Polariton Enhanced Free Charge Carrier Generation in DonorAcceptor Cavity Systems by a Second-Hybridization Mechanism Weijun Wu1Andrew E. Sifain1Courtney A. Delpo1and Gregory D. Scholes1

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Polariton Enhanced Free Charge Carrier Generation in Donor–Acceptor

Cavity Systems by a Second-Hybridization Mechanism

Weijun Wu,1Andrew E. Sifain,1Courtney A. Delpo,1and Gregory D. Scholes1

Department of Chemistry, Princeton University, Princeton, NJ 08540, U.S.

(*Electronic mail: gscholes@princeton.edu)

(Dated: 5 October 2022)

Cavity quantum electrodynamics has been studied as a potential approach to modify free charge carrier generation

in donor-acceptor heterojunctions because of the delocalization and controllable energy level properties of hybridized

light–matter states known as polaritons. However, in many experimental systems, cavity coupling decreases charge sep-

aration. Here, we theoretically study the quantum dynamics of a coherent and dissipative donor–acceptor cavity system,

to investigate the dynamical mechanism and further discover the conditions under which polaritons may enhance free

charge carrier generation. We use open quantum system methods based on single-pulse pumping to ﬁnd that polaritons

have the potential to connect excitonic states and charge separated states, further enhances free charge generation on an

ultrafast timescale of several hundred femtoseconds. The mechanism involves that polaritons with proper energy level

allow the exciton to overcome the high Coulomb barrier induced by electron-hole attraction. Moreover, we propose that

a second-hybridization between a polariton state and dark states with similar energy enables the formation of the hybrid

charge separated states that are optically active. These two mechanisms lead to a maximum of 50% enhancement of

free charge carrier generation on a short timescale. However, our simulation reveals that on the longer timescale of

picoseconds, internal conversion and cavity loss dominate and suppress free charge carrier generation, reproducing the

experimental results. Thus, our work shows that polaritons can affect the charge separation mechanism and promote

free charge carrier generation efﬁciency, but predominantly on a short timescale after photoexcitation.

I. INTRODUCTION

The excitonic character of organic semiconductors is cru-

cial for designing the architecture and geometry of organic

photovoltaic cells. A critical component is the donor-acceptor

heterojunction, where exciton goes dissociation and charge

transfer.1,2 The optimization of donor-acceptor materials’ ef-

ﬁciency for free charge carrier generation is based on the con-

trol of the basic ultrafast dynamics processes including optical

absorption and exciton formation3,4, exciton transport5–7, ex-

citon dissociation and charge separation8–10 and free charge-

carrier mobility11. However, the mechanism at the molecular

level and the coherent quantum-mechanical description of the

dynamics are still topics of debate and investigation.

It has already been theoretically proposed that the electron-

hole separation at the interface is signiﬁcantly inﬂuenced

by the contradictory relation between the Coulomb attrac-

tion barrier and vibronic charge transfer state.12 First, in-

creasing delocalization of charges can substantially reduce the

Coulomb barrier13 (FIG. 1(b)). Second, the excess energy of

the exciton-charge transfer transition can induce vibrational

excitations. The vibronic charge transfer state can be resonant

with charge separated states with high potential and thus over-

come the Coulomb barrier14–16 (FIG. 1(c)). It has also been

indicated that internal conversion and non-radiative transition

could modify the charge separation dynamics.17

Cavity quantum electrodynamics (cavity QED)18,19 has

been studied recently to modify the excitonic character

through light-dress, via embedding the materials between

two reﬂective mirrors (known as the Fabry–Pérot cavity).

Strong coupling between the polarization of a material and

a quantized electromagnetic ﬁeld conﬁned in a cavity forms

hybridized light-matter states known as polaritons. Polari-

tons have extraordinary delocalization in the molecule ba-

sis and modiﬁed energy levels. The collective nature of po-

laritons due to the coupling between many molecules and

a single global oscillator (i.e., photon) is central to ex-

traordinary delocalization phenomena including long-range

energy transfer,20–22 enhanced charge conductivity23–25 and

superconductivity26. The control and change of chemistry at

select energies has been a central topic including modiﬁca-

tion of ground27–30 and excited state chemical reactions,31–34,

singlet ﬁssion35,36 and spin state selectivity.37–40

The delocalization nature of the polaritons is predicted to

improve exciton migration and charge transport.19Studies on

charge transfer in cavities41 also show the potential of charge

transfer enhancement in a dimer system in the Marcus In-

verted Regime42 or under incoherent driving conditions43.

However, at the interface, there is a competition between

charge transfer from the polariton and the decay of the

polariton to the ground state. A recent spectroscopic

study on P3HT(poly(3-hexylthiophene)):PCBM(phenyl-C61-

butyric acid methyl ester) inside the cavity suggests that al-

though the rate of charge transfer from the lower polariton is

slowed relative to the rate of charge transfer from a bare P3HT

polymers, charge transfer from polariton to create free charge

carriers remains fast enough to compete with the decay of the

polariton to the ground state.44

In order to understand the competition between polariton

mediated free charge carrier generation and the dynamics of

dissipation and dephasing, we theoretically utilize a general

open quantum system method to study the coherent and dis-

sipative quantum dynamics in both bare and cavity donor-

acceptor systems. We elucidate the roles that polaritons may

play on the charge separation mechanism, analyze the dynam-

ical reason for the experimental negative effect of polaritons,

and furthermore, clarify the conditions of certain time-scale

arXiv:2210.01278v1 [physics.chem-ph] 3 Oct 2022

FIG. 1. (a) An illustration of the oligothiophene:fullerene hetero-

junction embedded in a Fabry–Pérot cavity. The fullerenes of 2

hexagon layers with 7 fullerenes per layer (7C60 ×2L) is displayed.

The fullerenes are treated as one super-molecule and the distance be-

tween fullerene super-molecule and a specﬁc layer of oligothiophene

polymer chain is labeled as ∆x. (b) The Coulomb attraction potential

between a hole on oligothiophene polymer chain and an electron on

fullerene super-molecule with a distance of ∆xfor different packing

types for fullerenes (zlayers with nfullerenes each layer, zC60 ×nL

in short). The data comes from Ref. 12 and 13 by Density Functional

Theory calculation. (c) Concept of charge separation mediated by

vibronic charge transfer states as explained in Ref. 12. The dashed

lines of each potential curves represent the vibrational states. Red

arrows represent population transfer.

and coupling strength under which the polaritons may posi-

tively enhance free charge carriers generation.

Here, we propose that adjusting the energy level of the po-

laritons may change the relation between the height of the

Coulomb barrier and the energy of vibronic charge transfer

state to promote exciton dissociation at the interface.12,13,17

Moreover, we propose a novel mechanism that a second-

hybridization (distinct from the light-matter hybridization that

forms polaritons) between polaritons and dark states can

create the polariton states with charge separation compo-

nents working as instantaneous charge generators on a short

timescale after photoexcitation. Our quantum dynamics sim-

ulation with Redﬁeld Theory for vibronic coupling and Lind-

blad Theory for cavity loss working on a phenomenological

Hamiltonian spanned by Frenkel exciton states and charge

separated states predicts that single-pulse-pumping polaritons

can signiﬁcantly enhance free charge carrier generation up to

50% comparing to the bare system on a short timescale of

several hundred femtoseconds, before internal conversion and

cavity loss dominates.

II. HAMILTONIAN AND DYNAMICS MODEL

We start by specifying a basis set of the Hilbert Space

to build the Hamiltonian as an extension of the model in

Ref. 17. To provide realistic parameters representing re-

gioregular polymer:fullerene heterojunctions, we consider the

parent compounds, oligothiophene:fullerene in FIG. 1(a),

as a model system. The fullerene acceptor is represented

by an effective, coarse-grained super-molecule, labeled as

A0, while the primary oligothiophene donor photoexcitations

in the self-assembled lamellae structures are interchain, H-

aggregate type excitons.45 Intrachain interactions are elimi-

nated by Fourier Transformation (see Supplementary Mate-

rial S1.1). Therefore, Noligothiophene chains are treated ex-

plicitly as individual sites labeled as Di, where i=1,2,...,N

represents the i-th site from fullerenes. Furthermore, we de-

ﬁne the localized Frenkel exciton conﬁgurations on oligoth-

iophene donor as |XTii=|D−

ii⊗|D+

iiand the non-localized

electron-hole pair with the one hole on oligothiophene and

one electron on fullerenes as |CSii=|A−

0i⊗|D+

ii. Since the

distinction between charge transfer state and charge separated

state is not rigorous, the charge transfer state is roughly de-

ﬁned as |CSi=1i. The quantized electromagnetic ﬁeld con-

ﬁned in the cavity with single mode approximation are written

in the Fock Space. For example, |1iis the single photon state.

A common ground state without any excitation, i.e. vacuum

state |vaci, is of zero energy and can be a sink for dissipation

processes.

A phenomenological Hamiltonian with N=13 oligothio-

phene sites can be represented within the basis set deﬁned

above after single excitation truncation

He=εXT ∑

i|XTiihXTi|+JXT ∑

(|XTiihXTi+1|+h.c.)

(1a)

+∑

εCS

i|CSiihCSi|+JCS ∑

(|CSiihCSi+1|+h.c.)

(1b)

+Jint (|XT1ihCS1|+h.c.)(1c)

+¯

hωc|1ih1|+g∑

(|1ihXTi|+h.c.)(1d)

The Frenkel Exciton states |XTiiare assumed to be spa-

tially invariant with the identical on-site energies εXT =2.3eV

as a typical value. The relative energies of the charge sep-

arated states εCS

i−εCS

1are deﬁned by the Coulomb bar-

rier in FIG. 1(b) with the inter-oligothiophene distance of

0.38 nm12,13. The potential energy surface for CS state is ap-

proximately unaffected by dipole self energy41 since the per-

manent dipole of CS states should be parallel to the stand-

ing wave vector, as shown in FIG. 1(a). The excess en-

ergy of the interfacial exciton-charge transfer transition that

induces a vibronic charge transfer state is expressed by the

energetic offset between the interfacial excitonic and charge

transfer states ∆Eoffset =εXT −εCS

1. The other intermolec-

ular coupling parameters are determined from ab-initio the-

ory calculation by Tamura and coworkers12,13,17,46,47: The

nearest-neighbour coupling within the H-aggregate excitonic

states and the charge separated states are JXT =0.1eV and

JCS =−0.12eV, respectively. Charge transfer process at the

interface can be described by the strength Jint =0.2eV.

The cavity photon mode frequency ωcand collective light-

matter strong coupling strength G=g√Nunder the rotating

wave approximation are adjustable by cavity preparation.48

Experimentally, one can change the inter-mirror distance to

modify ωcas the frequency of a standing wave, and change the

number of molecules in one oligothiophene chain Nyto mod-

ify single site-photon coupling strength because g∝pNy.

Nycan be large enough to have the system reach the strong

light-matter coupling regime in a typical Fabry–Pérot micro-

cavity (2 mirrors in the size of 1 inch by 1 inch and the dis-

tance between them of several hundred nanometers). In this

model, we assume cavity photon is single mode as a stand-

ing wave under long-wavelength limit, while other work re-

ports the cases where cavity photons have non-zero in-mirror-

plan momentum.49 Only XT states can effectively achieve

light-matter coupling, while CS are decoupled from the cav-

ity, since CS states have negligible transition strength, regard-

less of the cavity, because the transition dipole depends on

electron-hole overlap.

Vibronic coupling induced internal conversion and de-

phasing for the electronic density matrix ρe(t)is calculated

by Bloch-Redﬁeld Theory. Due to the weak electronic-

vibrational coupling HI17, Quantum Markovianity can deal

with the vibrational modes as a large thermal reservoir at

thermal-equilibrium ρv50. The quantum master equation in

the Dirac Picture (represented by tildes) reads

d˜

ρe(t)

dt =−1

h2Z∞

0dτTrv˜

HI(t),˜

HI(t−τ),˜

ρe(t)⊗ρv

(2)

Lindbladians γXLX[ρe] = γXXρeX†−1

2X†X,ρe are

added to Eq. (2) for cavity loss (X=|vacih1|,γX=κ=

20THz) and electron-hole recombination of Frenkel exciton

(X=|vacihXTi|,γX=γdecay =1GHz). The free charge car-

rier is described by the charge separation beyond |CSNiand its

generation is expressed as a Lindbladian with X=|vacihCSN|

and γX=γout =10THz. The observable of interest, the free

charge carrier generation rate, can be obtained by

IFC(t) = −γoutTrP

excitationL|vacihCSN|[ρe(t)](3)

where P

excitation =∑i|XTiihXTi|+∑i|CSiihCSi|+|1ih1|is

the projector operator to the single excitation subspace. The

details of the theory are in Supplementary Material S1.3.

This Hamiltonian and quantum dynamics framework pro-

vides a lens into the complex interplay between light-matter

coupling, delocalization, thermalization, dephasing and dissi-

pation.

III. RESULTS AND DISCUSSION

In section III A and III B, we mainly focus on the case of

fullerene packing type 7C60 ×1L (Coulomb barrier 0.25 eV),

interfacial energy offset ∆Eoffset =0eV (no initial vibronic ef-

fect), photon mode ωc=2.495eV and collective light-matter

coupling G=0.12eV, as an example to illustrate the effect

of polaritons. The cases for different parameters will be dis-

cussed in section III C.

A. Eigenspectrum Calculation

FIG. 2(a) depicts the eigenstates and absorption spectrum

of the bare system. A low-energy localized interface trap

state emerges, where an energy gap of 0.25 eV between the

interface trap and the lowest delocalized CS manifolds cor-

responds to the Coulomb barrier for fullerene packing type

7C60 ×1L. At higher energies, the delocalized excitonic man-

ifolds become mixed with delocalized charge separated states

below the effective top of the barrier, suggesting a tunneling

effect, which is beneﬁcial for free charge carrier generation.

However, the dark nature of these XT-CS mixed state prevents

populating from photoexcitation but internal conversion only.

The absorption spectrum demonstrates a bright state of the

H-aggregate at the upper edge of the XT manifolds, mixing

weakly with the CS manifolds. Above the bright state, there

are some delocalized and dark CS eigenstates. Thus by pump-

ing the bright state, free charge carrier generation mostly oc-

curs after population transfer downhill to dark XT-CS mixed

state or uphill to dark CS eigenstates.

FIG. 2(b) depicts the eigenstates and absorption spectrum

of the cavity system, where the photon mode ωc=2.495eV

is resonant with the bare bright state to achieve strong light-

matter hybridization. The upper polariton (EUP =2.61eV)

and lower polariton (ELP =2.36eV) show up with larger

broadening than the bare bright state, signifying inhomoge-

neous broadening (see Supplementary Material S2.2).

Within the absorption linewidth, the polaritons present

large CS components. As shown in FIG. 2, the CS fraction

for upper and lower polariton branchs are 51% and 34% re-

spectively, while the CS fraction for the bare bright state is

only 7%. In contrast to the bare bright state, the optically ac-

tive polaritons are direct and instantaneous free charge carrier

generators.

The inhomogeneous broadening and CS components fea-

tures indicate that polaritons |UP(+)/LP(−)iare not merely

a hybridization between photon |1iand bare bright state |Bi,

FIG. 2. Electronic excited eigenstate of the system with fullerene packing type 7C60 ×1L and ∆Eoffset =0eV for (a) Bare System and (b)

Cavity System (ωc=2.495eV and G=0.12eV). In the left panel, the ordinate deﬁnes the eigenvalues, while the abscissa deﬁnes a series

of basis states pertaining to the XT manifolds (|XTNi,... ,|XT2ifrom left to right), the subset of interfacial states (|XT1iand |CS1i) and the

CS manifolds (|CS2i,... ,|CSNifrom left to right). Photon is not displayed. The probabilities are represented as a density proﬁle. The right

panel is the absorption spectrum ﬁtted from the vacuum-to-eigen transition dipole. Bright state, interface trap, upper polariton (UP) and lower

polariton (LP) are marked.

but also involve mixing dark states |di, which could be under-

stood by ﬁrst-order perturbation theory (see Supplementary

Material S1.4)

|UP(+)/LP(−)i=1

√2(|1i±|Bi) + ∑

d/√2

EB±g0

B−Ed|di

(4)

where g0

B=~

ε·hB|~

µ|vaci ≈ Gand g0

d=~

ε·hd|~

µ|vaciare the

ﬁeld ~

ε- dipole ~

µinteraction for bright and dark state re-

spectively, while EBand Edare the unperturbed energy of

bright and dark state respectively. Second-hybridization oc-

curs when unperturbed polaritons have similar energy to cer-

tain unperturbed dark states with non-zero transition dipoles

(despite the transition dipoles are extremely tiny). The result-

ing perturbed polaritons contain signiﬁcant CS component,

which is beneﬁcial for free charge carrier generation.

B. Dynamics Simulation

Based on the eigenstates calculation (FIG. 2), Redﬁeld and

Lindblad Theory simulation for different cases are compared

in FIG. 3, while rates and population calculated by Eq. (3) for

free charge generation are shown in FIG. 4.

When the bare system is initially populated at the interface

trap (FIG. 3(a)), the population transfers to the delocalized XT

manifolds within 0.01 ps before transfering to the CS mani-

folds in 0.04 ps. If the bare bright state is pumped initially

(FIG. 3(b)), the population of delocalized bright exciton par-

tially transfers to the CS manifolds. However, on a longer

timescale of 0.3 ps, most of the population still gets trapped at

the interface and the XT population overwhelms the CS pop-

ulation.

On the contrary, when the polaritons are pumped for the

cavity system, second-hybridization between polaritons and

dark states populates the CS manifolds immediately. For the

lower polariton case (FIG. 3(c)), XT manifolds population

rises rapidly, while the CS manifolds population will transfer

to interface trap after 0.02 ps, corresponding to the population

transfer from lower polariton to XT-CS mixed state. For the

upper polariton case (FIG. 3(d)) there is signiﬁcant population

at the CS manifolds until 0.06 ps due to the downhill transfer

to dark CS eigenstates before deexcitation to XT manifolds.

XT manifolds population of upper polariton case is less than

that for lower polariton cases. In all the four cases, the sys-

tems seem to reach a quasi-steady state and the CS manifolds

populate more around CS1and CSN.

The free charge carrier generation for the four cases in

FIG. 4 better demonstrates the effect of cavity mediated

charge separation. In the ﬁrst 0.1 ps of FIG. 4(a), the rates

of the four cases are sorted in the order of the energy of the

states, meaning that the upper polariton can overcome the

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PolaritonEnhancedFreeChargeCarrierGenerationinDonorAcceptorCavitySystemsbyaSecond-HybridizationMechanismWeijunWu,1AndrewE.Sifain,1CourtneyA.Delpo,1andGregoryD.Scholes1DepartmentofChemistry,PrincetonUniversity,Princeton,NJ08540,U.S.(*Electronicmail:gscholes@princeton.edu)(Dated:5October2022)Cavityqu...

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Polariton Enhanced Free Charge Carrier Generation in DonorAcceptor Cavity Systems by a Second-Hybridization Mechanism Weijun Wu1Andrew E. Sifain1Courtney A. Delpo1and Gregory D. Scholes1

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