|  Multidimensional coherent spectroscopy of excitons in $\pi$-conjugated
polymer
---|---
|  Elizabeth Gutiérrez-Meza,a† Alejandro Vega-Flick,a† Eric R. Bittner,b,c and
|  Carlos Silva-Acuñaa,d,e∗
|  The photophysics of $\pi$-conjugated polymers has been of considerable
interest over the last three decades because of their organic semiconductor
properties. Primary photoexcitations, Frenkel excitons, can be probed
optically by means of numerous linear spectroscopies, providing a wealth of
information on the the strength of excitonic coupling, the exciton coherence
length, and on the nature of the disordered energy landscape. Nonetheless,
there are intrinsic limitations in the information that can be obtained with
linear spectroscopy compared to non-linear coherent techniques. Examples of
this are the separation of homogeneous and inhomogeneous broadening
contributions to the total exciton spectral lineshape, the detailed spectral
structure of exciton-vibrational coupling, and correlations between optical
excitations, including exciton-charge and exciton-exciton correlations. In the
present work, we discuss the role of two-dimensional coherent excitation
spectroscopy and review its applications towards unravelling the basic
photophysics of $\pi$-conjugated polymers as well as polymer:fullerene blends,
and we argue that these techniques are now valuable mainstream materials
optical probes.

##

††footnotetext: a School of Chemistry and Biochemistry, Georgia Institute of
Technology, 901 Atlantic Drive NW, Atlanta, Georgia 30332, United States.
E-mail<EMAIL_ADDRESS>b Department of Chemistry,
University of Houston, Houston, Texas 77204, United States††footnotetext: c
Center for Non-Linear Studies, Los Alamos National Lab, Los Alamos, New Mexico
87545, United States††footnotetext: d School of Physics, Georgia Institute of
Technology, 837 State Street NW, Atlanta, Georgia 30332, United States
††footnotetext: e School of Materials Science and Engineering, Georgia
Institute of Technology, North Avenue, Atlanta, GA 30332, United States
††footnotetext: † A.V.F. and E.G.M. are first co-authors of this manuscript.

### 10th Anniversary Statement

The influence of the Journal of Materials Chemistry C in the development of
the basic science of conjugated polymers is substantial, and C.S.A. has
published some of his key contributions to the photophysics of these materials
in the journal. It is his firm conviction that advanced optical probes are now
at the point of maturity that they are to be considered as valuable techniques
within the materials characterization toolbox, much like “standard” techniques
such as linear absorption and emission techniques. The readership of the
journal is broad, spanning all aspects of research on conjugated polymers,
ranging from synthesis, processing, physical characterization, and engineering
of their technologies, and this contribution to the 10th Anniversary Issue is
aimed at the entire community. The authors therefore hope to stimulate further
use of nonlinear coherent spectroscopies in the research of excitonic
properties of conjugated polymer materials and their devices by connecting the
ultrafast spectroscopy and materials science communities.

##  1 Introduction

Fig. 1: Schematic representation of a two-dimensional coherent spectroscopy
experimental arrangement. The geometry of the excitation pulse-train beam
pattern (red) and the resonant four-wave mixing signal ($S_{\mathrm{FWM}}$,
yellow-orange), detected by interference with a local oscillator (LO), is
shown in (a). This scheme uses a BOXCARS beam geometry, in which three pulse
trains (A, B, C) propagating along the corners of a square are focused onto
the sample with a common lens, defining incident wavevector $\vec{k}_{A}$,
$\vec{k}_{B}$, and $\vec{k}_{C}$. The LO beam, on the fourth apex of the
incident beam geometry, co-propagates with $S_{\mathrm{FWM}}$ with wavevector
imposed by the chosen phase matching conditions. The spectral interferogram of
$S_{\mathrm{FWM}}$ and the LO beam is recorded at every time step. (b) By
controlling the order of the pulse sequence with this beam geometry, we
measure two distinct $S_{\mathrm{FWM}}$ responses: the _rephasing_ signal with
wavevector $\vec{k}_{S_{\mathrm{FWM}}}=\vec{k}_{B}-\vec{k}_{A}+\vec{k}_{C}$
and _nonrephasing_ signal with wavevector
$\vec{k}_{S_{\mathrm{FWM}}}=\vec{k}_{A}-\vec{k}_{B}+\vec{k}_{C}$. The total
nonlinear response is then given by the sum of the rephasing and non-rephasing
spectra. (c) Pulse sequence for two-quantum (2Q) coherence measurement. Figure
taken with permission from Ref. 54. Fig. 2: Example of a 2D spectra of a
system with two excited levels $|a\rangle$, $|b\rangle$ and a ground state
$|g\rangle$. (a) The spectrum is symmetric along the diagonal since the states
share a common ground, (b) the states are decoupled, (c) coupling of the b and
a states leads to relaxation of b towards a, which translates to an asymmetric
amplitude of the off-diagonal peaks. Insets show the energy diagrams of the
excited state couplings. Modulus of a rephasing 2D spectrum for an
inhomogeneous system. (d) Each resonant frequency is decoupled from the others
and is represented by a red circle. The homogeneous width of each emitter,
represented by the size of the circle, is measurable along the anti-diagonal.
(e) The total signal forms an elongated peak along the diagonal (inhomogeneous
broadening). (f) Pulse sequence used to perform a MDCS measurement involving
3rd order polarization. Fig. 3: 2D amplitude lineshapes for rephasing signals.
(a-c) 2D frequency plots for a fixed value of homogeneous broadening with
increasing inhomogeneous broadening. The diagonal (red dashes) and cross-
diagonal (blue dots) lines are shown. The vertical scale for $\omega_{\tau}$
is negative because of phase-matching requirements, and increases (gets more
negative) going down. (d-f) Slices of the corresponding 2D frequency plots
along the diagonal (red) and cross-diagonal (blue) directions. The inset
compares cross-diagonal slices in the limits of strong homogeneous (dashes)
and inhomogeneous (dots) broadening. Figure extracted with permission from
Ref. 82.

The optical properties of $\pi$-conjugated polymers are governed by the nature
of their primary photoexcitations 1, which are Frenkel excitons 2. These are
strongly bound quasiparticles with binding energies $\gg 10\,k_{B}T$ at
ambient temperature 3. Excitonic properties in organic semiconductors in
general provide a window into their materials physics, and have always been
investigated as a key probe of their electronic landscape 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32, 33, 34. The solid-state absorption spectra of conjugated
polymers, while encoding signatures of photophysical aggregates of molecular
chromophores along the polymer backbone 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, are often dominated by static disorder, and the
large inhomogeneous linewidth obscures details of the rich spectral structure.
The seminal work of Spano on excitons in conjugated polymers, of which Refs.
19–34 are a subset, involved implementation of Holstein Hamiltonians that
treat excitonic coupling, vibronic coupling, energetic disorder, and charge-
transfer components when describing push-pull materials, incorporating all of
these contributions on equal footing. This has presented a formal way to
interpret absorption and photoluminescence lineshapes to quantify excitonic
parameters such as free-exciton bandwidth and the nature of their disordered
energy landscape, which has been tremendously impactful in semiconductor-
polymer science over the past decade. The main observables that enter the
linear lineshape interpretation are relative 0–0 and 0–1 peak intensities,
which quantify excitonic coupling in the absorption spectrum 19, 21, 22, 23,
with additional information on the correlated nature of the energetic disorder
via photoluminescence 24, 26. To go beyond this effective exciton model, more
detailed analysis of the spectral structure is desirable, and techniques such
as resonance Raman spectroscopy, for example, are valuable 35, 36, 37, 38.
Many details of the many-body excitonic structure that dominates solid-state
behaviour of conjugated polymers are not readily extracted from the broad,
convoluted linear optical lineshapes, and it is advantageous to measure
nonlinear lineshapes, which resolve rich structure within inhomogeneously
broadened spectra beyond linear lineshapes 39, 40, 41, 42, 43, 44.
Furthermore, many-body excitonic details central to the photophysical
aggregate behaviour are not readily identifiable from linear lineshapes;
although these states are localized within a chromophore, at sufficiently high
densities, exciton-exciton interactions start to dominate the optical
properties of organic solids 45. For example, the spontaneous formation of
strongly coupled light-matter quantum condensates in organic materials depends
fundamentally on the details of exciton-exciton interactions 46; how are two-
quantum interactions manifested in spectral lineshapes of Frenkel excitons? We
have found in early work that both charges and triplet excitons are produced
on ultrafast timescales due to multi-quantum interactions in conjugated
polymers 47, 48, 49, but no detail on these can be derived by linear
excitation lineshapes. Furthermore, we have found that prompt charge
separation occurs via what we have interpreted as charge-transfer states that
appear to be intrinsic in these systems 50, 51, 52, 53, and fundamental
electronic couplings that give rise to these is of fundamental relevance.

Our objective for this article is to discuss two-dimensional coherent
spectroscopy as a non-linear optical probe that contributes to the
understanding of excitons in $\pi$-conjugated polymer materials, to discuss
how this approach is now to be considered a mainstream materials
characterization tool in this field, and to review what we consider to be key
contributions of this class of techniques to the semiconductor polymer
materials science. We first discuss the information that can be discerned from
two-dimensional coherent optical lineshapes, and then we review contributions
to the understanding of exciton and biexciton structure in neat polymer films,
as well as charge separation in heterostructures with electron acceptors
designed for photovoltaic applications. We highlight how some photocurrent-
detected implementations are particularly valuable in optoelectronic devices.

##  2 Two-Dimensional Coherent Spectroscopy

In optical spectroscopy, the interaction between light and matter produces an
induced mesoscopic polarization $\vec{P}(t)$ that depends on time according to
the electric field component of the driving electromagnetic waves. According
to the laws of electrodynamics, this time-varying polarization serves itself
as a source of electromagnetic waves, resulting in emission of radiation
(measurable signal) with a well-defined wavevector ($\vec{k}_{S}$) that
respects momentum conservation of the incident excitation source, meaning it
has a distinct spacial orientation (see Fig. 1 and Ref. 54 for an example of
this experimental implementation). If the excitation source is coherent, due
to laser excitation, this signal has, in principle, a well defined phase
relationship with respect to the excitation source, and the dissipation of
this coherence limits the optical linewidth. Quantum mechanics establish that
the induced polarization is the expectation value of the quantum dipole
operator 55, in which the electric fields drive transitions between quantum
states of the system. It is common to expand the induced polarization in a
power series of the incident electric field, taking into account successive
light-matter interactions:

$\vec{P}(t)=\epsilon_{0}\biggl{(}\chi^{(1)}\vec{E}_{A}(t)+\chi^{(2)}\vec{E}_{A}(t)\cdot\vec{E}_{B}(t)\\\
+\chi^{(3)}\vec{E}_{A}(t)\cdot\vec{E}_{B}(t)\cdot\vec{E}_{C}(t)+\ldots\biggr{)},$
(1)

with the nonlinear susceptibilities $\chi^{(i)}$ (which are tensors),
$\epsilon_{0}$ the vacuum permittivity, and $\vec{E}_{j}(t)$ the electric
field of the wavepackets of the individual laser pulses that are used to
induce the successive light-matter interactions. In the first order of this
expansion, the polarization is proportional to the incident electric field,
which corresponds to the linear response of the sample. All higher orders are
considered to be the non-linear response of the material. The most common way
to explore the non-linear response is to have multiple electric field
interactions with the material. This involves using light pulses with strict
time control (inter pulse delay), well-defined phase, and frequency control
(pulse spectrum), with a typical experimental geometry as depicted in Fig. 1.
This is one implementation of multi-dimensional coherent spectroscopy (MDCS),
but we note that there are a diversity of MDCS; we direct the reader to Ref.
56 for an excellent overview of the diversity of techniques. The $n$ number of
light pulses employed corresponds to the $n^{\mathrm{th}}$ non-linear order,
e.g. interaction of three light pulses corresponds to the $3^{\mathrm{rd}}$
order non-linear response of the material, five light-matter interactions
correspond to the $5^{\mathrm{th}}$ order non-linear response, and so on. The
work on conjugated polymers has been limited to probing the third order
response, and we thus limit our discussion on this order of nonlinear
spectroscopy.

In MDCS, the non-linear signal contains information regarding ultrafast
dynamics of electronic and vibrational phenomena of a system 57, 58, 59, 60,
61. Some advantages over linear spectroscopy techniques are the ability to
resolve homogeneous and inhomogeneous broadening of the spectral features
independently from each other, the separation of interaction pathways as well
as the identification of correlations (coupling) between exited states of the
system 55, 62, 56, 63, 64, 39, 65. Additionally, the signal detection is not
limited only to coherent light 66, but in can also be implemented in order to
detect incoherent emission 67, 68, 69, photocurrent 68, 70, 71, 72, 57, 58,
73, steady-state photoinduced absorption 57 or transient absorption 74.

For the case of isotropic media the second order MDCS signal vanishes,
consequently the most common measurement is of third order susceptibility
$\chi^{(3)}$ in equation 1 55. Three light pulses are required in order to
obtain a third order non-linear response, with geometry as depicted in Fig.
1(a), and again in Fig. 2(f). The first pulse, creates a coherent
superposition between the ground and an excited state of the system, also
known as coherence. The phase of this coherence oscillates at the frequency
difference between the ground and excited states during a time ($\tau_{1}$),
and mesoscopically, a coherent polarization is generated. The second pulse
creates a population state that evolves during a time $\tau_{2}$. Depending on
the experiment, instead of population, the second pulse can create further
coherences between higher excited states. Finally, the third pulse creates a
new coherence between ground and excited states, which radiates the measured
signal after a time $\tau_{3}$ 60. This is represented in Figs. 1(b), 1(c),
and 2(f).

A MDCS experiment records the phase oscillations of the coherences created by
pulse 1 and measured with pulse 2, as a function of the delay time $\tau_{1}$.
Subsequently, a 2D spectrum is extracted using a Fourier transform along that
time variable. The axis related to the Fourier domain of $\tau_{1}$ is
referred to as _excitation_ (or _absorption_) energy. The spectral amplitude
and phase of the coherent signal emitted by the nonlinear material response is
measured with a spectrometer and heterodyne detection (spectral
interferometry) with the LO pulse, and this spectral axis is referred to as
the _emission_ energy, and these spectral interferograms are measured as a
function of $\tau_{1}$, usually at fixed $\tau_{2}$, but alternatively in the
case of two-quantum coherence measurements, at fixed $\tau_{1}$ and as a
function of $\tau_{2}$ (Fig. 1(a)). Fig. 2 shows a simulation of typical MDCS
2D maps considering a system of two excited states. The features along the
diagonal correspond to optical transition autocorrelations, which reproduce
the structure of a common linear absorption spectrum 65. Off diagonal features
(cross-peaks) are created when different excited states share a common ground
state (Fig. 2(a)), meaning that it is an intrinsic feature of the material and
not an indication of sample inhomogeneity. The absence of cross peaks (Fig.
2(b)) reflects two uncorrelated optical transitions. Variations in the
intensity and shape of these cross-peaks provides information on the coupling
(correlation) strength between the exited states (Fig. 2(c)).

Measurements can be classified as N-quantum depending on the difference in the
number of quanta separating the two states involved in the generated
coherences. The most common example is a one-quantum (1Q) measurement where
the ground and excited state are separated by one quantum of the incident
photon energy. In this case the MDCS 2D map probes directly the optical
transitions related to absorption and photoluminescence phenomena of the
sample 61.

To probe excited states lying two-quantum above the ground state arising from
e.g. biexcitons, it is necessary to perform two-quantum (2Q) measurements.
Depending on the MDCS experiment, when having a non-collinear beam
configuration, is possible to isolate two-quantum coherent signals by using a
specific pulse sequence as depicted in Fig. 1(c) 75, 76, 77, 54. While, for a
collinear beam configuration this can be achieved by doing a second harmonic
lock-in demodulation 78, 79, 80, this experiment has demonstrated to be
sufficiently sensitive to probe biexciton states in a polymeric system 80, 79.

The 2D spectra measured with MDCS provides us with the ability to separate the
homogeneous and inhomogeneous broadening contributions of the spectral
linewidth, in contrast to linear spectroscopy techniques, where the spectral
features are a combination of both types of broadening 55, 81. The homogeneous
linewidth is directly related to the time decay of the coherence generated by
the light-matter interaction (also known as dephasing time). Meanwhile, in the
case of inhomogeneous broadening, the linewidth is a result of the
contributions from the distribution of resonant frequencies, brought on by
sample inhomogeneities, defects, Doppler broadening, etc.66, 56, 82

Figs. 2(d), (e) show an example of a 2D spectra where a distribution of
decoupled emitters, each one with its own homogeneous width (red circles),
leads to the emergence of a spectral feature with inhomogeneous (homogeneous)
broadening corresponding to the diagonal (cross-diagonal) width of the
measured peak, in the limit of strong inhomogeneity. Typically, the way in
which the inhomogeneous (homogeneous) linewidths are obtained is by measuring
the peak width along a cut in the diagonal (cross-diagonal) of the 2D spectra.
However, this procedure is not entirely accurate in the limit in which
inhomogeneous broadening is not the dominant contribution. As Siemens et al.
showed 82, there is coupling between the diagonal and cross-diagonal spectral
lineshapes, i.e. an increase of the inhomogeneous broadening will also lead to
a slight elongation in the cross-diagonal direction, and vice versa. The
authors demonstrate a method, using projection-slice theorem, to extract the
absolute homogeneous and inhomogeneous linewidth of the spectra. They
considered a Markovian approximation 55 for the homogeneous broadening
(resulting in an exponential decay with a dephasing rate $\gamma$), and a
Gaussian envelope of with $\sigma$ for the inhomogeneous broadening. The
authors showed that in the case where the distribution of resonant frequencies
is much larger than the dephasing rate ($\sigma\gg\gamma$), the system is in
an inhomogeneous limit. The amplitude of a diagonal cut follows a Gaussian
form of width $\sigma$, whilst a cross-diagonal cut has a square root of a
Lorentzian shape with a width $2\gamma$. This is shown in Fig. 3(c) and (f),
respectively. In this limit, the linewidths can be extracted with minimal
error using the typical method previously mentioned. On the other hand, in a
purely homogeneous limit ($\gamma\gg\sigma$), the authors showed that both the
diagonal and cross-diagonal cuts have the same Lorentzian shape (width
$2\gamma$), as shown in Fig. 3(a) an (d). Finally, in the special case where
the homogeneous and inhomogeneous broadening are similar ($\sigma\sim\gamma$),
the diagonal and cross-diagonal amplitudes follow a more complicated shape, as
shown in Fig. 3(b) and (e). The diagonal amplitude is a convolution of a
Gaussian and a Lorentzian function, whilst the cross-diagonal amplitude takes
the shape of a complementary error function.

Conjugated polymers referred to in this work are typically in a inhomogeneous
broadening limit 83, 84. This means that the intrinsic homogeneous
(inhomogeneous) broadening can be obtained by performing a zero-time rephasing
experiment, and measuring the Lorentzian (Gaussian) width of the cross-
diagonal (diagonal) feature. In this type of measurement, the phase evolution
of the coherence during $\tau_{1}$ and $\tau_{3}$ are opposite, so any non-
intrinsic inhomogeneous dephasing that occurs during $\tau_{1}$ is cancelled
by a mirror-opposite rephasing during time $\tau_{3}$. If the population time
is set as $\tau_{2}=0$, then the measurement is referred to as photon-echo. We
will consider below an example of a 2D coherent spectral lineshape that is
_not_ indicative of dominant inhomogeneous broadening, although this would not
have been evident from linear spectral lineshape analysis.

We highlight that in spite of efforts by some members of the MDCS community to
establish conventions, there are currently none for displaying MDCS plots, as
different authors display such spectra in different ways. (Even the current
authors have adopted different conventions over time.) The published data
displayed in this review reflect that sociology.

##  3 Excitons in neat polymers

###  3.1 Linear optical signatures of excitons in photophysical aggregates

Fig. 4: (a) Absorption and photoluminescence (PL) spectra of a P3HT film cast
from a chloroform solution. (Insets) P3HT $\pi$-stack and Jablonski diagram
corresponding to the weakly coupled H-aggregate model. (b) Graphical depiction
of P3HT J- and H- aggregates. (c) Absorption and PL spectra of P3HT nanofibers
grown in toluene. The Figure was extracted with permission from Ref. 28.

The photophysics of conjugated polymer assemblies can be understood in the
context of Kasha theory 85, 28. In this model Kasha et al. describe the role
of optically generated excitations in the absorption and fluorescence of small
chromophore aggregates. Briefly, Coulombic coupling is mediated via dipole-
dipole interactions between polymer chromophores, leading to the formation of
excitonic states. If the coupling term is negative, the neighbouring polymer
chains have a head-to-tail orientation and are referred to as J-aggregates. On
the other hand, if the coupling term is positive, the chain arrangement is
side-by-side and the polymer is known as H-aggregate.

Excitons have different physical traits depending on the geometric arrangement
of the chains 30, 28. In J-aggergates (along the chromophore chain), excitons
are of Wannier-Mott characteristics, meaning they can be separated over
several repeated units. For H-aggregates (across the chains), the excitons
have more Frenkel-like properties, where the separation is limited to
neighbouring chains 86, 87, 25.

Extensions to the Kasha model include the effect of intramolecular vibrations
on the spectra of exciton states 19. These vibrations are often referred to as
progression-building due to the characteristic succession of peaks that appear
in the absorption/emission spectra, as shown in Fig. 4 for
poly(3-hexylthiophene) (P3HT) films. Because of this, P3HT has been used as
model platform to develop the physical understanding of vibrational coherence
and electron transfer dynamics in organic photovoltaics 84. Its semi
crystalline nature gives rise to the previous mentioned vibrational structure,
and its strong coupling with optical excitons are visible in the
absorption/emission spectra 84.

P3HT aggregates formed from good solvents are mostly H-like (see Fig. 4),
favoring interchain interactions. This is likely due to the abundance of
aggregates with low conjugation lengths, consequence of a high level of
disorder brought on by fabrication processes 19, 22, 23 (rapid solvent
evaporation, spin-casting films, etc) 28.

The underlying differences between J- and H-aggregates are reflected in the
features of the absorption and photoluminescence (PL) spectra of the polymers.
The clearest example is the Frank-Condon vibronic progression resulting from
coupling between the vinyl-stretching modes of the $\pi$-conjugated backbone
and the optical exciton $S_{0}\rightarrow S_{1}$ transition 30. Understanding
this vibrational structure of the absorption and PL spectra has been a key
approach to explain structure-property relations. Any distortion on the
vibronic progression upon aggregation helps predict the effect on the spectral
features due to disorder, temperature and molecular assembling 88, 89, 90, 91,
92, 93.

Regarding the PL spectrum, it is considered that the emission occurs from the
lowest energy exciton following Kashas rule 94. In the absence of disorder,
this corresponds to an exciton with wavevector equal to zero and vibrational
quanta $\nu=0$ (labeled as 0–0 in Figure 4). Subsequent spectral peaks are
referenced according to the total number ($\nu$) of vibrational quanta. The
relative intensity of the 0-0 peak with respect to the rest of the vibronic
progression is directly related to the exciton coherence length (distance over
which an exciton maintains wavelike behaviour) 95, 96, 26.

Specifically in J-aggregates, higher exciton coherence lengths lead to
constructive interference between the emitting dipoles resulting in an
enhancement of the 0-0 peak, whilst for H-aggregates, the interference is
destructive, causing a suppression of the 0-0 spectra peak 28. Therefore, an
ideal H-aggregate has a PL spectrum formed by a vibronic progression without
the 0-0 peak, whereas the PL spectrum of a J-aggregate is also a modified
Frank-Condon progression, dominated by a superradiant 0-0 peak 22, 97, 98.

The PL spectra is also sensitive to the amount of disorder in the conjugate.
In P3HT, this disorder comes in the form of structural (packing) defects,
chemical impurities (oxidized defects), torsional defects 99, 100, 101, 102.
In H-aggregates, where the 0-0 peak is suppressed, the introduction of
disorder allows for 0-0 emission. Meaning that the strength ratio between 0-0
and 0-1 emission is direct probe for disorder 103.

Another defining difference between J- and H- aggregates is observable in the
absorption spectrum. The ratio of the $A_{1}$ and $A_{2}$ absorption peaks is
related to the nearest-neighbor interchain Coulomb coupling, which is directly
proportional to the exciton bandwidth 23. In the case of J-aggregates, an
increasing ratio reflects an increase in the exciton bandwidth as well as a
redshift of the main peak ($A_{1}$). For H-aggregates, the behaviour is
opposite, i. e., the increase ratio of the $A_{1}$ and $A_{2}$ correlates to a
decrease in the exciton bandwidth and a blue-shift of the spectrum 21, 25.

Previous studies have found that the exciton bandwidth depends on the
interchain Coulomb coupling, which at the same time depends on the conjugate
chain length $L$. Once $L$ exceeds the intermolecular separation, the
interchain coupling goes down as a function of $L$, i. e. shorter chains lead
to stronger interchain interaction and more H- like behaviour 104, 105, 106,
107, 103, 23.

In an ideal case where the chains are infinitely long, the interchain
interaction vanishes. This indicates that the absorption spectra can directly
probe the interchain coupling and intrachain order. In practice, one has to
take into account concurrently inter- and intra-chain interactions in the
calculations. Therefore a realistic picture of a polymer aggregate is one that
has both a J- and H- characteristics, meaning the polymer is a 2D excitonic
system containing delocalized excitons along the polymer chain as well as
between the chains. Despite the valuable information that can be extracted
from absorption and photoluminescence spectra, the details on how excitons
interact with their environment cannot be obtained from the linear spectra.
This is mainly because the homogeneous and inhomogeneous line broadening
cannot be selectively extracted from the total spectral width. In this
context, MDCS has been an important tool to disentangle these broadening
contributions.

In the following sections we will discuss results regarding the photoexcited
dynamics of neat polythiopehenes, and their blends, as well as population
effects on the spectral linewidths probed via MDCS experiments.

###  3.2 2D coherent excitation lineshape of HJ aggregates

Fig. 5: (a) Absolute rephasing 2D coherent spectrum of PBTTT at zero-
population-waiting time, measured at 5 K. The 0–0 and 0–1 diagonal peaks, as
well as their cross peaks, are indicated in the figure. (b) Real and (c)
imaginary components of the spectrum in part (a). Top: absorption spectrum of
PBTTT and spectrum of the femtosecond pulses used in the measurement.
Previously unpublished data.

We start our discussion of spectral lineshapes by considering MDCS
measurements on a relatively rigid polythiophene derivative. The 2D coherent
rephasing spectrum of
[poly(2,5-bis(3-hexadecylthiophene-2-yl)thieno[3,2-b]thiophene)] 108, a
conjugated polymer in the polythiophene family, at zero population-waiting-
time, i.e. $\tau_{2}=0$ in Fig. 2(f), is displayed in Fig. 5. Part (a)
displays the norm of the spectrum, while parts (b) and (c) display the real
and imaginary spectra, respectively. In this sample, the linear lineshape (top
panel) is very similar to that of P3HT in Fig. 4(a), and is indicative of
dominant H-aggregate behaviour 28. The diagonal 0–0 feature in Fig. 5(a) is
suppressed with respect that corresponding to 0–1, consistent with the lineal
lineshape. Cross peaks between the 0–0 and 0–1 diagonal peaks are also
observed, as expected for a vibronic progression, which is the situation
depicted in Fig. 2(a). The real and imaginary spectra, however, show structure
in the (0–1/0–0) cross peak, with clear evidence of an excited-state
absorption superimposed with the cross peak, a feature not observed in the
(0–0/0–1) cross peak. This excited-state absorption is associated with two-
quantum transitions, which will be discussed in Section 4 below. Furthermore,
the norm of the spectrum, shown in Fig. 5(a), displays highly symmetric
lineshape of the 0–0 and 0–1 diagonal peaks, which is indicative of moderate
inhomogeneity, a regime depicted in Figs. 3(b) and diagonal and anti-diagonal
cuts in Fig. 3(e). From this lineshape, the homogeneous and inhomogeneous
linewidths have been extracted, and it was concluded that the importance of
the homogeneous contribution to the total linewidth is indicative of the
importance of dynamic disorder in this polymer system 79. The nonlinear
coherent lineshape thus reveals richer information than available via the
linear spectrum.

###  3.3 Homogeneous dephasing and exciton delocalization

Fig. 6: (a) Modulus of the measured 2D-PLE rephasing spectrum. The elongated
feature along the diagonal is characteristic of inhomogeneous broadening,
limited by the excitation laser spectrum, while the antidiagonal slice (blue
dashed line) reveals the homogeneously-broadened spectrum. (b) Antidiagonal
slice of the photon echo signal at 2.10 eV (blue open circles) corresponding
to the dashed blue line in part (a). The black continuous curve displays the
corresponding slice for the simulated spectra using $2\gamma=90\,\text{meV}$
and $\sigma=130$ meV (shown in Fig 7(c)). Figure extracted with permission
from Ref. 83 Fig. 7: Numerical simulations of the P3HT rephasing (left) and
nonrephasing response (right). The real part is displayed using $2\gamma=30$
meV (top) and $2\gamma=90$ meV (bottom), both with inhomogeneous broadening
$\sigma=130$ meV. Here $\gamma$ is the dephasing rate and $\sigma$ is the
inhomogeneous spectral width. Figure extracted with permission from Ref. 83

Single-molecule spectra of conjugated aggregates differs appreciably from its
solid state counterparts, in such manner that the spectral lineshape is not
due solely to a distribution of inhomogeneous resonators 109, 110, 111.
Previous studies on isolated P3HT chains at low temperature (4 K) showed that
the PL spectrum spans over a large spectral range in the visible region, with
a linewidth ranging from $\sim$10-30 meV and almost invariant line shapes 109.
Thiessen et al. 109 suggested that the inhomogeneous broadening of the PL
emission of solution P3HT arises from the conformational disorder at the
single chain level. They hypothesized that intrachain torsional disorder gives
rise to a distribution of chromophore energies through exciton delocalization
and electron-hole polarization 112. Consequently, this energetic disorder is
responsible for the variability in the spectral range of the emission spectra,
leading to a large inhomogeneous linewidth 109. This is in contrast to bulk
films, where the line shapes are very strongly dependent on the film
microstructure 26. The authors of Ref. 109 argued that the emission spectral
lineshape in the bulk are composed of a distribution of single-chain emission
spectra that are distributed throughout the entire visible range, and _not_
dictated by the aggregate mode of Spano 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34. That report highlighted the need to experimentally
isolate the homogeneous spectral linewidth of P3HT in the solid state.

In a subsequent study, Gregoire et al. 83 measured simultaneously the
homo/inhomogeneous excitation linewidth of a P3HT bulk film and compared them
to those obtained by single-molecule measurement, in order to determine the
validity of aggregate excitonic models and the effects of aggregation. They
employed an MDCS variant referred to as two dimensional photoluminescence
(2DPL), where the measured nonlinear response is the time-integrated PL
intensity. The authors demonstrated that the homogeneous linewidth was an
order of magnitude larger compared to single polymer chains (at 8 K). This can
be seen in Fig. 6(a), which presents the modulus of a 2D rephasing spectra.
Fig. 6(b) shows the extracted homogeneous linewidth (blue dots) compared with
simulations performed with a homogeneous dephasing rate of $2\gamma=90$ meV
(black line). The obtained dephasing rate is close to an order of magnitude
larger than the lower limit reported for isolated chains 109. This establishes
unabmiguously that the bulk spectral lineshape is _not_ due to a distribution
of non-interacting single-chain spectra, but by solid-state, aggregate
dispersion that results in more rapid homogeneous dephasing 83.

In addition, Gregoire et al. 83 reported numerical simulations of the 2D
rephasing and nonrephasing spectra, both with homogeneous dephasing rates of
$2\gamma=30$ meV and $2\gamma=90$ meV (see Fig 7). The authors demonstrated
that a homogeneous linewidth of $2\gamma=30$ meV, which would correspond to an
upper limit of the single-chain linewidth 109, leads to a significantly
narrower 2D spectra compared to the measurements. The simulations considered a
inhomogeneous linewidth of $\sigma=130$ meV which corresponds to the disorder
widths previously used to model absorption and PL lineshapes in P3HT 26. The
authors suggested that the additional homogeneous broadening seen in the bulk
film comes from interchain photophysical aggregate effects, which consists of
fluctuations in excitonic coupling between $\pi$ electrons across chains 27,
increasing the dephasing rate and promoting H-aggregate-like signatures in the
spectra.

###  3.4 Exciton population dynamics in P3HT

Fig. 8: Representative absorptive 2D spectra of P3HT film with a population
time $\tau_{2}=200$ fs, along with the linear absorption and photoluminescence
spectra to illustrate the peak positions. Figure extracted from Ref. 84 Fig.
9: (a) Real part of the MDCS 2D maps of a P3HT film at different population
times $\tau_{2}$. (b)-(d) Time traces of the amplitudes of one diagonal peak
(with excitation and detection wavelengths at 564 nm) and two cross peaks
(with excitation wavelength at 614 nm (or 564 nm) and detection wavelength at
564 nm (or 674 nm)). Time traces from the rephasing spectra (absolute value,
cross), nonrephasing spectra (absolute value, empty circle), and the 2D
spectra (absolute value, empty square) are shown. (e) and (f) Correspond to
the Fourier transforms of the time traces of two diagonal peaks (with
excitation and detection wavelengths at 564 nm (or 614 nm)) along the
population time axis. Figure extracted from Ref. 84.

MDCS has been shown to be a powerful method to study the sub picosecond time
evolution of electronic and vibrational coherences, specifically in conjugated
polymers, where strong correlation and population dynamics can affect exciton
delocalization 113, 114, 115, 116, 117, 118, vibrational relaxation 119, 120,
121, 122, electron transfer 123, 124, 125, etc. It has been shown that
analyzing the 2D spectra as a function of the population time is crucial to
estimate and isolate contributions to the homogeneous and inhomogeneous
broadening of the spectra due to excited and vibronic states 126, 127, 84.

Scholes et al. 126 performed measurements on N,N-bis-
dimethylphenyl-2,4,6,8-perylenetetra-carbonyl diamide (PERY) and
poly[2-(2’-ethylhexyloxy)-5-methoxy-1,4-phen-ylenevinylene] (MEH-PPV) in
toluene as a function of population time. The authors were able to demonstrate
the effect on the absorption line-shape due to Coulomb coupling between
energetically disordered sub units along the polymer backbone.

Song et al. 127 reported 2D electronic coherence measurements as a function of
population time $\tau_{2}$ on P3HT:PCBM, a fullerene derivative, nanoparticle
blends. The authors found a photoinduced absorption associated to hole
formation that appeared at a population time of $\sim 380$ fs, suggesting that
exciton relaxation precedes dissociation in neat P3HT nanoparticles. This is
consistent with the delayed photoluminescence findings in neat P3HT reported
by us previously 52, 53. In contrast, with the inclusion of PCBM, hole
formation was observed with a time constant of $\sim 24$ fs, indicating that
electron transfer occurs through dissociation of hot excitons. In addition,
Song et al. found underlining oscillations with a frequency of $\sim 1506$
cm-1 in the MDCS spectra, corresponding to the vibrational coherence from the
P3HT C=C stretching mode. They measured a dephasing time of $\sim 250$ fs for
the vibrational coherence which showed that the electron transfer process to
the fullerene domain occurs in a non-equilibrium state (specifically the
nuclear configuration and charge distribution). The authors mention the strong
amplitude of the vibrational coherence as an indication of similar
delocalization of the exciton and hole-polaron given the Huang-Rhys factor
would diminish otherwise due to exchange narrowing.

A subsequent study Song et al. 84 focused on P3HT films instead of
nanoparticle blends; the authors reported rephasing and non-rephasing MDCS
measurements on as a function of the population waiting time $\tau_{2}$. They
developed and proposed a method to identify and separate the signal from
vibrational coherences of ground and excited electronic states. Fig. 8 shows
the absorptive (real part) of the 2D spectra at a population time of
$\tau_{2}=200$ fs. The accompanying absorption and PL spectra was used by the
authors to identify the features in the 2D map, which are the following: (i)
Two diagonal peaks correspond to absorption/emission from electronic-
vibrational transitions $S_{0}^{\nu^{0}}\rightarrow S_{1}^{\nu^{1}}$ at 564
nm, and $S_{0}^{\nu^{0}}\rightarrow S_{1}^{\nu^{0}}$ at 614 nm, where $S_{0}$
and $S_{1}$ correspond to the ground and first excited electronic state,
respectively, and $\nu^{0}$ and $\nu^{1}$ represent the addition of 0 or 1
quantum of a vibrational excitation, respectively. (ii) Two cross-diagonal
peaks are present at the excitation/detection (detection/excitation)
wavelengths of 564 nm/614 nm (614 nm/564 nm) that correspond to stimulated
emission. (iii) Two features located at the detection wavelength 674 nm. The
position of these 6 main features are highlighted in Fig. 8 by the crossing of
horizontal and vertical dashed lines.

The authors found that all the six features in the 2D spectra have an
underlining oscillation with a period $\sim 20\,\text{fs}$
($1519\,\text{cm}^{-1}$). They corroborate this by presenting time traces and
their Fourier transforms as a function of the population time (see Fig.
9(b)-(f)). The oscillations correspond to the vibrational frequency of the C=C
stretching mode, indicating a coherent interaction with the P3HT electronic
states. The Fourier amplitudes of the rephasing and non-rephasing 2D maps (at
the frequency $1519\,\text{cm}^{-1}$) were used in conjunction with a four-
level model to successfully relate each spectral feature either to ground or
excited state vibrational coherences. For example, in the case of features
with a excitation wavelength of 614 nm, the nature of the vibrational
coherence changed from excited to ground state depending if the measurement
was rephasing or non-rephasing. The analysis resulted in direct evidence of a
faster dephasing time for excited state ($\sim 244$ fs) compared to the ground
state ($\sim 297$ fs) coherence, suggesting that electron transfer in P3HT
blends can be mediated via vibrational coupling 128, 129, 123.

##  4 Frenkel biexcitons

Fig. 10: Two-dimensional coherent photoluminescence excitation (2D-PLE)
spectroscopy. (A) Schematic of the experimental pulse sequence. Here
$\phi_{i}$ is the phase of pulse $i=1,2,3,4$. The inter-pulse delays are
$t_{21}$ (coherence time), $t_{32}$ (population waiting time), and $t_{43}$
(coherence time). The phase-modulation reference waveforms for phase-sensitive
detection are generated optically at frequencies $\Omega_{21}$ and
$\Omega_{43}$, at which the relative phase $\phi_{21}=\phi_{2}-\phi_{1}$ and
$\phi_{43}=\phi_{4}-\phi_{3}$ oscillate, respectively. In this work, time and
spectrally integrated photoluminescence (PL) intensity is demodulated by
phase-sensitive detection at the reference frequency
$f_{\mathrm{ref}}=\Omega_{43}+\Omega_{21}$ and $\Omega_{43}+2\Omega_{21}$ for
the one- and two-quantum correlation spectra, respectively, shown in Fig. 11.
We outline in Supplemental Material that the 2D-PLE lineshape can contain
contributions from nonlinear incoherent population dynamics over the entire
exciton lifetime, and that all spectral lineshapes presented in Fig. 11 are
free from this undesired contribution under the excitation conditions of this
experiment. (B) Double-sided Feynman diagrams of the two most important two-
quantum response terms that couple a ground state ($|0\rangle$), a single
exciton state ($|1\rangle$), and a two-exciton state ($|2\rangle$). (C)
Schematic representation of the 2D-PLE expected spectrum for two correlated
optical transitions at energies $\hbar\omega_{a}$ and $\hbar\omega_{b}$. The
spectral axes $\hbar\omega_{21}$ and $\hbar\omega_{43}$ are obtained by
Fourier transform of the 2D coherent PL decay function along time variables
$t_{21}$ and $t_{43}$ at fixed $t_{32}$. (D) Schematic representation of a
two-quantum 2D-PLE correlatios spectrum. The two-quantum energy
$\hbar\omega_{21,2Q}$ corresponds to the two quantum coherences involving
pulses 1 and 2, The diagonal line represents
$\hbar\omega_{21,2Q}=2\hbar\omega_{43}$.  Fig. 11:  2D-PLE spectra measured at
5 K and $t_{32}=30$ fs. (A) The real part of the one-quantum non-rephasing
($f_{\mathrm{ref}}=\Omega_{43}+\Omega_{21}$) spectrum. (B) The real part of
the two-quantum ($f_{\mathrm{ref}}=\Omega_{43}+2\Omega_{21}$)) spectrum. (C)
Spectral cuts along $\hbar\omega_{21,2Q}-\hbar\omega_{\mathrm{diag}}$ spectral
axis at fixed $\hbar\omega_{43}=2.06$ (blue), $1.99$ (fuscia), and $1.94$ eV
(red) for the spectrum in B. Here $\hbar\omega_{\mathrm{diag}}$ is the
$\hbar\omega_{21,2Q}=2\hbar\omega_{43}$ two-quantum diagonal energy,
corresponding to zero net two-quantum correlation energy (neither binding nor
repulsion). We chose to measure 2D coherent spectra with $t_{32}=30$ fs in
order to avoid ambiguous time ordering at $t_{32}=0$. Due to the highly
transient nature of the two-quantum coherence signal, it would not be possible
to measure with $t_{32}\gg 30$ fs.

Nonlinear coherent spectroscopies can unravel spectral structure beyond
excitons as reviewed in Section 3, but they can also resolve multi-particle
binding. Many-body electronic correlations play a strong role in determining
the optical properties of semiconductors 130. These correlations can give rise
to biexcitons, which are bound two-quantum quasiparticles. In organic
materials, these two-quantum excitonic states are of fundamental importance,
not only because they probe the electronic structure of the material but also
because they are expected to be key intermediate states that determine the
dynamics of photophysical processes, such as exciton bimolecular annihilation
47. A direct probe of Frenkel biexcitons in organic systems presents a
challenge because of the largely featureless nature and broad inhomogeneous
linewidths of Frenkel exciton spectra, in contrast to Wannier-Mott biexcitons
in semiconductor quantum wells 75, 131, which have been well characterized by
linear spectroscopy because of well defined spectral structure. An exception
to this are Frenkel biexcitons in solid argon 132. Despite the extensive
theoretical work predicting their existence in organic semiconductors 133,
134, 135, 136, 137, current experimental work in polymeric semiconductors has
shown only indirect evidence of two-quantum exciton states by sequential
ultrafast excitation 48, 47, 49.

###  4.1 Two-quantum coherence measurements in PBTTT

We have recently explored the spectral structure of Frenkel biexcitons in
PBTTT, with a nonlinear spectral lineshape already discussed in the context of
Fig. 5, by means of two-quantum coherence measurements 80. Such measurements
can directly identify biexciton resonances via two-quantum coherences. They
can therefore quantify the biexciton binding energies 75, 131. In that work,
we constructed the coherent two-dimensional excitation spectrum via incoherent
measurement of the time-integrated photoluminescence (PL) intensity due to a
fourth-order excited-state population arising from the interference of
wavepackets produced by a sequence of four light-matter interactions 67,
allowing the measurement of the spectral correlations between resonance
involving pulses 1 and 2 and those corresponding to pulses 3 and 4 in Fig.
10A. This is the same technique as was used for the homogeneous linewidth
measurements of P3HT 83, reported in Fig. 6. This technique uses phase
sensitive detection to construct the 2D coherent spectrum by modulating the
relative phase of each pulse pair in the four-pulse sequence, and demodulating
the total photoluminecence signal at a side band of the two phase modulation
reference waveforms. However, in Ref. 80, we implemented the demodulation at
the second harmonic of the signal relative to the phase modulation function of
the first two pulses. The two energy axes are constructed by Fourier
transformation of the two-dimensional coherence decay function along time
variables $t_{21}$ and $t_{43}$ at a fixed population time $t_{32}$.
Accordingly, the spectral correlation along resulting energy axes
($\hbar\omega_{21}$, $\hbar\omega_{43}$) involve single-quantum
($|0\rangle\rightarrow|1\rangle$) and two-quantum
($|0\rangle\rightrightarrows|2\rangle$) transitions, represented schematically
in the left of Fig. 10B. The two principal coherent pathways involving the
correlations between one- and two-quantum coherences under this detection
scheme are depicted by the double-sided Feynman diagrams in Fig. 10B. We also
display schematic diagrams of the one- and two-quantum 2D-PLE coherent spectra
for two correlated transitions in Figs. 10C and 10D respectively. These may be
correlated, for example, via a common ground state such as H-like and J-like
states in a HJ aggregate, each evident in the diagonal of the 2D-PLE spectrum
($\hbar\omega_{21}=\hbar\omega_{43}$). In the two-quantum correlation
spectrum, a signal along the two-quantum diagonal
($\hbar\omega_{21-2Q}=2\hbar\omega_{43}$) signifies two non-interacting
excitons, while a signal above or below the diagonal signifies binding
interactions with repulsive or attractive character, depending on the signs of
the excitonic coupling (H- or J-like coupling) 28, respectively. In Fig. 10D,
the lower energy diagonal peak displays 2J biexcitons (the two-quantum energy
is higher than twice the one-quantum energy), while the higher-energy
resonance displays structure corresponding to 2H biexcitons (in which the two-
quantum energy is less than twice the one-quantum energy).

The 2D spectrum in Fig. 11A is dominated by a symmetric diagonal peak centered
at $\hbar\omega_{21}=\hbar\omega_{43}\approx 2.06$ eV, corresponding to the
0–0 peak energy. We also observe a weak diagonal signal centered at $\sim
1.99$ eV, with even weaker structure identified at lower energy. These
features display intense cross peaks with the (0–0) resonance. The rich
spectral structure displayed at the low-energy tail of the 2D-PLE spectrum
demonstrates the existence of distinct states at the low-energy edge of the
0–0 absorption peak. We note that the energy of the weak diagonal feature
corresponds to the 0–0 absorbance peak energy found in PBTTT J-aggregates
induced when blending this semiconductor with a polar commodity plastic 138.
We thus hypothesize that these are weak signatures of J-aggregate
macromolecular conformations bearing effects of interactions with static
dipoles. We present in Ref. 80 the linear absorption spectrum of the same
batch of PBTTT in a blend with an ionic liquid, processed such that the
J-aggregate dominates the lineshape, which supports the assignment in Fig. 11.
The 0–0 absorption peak in that film is at 1.96 eV, consistent with the
spectrum reported in ref. 138, which supports the assignment of the weak, low-
energy diagonal feature as the origin of the J-aggregate progression.

We observe in Fig. 11B a broad distribution along the two-quantum energy axis
$\hbar\omega_{21,2Q}$ centered at the one-quantum energy
$\hbar\omega_{43}=2.06$ eV (the 0–0 excitation maximum), with the peak of the
distribution below the two-quantum diagonal, i.e., below the energy at which
$\hbar\omega_{21,2Q}=2\hbar\omega_{43}$. Thereby, the spectrum cuts through
the diagonal line, with a tail extending to the high-energy side of the
diagonal. We also observe two-quantum peaks for the features at lower energy
than the 0–0 peak in Fig. 11A. These are both centered at higher energy than
the diagonal; meaning that the two-quantum correlation is predominantly
repulsive for the low-energy features. In Fig. 11C we display cuts of the two-
quantum excitation spectrum at fixed one-quantum energies $\hbar\omega_{43}$.
The cuts are along $\hbar\omega_{21,2Q}$ relative to the two-quantum diagonal
energy $\hbar\omega_{\mathrm{diag}}$. This is a good reference because signal
on the diagonal corresponds to the energy of two excitons that coexist without
interaction. Importantly, we notice, on the one hand, that the 0–0 absorption
peak, assigned to the vibronic progression of an H aggregate 28, forms
biexcitons with attractive interactions with $E_{\mathrm{2H,b}}=-64\pm 6$ meV.
On the other hand, the low-energy resonances, which we hypothesize to
correspond to J-aggregate resonances, display predominantly repulsive two-
quantum correlations, peaked at $E_{\mathrm{2J,b}}=+106\pm 6$ and $+233\pm 6$
meV, respectively. This supports the hypothesis that both attractive and
repulsive biexciton binding coexist within the inhomogeneously broadened 0–0
linear absorption peak.

A key finding of Ref. 80 is that that _interchain_ H-like excitons are
associated with _intrachain_ exciton-exciton couplings. In contrast,
_intrachain_ J-like excitons are paired by _interchain_ exciton-exciton
couplings. In either case, the biexciton binding energy is related to the
exciton-exciton contact interaction and the inter-site hopping (excitonic
coupling) energy, which are intrinsic molecular parameters quantified by
quantum chemistry and that are expected to be controllable via chemical design
and polymer assembly.

###  4.2 Biexciton stability and exciton-backbone coupling

We generally think of bound states as arising from and attractive interaction
between two particles. However, in Ref. 80, as described in the preceeding
section, we reported both _attractive_ and _repulsive_ biexciton binding; we
reported further theoretical analysis in Ref. 139, where we discussed that in
J-aggregate systems, 2J-biexcitons can arise from repulsive dipolar
interactions with energies $E_{2J}>2E_{J}$ while in H-aggregates, 2H-biexciton
states $E_{2H}<2E_{H}$ corresponding to attractive dipole exciton-exciton
interactions. According to our model 139, Frenkel biexcitons mix J-like and
H-like character in terms of their collective quantum behavior with the
requirement that the ratio of the exciton/exciton interaction and the
perpendicular hopping term be $U/t>0$, with $t=-\hbar^{2}/2\mu$, $\mu$ the
effective mass of the exciton, and $U$ a contact interaction potential, which
gives rise to localized biexciton states in the perpendicular direction.

We an understand this in the context of a textbook-level model for a biexciton
state interacting with a deformable lattice. Adopting a relative coordinate
between the two excitons, we write the biexciton Schrödinger equation as

$\displaystyle t\psi^{\prime\prime}+U\delta(x)\psi=E\psi,$ (2)

where $U$ is the contact interaction between the two excitons. For bound
states, $\psi(x)$ must vanish as $x\to\pm\infty$, giving that

$\displaystyle\psi(x)=\left\\{\begin{array}[]{cc}\sqrt{\kappa}e^{-\kappa
x}&x>0\\\ \sqrt{\kappa}e^{+\kappa x}&x<0,\end{array}\right.$ (5)

where $\kappa=U/2t$ is a positive constant and $E=t\kappa^{2}$. In general, we
take $t=-\hbar^{2}/2\mu_{eff}$ and $U<0$ for an attractive potential giving
rise to a bound state energetically below the continuum for the scattering
states.

Generally, interactions with the lattice phonons produce an additional
stability to to the reorganization of the system about the bound state.
However, in the case of the 2J bi-exciton, reorganization of the lattice can
actually destabilize the state by pulling it back into the continuum of
unbound states. To see this, we append to the 1D impurity model a term
coupling the biexciton to the lattice as per the Davydov model, 140, 141, 142,
143, 144, 145, 146, 147, 148, 149 such that resulting equations of motion read

$\displaystyle i\hbar\dot{\psi}(x)$
$\displaystyle=(t\nabla^{2}+U\delta(x)+(E_{o}+2\chi\nabla u(x)))\psi(x)$
$\displaystyle\ddot{u}-\frac{k}{m}\nabla^{2}u$
$\displaystyle=2\frac{\chi}{m}\nabla|\psi|^{2},$ (6)

where $u(x)$ is the lattice deformation, $\chi$ is the linear coupling between
the biexciton and the lattice, $m$ is the mass of the lattice “atoms” and $k$
is the elastic modulus. The bound-states are invariant under Galilean
transformation and one can find a closure relation

$\displaystyle\nabla u=-\frac{2\chi}{k}|\psi|^{2}$ (7)

that gives us a non-linear Schrödinger equation

$\displaystyle i\hbar\dot{\psi}=(t\nabla^{2}+g|\psi|^{2}+U\delta(x))\psi,$ (8)

where $g=-4\chi^{2}/k$. Note that $E_{0}$ is a constant given by

$\displaystyle
E_{0}=E-2t+\frac{1}{2}\int_{-\infty}^{\infty}\left(m\ddot{u}^{2}+ku^{\prime\prime}\right)dx$
(9)

that we can ignore for purposes of this analysis. The $\delta$-function
potential implies the wave function should have the form in Eq. 5. Taking
$\kappa$ as a variational parameter and minimizing the total energy, one
obtains

$\displaystyle\kappa=\frac{U}{2t}+\frac{g}{8t}.$ (10)

$\kappa>0$ is necessary to produce a localized state and from above $U/t>0$
and $g<0$ from its definition above, we have a stability requirement that if
$U>0$ and $t>0$, then $-g<4U$. Solving for the binding energy,

$\displaystyle E_{B}=\frac{(4U+g)^{2}}{64t}.$ (11)

We obtain a straight-forward estimate of the contribution of both the lattice
and the exciton/exciton coupling to the biexciton binding.

Fig. 12: Biexciton binding energy (in reduced units) for the 2J and 2H cases
versus increasing exciton/lattice coupling. For the attractively bound 2H,
lattice reorganization is expected to stabilize the biexciton state while
destabilizing the 2J. The dashed line indicates the limit of stability for the
2J case. Fig. 13: (a) Structure of PCDTBT (left) and PCBM (right). (b)
Photocurrent excitation spectrum of the PCDTBT:PCBM solar cell under
investigation. (c) Current-voltage response of same cell under AM 1.5 G solar
illumination (right-axis) and under laser illumination (left-axis). Figure
extracted from Ref. 72. Fig. 14: Anti-diagonal slice of the 2D-PCE rephasing
signal. The solid line is a fit to a Lorentzian function. Figure extracted
from Ref. 72.

In Fig. 12 we plot the biexciton binding energy versus the non-linearity
parameter, $g$. For the attractively bound 2H, lattice reorganization is
expected to stabilize the biexciton state by further localizing the state
($\kappa$ increases as $g$ increases in magnitude). On the other hand, for the
2J state, increasing the magnitude of $g$ decreases $\kappa$ and destabilizes
the otherwise bound 2J state by effectively pulling it back into the continuum
of unbound (scattering) biexciton states. When $-g=2U$ the state is fully
delocalized and further increases in the lattice coupling lead to unbound
solutions. We rationalize the biexciton spectral structure observed in Fig. 11
by the dependence of the biexciton binding energy on $g$ as reported in Fig.
12.

##  5 Photocurrent-detected spectroscopy in operating diodes

Two-dimensional photocurrent excitation spectroscopy (2D-PCE) is an MDCS
experiment, where the detection mechanism probes the photocurrent generated by
a sequence of four ultrashort laser pulses, akin to the photoluminescence-
detected variant of the experiment in Fig. 11. 2D-PCE has been a key approach
in the research of photocharge generation dynamics, not only because of the
unique detection sensitivity but also due to the ability to probe the
correlation between optical excitations and photocarriers in polymer
semiconductors 72. These correlations are identifiable through cross-peaks in
2D spectral maps, and are of the utmost importance to study materials in which
photocarrier precursors, such as excitons in molecular semiconductors, are the
primary photoexcitations. A special case are materials for photovoltaic
applications, such as solar cells based on PCDTBT:PCBM blends (Fig. 13(a)).
This is a polymer:fullerence benchmark system that has been shown to yield
solar-power conversion efficiencies as high as 7% 150, 151, 72. The
photocurrent excitation spectrum of this system is diaplayed in Fig. 13(b).

Importantly, Vella et al. 72 demonstrated that illumination using an ultrafast
pulse sequence such as that depicted in Fig. 11, at sufficiently low fluence,
produces J-V curves that match those under standard solar (AM 1.5G)
illumination, shown in Fig. 13(c), where the current density (as a function is
applied voltage) is compared between both types of illumination. This result
suggests that the current densities probed by 2D-PCE experiments are
comparable to those of a functioning solar cell, which is critical in
establishing that these advanced optical probes probe dynamics in regimes that
are relevant in the optoelectronic device.

Additionally, as a result of 2D-PCE measurements, Vella et al. suggested a
tunneling process from an exciton state to a current producing charge-
separated state. Consequently, the authors stated that the timescale of this
process would be dominated by fluctuations due to interaction with the bath
ensemble, predicting dephasing times $\leq 20$ fs. In order to test their
prediction, the authors extracted the optical dephasing time by taking the
antidiagonal cut of the 2D spectra, as shown in Fig. 14. The measured peak was
fit to a Lorentzian function with a full-width-at-half-maximum (FWHM) of
100$\pm 1$ meV. This represents the homogeneous linewidth of the main exciton
transition from the donor polymer, and corresponds to a dephasing time of 13
fs 72.

Fig. 15: 2D PC measurements on a PCDTBT:PCBM photovoltaic diode (see Fig. 13)
at ambient temperature. Shown are the real (left column) and imaginary (right
column) parts of the rephasing (a, b), non-rephasing (c,d), and total
correlation function (e,f). The population time is 50 fs. Figure extracted
from Ref. 57.

A concurrent 2D-PCE study by Li et al. 57 on PCDTBT:PCBM, similar to that of
Vella et al. 72, but with larger bandwidth of the ultrafast pulse, revealed an
inhomogeneously broadened spectrum, and suggested structure of charge-transfer
excitons and photocarriers. Fig. 15 was extracted from this study, and it
shows the 2D maps corresponding to the rephasing (a,b), non-rephasing (c,d)
and total spectrum (e,f). In addition, the authors demonstrated that the 2D
spectra from photocurrent excitation (2D-PCE), photoluminescence excitation
(2D-PLE), and photoinduced absorption (2D-PIA) all yield similar features,
indicating that charge-transfer excitons, probed via photoluminescence,
photocarriers via photocurrent, and polarons generally via quasi-steady-state
photoinduced absorption, can all be probed using the same excitation
pulsetrain sequence. This work further confirmed the polaronic nature of the
photocarriers and charge-transfer excitons in PCDTBT:PCBM.

##  6 Perspective

Given the work that we have highlighted in this review, in this section we
provide out vision of where MDCS can make further impact in the understanding
of excitons in $\pi$-conjugated polymers.

###  6.1 Coherent optical lineshapes of photophysical aggregates

Besides disentangling the homogeneous and inhomogeneous contributions to the
total spectral width, and thereby quantifying the optical dephasing processes
in materials, MDCS can reveal a very well defined spectral structure as shown
in Fig. 5, which displays the absolute, real, and imaginary spectra of the
rephasing signal of a PBTTT film at 5 K and at a population waiting time
$\tau_{2}=0$ fs. If we observe carefully Fig. 5(b) and Fig. 5(c), The real
part of the 0–0 peak at $\sim 2.06$ eV displays absorptive lineshape in the
real component, but dispersive character in the imaginary component; a
diagonal peak in the real part and a derivative feature about the diagonal in
the imaginary part. This is what is expected for non-interacting chromophores.
However, the 0–1 diagonal feature at $\sim 2.21$ eV displays inverted
behaviour: the real part displays apparently dispersive lineshape while the
imaginary part displays apparently dispersive lineshape. This behaviour
evolves over the first few hundred femtoseconds (data not shown here) such
that this character inverts between 0–0 and 0–1 diagonal peaks: 0–0 in the
real part evolves from absorptive to dispersive character in the real spectrum
and vice-versa in the imaginary one. This behaviour encodes the quantum
dynamics of the photophysical aggregates. It is well understood now that the
0–0 feature encodes the many-body behaviour of the aggregate 19. It will be
highly insightful to describe this type of MDCS lineshape evolution by
simulation of the quantum dynamics that include all of the aggregate many-body
details, lattice reorganization, and energetic disorder. Our groups are
currently focused in this work.

It was also highlighted in Section 3.2 that the absolute spectrum in Fig. 5(a)
display highly symmetric diagonal 0–0 and 0–1 diagonal peaks. This reflects a
comparable values of the homogeneous and inhomogeneous linewidths, which we
interpret as indicative of dynamic disorder governing the disordered landscape
in PBTTT. On the other, P3HT is clearly in an inhomogeneously broadened limit,
as shown in Fig. 6. The linear absorption spectrum of both materials, however,
displays similar total width, and it would not have been possible to
differentiate between the two materials purely from linear spectroscopy. This
presents an opportunity to investigate more technologically relevant families
of conjugated polymers that have varying degree of backbone rididity, and that
span from flexible polymers such as P3HT to “hairy-rod” materials 152, for
example.

Another opportunity for MDCS is to address lineshapes in push-pull materials
34, 32, 33. There are already reports of these techniques applied to this
class of materials 153, 154. MDCS approaches are well suited to address the
structure of inter-chromophore coupling effects due to charge-transfer
interactions, and we consider that there is still much to understand in
structure-property relations in this realm.

###  6.2 Two-quantum spectral structure of Frenkel biexcitons

The rich spectral structure revealed in Fig. 11 is intriguing in that it
reveals the role of both attractive and repulsive binding in conjugated
polymer hybrid HJ aggregates. We rationalized this observation by considering
the geometry of two-quantum correlations along and between chains, but there
is tremendous scope to understand this further with microscopic detail, and we
consider that there is substantial scope for the electronic structure
community to push the development of first-principle methods that account for
many-body correlations to explore our reports of biexciton binding in PBTTT
80. On the same token, stochastic effects that drive many-body quantum
dynamics 155 need to be incorporated in the theoretical treatment of Frenkel
biexcitons. Finally, we consider that the series of materials families that
will be relevant for 2D coherent lineshape analysis in the context of the
discussion in Section 6.1 should be studied with the double-quantum coherence
techniques developed in Ref. 80, and the measurement should be executed beyond
two-quantum correlations — how many Frenkel excitons correlate in the series
of conjugated polymers of technological interest, and what does this imply for
their potential in classical and quantum photonics?

###  6.3 MDCS in operating optoelectronic devices

Fig. 13 addresses a general criticism of ultrafast spectroscopy, which affirms
that the excitation conditions and resulting excitation densities are beyond
the regimes that are relevant in the operation of polymer-based optoelectronic
devices. With photocurrent detection and using the phase-sensitive detection
techniques developed originally by Tekavec et al. 67, it is possible to
measure nonlinear responses of the polymer semiconductor in a diode with
current-voltage characteristics that are in the regime of solar power
conversion 72. We consider that there is ample scope to expand the studies to
address contemporary high-efficiency systems involving non-fullerene acceptor
systems, in which the spectral structure of interchromophore couplings will
play a critical role in the understanding of device operation. One unexplored
direction is in electrochemical cells. We have been able to carry out
incoherent ultrafast experiments in conjugated-polymer-based electrochemical
cells 156, and there are clear opportunities to extend such studies with the
photocurrent/photovoltage spectrsocopies discussed in this review.
Nevertheless, the role of unexpected incoherent nonlinear population dynamics
may be an issue in organic devices 58, 73, and methods to decouple this
background signal from the nonlinear coherent response will be required.
However, this also presents an opportunity to develop methods to measure
concurrently nonlinear population dynamics between multiple excitations, and
the nonlinear coherent response of the material.

## Author contributions

EGM and AVF gathered most of the material described in this review article,
and led the early drafts of the manuscript. CSA and ERB led the intellectual
development of the article. All authors participated in the redaction of the
manuscript.

## Conflicts of interest

There are no conflicts of interest to declare by any of the authors.

## Acknowledgements

The work at Georgia Tech was funded by the National Science Foundation
(DMR-1904293). CS acknowledges support from the School of Chemistry and
Biochemistry and the College of Science at Georgia Tech. The work at the
University of Houston was funded in part by the National Science Foundation
(CHE-2102506, DMR-1903785) and the Robert A. Welch Foundation (E-1337).

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