# Quantum Mechanical Assessment of Optimal Photovoltaic Conditions in Organic
Solar Cells

Artur M. Andermann  Department of Physics, Universidade Federal de Santa
Catarina, 88040-900, Florianópolis, Santa Catarina, Brazil  Luis G. C. Rego
Department of Physics, Universidade Federal de Santa Catarina, 88040-900,
Florianópolis, Santa Catarina, Brazil<EMAIL_ADDRESS>
###### Abstract

Recombination losses contribute to reduce $J_{SC}$, $V_{OC}$ and the fill
factor of organic solar cells. Recent advances in non-fullerene organic
photovoltaics have shown, nonetheless, that efficient charge generation can
occur under small energetic driving forces ($\Delta E_{DA}$) and low
recombination losses. To shed light on this issue, we set up a coarse-grained
open quantum mechanical model for investigating the charge generation dynamics
subject to various energy loss mechanisms. The influence of energetic driving
force, Coulomb interaction, vibrational disorder, geminate recombination,
temperature 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 $\Delta E_{DA}$
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.

Efficient charge generation has always been a major concern for organic
photovoltaics (OPV) due to the strong exciton binding energy in organic
materials.Clarke and Durrant (2010); Armin  _et al._ (2021); Cheng  _et al._
(2018); Hallermann  _et al._ (2008); Menke and Holmes (2014) Thus, fullerene
derivatives have been used as prototypical electron acceptor materials,
because of their remarkable 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
characteristics 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%,Armin  _et al._ (2021) as 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
$V_{out}=E_{CT}/q-\Delta V_{rec}^{r}-\Delta V_{rec}^{nr}$,Rosenthal  _et al._
(2019); Liu  _et al._ (2016); Hou  _et al._ (2018) where $E_{CT}/q$ is related
to the energetic driving forceLiu  _et al._ (2016); Perdigón-Toro  _et al._
(2020) and the voltage losses due to recombination comprise both radiative
($\Delta V_{rec}^{r}$) and non-radiative ($\Delta V_{rec}^{nr}$) 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.Rosenthal  _et al._ (2019); Liu  _et al._ (2016); Perdigón-
Toro  _et al._ (2020); Coropceanu  _et al._ (2019); Gao  _et al._ (2015);
Eisner  _et al._ (2019); Classen  _et al._ (2020); Chen  _et al._ (2021); Chan
_et al._ (2021); Benduhn  _et al._ (2017); Qian  _et al._ (2018); Hofinger
_et al._ (2021); Chen  _et al._ (2018)

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 interface. 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 experimental studies.Coffey  _et al._ (2012); Ward  _et al._
(2015); Rand  _et al._ (2007) 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
recombination processes, as

$H=H_{S}+H_{B}+H_{SB}~{}.$ (1)

The system Hamiltonian, $H_{S}$, 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, $H_{S}$ is a mixed quantum-classical Hamiltonian
(details provided in Support Information). The bath Hamiltonian, $H_{B}$,
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 ensemble of quantum harmonic oscillators. The system-bath
coupling ($H_{SB}$) is described within the framework of the Redfield theory.
Using the adiabatic representation for the system Hamiltonian,
$H_{S}|\varphi_{a}\rangle=E_{a}|\varphi_{a}\rangle$, the Redfield equation for
the reduced density matrix $\sigma$ reads

$\frac{\partial\sigma_{ab}}{\partial
t}=-i\omega_{ab}\sigma_{ab}+\sum_{c,d}R_{abcd}\sigma_{cd}$ (2)

where $\omega_{ab}=(E_{a}-E_{b})/\hbar$ designates the eigenfrequencies of the
electronic system and $R_{abcd}$ is the Redfield relaxation tensor.Redfield
(1957, 1965) 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.Andermann and Rego (2022) Later on we
will incorporate a term for the recombination effects into Eq. (2).

Figure 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.

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 $E_{CB}(A)$ and above $E_{VB}(D)$, respectively.

The coarse-grained Hamiltonian $H_{S}$, for either the electron and hole
photoexcited particles, is

$H_{S}\equiv
H^{\text{el/hl}}=\sum_{i}^{N}\left\\{E_{i}^{\text{el/hl}}+\varepsilon_{i}(t)-\Phi^{\text{el/hl}}_{i}(t)\right\\}|i\rangle\langle
i|+\sum_{i\neq j}^{N}V_{ij}(t)|i\rangle\langle j|~{},$ (3)

where $N$ denotes the total number of molecular sites in the lattice. The
orthogonal basis set {$|i\rangle\\}$ consists of diabatic states associated
with each of the molecular sites. In addition,
$E^{\textbf{el}}_{i}=E^{CB}_{i}$ and $E^{\textbf{hl}}_{i}=E^{VB}_{i}$ are the
on-site energies of the electron (conduction band) and hole (valence band);
$\varepsilon_{i}$ is the confinement energy associated with site $i$; and
$\Phi^{\text{el/hl}}_{i}$ describes the electron-hole electrostatic
interaction, given in the time-dependent mean-field approximation by

$\Phi^{\text{el/hl}}_{i}(t)=\xi_{bind}~{}\left(\sum_{j}\frac{P^{\text{hl/el}}_{j}}{(1+d_{ij})}\right)~{}.$
(4)

where $\xi_{bind}$ is the on-site electron-hole binding energy (refer to Table
1 for a complete list of model parameters), $P^{\text{el}}_{j}$
($P^{\text{hl}}_{j}$) is the time-dependent electron (hole) population on site
$j$ and $d_{ij}=|\vec{R}_{i}-\vec{R}_{j}|$ is the distance between molecular
sites $i$ and $j$. 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.Andermann
and Rego (2022) The last term in Eq. (S3),

$V_{ij}=V_{0}~{}F_{ij}=V_{0}\left[\frac{2\ell_{i}\ell_{j}}{\ell_{i}^{2}+\ell_{j}^{2}}\exp\Big{(}{\frac{-d^{2}_{ij}}{\ell_{i}^{2}+\ell_{j}^{2}}}\Big{)}\right]~{},$
(5)

describes the tunneling energy between lattice sites, where $V_{0}$ is the
bare electronic coupling and the form factor $F_{ij}$ is the time-dependent
overlap between lattice sites $i$ and $j$, with $\ell$ being the confinement
radius of the molecular site (see SI for details). The time dependence of the
energy parameters comprising Eq (S3), namely $\varepsilon_{i}(t)$ and
$V_{ij}(t)$ gives rise to the intra-molecular and inter-molecular vibrational
effects, respectively, that we associate with the Holstein and Peierls
couplings.Andermann and Rego (2022) The treatment includes dynamic disorder,
which is taken into account by the intra-molecular and inter-molecular
electron-phonon couplings in the system Hamiltonian ($H_{S}$). The dynamic
disorder is also taken into account in the system-bath Hamiltonian ($H_{SB}$)
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,Candiotto  _et al._ (2017) polaron formation,Andermann
and Rego (2022) 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 bimolecular 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.Credgington  _et al._ (2012) 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 proposed to describe recombination in radical
pairs,Haberkorn (1976) but later was used to describe the trapping and exciton
recombination in the Fenna-Matthews-Olson (FMO) protein complexRebentrost  _et
al._ (2009) and OSC models.Candiotto  _et al._ (2017) Herein we include the
vibrational dynamics, in addition to the electron-hole dynamics to extend the
plain Haberkorn formalism. Thus, consider a quantum state
$|\Psi^{\eta}(t)\rangle=\sum_{i}C^{\eta}_{i}(t)|i\rangle$ that comprises the
reduced density matrix $\sigma^{\eta}$, where $\eta=el,hl$ designates the type
of charge carrier. As in the Haberkorn model, the mutual annihilation of
electrons and holes is given by the dynamical equation

$\frac{d}{dt}|\Psi^{\eta}(t)\rangle=-\hat{\Gamma}^{\eta^{\prime}}_{R}|\Psi^{\eta}(t)\rangle$
(6)

where $\eta^{\prime}$ is the carrier type opposite to $\eta$. To include the
vibrational effects into the Haberkorn recombination model, we use the
operator $\hat{\Gamma}^{\eta^{\prime}}_{R}$, defined as

$\hat{\Gamma}^{\eta^{\prime}}_{R}=|\Psi^{\eta^{\prime}}(t)\rangle~{}\kappa\hat{F}~{}\langle\Psi^{\eta^{\prime}}(t)|~{},$
(7)

where the symmetric operator $\kappa\hat{F}=\sum_{i}\kappa
F_{ii}|i\rangle\langle i|+\sum_{i\neq j}\kappa F_{ij}|i\rangle\langle j|$
includes the molecular site vibrations, as defined in Eq. (S4). The parameter
$\kappa$ in Eq. (S10) (see Table 1) is a molecular decay rate constant.
Coropceanu and collaboratorsCoropceanu  _et al._ (2019); Chen  _et al._ (2021)
associate $\kappa F_{ii}$ with the Einstein spontaneous emission rate for the
radiative recombination term whereas the Marcus-Levich-Jortner (MLJ) formula
was used to define the non-radiative decay rate $\kappa F_{ij}$; herein we use
the same parameter $\kappa$ for both processes. A comparison between the
vibronic Haberkorn model and the three-state modelCoropceanu  _et al._ (2019);
Benduhn  _et al._ (2017); Qian  _et al._ (2018); Eisner  _et al._ (2019); Chen
_et al._ (2021) is presented as SI. Substitution of Eq. (S10) into Eq. (6)
yields the following equation for the coefficients of the wavepacket in the
molecular site basis

$\frac{d}{dt}C^{\eta}_{i}=-\kappa\left(\sum_{k,l}\left[C^{\eta^{\prime}}_{k}\right]^{*}F_{kl}~{}C^{\eta}_{l}\right)C^{\eta^{\prime}}_{i}~{}.$
(8)

Rewriting Eq. (8) in terms of the time derivative of the reduced density
matrix, $\sigma^{\eta}_{ij}=C^{\eta}_{i}\left[C^{\eta}_{j}\right]^{*}$, we get

$\frac{d}{dt}\sigma^{\eta}\Big{|}_{rec}=-\kappa\left(\sigma^{\eta^{\prime}}\hat{F}\sigma^{\eta}+\sigma^{\eta}\hat{F}\sigma^{\eta^{\prime}}\right)~{}.$
(9)

The quantum mechanical recombination rate given by Eq. (9), for geminate
recombination, can be associated with the classical law of mass action for the
rate of bimolecular recombination $R=knp$, where $n$ and $p$ are the
concentrations of free electrons and holes, respectively, and $k$ is an
effective recombination constant given, for instance, by the Langevin
model.Burke  _et al._ (2015) It is also worth noting that Eq. (9) gives the
recombination losses during the exciton diffusion and for the CT state at the
D:A interface.

Before adding this recombination rate term to the Redfield Master Equation,
Eq. (2), it is transformed to the adiabatic representation of basis states
$\\{|\varphi_{a}\rangle\\}$ by
$\dot{\sigma}^{\eta}_{rec}\rightarrow(\hat{U}^{\eta})^{-1}~{}\dot{\sigma}^{\eta}_{rec}~{}\hat{U}^{\eta}$,
where $\hat{U}^{\eta}$ is the unitary operator that diagonalizes
$\hat{H}^{\eta}$. The overall charge neutrality of the heterojunction is
conserved with Eq. (9). Based on Eq. (9), the energy lost by a molecular site
via recombination is calculated as
$W^{rec}_{i}=\sigma^{rec}_{ii}(t)E^{gap}_{i}$ where $\sigma^{rec}_{ii}$ is the
population lost by recombination and $E^{gap}_{i}$ is the corresponding CB-VB
gap. The total energy loss due to recombination is calculated as
$\sum_{i}W^{rec}_{i}(t_{f})$, where the sum runs over all the molecular sites
and $t_{f}$ is the final simulation time. Therefore, to obtain the voltage
loss due to geminate recombination ($\Delta V^{rec}$) during the charge
separation process we calculate

$\Delta V^{rec}=\frac{1}{e}\sum_{i}W^{rec}_{i}(t_{f})$ (10)

where $e$ is the electronic charge. Notice that this method accounts only for
the geminate electron-hole recombination. Bimolecular recombination processes
become important at open circuit conditions and high excitation intensities,
which are not described by the present simulations.

The model parameters used in the simulations are presented in Table 1. The set
of coupled differential equations produced by combining Eqs. (2) and (9) are
solved numerically by a fourth order adaptive step size Runge-Kutta method. We
simulate the process of free charge generation, including exciton
photoexcitation, excition diffusion, CT state formation and separation, for a
single el-hl pair in the presence of geminate recombination, during a time
frame of ten picoseconds.

Table 1: Values and description of the parameters used in calculations (refer to SI for details). | value | description
---|---|---
$N$ | 84 | number of sites
$\ell_{\text{o}}$ | 0.5 nm | confinement radius for the empty molecular site
$V^{nn}$ |  $-40$ meV | nearest-neighbor electronic coupling
$d_{ij}$ | 1.3 nm | distance of nearest neighbor sites
$\xi_{\text{bind}}$ | 0.2 eV | electron-hole coupling constant
$\lambda$ | 50 meV | reorganization energy
$\omega_{c}$ | 1334 $\text{cm}^{-1}$ (165 meV) | cutoff frequency
$\Omega$ | 272 $\text{cm}^{-1}$ (34 meV) | classical normal mode frequency
$\gamma^{-1}$ |  $0.1$ ps | phonon-phonon relaxation time
$\kappa$ | 1 ps-1 | molecular site decay rate constant

We consider that a Frenkel exciton is photoexcited in the center of the donor
region, as shown in Figure 1. The coarse-grained model consists of a 2D-square
lattice with nearest neighbor (nn) distance $a$ = 1.3 nm and equal D and A
domain lengths of approximately 7.5 nm. The effective electronic coupling
between nearest neighbors is set to be $V^{nn}=-40$ meV, as given by Eq. (S4),
which is in agreement with other molecular crystal models.Lee  _et al._
(2015); Coropceanu  _et al._ (2007); Mozafari and Stafström (2013);
Kocherzhenko  _et al._ (2015); De Sio and Lienau (2017); Troisi and Orlandi
(2006); Candiotto  _et al._ (2017) The low frequency vibration mode $\Omega$
is treated classically. It modulates the radii $\ell_{i}(t)$ of the molecular
sites and, thereby, the electronic coupling $V_{ij}$, as given by (S4). The
classical vibrations are self-consistently coupled to the el-hl quantum states
by semiclassical Ehrenfest forces (Supporting Information). The high energy
quantum vibrations are incorporated into the Redfield relaxation tensor, up to
the cutoff frequency $\omega_{c}$ = 1334 cm-1 that corresponds to the
localized C$=$C stretching mode. For the system-bath coupling we use a
reorganization energy parameter of $\lambda$ = 50 meV for either the electron
and hole density matrices. The magnitude of $\lambda$ is commonly found to be
in the range of 100-300 meV for organic blends.Vandewal  _et al._ (2017)
However, it has been shown that the values of $\lambda$ obtained from
experiments tend to be larger than the bare values obtained by static
calculations. Vanderwal et al.Vandewal  _et al._ (2017) derived the empirical
relationship $\lambda_{exp}=\lambda_{calc}+\Delta$, where $\Delta\approx$ 95
meV is a correction due to dynamic disorder (disregarding the static
disorder). Additionally, McGehee et al.Burke  _et al._ (2015) derived another
relation in which $\lambda_{exp}=\lambda_{calc}+\sigma^{2}/(2kT)$, where
$\sigma$ is the static energetic disorder parameter and
$\sigma^{2}/(2kT)\approx$ 108 meV at room temperature, assuming $\sigma$ = 75
meV. $\lambda_{exp}$ can be associated with the CT emission spectral
broadening that is caused by disorder. In our model we ascribe the effects of
disorder fully to the molecular vibrations. Therefore, the effective
reorganization energy affecting the el-hl dynamics in our simulations can be
estimated as $\lambda_{eff}\approx 50meV+100meV=150meV$. This is in good
agreement with the optimal charge generation conditions obtained in our
simulations, which is given $E_{DA}\approx\lambda_{eff}$ that occurs between
100 and 200 meV, as shown ahead. As for the recombination rate constant,
$\kappa$, its value is defined based on the characteristics of the coarse-
grained model. Experiments have determined that the exciton diffusion length
in organic semiconductors and polymers is within the range of 5-20
nm,Mikhnenko  _et al._ (2012); Firdaus  _et al._ (2020) and the diffusion
length is limited by the exciton lifetime. In our coarse-grained model the
length of the donor (D) domain is about 7.5 nm, which should correspond
approximately to an exciton lifetime. To follow this constraint and to evince
the effects of the recombination within the span of the simulation time (10
ps), we use $\kappa$ = 1 ps-1. For the sake of comparison, Figure S4 in the
Supporting Information presents simulation results with and without
recombination. Finally, in accordance with the model (Figure 1), we define the
energetic driving force of the D:A interface as $E_{DA}\equiv
E^{{CB}}(D)-E^{{VB}}(A)$.

Figure 2: Collected electronic population ($N_{e}$) subjected to recombination
losses, for various bath temperatures (TB) and donor-acceptor energy offsets:
a) $E_{DA}$ = 0 eV, b) $E_{DA}$ = 0.1 eV and c) $E_{DA}$ = 0.3 eV. The
illustration at the top-right corner is a reference to the Marcus Theory,
depicting electron transfer in the normal regime (a), the activationless
regime (b), and the inverted regime (c) as the energetic driving force
$E_{DA}$ increases.

The evaluation of the activation energy (Ea) required for el-hl pair
separation has been open to debate,Gao  _et al._ (2015); Dong  _et al._
(2019); Perdigón-Toro  _et al._ (2020); Rosenthal  _et al._ (2019); Coffey
_et al._ (2012); Ward  _et al._ (2015); Unger  _et al._ (2017) given that the
obtained values may vary depending on the technique and the time scale of the
measurements. From the theoretical viewpoint, considering the kinetic models,
Ea is an effective parameter that is influenced by Coulombic dissociation and
recombination rates.Gao  _et al._ (2015); Dong  _et al._ (2019); Perdigón-Toro
_et al._ (2020) Figure 2 shows the collected electronic population after
recombination, for various bath temperatures (TB) and energetic driving forces
($E_{DA}$), that is $N_{e}(E_{DA},T_{B})$, without electric field bias. The
simulations show, as a whole, that the charge generation rate increases with
temperature for all D:A offsets. In addition, the effect of TB on $N_{e}$
reveals three distinct electron separation regimes, as depicted in the inset
of Figure 2: a) the normal electron separation for negligible $E_{DA}$ = 0 eV;
b) the activationless regime for $E_{DA}\approx$ 0.1 eV; and c) the inverted
Marcus regime for $E_{DA}$ = 0.3 eV. The highest internal quantum efficiency
(IQE) is obtained for the activationless regime, $E_{DA}\approx$ 0.1 eV,
whereby the charge generation process is almost independent of the
temperature. The maximum IQE is obtained for $E_{DA}\approx\lambda_{eff}$,
where $\lambda_{eff}$ corresponds to the effective reorganization energy at
the heterojunction, corroborating experimental studies that show the existence
of such an optimal driving force.Coffey  _et al._ (2012); Ward  _et al._
(2015); Rand  _et al._ (2007) However, for vanishing energetic driving force,
$E_{DA}\approx$ 0, and no electric fields (to be discussed next), the charge
generation is very sensitive to the temperature. Lastly, if the energy offset
becomes excessive, i.e. $E_{DA}\gg\lambda$, the charge generation process
enters the inverted region described by the Marcus-Jortner-Levich theory
Jortner (1976); Barbara  _et al._ (1996); Chaudhuri  _et al._ (2017), where
charge transfer is assisted by quantum vibronic effects. A detailed account of
the impact produced by vibronic effects (such as polaron formation) and
internal electric fields on the free charge carrier generation is was
presented by Andermann and Rego elsewhere.Andermann and Rego (2022) Figures S2
and S3 of the SI evince the importance of electron-hole interaction and
vibrational dynamics to recombination, respectively.

Figure 3: a) Geminate recombination induced voltage-loss, $\Delta
V^{rec}(E_{DA},T_{B},\mathcal{E})$, as calculated with Eq. (10), without bias
field ($\mathcal{E}$=0), for TB = 100, 200, 300 and 400 K. b) Same as in a)
with a reverse bias field $\mathcal{E}$ = -10 V/$\mu$m. The dashed lines
indicate the approximate position of the optimal charge generation regime,
which does not change appreciably under the influence of temperature or
reverse bias. c) Temperature and bias field dependence of geminate
recombination voltage-loss without D:A driving force, $\Delta
V^{rec}(E_{DA}=0,T_{B},\mathcal{E})$.

In addition to the energetic driving force for charge separation, the voltage
losses in the OSCs also include the contributions from recombination
processes, both radiative and non-radiative. Figure 3 shows the effect of
driving force ($E_{DA}$), temperature ($T_{B}$) and electric field bias
($\mathcal{E}$) on the voltage loss due to geminate recombination, $\Delta
V_{rec}(E_{DA},T_{B},\mathcal{E})$, as given by Eq. (10). The simulations show
that $\Delta V_{rec}$ is practically insensitive to temperature and reverse
bias field if the energetics of the heterojunction is set to the
activationless regime of charge separation, in this case $E_{DA}$ = 0.1 eV.
Out of this point the losses due to geminate recombination at the D:A
interface increase as the temperature decreases, particularly so for vanishing
D:A offset in the absence of reverse bias (Figure 3-a). The influence of
temperature on the geminate recombination losses during charge carrier-
generation has not been extensively investigated because it is difficult to
measure geminate recombination at low illumination intensities, since
parasitic leakage currents can affect the results.Proctor and Nguyen (2015) To
study this issue, Gao et al.Gao  _et al._ (2015) used low excitation
intensities to observe a decreasing VOC at low temperatures, which was
ascribed to reduced carrier generation. Here, it is important to distinguish
the nonequilibrium carrier generation process that is observed at low
illumination intensities from the quasi-equilibrium conditions attained at
open-circuit with moderate-to-high illumination. In the first case, the
efficiency of charge generation is determined by the competition between
photogeneration of free charge carriers and geminate recombination, as
described in Figure 3. In the second case, bimolecular recombination kinetics
is the main cause of energy loss, so that at the steady-state
$qV_{OC}=E_{g}-k_{B}T\ln(N_{c}N_{v}/np)$, where $n$ and $p$ are the electron
and hole stationary free carrier densities.Burke  _et al._ (2015); Gao  _et
al._ (2015); Rosenthal  _et al._ (2019) Thus, at high excitation densities $n$
and $p$ are nearly temperature independent and $V_{OC}$ decreases linearly
with temperature.Gao  _et al._ (2015); Rauh  _et al._ (2011) At low
excitations, however, the density of free carriers become temperature
dependent, $n(T)$ and $p(T)$,Gao  _et al._ (2015); Rauh  _et al._ (2011); Rand
_et al._ (2007); Chan  _et al._ (2021) thereby changing the behavior of
$V_{OC}(T)$. Chan et al.Chan  _et al._ (2021) reported that as the blends
P3TEA:SF-PDI2 and PM6:Y6 are cooled, the reduced charge separation rate leads
to increasing charge recombination, either directly at the D:A interface or
via the LE state. Thus, in this case $V_{OC}$ can exhibit a sublinear
dependence on temperature,Vandewal  _et al._ (2010); Rand  _et al._ (2007);
Rauh  _et al._ (2011) or even increase with temperatureGao  _et al._ (2015) if
the carrier-generation process dominates $V_{OC}$. Notice, though, that the
application of a reverse bias at the heterojunction should promote the free
carrier generation,Proctor  _et al._ (2014) thereby decreasing the losses by
geminate recombination and improving $V_{OC}$, as shown in Figure 3-b. The
effect of a reverse bias is particularly strong in the absence of driving
force ($E_{DA}$=0), as shown in Figure 3-c, where it is shown that a moderate
bias reduces recombination losses substantially.

The various recombination pathways contribute to reduce $J_{SC}$, $V_{OC}$ and
the fill factor (FF) of the OSCs, thereby limiting its power conversion
efficiency. In regard to free carrier generation, this work indicates that the
best device performances should be realized at the optimal offset
$E_{DA}\approx\lambda_{eff}$, which produces low overall energy losses for
free carrier generation and superior $J_{SC}$ and FF.

Acknowledgements. AMA is grateful for the financial support from FAPESC
(Fundação de Amparo à Pesquisa do Estado de Santa Catarina). LGCR acknowledges
support by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior Brasil
(CAPES) - Finance Code 001, by the Brazilian National Counsel of Technological
and Scientific Development (CNPq) and the National Institute for Organic
Electronics (INEO). The authors acknowledge support by allocation of
supercomputer time from Laboratory for Scientific Computing (LNCC/MCTI,
Brazil).

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Supporting Information for
Quantum Mechanical Assessment of Optimal Photovoltaic Conditions in Organic
Solar Cells

##  I Semiclassical Electron-Phonon Hamiltonian

The paraboloids depicted in Figure S1-a) describe the local confinement
potential felt by the electron and the hole at a given molecular site. We
assume that the parabolic confinement potential undergoes a vibrational
reorganization whenever charge is transferred in or out of the molecular site.
This mechanism gives rise to coherent vibrational effects that we describe
within the framework of the Ehrenfest method. The density profile of the
electron and hole in the molecular site is given by the gaussian

$g(\vec{r}-\vec{R})=\sqrt{\frac{2}{\pi\ell^{2}}}\exp\left[-\Big{(}\frac{\vec{r}-\vec{R}}{\ell}\Big{)}^{2}\right],$
(S1)

with $\vec{R}$ designating the position of the molecular site, and $\ell$ is
the confinement radius of the molecular site, given by

$\ell=\sqrt{2\hbar^{2}/(m_{e}\varepsilon)}$ (S2)

for a gaussian wavepacket of on-site energy $\varepsilon$.

To write the system Hamiltonian, $H_{S}$, we assume an orthogonal basis set
comprised of diabatic electronic states $\\{|i\rangle\\}$ associated with the
molecular sites, as described in the paper,

$H_{S}\equiv
H^{\text{el/hl}}=\sum_{i}^{N}\left\\{E_{i}^{\text{el/hl}}+\varepsilon_{i}(t)-\Phi^{\text{el/hl}}_{i}(t)\right\\}|i\rangle\langle
i|+\sum_{i\neq j}^{N}V_{ij}(t)|i\rangle\langle j|~{}.$ (S3)

The coupling $V_{ij}$ between lattice sites is given by

$V_{ij}=V_{0}~{}F(\ell_{i},\ell_{j},d_{ij})=V_{0}\left[\frac{2\ell_{i}\ell_{j}}{\ell_{i}^{2}+\ell_{j}^{2}}\exp\Big{(}{\frac{-d^{2}_{ij}}{\ell_{i}^{2}+\ell_{j}^{2}}}\Big{)}\right],$
(S4)

where $V_{0}$ is the bare electronic coupling and the form factor $F$ results
from the overlap between gaussian wavepackets located at lattice sites $i$ and
$j$

$F(\ell_{i},\ell_{j},d_{ij})=\int
g_{i}(\vec{r}-\vec{R}_{i})~{}g_{j}(\vec{r}-\vec{R}_{j})dxdy.$ (S5)

The time dependence of $\varepsilon_{i}(t)$ and $V_{ij}(t)$, which comprise
Eq. (S3), gives rise to intra-molecular and inter-molecular vibrational
couplings that we associate with the Holstein and Peierls couplings,
respectively.

Figure S1: a) Schematics of the electronic relaxation. The parabola describes
the confinement potential felt by the electron (or hole) with its gaussian
wavefunction profile. On the left-hand side we show an empty molecular site,
with confinement energy $\varepsilon_{\text{o}}$ and the corresponding
confinement length $\ell_{\text{o}}$. On the right-hand side we show an
occupied molecular site, with on-site energy $\varepsilon$ and $\ell$. The
electronic relaxation energy is
$\Delta\varepsilon=\varepsilon_{\text{o}}-\varepsilon$. b) The semiclassical
electron-phonon system is equivalent to a spring-damper hybrid quantum-
classical system.

We describe the coherent coupling between the electron-hole states and the
fundamental intra-molecular vibrational mode via the Ehrenfest method. In this
model the semiclassical electron-phonon system is equivalent to a spring-
damper hybrid quantum-classical system, as shown in Figure S1-b) The classical
Hamiltonian $h_{i}$ describing the vibrational mode of site $i$ is given by

$h_{i}=\frac{p_{i}^{2}}{2\mu}+\frac{\mu\Omega^{2}}{2}(\ell_{i}-\ell_{\text{o}})^{2}+W^{eh}+Q^{thermo},$
(S6)

where $\ell_{i}$ is the vibrational coordinate associated with the molecular
site $i$ and $p_{i}$ is the conjugate momentum. In addition, $\ell_{\text{o}}$
is the confinement length for an empty (that is, neutral) site, $\mu$ is the
effective mass of the vibrational mode and $\Omega$ is the frequency of the
relevant normal mode. Refer to Table 1 in the paper for a complete list of
model parameters. The energy $W^{eh}$ is the work exchanged with the
electronic degrees of freedom via the Ehrenfest force $F^{eh}$ and
$Q^{thermo}$ is the heat exchanged with the classical thermostat. A similar
hamiltonian has been considered by Egorova et al.Egorova  _et al._ (2001,
2003) for quantum intra-molecular degrees of freedom. The Ehrenfest force that
acts on the vibrational coordinate of a given molecular site $k$ is

$F^{eh}_{k}=F_{k}^{\text{el}}+F_{k}^{\text{hl}}=-\frac{\partial
U^{\text{el}}}{\partial\ell_{k}}-\frac{\partial
U^{\text{hl}}}{\partial\ell_{k}},$ (S7)

with $U^{\text{el}}=Tr[\sigma^{\text{el}}H^{el}]$ and
$U^{\text{hl}}=Tr[\sigma^{\text{hl}}H^{hl}]$. Solving equation (S7) we
obtainAndermann and Rego (2022)

$F^{eh}_{k}=-(\sigma^{\text{el}}_{kk}+\sigma^{\text{hl}}_{kk})\frac{d\varepsilon_{k}}{d\ell_{k}}-\sum_{i\neq
j}(\sigma^{\text{el}}_{ij}+\sigma^{\text{hl}}_{ij})\frac{\partial
V_{ij}}{\partial\ell_{k}}.$ (S8)

The first term on the right-hand side (RHS) of Equation (S8) is responsible
for the relaxation of the molecular site due to its charge occupation; this is
the classical work exchanged between electronic and vibrational degrees of
freedom. The second term gives rise to coherent vibronic effects, for it is
proportional to the off-diagonal elements of $\sigma$ and couples the
electronic and nuclear coordinates via $\partial
V_{ij}/\partial\ell_{k}$.Sachtleben  _et al._ (2017)

The classical equations of motion derived from Eq. (S6) are solved with the
velocity Verlet algorithm, coupled to a classical bath, described by the
Berendsen thermostat,Berendsen  _et al._ (1984) with relaxation constant
$\gamma^{-1}=0.1$ ps. The perturbing forces change the equilibrium
configuration of the parabolic confining potential, which reacts with a
restoring force $F^{\text{res}}=-\mu\Omega^{2}(\ell-\ell_{\text{o}})$.

##  II Vibronic Recombination Model

In order to compare the vibronic recombination model with the three-state
recombination model,Coropceanu  _et al._ (2019); Benduhn  _et al._ (2017);
Qian  _et al._ (2018); Eisner  _et al._ (2019); Chen  _et al._ (2021) let us
consider the dynamical equation for recombination, Eq. (6), in the paper

$\frac{d}{dt}|\Psi^{\eta}(t)\rangle=-\hat{\Gamma}^{\eta^{\prime}}_{R}|\Psi^{\eta}(t)\rangle~{},$
(S9)

where

$\hat{\Gamma}^{\eta^{\prime}}_{R}=|\Psi^{\eta^{\prime}}(t)\rangle~{}\kappa\hat{F}~{}\langle\Psi^{\eta^{\prime}}(t)|~{},$
(S10)

so that we obtain the following set of coupled equations

$\displaystyle\frac{d}{dt}|\Psi^{el}(t)\rangle$ $\displaystyle=$
$\displaystyle-\left(\langle\Psi^{hl}(t)|\kappa\hat{F}|\Psi^{el}(t)\rangle\right)|\Psi^{hl}(t)\rangle$
(S11) $\displaystyle\frac{d}{dt}|\Psi^{hl}(t)\rangle$ $\displaystyle=$
$\displaystyle-\left(\langle\Psi^{el}(t)|\kappa\hat{F}|\Psi^{hl}(t)\rangle\right)|\Psi^{el}(t)\rangle~{}.$
(S12)

For the sake of simplifying the argument we are assuming here that the
electron and hole states are described by wavefunctions. However, in our
treatment the electron and hole states are described by reduced density
matrices because of the system-bath interaction. So, let us consider the
matrix element $\langle\Psi^{hl}(t)|\kappa\hat{F}|\Psi^{el}(t)\rangle$ that
gives rise to the recombination of electrons due to the operator

$\kappa\hat{F}=\kappa\sum_{i}F_{ii}|i\rangle\langle i|+\kappa\sum_{i\neq
j}F_{ij}|i\rangle\langle j|~{}.$ (S13)

Substituting Eq. (S13) into the previous matrix element and writing the
electron and hole wavepackets in the molecular site basis,
$|\Psi^{\eta}(t)\rangle=\sum_{i}C^{\eta}_{i}(t)|i\rangle$, we get

$\displaystyle\langle\Psi^{hl}(t)|\kappa\hat{F}|\Psi^{el}(t)\rangle$
$\displaystyle=$
$\displaystyle\kappa\sum_{i}\left[C_{i}^{hl}(t)\right]^{*}F_{ii}~{}C_{i}^{el}(t)+\kappa\sum_{i\neq
j}\left[C_{i}^{hl}(t)\right]^{*}F_{ij}~{}C_{j}^{el}(t)$ (S14) $\displaystyle=$
$\displaystyle\kappa\sum_{i}\left[C_{i}^{hl}(t)\right]^{*}C_{i}^{el}(t)+\kappa\sum_{i}\left(\sum_{j}^{(j\neq
i)}\left[C_{i}^{hl}(t)\right]^{*}F_{ij}~{}C_{j}^{el}(t)\right),$ (S15)

where $F_{ii}=1$ as given by Eq. (5) of the paper. This expression can be
compared with the three-state model. The first term describes the radiative
recombination taking place on molecular site $i$, and $\kappa$ can be
associated with the radiative decay constant, since it does not depend on
temperature or vibrational motion. The second term couples distinct molecular
sites. For instance, a hole in the donor site $i$ is coupled to electrons in
both donor and acceptor sites via $F_{ij}$, which is the vibronic coupling
term that depends on the temperature. This term is a generalization of the LE-
CT hybridization considered in the three-state model. However, the choice of a
single recombination parameter $\kappa$, associated with the el-hl lifetime,
for both contributions is clearly an oversimplification. Previous studies have
treated $\kappa$ differently for radiative and non-radiate recombination
processes. For instance, Coropceanu and collaboratorsCoropceanu  _et al._
(2019); Chen  _et al._ (2021) associate $\kappa_{r}$ with the Einstein
spontaneous emission rate for the radiative recombination term whereas the
Marcus-Levich-Jortner (MLJ) formula was used to define the non-radiative decay
rate $\kappa_{nr}$. The interplay of charge transfer and non-radiative decay
was investigated by atomistic non-adiabatic molecular dynamics simulations in
dye-sensitized semiconductor interfaces,Rego and Bortolini (2019) where it was
revealed a rate competition between the two mechanisms.

Finally, it is worth noting that Eq. (S15) describes the recombination losses
during the exciton diffusion and during the life-time of the CT state at the
D:A interface.

##  III Supporting Simulation Results

###  III.1 Electron-Hole Interaction and Recombination

Figure S2: Effect of electron-hole interaction
$\Phi^{\text{el/hl}}_{i}(t)|i\rangle\langle i|$, from Equation (4) of the
paper, on the geminate recombination rate during charge separation. The bath
temperature is TB = 300 K. The energetic driving force $E_{DA}$ is: 0 (a), 0.1
eV (b), 0.2 eV (c), 0.3 eV (d), and 0.4 eV (e).

###  III.2 Vibrational Dynamics and Recombination

Figure S3: Effect of vibrational dynamics (also considered as dynamical
disorder) on the geminate recombination rate during charge separation. The
bath temperature is TB = 300 K. The energetic driving force $E_{DA}$ is: 0
(a), 0.1 eV (b), 0.2 eV (c), 0.3 eV (d), and 0.4 eV (e).

###  III.3 Charge Generation with and without Recombination

Figure S4: Calculations of time-dependent charge collection, a) and b), and
energy collection, c) and d). Left-hand-side (LHS) panel: no recombination
effects. Right-hand-side (RHS) panel: with recombination effects. Calculations
performed at TB = 300 K for various LUMO level offsets, EDA, as indicated by
the legend.

By comparing the LHS (no recombination) with the RHS (with recombination)
panels of Figure S4, we observe that both the charge separation and energy
conversion efficiencies decrease as a consequence of recombination for all
values of the driving force $E_{DA}$. For each case, the photovoltaic energy
conversion decreased by: 19% ($\Delta E^{rec}\approx$ 190 meV) for $E_{DA}$ =
0 (black), 12% ($\Delta E^{rec}\approx$ 120 meV) for $E_{DA}$ = 0.1 eV (red),
16% ($\Delta E^{rec}\approx$ 140 meV) for $E_{DA}$ = 0.2 eV (green), 21%
($\Delta E^{rec}\approx$ 160 meV) for $E_{DA}$ = 0.3 eV (blue) and 24%
($\Delta E^{rec}\approx$ 150 meV) for $E_{DA}$ = 0.4 eV (magenta). The results
show that for the activationless regime ($E_{DA}$ = 100 meV $\approx\lambda$)
the losses due to recombination are minimized, since charge separation is more
efficient in this regime.Andermann and Rego (2022) As the driving force
increases, for $E_{DA}$ = 0.2, 0.3 and 0.4 eV, and the system enters the
Marcus inverted regime for charge separation, the losses due to recombination
increase. We ascribe this effect to reduced charge separation, as the driving
force exceeds to the reorganization energy at the interface. Here the
combination of big energetic driving forces and increased recombination
renders the overall energy loss for $E_{DA}$ = 0.4 eV the highest of all
cases, followed by $E_{DA}$ = 0.3 and 0.2 eV, respectively. However, the
biggest energy loss due exclusively to recombination (in absolute value)
occurs for $E_{DA}$ = 0, because of the much slower electron-hole pair
separation rate without the energetic driving force. Internal electric fields
should improve considerably the power conversion efficiency at vanish driving
forces. The simulation results indicate that the condition
$E_{DA}\approx\lambda$ has the effect of minimizing the effects of the
geminate recombination.