#
Prediction of Strong Transversal s(TE) Exciton–Polaritons in C_{60} Thin Crystalline Films

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## Abstract

**:**

## 1. Introduction

**do not**exist in the non-retarded ($c\to \infty $) limit, i.e., they appear only as the coexistence of a photon and an exciton. The extent of the photon’s participation in the s(TE) exciton–polariton is determined from the bending of the horizontal exciton branch (${\omega}_{\mathrm{ex}}$) at the exciton–photon crossing (${\omega}_{\mathrm{ex}}={Q}_{\mathrm{ex}}c/\sqrt{\u03f5}$, where ${Q}_{\mathrm{ex}}$ is the photon wave vector at the exciton–photon crossing point), which we call the Rabi splitting $\Omega $, to keep the terminology compatible with the cavity systems. Even though the s(TE) surface or 2D polaritons do exist [40] for some conditions, there is still no experimental evidence of such modes. However, the hybridization between the s(TE) Bloch surface waves (BSWs) (i.e., the photons confined between a truncated photonic crystal and a semi-infinite dielectric), and the excitons has been experimentally demonstrated in both inorganic (quantum well and TMD monolayer) and organic systems [48,49,50,51].

## 2. Theoretical Formulation

#### 2.1. Calculation of Electric Field Propagator $\mathcal{E}$

#### 2.2. Calculation of the Optical Conductivity of a Single Molecule

#### 2.3. Optical Conductivity in a Molecule Physisorbed at a Dielectric Surface

#### 2.4. Computational Details

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) C${}_{60}$ molecules upon deposition on the surface self-assemble in a regular FCC structure forming a (111) surface. The C${}_{60}$ layers, occupying $z>0$ half-space, are immersed in a dielectric medium described by a dielectric constant ${\u03f5}_{0}$. The dielectric response of the substrate, occupying $z<0$ half-space, is approximated by a local macroscopic dielectric function ${\u03f5}_{\mathrm{M}}\left(\omega \right)$. The FCC lattice constant is ${a}_{3\mathrm{D}}=14$ Å so that the separation between the layers is $\Delta ={a}_{3\mathrm{D}}/\sqrt{3}=8.1$ Å. (

**b**) Each crystal plane forms a 2D-hexagonal Bravais lattice with lattice constant ${a}_{2\mathrm{D}}={a}_{3\mathrm{D}}/\sqrt{2}=9.9$ Å.

**Figure 2.**(

**a**) The character of the electromagnetic modes at the dielectric/vacuum (${\u03f5}_{\mathrm{M}}/{\u03f5}_{0}$) interface. In the region $\omega >Qc$, the electromagnetic modes are entirely radiative (both in vacuum and in the dielectric), in the region $Qc/\sqrt{{\u03f5}_{\mathrm{M}}}<\omega <Qc$, they are radiative in the dielectric and evanescent in vacuum, and in the $\omega <Qc/\sqrt{{\u03f5}_{\mathrm{M}}}$ region, they have fully evanescent character. In the latter region, the photon and molecular exciton (${\omega}_{\mathrm{ex}}$) hybridize, and an exciton–polariton (${\omega}_{\mathrm{ex}-\mathrm{pol}}$) occurs. The measure of the coupling strength between the exciton ${\omega}_{\mathrm{ex}}$ and the photon is given by Rabi splitting $\Omega $. (

**b**) The evanescent electric field ${E}_{\mu}\left(z\right)$ produced by an exciton–polariton in the ${\mathrm{C}}_{60}$ film.

**Figure 3.**The fullerene molecule C${}_{60}$ centered at $z={z}_{i}$ is physisorbed at the supporting crystal occupying the half-space $z<0$, with the dielectric properties approximated by the macroscopic dielectric function ${\u03f5}_{\mathrm{M}}\left(\omega \right)$.

**Figure 4.**(

**a**) The optical conductivities ${\tilde{\sigma}}_{xx}\left(\omega \right)$ in the ${\mathrm{C}}_{60}$ single layer in vacuum (black solid), in the ${\mathrm{C}}_{60}$ single layer at the Al${}_{2}$O${}_{3}$ dielectric surface (cyan dashed) (where ${z}_{0}=6.5$ Å), and the experimental optical absorption in the gas phase fullerene C${}_{60}$ (red circles). (

**b**) The lower and upper exciton–polariton branches, LPB and UPB, respectively, in the self-standing ${\mathrm{C}}_{60}$ film (blue dots) and the ${\mathrm{C}}_{60}$ film deposited at the Al${}_{2}$O${}_{3}$ surface (red dots). The LPB corresponds to the dispersion relation of the exciton–polariton ${\omega}_{\mathrm{ex}1-\mathrm{pol}}\left(Q\right)$ appearing in Equation (53). The number of ${C}_{60}$ layers is $N=6$. (

**c**) The spectra of the s(TE)-polarized electromagnetic modes $S({Q}_{ex1},\omega )$ in the self-standing ${\mathrm{C}}_{60}$ films for ${Q}_{\mathrm{ex}1}^{0}=0.02\phantom{\rule{3.33333pt}{0ex}}{\mathrm{nm}}^{-1}$ (blue solid) and in the ${\mathrm{C}}_{60}$ films at the Al${}_{2}$O${}_{3}$ surface for ${Q}_{\mathrm{ex}1}=0.035\phantom{\rule{3.33333pt}{0ex}}{\mathrm{nm}}^{-1}$ (red dashed). The number of the single layers is $N=0,1,2,3,\cdots ,10$, where the case $N=0$ corresponds to the spectrum of the free photons in vacuum or at the vacuum/Al${}_{2}$O${}_{3}$ interface (the photon continuum). All spectra for the supported films are multiplied by factor 2.

**Figure 5.**The spectral intensity of the s(TE)-polarized electromagnetic modes $S(Q,\omega )$ in the self-standing ${\mathrm{C}}_{60}$ films for (

**a**) $N=3$, (

**b**) $N=6$, and (

**c**) $N=9$. Figures (

**d**–

**f**) show the same for the ${\mathrm{C}}_{60}$ films deposited at the Al${}_{2}$O${}_{3}$ surface. In both cases, the strong electromagnetic modes (${\omega}_{\mathrm{ex}1-\mathrm{pol}}$, ${\omega}_{\mathrm{ex}2-\mathrm{pol}}$, and ${\omega}_{\mathrm{ex}3-\mathrm{pol}}$) occur in the evanescent regions $\omega <Qc$ and $\omega <Qc/{\u03f5}_{\mathrm{M}}\left(\omega \right)$.

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**MDPI and ACS Style**

Despoja, V.; Marušić, L.
Prediction of Strong Transversal s(TE) Exciton–Polaritons in C_{60} Thin Crystalline Films. *Int. J. Mol. Sci.* **2022**, *23*, 6943.
https://doi.org/10.3390/ijms23136943

**AMA Style**

Despoja V, Marušić L.
Prediction of Strong Transversal s(TE) Exciton–Polaritons in C_{60} Thin Crystalline Films. *International Journal of Molecular Sciences*. 2022; 23(13):6943.
https://doi.org/10.3390/ijms23136943

**Chicago/Turabian Style**

Despoja, Vito, and Leonardo Marušić.
2022. "Prediction of Strong Transversal s(TE) Exciton–Polaritons in C_{60} Thin Crystalline Films" *International Journal of Molecular Sciences* 23, no. 13: 6943.
https://doi.org/10.3390/ijms23136943