Dipolar Coupling at Interfaces of Ultrathin Semiconductors, Semimetals, Plasmonic Nanoparticles, and Molecules [original]

Dipolar Coupling at Interfaces of Ultrathin

Semiconductors, Semimetals, Plasmonic Nanoparticles,

and Molecules

Lara Greten,* Robert Salzwedel, Manuel Katzer, Henry Mittenzwey,

Dominik Christiansen, Andreas Knorr,* and Malte Selig

1. Introduction

Monolayers of transition metal dichalcogenides (TMDCs) exhibit

an extremely strong light–matter interaction

[1]

with absorption

rates up to 10% in the visible spectrum.

[2,3]

This fact is even more

remarkable considering their 2D nature with less than 1 nm

thickness. In addition to a direct bandgap, their 2D structure

leads to a reduced screening of the Coulomb interaction in

the surrounding material and consequently to the formation

of stable, bound electron–hole pairs, namely Wannier-type exci-

tons, which dominate the optical properties.

[4–13]

Moreover, the

atomical thickness provides a strong sensitivity to the environ-

ment: atomically thin materials can be manipulated by the choice

of the embedding material and its geome-

try

[14]

as well as by defects

[15]

or functional-

ization.

[16]

When coated in leaky cavities,

2D TMDCs exhibit an interesting interplay

of phonon and photon dissipation, leading

to anomalous dispersion and negative mass

effects.

[17]

Due to the variety of feasible

material combinations, functionalization

proves to be a powerful tool for tailoring

electronic and optical properties on the

nanoscale. The scope of this work is to pres-

ent a theoretical framework to analyze

hybrids of a TMDC layer parallel to the

xy-plane functionalized with other nano-

materials as displayed in Figure 1.We

focus on interactions that are mediated

by the electric near-ﬁeld, starting from

the setup of the general light–matter

Hamiltonian valid for all discussed hybrids in Section 2. The cor-

responding matrix elements are speciﬁed for molecules, metals,

graphene, and TMDCs in Section 2.1–2.3, where we also develop

the Heisenberg equations of motion for excitations in these

TMDCs as the overall substrate. Subsequently, we apply the for-

malism on hybrids of a TMDC monolayer functionalized with

molecules, metal nanoparticles (MNP) or graphene, and a

TMDC heterobilayer. In this work, we focus on coupling effects

that are mediated by the mutual electric ﬁelds between the

constituents:

1) organic molecules show highly tunable transition energies

and provide large optical dipole moments as the respective exci-

tons are of the Frenkel type,

[18–26]

Section 3. We discuss a spec-

trally resolved Förster rate in Section 3.2, allowing to directly

determine momentum-dark excitonic states in the TMDC, not

observable in coherent optical experiments.

[11,27–32]

2) MNPs provide an impressive ampliﬁcation of the electric

near-ﬁeld,

[33,34]

Section 4. The interaction of TMDC excitons with

the MNP plasmons features exciton localization, Section 4.1, and

allows to enter the strong coupling regime,

[35–44]

Section 4.2.

3) Other widely considered heterostructures are interfaces of

TMDC monolayers and graphene, where many studies have

focused on the individual role of charge and energy transfer

between both layers.

[45–51]

We discuss a new type of interlayer

energy transfer in Section 5, namely the Meitner–Auger

coupling.

[51]

Here, the energy is transferred from the TMDC

layer to graphene via non-radiative recombination of a TMDC

exciton which excites an intraband plasmon in graphene.

L. Greten, R. Salzwedel, M. Katzer, H. Mittenzwey, D. Christiansen,

A. Knorr, M. Selig

Nichtlineare Optik und Quantenelektronik

Institut für Theoretische Physik

Technische Universität Berlin

10623 Berlin, Germany

E-mail: lara.greten@tu-berlin.de; [email protected]

The ORCID identiﬁcation number(s) for the author(s) of this article

can be found under https://doi.org/10.1002/pssa.202300102.

science published by Wiley-VCH GmbH. This is an open access article

under the terms of the Creative Commons Attribution License, which

permits use, distribution and reproduction in any medium, provided

the original work is properly cited.

DOI: 10.1002/pssa.202300102

Recent progress in growth techniques has enabled the fabrication of stacks of

transition metal dichalcogenide monolayers combined with different nano-

structures ranging from other 2D layers over dye molecules to even plasmonic

nanoparticles. Such structures promise to combine the optoelectric properties of

the constituents allowing to design structures with desired properties. For all of

these examples, a detailed knowledge of the coupling among different constit-

uents is crucial. In this article, a uniﬁed description is presented based on

Maxwell Bloch equations to describe dipolar interactions among different types of

heterostructures. Exemplary, Förster-type energy transfer from dye molecules to

MoS

monolayers, strong coupling at MoSe

–metal nanoparticle interfaces,

Meitner–Auger-like interlayer coupling in WSe

–graphene stacks, and relaxation

processes of hot interlayer excitons in MoSe

–WSe

heterobilayers are discussed.

RESEARCH ARTICLE

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4) Finally, we apply the framework on TMDC MoSe

–WSe

heterobilayers in Section 6. They exhibit unique optical proper-

ties compared to their monolayer constituents.

[52–56]

Due to

their type-II band alignment,

[57,58]

interlayer excitons

[59]

emerge

after charge transfer by optically pumping of intralayer

excitons.

[60,61]

In this work, we analyze their cooling and thermal-

ization dynamics.

2. Heterostructure Hamiltonian

The general heterostructure Hamiltonian considered throughout

this work reads

H¼X

εii†iþ1

i,j,i0,j0

Vijj0i0i†j†j0i0X

i,i0

Ωii0i†i0(1)

with annihilation (creation) operators ið†Þfor electrons with com-

pound quantum number i, which will be speciﬁed in the next

subsections to distinguish the different applications (i–iv). The

ﬁrst term in Equation (1) accounts for the dispersion of the elec-

trons of all constituents of the hybrid. The second term incorpo-

rates monopole contributions of electron–electron interactions

Vijj0i0¼Zd3rΨ

iðrÞΨ

jðr0ÞVðr,r0ÞΨj0ðr0ÞΨi0ðrÞ(2)

with Coulomb potential Vðr,r0Þand electron wave functions

ΨiðrÞ. The third term accounts for the electric ﬁeld–matter cou-

pling in dipole approximation with the Rabi energy

Ωii0ðtÞ¼qZd3rΨ

iðrÞEðr,tÞ⋅rΨi0ðrÞ(3)

including the charge qof the carriers and the electric ﬁeld Eðr,tÞ.

In course of evaluating the Rabi energy for the different materi-

als, we approximate the electric ﬁeld to be constant within one

unit cell. This allows to separate electronic- and electric-ﬁeld

coordinates and to identify electronic intra- and inter-band

transitions. We specify the matrix elements Vijj0i0and Ωii0, for

the molecule, metal, and semiconductor/semimetal electrons

in Section 2.1–2.3. The framework allows to derive diverse

energy-transfer mechanisms between the different material sys-

tems on the same footing by considering the respective contri-

butions in the Hamiltonian when deriving Heisenberg’s

equation of motion as illustrated in Sections 3–6. For each

hybrid, the electric ﬁeld is determined by Maxwell’s equations,

which can be accessed in the Green’s function formalism

[16,62]

EQkðz,ωÞ¼ X

l¼0;1fg

GQkðz,zl,ωÞ⋅Pl

QkðωÞþE0

Qkðz,ωÞ(4)

with the 2D polarization densities Pl

Qkfor layer land the incident

external electric ﬁeld E0

Qkas a source. To achieve this algebraic

form for the electric ﬁeld, we assumed the individual layers to

act as inﬁnitesimally thin.

[63]

The Green’s dyadic GQkðz,zl,ωÞ

transfers the polarization densities located at zlto the observers’

position z. It depends on the scalar Green’s function GQkðz,z0,ωÞ

with Qk¼jQkjand

GQkðz,zl,ωÞ

¼ω21

ϵ0c2þQk⊗Qk

ϵ0ϵðzÞ

iQk

ϵ0ϵðzÞ

∂

iQkT

ϵ0ϵðzÞ

∂

z0ω2

ϵ0c21

ϵ0ϵðzÞ

∂

AGQkðz,z0,ωÞjz0¼zl

(5)

The Green’s functions for the chosen geometries (additional

dielectric interfaces in z-direction) are given in the appendix.

Each considered heterostructure, see Figure 1, consists of two

layers ðl¼0, 1Þat z¼z0,z1, respectively. To solve the electric

near-ﬁeld, that is responsible for the coupling between the layers,

we apply the quasi-static approximation, setting ω!0 in the

Green’s dyadic, Equation (5). This corresponds to neglecting

propagation effects and is valid as long as the layer distance

Δz¼jz0z1jλis small compared to the wavelength of

the exciting light ﬁeld E0

Qk.

[64]

In contrast, we can simplify

Qk!0to derive the far-ﬁeld of the heterostructure for perpen-

dicular plane-wave excitation.

2.1. Molecules

In the case of molecules, the compound quantum number i

in Equation (1) accounts for the molecular orbitals λ¼H,L

(highest occupied molecular orbital (HOMO), lowest unoccupied

molecular orbital (LUMO)). The single-particle energies are

already treated as effective excitonic energies. The corresponding

Hamiltonian reads

Figure 1. Examples of hybrid nanomaterials with one transition metal dichalcogenide (TMDC) monolayer. From left to right: the TMDC monolayer is

functionalized with molecules, a metal nanoparticle (MNP), a 2D plasmonic crystal (PC), graphene, and another TMDC monolayer.

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H¼X

ελλ†λX

dλλ ⋅Eðr1,tÞλ†λ(6)

The ﬁrst term represents the energy ελof the molecular

orbitals,

[65]

described as fermions, a sufﬁcient approach in linear

optics.

[66]

The molecular transition energy reads EHL ¼εHεL,

see Figure 2. For molecules, the electron–electron interactions in

the general light–matter Hamiltonian, Equation (1), are already

included in the effective two-level energies ελ. The second term

accounts for the light–matter coupling where we have assumed a

point-like molecule, located at the center of charge r¼r1,

exhibiting a dipole moment dλλ. For the plot, we assume exem-

plary a dichloromethane (DCM) molecule. The strength of the

molecular dipole moment, dλλ ¼dHL is taken from ab initio

calculations.

[67]

2.2. Metals

Metals have a partially occupied conduction band that allows for

intraband as well as interband transitions. Depending on the

spectral range of interest, the optical response in metal structures

is either dominated by intraband transitions, known as plasmons,

or interband transitions.

[68]

Accordingly, the compound quantum

number iin the Hamiltonian in Equation (1) contains the band

index λ, constituting for valence ðvÞand conduction ðcÞband, and

the electronic momentum k. The Hamiltonian is given by

H¼X

λ,k

ελkλ†

kλkX

λ,k,Q

dλλk⋅EQðtÞλ†

kþQλk

iq

λ,k,Q

EQðtÞ⋅∇

Qðλ†

kþQλkÞ

(7)

The band structure is accounted for by the ﬁrst term, which

includes the electronic dispersion ελk, renormalized due to

electron–electron interactions. The second term covers interband

transitions between different bands λ6¼ λ, with the transition

probability determined by the optical dipole element dλλk. The

last term describes intraband transitions where momentum Q

is transferred from the electric ﬁeld to an electron within one

band. The probability of this process is given by the Fourier trans-

form of the electric ﬁeld EQand the electron charge q¼enor-

malized by the volume V. Only the conduction band (λ¼c)

needs to be considered for intraband transitions as the valence

band is fully occupied.

The permittivity of gold, ϵAu, can be derived using the

Heisenberg equation of motion framework. The intraband con-

tribution results in the Drude susceptibility for a quasi-free elec-

tron plasma in the gold conduction band,

[69,70]

while the

interband transitions can, in its most simple form, be approxi-

mated by Lorentz-shaped contributions around the respective

transition energies. In gold, interband transitions become rele-

vant above 2.4 eV

[68]

and while the correct allocation of the visible

interband transitions in the gold band structure is still controver-

sial, there are several models that ﬁt the experimental data in

refs. [71,72], incorporating multiple interband transitions.

[73–76]

In this paper, we use the permittivity provided by ref. [74].

ϵAuðωÞ¼ϵ∞ω2

ω2þiΓωþX

a¼1, 2

Aaωa

eiϕa

ωaωiΓaþeiϕa

ωaþωþiΓa



(8)

The ﬁrst two terms resemble a standard Drude model that

incorporates dielectric screening to account for strongly polar-

ized d-orbitals. Damping rates Γ,Γaare added phenomenologi-

cally to ﬁt experimental curves measured from bulk material. All

parameters are given in the appendix.

2.2.1. Plasmonic Crystals

The optical response of MNPs is given in Mie theory similar to

the electric ﬁeld of a point dipole.

[77,78]

In the case of a 2D plas-

monic crystal (PC) consisting of periodically arranged MNPs, the

particle and lattice geometry is as important for the optical prop-

erties as the material. The macroscopic polarization density is

given by

[79,80]

PMNP

QkðωÞ¼X

αðωÞ⋅EQ0

kðz1,ωÞ(9)

with an effective PC polarizability αðωÞ, taking MNP interac-

tions and Umklapp processes into account.

[79]

For noninteracting

MNPs, Equation (9) is also applicable with MNP polarizability

αðωÞin Mie–Gans theory

[78,81]

that depends on the surrounding

and MNP permittivity ϵAuðωÞ, Equation (8).

[74]

We provide the

MNP polarizability in the appendix.

2.3. 2D Materials

In the case of 2D semiconductors and semimetals, for example,

TMDCs and graphene, the compound quantum numbers of the

electrons constitute the layer l, the band λ¼c,v(conduction

band, valence band) and the momentum ξþkwhere ξdenotes

the considered high symmetry points, in our case K=K0with the

2D wave number ktherein, that is, i¼ðl,λ,ξ,kÞ. The semicon-

ductor/semimetal Hamiltonian takes the form

Figure 2. Approximated valence (blue) and conduction (red) bands ελ¼v=c

of molecules, graphene, and a TMDC. The molecular transition energy EHL

is given by the energy of the HOMO and the LUMO. In graphene, valence

and conduction bands touch at the Kpoints and are approximately linear

in their vicinity. TMDC monolayers feature a direct bandgap at the K=K’

points; their electronic dispersion is approximated parabolic. For simplic-

ity, the TMDC is illustrated for one spin direction.

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H¼X

l,λ,ξ,k

εlξ

λkλlξ†

kλlξ

kþ1

l1,l2,

λ1,λ2,λ3,λ4,

ξ1,ξ2,k1,k2,q

Vqλl1

ξ1†

k1þqλl2

ξ2†

k2qλ3

l2ξ2

k2λ4

l1ξ1

X

l,λ,ξ,k,Qk

dlξ

λλ⋅EQkðzl,tÞλlξ†

kþQkλlξ

(10)

The ﬁrst term accounts for the energy of electrons with the

dispersion εlξ

λk. The second term describes monopole,

Coulomb electron–electron interactions.

[82]

As we are only inter-

ested in energy-transfer mechanisms, Equation (10) disregards

electronic tunnel processes between the constituents. The third

term accounts for light–matter interaction. It is treated in thin

ﬁlm

[63]

and low wave number approximation, such that the

momentum dependence of the dipole moment dlξ

λλdrops.

[83]

Further, EQkðzl,tÞ¼∫d2reiQk⋅rkEðrk,zl,tÞaccounts for the

Fourier component of the electric ﬁeld inside the semiconduc-

tor/semimetal layer z¼zlwith respect to the in-plane compo-

nent. Equation (10) is applicable for mono- and multilayer. We

concentrate on the description of monolayer (MoSe

, MoS

) and

heterobilayer (WSe

/graphene, MoSe

/WSe

), where the layer

number lalso speciﬁes the layer material.

2.3.1. Graphene

In the case of graphene, we concentrate on the reciprocal space

near the Kpoints where the valence and conduction bands touch

and form the Dirac cones. Since graphene does not exhibit a

bandgap, excitonic effects are typically of minor importance in

the optical range and we use the free particle band structure.

As displayed in Figure 2, the graphene dispersion is linear close

to the Kpoint: εk¼ℏvFjkjfor the conduction (þ) or valence

ðÞ bands, respectively.

[51]

2.3.2. TMDCs as Substrates

In the monolayer limit, TMDCs facilitate an optically accessible

direct bandgap at the K=K0points. In their vicinity, we approxi-

mate the dispersion εξ

λkto be parabolic, parameterized from DFT

calculations,

[84]

see Figure 2. Furthermore, TMDCs feature a

spin-splitting of the electronic valence and conduction bands.

Their dispersion εξ

λkdepends on the valley but also on the spin

index swhich we absorb in a compound quantum number ξfor

valley and spin. The optical dipole moment dξ

cv of the TMDC

inter-band transition is taken from ab initio electronic structure

calculations.

[85]

For the hybrid materials considered in this man-

uscript, see Figure 1, a TMDC monolayer serves as the substrate.

Therefore, we derive already at this point the Heisenberg equa-

tion of motion for the excitonic transition

[86]

which all hybrids

have in common

pξν

Qk¼ˆ

pξν

DE (11)

given by the expectation value of the exciton operator

pξν

Qk¼X

φξν

qkvξ†

qkβξQkcξ

qkþαξQk(12)

φξν

rk=qkdenotes the excitonic wave function in real and momen-

tum space, respectively. αξand βξare the ratios of the effective

masses for the respective excitonic conﬁguration ξas deﬁned in

ref. [86].

The relative motion qk, arising from the electron–electron

Coulomb interactions in Equation (1), is responsible for the for-

mation of bound exciton states with index ν. The remaining

degree of freedom is the excitonic center-of-mass motion Qk.

Using the center-of-mass momentum Qkinstead of the full

set of electron/hole momenta, see the k-axis in Figure 2, corre-

sponds to the exciton picture. As we perform the calculations in

the exciton picture, we also give schematic dispersions in further

sections in the Qk-space. For the derivation of the Heisenberg

equation, we restrict ourselves to low exciton densities, that is,

to a regime where Pauli blocking and exciton–exciton scattering

is negligible and excitons can be treated as bosonic particles.

Screening would occur on the same order as the neglected fer-

mionic contributions for the exciton operators ˆ

pξν

Qk,

[86,87]

which

modify the linear equations of motion. To describe excitonic

screening, principle value integrals in the scattering contribution

could be summed up to provide excitonic screening.

[88–90]

The

main effect of screening would be a reduction of the interaction

strength between and in the layers. We arrive at the excitonic

Bloch equation

[82,86,91]

ðℏωEξν

QkþiγξνÞpξν

QkðωÞ¼φξν

rk¼0dξ⋅EQkðzl,ωÞ(13)

A phenomenological damping constant γξν mainly stemming

from electron–phonon scattering is introduced as a dephas-

ing.

[92]

The excitonic dispersion reads in parabolic approximation

Eξν

Qk¼Eξν þ

ℏ2Q2

2Mξ(14)

with exciton transition energy Eξν and Mξas the exciton mass at

valley ξ. The spin splitting of the electronic bands results in two

types of excitons with different transition energies corresponding

to their spins, namely the Aand the Bexciton. The macroscopic

polarization within the TMDC layer lin the excitonic picture

reads

Pl,TMDC

QkðωÞ¼X

ξν

dξφξν

rk¼0pξν

QkðωÞþc:c:. (15)

The macroscopic polarization density connects the exciton

dynamics to the generated electric ﬁeld EQkvia Equation (4), that

mediates the interaction between the constituents of the hybrids

and that has to be solved self-consistently as illustrated in the

following sections.

The excitonic transition pξν

Qkis the crucial microscopic quantity

to describe linear optical experiments as in Sections 3 and 4, for

example, direct transmission, reﬂection, and absorption.

However, if it comes to photoluminescence or nonlinear optical

spectroscopy, one further needs to consider the excitonic

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occupation Nξν

Qk, that is deﬁned as the correlated (c) expectation

value of an electron–hole pair

[93,94]

Nξν

Qk¼ˆ

pξν†

pξν

c(16)

From the Heisenberg equation of motion, it is possible to

derive the Boltzmann equation for the excitonic occupations.

[28]

Here, we provide a general form of the Boltzmann equation for

low exciton densities Nξν

Qk1, allowing to neglect contributions

that are nonlinear in Nξν

∂

tNξν

Qk¼Γξν,in

QkðNξ0ν

kÞΓξν,out

QkNξν

Qk(17)

Equation (17) describes scattering that can, for example,

emerge due electromagnetic coupling, see Section 5, or exci-

ton–phonon coupling, see Section 6. The decay of the excitonic

occupation with center-of-mass momentum Qkdepends on the

total decay rate Γξν,out

Qk, which is given by a summation over all

possible target states, in example, center-of-mass momenta

QkþKk, of the microscopically calculated out-scattering rates

Γξν;out

Qk,QkþKk

Γξν,out

Qk¼X

Γξν,out

Qk,QkþKk(18)

The buildup of the excitonic occupation in Equation (17) is

caused by the total in-scattering rate Γξν,in

Qk. This rate typically

depends on the respective scattering process and on other occu-

pations, such as occupations in adjacent materials, see Section 5,

or excitonic occupations with different momenta Nξν

k, see

Section 6, as well as excitonic transitions pξν†

Qkpξν

QkδQk,0. It is fur-

thermore common to theoretically divide Nξν

Qkinto an incoherent

and a coherent part: ðNξν

QkÞfull ¼Nξν

Qkþjpξν

Qkj2, which allows to

couple it to Equation (13). However, in most cases, it is sufﬁcient

to take into account pξν

Qk¼0(lightcone). This allows to accurately

describe excitation entering the equation by penetration with

light, both resonantly

[7,28]

and off-resonant.

[12]

3. Dipolar Coupling in TMDC Molecule

Heterostructures

We assume one molecular layer l¼1 at position z¼z1on top

of a monolayer TMDC ðl¼0Þ, see Figure 3a. We omit the

respective layer index lin the following. We assume a dielectric

environment as given in Figure 3a. For the TMDC, placed on a

hBN substrate, we assume a dielectric constant due to the off-res-

onant transitions

[95]

and we consider the molecules to be in vac-

uum. All parameters can be found in the appendix.

3.1. Bloch Equations

In frequency domain ω, from Equations (4), (6), and (13), we can

derive a set of Bloch equations for the interaction between mole-

cule and TMDC mediated by the electric ﬁeld, for the HOMO–

LUMO transition σHL ¼H†Lhiin the molecule and the excitonic

transition pξν

Qk. We compute the TMDC Bloch equation as given

in Equation (13), also analogously for the molecule, and self-

consistently solve the set of equations with the electric ﬁeld,

Equation (4), for details see ref. [16]. We ﬁnd a coupled set of

equations

iℏ

∂

tσHL ¼½EHL iγHLσHL þX

ξνQk

VT!M

Qkξν ðz0,z1Þpξν

dHL ⋅E0ðr1,tÞ

(19)

iℏ

∂

tpξν

Qk¼½Eξν

Qkiγξνpξν

QkþVM!T

Qkξν ðz1,z0ÞσHL

ðdξφξν

rk¼0Þ⋅E0

Qkðz0,tÞ(20)

Figure 3. a) Sketch of a TMDC (MoS2) monolayer at z0with thickness ΔTMDC 0.6nm, which is screened by its off-resonant transitions ϵT¼13.36,

[95]

a hBN substrate with permittivity ϵ1¼4.5

[95]

covered by a molecular layer at z1in vacuum environment with ϵ2¼1. b) Illustration of a Förster-type energy

transfer between TMDC excitons (illustration in the exciton picture) and molecular excitation. c) Förster rate for a DCM molecule on a monolayer MoS

,at

a distance of Δz¼1nm, with an hBN substrate underneath, at low temperatures, that is, γξν 0. The absorption of the pristine TMDC is given in light

gray for comparison. Figure adapted from ref. [16].

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The coupling between Equations (19) and (20) occurs in the

near-ﬁeld via the stationary version (ω!0) of the Green´s

dyadic in Equation (5), that yields, VT!M

Qkξν ðz0,z1Þ¼

AϵTϵ0dHL ⋅GQkðz0,z1Þ⋅dξφξν

rk¼0(and VM!T

Qkξν analogously).

[16]

3.2. Förster Rates

From Equations (19) and (20), a rate for the Förster process can

be derived occurring as a decay channel for excitations σHL of the

molecular HOMO–LUMO transition. In frequency space, we

ﬁnd

ℏωσHL ¼½EHL iγHL ΣFðωÞσHL dHL ⋅E0ðz1,ωÞ(21)

The rate is given by the imaginary part of the self-energy

γF¼ImðΣFðωÞÞ, and can be computed analytically in the limit

γξν !0, which nicely approximates more exact results with

γξν 6¼ 0.

[92]

In this limit, the rate depends on the transition energy

EHL of the molecule with respect to the exciton energy Eξν, and is

only nonzero when the energy of the molecular transition is larger

than the bright TMDC 1s resonance (energy conservation)

[16]

γF¼X

νξ

ðdkφξν

rk¼0Þ2ðdHLÞ2M

8ϵ2

0ðϵ2þϵTÞ2ℏ2ΘðQk

0Þe2Qk

0Δz

ðQk

0Þ2ð1þδ1eQk

0ðΔzþ1

2ΔTMDCÞÞ2

ð1δ1δ2e4Qk

0ΔTMDC Þ2

(22)

In Equation (22), the in-plane momentum Qk

0ðEHLÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ℏ2ðEHL EξνÞ

qresembles the momentum for which the exci-

ton Eξν

Qkis in resonance with the molecular transition energy EHL,

see Figure 3b. Obviously, at this point, the rate strongly depends

on the molecular transition energy inducing a spectral depen-

dence in the rate. The dielectric environment is taken into

account via a Rytova–Keldysh-type approach,

[96,97]

and is

expressed here with the abbreviations δ1¼ϵTϵ1

ϵTþϵ1and

δ2¼ϵTϵ2

ϵTþϵ2, respectively. dkstands for the absolute value of the

in-plane part of the optical dipole moment of the TMDC.

[85]

It can be seen from Equation (22) that the Förster transfer

depends on the squares of the optical dipole matrix elements

dkφξν

rk¼0,dHL of both constituents, as it is typical for this energy

transfer.

[50,98,99]

Screening due to the dielectric environment

occurs due to the corrections δ1,δ2. This includes the case of

homogeneous dielectric environment ϵ1¼ϵ2¼ϵT≡ϵ, where

δ1¼0¼δ2results in the usual 1

ϵ-dependence of the Förster rate.

For small distances, the rate shows a combination of exponential

and power law decay as a function of the distance Δzbetween

molecule and TMDC sheet. The dependence develops into the

standard power law ðΔz4Þfor larger distance, in agreement with

earlier work on 0d–2d interfaces.

[50,98–100]

Figure 3c shows the Förster rate for negligible γξν 0, which

is a good approximation for low temperatures below the phonon

threshold

[92]

for DCM molecules evaporated over a MoS

mono-

layer placed on a hBN substrate. The rate is plotted over the

molecular transition energy EHL. It is strongest directly above

the energy A and B resonances in the TMDC. As seen from

Equation (22), ﬁnite-exciton momenta (constituting dark exci-

tons) are necessary to allow for the energy-transfer process

(although this is smeared out slightly for nonzero temperatures).

Due to the ability of this near-ﬁeld effect occurring from the

momentum distribution of the molecular dipole which activates

excitonic states with nonzero momentum, it becomes possible to

resolve these momentum-dark states of the TMDC in the far-

ﬁeld. The signatures of dark excitons are directly experimentally

accessible via the photoluminescence of the functionalizing mol-

ecules, see ref. [16].

4. Dipolar Coupling in TMDC MNP

Heterostructures

In this section, we discuss the interaction between TMDC exci-

tons and MNP plasmons that facilitates strong coupling as

shown in a plethora of experimental literature demonstrating

the strong exciton–plasmon coupling.

[37–44]

These observations

motivate the development of a microscopic theory and its means

to explore exciton localization near the nanoparticles. A sche-

matic of a TMDC–MNP and a TMDC–PC heterostructure is

given in Figures 4a and 5a, respectively. All parameters can

be found in the appendix. As we are interested in the basics

of resonant 1s exciton–plasmon interaction, we neglect higher

excitonic states in the excitonic Bloch Equation (13), setting

ν¼1s. This is valid since the excitonic states are energetically

separated

[6]

further than the plasmon broadening.

Furthermore, we disregard the valley dependency of the exci-

tonic dispersion Eξ,ν¼1s,l¼0≡E1s. To self-consistently solve the

exciton–plasmon interaction in TMDC–MNP hybrids, we apply

the solution of Maxwell’s equations, Equation (4), on the dynam-

ics of excitons, Equation (13), and plasmon, Equation (9).

This way, we obtain a direct as well as a plasmon-mediated

inter-valley coupling between Kand K0-excitons in the excitonic

Bloch equation. In the eigenbasis of the in-plane contribution of

the Green’s dyadic, Equation (5), with momentum-dependent

eigenvectors eU

Qk,eV

Qk,ez, the inter-valley coupling, induced by

the vector-mixing self-consistent electric ﬁeld, is decoupled.

[101]

Moreover, the electric near-ﬁeld is only driving the V-component

of the excitonic transition pV

Qk¼1

ﬃﬃ2

pðpK1seiΦþpK01seiΦÞwith the

polar angle Φof the momentum Qk. The corresponding Bloch

equation reads

ℏωE1s 

ℏ2Q2

2MþiγþXQkðz0Þ

QkðωÞ

þX

VQkQ0

kðz0,z1,ωÞpV

kðωÞ¼SQkðz0,z1,ωÞ

(23)

The ﬁrst term in the parentheses describes the exciton disper-

sion and includes direct exciton–exciton (exchange) interactions,

XQk. The second term incorporates effective, plasmon-induced

exciton–exciton interactions, VQkQ0

k. The external ﬁeld E0

Qkat

the TMDC as well as the MNP positions ðz0=z1Þis included

in the source term SQkon the right-hand side of

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Equation (23), providing a direct and a plasmon-mediated exci-

tation for TMDC excitons.

4.1. Localization of Excitons

Considering the coupled Equation (23), we ﬁnd that a set of

eigenfunctions ψR,μ

Qkfor the center-of-mass motion can be

deﬁned via

ℏ2Q2

2MXQk

ψR,μ

QkX

VQkQ0kψR,μ

k¼EμψR,μ

Qk(24)

to extract the plexcitonic eigenvalues Eμ. It describes the excitonic

motion under the inﬂuence of an exchange coupling XQk

[82]

in the

plasmon-induced potential VQkQ0

k. For further investigation, the

full excitonic Bloch Equation (23) will be mapped on these eigen-

states using the projection

Qk¼X

ψR,μ

Qkpμ(25)

where the wave functions are left- and right-handed solutions of

the non-Hermitian eigenvalue problem in Equation (24). These

eigenfunctions can be used to derive the polarization induced

by the plasmon in real space that we display in Figure 4b.

4.2. Strong Coupling

To connect the exciton localization to related optical experiments

that observe strong exciton–plasmon coupling,

[41,102–105]

trans-

mission T,reﬂection R, and true absorption Aof TMDC–PC

hybrids can be evaluated for a 2D PC, see Figure 5a, speciﬁed

Figure 4. a) Sketch of a single MNP with semiaxis ðrx,ry,rzÞat z1in a surrounding permittivity ϵ2on top of a TMDC monolayer at z0placed on a substrate

with ϵ1. A spacer dielectric with permittivity ϵ1is inserted between the TMDC and the MNP. b) Plasmon- and light-induced polarization within TMDC layer

in real space.

Figure 5. a) Sketch of a square 2D PC with lattice constant a¼3rxconsisting of oblate MNPs (rx¼ry¼10nm, rz¼5nm) on top of a TMDC (MoSe2)

monolayer. The vertical conﬁguration is similar to Figure 4a with surrounding permittivity ϵ1¼ϵ2¼2.4 (silicon dioxide). b) Rabi-splitting for different

interlayer distances Δz∈½rz,3rz¼½5, 15nm extracted as energetic separation between the plexcitonic absorption maxima (inset). The inset displays the

absorption of the TMDC–PC hybrid, where undisturbed exciton and plasmon resonance energies coincide, over the detuning Δ¼ℏωE1s of the exciting

ﬁeld E0.

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for the case of perpendicular incident plane-wave excitation

Qk¼δQk,0E0

Qk. In the course of this, an extension of the

quasi-static approximation is incorporated.

[106]

The absorption

of a MoSe

–PC hybrid at room temperature is displayed in

Figure 5b for different TMDC–PC distances Δz. Since the dis-

tance changes the interaction strength between excitons and plas-

mons, Figure 5b displays the transition between weak (red) and

strong (blue) coupling.

For a close stack Δz¼rzþ1nm, Figure 5b shows a typical

strong coupling spectrum with a Rabi-splitting Ωthat is several

tens of meV. The splitting, which is a measure of the coupling

strength, shrinks signiﬁcantly with increasing distance between

TMDC and PC. The trend is elaborated in the inset of Figure 5b,

showing a Ω∝ðΔzÞ2dependency. This behavior differs from

expectations for two spatially localized coupled dipoles, but is

characteristic for a single dipole interacting with the polarization

of a 2D plane. However, when maximizing the coupling strength

in an experimental setup, for interlayer gaps ðΔzrzÞsigniﬁ-

cantly smaller than 1 nm,

[42]

one has to additionally consider

tunneling processes respectively charge transfer between the

constituents which would suppress the strong exciton–plasmon

interaction.

In summary, the developed framework allows to estimate the

Rabi-splitting occurring for strong exciton–plasmon coupling.

Since the equations are based on a microscopic description of

the exciton dynamics in TMDCs, we can connect the strong cou-

pling with the appearance of exciton localization. Thus, the

theory complements previous models based on quasi-normal

modes,

[35,107,108]

that focus more on a detailed description for

the plasmonic nanoparticles photonics modes.

5. Dipole-Monopole Coupling of TMDC Graphene

Heterostructures

So far, we investigated the dipolar coupling of atomically thin

semiconductors with molecules and MNPs. In this section, we

focus on the coupling between TMDC and graphene, illustrated

in Figure 6a. The dielectric environment of the TMDC–graphene

stack is assumed as displayed in Figure 6a. All parameters can be

found in the appendix. We note that in addition to energy trans-

fer between the sheets due to Coulomb coupling, also charge

transfer via electron tunneling between the layers due to

ﬁnite-electronic-wave function overlaps occurs. This electron

tunneling is assisted by phonon scattering to match energy

and momentum conservation during the tunneling event. The

time scale for the tunneling transfer is on the order of about

100 fs.

[51]

As a new effect beyond the electron-tunneling process,

the combination of a gapped semiconductor and a semimetal

leads to the intriguing interfacial Meitner–Auger energy transfer

corresponding to a dipole–monopole coupling, as observed in

ref. [51]. The process we address is sketched in Figure 6b.

Here, the excitation energy of an optically excited exciton in

the TMDC sheet is transferred to the graphene layer inducing

an intraband transition, which creates hot-hole distributions.

This kind of coupling is only possible for either transiently

excited graphene leading to vacancies slightly below the Fermi

level or p-doped graphene. We consider our description as a ﬁrst

step to illustrate the basic transfer mechanism. In principle, for

large hole occupation difference, screening must be taken into

account. While we discuss here the intraband excitation in the

valence band, the mirrored process is possible also in the con-

duction band in case of n-doped graphene.

To study the Meitner–Auger energy transfer, Figure 6b, we

derive the equation of motion for the incoherent (Qk6¼ 0) exciton

occupation in the K-valley NQk≡NK1s

Qk, see Equation (16)

[51]

∂

tNQk¼2π

ℏX

kjWQkj2ðfkð1fkQkÞNQkðfkQkfkÞ

δðεkεkQkEQkÞ

(26)

The graphene valence band occupation is denoted by fkwith

momentum k, the linear graphene dispersion by εkand the par-

abolic exciton dispersion by EQkas introduced in Section 2.3.

Equation (26) consists of an out-scattering from the TMDC,

an in-scattering of graphene and the matrix element

WQk¼VQkdK1s ⋅Qkφ1s

rk¼0=e. The occurrence of a single-dipole

moment and the Coulomb potential VQk, see Equation (10),

shows that a single-dipole–monopole transition occurs. For a pre-

dominantly TMDC excitation as the initial occupation, compar-

ing Equation (26) with the Boltzmann Equation (17) allows to

identify the decay rate of the excitonic occupation as

ΓQk¼8πX

kjWQkj2ðfkQkfkÞδðεkεkQkEQkÞ(27)

Figure 6. a) Sketch of a TMDC–graphene heterostructure with interlayer

distance Δzand environment dielectric constants ϵ1,ϵ2,ϵ3. b) Dispersion

of the TMDC in the exciton picture depending on the excitonic center-of-

mass momentum Qk(left) and electronic graphene dispersion over the

electron momentum k(right). The decay of an exciton in the TMDC indu-

ces an intraband transition in the graphene valence band.

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where the additional factor of 4 accounts for the valley and spin

degree of freedom in graphene. Evaluating the energy conserving

delta function, we ﬁnd that kaccounts for electrons close to the

Dirac point and kQkfor electrons deep in the valence band. To

obtain an analytical expression, we assume that electrons close to

the Dirac point have much smaller momenta than electrons deep

in the valence band, that is, kQkand jkQkjQk. A sub-

sequent analytical evaluation of the sum yields

ΓQk¼jWQkj2fQkfQkEQk=ℏvF



DOSðQkÞ(28)

with the density of states in graphene

DOSðQkÞ¼ 4

ℏvF

QkEQk

ℏvF



θQkEQk

ℏvF

 (29)

We can see that the rate depends on the matrix element WQk

of the Meitner–Auger transfer, on the density of states in gra-

phene (via the k-sum), and on the occupation difference of

the occupied states fQkin graphene, which accounts for Pauli

blocking. The Heaviside function accounts for the fact that dur-

ing the interlayer transfer a minimal center-of-mass momentum

is required to fulﬁll the energy and momentum conservation

between the strongly different band curvature of a TMDC and

graphene.

Figure 7a shows the calculated Meitner–Auger rate for a

WSe

/graphene interface as function of center-of-mass momen-

tum Qk, Equation (27), for different chemical potentials of the

graphene occupation. As can be observed from Figure 7a due

to the matrix element WQk, the interfacial coupling vanishes

for zero-excitonic center-of-mass momentum. Therefore, only

incoherent excitons NQk6¼0contribute to this type of coupling

mechanism in contrast to coherent excitons pQk¼0,

Equation (13)

[28,92,109,110]

optically generated at Qk¼0. For

momenta close to 1.5 nm

1

, we observe a substantial transition

rate of 1 meV, which increases with increasing chemical poten-

tial. A decreasing chemical potential leads to larger hole

occupations close to the Dirac point, favoring the occupation dif-

ference in Equation (28), and therefore an increase of the transi-

tion rate. As discussed at the beginning, the chemical potential

can either be set initially by doping or induced transiently due to

the pulsed excitation.

Finally, we want to have a closer look at the minimal center-of-

mass momentum necessary to obtain a ﬁnite-transfer rate. The

amount of minimal required center-of-mass momentum can be

controlled via the steepness of the graphene dispersion, that is,

the Fermi velocity, and the TMDC optical bandgap, that is, the

exciton resonance. For the chosen parameters (see appendix),

Figure 7a shows a minimum required center-of-mass momen-

tum of about 1.5 nm

1

. This corresponds to a kinetic energy

of 100 meV, which far exceeds the mean kinetic energy at room

temperature. However, the required momentum can also be

provided by a hot TMDC exciton occupation NQk6¼0induced

for nonresonant optical pumping of the TMDC exciton, see

Equation (26).

[12]

With increasing detunings above the exciton

energy, the amount of injected excitons decreases, but they

occupy larger momenta due an efﬁcient scattering with acoustic

and optical phonons.

[12]

Also for detuning below the optical

bandgap, a substantial amount hot excitons with large center-

of-mass momentum is formed, which can contribute to the inter-

facial Meitner–Auger energy transfer.

[12,51]

Additionally, we investigate the inﬂuence of the interlayer

distance to the Meitner–Auger transfer and compare with the

Förster coupling. To evaluate the z-dependence of the

Meitner–Auger rate, we perform a thermal average

[50]

ΓT¼2∫d2QkeEQk=ðkBTÞΓQk=ℏZwith Z¼∫d2QkeEQk=ðkBTÞof

the out-scattering rate. To evaluate the scattering rate,

Equation (28), we set the out-scattering and in-scattering occupa-

tions to one and zero, respectively. Although, assuming thermal-

ized electron occupations in graphene for a transient coupling

mechanism is a strong approximation, this approach enables

analytic access to the interlayer distance dependence of the

Meitner–Auger coupling mechanism. Figure 7b shows the Δz-

dependence of Meitner–Auger transfer and for comparison also

the result for the Förster transfer. First of all, we see that both

Figure 7. a) Meitner–Auger transfer rate for different transient chemical potentials at room temperature. b) Distance dependence of Meitner–Auger

transfer and Förster. We use vF¼1.8 nm=fsand E1s ¼1.65 eV and ﬁnd a minimum required center-of-mass momentum of around 1.5 nm

1

with an

effective exciton mass of M¼0.65me.

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coupling mechanisms decrease exponentially in the near-ﬁeld.

However, in the far-ﬁeld, the Meitner–Auger transfer continues

to decrease exponentially, while the Förster transfer changes to a

Δz4dependence. Interestingly, the different far-ﬁeld behavior

can be traced back to the linear band structure of graphene,

reﬂected by the Heaviside function in the scattering rate, and

not the difference in the type of dipole–dipole/dipole–monopole

interaction.

In this section, we presented a, to our knowledge, so far

unstudied excitation transfer mechanism (Meitner–Auger pro-

cess) uniquely occurring at semiconductor–semimetal interfa-

ces.

[51]

In contrast to well-known energy-transfer mechanisms

such as Förster or Dexter transfer or carrier transfer such as

tunneling, the interfacial Meitner–Auger energy transfer enables

an excitation transfer from the semiconductor to the semimetal

inducing hot-carrier distributions in the semimetal. This unique

coupling could advance conceptual devices as hot-carrier-based

photocells.

[111]

6. Interlayer Exciton Relaxation in a MoSe

–WSe

Heterostructure

In this section, we investigate the relaxation process of hot

(Qk6¼ 0) interlayer excitons in a TMDC heterostructure, which

consists of a MoSe

layer and a WSe

layer vertically stacked

on top of a SiO

substrate, see Figure 8a. All parameters can

be found in the appendix. Hot interlayer excitons typically

emerge by either pumping the MoSe

intralayer excitonic tran-

sition (favoring a subsequent hole transfer)

[60]

or by pumping the

WSe

excitonic transition (electron transfer)

[61]

due to the type-II

band alignment, see Figure 8b. They also form a microscopic

dipolar ﬁeld due to the interlayer electron–hole separation.

We employ the semiconductor Hamiltonian from

Equation (10) starting from an initially injected hot exciton occu-

pation and add the phonon Hamiltonian

[112]

adjusted to inter-

layer excitons

Hphon ¼X

Kk,α,l

ℏΩl

Kkαbl†

Kkαbl

Kkα

þX

Kk,α,Qk,lðbl

Kkαþbl†

KkαÞGl

Kkα

pIL†

QkþKk

pIL

(30)

Here, the ﬁrst term denotes the free phonon Hamiltonian

with the phonon dispersion ℏΩl

Kkαwith momentum Kk, mode

α, layer l, and the phonon annihilation (creation) operators

bð†Þ. The second term denotes the exciton–phonon interaction

with the excitonic transition operators given in Equation (11)

and the exciton–phonon matrix element Gl

Kkα, given by

GMðWÞ

Kkα¼P

gcðvÞMðWÞ

KkαφIL

qkφIL

qkβILðþαILÞKk(31)

Here, gcðvÞ,MðWÞ

Kkαdenotes the interaction matrix elements for

electron/hole scattering (c=v). The compound index IL denotes

IL ¼flh¼M,ξh¼K,le¼W,ξe¼Kg. As depicted in

Figure 8b, we consider interlayer excitons with holes located

at the Kvalley in the WSe2layer and electrons located at the

Kvalley in the MoSe

layer. The phonon-assisted scattering pro-

cess of electrons and holes as constituents of interlayer excitons

is schematically displayed in Figure 8b. The excitonic wave func-

tions φIL are obtained by solving the Wannier equation for a static

Coulomb potential Vqkin Equation (10) in a ﬁve-layer geometry,

see Figure 8a, as in ref. [113]. With Equations (10) and (30), we

are able to obtain the Boltzmann equation

∂

tNIL

Qk¼X

Γin

Qk,QkþKkNIL

QkþKkX

Γout

Qk,QkþKkNIL

Qk(32)

for the excitonic interlayer occupations NIL

Qk¼ˆ

pIL†

pIL

[93]

see

Equation (17). The in-scattering rates are given by

Γin

Qk,QkþKk¼2π

ℏX

l,α,jGl

Kk,αj2nl

Kk,αþ1

21



δEIL

QkEIL

QkþKkℏΩl

Kk,α



(33)

where nl

Kkαdenotes the phonon occupation. The signs reﬂect

phonon emission (þ) and absorption () processes. The out-

scattering rates can be obtained via the symmetry relation

Figure 8. a) Sketch of a TMDC heterobilayer consisting of one MoSe

layer

(ϵM¼17.4) and one WSe2layer (ϵW¼15.6)

[114]

on top of a SiO2substrate

(ϵ1¼3.9). The gap and the region above the heterostructure are assumed

as vacuum (ϵ2¼ϵ3¼1). Δzdenotes the distance between both layers.

b) Sketch of the phonon-assisted relaxation processes of interlayer occu-

pations (electron in MoSe

, hole in WSe

), described by Equation (30). The

parabolas denote the electron () and hole (þ) dispersion at the Kvalley

in the respective layer over their momenta k. Hot excitons, that are,

Coulomb-bound electron–hole pairs at high momenta, scatter down via

optical and acoustic phonons to cold excitons, that are, Coulomb-bound

electron–hole pairs at low momenta. c) Similar process as in (a), illus-

trated in the exciton picture for interlayer excitons.

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Γout

Qk,QkþKk¼Γin

QkKk,Qk(34)

We solve Equation (32) by assuming a Boltzmann distribution

at the initial condition NIL

Qkðt¼0Þ¼NIL

0eEIL

Qk=ðkBT0Þ=Znormal-

ized with Z¼PQkeEIL

Qk=ðkBT0Þ, where NIL

0is the total exciton

density within the linear regime, where phonon-assisted scatter-

ing dominates. Since we want to model hot interlayer excitonic

occupations, we set the initial exciton temperature to T0¼300K

and assume a lattice temperature of T¼77K. This temperature

mismatch leads to a relaxation process mediated by optical and

acoustic phonons until the temperature of the interlayer occupa-

tions equals the lattice temperature in equilibrium. In Figure 9a,

we show the thermalization process of initially injected hot exci-

tonic occupations. Within about two picoseconds (bright lines in

Figure 9a), the excitonic occupations at higher momenta are

depleted down to momenta or kinetic energies of about

1nm

1or 40 meV, respectively. Then, the scattering process sig-

niﬁcantly slows down until a Boltzmann distribution, whose tem-

perature matches the lattice temperature of 77 K, is formed (dark

line in Figure 9a). The entire thermalization process takes about

20–40 ps. The peculiar characteristics of the thermalization pro-

cess will be examined in the following.

In Figure 9b, we display the decay rates Γout

Qk¼PKk

Γout

QkKkof

the thermalization process of the excitonic interlayer occupations

NIL

Qkin Figure 9a. We observe two features.

First, the decay rates are strongly non-monotonic and can be

divided into two parts depending on the center-of-mass momen-

tum Qkor the kinetic energy ℏ2Q2

2MIL: at high center-of-mass

momenta Qk, the decay rate exhibits a moderate value, which

increases with decreasing momentum until about 1 nm1or

40 meV, where a sharp edge at a high value is formed, after that

the decay rate exhibits a substantial drop. Then, the rate increases

again up to a moderate value at zero momentum. This edge con-

stitutes an optical phonon bottleneck, that is, states of momenta

smaller than the edge location that cannot be reached by optical

phonons due to their large energy transfer. Thus, the scattering

processes on the high-momentum side of the edge are domi-

nated by optical phonons (fast downscattering in Figure 9a),

whereas the scattering in the low-momentum range of the edge

is solely caused by acoustic phonons, which are slow and exhibit

a very small energy transfer (slow thermalization in Figure 9a).

The layer spacing Δz, see Figure 8a, is an important feature

also for TMDC heterostructures. Since the exciton–phonon

matrix elements in Equation (31) depend on the excitonic inter-

layer wave functions φIL, the Δzdependence enters through the

localization of the wave functions in qk-space: a decreasing layer

spacing Δzweakens the localization of the interlayer wave func-

tion φIL

qkand vice versa, since an increasing layer spacing leads to

an increasing charge separation and thus to a weakening of the

Coulomb bonding of the interlayer exciton. In Figure 9b, we

observe that the overall magnitude of the decay rate decreases

monotonous by increasing layer spacing Δz. The cause is the

localization of the excitonic interlayer wave function in momen-

tum space: an increasing localization, caused by a growing layer

spacing, sharpens the exciton–phonon matrix element in

momentum space, see Equation (31), and therefore reduces

the range of scattering partners. Thus, the larger the layer spac-

ing, the longer the duration of the thermalization process dis-

played in Figure 9a becomes.

In this section, we discussed the layer spacing–dependent cool-

ing and thermalization time of hot interlayer excitons, where the

phonon-assisted decay time of hot interlayer excitons increases

with the layer spacing. This can be important, if the TMDC het-

erostructure sample exhibits additional layer spacer. In addition,

the relevant exciton–phonon coupling determines the relaxation

of interlayer excitons into their ground state after their formation

due to electron/hole transfer from intralayer excitons.

7. Conclusion

In conclusion, we have presented a general Maxwell Bloch equa-

tion framework to describe dipolar interactions at interfaces of

Figure 9. a) Snapshots of the interlayer occupations NIL

Qkranging from 0 fs

to 40 ps, displaying the thermalization process of hot excitons, see

Equation (32), for different interlayer spacings. The initial interlayer occu-

pation is assumed as a hot Boltzmann distribution with a temperature of

T0¼300K, whereas the lattice temperature is assumed as T¼77K.

b) Decay rates Γout

Qk¼PKk

Γout

QkKk, see Equations (33) and (34), of the ther-

malization process of the interlayer occupation in a MoSe

–WSe

heterostructure.

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published by Wiley-VCH GmbH

TMDC monolayers with a functionalization, for example, dye

molecules, plasmonic nanoparticles, arrays of plasmonic nano-

particles, graphene, and other TMDC monolayers on the same

footing. To focus on the electromagnetic near-ﬁeld coupling,

electron/hole-tunneling processes are not taken into account.

We have demonstrated that Förster interaction manifests an

important relaxation pathway of optical dye–molecule excitation:

tightly bound excitons in the TMDC monolayer modulate the

dependence on the molecular transition energy. In particular,

momentum-dark excitons induce sharp spectral features in

the Förster rate, and therefore the functionalization with mole-

cules makes these states observable in the far-ﬁeld.

We further have revealed the appearance of bound exciton–

plasmon states in stacks of TMDC monolayers and plasmonic

nanoparticles which arise due to dipole–dipole coupling of exci-

tons and plasmons, so-called plexcitons. The framework, based

on microscopic electron dynamics, allows us to describe the exci-

ton localization near the plasmonic nanoparticles. At the same

time, strong coupling in optical spectra is a ﬁngerprint for exci-

ton localization. Moreover, in stacks of TMDC monolayers with

2D PCs, we ﬁnd strong coupling leading to a hybridization of

exciton and plasmon dispersion with mode splittings on the

order of some tens of meV. Due to the periodicity of the struc-

ture, this splitting can be observed in far-ﬁeld detection.

Next, we have studied the energy transfer from WSe

excitons

to graphene plasmon hole excitations, where we have proposed a

new type of interlayer transfer, which is based on the Meitner–

Auger interlayer coupling: here, an exciton in the WSe

layer

decays via excitation of an intraband hole plasmon in graphene.

Following our microscopic evaluation, this transfer process is

fast compared to conventional energy-transfer mechanisms such

as Dexter and Förster coupling in the considered structure and

allows an excitation transfer from the TMDC inducing hot-

carrier distributions in the graphene layer.

Last, we study the phonon-assisted relaxation dynamics of hot

interlayer excitons in a TMDC heterobilayer with type-II band

alignment: we ﬁnd that the duration of the thermalization

process of hot interlayer excitons grows by an increasing layer

spacing. This behavior can be traced back to the layer spacing-

dependent Coulomb bonding of interlayer excitons and, thus,

can be of crucial importance in heterostructure samples with

additional layer spacer.

In summary, we have presented a general theoretical frame-

work for the investigation of electromagnetic interactions in

hybrid structures and we provided examples to demonstrate

its capabilities. While the structures presented here are by no

means exhaustive, they illustrate the potential of the framework

to explore hybrid systems from different perspectives. Future

work will involve more detailed investigations of the structures

presented here, as well as the application of the framework to

different hybrid structures or excitons at elevated densities.

Acknowledgements

The authors acknowledge 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,

and Project KN 427/11-2 (H.M. and A.K.) Project No. 420760124.

Open Access funding enabled and organized by Projekt DEAL.

Conﬂict of Interest

The authors declare no conﬂict of interest.

Data Availability Statement

The data that support the ﬁndings of this study are available from the

corresponding author upon reasonable request.

Keywords

dipolar coupling, excitons, Förster coupling, Meitner–Auger coupling,

metals, plasmons, semiconductors

Received: February 15, 2023

Revised: May 11, 2023

Published online: July 20, 2023

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