Quantum Mechanical Assessment of Optimal Photovoltaic Conditions in Organic Solar Cells Artur M. Andermann and Luis G. C. Rego

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Quantum Mechanical Assessment of Optimal Photovoltaic

Conditions in Organic Solar Cells

Artur M. Andermann and Luis G. C. Rego

Department of Physics, Universidade Federal de Santa Catarina,

88040-900, Florian´opolis, Santa Catarina, Brazil

Abstract

Recombination losses contribute to reduce JSC ,VOC and the ﬁll factor of organic solar cells. Re-

cent advances in non-fullerene organic photovoltaics have shown, nonetheless, that eﬃcient charge

generation can occur under small energetic driving forces (∆EDA) and low recombination losses.

To shed light on this issue, we set up a coarse-grained open quantum mechanical model for investi-

gating the charge generation dynamics subject to various energy loss mechanisms. The inﬂuence of

energetic driving force, Coulomb interaction, vibrational disorder, geminate recombination, tem-

perature and external bias are included in the analysis of the optimal photovoltaic conditions

for charge carrier generation. The assessment reveals that the overall energy losses are not only

minimized when ∆EDA approaches the eﬀective reorganization energy at the interface but also

become insensitive to temperature and electric ﬁeld variations. It is also observed that a moderate

reverse bias reduces geminate recombination losses signiﬁcantly at vanishing driving forces, where

the charge generation is strongly aﬀected by temperature.

arXiv:2210.12537v1 [cond-mat.mtrl-sci] 22 Oct 2022

Eﬃcient charge generation has always been a major concern for organic photovoltaics

(OPV) due to the strong exciton binding energy in organic materials.1–5 Thus, fullerene

derivatives have been used as prototypical electron acceptor materials, because of their re-

markable electron-accepting capabilities. However, the eﬃciency of fullerene-based organic

solar cells (OCS) has stagnated around 13% as a consequence of the excessive energy losses

undergone to achieve electron-hole (e-h) charge separation, and the very poor optical char-

acteristics of the C60 molecule that does not contribute to charge generation. Meanwhile, a

new class of non-fullerene acceptor (NFA) materials have been developed, which revealed a

new paradigm for OPV, one of eﬃcient charge generation and high output voltage despite

lower energetic gradients at the heterojunction. Nowadays, the conversion eﬃciency of the

NFA devices is approaching the barrier of 20%,2as a result of the high photo-absorption

of non-fullerene small molecules; the push-pull design of the NFA molecules that facilitate

the electron-hole pair separation upon photoexcitation; the complementary light-harvesting

roles played by large band-gap donor materials and low-bandgap NFA materials; besides

the beneﬁcial morphological properties of the NFA molecules (e.g., miscibility, stability, and

planarity). For OSCs the output voltage is given by Vout =ECT /q −∆Vr

rec −∆Vnr

rec,6–8 where

ECT /q is related to the energetic driving force7,9 and the voltage losses due to recombina-

tion comprise both radiative (∆Vr

rec) and non-radiative (∆Vnr

rec) pathways. In NFA-OSCs

the voltage loss mechanisms have been empirically reduced whereas the interplay among the

relevant physical processes at the D:A interface remains a subject of study.6,7,9–19

In this paper we combine the Ehrenfest and Redﬁeld methods in a coarse-grained open

quantum mechanical model to investigate, from a fundamental point of view, the energy

loss eﬀects that occur during the free charge carrier generation process at the D:A inter-

face. We study the inﬂuence of the energetic driving, temperature and external bias on

the charge generation process. The simulation results demonstrate that the overall energy

losses are minimized at the activationless regime of the charge separation, where driving force

matches the eﬀective reorganization energy of the heterojunction, corroborating experimen-

tal studies.20–22 Moreover, the optimal photovoltaic condition is insensitive to variations of

temperature and reverse bias.

We partition the total Hamiltonian, responsible for the charge generation and recombi-

nation processes, as

H=HS+HB+HSB .(1)

The system Hamiltonian, HS, comprises the degrees of freedom of the electron and the hole

and a subset of vibrational reorganization modes that are described as classical modes within

the framework of the Ehrenfest self-consistent method. Thus, HSis a mixed quantum-

classical Hamiltonian (details provided in Support Information). The bath Hamiltonian,

HB, accounts for the degrees of freedom of the environment, namely the vibrational degrees

of freedom and ﬂuctuations of the dielectric background, which we describe as an ensem-

ble of quantum harmonic oscillators. The system-bath coupling (HSB ) is described within

the framework of the Redﬁeld theory. Using the adiabatic representation for the system

Hamiltonian, HS|ϕai=Ea|ϕai, the Redﬁeld equation for the reduced density matrix σ

reads

∂σab

∂t =−iωabσab +X

c,d

Rabcdσcd (2)

where ωab = (Ea−Eb)/~designates the eigenfrequencies of the electronic system and Rabcd

is the Redﬁeld relaxation tensor.23,24 The ﬁrst term on the right-hand-side (RHS) of Eq. (2)

describes the coherent quantum dynamics of S, the second term describes the interaction

of S with the environment. The implementation of the Redﬁeld equations for this model

has been described elsewhere.25 Later on we will incorporate a term for the recombination

eﬀects into Eq. (2).

The D:A interface is modelled as a two-dimensional (2D) coarse-grained lattice, with

energy proﬁle corresponding to a staggered (type II) interface, as described in Figure 1. The

spatial arrangement of the 2D-D:A interface is shown at the bottom of the ﬁgure, where each

site of the lattice represents a molecular site in the coarse-grained model. Periodic boundary

conditions are applied along the transverse direction. In the simulations, we consider the

photoexcitation of a molecular site of the donor material, but the obtained results are equally

valid for a photoexcitation in the acceptor material. After charge separation, the electron

and hole are collected at the respective collecting layers at the border of the 2D lattice (see

Figure 1). The electron and hole collecting layers in the model do not represent the actual

cathode and anode terminals of the device, they are simply used as a theoretical tool to

avoid the reﬂection of the wavepackets at the left and right boundaries; they are positioned

below ECB(A) and above EV B(D), respectively.

The coarse-grained Hamiltonian HS, for either the electron and hole photoexcited parti-

FIG. 1. Model for the donor-acceptor interface. The ﬁgure on top shows the energetics of the D:A

interface, including the electron and hole collectors. The gaussian envelopes depict the electron-hole

excitation dynamics and charge separation. The upward arrow indicates the photoexcitation of the

electron-hole pair (black) whereas the downward dashed arrows describe overall radiative and non-

radiative recombination decays. At the bottom, the spatial conﬁguration of the two-dimensional

model of the D:A interface. Each molecule is described as a site in a 2D lattice.

cles, is

HS≡Hel/hl =

inEel/hl

i+εi(t)−Φel/hl

i(t)o|iihi|+

i6=j

Vij (t)|iihj|,(3)

where Ndenotes the total number of molecular sites in the lattice. The orthogonal basis

set {|ii} consists of diabatic states associated with each of the molecular sites. In addition,

Eel

i=ECB

iand Ehl

i=EV B

iare the on-site energies of the electron (conduction band)

and hole (valence band); εiis the conﬁnement energy associated with site i; and Φel/hl

describes the electron-hole electrostatic interaction, given in the time-dependent mean-ﬁeld

approximation by

Φel/hl

i(t) = ξbind X

Phl/el

(1 + dij )!.(4)

where ξbind is the on-site electron-hole binding energy (refer to Table I for a complete list

of model parameters), Pel

j(Phl

j) is the time-dependent electron (hole) population on site j

and dij =|~

Ri−~

Rj|is the distance between molecular sites iand j. The treatment accounts

for the decrease of the el-hl Coulomb barrier induced by the charge delocalization, besides

describing the inﬂuence of temperature in the el-hl separation, as shown elsewhere.25 The

last term in Eq. (S3),

Vij =V0Fij =V02`i`j

i+`2

exp −d2

i+`2

j,(5)

describes the tunneling energy between lattice sites, where V0is the bare electronic coupling

and the form factor Fij is the time-dependent overlap between lattice sites iand j, with `

being the conﬁnement radius of the molecular site (see SI for details). The time dependence

of the energy parameters comprising Eq (S3), namely εi(t) and Vij (t) gives rise to the intra-

molecular and inter-molecular vibrational eﬀects, respectively, that we associate with the

Holstein and Peierls couplings.25 The treatment includes dynamic disorder, which is taken

into account by the intra-molecular and inter-molecular electron-phonon couplings in the

system Hamiltonian (HS). The dynamic disorder is also taken into account in the system-

bath Hamiltonian (HSB ) that gives rise to the Redﬁeld relaxation tensor and, lastly, through

the action of the classical Berendsen thermostat that acts on the molecular sites (describe

in SI). These interactions give rise to localization of the electron and hole wavepackets,26

polaron formation,25 and temperature dependence of the charge generation rate. The model

does not include static disorder.

Recombination is the main eﬃciency loss mechanism in photovoltaics, even more so in

OSCs due to the convoluted charge separation process in organic devices. Recombination

processes can be separated in two types: geminate and non-geminate (including bimolec-

ular and trap-assisted processes). The former designates the annihilation of a correlated

electron-hole pair generated by the same photoexcitation event, or the recombination of

a charge separated (CT) state before complete dissociation. Experiments have shown that

geminate recombination is more important at short-circuit and low illumination conditions.27

Bimolecular recombination, on the other hand, involves fully dissociated carriers generated

by independent photoabsorption events. It is more relevant at open-circuit and/or high

excitation conditions. A quantum mechanical model that is well suited for describing the

geminate recombination in molecular systems is the Haberkorn model. It was initially pro-

posed to describe recombination in radical pairs,28 but later was used to describe the trap-

ping and exciton recombination in the Fenna-Matthews-Olson (FMO) protein complex29

and OSC models.26 Herein we include the vibrational dynamics, in addition to the electron-

hole dynamics to extend the plain Haberkorn formalism. Thus, consider a quantum state

|Ψη(t)i=PiCη

i(t)|iithat comprises the reduced density matrix ση, where η=el, hl des-

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QuantumMechanicalAssessmentofOptimalPhotovoltaicConditionsinOrganicSolarCellsArturM.AndermannandLuisG.C.RegoDepartmentofPhysics,UniversidadeFederaldeSantaCatarina,88040-900,Florianopolis,SantaCatarina,BrazilAbstractRecombinationlossescontributetoreduceJSC,VOCandthellfactoroforganicsolarcells.Re-ce...

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Quantum Mechanical Assessment of Optimal Photovoltaic Conditions in Organic Solar Cells Artur M. Andermann and Luis G. C. Rego

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