Antonios M Alvertis

Raj Pandya

Claudio Quarti

Laurent Legrand

Thierry Barisien

Bartomeu Monserrat

Andrew J Musser

Akshay Rao

Alex W Chin

David Beljonne

First principles modeling of exciton-polaritons in polydiacetylene chains

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Excitons are bound electron-hole pairs that appear in semiconductors, and are critical to the optoelectronic response of these materials. These quasiparticles are connected to a finite oscillating polarization that emits an electromagnetic wave, which can act on an incident wave, leading to a light-matter interplay. Once a photon is close to resonance with an exciton, light-matter coupling can become strong, and the two cannot be treated independently, as it was first realized in the pioneering works of Pekar [START_REF] Pekar | The Theory of Electromagnetic Waves in a Crystal in which Excitons Are Produced[END_REF][START_REF] Pekar | Identification of excitons with light waves in a crystal[END_REF] and Hopfield 3 more than six decades ago. This coupling between polarization and photons is captured by introducing the quanta of a new quasiparticle referred to as a polariton, which describes the propagation of electromagnetic energy in matter [START_REF] Kalt | Semiconductor Optics[END_REF] .

Organic semiconductors are a class of materials the lightmatter interactions of which are relevant to a number of promising optoelectronic applications, making the understanding of their polariton properties interesting from a technological point of view. Their applications include lightemitting diodes (LEDs) [START_REF] Reineke | White organic light-emitting diodes with fluorescent tube efficiency[END_REF] , solar cells [START_REF] Meng | Organic and solution-processed tandem solar cells with 17.3% efficiency[END_REF] and room temperature polariton lasers [START_REF] Strashko | Organic Polariton Lasing and the Weak to Strong Coupling Crossover[END_REF][START_REF] Kéna-Cohen | Room-temperature polariton lasing in an organic single-crystal microcavity[END_REF] . Excitons in organic semiconductors are typically strongly bound [START_REF] Knupfer | Exciton binding energies in organic semiconductors[END_REF][START_REF] Köhler | Triplet states in organic semiconductors[END_REF] Frenkel-like excitons [START_REF] Beljonne | Charge-transfer excitations steer the davydov splitting and mediate singlet exciton fission in pentacene[END_REF][START_REF] Berkelbach | Microscopic theory of singlet exciton fission. II. Application to pentacene dimers and the role of superexchange[END_REF][START_REF] Shane R Yost | A transferable model for singlet-fission kinetics[END_REF][START_REF] Coto | Low-lying electronic excited states of pentacene oligomers: A comparative electronic structure study in the context of singlet fission[END_REF] , and their dispersion can be considered to be approximately flat [START_REF] Litinskaya | Exciton-polaritons in organic microcavities[END_REF] compared to the strong dispersion of light, as visualized in Figure 1. For an incident photon close to resonance with an a) Electronic mail: ama80@cam.ac.uk b) Electronic mail: David.BELJONNE@umons.ac.be exciton state of an organic semiconductor, strong coupling leads to two new polariton states that are referred to as upper polariton (UP) and lower polariton (LP) respectively. As depicted in Figure 1, this results in a stop-band (shaded region), where no light propagation takes place within the material. The width ∆ of the stop-band depends on the magnitude of the exciton-photon interaction. The stop-band is also related to, but is distinct from, the so-called Rabi splitting that is commonly used to quantify light-matter interactions, and which is defined as the maximal vertical distance between the upper and lower polariton branches [START_REF] Lidzey | Strong exciton-photon coupling in an organic semiconductor microcavity[END_REF] . Polaritons in organic semiconductors have often been studied for molecular systems placed between highly reflecting mirrors that confine incident light [START_REF] Kéna-Cohen | Room-temperature polariton lasing in an organic single-crystal microcavity[END_REF][START_REF] Polak | Manipulating molecules with strong coupling: Harvesting triplet excitons in organic exciton microcavities[END_REF][START_REF] Agranovich | Cavity polaritons in microcavities containing disordered organic semiconductors[END_REF][START_REF] Canaguier-Durand | Non-Markovian polariton dynamics in organic strong coupling[END_REF][START_REF] Ebbesen | Hybrid Light-Matter States in a Molecular and Material Science Perspective[END_REF] . This microcavity effect leads to a standing electromagnetic wave that interacts with the material system. However, here we focus on the case of unconfined photons, and not on such microcavity polaritons.

The building block of organic semiconductors is often organic molecules, the large size and weak intermolecular van der Waals interactions of which lead to phonon modes with low frequencies, hence strongly activated at room temperature. These slow nuclear motions lead to large variations of the electronic interactions between neighboring molecules [START_REF] Troisi | Electronic interactions and thermal disorder in molecular crystals containing cofacial pentacene units[END_REF][START_REF] Troisi | Dynamics of the intermolecular transfer integral in crystalline organic semiconductors[END_REF] . Since excitons can spatially extend over several molecules [START_REF] Cudazzo | Exciton dispersion in molecular solids[END_REF][START_REF] Refaely-Abramson | Origins of Singlet Fission in Solid Pentacene from an ab initio Green's Function Approach[END_REF] , such low-energy phonons can have a strong influence on their properties [START_REF] Antonios | The impact of exciton delocalization on exciton-vibration interactions in organic semiconductors[END_REF] . Similarly, organic semiconductors based on conjugated polymers exhibit very lowfrequency phonon modes that are strongly coupled to their excitons [START_REF] Mahrt | Light and heavy excitonic polarons in conjugated polymers[END_REF] . However, it is not only vibrations of low energies that couple strongly to excitons, as in organic systems highfrequency motions such as carbon-carbon stretches have often The dispersion of a photon in a material intersects with that of an exciton, and their interaction leads to two new hybrid polariton states. The upper and lower polaritons are separated by an energy gap of width ∆ (shaded region), which is referred to as the stop-band.

been discussed to dominate this interaction [START_REF] Frank | The spectral signatures of frenkel polarons in H-And J-aggregates[END_REF][START_REF] Yamagata | Vibronic coupling in quantum wires: Applications to polydiacetylene[END_REF] . This strong coupling of excitons to the vibrations of organic semiconductors is expected to also have an impact on exciton-photon coupling, and indeed it has been shown that vibrations have a strong effect on the dynamics of polariton states [START_REF] Somaschi | Ultrafast polariton population build-up mediated by molecular phonons in organic microcavities[END_REF][START_REF] Wu | When polarons meet polaritons: Exciton-vibration interactions in organic molecules strongly coupled to confined light fields[END_REF][START_REF] Coles | Vibrationally assisted polaritonrelaxation processes in strongly coupled organic-semiconductor microcavities[END_REF][START_REF] Del Pino | Tensor network simulation of polaronpolaritons in organic microcavities[END_REF][START_REF] Del Pino | Tensor network simulation of non-Markovian dynamics in organic polaritons[END_REF] . Therefore, understanding the properties of polaritons requires accounting not only for exciton-photon, but also exciton-vibration interactions, through a Hamiltonian operator of the form:

H = H exc-ph + H exc-vib , (1)
where the operator H exc-vib describes the exciton-vibration interactions. For a single exciton state with energy ω exc , coupled to a photon of energy ω ph , the exciton-photon Hamiltonian can be written within the so-called resonance approximation [START_REF] Haug | Quantum Theory of the Optical and Electronic Properties of Semiconductors[END_REF] :

H exc-ph = ω ph â † â + ω exc ĉ † ĉ -ig( â ĉ † -â † c). (2)
Here a † (a) is the creation (annihilation) operator of the photon state, and c † (c) the creation (annihilation) operator of the exciton. The constant g quantifies the magnitude of the exciton-photon interaction and is related to the width of the stop-band ∆ of Figure 1 via the relationship: g = 1 4

ω ph ∆. While first principles approaches have been used in several cases to parametrize the exciton-vibration interactions encoded in H exc-vib [START_REF] Florian | Tensor network simulation of multienvironmental open quantum dynamics via machine learning and entanglement renormalisation[END_REF][START_REF] Antonios | Nonequilibrium relaxation of hot states in organic semiconductors: Impact of mode-selective excitation on charge transfer[END_REF] , the same cannot be said for the excitonphoton Hamiltonian of equation 2. To the best of our knowledge, a comprehensive ab initio method for modeling the stopband of organic systems is thus far missing. Naturally, experiment can inform computational models on the range of values of ∆, however this does not allow for a truly predictive approach of modeling light-matter interactions in these systems. It also remains unclear how vibrational effects affect the stopband.

Here we develop a theoretical framework for simulating exciton-photon interactions in organic systems by combining first principles calculations for excitons with the macroscopic theory of polaritons from classical electrodynamics. We thus extract the stop-band width ∆ that appears in the microscopic Hamiltonian 2. We exemplify our approach by studying variants of the conjugated polymer polydiacetylene (PDA) (Figure 2), and we show that the conjugation length of the polymer chains has a large impact on the magnitude of excitonphoton interactions. Exciton properties of PDA are obtained from first principles by using accurate Green's function based methods (GW -BSE) [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF][START_REF] Rohlfing | Electron-hole excitations in semiconductors and insulators[END_REF][START_REF] Rohlfing | Electron-hole excitations and optical spectra from first principles[END_REF] , which are known to be appropriate for the study of conjugated polymers [START_REF] Rohlfing | Optical Excitations in Conjugated Polymers[END_REF] , due to successfully capturing the large intrinsic correlation length of the electrons and holes that participate in exciton states. We compare the simulated results to experimental reflectivity measurements of the stop-band width, and we find the two to be in reasonable agreement. However, by only considering the effect of the main exciton peak on incident photons, the internal structure of the stop-band as it appears in reflectivity measurements is not reproduced in the calculation. By coupling GW -BSE calculations for the excitons, to non-perturbative finite differences methods for the vibrations [START_REF] Kunc | Ab initio force constants of GaAs: A new approach to calculation of phonons and dielectric properties[END_REF][START_REF] Monserrat | Electron -phonon coupling from finite differences[END_REF] , we show that phonon effects have an important effect on the internal stop-band structure.

The structure of this paper is as follows. Section II provides an overview of our theoretical approach for describing light-matter interactions. We apply our methodology to disorder-free PDA in section III. An overview of the studied systems is also provided in this section, alongside a discussion of their exciton properties and the experimental determination of the magnitude of light-matter interactions. We then proceed to discuss the role of static disorder in the sense of limiting the infinite conjugation length of disorder-free chains, in section IV. The impact of dynamic disorder (vibrations) on light-matter interactions is discussed in section V, along with a detailed outline of the employed methodology for accounting for its effect. We summarize the entirety of our results and discuss potential future avenues in the Conclusions of section VI.

II. THEORETICAL BACKGROUND

Consider an electromagnetic wave E, B interacting with a material of dielectric ε(ω). The wave equation for the electric field in frequency space reads:

∇ 2 E(r, ω) + ω 2 c 2 ε(ω)E(r, ω) = 0. ( 3
)
For the moment, and for the sake of simplicity, let us consider a bulk isotropic material, for which the electric field can be treated as a plane wave:

E(r, ω) = E • e i(k•r-ωt) (4)
By substituting equation 4 into 3 we obtain:

[-k 2 + ω 2 c 2 ε(ω)]E • e i(k•r-ωt) = 0. ( 5
)
The transverse eigenmodes of the system satisfy the condition E = 0, therefore:

c 2 k 2 = ω 2 ε(ω) (6)
It thus becomes obvious that if one obtains the dielectric response ε(ω) of a material, the dispersion of the light field in the material, i.e. the polariton dispersion, can be obtained by solving equation 6. This method is agnostic to the level at which the electronic structure is computed, and our GW -BSE methodology provides but one possibility.

Let us now consider the so-called resonance approximation to equation 6, as a useful way of gaining physical intuition on the origins of the stop-band. However, we emphasize that we do not make use of this approximation to obtain the simulated results presented in this paper, and it only serves as a way of developing intuition. Unless we are at resonance with an excited state ω = ω o , the imaginary part of the dielectric response is negligible, therefore equation 6 can be approximated as:

c 2 k 2 = ω 2 Re(ε(ω)). ( 7
)
Equation 7 shows that there will only exist real solutions for the wavevector k once Re(ε(ω)) is positive. While this is generally the case, Re(ε(ω)) can become negative in the vicinity of an optical transition i with a high oscillator strength f i . This can be seen within a Lorentz oscillator model of the dielectric response [START_REF] David | Classical electrodynamics[END_REF] :

ε(ω) = ε 0 + Ne 2 m ∑ i f i ω 2 i -ω 2 -iωγ i . ( 8
)
The real part can be written as:

Re(ε(ω)) = ε 0 + Ne 2 m ∑ i f i ω 2 i -ω 2 (ω 2 i -ω 2 ) 2 + ω 2 γ 2 i ( 9
)
where ε 0 is the background dielectric response, N the number of molecules per unit volume, m the electron mass and γ i the dephasing rate. In the vicinity of a transition i, ω = ω i + δ ω, and only keeping terms in first order of δ ω we obtain the condition for Re(ε(ω)) < 0 to be:

f i > (1 + ω i 2δ ω ) • γ 2 i ω 2 p , (10)
where ω 2 p = Ne 2 ε 0 m the plasma frequency of a material. The width of the frequency region over which condition 10 holds and thus Re(ε(ω)) < 0, defines the stop-band width ∆.

Therefore, the brighter an optical transition, the more likely it is that it will lead to a forbidden region for light propagation in the material. Since the oscillator strength of a material satisfies the sum rule:

∑ i f i = Z, (11)
with Z the number of electrons per molecule, we conclude that materials with fewer bright optical transitions are more likely to satisfy condition 10 and lead to strong light-matter interactions. This provides the motivation for studying extended one-dimensional systems like PDA, as elaborated in the following section. Before proceeding to the results section, we would like to highlight two points regarding the range of applicability of the presented methodology. First of all, this has been discussed in the context of a general Lorentz oscillator, and the frequencies ω i appearing in equation 8 need not correspond to exciton states. While here we focus on exciton-polaritons, optical phonons and plasmons can also form hybrid states with light [START_REF] Dai | Tunable phonon polaritons in atomically thin van der waals crystals of boron nitride[END_REF][START_REF] Zayats | Nanooptics of surface plasmon polaritons[END_REF] , and the presented theoretical framework could be used to describe such phenomena. Secondly, the polariton properties we calculate correspond to the bulk of an infinite material structure. However, these states can also form at the surface of a material, only propagating along the interface between different media. These states are referred to as surface polaritons, and have a finite dispersion within the stop-band of the infinite 'bulk' polariton states that we study here [START_REF] Lagois | Chapter 2 -Surface Exciton Polaritons from an Experimental Viewpoint[END_REF] . Our theoretical framework does not capture such surface effects, but could be extended in that direction in the future.

III. APPLICATION TO DISORDER-FREE PDA

Polydiacetylene (PDA) is a conjugated polymer with an alternating sequence of single-double-single-triple carboncarbon bonds, as visualized in Figure 2. These materials are known to give very high-quality crystals, with an almost perfect alignment of the polymer chains parallel to each other and free from conformational defects, due to the topochemical reaction they originate from [START_REF] Schott | Chapter III-1 -Basic Structural and Electronic Properties of Polydiacetylenes[END_REF] . Variants of PDA are traditionally named after the color of the transmitted light through a film of the material. Here we study two PDA variants shown in Figure 3, alongside their absorption and reflection spectra. We refer to the structure of Figure 3a that absorbs at lower energies as 'blue' PDA [START_REF] Spinat | Determination Structurale a 205 K du Poly { bis [( pbromophenylcarbamoyloxy ) -4 butyl ] -1 , 2 butene-1 yne-3 ylene }, ( C 26 H 26 Br 2 N 204 ) n[END_REF] and the one of Figure 3b that absorbs at higher energies as 'red' [START_REF] Enkelmann | Poly[1,2-bis(diphenylaminomethyl)-1buten-3-ynylene[END_REF][START_REF] Barisien | Exciton spectroscopy of red polydiacetylene chains in single crystals[END_REF] . It is important to note that the two types of PDA chains and their optical responses should in no way be associated to varying levels of disorder in their structures but are largely determined by their backbone geometries (see below) [START_REF] Schott | The colors of polydiacetylenes: A commentary[END_REF] .

We determine the stop-band of the two PDA variants by measuring their reflectivity spectra presented in Figure 3 measurements are given in Appendix A. If we were to disregard the absorption from the exciton over a finite region of frequencies, then every incident photon with an energy within the stop-band region would be reflected, giving a reflectance of R = 1 [START_REF] Michael | Optical Reflection Spectroscopy of Organic Solids[END_REF] , since no propagation of light can take place within the forbidden region. However, the finite damping γ that appears in the dielectric function of equation 8 due to the absorption of an exciton, leads to weaker reflectance and a 'gap' within the forbidden region of the stop-band [START_REF] Kalt | Semiconductor Optics[END_REF][START_REF] Stefan | Reflection of light by anisotropic molecular crystals including exciton-polaritons and spatial dispersion[END_REF] (as also seen in Figure 7b for a model PDA system). Further gaps to the high reflectivity of the stop-band can appear due to disorder or additional bright exciton peaks. Therefore, it is not always the case that one can identify the stop-band as a region of high reflectance, and its internal structure can be rich. This is the case with the reflectivity spectra of Figure 3. Nevertheless, we can still identify the width of the stop-band as the distance of two peaks in the reflectivity spectra: the one which appears on the low-energy side of the dip due to the absorption from an exciton state, and the one for which the reflectance reaches values at least as large as the maximum of the first peak. This way of determining the stop-band is similar to previous work [START_REF] Stefan | Reflection of light by anisotropic molecular crystals including exciton-polaritons and spatial dispersion[END_REF] , and strong light-matter coupling in squarylium [START_REF] Tristani-Kendra | Strong coupling in the optical spectra of polymorphs of a squarylium dye[END_REF] provides a particularly good example of using the reflectivity measurements to that end.

Our reflectivity measurements show that the stop-band width is 2.64 eV and 1.16 eV for the blue and red chains respectively. We discuss the origin of this large difference between the red and blue chains with the aid of simulations below. The measured values are within the expected range, as the stop-band width has been shown to surpass values of 1 -2 eV in several organic materials [START_REF] Stefan | Reflection of light by anisotropic molecular crystals including exciton-polaritons and spatial dispersion[END_REF] . In particular, it has been shown that for near-normal incidence of light on a lattice of aligned dipoles, values of this order of magnitude might be obtained [START_REF] Stefan | Reflection of light by anisotropic molecular crystals including exciton-polaritons and spatial dispersion[END_REF] . Given the close-to-perfect alignment of the polymer chains in PDA, as well as the large oscillator strength of their lowest bright exciton state, it is not surprising that lightmatter coupling is strong.

Let us now examine some of the details of the experimental reflectivity spectra. For the blue chains, the reflectance dips around the energy of the bright singlet exciton state at 1.97 eV, and a vibronic peak is also weakly resolved at 2.15 eV. Fur-thermore, a small dip appears at 1.85 eV, and this has been discussed to be due to minority chain populations with different geometries, which are present within the samples [START_REF] Schott | The colors of polydiacetylenes: A commentary[END_REF] . Additional structure appears at energies of 2.72 eV and 3.06 eV. It is currently not clear what these peaks correspond to, however the side-chains that are connected to the conjugated PDA structure can host exciton states of their own, potentially leading to some of the observed features. At higher energies, the reflectance shows an increase at approximately 3.5 eV and then decreases again. This is due to exciton states of unpolymerized diacetylene monomers which are known to appear in this energy range [START_REF] Schott | The colors of polydiacetylenes: A commentary[END_REF] . Finally, there is a stark increase of the reflectance as photon energies approach 4.4 eV, which is then followed by a continuous drop in its value, behavior characteristic of the stop-band edge. For the red PDA, a large dip in the reflectance appears at 2.48 eV where the bright exciton state of this structure absorbs, and a vibronic peak is then present at 2.68 eV. The reflectance then increases steadily until reaching the edge of the stop-band. On the low-energy side of the main exciton peak, some additional structure is present, and similar to the case of blue PDA, we attribute this to minority chain populations.

The PDA variants of Figure 3 contain a very large number of atoms in their unit cells, due to the size of the side-chains connected to the conjugated structure (highlighted with a red rectangle in Figure 3). In order to understand the role of the side-chains, we resort to studying two simplified model PDA systems, where the side-chains have been substituted by hydrogen atoms, as visualized in Figure 4. The small size of the model systems has the advantage of making the computational study of exciton-phonon interactions feasible, a fact that we return to later. When viewing the model PDA chains sideways as on the left of Figure 4 these systems look identical. However, by looking into the direction along which the chains extend, it becomes clear that the model red structure is not planar, and there exists a finite torsional angle of 14 o between subsequent diacetylene monomers. Overall, since the interatomic distances within the polymer backbone of the four studied PDA systems are taken to be identical, the differences in their optoelectronic properties are purely due to the effects of the side-groups and the (lack of) torsion between subsequent diacetylene monomers.

The effect of this torsion has been discussed in detail previously [START_REF] Sébastien Filhol | Polymorphs and colors of polydiacetylenes: A first principles study[END_REF] , and shown to lead to significant differences in the optoelectronic response of the two systems. In crystalline PDAs the side groups, through their mutual organization, pilot the chain conformation that in turn governs the electronic properties. However, the side groups themselves do not participate to the conjugated system, justifying the substitution by hydrogen atoms. Nevertheless, this simplification leads to a renormalization of exciton energies. This is due to the the strong dielectric mismatch introduced between the conjugated chain and the outside medium, as discussed below.

The absorption of the model PDAs and real systems as computed from GW -BSE is given in Figure 5. The computational details for the electronic structure calculations are given in Appendix B. The first thing to observe from Figure 5 is that there is only a single bright transition for all the stud-a b ied systems, at least in the idealized case of the infinite periodic material. Therefore, PDA provides an ideal system for satisfying condition 10 and observing strong light-matter interactions, since the oscillator strength is not shared between several transitions according to the sum rule of equation 11. It should however be noted that the real systems have some additional exciton states with small but non-vanishing oscillator strength at energies above 3 eV, which are associated with the side-chains. Moreover, it is interesting to note the difference in the ex- citon energy between the real and the respective model structures. Since the polymer backbone is identical in both cases, it is only the side-chains that can be causing the observed difference. The side-chains of the real red PDA consist of two phenyl rings, which are highly polarizable and hence provide significant screening to the Coulomb interaction that binds electron-hole pairs together. In contrast, in the model system the field lines of this Coulomb interaction are not screened by a medium, since the side-chains have been substituted by hydrogen atoms. Consequently, the exciton is more strongly bound in the model chain, further stabilizing its energy. For the real blue system, the side-groups largely consist of nonpolarizable alkyl chains, hence the screening they provide is much weaker compared to the red PDA. This is reflected by the fact that the exciton energies of the model and real blue systems are close-to-identical, closely reproducing the experimental value from the absorption spectrum of Figure 3a. Table I summarizes the values for the fundamental gaps of the chains, along with the energy of the excitons and their binding energy. The exciton binding energy of the real red and blue systems is equal to 0.52 eV and 1.36 eV respectively, while for both model chains it is greater than 2 eV, emphasizing the large impact of the screening provided by the polarizable side-chains. This effect can be further emphasized by visualizing the exciton wavefunctions of the real and model red chains in Figure 6. The electron density is visualized in red, for a hole fixed at the position indicated by a blue sphere. As expected, the lack of side-chain screening of the Coulomb interaction between the electron and hole in the case of the model chain leads to an exciton that is spatially more confined than in the real system. As far as the relative electronhole motion is concerned, we also reach a good agreement with the results from electroabsorption studies that provide an exciton bohr radius of several nanometers, corresponding to an exciton covering a few monomer units in the blue and red conformations [START_REF] Barisien | Exciton spectroscopy of red polydiacetylene chains in single crystals[END_REF][START_REF] Horvath | Wannier excitons and Franz-Keldysh effect of polydiacetylene chains diluted in their single crystal monomer matrix[END_REF] .

Having obtained ε(ω) from GW -BSE electronic structure calculations of the three materials, we proceed to obtain the polariton dispersion. Due to the almost perfect alignment of PDA chains along one axis (x), we use the polariton dispersion relation [START_REF] Stefan | Reflection of light by anisotropic molecular crystals including exciton-polaritons and spatial dispersion[END_REF] :

c 2 k 2 = ω 2 ε xx (ω), (12)
Here ε xx (ω) is the only non-zero component of the dielectric tensor of the material, and in the y and z directions, the vacuum dispersion holds:

ω 2 = c 2 k 2 .
We obtain the component ε xx (ω) of the dielectric tensor, by solving the Bethe-Salpeter equation for incident light normal to the chains, which approximately emulates the experimental conditions used to obtain the reflectivity spectra of Figure 3. Figure 7a visualizes the computed dispersion for the example case of the model blue chains, highlighting the resulting stop-band of 2.23 eV.

The ∆ values obtained for all studied systems are summarized in Table II, and the experimental values are given for comparison. It should be emphasized that the effect of exciton dispersion on the polariton states of these systems is negligible. As outlined in Appendix B, the bandwidth of the exciton is equal to 1.33 eV from the zone center Γ (|k|= 0) to the band-edge X (|k|= π α = 6.43 nm -1 ) of the first Brillouin zone in the direction of the polymer chains. Within the same distance in momentum space, the energy of a photon changes by ∆E = hc π α -1.87 = 1, 268 eV, a change which is three orders of magnitude greater than that of the exciton energy.

We see from Table II that the model blue chains have significantly larger stop-band width than the model red system. Since the only difference between these structures is the torsional angle between subsequent diacetylene monomers (see Figure 4), we conclude that the magnitude of this torsion has a strong effect on the magnitude of light-matter interactions. Indeed the finite torsion of the model red system leads to deviations from the perfect dipole alignment of model blue PDA, and which is crucial to facilitating strong coupling to photons. The computed values for the stop-band width are in reasonable agreement with the experimental values of Figure 3, and the large difference between the blue and red structures is captured.

Furthermore, the comparison of the predicted ∆ values for the real and model red systems suggests that the introduction of polarizable side-chains leads to weaker exciton-photon interactions, closing the stop-band by almost 0.4 eV. This is likely due to two reasons. Firstly, the dielectric screening that they provide to electron-hole pairs (Figure 6) leads to a more weakly bound exciton, which has a higher transition energy compared to the model system. Hence the condition of equation 10 becomes more difficult to satisfy. This argument only holds for polarizable side-chains that provide significant screening, which is not the case for the real blue PDA. Indeed the computed values of ∆ for the real and model blue systems differ by a mere 50 meV. Moreover, upon the introduction of side-groups, a few exciton states with small but non-vanishing oscillator strength appear at energies above 3 eV, hence reducing the brightness of the main exciton peak according to the sum rule of equation 11, and also making the frequency range that satisfies the condition of equation 10 narrower.

The values predicted by the calculations for the stop-band width are in reasonable agreement with experiment, as summarized in Table II. We believe that the differences are largely due to our modeling of a perfect periodic structure and ignoring exciton resonances which are present in a realistic film, for example due to unpolymerized diacetylene monomers and minority chain populations at different geometries. Each of these weak transitions couples to photons and has a (narrow) stopband of its own. The overlap of the different stop-bands due to the various weak exciton resonances with that of the main exciton peak leads to an overall wider stop-band [START_REF] Kalt | Semiconductor Optics[END_REF] . In agreement with this line of thought, the predicted values underestimate experiment for both materials. It is however worth pointing out that while the overall stop-band is wider due to the presence of additional resonances, the individual stop-bands become narrower. This effect is discussed later and visualized in Figure 12 for the main exciton peak in the presence of vibronic resonances.

For normal incidence of light, the reflectance can be calculated as:

R(ω) = (n(ω) -1) 2 (n(ω) + 1) 2 , (13)
where n(ω) the complex refractive index of the material. For the model blue chains this is plotted in Figure 7b, where the stop-band is clearly resolved. The absorption from the exciton state results in a low-reflectance gap within the stop-band, which is in agreement with experiment. However, the computed reflectivity spectrum looks qualitatively very different from the experimental one of Figure 3a. This is due to entirely ignoring the effects of disorder in the calculation. The presence of minority chain populations with different geometries, as well as unpolymerized diacetylene monomers are ignored, and the exciton states associated with these would lead to more gaps in the reflectivity. For the model chains we ignore excitons associated with the side-chains, which would also introduce additional structure within the stop-band. We will see later that dynamic disorder is another important factor towards determining the internal structure of the stop-band. Before moving on to consider the effects of disorder, we would like to comment on the speed of energy transfer in these materials, as predicted by the results of this section. First, let us consider the case of purely excitonic motion. The quantity that we need to compute in order to quantify the speed of band-like exciton motion is the exciton group velocity:

v g,exc = 1 h • ∂ E exc ∂ q , (14)
where E exc the exciton energy and q the exciton wavenumber. Following photoexcitation, a wavepacket is generated on the exciton potential energy surface, the momentum of which is narrowly distributed around q = 0, since light induces closeto-vertical transitions within the Franck-Condon approximation. The center of mass of this Γ-point exciton will then travel with the velocity of equation 14. We are interested in energy transfer in the direction of the chains, which we define to lie along the x axis, we thus approximate v g,exc as the average along the respective dimension of reciprocal space. To this end, we calculate the exciton energy at the band-edge X, using a supercell calculation that is outlined in Appendix B, and we obtain:

v g,exc ≈ 1 h • E exc (X) -E exc (Γ) q(X) -q(Γ) = 3 • 10 5 m/s. ( 15
)
Now let us consider the group velocity of a polariton v g,pol close to the exciton resonance, which can be computed from the solution of equation 12 for different energies of the incident light. The result for the model blue chains is given in Figure 8a; at low frequencies, the speed of light propagating along the PDA chains is c = c o /n xx . However, as we approach the energy of the exciton transition, the exciton content of the lower polariton state becomes larger, reducing the speed of propagation. Figure 8b provides a closer view to the region around resonance, and we find that the polariton propagates with a speed of the order of 10 6 m/s, which is an order of magnitude faster than purely excitonic band-like propagation.

One might observe that very close to the exciton resonance the group velocity obtains negative values, which is unphysical. This is due to the fact that in this narrow region, the material exhibits anomalous dispersion and the group velocity is generally not a useful concept in these cases [START_REF] David | Classical electrodynamics[END_REF] . FIG. 9: A hydrogen-terminated trimer of diacetylene.

IV. STATIC DISORDER: THE EFFECT OF FINITE CHAIN LENGTH

In a realistic film of PDA, the system deviates from the perfectly periodic behavior in more than one ways. As we saw when discussing the experimental reflectivity spectra, unpolymerized chains of diacetylene can be present, as well as minority populations of chains at different conformations. Imperfections in the chain polymerization can lead to breaks in the conjugation of the system, or even to the termination of the polymer structure after a finite number of monomers. Here we consider such effects limiting the chain conjugation, and we ignore atomic defects and vacancies, due to the pure character of the samples as discussed previously.

In order to understand the impact of finite conjugation lengths on the exciton and polariton properties, we study five different finite systems consisting of one to five diacetylene monomers, which we terminate on both sides with hydrogen atoms. For simplicity, we choose to terminate the chains after the occurrence of a triple bond, so that the carbon atoms at the boundaries only need to bond with a single hydrogen. As an example, Figure 9 visualizes a trimer of diacetylene.

We perform GW -BSE calculations on these oligomers, introducing a vacuum of 20 Å between their periodic images. We thus obtain ε(ω) for each of these, and similar to the previous section, compute the energy of the first singlet exciton E(S 1 ) and the stop-band width ∆, which we visualize in Figures 10a and 10b respectively. We have chosen to construct the oligomer chains so that they are planar, and hence by increasing the number of monomers, we approach the periodic limit of the model blue PDA chains of the previous section, which is also noted on Figure 10.

First of all, it is clear from Figure 10a that as we add more monomers to the chain, the exciton energy is stabilized and converges to the periodic limit. Due to the large computational cost, we were unable to add the required number N of diacetylene monomers in order to reach the periodic limit, however by performing model fits to the first five datapoints of Figures 10a and10b (dashed line), we estimate that the periodic limit is obtained for N = 13 -15 monomers, in very close agreement with a previous experimental study [START_REF] Wudl | Synthesis and Some Properties of Poly(diacety1ene) (Polyenyne) Oligomers[END_REF] . Furthermore, we find that the stop-band width increases with a larger number of monomers in Figure 10b, and the periodic limit of an infinite chain provides the maximal possible magnitude for light-matter interactions. This can be rationalized from the fact that the exciton frequency appearing in the oscillator strength condition 10 decreases with increasing number of diacetylene monomers.

We therefore conclude that the presence of static disorder, in this case in the form of finite chain lengths, reduces the magnitude of light-matter interactions. However, the large measured stop-band width of ∆ = 2.64 eV for the blue chains suggests that conjugation-limiting effects are not important in PDA. Naturally, breaks in the conjugation and imperfect polymerization do occur, however the results suggest that the vast majority of chains extend beyond 15 diacetylene monomers, for which number the periodic limit is reached, and the maximal light-matter coupling is obtained. Here we do not compute the effect of static disorder on the reflectance of PDA, as this would require explicit inclusion of e.g. unpolymerized diacetylene monomers within a supercell of PDA, making the cost of these calculations prohibitive. We now proceed to discuss the effects of dynamic disorder on light-matter coupling. for chains consisting of a different number of diacetylene monomers N. Since the constructed oligomers are planar, the periodic limit of the model blue chains is approached for a larger number of monomers.

V. DYNAMIC DISORDER: THE EFFECT OF VIBRATIONS

A purely electronic structure calculation predicts that there is only a single bright exciton state in PDA, the vertical transition peak in the computed absorption spectrum of Figure 5. In reality the ground-and excited-state potentials of the PDA chains are displaced along coupled vibrational normal modes, and electronic excitations involve multiple ending vibrational levels (and a thermalized distribution of starting vibrational levels). This is evident in the experimental absorption spectra of Figure 3, where one can resolve vibrationally dressed 0 -1 and 0 -2 phonon side bands on top of the purely 0 -0 electronic transition. Intuitively, it is obvious that these phonon side bands in the structure of Im(ε(ω)), will give rise to more structure in Re(ε(ω)) compared to the case of a single exciton peak, according to the Kramers-Kronig relation:

Re(ε(ω)) = P +∞ 0 dν π Im(ε(ν)) 2ν ν 2 -ω 2 (16)
This effect reduces the width of the region in which Re(ε(ω)) < 0 following an excitonic transition, as we show in Figure 11b. We call the width of this region ∆ vib , and the fact that it is smaller than ∆ means that a gap in the high reflectivity of the stop-band is introduced (see Figure 11c below).

We therefore proceed to study the effects of exciton-phonon coupling, as a means of understanding the internal structure of the stop-band. To this end, we employ the linear-vibronic Hamiltonian [START_REF] Frank | Absorption and emission in oligo-phenylene vinylene nanoaggregates: The role of disorder and structural defects[END_REF] :

H = E(S 1 ) • I + ∑ k hω k λ k (b † k + b k ) + ∑ k hω k b † k b k , (17)
where I is the identity, k runs over the phonon modes of the system, which have frequencies of ω k . The energy of the first singlet exciton refers to the infinite periodic limit. The Huang-Rhys factors are calculated as:

λ 2 k = ( V ep,k hω k ) 2 , (18)
where we approximate the first derivative V ep,k of the exciton energy along mode k using the finite difference formula:

V ep,k = E(S 1 ) +δ u(k) -E(S 1 ) -δ u(k) 2δ u(k) , (19)
with δ u(k) being a small dimensionless displacement along mode k. At a practical level, the calculation of V ep,k requires two GW -BSE calculations at the displaced configurations of each mode, and constitutes a very computationally intensive process. We were therefore only able to obtain a full parametrization of the Hamiltonian 17 for the two model chains, which only have a small number of phonon modes due to the small size of their unit cell (18 and 36 modes for the model blue and red chains respectively). The existence of the side-chains for the real PDAs, for which we previously calculated the exciton properties, makes the task of computing exciton-phonon interactions a very formidable one, and we thus limit the following analysis to the model chains.

When computing the Huang-Rhys factors of PDA, one needs to account for the fact that the exciton can travel a finite distance of N monomers coherently, before being scattered by a phonon. Therefore if λ k,unit is the value obtained within a unit cell exciton calculation where the phonon mode k has been displaced, then the actual value of the Huang-Rhys parameter is [START_REF] Frank | Vibronic coupling in J-aggregates and beyond: A direct means of determining the exciton coherence length from the photoluminescence spectrum[END_REF] :

λ 2 k = λ 2 k,unit N . (20)
We estimate the exciton coherence length N as:

N = v g,exc • τ scatter L DA , (21)
where v g,exc the exciton group velocity from section III, τ scatter the timescale for the exciton to be scattered by a phonon, and L DA the length of a single diacetylene monomer. Since we find that the excited state surface of the double-bond stretch is the steepest, i.e. has the largest value of V ep,k , we approximate τ scatter by the period of 23 fs of this mode, giving us an exciton coherence length of N = 15 monomers for both model chains, in good agreement with previous estimates on similar systems [START_REF] Yamagata | Vibronic coupling in quantum wires: Applications to polydiacetylene[END_REF] . We write the i th eigenstate of the Hamiltonian of equation 17 as:

Ψ (i) = M ∑ k=1 N q ∑ ṽ c (i) ṽk |1, ṽk (22)
where there are M vibrational modes in total. The basis state |1, ṽk is the short-hand notation for |1, ṽk = |1, 0, 0, ..., ṽk , 0, ..., 0 ,

i.e. the state where the exciton is populated and the first index set to one, and where among all vibrational modes, only mode k is populated by ṽk quanta. We thus only allow a single mode to be excited at one time, as a means of simplifying the problem. In principle, every phonon mode can admit an infinite number of quanta, however in practice we need to truncate the associated Hilbert space to only allow for up to N q quanta in each mode. We find that for the model PDA chains N q = 3 converges the results for the dielectric response. Similar to previous works [START_REF] Frank | Absorption and emission in oligo-phenylene vinylene nanoaggregates: The role of disorder and structural defects[END_REF] , we include the effect of vibrations in the absorption spectrum by taking:

Im(ε(ω)) = (c (1) 0 ) 2 • Im(ε(ω)) 0-0 + | ∑ k ∑ ṽ c (1) ṽk λ 2 ṽk k e -λ 2 k ṽk ! | 2 W (ω -ω S 1 -ω k ), (24)
where W (ωω S 1ω k ) a Lorentzian centered around ω S 1 + ω k , for which we use the same width as for the peaks of the purely electronic part Im(ε(ω)) 0-0 of the dielectric response, as obtained from GW -BSE calculations detailed in Appendix B. Moreover, in Appendix B, we show for the example case of the model blue chains, that the result for the width ∆ vib is largely independent of the peak width γ, for values of γ between 5 meV and 50 meV. The coefficients c [START_REF] Pekar | The Theory of Electromagnetic Waves in a Crystal in which Excitons Are Produced[END_REF] are obtained by diagonalizing the Hamiltonian 17, with c

(1) 0 denoting the contribution from the purely excitonic basis state |1, 0, 0, ..., 0 . Here we reside to approximating the absorbance α(ω) with Im(ε(ω)). This is a reasonable approximation for determining ∆ vib , since this quantity depends on the intensity ratio of the 0 -0, 0 -1, 0 -2 peaks etc. The absorbance α(ω) and imaginary dielectric response Im(ε(ω)) are proportional to each other, which suggests that the error we introduce is small.

For the example case of the model blue chains, we visualize the result of including phonon effects on Im(ε(ω)) in Figure 11a, while Figure 11b shows the corresponding Re(ε(ω)) as obtained by using the Kramers-Kronig relation 16. For Re(ε(ω)) we also draw a comparison to the case without phonons. The peaks that feature prominently in the absorption spectrum are due to the double-and triple-bond carbon stretching motions, with energies of 0.18 eV and 0.26 eV respectively. Indeed in the experimental absorption spectrum of the blue PDA in Figure 3a, two vibronic peaks feature prominently at energies of 0.19 eV and 0.25 eV above the main exciton peak, however their relative intensity is opposite to the computed one. Intuitively, one can understand why the double-bond and triple-bond vibrations are most strongly coupled to the exciton from the fact that they transiently lead to structures with a smaller bond-length alternation, i.e. deviating from the Peierls-distorted structure that gives conjugated polymers such as PDA their semiconducting character, and bring the material closer to the metallic limit [START_REF] Kivelson | First-Order Transition to a Metallic State in Polyacetylene: A Strong-Coupling Polaronic Metal[END_REF] . It is hence not surprising that these motions have a large effect on the optical properties of the studied PDAs. This also suggests that the coupling to these motions is extremely sensitive to the precise bond-length alternation of the PDA backbone, potentially accounting for the discrepancy between the theoretical and experimental absorption spectra when it comes to their relative importance. For the computed exciton coherence length of N = 15, we obtain ∆ vib values of 137 meV and 123 meV for the model blue and the model red chains respectively. We compute the associated reflectivity spectra in Figure 11c, where we find that vibronic peaks introduce 'gaps' within the high reflectance region of the stop-band.

Therefore, accounting for vibrational effects allows one to predict part of the internal structure of the stop-band of an exciton that is strongly coupled to a photon. However, the precise form of the reflectivity, depends very sensitively on the exciton coherence length N, as shown in Figure 12, where we use the width ∆ vib as a proxy for the size of phonon-induced gaps in the reflectivity. This dependence of ∆ vib on N enters through the Huang-Rhys factors of the extended system, see equation 20. In turn, the exciton coherence length depends on the exciton group velocity and the period of phonon modes, as seen in equation 21. Hence the interplay of phonons with excitons is established to be crucial for the interaction of the latter with light.

VI. CONCLUSIONS

In this work, we have presented a theoretical framework for obtaining the properties of exciton-polaritons in organic materials. Our approach requires input from a calculation of the material's dielectric response, and is agnostic to the methodology used to obtain it. By employing GW -BSE calculations for the electronic structure of polydiacetylenes, we obtain their stop-band width, which characterizes the strength of lightmatter interactions. The results are in reasonable agreement with experiment, and we attribute the differences to our treatment of the materials as perfectly periodic, neglecting effects of disorder and additional excitons that weakly couple to light. We compute the speed of propagation of a polariton close to the resonance of light with an exciton, showing that this is an order of magnitude faster than purely excitonic motion. We find the conjugation length of polymer chains to be a critical parameter towards determining the magnitude of light-matter interactions, highlighting one manifestation of how static disorder can lead to weaker exciton-photon coupling. This effect is not important for the polydiacetylenes studied here, due to The main extra contributions to the structure of ε(ω), compared to the disorder-free case, arise due to the effect of the carbon-carbon double-and triple-bond stretch. Comparing the structure of Re(ε(ω)) for the cases with and without phonons, we find that dynamic disorder leads to 'breaks' within the region of Re(ε(ω)) < 0. This leads to a richer structure for the reflectance in panel c, compared to the disorder-free case of Figure 7.

the purity of these materials. It could however be of relevance for different systems. Furthermore, we discuss the important role of dynamic disorder and show that vibrational effects introduce gaps within the frequency range of large reflectance that characterizes polaritons. We also discuss the screening of the Coulomb interaction provided by the side-chains of polydiacetylene as another potentially important contribution towards determining the magnitude of light-matter interactions. Thus all of the aforementioned factors appearing in realistic materials are important towards an accurate description of exciton-photon coupling. In principle, one needs to account for the above factors at the same time and self-consistently, something which is computationally challenging and will be the subject of a future work.

and transmittance were calculated using a silver mirror (Thor-Labs, PF10-03-P01) and the glass substrate as references respectively.

Appendix B: Computational details Electronic structure calculations

We perform DFT calculations using the Quantum Espresso software package [START_REF] Giannozzi | QUANTUM ESPRESSO: A modular and opensource software project for quantum simulations of materials[END_REF] , with the PBE functional together with a 60 Ry plane wave cutoff energy. For the model blue and model red chains, we employ 8 × 4 × 2 and 4 × 4 × 2 k-point grids respectively. For the real red chains, we use 4 × 4 × 2, and for the real blue system 4 × 2 × 4. For the model PDAs, we introduce a vacuum of 10 Å between neighboring polymer chains, in order to minimize inter-chain interactions.

We perform energy self-consistent GW calculations (i.e. GW 0 ) using the YAMBO code [START_REF] Sangalli | Many-body perturbation theory calculations using the yambo code[END_REF] . For the real chains, we find that including 400 Kohn-Sham states, 300 bands for the calculation of the polarization function, and a 7 Ry cutoff for the dielectric matrix, leads to converged values for the quasiparticle bandgaps. For the model blue (red) chains, we use 40 (80) Kohn-Sham states, 40 (80) bands for the calculation of the polarization function, and a 3 Ry cutoff for the dielectric matrix.

We solve the Bethe-Salpeter equation for the model blue (red) PDA using 4 (8) occupied and 4 (8) unoccupied bands, converging the position of the first exciton peak. For the exchange term in the Bethe-Salpeter kernel, the cutoff is set to 60 Ry. For the real chains, we use 10 occupied and 10 unoccupied bands, and set the exchange to 40 Ry. In order to emulate the experimental conditions, the incident electric field is chosen to lie in the direction of the polymer chains. The imaginary part of ε(ω) is obtained by fitting a Lorentzian of 10 meV width to the computed excitonic eigenvalues of the BSE Hamiltonian. The broadening of the absorption spectrum due to vibrational effects as presented in Figure 11 is then added on top of that, along with the additional peaks.

Due to the one-dimensional character of the model PDAs, divergences can occur at small values of q (i.e. at long distances) in the Coulomb term that appears in the many-body calculations. In order to avoid this problem, we use the random integration method, as implemented in YAMBO. We converge the calculations using 10 5 random q-points in the first Brillouin zone, and by setting the cutoff for the real space components of the Coulomb interaction to 20 Ry.

For the model PDAs, we estimate the dispersion of the exciton band along Γ -X as follows. We generate a supercell of size 2 × 1 × 1, thus halving the size of the first Brillouin zone. As a result, the zone-boundary X point folds onto Γ, which appears as an extra exciton state once the BSE Hamiltonian of the 2 × 1 × 1 system is diagonalized. The distance of this new band from the exciton value at Γ gives the width of the exciton band, which we find to be equal to 1.33 eV.

The aforementioned electronic structure calculations are performed on the PDA structures as obtained from X-ray diffraction (for the real systems). For the model systems, cal- culations are performed on the structures as obtained from X-ray diffraction and following the substitution of the sidechains with hydrogen atoms. Due to the extreme sensitivity of the computed exciton properties on the bond-length alternation (BLA) of the polymer chain, we set the BLA of the blue chains to be exactly identical to that of the red chains, in order to focus on the effects of the relative torsion between diacetylene monomers and on the screening effect provided by the side-chains. A geometry optimization of these structures is not carried out, as the BLA of these one-dimensional carbon chains is known to be extremely sensitive to the fraction of exact exchange that the employed DFT functional includes [START_REF] Mostaani | Quasiparticle and excitonic gaps of one-dimensional carbon chains[END_REF] . It has been shown that while hybrid functionals perform better than GGA [START_REF] Mostaani | Quasiparticle and excitonic gaps of one-dimensional carbon chains[END_REF] , it is necessary to use very accurate methods such as Quantum Monte Carlo in order to obtain geometries with realistic BLAs, which is beyond the scope of this work. Since the exciton energy is extremely sensitive to the precise value of the BLA, we limit ourselves to obtaining exciton energies for unoptimized structures, and only optimize the geometry of the studied structures prior to phonon calculations.

Phonon calculations

We calculate the lattice dynamics of the different PDA variants in the harmonic approximation using finite differences [START_REF] Kunc | Ab initio force constants of GaAs: A new approach to calculation of phonons and dielectric properties[END_REF][START_REF] Monserrat | Electron -phonon coupling from finite differences[END_REF] . We calculate the dynamical matrix on a 1 × 1 × 1 coarse q-point grid. Therefore, we only account for the effect of Γ-point phonons, which provides a good approximation, since these are most strongly coupled to the zero-momentum exciton states that result from photoexcitation. Prior to phonon calculations, the geometry of the different PDAs is optimized using the PBE functional.

Dependence of polariton splitting on peak width Figure 13 visualizes the value of ∆ vib of the model blue PDA, within a range of values for the width of the exciton and vibronic peaks, ranging from 5 meV to 50 meV. We find that ∆ vib does not vary by more than 10 meV within this range of widths. Here an exciton coherence length N = 15 has been used.

  photon

FIG. 2 :

 2 FIG.2:The structure of a diacetylene monomer, which generally coincides with the unit cell of an infinite and periodic crystal of polydiacetylene.

  FIG.3: Experimental reflection and absorption spectra of two studied variants of polydiacetylene: a blue chains and b red chains. The stop-band width of the two structures is approximately determined from the reflectivity spectra. The polymer structures are given alongside the experimental plots, with the polymer backbone highlighted in red rectangles, in order to distinguish it from the side-chains. Carbon atoms are denoted in light gray, hydrogen in white, oxygen in red, nitrogen in light blue and bromine in brown.

FIG. 5 :

 5 FIG. 5: Absorption of the real and model PDA chains.

FIG. 6 :

 6 FIG. 6: Exciton wavefunction of a the real red PDA chains and b the model red chains. The electron density is visualized in red for a hole fixed at the position indicated by a blue sphere.

FIG. 7 :FIG. 8 :

 78 FIG. 7: Calculated a polariton dispersion and b reflectance of the model blue PDA chains, highlighting the stop-band of width ∆ = 2.23 eV.

FIG. 10 :

 10 FIG.10:The exciton energy (panel a) and stop-band width (panel b) for chains consisting of a different number of diacetylene monomers N. Since the constructed oligomers are planar, the periodic limit of the model blue chains is approached for a larger number of monomers.

FIG. 11 :

 11 FIG.11:The imaginary (panel a) and real part (panel b) of the dielectric response of the model blue chains, once the effect of phonons is accounted for. The main extra contributions to the structure of ε(ω), compared to the disorder-free case, arise due to the effect of the carbon-carbon double-and triple-bond stretch. Comparing the structure of Re(ε(ω)) for the cases with and without phonons, we find that dynamic disorder leads to 'breaks' within the region of Re(ε(ω)) < 0. This leads to a richer structure for the reflectance in panel c, compared to the disorder-free case of Figure7.

FIG. 12 :

 12 FIG.12:The dependence of the width ∆ vib of the two model PDA chains, on the exciton coherence length.

FIG. 13 :

 13 FIG.13:The dependence of the width ∆ vib of the first region with Re(ε(ω)) on the width γ of the exciton and vibronic peaks.

TABLE I :

 I Values for the first exciton energy E(S 1 ), energy of the fundamental gap E g and exciton binding energy E binding of the four PDA chains.

	structure	∆ (eV) -theory	∆ (eV) -experiment
	real blue	2.28	2.64
	model blue	2.23	-
	real red	1.06	1.16
	model red	1.45	-

TABLE II :

 II Values for the stop-band width of the four disorder-free PDA chains.

ACKNOWLEDGMENTS

The authors acknowledge useful discussions with Richard Phillips (Cavendish) and Girish Lakhwani (University of Sydney). We thank Gianni Jacucci and Silvia Vignolini (Cambridge) for assistance with reflectivity measurements. We acknowledge the support of the Winton Programme for the Physics of Sustainability. A.M.A. acknowledges the support of the Engineering and Physical Sciences Research Council (EPSRC) for funding under grant EP/L015552/1. A.W.C. acknowledges financial support from Agence Nationale de la Recherche (Grant N. ANR-19-CE24-0028). The work in Mons is supported by the Belgian National Fund for Scientific Research (FRS-FNRS). Computational resources were provided by the Consortium des Équipements de Calcul Intensif (CECI). D.B. is FNRS Research Director.

DATA AVAILABILITY

The data underlying this publication can be found in [URL added in proof].

Appendix A: Experimental reflectivity measurements

Microscopic transmission and reflectance measurements were performed using a Zeiss AxioScope optical microscope in Köhler illumination equipped with a 100× objective (Zeiss EC Epiplan-APOCHROMAT 0.95 HD DIC) coupled to a spectrometer (Avantes HS2048) via an optical fibre (Thorlabs, FC-UV50-2-SR). Five spectra were collected for each sample using an integration time of 10 ms and 20 ms for reflection and transmission measurements, respectively. The reflectance


Joséjh Videira

Keith Wj Barnham

Anna Hankin

James P Connolly

Matthew Leak

Jonathan Johnson

Geoff H Kelsall

Ken Kennedy

John S Roberts

Alex J Cowan

Amandaj Chatten

Introducing novel light management to design ahybrid high concentration photovoltaic/water splitting system

Keywords: geometrical optics, luminescent devices, photoluminescence, quantum wells, ray tracing, solar cells

We present a novel way to utilize high-concentration photovoltaic (HCPV) radiativelossesand diffuse light, otherwise unused in conventional HCPV systems, to poweran Imperial College designed photoelectrochemical reactor (PECR) producing H2 fuel through water splitting. Ahigh efficiency photovoltaic (HEPV) isembedded inside a Luminescent Solar Concentrator (LSC). Edge emission from the radiative recombination loss mechanism in the HEPV is guided withinthe LSCto thePECR photocathode, whilst the LSCemitted light is guided to the photoanode. The photon streams can be independently optimised in intensityand wavelength. We demonstrate howphoton streams with balanced intensity can be achieved.

I. IN T RO D U C T I ON

The extraction of hydrogen from water using sunlight was first proposed30 years ago [START_REF] Fujishima | Electrochemical photolysis of water at a semiconductor electrode[END_REF]. Recentlyinterest in the subject is fueled by the need to finda renewable, sustainable and environmentally safe alternative form of energy storage. The main challenge inthis field of researchisfinding a material that can match the energetic and stability requirements of spontaneous photo-water splitting [START_REF] Minggu | An overview of photocells and photoreactors for photoelectrochemical water splitting[END_REF], and a breakthrough in this would be thesurest way toreach efficiencies high enough to seriouslyconsidercommercialisation [START_REF]Hydrogen, fuel cells &infrastructure technologies program[END_REF].

The ultimate aimof this project, in conjunction with the Chemical Engineering department atImperial CollegeLondon, is to demonstrate photoelectrochemical (PEC) water splitting using photoanodes and photocathodes (Figure 1) that are independently illuminated by twophotonstreams that are conventionally regarded as waste from HCPV operation.

A further advantage oftheschemeisthatthe intensity and wavelength of each photon stream can beindependently optimised. The system wouldproduce zero-carbon hydrogen forenergy storage and transport as aby-product of HCPV electrical power generation. These two waste photon streams are 1) light that is emitted due to dominant radiative recombination mechanisms athigh celloperating bias and 2) entirelyunused diffuse light.

We present a concept that merges conventional CPV and LSC technologyinto ahybridsystemthat can both generate electricity and generate hydrogenthrough PECwatersplitting. The challenge of directing two separate photon streams to individual photoelectrodes can be solved with a modifiedluminescent solar concentrator (LSC). The LSC, first introduced in the 1970s, inits basic design is a transparent waveguide structure, either homogeneously doped with, or coatedwith,luminescent material (luminophores). The luminophores absorb incident light at or above their absorption threshold, and re-emit it with acertain redshift with ideally near unity luminescent quantum yield (LQY). A large fraction of this emitted light is waveguidedby total internal reflection(TIR)tothestructure's edges, where it is collectedby opticallycoupledoptoelectronic devices. The main benefit ofusing a LSC is the abilityto tune its absorption/emission spectra according to different needs, by usingdifferentluminophorespecies. TheLSC willprovide one of the photon streams by absorbing diffuse light, re-emitting it and waveguiding it into the photoanode of the PECR.

For the other photon stream, we will use high efficiency photovoltaics (HEPV).HCPV modulesutilise low-cost optical elementsto reduce the amount ofsemiconductor needed by concentrating direct sunlight onto these very small but highefficiencycells.This replacement ofexpensivesemiconductor withcheaper opticsprovides a routetowards reducing the costs of solar electricity generation. A conventional HCPV module consists ofthree elements: Theconcentrating optics, the solar cells and the housing. The aimis to concentrate light as much as possible, onto a HEPV, and to dissipate heat as much as possible.

Any high quality HEPV at high cell operating bias will experience radiatively dominatedrecombinationlosses.Therefore at its maximum power point, when forward biased by theabsorption of concentratedsunlight, a HEPV can act like an edge emitting LED. However, quantum well solar cells (QWSCs) are particularly suitable for this application for three reasons: 1) As will be discussed, their radiative efficiency canbesimply established from measurements of quantum efficiencyand dark current. 2) The radiative recombination takes place at the bottom ofthe quantum well. Quantum wells are acommon feature of LEDsbecause oftheir high radiative efficiency and because the barrier and substrate materialsl is transparent to the radiation. 3) The wavelength and intensity of the emission can beoptimised for agiven catalyst or photoelectrode simply bythe choice of well depth.

The HEPV, which will have antireflection coating on its edges, isembedded in andoptically coupled to, an undoped waveguide whichforms the transparent substrate of a square LSC with thin-film luminescent coating on the topsurface, as in Figure 2.

AFresnel lens concentrates direct light onto aquantum well solar cell (QWSC) to generate aphotocurrent, as per conventional HCPV functioning. Dependingon the QWSC material structure and operating conditions, acertain fraction ofthe light from radiativerecombinationwill bewaveguided to the edges andbe coupled into the LSC, where it gets trapped due to total internal reflection, and waveguided to the photocathode. The majority of the edge emission from the QWSC will be at an angle greater than the escape cone from the top and bottomsurfaces of thewaveguide [START_REF] Sze | [END_REF]. The emission willalso be longerwavelength that theluminescence from theLSC coating.Hence the thin-film coating of the LSC will not absorb the radiation from the QWSC.

Inaddition to theessentially isotropicdiffuse contribution of light to the LSC, there is also a contribution from the circumsolar sunlight due to the scattering of sunlight by atmospheric aerosols. The intensity of the short-wavelength photon streamcan thereforebe optimisedfor given spectral conditions by adjusting theacceptance angleof the CPV system.

Wavelength selective mirrors, which are coupled in between the LSC and the photoelectrodes, provide the means to direct the two photonstreams to their desired locations. The photoanode, which in our scheme isilluminated by short wavelength light is coupled to thewaveguide through afilterthat transmits light in the visible range, and reflects it in theinfrared range. The photocathode which on our schemeabsorbs infrared light iscoupledtothe waveguide byafilterthat transmits light in theinfraredand reflectsit inthe visible range.

The two other edges as well asthe undersideof the LSChave Lambertian high reflectivity air-gap reflectors to minimise escape losses.

Thewavelengthsofthe photonstreamsareengineered to be appropriate for the electrodematerials thus allowingthe water splittinghalf reactions that produceH 2 to occurwithin the PECRs. As shown in Figure 3, the system is designed to be modular, with a view of creating a series of arraysof LSCs and PECRs. Figure 4 shows how the reflectivity spectra for the mirrors correspondwell with thespectra for both photon streams. I II. TES TI NG TH E PR IN CI P LE OF T HE QUA N T UMWELL SO LAR CEL L AC TI NG AS LU M I N ES CE NT SO LAR CO N C E N T R AT OR

A. Premise

The starting point of system optimisation is the initial calculation of the powerdensity that can be achieved bythe LSC and the HEPV. Inthis section wepresent the calculations for bothpower densities when the system is illuminated under ASTM G-173 conditions [START_REF] Gueymard | Proposed reference irradiancespectra for solar energy systems testing[END_REF] and the HEPV is being operated at 500x concentration.

B. Luminescent Solar Concentrator

For our calculations we use a thin-film luminescent solar concentrator (LSC), witha luminophore/poly(methyl methacrylate) (PMMA) film blade coated onto a UVtransparent glass substrate. The photoanode absorbs light at wavelengths shorter than 500nm. Therefore we utilise Lumogen FViolet 570 (LV570), a naphthalimide-based luminophore. It absorbs light in the 300nm-410nm region, and emits inthe 390nm-550nm regionwith a luminescent quantum yield of around 85%. The optical concentrationC opt, defined Fig. 6. Measured(black dots) andfitted (black line)dark-IV curve ofa p-i-n strain balanced quantum well solar cell. The radiativecontribution (magenta) is a prediction based on the quantum well absorption aobratined from the quantum efficiency fit in Figure 5. The SRH contribution (blue) is aone parameter fit as the ratio of photon flux coupledfrom the edges to the incident photon flux, can be manipulated by changing the geometry of the LSC. We use a combination of our in-house raytracerPVTrace [START_REF] Farrell | pvtrace Optical ray tracing for photovoltaic devices and luminescent materials[END_REF]and theSMARTSsky spectralirradiance calculator [START_REF] Gueymard | SMARTS, A Simple Model of the Atmospheric Radiative Transfer of Sunshine: Algorithms and Performance Assessment[END_REF] to give performance characteristicsof the LSC in ASTM G-173 conditions.

The LV570 luminophore is mixed with PMMA at a dye load concentration of 1.5% and given dimensions of 200x 200 x 5mm. With PVTrace wesimulatedthe performanceof this LSC setup under incident light conditions of an isotropic hemispherical light source with thespectrumof the diffuse component of ASTM G-173. The calculated optical efficiency of this setup was 2%, giving a photon output of 1.9x10 17 s -1 .

C. Quantum Well Solar Cell

QWSCs, first invented by Barnham and Duggan [START_REF] Barnhamand | A new approach tohigh-efficiency multiband gap solar cells[END_REF], offer an way toimprove multi-junction solar cells by theability to tune their bandgaps, as well as enhancing photon coupling due to recombination beingdominated byradiativeprocesses at high concentration (in excess of 200suns) [START_REF] Johnson | Obervationof photon recycling instrainbalanced quantum well solarcells[END_REF] - [START_REF] Ekins-Daukes | Controlling radiative loss in quantum well solar cells[END_REF].

We demonstrate the principle of a QWSC acting as an LED with anumber of square photoluminescence samples of different sizesfrom a wafer QT1604 grown by MOVPE at the EPSRC National Centre for III-V Technologies at Sheffield. It consists of 22 strain-balanced GaAsP/InGaAs quantumwells in aGaAs p-i-n structure. There was an AlGaAs window below the p-doped GaAs emitter layer thatcan producesome limited transverse waveguiding due to itscomparativelylow refractive index. The edges of the photoluminescence sample were anti-reflection coated.

Inorder to determine the power density coming outof the edges of the devicewhenilluminated with 500xconcentration, we must determine a) its radiative efficiency for a given incident power density and b) its waveguiding capabilities. This is done by a combinationof modelling and edgephotoluminescence measurements.

The radiative efficiency of a given QWSC can be determined by fitsto the quantum efficiency and dark current as demonstrated in Figure 5and Figure 6, with a fitting model, SOL [START_REF] Paxman | Modeling the spectral response of the quantum well solar cell[END_REF]. The radiative efficiency is expressed as:

ηr ad = J rad J rad +J S RH (1)
WhereJ rad isthe radiativerecombinationcurrent contribution andJ SRH is theShockley-Read-Hallnon-radiative recombination current contribution. This efficiency is demonstrated in Figure 6 as the red line.The lowbias darkcurrent is fitted by Shockley-Read-Hall recombination with oneparameter: the effective lifetime for carriertrapping. The radiative contribution to the dark current (magenta circles in Figure 6), whichresults inphotonemissionfromaround thefirst excitoninFigure5, is an absolute prediction based on detailed balance assuming photon absorption isgivenbythefit totheQE around the first exciton feature.This procedure is similar to thedetermination of the secondary emission from a luminescent species ina LSC.

The merit of usinga non-metalised photoluminescence sample is that it functions under open-circuit voltageV OC, and without the complication of series resistance that causes the dark-current to curve at highbias in Figure 6. UnderV OC conditions, the current JV in the cell is zero, therefore the short circuit currentJ SC equalsthe darkcurrentJ D dueto the expression:

J V =J S C -J D (2)
The short circuit currentJ SC isexpressed as [START_REF] Sze | [END_REF]:

J SC =q η Popt hν (3)
whereP opt isthe incident optical power density and η is the quantum efficiency Atopen circuitJ SC =J D andfrom Figure6 theradiativeefficiency andhence the radiative contribution to thedark current density can bedetermined. The radiative current through the cell is the product of this radiative current density and the sample area. The current is converted into a photon flux by dividing bythe electronic charge. The photon fluxat the cell edge depends on the cell waveguide efficiency ηC. The photon flux at the PEC depends on the LSC waveguide efficiency at the wavelength of the emission.

QT1604 was uniformly illuminated with a 532nmbeam from aMillennia V laser with two different power settings. A shield blocked all incident and top surface PL light from entering the detector. A calibrated fibre spectrometer was coupled onto an edge of QT1604 to give an absolute spectrum 2.55x10 4 3.46x10 4 Irad (A) 6.37x10 -1 8.66x10 -1 Maximum photon count (s -1 ) 3.98x10 18 5.35x10 18 Measured edge photon flux (m -2 s -1 ) 7.77x10 19 1.11x10 20 Measured edge photon count (s -1 ) 7.75x10 14 1.10x10 15 Waveguide efficiency ηc (%) 0.02 0.02 Measuring QT1604 with dimensions 5mm x5mm x0.5mm with two different laser power settings. The maximum photon flux is calculated using Equation 3. The radiative efficiency is 0.85 and the EQE is 0.92 at incident wavelength of 532nm.

Preliminary studies with a Monte Carlo model suggest this very small cell waveguiding efficiencyisdue to the free carrier absorption in the highly doped substrate. Though small, ηc can be used to calibrate the Monte Carlo model using the edge measurements.

Using SOL, wecan calculate thenecessary QWSCwaveguiding efficiency needed forASTM G-173,500xconcentration light conditionswhen the QWSC is at maximumpower point. An LSC with the parameters mentioned above under diffuse illumination will produce on the edges a photon count of 1.9x10 17 s -1 .

Table II shows that this photon flux from the diffuse radiationabsorbed by the LSC canbe matched with the waste radiativeemission from the QWSC even withthe high resistance cell in Figure 6 as long as the cell waveguide efficiency is around 62%. The second column ismore typical of an optimised concentrator cell. Methods of reaching these cell waveguide efficiencies is discussed in section IV. Assuming a QT1604 solar cell dimension of7mm x 7mm to match the LSC photon output of 1.9x10 17 s -1 atthe LSCedge, using calculations for voltage at maximum power point and radiative current from SOL. The QWSC is calculated to be operating at maximum power point MPP, at 500x concentration in AM1.5D light conditions.

IV. A B R IE F D I SC U SS I ON ON OP TI M I S AT I ON

The calibratedMonte Carlo model willbe used tooptimise the structure of the QWSC toenhance the waveguideefficiency to the valuesin Table II without loss of cell power conversion efficiency.Themost obvious improvement would be to remove the substrate as couldbeachieved in practicein a lift off process. This wouldbe predicted to give a significant increase in edge emission due to the removal of what is essentially aphoton sink. Solar cell structures could be optimised to further improve its waveguiding capabilities. Due to having a highrefractive index, asolar cell has the ability toact as a good LSC if other free carrier absorption is minimised. For example, adding asecond lower-refractiveindexwindow layers into a structure would enhance transversewaveguiding to the edges. If these optical confining structures are dopedfor as to providethebuilt-infield thereneednot beany intentional dopingin theGaAs and quantum wellactive region.Lessening the dimensions of the solar cell would further reduce the free carrier absorption rate. Further studies on the Monte Carlo model will lead tothe identification of better waveguide designs.

To increase the optical efficiency of an LSC, progress needs to be made on novel luminescent materials. Recentchallenges involve striking abalance between attempting to engineera larger Stokes shift between the absorption and emissionspectra ofthe luminophoresin ordertodecrease reabsorption losses, without decreasing the luminescent quantum yields.

Wavelength selective mirrors present a bottleneck in the system performance due to the non-ideal reflection behaviour that occurs at large angles of incidence. Apoint to note is that this could be mitigated in part by the development of novel luminescent materials, especially in the design of anisotropic emission. Research is currentlybeingundertaken inthis area [START_REF] Verbunt | Anisotropic lightemissions inluminescent solarconcentratorsisotropic systems[END_REF]. Hot and cold mirrors from EdmundOptics were purchased and a characteristion was performed oftheir reflectivities as a function of angle and wavelength. Results are shown in Figure 8.

V. CO N CL U SI O NS Theauthorshavepresenteda novelwaytomanage radiative losses from a quantumwell solar cell (QWSC) and diffuse light, with the objective being topower a doubly-illuminated photoelectrochemical reactor to produce H2. A QWSC is embedded in aluminescent solar concentrator (LSC), where light generatedfrom radiative recombinationescapes from the edges of the QWSC and trapped in the LSC through total internal reflection. The LSC absorbs diffuselight and traps re-emitted light. Wavelengthselective mirrors are used todirect thesetwo photonstreamsto theirdesired output edge. The design oftheQWSC is acritical factor inenabling the efficicient waveguidingand transmission of the radiative light out of its edges. Raytrace modelling from an in-house raytracer willbe used to determinethe performance of the whole optical setup.

REF ER E NC ES

Fig. 1 .

 1 Fig. 1.Schematic of adoubly illuminated photoelectrochemical reactor (PECR). The photoanode (red) produces oxygen molecules through oxidation of water whilst the photocathods (blue) produces hydrogen molecules through the reduction of hydrogen ions.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Summaryof thehybridsystemoptics.1)Directlightisconcentrated onto ahigh efficiency PV cell by Fresnel lens. 2) Emission from radiative recombination escapes from the PV and is waveguided into the reactor photocathode by (a) direct from PV edge, or (b) via hot mirror reflection. 3) Diffuse light is absorbed, re-emitted and waveguided into thereactor photoanode a) directly from luminescent re-emission, or b) via cold mirror reflection.

Fig. 4 .

 4 Fig. 4. Reflectivityof the cold(black) and hot (red) mirror as a function of wavelength at 0°angle of incidence. The LSC emitted light (blue) gets transmittedby the hot mirror and reflectedby thecold mirror, whereas the opposite happens for the HEPV edge emission (magenta).

Fig. 5 .

 5 Fig. 5. Measuredandfitted external quantum efficiency (EQE) of a p-i-n strain-balanced quantum well solar cell, with its dark current in Figure6

Fig. 7 .

 7 Fig. 7. Spectral irradiance of the edge emission from QT1604, aquantum well solar cell PL sample

Fig. 8 .

 8 Fig. 8. Angularand wavelengthdependence ofthereflectivity ofa hotmirror. Note the degradation of performance after the angle of incidence exceeds 60°.