thiophene might exhibit polaritons effects and may serve as a model system. Previous studies on monolayer optical properties have included Langmuir Blodgett layers at the air water interface^32,33 and submonolayers of sexithiophene deposited by vacuum sublimation.^34,35 Coupling of the excitations with the electromagnetic field is evidenced by the observation of a reflection band at frequencies close to excitonic transitions of the coupled chromophoric groups. In the analysis of the spectra, coupling with vibrations was neglected.^28,36,37

In molecular exciton theory, the coupling between the electromagnetic field and the excitons is treated perturbatively, in first order. Absorption and scattering of light by small aggregates can readily be calculated but reflection of light by extended monolayers is more difficult. No methodology exists to include all three important couplings, i.e., interchromophric, electron-phonon, and electron-photon, simultaneously.

This paper is organized as follows. We first introduce the scattering matrix formalism. The total scattering or S matrix is obtained from the K matrix that describes kinetic constants for interconversion of various quasi-particles. The incoming photon of the incident beam is decomposed as a superposition of an upper and a lower polariton branch. Perturbation of the polaritons by molecular vibrations contributes to the reflection and transmission of light which is described by the scattering matrix. Next, in Sec. III, we describe the nature of the polaritons as arising from coupling between the electromagnetic field and Frenkel excitons in the monolayer. Next the elements of the scattering matrix are calculated. Finally, we calculate extinction and reflection spectra from the scattering matrix and compare the calculated with experimental spectra.

In order to model the optical properties of the monolayer, we imagine a layer of large but finite extent enclosed in a rectangular volume. Due to intermolecular interactions within the tightly packed layer, optical excitations in the monolayer are not localized on individual molecules but form delocalized Frenkel excitons. The 2D Frenkel excitons with small momentum (k ≈ 0) will interact with the electromagnetic modes in the 3D volume with the same in-plane momentum. This leads to formation of hybrid exciton-photon quasi-particles called polaritons. We note that the majority of the excitons has a momentum that is too large to interact directly with a photon and will not hybridize. These excitons we will refer to as “dark” excitons.

Perfect crystalline order with infinitely long range correlation is impossible in 2D structures at non-zero temperatures according to the Mermin-Wagner theorem.³⁸ The disorder in the monolayer implies a finite coherence length L_coh, for the exciton in the monolayer. Within the limited spatial range of coherence we then assume that there is a single exciton with k = 0 that couples to the electromagnetic field. The pair of interacting photon and exciton gives rise to two polaritons, one on the upper and one on the lower polariton branch (Fig. 2(b)). The splitting between the branches is determined by the strength of the interaction v between exciton and photon. The coupling energy is equal to the product of the electric field of the photon and the transition dipole moment of the exciton and the ground state. The magnitude of the transition dipole moment, μ, from the transition between the k = 0 exciton state and the ground state depends on the exciton coherence length, L_coh.²⁹ We take μ ∝ L_coh/a with a the average distance between neighboring molecules. In the calculations we treat the coherence length as an adjustable parameter. The coherence length sets the strength of the exciton-photon coupling, v.

A photon coming from outside the volume enclosing the monolayer will excite two polariton modes (Fig. 2(b)). The propagation of the polaritons will be perturbed by nuclear motion in the quinque-thiophene molecules, through the exciton-phonon coupling. Finally, transmission and reflection of light are determined by the coherent scattering of the polaritons. The scattering or S matrix can be used to describe the scattering of the polaritons and thus the optical properties of the monolayer.

In order to calculate the S matrix, we use the K-matrix formalism.^39–42 The K-matrix formalism was originally developed to compute transmission spectra of slow neutrons through matter. The neutrons can excite the atomic nuclei to unstable, intermediate states. The K-matrix method allows one to describe multiple overlapping resonances. Conceptually this problem from nuclear physics is similar to our case where the polariton can be in resonance with several exciton states containing either one, two, or even more phonons. In the K-matrix formalism, the S matrix is expressed as

where K is the reaction matrix operator. The unitarity of the S matrix is ensured by demanding K to be Hermitian. K can be related directly to parts of the Hamiltonian of the system

where V represents the operator describing the perturbation, in our case the coupling between electronic and vibrational excitation. Pv indicates that when taking the integral over energy E to calculate the magnitude of a matrix element, the principal value should be taken. Expression (2) for K can be iterated to yield a perturbation series for K.

If we now want to calculate transmission and reflection of radiation with frequency ω, we first need to choose a set of quantum mechanical basis states that describe the initial and possible final and intermediate states of the system. For transmission and reflection the basis should at least contain a state i describing the incoming radiation and a state r describing the reflected light. Given the complete basis, one can then calculate the K matrix which has a unique element for each possible pair of basis states describing the probability of interconversion between these two states. From the K matrix one can then compute the corresponding S matrix. Finally the probability for transmission is obtained by taking the square modulus of the diagonal element of the S matrix between state i and itself, i.e., |S_i,i(ω)|², i.e., minus the logarithm of transmission. The extinction spectrum is obtained from A(ω) = −¹⁰log(|S_i,i(ω)|²). The probability for reflection, or reflectance, is obtained by taking |S_i,r(ω)|². Because the S matrix is unitary, all the probabilities calculated are properly normalized and both absorption and reflection of the incoming light contribute to calculated extinction.

In Sec. III we investigate the lowest electronically excited states of the monolayer and their coupling to the electromagnetic field. The strategy we adopt in this computation is as follows. We assume a dipole oscillator model for the lowest optical transition on the individual chromophores. Then we take into account the excited state electronic interactions between the chromophoric groups in the monolayer. The energy and density of excitons states in the monolayer are calculated. This involves diagonalization of the Hamiltonian describing the interchromophoric coupling assuming transition dipole–transition dipole interaction. At this stage, we neglect the coupling of the excitation with nuclear vibrations and assume separability of nuclear and electronic motion (Born-Oppenheimer approximation, Simpson’s strong interchromophoric coupling case⁴³). Secondly, we take into account the coupling between photons and excitons, yielding polaritons. In Secs. IV-VI we take into account the coupling between the polaritons and nuclear vibrations in a perturbation approach. A rationale for this particular approach is that it allows us to introduce non-radiative transitions induced by the coupling of electronic and nuclear degrees of freedom. The non-radiative transitions are responsible for the energy dissipation in the system.

For simplicity, we assume here that the molecules in the monolayer pack in a square lattice with lattice constant a₀ = 0.45 nm. This value is obtained by taking the geometric mean √((ab)/2) of the lattice constants a and b determined from X-ray scattering on the monolayers,¹⁷ taking into account that the unit cell contains two molecules. From the transition dipole moment and the lattice constant we can calculate a characteristic energy for the dipole-dipole interaction between neighboring molecules, V₀ = μ²/4πε₀a₀³, with a characteristic order of magnitude of 10⁻¹ eV. Here we adopt the Coulomb gauge and calculate the exciton energies using the unretarded dipole-dipole interaction.⁴⁴ Radiation corrections will be included later. The dipolar coupling between transition dipoles of molecules m and n both oriented along the normal to the surface is given by

where the distance r between m and n is expressed in units of the lattice constant, a₀.

The energy of the exciton states resulting from the interacting molecular excited states can be obtained as the 2D cosine transform of the dipolar interaction

where k_x and k_y are the wavevectors characterizing the exciton. E_S1 is the energy of the excited state of the isolated molecule, here taken equal to 2.75 eV.

The relation (4) is illustrated in Fig. 3(a). Using Eq. (4), we can now numerically calculate excitonic energy levels for a large grid of 2000 × 2000 molecules, assuming periodic boundary conditions. The shape of the density of states is show in Fig. 3(b) as the black line. Excitons with momentum k≠0 cannot interact directly with the electromagnetic radiation due to the mismatch in momenta of the photon and the exciton. In the following discussion we will refer to the excitons with k≠0 as “dark” excitons as they can neither be generated directly by a photon nor decay directly via photon emission. We note that because the relevant transition dipole moments of the two molecules in the unit cell are parallel, there is only a single allowed optical transition from the ground state to the exciton state in which all the transition dipole moments are coupled exactly in phase. The energy of this (k_x = 0, k_y = 0) exciton is above that of the isolated molecule. This explains the experimentally observed blue shift of the absorption maximum for the monolayer to a photon energy of 3.5 eV. The absorption of the isolated molecules peaks near 2.75 eV. In order to match the energy of the highest exciton state with the experimentally determined energy of 3.5 eV, we need to set V₀ to 0.083 eV. This interaction energy is smaller than the energy calculated for interaction transition dipoles at a distance of a₀. It is well-known however that at short distances the dipole-dipole interaction overestimates the interchromophoric coupling between neighboring oligothiophene molecules. This is because the lateral distance between two neighboring oligothiophene molecules is much smaller than the length of the 5T chromophore and so the dipole-dipole approximation is inaccurate for calculating interaction energies. With V₀ equal to 0.083 eV we find exciton states based on the S₁ electronic excitation in the range from 2.5 to 3.5 eV. This range is consistent with advanced quantum chemical calculations of exciton energies in a layer of oligothiophene molecules.⁴⁵ In addition also the gas-to-crystal shift of the excited state energy of the individual molecule which arises from the more polarizable environment of the molecule in the crystal is a factor that can contribute to the exact position of the exciton levels. In brief, we treat the energy of the k = 0 exciton level as an adjustable parameter.

The next step in calculating the optical properties of the monolayer involves the coupling between the electromagnetic field of the incoming light and the k ≈ 0 exciton (see Fig. 2). This coupling leads to the formation of a new quasi-particle, the polariton. To obtain the strength of the coupling we consider the electric field ε of a photon with radial frequency ω confined to a volume V

A natural choice for the volume is to take the cube of the coherence length of the light, L_opt,

The light interacts with the total transition dipole moment of the excitons in the monolayer. Because the wavelength of light is much longer than the intermolecular distance only exciton states with k ≈ 0 interact with an interaction energy v equal to

where θ_i is the angle of incidence of the incoming light. The square of the total transition dipole moment of the exciton, μ², is proportional to the total number of molecules over which the exciton is coherently delocalized, N_coh. This is well-known for 1D aggregates⁴⁶ and also takes into account the normalization of the exciton state. The number N_coh determines the exciton coherence length, L_coh = a₀√N_coh. The square of the transition dipole moment can now be expressed as

Combining Eqs. (7) and (8) we then arrive at

We note that the Mermin-Wagner theorem,³⁵ which asserts that perfect crystalline order with infinitely long range correlation is impossible in two-dimensional structures at non-zero temperatures, also provides a compelling argument for introducing a finite coherence length for the exciton in the monolayer.

With the strength of the coupling v between photon and exciton with corresponding momentum known, we proceed by determining the eigenmodes of the coupled system. The eigenmodes or polaritons are obtained from diagonalizing the Hamiltonian for exciton-photon coupling

where λ denotes the energy of the polariton. Fig. 4 illustrates the polariton energies calculated from Eq. (10) as a function of photon energy. Importantly, as can be seen from Figs. 2(b) and 4, for a particular photon energy there are two polaritons, a polariton on the upper polariton branch with eigenvalue λ₁ and a polariton on the lower polariton branch corresponding to the second eigenvalue λ₂ of Eq. (10). With the polariton energies known, also the eigenvectors describing the composition of the polariton in terms of photon and exciton component can be obtained. We denote the square of the coefficient describing the exciton contribution to the polariton with energy E as a_n²(E). This coefficient varies from essentially zero for polaritons made up from mixing a photon and exciton with energy difference much larger than the coupling v, to a coefficient of √0.5 for a polariton that arises from a photon with energy exactly equal to the exciton energy.

The polaritons and dark excitons discussed above can be considered as quasi-particles that interact with molecular vibrations in the monolayer. In Secs. IV-VI we take into account the coupling between the polaritons, dark excitons, and phonons in a perturbation approach. Diagrams will be used to summarize the perturbation integrals and Fig. 5 introduces the symbols used in these diagrams. An open circle denotes a coupling between a dark exciton and a phonon, while a filled circle indicate the interaction between a polariton and a phonon.

The K matrix describes for the group of basis states chosen, the complete set of conversion coefficients of parallel reaction pathways. A first step in the calculation is to decide which basis states to include. For calculation of reflection and transmission we need to include the incoming and reflected polariton as basis states. These will be indicated by i and r, respectively, see Fig. 6. In addition, we need to include vibronic states of the monolayer comprising a dark exciton combined with vibrational excitations. In our description of the vibronic motion in the excited state, we only include high frequency intramolecular vibrations that couple strongly with the excitation. These modes appear in the absorption and emission spectra of the isolated molecule in frozen solution, see Fig. 1. We neglect intermolecular vibrational modes both of acoustic and optical kind, because these have generally low frequency in crystals of aromatic molecules (<0.03 eV).⁴⁷ We simplify the dynamics by taking into account only a single totally symmetric, intramolecular active mode with a frequency of 0.17 eV. Furthermore we neglect any interactions between the intramolecular vibrations on different molecules. This degeneracy allows one to combine the intramolecular vibrations into optical phonon modes⁴⁸ adapted to the translational symmetry of the monolayer. Due to momentum conservation, vibronic states can only interact with states whose combined momentum of dark exciton and vibrational excitation equals the momentum of the incoming polariton. The vibronic states with momentum equal to the incoming polariton will be indicated by e0, e1, …, en with the numeral n indicating the number of vibrational quanta.

When the exciton is uniformly delocalized over N_coh molecules, then in the strong interchromophoric coupling case, the exciton-phonon coupling constant γ scales as γ₀/√N_coh with γ₀ the coupling constant for an isolated molecule. In the manifold of vibronic states containing a dark exciton and a single vibrational quantum, for each dark exciton with momentum k_e there is exactly one delocalized vibrational state with momentum k_q such that k_e + k_q = k_p with k_p ≅ 0 indicating the momentum of the incoming polariton. The density of states e1 thus equals the density of dark exciton states D0. The density of states D0 is normalized such that its integral value equals N_coh. The density of states of type en with n > 1 is denoted as Dn, see Fig. 3. Dn is normalized such that its integral value ∫ Dn scales as (N_coh)ⁿ⁻¹∫ D0.

In a first minimalistic attempt we include dark excitons combined with just one quantum of vibration, e1. Later we will extend the range of the number of vibrational quanta from one up to four, as indicated by e1-e4. We neglect inelastic scattering processes of the polaritons. The K matrix can now be expressed as

Individual elements of the K matrix are calculated as^49,50

where ρ_i represents the density of states of type i. The density of states of the bare excitons is taken from the numerical evaluation of Eq. (4). The density is normalized such that its integral value over energy equals N_coh. The density of polariton states is set equal to unity. In order to take into account the number of independent interactions that a photon with coherence length L_opt can have with excitons with coherence length L_coh, we introduce an additional factor L_opt²/L_coh²cosθ_i in all matrix elements of K that involve interaction between a polariton and a dark exciton.

The matrix elements K_ij can be represented graphically in Feynman-like diagrams by realizing that V contains both an annihilation and creation operator for vibrations. Additionally, 1/(E − H₀) represents a propagator. If we limit the perturbation series in V to first order, we can calculate a non-trivial value for the element K_i,e1 of the ³K matrix. The Feynman diagram for K_i,e1 is depicted in Fig. 2(d). In first order, values for the elements K_i,i describing transmission, K_i,r describing reflection, and K_e1,e1 are trivially zero. Second order perturbation terms are needed to arrive at nonvanishing values for these matrix elements as illustrated by the Feynman type diagram for K_i,i in Fig. 2(c). The computation of these second order terms involves an integral over virtual, intermediate states. In Fig. 7 we illustrate the computation of the simplest possible matrix element K_e0,e0 in more detail The upper part of Fig. 7 describes evaluation of the second order contribution to the matrix element K_e0,e0. The dark exciton e0 can scatter with e1 as intermediate quantum state. In the calculation one need to integrate over all possible energies of the intermediate state. The complete expression for the matrix element reads

The integrand in (13) diverges for E₁ approaching E₀. Yet by taking the principle value of the integral, indicated by Pv, a finite value is obtained. For the calculation of the matrix element K_i,i, K_i,r, and K_i,e1, we need to take into account polaritons, dark excitons and phonons, and their coupling, see also Fig. 5 for an overview. We take the strength of the polariton-phonon coupling proportional to the coefficient describing the exciton character of the polariton, γ_{polariton−phonon} as γ a_n. Analytic expression for the matrix elements can be obtained analogously to Eq. (13) above. With a complete expression for the matrix ³K we can compute the corresponding ³S matrix using relation (1) and extinction and reflection can be computed as function of the frequency of the radiation incident on the monolayer. Results will be presented and discussed in Sec. V. In the remainder of this section we discuss the technical details of extending the K matrix to include also vibronic states with more than one quantum of vibration.

First we formally write down the elements of the K-matrix including vibronic states with up to four quanta of vibrational energy, e1-e4,

To get a non-zero value for, e.g., the matrix element K_i,e4, we need to include perturbation terms up to fourth order. The inclusion of such higher order terms introduces divergencies. This is illustrated in the middle part of Fig. 7 by considering the fourth order contribution to K_e0,e0. The algebraic expression of the fourth order contribution reads

In this fourth order contribution to K_e0,e0, the integral no longer converges. As is well known in perturbation theory, the integrals need to be renormalized in order to get finite values. In the lower part of Fig. 7, we illustrate our renormalization procedure in the calculation of K_e0,e0. We first introduce

The contributions from all possible order to K_e0,e0 can now be expressed as

The higher order contributions can be summed up in a Dyson series to yield a renormalized density of states $D ̃ 0$

This procedure can be generalized to dark exciton states with one and more phonons. In this way we obtain a recursive relation. The renormalized density of exciton states with n phonons, $D ˜ n$⁠, has a functional relation with the density of exciton states with n + 1 phonons, $D ˜ n+1$⁠, that can be expressed symbolically as

The set of recursive relations Eq. (19) is now applied as follows. In our computation of the matrix ⁶K we want to include exciton states with up to four phonons with density D4. Below in Fig. 8 we have listed all Feynman diagrams and corresponding convergent analytical expressions involving the renormalized density of states. As can be seen the highest order of the density that appears in these expressions is $D ̃ 5$⁠. We then take the bare density of states for dark excitons with 6 phonon, D6, and use this density to calculate a renormalized density $D ̃ 5$⁠. With this renormalized density of states we can then subsequently calculate $D ̃ 4, D ̃ 3, D ̃ 2, D ̃ 1$⁠, and $D ̃ 0$⁠. The renormalized densities are illustrated in Fig. 3(c). As can be seen, the renormalization induces a sort of “compartmentalization” in the energy dimension, where the overlap of the various densities of states in energy is strongly suppressed. Also a renormalized polariton density of states can be obtained in this way. In this contribution we neglect renormalization of the coupling constants γ.

In order to calculate transmission and reflection we first consider the incident beam at a distance far from the monolayer. Here photons can be described as the coherent superposition of a lower polariton and an upper polariton (see Fig. 2(b)). The K matrix calculated above can then be used to calculate how these two polaritons propagate through the monolayer. Once propagation of the two polaritons has been calculated, the probability for transmission of a photon can be obtained by calculating the photon component in the set of the two propagated polaritons. Incidentally, we note that in the case of a pulsed light source, the coherent excitation of two polariton modes could induce quantum beats in the transmission that are well-known experimental characteristics of polaritons.⁵¹ 

The propagation of the polaritons is perturbed by the phonons in the molecular layer. The coupling of electronic and nuclear motion with strength γ can transform a polariton with k ≈ 0 into a pair of a dark exciton with in plane momentum k_e and a phonon with momentum k_q. Conservation of momentum requires k_e + k_q to be equal to zero. This scattering process is illustrated in a Feynman-type diagram in Fig. 2(d). The matrix element associated with the diagram of 2d describes in first order the conversion of the polariton into a dark exciton plus phonon. Fig. 2(c) shows a second order diagram that is important in calculating the transmission of the polariton. Here the incoming polariton is converted into a dark exciton plus phonon that recombine again into a polariton.

Fig. 9 shows calculated p-polarized extinction and reflection spectra for the monolayer using the K matrix. In the calculation, we assume a single active molecular vibration with frequency of 0.17 eV. The transition dipole moment of the allowed optical transition in the individual quinque-thiophene molecule is oriented along its long axis.²⁰ The quinque-thiophene molecules in the monolayer are oriented parallel to the normal of the layer. In the arrangement of cofacially packed chromophoric groups only the transition between the ground state and the delocalized excited state with all constituent molecule transition dipole combined perfectly in phase is optically allowed. The latter excited level is the Frenkel exciton with k = 0. The total transition dipole moment for the transition between the ground state and the k = 0 Frenkel level is oriented perpendicular to the plane. Adopting the dipole approximation for the excited state intermolecular coupling, the k = 0 exciton occurs at the highest energy within the band of exciton states, at 3.5 eV.

The calculated spectra can be compared to the experimental data for the quinque-thiophene monolayer shown in Figs. 9(e) and 9(f). The monolayers show an extinction band with maximum at 3.5 eV. The maximum extinction increases with increasing angle of incidence, yet the band shape of the extinction does not vary with the degree of surface coverage in samples with incomplete monolayers.^19,52 At 4.6 eV another electronic transition contributes to the spectrum with polarization in the plane of the monolayer. This high energy transition will not be analyzed further here.

Calculations based on minimal ³K and associated ³S matrix are depicted in Figs. 9(a) and 9(b). Results for more extended ⁶K matrix from Eq. (14) are shown in Figs. 9(c) and 9(d). Experimental extinction and reflection data for the monolayer are shown in Figs. 9(e) and 9(f). The calculations reproduce the main band observed in the extinction spectra at 3.5 eV. The experimental extinction spectra show a second electronic transition around 4.6 eV which is derived from an electronic transition different from the S₀–S₁ under consideration here. The calculated spectra therefore do not reproduce this second band at 4.6 eV. Predictions based on the ³S and ⁶S matrix are quite similar. The band predicted by the ⁶S matrix is somewhat broader due of the presence of vibronic side bands. Because in the ⁶S matrix a larger number of possible vibronic states are included, the additional side bands must be related to resonance of the incoming polariton with states of the type e3 or e4 acting as either intermediate or final state. When taking L_coh = 0.83 nm, the calculations for an angle of incidence θ_i = 45° predict a maximum extinction of 0.045 using the ³S matrix and 0.047 with ⁶S. These values are about a factor of 4 higher than the experimental value. If we however compare the value for the extinction integrated over the spectral range corresponding to the S₀–S₁ electronic transition, we find a closer correspondence. The experiment yields 0.91 eV while ³S and ⁶S matrix calculations yield 0.71 and 0.73 eV, respectively, with a coherence length L_coh = 0.83 nm, see Fig. 10. The dependence of the integrated extinction on the angle of incidence is also shown in Fig. 10. The calculations predict a decrease in the integrated extinction with decreasing angle of incidence, in qualitative agreement with the experimental data.

Comparing the band shape of the calculated and experimental extinction spectra, we note that the calculations do not fully account for the bandwidth in the experimental data. Furthermore, although the calculated band shape shows some asymmetry tailing towards the low frequency, the asymmetry in the experimental spectra is more pronounced. In Fig. S2 of the supplementary material⁵³ we show that by taking into account statistical fluctuations in the intermolecular excited state interaction energies V due to, e.g., low frequency intermolecular vibrational motion, the agreement between predicted and observed band shape can be improved. In addition, this mechanism can provide a microscopic description of the limited coherence length of the excitons, see Fig. S1.⁵³ 

For the monolayer reflection spectra were measured using a Wvase instrument from Woollam. For small angles of incidence, the reflection from the monolayer is below our detection limit. At a high angle of incidence, θ_i = 85°, a weak p-polarized band can be detected, see Fig. 9(f). When subtracting a baseline from the measured reflectance spectra to account for reflection by the substrate, we find a reflectance at the maximum of the reflectance band at 3.5 eV of only a few percent.

The calculations correctly predict a reflection band at 3.5 eV, see Figs. 9(b), 9(d), and 9(f). The maximum reflectance is about an order of magnitude lower than the experimental value. The width predicted for the reflection band is too narrow in comparison with the experiment. Predictions based on the ⁶S matrix yield additional reflection bands related to the vibronic side bands that occur in the extinction spectra. Here these additional bands result from vibronic states of the type e3 and e4 acting as intermediate state in the reflection process. As also mentioned gain calculations using the ³S or ⁶S matrix predict that reflectance decreases with decreasing angle of incidence. This is in qualitative agreement with the experiment where only for large angles of incidence, θ_i ≥ 80° significant reflection could be detected. It should be noted that once the exciton coherence length is fixed, the calculation of the reflectance does not involve any adjustable parameters. Using the ³S matrix the reflectance is predicted to slightly decrease with increasing coherence length while the extinction increases. For the ⁶S matrix, the calculated reflectance rapidly increases, approaching the limiting value of unity in the 3–4 eV spectral range for L_coh > 2 nm. Correspondingly, also the extinction in the 3–4 eV range calculated from the ⁶S matrix rises sharply with increasing coherence length. The difference in the predictions based on the first order ³S matrix and the fourth order ⁶S matrix together with their dependencies on coherence length indicate that the higher reflectance for the higher order approximation and longer coherence length results from a larger number of vibronic states close in energy to the k ≅ 0 exciton level that can act as intermediate states in the reflection process. In analogy to the Breit-Wigner relation⁵⁴ from nuclear physics, the large width of the reflectance band for high coherence length might be interpreted in terms of short lifetime of the polariton state induced by the large number of available channels. We note that experiment yields typically narrow bands for the spectra of J-aggregates, while in contrast spectra of H-aggregates show considerably broader bands.^29,55 Also here the much larger density of dark states near the k ≅ 0 exciton in the H-aggregates compared to their counterpart J-type molecular assemblies could provide an interpretation for the experimental findings.⁵⁶ 

In summary, by comparing extinction and reflection spectra using the K-matrix formalism with experimental data, we find the scattering method can account on a semi-quantitative for the experimentally observed band positions and intensities assuming a coherence length for the exciton on the order of a nanometer. Considering the fact that for the coherence length assumed, the number of chromophores accommodating the electronic excitation is roughly on the order of 10, one could argue that there may still be considerable variation in the energy of the k = 0 exciton. This could induce additional inhomogeneous broadening of the bands in the extinction and reflection spectra. We note that in principle the K-matrix method introduced here might be extended to take into account randomly distributed, localized exciton states. The K-matrix method can be augmented to cover also multiple scattering of light between different regions in the monolayer film.

We demonstrate that the K-matrix formalism, developed originally to describe neutron scattering spectra in nuclear physics, can be used to calculate reflection, absorption, and extinction of light by 2D self-assembled monolayers of molecules containing quinque-thiophene chromophoric groups. This perturbation approach to obtain the scattering matrix for the monolayer takes into account both exciton-photon and exciton-phonon coupling and allows one to go beyond molecular exciton theory. Transmission and reflection can be calculated directly without the need of introducing a bulk response function such as a dielectric constant or refractive index. In this way, no additional boundary condition (ABC) for the electrical polarization of the molecular material needs to be specified. Thus, in contrast to 3D systems, where the boundary conditions for the polariton and momentum of the electromagnetic waves inside the solid are troublesome issues, the spectroscopic properties of 2D layers can be modeled in a straightforward manner adopting a system of dipole oscillators.

The calculations on the monolayer predict a reflection band whose intensity depends on the coherence length of the delocalized electronic excitation in the monolayer. Comparison of the predictions and the experimental observations indicate a coherence length for the electronic excitation on the order of 1 nm. Finally, we note that in analogy to nuclear physics and the Breit-Wigner relation,⁵⁰ the spectral width of the extinction band of the oligothiophene based monolayer can be interpreted as the lifetime of the intermediate polariton state. This lifetime is influenced by non-radiative relaxation processes and decreases with increasing coherence length due to enhanced probability of dark exciton formation and associated phonon emission. Whereas for J aggregates the width of the absorption band narrows with increasing delocalization,^29,53 for 2D H-type aggregates, the width of the extinction band increases with increasing coherence length.⁵⁷ 

In this appendix we investigate the limit of the lattice constant going to infinity. In this limit, assuming dipole-dipole interactions between the molecules, intermolecular interactions can be neglected. Furthermore the coherence length of the excitations can be restricted to one molecule. The optical coherence length of the light may be taken as smaller than the lattice constant. Under these conditions the K-matrix formalism for the monolayer should reduce to the well-known case of absorption of light by isolated molecules.

For L_coh = a₀ the strength of the exciton photon coupling given by Eq. (9) reduces to

The interaction between an incoming photon and an isolated molecule is very weak. In such a case the eigenvalue problem of finding the polariton states can be treated perturbatively. The coefficient describing the quantum mechanical admixture of pure excited state with energy E_exc into a perturbed photon state with energy E_p up to first order in v reads

The perturbed photon state can now be expressed as

where |i〉₀ and |exc〉₀ denote the zero order photon and molecular exciton states, respectively. The zero order states can be expressed as a product of the electromagnetic field (with $1 ¯$ or $0 ¯$ photons), the electrons on the molecules (either in the singlet electronic ground state S₀ or in the lowest electronic excited singlet state S₁), and the 0.17 eV nuclear vibration on the molecule (here with $0 ̄$ quanta of vibrational energy).

In analogy to the traditional treatment of optical absorption by dispersed molecules or atoms, we neglect all intramolecular interconversion processes between various vibronic states. Also re-emission of photons is neglected. With these simplifications, the K matrix can now be expressed as

Assuming non overlapping vibronic bands, the T matrix defined as

Individual elements of the K matrix are calculated as^58,59

where ρ_i represents the density of states of type i. These densities can be delta functions. Because the vibrations are now also localized on the isolated molecules, all the densities of vibronic states have the same integral value. The concentration c of the molecules can be included in the normalization of these densities

Hamiltonian describing the nuclear vibrational motion in the S₁ excited state in units of hν_vib has the form

with H⁰ describing the vibrational motion in the ground state

and with V representing the perturbation of the vibrational motion by the electronic excitation

where a and a⁺ denote the ladder operators.

The elements of the K matrix can now be worked out

Similarly

If we know that the square of these K matrix elements, we get

In this expression, ρ symbolized the liner dependence on the intensity of the incoming light, c the dependence on concentration, and v² contains the dependence on the square of the transition dipole moment. The remaining factors can be recognized as a part of the Huang-Rhys factor S_n describing the relative intensity of the 0-n vibronic transition in absorption based on a totally symmetric vibrational mode

Obviously the exponential factor e^−γ2 is still missing in Eq. (A14) when compared to (A14). Furthermore the next element K_i,e3 is not of the form (A15)

Finally, the matrix element K_i.e0 cannot yet be calculated because the related expressions are ill-defined. For instance, in first order we have

The matrix element of the ladder operators gives zero. However, the first in (A17) diverges for energies of the incoming photon approaching the energy of the exciton state e0, prohibiting a numerical evaluation of the matrix element.

To remedy these deficiencies we need to apply renormalization. We follow the same procedure as outline in the main text. Using Eq. (18) and assuming discrete vibronic levels for the isolated molecules, we get for the case γ ∼ 1

With these renormalized densities we get

This expression is indeed conform Eq. (18) from the main manuscript. Finally to repair the other deficiencies we need to consider the renormalization of the pure exciton state that is involved in the calculation of the exciton–phonon interaction.

We first calculate the self-energies of the exciton state and use these to correct the energy of the exciton state, see also the Feynman type diagrams in Fig. 11. In first order, the self-energy of the exciton state vanishes

The second order correction reads

With this correction of the exciton energy, the divergence of (A17) for E_i approaching E_e0 is removed because E_Exc no longer equals E_e0. It then follows that in first order K_i,e0 vanishes, at variance with (A15). The second order contribution to K_i,e0 incorporating the renormalized densities $D ˜ 0$ and $D ˜ 1$ reads

Finally, we need to renormalize the probability density of exciton state involved in the calculation of v. We note that for perturbation up to order n we have

where adopting the crude Born-Oppenheimer approximation, |S₁〉 denotes the electronic wavefunction of the S₁ excited state and $n ̄$ the vibrational wavefunction with n quanta. In order to ensure proper normalization of the exciton state we need to introduce a normalization factor N_norm

Inclusion of the normalization factor N_norm in the exciton-photon interaction energy v introduces the exponential factor in the expressions for the intensities, in full agreement with the Huang-Rhys expressions S_n.

Using the ⁶S matrix we have evaluated the predicted optical response of the monolayer in the limit of a long exciton coherence length. Results of the numerical calculations are shown in Fig. 12. We find that for long coherence length, the reflectivity approaches unit near the resonance. Exactly at resonance, the reflection spectra show a dip for long coherence length. Such dips have also been found experimentally in the reflection spectra of molecular crystals and can be accounted for taking into account polaritons.^21,60

In summary, the prediction from the K matrix method for the reduced transmission of light due to absorption for a film of dispersed, non-interacting molecules are consistent with the absorption spectrum for dilute solutions include a vibronic progression with intensities distributed according to the Huang-Rhys expression.