# Strong Coupling of Two-Dimensional Excitons and Plasmonic Photonic Crystals:
Microscopic Theory Reveals Triplet Spectra

Lara Greten<EMAIL_ADDRESS>Nichtlineare Optik und
Quantenelektronik, Institut für Theoretische Physik, Technische Universität
Berlin, 10623 Berlin, Germany  Robert Salzwedel  Nichtlineare Optik und
Quantenelektronik, Institut für Theoretische Physik, Technische Universität
Berlin, 10623 Berlin, Germany  Tobias Göde  Nichtlineare Optik und
Quantenelektronik, Institut für Theoretische Physik, Technische Universität
Berlin, 10623 Berlin, Germany  David Greten  Formerly: Institut für
Theoretische Physik, Technische Universität Berlin, 10623 Berlin, Germany
Current address: Fritz Haber Institute of the Max Planck Society, Theory
Department, 14195 Berlin, Germany  Stephanie Reich  Experimentelle
Festkörperphysik, Freie Universität Berlin, 14195 Berlin, Germany  Stephen
Hughes  Department of Physics, Engineering Physics and Astronomy, Queen’s
University, Kingston, Ontario K7L 3N6, Canada  Malte Selig  Nichtlineare Optik
und Quantenelektronik, Institut für Theoretische Physik, Technische
Universität Berlin, 10623 Berlin, Germany  Andreas Knorr  Nichtlineare Optik
und Quantenelektronik, Institut für Theoretische Physik, Technische
Universität Berlin, 10623 Berlin, Germany

###### Abstract

Monolayers of transition metal dichalcogenides (TMDC) are direct-gap
semiconductors with strong light-matter interactions featuring tightly bound
excitons, while plasmonic crystals (PCs), consisting of metal nanoparticles
that act as meta-atoms, exhibit collective plasmon modes and allow one to
tailor electric fields on the nanoscale. Recent experiments show that TMDC-PC
hybrids can reach the strong-coupling limit between excitons and plasmons
forming new quasiparticles, so-called plexcitons. To describe this coupling
theoretically, we develop a self-consistent Maxwell-Bloch theory for TMDC-PC
hybrid structures, which allows us to compute the scattered light in the near-
and far-field explicitly and provide guidance for experimental studies. Our
calculations reveal a spectral splitting signature of strong coupling of more
than $100\,$meV in gold-MoSe2 structures with $30\,$nm nanoparticles,
manifesting in a hybridization of exciton and plasmon into two effective
plexcitonic bands. In addition to the hybridized states, we find a remaining
excitonic mode with significantly smaller coupling to the plasmonic near-
field, emitting directly into the far-field. Thus, hybrid spectra in the
strong coupling regime can contain three emission peaks.

##  I Introduction

The light-matter interaction strength in transition metal dichalcogenide
(TMDC) monolayers has been reported to be extremely strong [1], e.g., as
demonstrated by absorption rates of up to 10% in the visible spectrum [2, 3].
Such a high absorption is particularly noteworthy given the two-dimensional
nature of these materials, which possess a thickness of less than $1\,$nm. In
addition to featuring a direct bandgap [2], TMDC monolayers support in-plane
exciton formation due to their two-dimensional structure [4]. Excitons (bound
electron-hole pairs) therefore dominate the optical spectrum below the band
edge [5]. In addition, the remarkably thin nature of TMDC monolayers results
in an increased sensitivity to surrounding materials. Consequently, the
atomically thin materials can easily be influenced by various factors, such as
the choice of the substrate material, defects [6, 7], and functionalization
[8], e.g., with molecules [9, 10] and heterostructure configurations [11, 12].

Figure 1: Sketch of the hybrid system: the 2D semiconductor (TMDC) is covered
by a square-structured 2D PC of metal nanodisks with the example of gold. The
structure is periodic and infinite in the $xy$-plane and embedded in a
surrounding medium with constant permittivity $\varepsilon$.

In contrast, the optical response of metal nanoparticles (MNPs) is dominated
by localized plasmons which are collective electron oscillations formed within
the metal conduction band [13]. A special feature of MNPs is a significant
amplification of the electric near-field, which additionally allows for
manipulating the electric field on dimensions far below the diffraction limit
[14, 15]. Arranging MNPs as meta-atoms in a crystal structure yields a
plasmonic crystal (PC) with extraordinary strong light-matter interaction. The
localized plasmons couple with the electric field and form plasmon-polaritons
which can propagate within the crystal [16, 17] and sharpen the single
particle plasmon resonance. The optical properties of PCs strongly depend on a
variety of parameters like lattice structure and nanoparticle shape. By
manipulating these, it is possible to tune the optical properties of the
crystal over a wide range [16, 18, 19, 20]. The strong tunability and
enhancement of the electric field make periodic plasmonic structures appealing
for light harvesting and non-linear optics, yielding, e.g., applications for
nanoscale lasing [21, 19] and advanced optical spectroscopy [22, 23].

It has also been shown that the absorption of graphene is significantly
enhanced by depositing plasmonic nanostructures near the graphene layer [24,
25]. An interaction between localized surface plasmons of a single MNP and
excitons in the semiconductor plane can reach the strong coupling regime [26,
27, 28, 29, 30, 31, 32, 33] and absorption rates up to $90\%$ [34]. This
motivates that combining the particular properties of PCs with the
environment-sensitive semiconductor monolayer promises a further increase of
the light-matter-interaction of the excitons in the TMDC.

In the context of our study, we adopt a definition of “strong coupling” where
the interaction strength between the two systems[35, 36] exceeds the total
losses of the system, where the former is quantified by an effective Rabi
splitting $\Omega_{\text{eff}}\approx
2g_{\text{eff}}>\gamma_{\text{ex}}+\gamma_{\text{pl}}$ [32, 31, 37] with
$g_{\text{eff}}$ an effective exciton-plasmon interaction strength. In a
classical theory, this is usually termed normal mode splitting, though the
underlying dissipative modes are quasinormal modes with complex
eigenfrequencies, which are well characterized by Maxwell’s equations [27].
This definition is particularly relevant for exciton-plasmon coupling as the
losses in the plasmonic components are significantly larger than the losses in
the excitonic component [27]. In this work, we study the impact of a two-
dimensional (2D) PC on the excitonic dynamics in TMDCs. A sketch of the hybrid
system is depicted in Fig. 1. The semiconductor is located parallel to the
$xy$-plane. The 2D PC of gold disks (square array) is placed on top of the
TMDC. Experimentally, a similar configuration for relatively large
nanoparticles with an in-plane radius of more than $50$ nm was already
realized in Refs. [38, 39, 40, 41], where TMDCs are coupled to 2D PCs with
different lattice structures.

To theoretically study the exciton-plasmon coupling, we develop a self-
consistent Maxwell-Bloch theory for the hybrid structure in Sec. II. We first
give a short review on the solution of the Maxwell’s equations (Sec. II.1),
the description of the TMDC monolayer with the excitonic Bloch equation (Sec.
II.2) and the 2D PC using Mie theory (Sec. II.3). This allows us in Sec. II.4
to use the plasmonic dipole density within the PC as a source for the
excitonic dipole density of the TMDC, by inserting it into the excitonic Bloch
equations. The resulting formulas are Bloch equations that describe the
dynamics of excitons in the presence of an electric field and the field-
mediated interaction. We highlight the qualitatively different behavior of
momentum-dark excitons, that couple to the plasmon-enhanced electric near-
field, and originally bright excitons with vanishing center-of-mass momentum.
Finally, an expression for the emitted electric field is derived from which we
can deduce transmission, reflection, and absorption of the hybrid structure
when using a plane wave excitation.

Using this theoretical framework, we perform numerical calculations to obtain
the absorption spectra of the hybrid system in Sec. III. Our analysis
resembles the broad range of coupling regimes, varying from weak to strong. In
the strong coupling case, we find an additional peak at the unperturbed
exciton energy, which exhibits a strong dependence with the temperature and
coupling strength. Our theory suggests that this additional mode stems from
the originally bright excitons that do not participate in the strong coupling
to the plasmon-enhanced electric near-field. We find that this excitonic mode
is uniformly distributed in real space, thus also located in the immediate
vicinity of the MNP. The observed strong coupling between TMDCs and MNPs or
PCs is based on the coupling between plasmon and momentum-dark excitons. We
note that there is ongoing discussion regarding exciton-plasmon coupling in
nanoshells, with some studies also suggesting the existence of an undisturbed
excitonic mode [42]. To the best of our knowledge, this mode has not been
observed in experiments for nanoshells [43]. However, our results agree with
recent experiments [40] which observe the presence of such an additional
excitonic peak for TMDC-PC hybrids.

##  II Theoretical Model

In this section, we develop a theoretical framework that describes the
interaction between TMDC excitons and PC plasmons via the radiation field.

###  II.1 Maxwell’s Equations

Starting from Maxwell’s equations, assuming a non-magnetic and isotropic
medium, the wave equation for the electric field reads

$\displaystyle\left(\bm{\nabla}^{2}-\frac{\varepsilon}{c^{2}}{\partial_{t}^{2}}\right)\mathbf{E}(\mathbf{r},t)$
$\displaystyle=\mathbf{S}(\mathbf{r},t),$ (1)

with the source

$\displaystyle\mathbf{S}(\mathbf{r},t)$
$\displaystyle=\left(\frac{\partial_{t}^{2}}{\varepsilon_{0}c^{2}}-\frac{1}{\varepsilon_{0}\varepsilon}\bm{\nabla}\bm{\nabla}\cdot\right)\mathbf{P}(\mathbf{r},t),$
(2)

which is valid for a freestanding sample with dipole density
$\mathbf{P}(\mathbf{r},t)$ embedded in a homogeneous, isotropic and non-
dispersive dielectric environment with permittivity $\varepsilon$. A solution
of Eq. (1) can be formally obtained via the scalar Green’s function
$G(\mathbf{r},\mathbf{r^{\prime}},t-t^{\prime})$,

$\displaystyle\mathbf{E}(\mathbf{r},t)=\int dt^{\prime}\int
d^{3}r^{\prime}\,G(\mathbf{r},\mathbf{r^{\prime}},t-t^{\prime})\,\mathbf{S}(\mathbf{r}^{\prime},t^{\prime}),$
(3)

where the scalar Green’s function depends on the boundary conditions. Both
layers of the hybrid structure, cf. Fig. 1, are assumed to be aligned parallel
to the $xy$-plane with a discrete translation invariance for the PC. We treat
the $z$-direction separately and apply a Fourier transform in the in-plane
coordinates $x$, $y$ and time $t$, where we introduced the new Green’s dyadic
$\mathbcal{G}_{\mathbf{q}_{\parallel}}(z,z^{\prime},\omega)$ that converts the
dipole density at the position $z^{\prime}$ into an electric field and
describes its propagation from $z^{\prime}$ to the observation position $z$.
The term $\mathbcal{G}_{\mathbf{q}_{\parallel}}$ can be derived analytically
[44, 45],

$\displaystyle\mathbcal{G}_{\mathbf{q}_{\parallel}}(z,z^{\prime},\omega)=$
$\displaystyle\ \
\begin{pmatrix}-\frac{\omega^{2}}{\varepsilon_{0}c^{2}}\mathbb{1}+\frac{\mathbf{q}_{\parallel}\otimes\,\mathbf{q}_{\parallel}}{\varepsilon_{0}\varepsilon}&\frac{i\mathbf{q}_{\parallel}}{\varepsilon_{0}\varepsilon}\partial_{z^{\prime}}\\\
&\\\
\frac{i\mathbf{q}_{\parallel}^{T}}{\varepsilon_{0}\varepsilon}\partial_{z^{\prime}}&-\frac{\omega^{2}}{\varepsilon_{0}c^{2}}-\frac{1}{\varepsilon_{0}\varepsilon}\partial_{z^{\prime}}^{2}\end{pmatrix}G_{q_{\parallel}}(z,z^{\prime},\omega).$
(4)

Equation (4) gives the Green’s dyadic in Cartesian coordinates regarding the
$z$-direction whereas the in-plane components are expressed independent of a
special basis for $\mathbf{q}_{\parallel}$, which will be specified in Sec.
II.4. The scalar Green’s function for a constant surrounding permittivity is

$\displaystyle G_{q_{\parallel}}(z,z^{\prime},\omega)$
$\displaystyle=\frac{-i}{2k}e^{ik_{\mathbf{q}_{\parallel}}|z-z^{\prime}|},\hskip
5.69046pt\hskip
5.69046ptk_{\mathbf{q}_{\parallel}}=\sqrt{\frac{\varepsilon}{c^{2}}\omega^{2}-q_{\parallel}^{2}},$
(5)

where we abbreviate the absolute value of a vector by, e.g.,
$|\mathbf{q}_{\parallel}|=q_{\parallel}$.

Next, using the thin film approximation for the individual layers [46, 47],
i.e., with TMDC (ex) and PC (pl), then

$\mathbf{P}_{\mathbf{q}_{\parallel}}(z,\omega)=\sum_{\ell=\text{\\{ex,pl\\}}}\mathbf{P}^{2\text{D},\ell}_{\mathbf{q}_{\parallel}}(\omega)\,\delta(z-z_{\ell})$
(6)

allows to find an algebraic solution of Maxwell’s equations in Fourier space.
Adding the incident field $\mathbf{E}^{0}_{\mathbf{q}_{\parallel}}(z,\omega)$
as a solution of the homogeneous wave equation, Eq. (1), yields

$\mathbf{E}_{\mathbf{q}_{\parallel}}(z,\omega)=\sum_{\ell=\text{\\{ex,pl\\}
}}\mathbcal{G}_{\mathbf{q}_{\parallel}}(z,z_{\ell},\omega)\cdot\mathbf{P}^{2\text{D},\ell}_{\mathbf{q}_{\parallel}}(\omega)+\mathbf{E}^{0}_{\mathbf{q}_{\parallel}}(z,\omega),$
(7)

which allows us to compute the transmission and reflection. Thus, we can
connect the dipole density to experimentally measurable far-field signals.

We assume a perpendicular plane wave excitation that is propagating in the
positive $z$-direction,

$\displaystyle\mathbf{E}^{0}_{\mathbf{q}_{\parallel}}(\omega,z)=e^{ikz}\delta(\mathbf{q}_{\parallel})\,\mathbf{E}^{0}(\omega),$
(8)

with an amplitude perpendicular to the $z$-axis
$\mathbf{E}^{0}\perp\mathbf{e}_{z}$. In the far-field limit, only a
$q_{\parallel}=0$ Fourier component occurs, see Eq. (38) . The transmission is
given by

$\displaystyle
T(\omega)=\frac{|\mathbf{E}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega)|^{2}}{|\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega)|^{2}},\hskip
5.69046pt\,z\rightarrow\infty,$ (9)

and the reflection is

$\displaystyle
R(\omega)=\frac{|\mathbf{E}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega)-\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega)|^{2}}{|\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega)|^{2}},\hskip
5.69046pt\,z\rightarrow-\infty.$ (10)

The absorption is then easily obtained from

$\displaystyle A(\omega)=1-T(\omega)-R(\omega).$ (11)

###  II.2 Excitonic Dipole Density

To describe the response of the TMDC excitons to the electric field, we define
the macroscopic 2D dipole density [48, 46]

$\mathbf{P}^{2\text{D,ex}}_{{\mathbf{q}_{\parallel}}}(\omega)=\sum\limits_{\xi}\mathbf{d}^{\xi}\varphi_{\mathbf{r}_{\parallel}=\mathbf{0}}\,p^{\xi}_{{\mathbf{q}_{\parallel}}}(\omega),$
(12)

with the dipole moment $\mathbf{d}^{\xi}$, $\xi=+/-$ corresponding to the
$K^{+}$ and $K^{-}$ valley, carrying the circular dichroism [49]. The strength
of the dipole moment $\mathbf{d}^{\xi}$ is taken from DFT calculations [49];
the term $\varphi_{\mathbf{r}_{\parallel}=\mathbf{0}}$ accounts for the value
of the 1s excitonic wave function in real space at
$\mathbf{r}_{\parallel}=\mathbf{0}$ and is obtained from the solution of the
Wannier equation [50, 51], incorporating the dielectric environment in the
Rytova-Keldysh approximation [52]. We concentrate on the lowest 1s excitonic
state, since it is energetically separated from higher transitions [5]. A
table with all parameters for MoSe2 can be found in Appendix A. The excitonic
transition amplitude is denoted as $p^{\xi}_{{\mathbf{q}_{\parallel}}}$ and
obtained via its Bloch equation in the Fourier domain [53],

$\left(\hbar\omega-\mathcal{E}_{\mathbf{q}_{\parallel}}+i\gamma(T)\right)\mathbf{p}_{{\mathbf{q}_{\parallel}}}(\omega)=-{\mathbcal{d}}^{*}\cdot\mathbf{E}_{{\mathbf{q}_{\parallel}}}(z_{\text{ex}},\omega),$
(13)

with the dipole tensor

$\displaystyle\mathbcal{d}=\varphi_{\mathbf{r}_{\parallel}=\mathbf{0}}\left(\mathbf{d}^{+},\mathbf{d}^{-},\mathbf{0}\right),$
(14)

and the excitonic transition in vector notation,

$\displaystyle\mathbf{p}_{{\mathbf{q}_{\parallel}}}(\omega)=\left(\begin{array}[]{c}p^{+}_{{\mathbf{q}_{\parallel}}}(\omega)\\\
p^{-}_{{\mathbf{q}_{\parallel}}}(\omega)\\\ 0\end{array}\right).$ (18)

The zero in the $z$ component is added for convenience, to only allow for the
appearance of square matrices.

A detailed derivation is provided in Ref. [54], where the rotating wave
approximation (RWA) is utilized. This approximation imposes a constraint on
the light-matter interaction strength, which must be significantly smaller
than the system energies. Therefore, the range of exciton-plasmon coupling
strength that can be accurately described in this work is limited to
$g_{\text{eff}}/E^{\text{pl}}<0.1$ which is below the ultra-strong coupling
regime [55, 56, 17].

The temperature-dependent dephasing rate $\gamma(T)$ accounts for non-
radiative decay, which typically results from exciton-phonon interactions and
is calculated microscopically [51],

$\displaystyle\gamma(T)=c_{1}T+\frac{c_{2}}{\frac{\Omega}{e^{k_{B}T}}-1},$
(19)

where $c_{1}$, $c_{2},$ and the averaged phonon-energy $\Omega$ are given in
Appendix A. All radiative corrections such as lineshift/splitting and
broadening are incorporated via the self-consistently calculated electric
field $\mathbf{E}_{{\mathbf{q}_{\parallel}}}(z_{\text{ex}},\omega)$ given in
Eq. (7).

The left-hand side of Eq. (13) accounts for the dispersion of excitons

$\displaystyle\mathcal{E}_{\mathbf{q}_{\parallel}}=E^{\text{ex}}+\frac{\hbar^{2}\mathbf{q}_{\parallel}^{2}}{2M},$
(20)

with center-of-mass wave number ${\mathbf{q}_{\parallel}}$ (also referred to
as in-plane momentum), the excitonic mass $M$, the excitonic transition energy
$E^{\text{ex}}$ of 1s-excitons with vanishing momentum. The parabolic exciton
dispersion is illustrated in Fig. 2. In Eq. (13), an excitonic transition
$\mathbf{p}_{{\mathbf{q}_{\parallel}}}$ can be excited by an electric field
$\mathbf{E}_{{\mathbf{q}_{\parallel}}}$ with the same in-plane momentum. The
excitonic transition $\mathbf{p}_{{\mathbf{q}_{\parallel}}}$ is called bright
if it can be excited from the far-field. However, propagating solutions of the
wave equation for the electric field, Eq. (1), are only possible for real
values of $k_{\mathbf{q}_{\parallel}}$, cf. Eq. (5), meaning for small momenta
$q_{\parallel}$. We refer to excitons with higher momenta as momentum-dark, as
they are inaccessible via far-field illumination.

To connect the solution of Eq. (13) with the Green’s dyadic formalism solving
Maxwell’s equations in the circular basis $(+,-)$, we adjust the notation of
Eq. (12) to

$\mathbf{P}^{2\text{D,ex}}_{{\mathbf{q}_{\parallel}}}(\omega)=\mathbcal{d}\cdot\mathbf{p}_{{\mathbf{q}_{\parallel}}}(\omega).$
(21)

Due to the circular dichroism of TMDCs, the dipole tensor is diagonal in a
circular polarized basis, $\mathbcal{d}^{\sigma\xi}=d\delta_{\sigma\xi}$.

Figure 2: Parabolic exciton dispersion over center-of-mass momentum
${\mathbf{q}_{\parallel}}$. A bright and a momentum-dark exciton are indicated
exemplary. The schematic “sun” represents a radiative excitation from the far-
field.

###  II.3 Plasmonic Polarizability

To describe the response of MNPs, we use Mie-Gans theory [57, 58], providing a
parametric frequency dependent polarizability $\bm{\alpha}(\omega)$. The
optical response of every MNP is approximated by a point dipole
$\mathbf{p}^{j}$ at the lattice position $\mathbf{R}^{j}$,

$\mathbf{P}^{2\text{D,pl}}(\mathbf{r}_{\parallel},\omega)=\sum_{j}\mathbf{p}^{j}(\omega)\,\delta(\mathbf{r}_{\parallel}-\mathbf{R}^{j}_{\parallel}),$
(22)

with

$\mathbf{p}^{j}(\omega)=\bm{\alpha}(\omega)\cdot\mathbf{E}(\mathbf{R}^{j},\omega),$
(23)

where $\mathbf{E}(\mathbf{R}^{j},\omega)$ is the electric field, excluding the
field generated by the MNP itself which is already incorporated in
$\bm{\alpha}(\omega)$. The quasi-static polarizability
$\bm{\alpha}^{\text{qs}}$ for spheroids is given by, e.g., Mie-Gans theory
[58, 59]. It can be used for a quasi-static description, if the extensions of
the MNPs are significantly smaller than the wavelength of the exciting light-
field $r_{i}\ll\lambda$. In Cartesian coordinates, with the MNP axes along the
corresponding semi-axes $(i=\\{x,y,z\\})$, the quasi-static polarizability is
diagonal with the entries

$\alpha^{\text{qs}}_{ii}=4\pi\varepsilon_{0}\varepsilon\,r_{x}r_{y}r_{z}\frac{\varepsilon_{\text{Au}}(\omega)-\varepsilon}{3\varepsilon+3L_{i}(\varepsilon_{\text{Au}}(\omega)-\varepsilon)},$
(24)

where $r_{i}$ is the half-axis of the MNP along the corresponding direction,
$\varepsilon_{\text{Au}}(\omega)$ is the permittivity of gold, $\varepsilon$
the surrounding permittivity and $L_{i}$ the shape factor for oblate
spheroids,

$\displaystyle L_{x}$
$\displaystyle=L_{y}=\frac{1}{2e_{0}}\left[\frac{\sqrt{1-e_{0}^{2}}}{e_{0}}\arcsin(e_{0})-(1-e_{0}^{2})\right],$
(25) $\displaystyle L_{z}$ $\displaystyle=1-2L_{x},$ (26)

with the eccentricity,

$e_{0}=1-\frac{r_{z}^{2}}{r_{x}^{2}},\hskip 5.69046pt\text{if}\hskip
5.69046ptr_{x}=r_{y}.$ (27)

To construct the optical response of the MNP-based 2D PC depicted in Fig. 1,
we need to take the radiative coupling between the nanoparticles into account.
Therefore, we consider corrections to the quasi-static Mie-Gans solution of
the single particle polarizability in the modified long-wavelength
approximation (MLWA) including correction terms for dynamic depolarization and
radiation damping [60, 61, 62],

$\alpha^{\text{MLWA}}_{ii}=\frac{\alpha^{\text{qs}}_{ii}}{1-\frac{2}{3}i\,k_{{\mathbf{q}_{\parallel}}=0}^{3}\frac{\alpha^{\text{qs}}_{ii}}{4\pi\varepsilon_{0}\varepsilon}-\frac{k_{{\mathbf{q}_{\parallel}}=0}^{2}}{r_{i}}\frac{\alpha^{\text{qs}}_{ii}}{4\pi\varepsilon_{0}\varepsilon}},$
(28)

with the radiative wavevector
$k_{{\mathbf{q}_{\parallel}}=0}=\sqrt{\epsilon}\frac{\omega}{c}$, compare for
Eq. (5). A full self-consistent solution would inherently include the
radiation damping. However, in Eq. (23) the electric field generated by the
MNP itself is excluded, making this correction necessary. Whether dynamic
depolarization and radiation damping are important for a single MNP depends on
the size of the MNP: for particle diameters significantly smaller than
$\lambda/2\pi$, the static Mie-Gans theory, Eq. (24), would be a sufficient
description. In Sec. III, we evaluate the theory for MNPs with in-plane radii
of $30\,$nm, where corrections due to the finite extent become essential for
the description of their optical properties.

For the gold-MNP permittivity $\varepsilon_{\text{Au}}(\omega)$, as occurring
in Eq. (24), a heuristic, analytical model that fits experimental data for
bulk [63] is parameterized by [64],

$\displaystyle\varepsilon_{\text{Au}}(\omega)=$
$\displaystyle\,\varepsilon_{\infty}-\frac{\omega^{2}_{\text{p}}}{\omega^{2}+i\Gamma(\omega,T)\omega}$
$\displaystyle+\sum_{j=1,2}A_{j}\omega_{j}\left(\frac{e^{i\phi_{j}}}{\omega_{j}-\omega-i\Gamma_{j}}+\frac{e^{-i\phi_{j}}}{\omega_{j}+\omega+i\Gamma_{j}}\right).$
(29)

In the high-frequency limit, the permittivity $\varepsilon_{\infty}$ of gold
differs from unity since $d$-bands in noble metals are filled and provide a
high residual polarization [13]. Equation (29) incorporates a Drude-like
intraband response of the conduction band and two interband transitions with
the possibility to fit asymmetric line shapes in linear bulk spectra (last two
terms in Eq. (29)). We combine the room-temperature permittivity given in Ref.
[64] with a temperature and spectrally dependent linewidth [65, 66, 67, 68]
via the damping term $\Gamma(\omega,T)$ as a sum of electron-electron and
electron-phonon scattering using a Debye model,

$\displaystyle\Gamma(\omega,T)=\Gamma_{\text{el-
el}}(\omega,T)+\Gamma_{\text{el-ph}}(T),$ (30)

with [65, 66, 67, 68]

$\displaystyle\Gamma_{\text{el-el}}(\omega,T)$
$\displaystyle=b\left[(k_{B}T)^{2}+(\hbar\omega/2\pi)^{2}\right],$ (31)
$\displaystyle\Gamma_{\text{el-ph}}(T)$
$\displaystyle=\gamma_{0}\left[\frac{2}{5}+4\left(\frac{T}{\Theta}\right)^{5}\int_{0}^{\Theta/T}\frac{z^{4}}{e^{z}-1}dz\right].$
(32)

A table with all parameters, including the Debye temperature $\Theta$, for the
permittivity of gold is given in Appendix A. We consider the temperature
dependence of the linewidth only for the Drude contribution in Eq. (29), since
it is the main contribution in the well-studied energy regime below the energy
range where interband transitions become relevant, $\hbar\omega<2.4\,$eV [68].
For simplicity, we disregard a subtle redshift of $\omega_{\text{p}}$ for low
temperatures. Equation (30) does not consider radiative damping, since it is
included by self-consistently solving Maxwell’s equations. Equations (23)-(29)
fully describe the response of single nanoparticles to a self-consistently
calculated electric field $\mathbf{E}$, Eq. (7).

In contrast to the single MNP properties, in a PC, the field generated by all
other nanoparticles has to be considered self-consistently. The discrete
translational invariance in the $xy$-plane allows to collect the interactions
with the other MNPs in a modified polarizability
$\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}(\omega)$ in Fourier space. A self-
consistent solution for the two-dimensional dipole density of the 2D PC in
coupled dipole approximation is given by Ref. [69] and experimentally
confirmed for gold nanodisks with $r_{x}=r_{y}\approx 60\,$nm and a lattice
constant $a\approx 500\,$nm in Ref. [61]. The corresponding dipole density
reads

$\mathbf{P}^{2\text{D,pl}}_{{\mathbf{q}_{\parallel}}}(\omega)=\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}(\omega)\cdot\sum\limits_{\mathbf{g}_{\parallel}}\mathbf{E}_{\mathbf{q}_{\parallel}+{\mathbf{g}_{\parallel}}}(z_{\text{pl}},\omega),$
(33)

where the sum includes all reciprocal lattice vectors
${\mathbf{g}_{\parallel}}$ corresponding to Umklapp processes. The effective
polarizability tensor $\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}$ contains
corrections due to interactions between the nanoparticles in the coupled
dipole approximation restricted to MNP center-to-center distances $a\geq
3r_{xy}$. In this limit [70], the effective polarizability is given by

$\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}(\omega)=\frac{1}{A^{\text{pl}}_{\text{UC}}}\left(\left(\bm{\alpha}^{\text{MLWA}}(\omega)\right)^{-1}-\frac{1}{4\pi\epsilon\epsilon_{0}}\mathbcal{F}_{\mathbf{q}_{\parallel}}(\omega)\right)^{-1},$
(34)

normalized by the unit cell area $A^{\text{pl}}_{\text{UC}}$ of the PC. The
form factor

$\displaystyle\mathbcal{F}_{\mathbf{q}_{\parallel}}(\omega)=$ (35)
$\displaystyle\sum_{j\neq
0}e^{i\left(k_{{\mathbf{q}_{\parallel}}=0}|\mathbf{R}_{\parallel}^{j}|-{\mathbf{q}_{\parallel}}\cdot\mathbf{R}_{\parallel}^{j}\right)}\left[\left(\frac{k_{{\mathbf{q}_{\parallel}}=0}^{2}}{|\mathbf{R}_{\parallel}^{j}|}+\frac{i\,k_{{\mathbf{q}_{\parallel}}=0}}{|\mathbf{R}_{\parallel}^{j}|^{2}}-\frac{1}{|\mathbf{R}_{\parallel}^{j}|^{3}}\right)\mathbb{1}\right.$
$\displaystyle-\left.\frac{1}{|\mathbf{R}_{\parallel}^{j}|^{3}}\left(k_{{\mathbf{q}_{\parallel}}=0}^{2}+\frac{3i\,k_{{\mathbf{q}_{\parallel}}=0}}{|\mathbf{R}_{\parallel}^{j}|}-\frac{3}{|\mathbf{R}_{\parallel}^{j}|^{2}}\right)\begin{pmatrix}\mathbf{R}_{\parallel}^{j}\otimes{\mathbf{R}_{\parallel}^{j}}&0\\\
&\\\ 0&0\end{pmatrix}\right],$

accounts for particle interactions, with the $3\times 3$ unity matrix
$\mathbb{1}$. The integer $j$ indexes the nanoparticles at the in-plane
lattice vector positions of the 2D PC. Therefore, the form factor depends on
the lattice structure and lattice constant $a$. If the momentum
${\mathbf{q}_{\parallel}}$ equals $\mathbf{0}$ or a reciprocal lattice vector
${\mathbf{g}_{\parallel}}$, the form factor becomes diagonal with the entries
[71],

$\displaystyle\mathbcal{F}_{{\mathbf{q}_{\parallel}}=\mathbf{0},ii}(\omega)=\mathbcal{F}_{{\mathbf{g}_{\parallel}},ii}(\omega)=\sum_{j\neq
0}e^{ik_{{q_{\parallel}}=0}|\mathbf{R}_{\parallel}^{j}|}\,\,\times$ (36)
$\displaystyle\times\left(\frac{(1-ik_{{q_{\parallel}}=0}|\mathbf{R}_{\parallel}^{j}|)(3X^{j}_{i}-1)}{|\mathbf{R}_{\parallel}^{j}|^{3}}+\frac{k_{{\mathbf{q}_{\parallel}}=0}^{2}(1-X^{j}_{i})}{|\mathbf{R}_{\parallel}^{j}|}\right).$

The angle dependency of the summands is given by
$X_{i}^{j}=(\cos^{2}\theta^{j},\sin^{2}\theta^{j},0)$ for the entries
$i=x,y,z$, respectively, with the polar angle $\theta_{j}$ of
$\mathbf{R}_{\parallel}^{j}$. In the numerical evaluation of the form factor,
Eq. (36), the sum is only slowly converging, leading to spurious oscillations.
This unphysical result can be circumvented using Ewald’s onefold integral
transform [72], yielding a fast converging expression for the form factor.

The Umklapp processes in Eq. (33) are considered in the near-field of the
sample, but not in the far-field $(|z-z_{\text{pl}}|\rightarrow\infty)$. This
is appropriate since dipole densities with in-plane momentum
$\mathbf{q}_{\parallel}$ provide purely evanescent electric fields if the wave
number $k_{\mathbf{q}_{\parallel}}$ is imaginary, cf. Eq. (5). It follows that
the ${\mathbf{g}_{\parallel}}=0$ summand is the only propagating contribution
if

$\displaystyle\real(k_{\mathbf{g}_{\parallel}\neq
0})=0\Leftrightarrow\real(k_{\mathbf{g}_{\parallel}^{\text{min}}\neq 0})=0.$
(37)

The absolute value of the minimal non-trivial reciprocal lattice vector is
$|\mathbf{g}_{\parallel}^{\text{min}}|=2\pi/a$ with the lattice constant $a$,
as depicted in Fig. 1. Consequently, Eq. (37) can be rewritten as a condition
for the lattice constant $a$ compared to the wavelength $\lambda$ of the
incoming light,

$\displaystyle a<\frac{\lambda}{\sqrt{\epsilon}}.$ (38)

If we consider the sum over reciprocal lattice vectors in Eq. (33) as a
diffraction phenomenon, Eq. (38) states that the lattice constant $a$ is small
enough that the diffraction pattern in the far-field only exhibits the main
maximum and not any additional side lobes. However, for the self-consistent
solution, we have to consider all Umklapp processes in Eq. (33), since both
propagating and evanescent electric fields couple to the TMDC excitons which
are located in the near-field of the PC.

###  II.4 Plexcitons

In the following, the electric field, Eq. 7, and the dipole densities, Eq. 12
and 33, are solved self-consistently to provide the near-field and far-field
response of 2D TMDC-PC hybrids to an initially incident plane wave. To solve
the set of equations, we insert the dipole densities of the individual layers,
Eqs. (21) and (33), into the solution of the electric field, Eq. (7), and
altogether in the excitonic Bloch equation, Eq. (13). From this procedure, we
obtain a Bloch equation for the excitonic transition
$\mathbf{p}_{\mathbf{q}_{\parallel}}$,

$\displaystyle\left(\hbar\omega-\mathcal{E}_{\mathbf{q}_{\parallel}}+i\gamma\right)\mathbf{p}_{\mathbf{q}_{\parallel}}(\omega)=-\mathbcal{d}^{*}\cdot$
$\displaystyle\left(\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}}(z_{\text{ex}},\omega)\right.$
$\displaystyle+\mathbcal{G}_{{\mathbf{q}_{\parallel}}}(z_{\text{ex}},z_{\text{ex}},\omega)\cdot\mathbcal{d}\cdot\mathbf{p}_{{\mathbf{q}_{\parallel}}}(\omega)$
$\displaystyle+\mathbcal{G}_{{\mathbf{q}_{\parallel}}}(z_{\text{ex}},z_{\text{pl}},\omega)\cdot\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}(\omega)\cdot\sum\limits_{\scriptscriptstyle\mathbf{q_{\parallel}^{\prime}}={\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}\mathbf{E}^{0}_{\mathbf{q}_{\parallel}^{\prime}}(z_{\text{pl}},\omega)$
$\displaystyle+\left.\mathbcal{G}_{{\mathbf{q}_{\parallel}}}(z_{\text{ex}},z_{\text{pl}},\omega)\cdot\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}(\omega)\cdot\sum\limits_{\scriptscriptstyle\mathbf{\mathbf{q}_{\parallel}^{\prime}}={\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}\mathbcal{G}_{\mathbf{q}_{\parallel}^{\prime}}(z_{\text{pl}},z_{\text{ex}},\omega)\cdot\mathbcal{d}\cdot\mathbf{p}_{\mathbf{q}_{\parallel}^{\prime}}(\omega)\right),$
(39)

where the left-hand side describes the free propagation of the excitons, cf.
Eq. (13), and on the right-hand side the self-consistent electric field occurs
as a function of the dipole densities.

The first term appearing on the right-hand side of (39), after the left
parentheses,
$\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}}(z_{\text{ex}},\omega)$, accounts
for the undisturbed incoming electric field contribution at the TMDC position;
the second term carries the renormalization of the exciton line due to
radiative self-interactions; the third term accounts for the incoming field
$\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}}(z_{\text{pl}},\omega)$ scattered at
the PC; and the fourth (final) term, is the electric field generated by the
excitons and back-scattered from the PC. This term is a radiative near-field
interaction that couples excitonic transitions with momenta
${\mathbf{q}_{\parallel}}$ to those ${\mathbf{q}_{\parallel}^{\prime}}$ which
are shifted by reciprocal lattice vectors $\mathbf{g}_{\parallel}$ of the PC
compared to the excitonic momentum on the left-hand side. This coupling
results from the discretized translational invariance and will later be
interpreted as a binding potential for originally spatially extended excitons.
The terms two and four give rise to an intervalley coupling in Eq. (39), since
their prefactor matrices are non-diagonal in the valley index.

Since for the solution of Eq. (39), we need to consider the observables near
the sample, i.e., in a near-field coupling, we apply a quasi-static
approximation in Eq. (4). We neglect propagation effects by setting $\omega$
to zero and otherwise only consider a parametric $\omega$-dependency, e.g., in
the permittivity $\varepsilon_{\text{Au}}(\omega)$, the polarization densities
or the electric field. This approximation is valid as long as [73]

$\displaystyle\lambda\gg|z_{\text{pl}}-z_{\text{ex}}|,$ (40)

where the wavelength $\lambda$ of the incoming light is compared to the
distance between the two lattices. Thus, we obtain the simplified (quasi-
static) scalar Green’s function

$\displaystyle G_{q_{\parallel}}(z,z^{\prime})$
$\displaystyle=\frac{-1}{2q_{\parallel}}e^{-q_{\parallel}|z-z^{\prime}|},$
(41)

which shows an exponential decrease that depends on the distance between the
source of the electric field at $z^{\prime}$ and the observer with position
$z$. We indicate the quasi-static Green’s function by dropping the dependence
on the frequency $\omega$. However, the Green’s dyadic, Eq. (4), shows that
the quasi-static approximation corresponds to neglecting $\omega^{2}$ compared
to the in-plane momentum $q_{\parallel}^{2}$ which is only reasonable if

$\displaystyle\frac{\omega^{2}}{c^{2}}<\frac{q_{\parallel}^{2}}{\epsilon}.$
(42)

In the following, we refer to electric fields and dipole densities that
fulfill the condition in Eq. (42) as outside the light-cone. No propagating
solution of the wave equations is possible, since the electric field is damped
exponentially with the distance, cf. Eqs. (5) and (41). In contrast, in-plane
momenta that do not fulfill the condition are inside the light-cone. The
condition (42) is only valid for excitonic momenta ${\mathbf{q}_{\parallel}}$
belonging to originally momentum-dark excitonic transitions. However, an
inspection of Eqs. (7) and (33) shows that, for a perpendicular plane wave
excitation, cf. Eq. (8), the plasmonic dipole density is only non-zero for in-
plane momenta equal to a reciprocal lattice vector $\mathbf{g}_{\parallel}$ of
the PC. Since only the zeroth order scattering momentum
$\mathbf{g}_{\parallel}=0$ is a propagating solution (inside the light cone),
it is allowed to apply the quasi-static approximation for all momenta
${\mathbf{q}_{\parallel}}$ of the excitonic transition, except for
${\mathbf{q}_{\parallel}}=\mathbf{0}$. Therefore, we split Eq. (39) into a
near-field Bloch equation ($q_{\parallel}>0$) and a radiative contribution
($q_{\parallel}=0$) and discuss them separately in the following (Sec. II.4.1,
II.4.2, respectively). In particular, by inserting the Green’s function for a
homogeneous environment permittivity in quasi-static approximation, Eq. (41),
one observes that the right-hand side of Eq. (39) is proportional to the in-
plane momentum. The momentum-dark excitonic transition
$\mathbf{p}_{{\mathbf{q}_{\parallel}}\neq\mathbf{0}}(\omega)$ (Sec. II.4.1)
does not couple to the radiative excitonic transition
$\mathbf{p}_{{\mathbf{q}_{\parallel}}=0}(\omega)$ (Sec. II.4.2) and can be
solved independently.

####  II.4.1 Near-Field Exciton-Plasmon Interaction

The resulting Bloch equations for $q_{\parallel}\neq 0$ are identical to Eq.
(39), with the quasi-static approximation applied to every Green’s dyadic. To
account for the circular dichroism of the TMDC excitons, we choose a circular
polarized basis $(\mathbf{e}^{+},\mathbf{e}^{-},\mathbf{e}^{z})$. Any quantity
in Cartesian coordinates is transformed to the circular polarized basis by
multiplying with the unitary (change-of-basis) matrix
$\mathbcal{T}^{\text{circ}}$,

$\displaystyle\begin{pmatrix}E^{+}\\\ E^{-}\\\
E^{z}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&-i&0\\\ 1&i&0\\\
0&0&\sqrt{2}\end{pmatrix}\cdot\begin{pmatrix}E^{x}\\\ E^{y}\\\
E^{z}\end{pmatrix}\equiv\mathbcal{T}^{\text{circ}}\cdot\begin{pmatrix}E^{x}\\\
E^{y}\\\ E^{z}\end{pmatrix}.$ (43)

In the quasi-static limit, a diagonalization of Eq. (39) in circular polarized
basis corresponding to the valleys $K^{+}$ and $K^{-}$ is given by a
transformation $T_{\phi}$, Eq. (44), that depend on the polar angle $\phi$ of
the in-plane momentum ${\mathbf{q}_{\parallel}}$,

$\displaystyle\begin{pmatrix}p^{U}_{\mathbf{q}_{\parallel}}\\\
p^{V}_{\mathbf{q}_{\parallel}}\\\
p^{z}_{\mathbf{q}_{\parallel}}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}-e^{i\phi}&e^{-i\phi}&0\\\
e^{i\phi}&e^{-i\phi}&0\\\
0&0&\sqrt{2}\end{pmatrix}\cdot\begin{pmatrix}p^{+}_{\mathbf{q}_{\parallel}}\\\
p^{-}_{\mathbf{q}_{\parallel}}\\\
p^{z}_{\mathbf{q}_{\parallel}}\end{pmatrix}\equiv\mathbcal{T}_{\phi}\cdot\begin{pmatrix}p^{+}_{\mathbf{q}_{\parallel}}\\\
p^{-}_{\mathbf{q}_{\parallel}}\\\
p^{z}_{\mathbf{q}_{\parallel}}\end{pmatrix}.$ (44)

The transformation $\mathbcal{T}_{\phi}$ corresponds to an in-plane rotation
of the momentum space, orienting the momentum ${\mathbf{q}_{\parallel}}$ along
the $V$-axis. We find that the transformation into the
$(\mathbf{e}^{U},\mathbf{e}^{V},\mathbf{e}^{z})$ basis is the diagonalization
of the in-plane contribution of the quasi-static Green’s dyadic,

$\displaystyle\mathbcal{G}^{U/V}_{{\mathbf{q}_{\parallel}}}(z,z^{\prime})=$
$\displaystyle\frac{1}{\varepsilon_{0}\varepsilon}\begin{pmatrix}0&0&0\\\
0&q_{\parallel}^{2}&iq_{\parallel}\partial_{z^{\prime}}\\\
0&iq_{\parallel}\partial_{z^{\prime}}&-\partial_{z^{\prime}}^{2}\end{pmatrix}G_{q_{\parallel}}(z,z^{\prime}).$

Applying the rotation to the Bloch equations (39) finally allows a
diagonalization process,

$\displaystyle\left(\hbar\omega-\mathcal{E}_{\mathbf{q}_{\parallel}}+i\gamma\right)p^{U}_{{\mathbf{q}_{\parallel}}\neq\mathbf{0}}(\omega)=0,$
(45)
$\displaystyle\left(\hbar\omega-\mathcal{E}^{V}_{\mathbf{q}_{\parallel}}+i\gamma\right)p^{V}_{{\mathbf{q}_{\parallel}}\neq\mathbf{0}}(\omega)+\sum\limits_{\scriptscriptstyle\mathbf{q_{\parallel}^{\prime}}={\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}C^{\text{ex}\leftrightarrow\text{pl}}_{{\mathbf{q}_{\parallel}}\mathbf{q_{\parallel}^{\prime}}\omega}\,p^{V}_{\mathbf{q}_{\parallel}^{\prime}}(\omega)$
$\displaystyle=-\frac{d^{*}}{\epsilon_{0}\epsilon}\mathbf{S}^{*\text{ex}\leftarrow\text{pl}}_{{\mathbf{q}_{\parallel}}}(\omega)\cdot\sum\limits_{\scriptscriptstyle\mathbf{q_{\parallel}^{\prime}}={\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}\mathbf{E}^{0}_{\mathbf{q}_{\parallel}^{\prime}}(z_{\text{pl}},\omega),$
(46)

where Eq. (45) possesses the undisturbed, parabolic ($U$-shaped) exciton
dispersion for $p^{U}_{\mathbf{q}_{\parallel}}$. In contrast, we find

$\displaystyle\mathcal{E}^{V}_{\mathbf{q}_{\parallel}}=E^{\text{ex}}+\frac{\hbar^{2}q_{\parallel}^{2}}{2M}-\frac{|d|^{2}}{\epsilon_{0}\epsilon}q_{\parallel}^{2}\,G_{q_{\parallel}}^{\text{ex}-\text{ex}}\approx
E^{\text{ex}}+\beta q_{\parallel},$ (47) $\displaystyle\text{with}\hskip
14.22636pt\beta=\frac{|d|^{2}}{2\epsilon_{0}\epsilon},$

yielding a conic ($V$-shaped) dispersion for $p^{V}_{\mathbf{q}_{\parallel}}$
similar to results reported in Ref. [74]. The excitonic self-interaction term,
$q_{\parallel}^{2}\,G_{q_{\parallel}}^{\text{ex}-\text{ex}}$, dominates the
kinetic energy contribution for small $q_{\parallel}$. The optical sources of
$p^{V}_{\mathbf{q}_{\parallel}}$, cf. Eq. (46), occur on the right. Obviously,
the near-field interaction of exciton and plasmon is fully encoded in
$p^{V}_{\mathbf{q}_{\parallel}}$. The interaction with the PC is denoted by
the contributions $(\text{ex}\leftrightarrow\text{pl})$ and provides a
momentum and frequency dependent PC-induced excitonic self-interaction
$C^{\text{ex}\leftrightarrow\text{pl}}_{{\mathbf{q}_{\parallel}}\mathbf{q_{\parallel}^{\prime}}\omega}$,
namely, a plasmon mediated effective exciton-exciton interaction. For a
concise notation without loss of information, we indicate the $z$-dependencies
of the prefactor Green’s functions in their superscripts. On the right-hand
side of Eq. (46), the existence of an incoming field
$\mathbf{E}^{0}_{\mathbf{q}_{\parallel}}$, that was scattered at the PC,
results in the excitation of the conic excitonic transition. The interaction
with the PC provides a coupling to in-plane momenta shifted by a reciprocal
lattice vector $\mathbf{g}_{\parallel}$ of the PC due to Umklapp processes
depending on the PC lattice structure. The PC-induced excitonic self-
interaction reads

$\displaystyle
C^{\text{ex}\leftrightarrow\text{pl}}_{{\mathbf{q}_{\parallel}}\mathbf{q_{\parallel}^{\prime}}\omega}=$
$\displaystyle\left(\cos\phi\cos\phi^{\prime}\alpha^{*}_{xx}(\omega)+\sin\phi\cos\phi^{\prime}\alpha^{*}_{yx}(\omega)\right.$
$\displaystyle+\cos\phi\sin\phi^{\prime}\alpha^{*}_{xy}(\omega)+\sin\phi\sin\phi^{\prime}\alpha^{*}_{yy}(\omega)$
$\displaystyle+\left.\alpha^{*}_{zz}(\omega)\right)\frac{|d|^{2}}{(\epsilon_{0}\epsilon)^{2}}q_{\parallel}^{2}{q_{\parallel}^{\prime}}^{2}G_{q_{\parallel}}^{\text{ex-
pl}}G_{q_{\parallel}^{\prime}}^{\text{pl-ex}},$ (48)

and the source term that arises from the scattering of the incoming field at
the PC becomes

$\displaystyle\mathbf{S}^{\text{ex}\leftarrow\text{pl}}_{{\mathbf{q}_{\parallel}}}(\omega)=$
$\displaystyle\,q_{\parallel}^{2}G_{q_{\parallel}}^{\text{ex}-\text{pl}}\begin{pmatrix}\alpha^{*}_{xx}(\omega)\cos\phi+\alpha^{*}_{yx}(\omega)\sin\phi\\\
\alpha^{*}_{xy}(\omega)\cos\phi+\alpha^{*}_{yy}(\omega)\sin\phi\\\
i\alpha^{*}_{zz}(\omega)\text{sgn}(z_{ex}-z_{pl})\end{pmatrix},$ (49)

if we express
$\mathbf{E}^{0}_{\mathbf{q}_{\parallel}^{\prime}}(z_{\text{pl}},\omega)$ in
Cartesian coordinates. To achieve these expressions, derivatives of the
Green’s function with respect to $z^{\prime}$ have been carried out, cf. Eq.
(4).

Since we were able to decouple the valleys of the excitonic transition
$\\{p^{+}_{\mathbf{q}_{\parallel}},p^{-}_{\mathbf{q}_{\parallel}}\\}$ in Eq.
(39) into a conic (optically driven $p^{V}_{\mathbf{q}_{\parallel}}$, Eq.
(46)) and an undisturbed parabolic ($p^{U}_{\mathbf{q}_{\parallel}}$, Eq.
(45)) contribution, we can now solve them independently. The parabolic Bloch
equation for $p^{U}_{\mathbf{q}_{\parallel}}$ immediately yields

$\displaystyle p^{U}_{\mathbf{q_{\parallel}\neq 0}}(\omega)=0.$ (50)

In contrast, in Eq. (46), the conic Bloch equation for
$p^{V}_{\mathbf{q}_{\parallel}}$ contains a coupling of excitons with
different center-of-mass momenta, mediated by the PC. We project the excitonic
transition $p^{V}_{\mathbf{q}_{\parallel}}$ onto a set of eigenstates of the
conic dispersion,

$\displaystyle p^{V}_{\mathbf{q_{\parallel}\neq
0}}(\omega)=\sum_{\lambda}v^{R\lambda}_{\mathbf{q}_{\parallel}}(\omega)p^{\lambda}(\omega).$
(51)

The corresponding symmetric, non-Hermitian eigenvalue problem reads similar to
Ref. [26],

$\displaystyle\mathcal{E}^{V}_{\mathbf{q}_{\parallel}}v^{R\lambda}_{{\mathbf{q}_{\parallel}}}(\omega)-\sum\limits_{\scriptscriptstyle\mathbf{q_{\parallel}^{\prime}}={\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}C^{\text{ex}\leftrightarrow\text{pl}}_{{\mathbf{q}_{\parallel}}\mathbf{q_{\parallel}^{\prime}}\omega}\,v^{R\lambda}_{\mathbf{q}_{\parallel}^{\prime}}(\omega)=E^{\lambda}(\omega)v^{R\lambda}_{\mathbf{q}_{\parallel}}(\omega).$
(52)

As a consequence of the non-Hermiticity we have to distinguish between left
and right eigenstates
$v^{L\lambda}_{\mathbf{q}_{\parallel}},\,v^{R\lambda}_{\mathbf{q}_{\parallel}}$.
To justify the projection onto these states, we verify the existence of
solutions of Eq. (52) and their completeness numerically. By applying a
suitable normalization, we ensure orthonormality using

$\displaystyle\sum_{\mathbf{g}_{\parallel}}\left(v^{L\lambda}_{{\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}(\omega)\right)^{*}\left(v^{R\lambda^{\prime}}_{{\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}(\omega)\right)=\delta_{\lambda\lambda^{\prime}}.$
(53)

With this expression, we project the conic excitonic transition in the Bloch
equation (46) on the new states according to Eq. (51). Substituting the
momentum dependencies with the corresponding eigenvalue in Eq. (52) and taking
advantage of the orthonormality condition between the eigenstates finally
allows us to give a solution for the conic contribution of the excitonic
transition

$\displaystyle p^{V}_{\mathbf{q_{\parallel}\neq 0}}(\omega)$
$\displaystyle=-d^{*}\sum\limits_{\lambda,\mathbf{q}_{\parallel}^{\prime}={\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}\frac{v^{R\lambda}_{{\mathbf{q}_{\parallel}}}(\omega)v^{L\lambda^{*}}_{{\mathbf{q}_{\parallel}}^{\prime}}(\omega)}{\hbar\omega-E^{\lambda}(\omega)+i\gamma}\times$
(54)
$\displaystyle\times\left(\frac{1}{\epsilon_{0}\epsilon}\mathbf{S}^{*\text{ex}\leftarrow\text{pl}}_{\mathbf{q}_{\parallel}^{\prime}}(\omega)\cdot\sum\limits_{{\mathbf{g}_{\parallel}}}\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}(z_{\text{pl}},\omega)\right).$

To connect the excitonic transition with observables, we insert it in the
dipole density of the TMDC layer. The contribution arising from the conic part
becomes

$\displaystyle{P^{2\text{D},\text{ex}}_{{\mathbf{q}_{\parallel}}\neq\mathbf{0}}\hskip
1.42271pt}^{\pm}(\omega)=d\frac{e^{\mp
i\phi}}{\sqrt{2}}\sum\limits_{\lambda}v^{R\lambda}_{{\mathbf{q}_{\parallel}}}(\omega)p^{\lambda}(\omega).$
(55)

The full dipole density stemming from momentum-dark excitons trapped in the
plasmonic potential is finally given by

$\displaystyle{P^{2\text{D},\text{ex}}_{{\mathbf{q}_{\parallel}}={\mathbf{g}_{\parallel}}\neq\mathbf{0}}}^{\pm}\hskip
1.42271pt(\omega)$
$\displaystyle=-\frac{|d|^{2}}{\sqrt{2}{\epsilon\epsilon_{0}}}e^{\mp
i\phi}\times$ (56)
$\displaystyle\times\sum\limits_{\lambda{\mathbf{g}_{\parallel}}}$
$\displaystyle\frac{v^{R\lambda}_{{\mathbf{q}_{\parallel}}}(\omega)v^{L\lambda^{*}}_{{\mathbf{g}_{\parallel}}}(\omega)}{\hbar\omega-E^{\lambda}+i\gamma}\mathbf{S}^{*\text{ex}\leftarrow\text{pl}}_{\mathbf{g}_{\parallel}^{\prime}}(\omega)\cdot\mathbf{\tilde{E}}^{0}_{{\mathbf{g}_{\parallel}}}(z_{\text{pl}},\omega).$

For momenta that do not correspond to the wave number of the incoming field
shifted by a reciprocal lattice vector, the TMDC polarization vanishes due to
the absence of a suitable excitation.

####  II.4.2 Radiative Exciton-Plasmon Interaction

In the previous section, the excitonic dipole density outside the light-cone
was derived. However, to determine the far-field response, we need an
expression for the dipole density within the light cone which is a source for
propagating solutions of the electric field. Therefore, we evaluate the Bloch
equation (39) for $q_{\parallel}=0$, i.e., the radiative Bloch equations,

$\displaystyle\left(\hbar\omega-E^{\text{ex}}+i\gamma\right)\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}$
$\displaystyle=$
$\displaystyle-\biggl{(}\mathbcal{d}^{*}\cdot\mathbcal{G}(z_{\text{ex}},z_{\text{pl}},\omega)\cdot\alpha^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(\omega)\cdot\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z_{\text{pl}},\omega)$
$\displaystyle+$
$\displaystyle\mathbcal{d}^{*}\cdot\mathbcal{G}(z_{\text{ex}},z_{\text{pl}},\omega)\cdot\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(\omega)\cdot\sum\limits_{\scriptscriptstyle\mathbf{q_{\parallel}^{\prime}}={\mathbf{q}_{\parallel}}+\mathbf{g}_{\parallel}}\mathbcal{G}_{\mathbf{q}_{\parallel}^{\prime}}(z_{\text{pl}},z_{\text{ex}})\cdot\mathbcal{d}\cdot\,\mathbf{p}_{\mathbf{q}_{\parallel}^{\prime}}$
$\displaystyle+$
$\displaystyle\mathbcal{d}^{*}\cdot\mathbcal{G}(z_{\text{ex}},z_{\text{pl}},\omega)\cdot\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(\omega)\cdot\mathbcal{G}(z_{\text{pl}},z_{\text{ex}},\omega)\cdot\mathbcal{d}\cdot\,\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}$
$\displaystyle+$
$\displaystyle\mathbcal{d}^{*}\cdot\mathbcal{G}(z_{\text{ex}},z_{\text{ex}},\omega)\cdot\mathbcal{d}\cdot\,\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}\biggr{)}$
$\displaystyle-\mathbcal{d}^{*}\cdot\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z_{\text{ex}},\omega).$
(57)

For clarity, we suppress the index of the vanishing in-plane momentum
${\mathbf{q}_{\parallel}}=\mathbf{0}$ in the notation for the radiative
Green’s dyadic. The previous provides us with all the necessary tools to solve
this equation. Simplifying the individual terms on the right-hand side is
demanding but straight-forward by explicitly evaluating the matrix products
between the dipole elements, Eq. (14), the Green’s dyadic, Eq. (4) and the PC
polarizability $\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}(\omega)$. Finally, we
collect all summands containing
$\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}$ on the left-hand side and
multiply with the inverse of its prefactor. This procedure yields

$\displaystyle\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}=\left[\left(\hbar\omega-E^{\text{ex}}+i\gamma-|d|^{2}\frac{\omega^{2}}{\varepsilon_{0}c^{2}}G_{q_{\parallel}=0,\omega}^{\text{ex-
ex}}\right)\mathbb{1}+|d|^{2}\,\left(\frac{\omega^{2}}{\varepsilon_{0}c^{2}}\right)^{2}\,G_{q_{\parallel}=0,\omega}^{\text{ex-
pl}}\,G_{q_{\parallel}=0,\omega}^{\text{pl-
ex}}\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(\omega)\right]^{-1}\cdot$
(58)
$\displaystyle\cdot\left[-d^{*}\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z_{\text{ex}},\omega)+d^{*}\frac{\omega^{2}}{\varepsilon_{0}c^{2}}G_{q_{\parallel}=0,\omega}^{\text{ex-
pl}}\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}\cdot\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z_{\text{pl}},\omega)+|d|^{2}\frac{\omega^{2}}{\varepsilon_{0}c^{2}}\sum\limits_{\scriptscriptstyle\mathbf{q_{\parallel}^{\prime}}=\mathbf{g}_{\parallel}}\frac{{q_{\parallel}^{\prime}}^{2}}{\varepsilon_{0}\varepsilon}G_{q_{\parallel}=0,\omega}^{\text{ex-
pl}}G_{q_{\parallel}^{\prime},\omega=0}^{\text{pl-
ex}}\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}\cdot\mathbf{e}_{\mathbf{q}_{\parallel}^{\prime}}\,p^{V}_{\mathbf{q}_{\parallel}^{\prime}}\right],$

with the unity vector in polar coordinates

$\displaystyle\mathbf{e}_{\mathbf{q}_{\parallel}^{\prime}}=\begin{pmatrix}\cos{\phi^{\prime}}\\\
\sin{\phi^{\prime}}\\\ 0\end{pmatrix},$

when we express the excitonic transition
$\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}$ and the PC polarizability
$\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}}$ in a linear polarized basis.

The denominator of Eq. (58) corresponds to harmonic oscillators at frequency
$\omega$ with renormalized eigenenergies

$\displaystyle
E^{\text{ex}}\,\mathbb{1}-|d|^{2}\,\left(\frac{\omega^{2}}{\varepsilon_{0}c^{2}}G_{q_{\parallel}=0,\omega}^{\text{ex-
pl}}\right)^{2}\,\real\textnormal{e}(\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(\omega)),$
(59)

and the damping

$\displaystyle\gamma\,\mathbb{1}$
$\displaystyle+i|d|^{2}\frac{\omega^{2}}{\varepsilon_{0}c^{2}}G_{q_{\parallel}=0,\omega}^{\text{ex-
ex}}\,\mathbb{1}$
$\displaystyle+|d|^{2}\left(\frac{\omega^{2}}{\varepsilon_{0}c^{2}}G_{q_{\parallel}=0,\omega}^{\text{ex-
pl}}\\!\right)^{2}\\!\\!\imaginary\textnormal{m}(\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(\omega)).$

The ex-ex term accounts for the radiative dephasing of the TMDC excitons
whereas the last term governs radiative interference phenomena between the
TMDC and the PC. For a resonant TMDC-PC interaction
($E^{\text{ex}}=E^{\text{pl}}$), the real part of the PC polarizability can be
neglected
$\real\textnormal{e}(\bm{\alpha}^{*}_{\mathbf{q}_{\parallel}}(\hbar^{-1}E^{\text{pl}}))\approx
0$. This yields an unchanged excitonic eigenenergy and the coupling to the PC
mainly modifies the damping. The numerator includes from left to right: the
direct excitation of excitonic transitions via the incident electric field,
the plasmon mediated excitation from the incoming field that was first
scattered at the PC, and finally the plasmon-mediated influence of momentum-
dark excitonic transitions. From the derived excitonic transition, we deduce
the dipole density via Eq. (21).

####  II.4.3 Scattered Electric Field

To give explicit results for the electric field emitted by the TMDC layer
$\mathbf{E}_{{\mathbf{q}_{\parallel}}}^{\text{ex}}$, we multiply the dipole
density, that was derived in the previous section, with the Green’s dyadic. We
find the momentum-dark dipole density to be an eigenstate of the quasi-static
Green’s dyadic, cf. Eq. (4). Therefore, we find

$\displaystyle\mathbf{E}^{\text{ex}}_{{\mathbf{q}_{\parallel}}\neq\mathbf{0}}(z,\omega)$
$\displaystyle=\frac{q_{\parallel}^{2}}{\epsilon\epsilon_{0}}G_{{q_{\parallel}}}(z,z_{\text{ex}},\omega=0)\,\mathbf{P}^{2\text{D},\text{ex}}_{{\mathbf{q}_{\parallel}}\neq\mathbf{0}}$
(60)
$\displaystyle\mathbf{E}^{\text{ex}}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega)$
$\displaystyle=-\frac{\omega^{2}}{\epsilon_{0}c^{2}}G_{{q_{\parallel}}=0}(z,z_{\text{ex}},\omega)\,\mathbf{P}^{2\text{D},\text{ex}}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}.$
(61)

In fact, also a $z$-component of the electric near-field
$E^{\text{ex}^{z}}_{{\mathbf{q}_{\parallel}}\neq\mathbf{0}}(z,\omega)$ occurs.
However, it only causes a non-vanishing $z$-component of the PC polarization
since $\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}}$ is assumed to be diagonal
regarding its $z$-entries. This $z$-polarization of the PC is not transferred
to the far-field for ${\mathbf{q}_{\parallel}}=\mathbf{0}$. To obtain the
contribution of the PC to the far-field,
$\mathbf{E}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}^{\text{pl}}$, we deduce the
dipole density of the 2D PC, according to Eq. (33). It is

$\displaystyle\mathbf{P}^{\text{2D},\text{pl}}_{{\mathbf{q}_{\parallel}}}(\omega)$
$\displaystyle=\frac{\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}}(\omega)}{A_{\text{UC}}^{\text{pl}}}\cdot$
(62)
$\displaystyle\sum\limits_{{\mathbf{g}_{\parallel}}}\left(\mathbf{E}^{0}_{{{\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}}(z_{\text{pl}},\omega)+\mathbf{E}_{{\mathbf{q}_{\parallel}}+{\mathbf{g}_{\parallel}}}^{\text{ex}}(z_{\text{pl}},\omega)\right).$

As already justified above for dense PCs, cf. Eq. (38), there is no
propagating solution of Maxwell’s equations with an in-plane wavenumber equal
to a non-trivial reciprocal lattice vector. Therefore, we may drop the sum
over ${\mathbf{g}_{\parallel}}\neq 0$ for the exciting field. As for the
radiative TMDC contribution, the dyadic Green’s function has to contain the
full time dependency. Furthermore, since $z\neq z^{\prime}$, the $z$-entries
of the Green’s dyadic vanish. The in-plane far-field plasmon contribution
reads

$\displaystyle{\mathbf{E}}^{\text{pl}}_{{{\mathbf{q}_{\parallel}}=\mathbf{0}}}(z,\omega)=\frac{-\omega^{2}}{\epsilon_{0}c^{2}}G_{q_{\parallel}=0}(z,z_{\text{pl}},\omega)\,\bm{\alpha}^{*}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(\omega)\cdot$
(63)
$\displaystyle\cdot\left(\mathbf{E}^{0}_{{{\mathbf{q}_{\parallel}}}=\mathbf{0}}(z_{\text{pl}},\omega)+\mathbf{E}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}^{\text{ex}}(z_{\text{pl}},\omega)+\sum\limits_{\mathbf{g}_{\parallel}\neq\mathbf{0}}\mathbf{E}_{\mathbf{g}_{\parallel}\neq\mathbf{0}}^{\text{ex}}(z_{\text{pl}},\omega)\right).$

Adding all contributions of the electric field according to Eq. (7), results
in

$\mathbf{E}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega)=\mathbf{E}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}^{\text{ex}}(z,\omega)+\mathbf{E}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}^{\text{pl}}(z,\omega)+\mathbf{E}^{0}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}(z,\omega).$
(64)

####  II.4.4 Overview of the Theory

Figure 3: Schematic of the theory: the PC couples to the bright and momentum-
dark TMDC excitons which are illustrated on the parabolic dispersion curve.
The sun represents the far-field excitation. The gray arrows account for the
exciton-plasmon coupling, with the indicated coupling strengths.

With the theoretical framework now fully established, our attention turns to
shedding some light on the physical meaning of the derived equations. Figure 3
provides a schematic of the theory.

We excite the TMDC-PC hybrid via far-field illumination
$\mathbf{E}^{0}_{\mathbf{q}_{\parallel}}$, Eq. (8), represented by the sun-
like schematic in Fig. 3. It directly acts on the PC plasmons, Eq. (62), and
the bright excitons $\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}$, Eq.
(58). The scattering respectively Umklapp processes of the incoming light at
the PC provide access to momentum-dark excitonic transitions, Eq. (46), via
the plasmon-enhanced electric near-field. These interactions are typical
dipole-dipole near-field interactions, proportional to
$q_{\parallel}e^{-{q_{\parallel}}\delta z}$, which can be seen by explicitly
inserting the quasi-static Green’s function, Eq. (41), into the conic Bloch
equation, Eq. (46). The coupling between the plasmon and the bright exciton
behaves qualitatively different with the coupling strength significantly
reduced, proportional to ${\omega}/{c}$, cf. Eq. (58). The implications of
these different coupling mechanisms are numerically evaluated in Sec. III.
Finally, we note that, in the limit $a\rightarrow\infty$, the developed theory
is also applicable for a TMDC coupled to a single MNP without qualitatively
changing the results.

Figure 4:  Absorption for different distances $\delta
z=|z_{\text{pl}}-z_{\text{ex}}|$ between TMDC and PC at room temperature,
corresponding to a scan over an effective coupling strength
$g_{\mathrm{eff}}$. (a) Absorption spectra for $\delta z=[11,17,30]\,$nm
belonging to different interaction regimes, from weak
$(2g_{\mathrm{eff}}<\gamma_{\mathrm{ex}}+\gamma_{\mathrm{pl}})$ to strong
$(2g_{\mathrm{eff}}>\gamma_{\mathrm{ex}}+\gamma_{\mathrm{pl}})$ coupling. For
comparison, the artificially uncoupled case (gray), is the sum of single TMDC
(blue) and PC (yellow) absorption in the inset. (b) Color map of the
absorption depending on the distance $\delta z$ and the Detuning $\Delta$. The
vertical lines indicate the spectra shown in panel a). Inset: Effective Rabi
energy $\Omega_{\text{eff}}$ over $\delta z$. It reaches values significantly
larger than $100\,$meV. The peak splitting extracted from the full theory is
compared to a fit $b/(\delta z)^{2}$ with $b=17.24\,\text{nm}^{2}\text{eV}$.

##  III Numerical Results and Discussion

We numerically evaluate the key equations (63), (61), (60), (56) and (52) to
calculate the absorption, Eq. (11), of the hybrid structure. If not stated
differently, we perform calculations for a MoSe2-PC stack with the geometry
specified in Table 1. The given parameters yield a PC with its resonance
energy at $E^{\text{pl}}=2.03\,$eV. We choose the lattice constant $a$ in a
way that the collective PC resonance sharpens the single MNP response that is
governed by the bulk metal parameters and aspect ratio $r_{z}/r_{x}$. Material
specific parameters for gold and MoSe2 are given in Appendix A.

Table 1: Geometry of TMDC-PC hybrid $r_{x}$ |  $30\,$nm | $a$ | $10\,r_{x}$
---|---|---|---
$r_{y}$ |  $30\,$nm | $\delta z$ |  $11\,$nm
$r_{z}$ |  $10\,$nm | $\varepsilon$ | $2.4$

###  III.1 Coupling Regimes at Room Temperature

To study different coupling regimes, the modification of the layer distance
$\delta z$ allows to directly access the strength of the near-field mediated
interaction. Figure 4 shows the absorption of the hybrid over the detuning
$\Delta=\hbar\omega-E^{\text{pl}}$ between the electric field and the plasmon
for the resonant case for exciton and plasmon $(E^{\text{ex}}=E^{\text{pl}})$
for different exciton-plasmon distances $\delta z$. Modifying the distance
$\delta z$ qualitatively changes the hybrid’s spectrum. All cases shown in
Fig. 4 have been observed in experiments with TMDC excitons coupled to MNPs or
PCs, yet lacking a microscopic theory, which we provide from the excitonic
perspective.

####  III.1.1 Weak Coupling ($\delta z=30\,\mathrm{nm}$, $\delta
z=17\,\mathrm{nm})$)

For MNP-TMDC distances where $\delta z$ is approximately equal to or exceeds
the effective extensions of the MNP, the exciton-plasmon coupling is weak.
Thus, for $\delta z\geq 30\,$nm, the line shape is qualitatively preserved
compared to the uncoupled case. However, the interaction induces an additional
damping, increasing the linewidth and reducing the maximal absorption. This is
because the interaction terms in the denominators of Eqs. (56) and (58) are
imaginary, thus effectively acting like an additional damping to the phonon
damping $\gamma$. This behavior is analogous to the classical coupled
oscillator model [75, 76] and has been experimentally observed between TMDC
excitons and MNP plasmons in Refs. [29, 77].

With further decreasing TMDC-PC distance, the absorption line shape changes.
The absorption of the hybrid develops two maxima, while the absorption at the
original, non-perturbed, resonance energy shrinks. The splitting between the
formed maxima can be identified with an effective Rabi energy
$\Omega_{\text{eff}}$. Its distance dependency is depicted in the inset of
Fig. 4b, illustrating a $(\delta z)^{-2}$ proportionality. To assign the
hybrids to a particular coupling regime, the effective Rabi energy
$\Omega_{\text{eff}}\neq 0$ is compared with the linewidth
$\gamma_{\text{ex}}$ and $\gamma_{\text{pl}}$ of the individual constituents
[78, 76].

A coupling strength leading to a peak splitting that is small compared to the
linewidths as e.g. in the $\delta z=17\,$nm case is still assigned to weak
coupling and has been observed in experiments with TMDCs coupled to a MNP or a
PC in Refs. [38, 28, 79, 40].

Figure 5: (a) Absorption spectrum at liquid nitrogen temperature $T=77\,$K of
the hybrid structure (green) compared to the uncoupled absorption of the TMDC
(blue) and the 2D PC (yellow) for $E^{\text{ex}}=E^{\text{pl}}$. The effective
Rabi energy $\Omega_{\text{eff}}\approx 140\,$meV is identified as the
energetic separation between the two outer plexcitonic resonances. (b)
Dispersion of the plexciton branches depending on the excitonic resonance
energy $E^{\text{ex}}$. The upper and lower branches are highlighted with
solid lines as a guide to the eye. The hybrid spectrum shows an exciton-
plasmon splitting, where plasmon (yellow) and exciton (blue) hybridize to
plexcitonic modes (green). The horizontal and main diagonal mark the
undisturbed plasmon (yellow) respectively exciton (blue) resonances. Inset:
Splitting $\Omega_{\text{eff}}$ (green) between plexcitonic branches depending
on the difference between the resonance energies
$E^{\text{ex}}-E^{\text{pl}}$. The gray linear plot indicates the splitting
without any interaction.

####  III.1.2 Strong Coupling ($\delta z=11\,{\rm nm}$)

Further decreasing the distance of TMDC and PC allows one to enter the strong
coupling regime as $\Omega_{\text{eff}}>\gamma_{\text{pl}}+\gamma_{\text{ex}}$
with peak splittings up to $\Omega_{\text{eff}}\approx 140\,\text{meV}$. In
this case, mathematically, in Eqs. (52-56), a real part is added to the
excitonic eigenenergy, thus changing the root of the denominator in Eq. (56),
and consequently the plexcitonic resonance energies. Experimentally, strong
coupling has been observed up to room temperature [29, 80, 30, 81, 31, 77, 82,
32, 41] for TMDC-MNP hybrids with effective Rabi energies similar to the
values shown in Fig. 4.

It would be worth also considering the possibility of ultra-strong coupling in
a similar platform. However, this limit cannot be addressed by the developed
theory in this work due to the constraint
$g_{\mathrm{eff}}/E^{\mathrm{pl}}<0.1$, since the rotating wave approximation
is applied (Sec. II).

###  III.2 Low Temperatures

To study low-temperature effects, Fig. 5a depicts the absorption spectrum in
the strong coupling case for coinciding exciton and plasmon resonance
$E^{\text{ex}}=E^{\text{pl}}$ at liquid nitrogen temperature $T=77\,$K. In
addition to the typical strong coupling spectrum, we find a third peak at the
undisturbed exciton resonance. At room temperature (Fig. 4a) this peak is not
well resolved due to the increased damping. To investigate the dispersion of
the three-peak spectrum, we numerically vary the exciton energy around the
plasmon resonance in Fig. 5b. The color coding breaks the full absorption of
the hybrid down to exciton (blue), plasmon (yellow) and the hybridized
plexciton contribution (green). The individual plasmon and exciton dispersions
are indicated by the horizontal line and the main diagonal, respectively. The
avoided-crossing behavior of the upper and lower branch substantiates that the
system is indeed in the strong coupling regime. However, the spectral position
of the additional middle peak coincides with the unperturbed exciton energy
$E^{\text{ex}}$. The inset depicts the energy separation of the hybrid
branches, where the minimum is the effective Rabi energy of exciton and
plasmon. For comparison, the linear gray plot indicates the energy separation
between exciton and plasmon without any interaction.

####  III.2.1 Interpretation of the Bright Excitonic Mode

The additional peak in Figs. 5a and 5b stems from the bright excitonic
transition $\mathbf{p}_{{\mathbf{q}_{\parallel}}}$ with in-plane momenta
${\mathbf{q}_{\parallel}}$ inside the light cone, see Eq. (42) and Fig. 3. It
is caused by the qualitative difference between the exciton-plasmon coupling
strength for bright and momentum-dark excitons that was already discussed in
Sec. II.4.4. With our restriction to PCs that fulfill Eq. (38), the only non-
vanishing contribution to the third peak is
$\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}$, Eq. (58), meaning it
constitutes a weakly coupled bright excitonic mode.

The appearance of the bright excitonic mode in Fig. 5a can be traced back to a
qualitative difference of our description to previous models [27, 83, 33, 75].
It results from the geometry of the considered subsystems: the PC consists of
discrete dipoles that feature scattering processes. This allows the plasmon to
couple to excitonic transitions outside the light cone via the plasmon-
enhanced electric near-field. In contrast, the TMDC facilitates a continuous
translational invariance. Therefore, it is necessary to distinguish between
excitonic transitions with in-plane momenta in- and outside the light cone,
providing that $\mathbf{p}_{{\mathbf{q}_{\parallel}}=\mathbf{0}}$ does not
couple to the plasmon-enhanced near-field but only to its
$\mathbf{q}_{\parallel}=\mathbf{0}$ Fourier mode.

As plasmons are popular for the amplification of the electric field in their
vicinity, the near-field exciton-plasmon interaction is much stronger than the
radiative (far-field) contribution. However, the
${\mathbf{q}_{\parallel}}=\mathbf{0}$ Fourier component is the propagating
mode, also transferring the plasmon response to the far-field. It is
physically intuitive, due to the argument of energy conservation, that this
mode cannot provide a field enhancement and thus yields a reduced coupling
strength to the TMDC excitons.

In Fig. 5a, we find a subtle damping and broadening due to the radiative
exciton-plasmon interaction for the excitonic mode compared to the unperturbed
excitonic TMDC absorption (blue), similar to the weak exciton-plasmon coupling
regime.

####  III.2.2 Visibility of the Bright Excitonic Mode

Comparing Fig. 4 and Fig. 5, we find that the visibility of the weakly coupled
excitonic mode strongly depends on the temperature. This trend is analyzed in
Fig. 6. It illustrates the hybrid absorption spectrum for several temperatures
between $T=50\,$K and $T=300\,$K with the excitonic mode clearly visible for
low temperatures. With increasing temperature, the absorption peak shrinks
until the increasing exciton damping reaches values of approximately half the
peak splitting at room temperature. The larger the peak splitting
$\Omega_{\text{eff}}$, that in turn characterizes the coupling strength, the
smaller is the plexcitonic absorption contribution at $\Delta=0$, cf. Fig. 4,
which improves the visibility of the excitonic mode.

The bright excitonic mode, beyond the usually found peak splitting [84, 38,
79, 28, 29, 30, 31, 32, 33], was recently observed in Ref. [40] at cryogenic
temperatures. Measured was the differential reflection of a WSe2-monolayer,
encapsulated in hBN, on top of a PC with lattice constant $a=300\,$nm
consisting of gold nanodisks with radii $r_{x}=r_{y}\approx 60\,$nm,
$r_{z}\approx 8\,$nm and TMDC-PC distance $\delta z=12\,$nm. To the best of
our knowledge, this was the first observation of the additional excitonic
mode, since most experimental works operate either at room temperature or with
the interaction strength too small.

####  III.2.3 Comparison to Previous Models

Previous theoretical analysis was restricted to a phenomenological coupled
oscillator model [75] and the Jaynes-Cummings model [76] applied on TMDC-MNP
and PC hybrids [28, 30, 84, 38, 82, 81, 29, 39, 77, 40, 32, 41] or to near-
field excitation involving quasinormal modes [27, 83, 85]. Numerical
approaches using Maxwell solvers have been used to confirm theoretical and
experimental results at room temperature [27, 28, 29, 30, 32, 39, 40, 86, 31,
87, 41]. The theory developed in this contribution provides a solution for the
electric near- and far-field explicitly and exceeds the limits of
phenomenological models by giving a microscopic description of the TMDC
excitons by including the exciton center-of-mass momentum to derive the
coupling strength.

When included in the coupled oscillator model, the additional excitonic mode
was described as a third oscillator and assigned to excitons spatially
separated from the electric field hot spots caused by a single MNP [30] or a
PC [40]; this phenomenological explanation is consistent with the developed
microscopic theory, since the strong coupling is restricted to
${\mathbf{q}_{\parallel}}\neq\mathbf{0}$ Fourier modes, corresponding to
spatially localized states [26]. However, the ${\mathbf{q}_{\parallel}}=0$
mode, responsible for the bright excitonic mode, constitutes a constant
spatial distribution in real space. Our analysis therefore shows that the
additional excitonic mode $P^{\text{ex}}_{{\mathbf{q}_{\parallel}}=0}$ is
uniformly distributed also in the vicinity of the MNPs.

Another interesting feature of the absorption spectra is the asymmetry of the
spectral peaks line shape with respect to width and height. The asymmetry is
also covered by phenomenological models [75, 76] and is consistently observed
in experiments [82, 30, 29, 39, 28, 77, 31]. The reason for the asymmetry are
different line shapes, line widths and dipole moments of exciton and plasmon.
Within our microscopic model, another factor contributes to the observed
asymmetry. The momentum-dark excitons responsible for the peak-splitting
exhibit energy distributions up to $50\,$meV on the exciton dispersion, Eq.
20, well above the bright exciton state. Consequently, these excitons, that
facilitate the strong exciton-plasmon coupling, are detuned from the joint
plasmon and bright exciton resonance. The magnitude of this energy difference
depends on the dielectric environment, cp. Eq. 20, and the distance $\delta
z$, as discussed in sec. II.4.4. The detuning leads to an incomplete
hybridization between plasmons and excitons. Consequently, the peak at lower
energies effectively contains more plasmonic contributions which results in a
stronger absorption.

Figure 6: Absorption for different temperatures $T$ between 50 and 300 K in
the case of coinciding exciton and plasmon resonance. For room temperature
$T=300\,$K (red), the hybrid features a typical strong coupling spectrum.

##  IV Conclusions

We have introduced a self-consistent theory that highlights the extraordinary
optical properties that can be achieved by combining ultrathin semiconductors,
such as transition metal dichalcogenides (TMDCs), with plasmonic crystals
(PCs) composed of metal nanoparticles. Our approach yields new analytical
insights into exciton-plasmon coupling and allows to significantly reduce the
numerical costs compared to general Maxwell solvers, such as ANSYS Lumerical
(finite-difference time-domain) or COMSOL (finite element).

We developed a theoretical formalism to describe the coupling between a TMDC
and a 2D PC, mediated by the self-consistently solved electric field. More
explicitly, we considered the optical response of collective plasmons of 2D
PCs within the excitonic Bloch equation. The arising effective intra- and
intervalley exciton-exciton interactions for momentum-dark excitons have been
decoupled in the eigenbasis of the quasi-static Green’s dyadic into excitonic
branches with conic respectively parabolic dispersion. Any near-field effects
appeared to be incorporated via the conic contribution. In contrast, momentum-
bright excitons show a qualitatively different behavior, as the coupling
strength is proportional to the negligible center-of-mass momenta.
Nevertheless, signatures of the momentum-dark excitons are visible in the far-
field signal due to a second scattering process at the PC.

By evaluating the absorption for different TMDC-MNP separations, we were able
to compare different coupling strengths and observe weak and strong coupling
line shapes, the latter obeying an avoided-crossing dispersion behavior. We
also find a new feature in the hybrid spectrum at low temperatures beyond the
phenomenological model of two coupled oscillators: the momentum-bright TMDC
excitons do not participate in the strong coupling due to their reduced
coupling strength but emit undisturbed into the far-field.

###### Acknowledgements.

The authors thank financial support from the Deutsche Forschungsgemeinschaft
(DFG) through SFB 951 Project No. 182087777, Project SE 3098/1-1 (R.S. and
M.S.) Project No. 432266622, DPG Sachbeihilfe No. 504656879 (S.R.), the
European Commission through the Consolidator Grant DarkSERS (772108) and the
Natural Sciences and Engineering Research Council of Canada. We also
acknowledge support from the Alexander von Humboldt Foundation through a
Humboldt Research Award (S.H., A.K.).

##  Appendix A Material and Model Parameters

The parameters for numerical computations are mainly given in semiconductor
units to improve the numerical accuracy which use fs and nm as a measure of
time and length, eV for energy and the elementary charge $e$.

Table 2: universal constants in semiconductor units $c$ |  $299.7925$ nm/fs
---|---
$\hbar$ |  $0.658212196$ eV fs
$k_{B}$ |  $0.0861745\,$meV/K
Table 3: Parameters for MoSe2 $d$ |  $\varphi_{\mathbf{r_{\parallel}}=\mathbf{0}}\cdot 0.25\,e\,$nm [49]
---|---
$\varphi_{\mathbf{r_{\parallel}}=\mathbf{0}}$ |  $0.46\,$nm-1
$M$ |  $6.1\,\text{eV\,fs}^{2}\text{nm}^{-2}$ [53]
$c_{1}$ |  $0.091\,$meV/K [51]
$c_{2}$ |  $15.6\,$meV [51]
$\Omega$ |  $30\,$meV [51]

The excitonic wavefunction $\varphi_{\mathbf{r_{\parallel}}=\mathbf{0}}$
appears as the solution of the Wannier equation, similar to Refs. [50, 51] for
the chosen dielectric environment.

Table 4: Parameters for the permittivity of gold taken from Ref. [64] (converted to semiconductor units) $\epsilon_{\infty}$ | $1.53$
---|---
$\omega_{\text{p}}$ |  $12.99\,$fs-1
$\omega_{1}$ |  $4.02\,$fs-1
$\omega_{2}$ |  $5.69\,$fs-1
$\Gamma_{1}$ |  $0.82\,$fs-1
$\Gamma_{2}$ |  $2.00\,$fs-1
$A_{1}$ | $0.94$
$A_{2}$ | $1.36$
$\phi_{1}$ | $-\pi/4$
$\phi_{2}$ | $-\pi/4$
Table 5: Parameters for temperature dependent linewidth of gold taken from Ref. [65] (converted to semiconductor units) $b$ | $0.6329\,\text{eV}^{-1}$
---|---
$\gamma_{0}$ |  $0.0219\,$eV
$\Theta$ |  $185\,$K

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