\chapter{PEDOT:PSS}  \label{cha:pps}


\textit{This Chapter presents content first published in \textcite{HorakErikH2018a}.
  The authors are:
  \begin{denumerate}
    \item Erik H. Horak
    \item Morgan T. Rea
    \item Kevin D. Heylman
    \item David Gelbwaser-Klimovsky
    \item Semion K. Saikin
    \item Blaise J. Thompson
    \item Daniel D. Kohler
    \item Kassandra A. Knapper
    \item Wei Wei
    \item Feng Pan
    \item Padma Gopalan
    \item John C. Wright
    \item Alán Aspuru-Guzik
    \item Randall H. Goldsmith
  \end{denumerate}
  This Chapter focuses on my contribution.  %
}

\section{Introduction}  % =========================================================================

Poly(3,4-ethylenedioxythiophene)-poly(styrenesulfonate) (PEDOT:PSS) is a transparent, electrically
conductive (up to 4380 S cm$^{-1}$ \cite{KimNara2013a}) polymer.  %
It has found widespread use as a flexible, cheap alternative to inorganic transparent electrodes
such as indium tin oxide.  %

As a polymer, PEDOT:PSS implicitly contains a large amount of structural inhomogeneity.  %
On top of this, PEDOT:PSS is a two component material, composed of PEDOT (low molecular weight,
p-doped, highly conductive) and PSS (high molecular-weight, insulating, stabilizing).  %
These two components segment into domains of conductive and non-conductive material, leading to
even more structural inhomogeneity.  %
Nonlinear spectroscopy may be able to shed light on the microscopic environment of electronic
states within PEDOT:PSS.  %

\section{Background}  % ===========================================================================

Complex microstructure:
\begin{enumerate}
  \item PEDOT oligomers (6---18-mers)
  \item these oligomers $\pi$-stack to form small nanocrystalites, 3 to 14 oligomers for pristine
    films to as many as 13---14 oligomers for more conductive solvent treated films
  \item nanocrystallites then arrange into globular conductive particles in a pancakge-like shape
  \item these particles themselves are then linked via PSS-rich domains and assembled into
    nanofibril geometry akin to a string of pearls
  \item nanofibrils interweave to form thin films, with PSS capping layer at surface
\end{enumerate}

Prior spectroscopy (absorption anisotropy, X-ray scattering, condutivity).  %

% TODO: absorption spectrum of thin film

Broad in the infrared due to midgap states created during doping from charge-induced lattice
relaxations.  %
These electronic perturbations arise from injected holes producing a quinoidal distortion spread
over 4-5 monomers of the CP aromatic backbone, collectively called a polaron.  %
Energetically favorable to be spin-silent bipolaron.  %

\section{Methods}  % ==============================================================================

PEDOT:PSS (Orgacon Dry, Sigma Aldrich) was dropcast onto a glass microscope slide at 1 mg/mL at a
tilt to ensure homogeneous film formation.  %
The sample was heated at 100 $^\circ$C for $\sim$15 min to evaporate water.  %

An ultrafast oscillator (Spectra-Physics Tsunami) was used to prepare $\sim$35 fs seed pulses.  %
These were amplified (Spectra-Physics Spitfire Pro XP, 1 kHz), split, and converted into 1300 nm 40
fs pulses using two separate optical parametric amplifiers (Light Conversion TOPAS-C): ``OPA1'' and
``OPA2''.  %
Pulses from OPA2 were split again, for a total of three excitation pulses: $\omega_1$, $\omega_2$
and $\omega_{2^\prime}$.  %
These were passed through motorized (Newport MFA-CC) retroreflectors to control their relative
arrival time (``delay'') at the sample: $\tau_{21} = \tau_2 - \tau_1$ and $\tau_{22^\prime} =
\tau_2 - \tau_{2^\prime}$. The three excitation pulses were focused into the sample in a $1^\circ$
right-angle isoceles triange, as in the BOXCARS configuration. \cite{EckbrethAlanC1978a}  %
Each excitation beam was 67 nJ focused into a 375 $\mathsf{\mu m}$ symmetric Gaussian mode for an
intensity of 67 $\mathsf{\mu J / cm^2}$.  %
A new beam, emitted coherently from the sample, was isolated with apertures and passed into a
monochromator (HORIBA Jobin Yvon MicroHR, 140 mm focal length) with a visible grating (500 nm blaze
300 groves per mm).  %
The monochromator was set to pass all colors (0 nm, 250 $\mathsf{\mu m}$ slits) to keep the
measurement impulsive.  %
Signal was detected using an InSb photodiode (Teledyne Judson J10D-M204-R01M-3C-SP28).  %
Four wave mixing was isolated from excitation scatter using dual chopping and digital signal
processing.  %

\section{Transmittance and reflectance}  % ========================================================

\autoref{fig:PEDOTPSS_linear} shows the transmission, reflectance, and extinction spectrum of the
thin film used in this work.  %

\clearpage
\begin{figure}
  \centering
	\includegraphics[width=0.5\linewidth]{"PEDOT_PSS/linear"}
	\caption[PEDOT:PSS transmission and reflectance spectra.]{
    Thin film spectra.
    Transmission, reflectance, and extinction spectrum of the thin film used in this work.  %
    Extinction is $\log_{10}{\mathsf{(transmission)}}$.  %
  }
	\label{fig:PEDOTPSS_linear}
\end{figure}
\clearpage

\section{Three-pulse echo spectroscopy}  % ========================================================

Two dimensional $\tau_{21}, \tau_{22^\prime}$ scans were taken for two phase matching
configurations: (1) $k_{\mathsf{out}} = k_1 - k_2 + k_{2^\prime}$ (3PE) and (2) $k_{\mathsf{out}} =
k_1 + k_2 - k_{2^\prime}$ (3PE*).  %
The rephasing and nonrephasing pathways exchange their time dependance between these two
configurations.  %
Comparing both pathways, rephasing-induced peak shifts can be extracted as in 3PE. [CITE]  %
All data was modeled using numerical integration of the Liouville-von Numann equation.  %

Continuously variable ND filters (THORLABS NDC-100C-4M, THORLABS NDL-10C-4) were used to ensure
that all three excitation pulse powers were equal within measurement error.  %

\autoref{fig:PEDOTPSS_mask} diagrams the phase matching mask used in this set of experiments.  %

\begin{figure}
	\includegraphics[width=0.5\linewidth]{"PEDOT_PSS/mask"}
	\caption[PEDOT:PSS 3PE phase matching mask.]{
    Phase matching mask used in this experiment.
    Each successive ring subtends 1 degree, such that the excitation pulses are each angled one
    degree relative to the mask center.
    The two stars mark the two output poyntings detected in this work.
  }
	\label{fig:PEDOTPSS_mask}
\end{figure}

\autoref{fig:PEDOTPSS_raw} shows the ten raw 2D delay-delay scans that comprise the primary dataset
described in this section.  %
The rows correspond to the two phase matching conditions, as labeled.  %

\begin{figure}
	\includegraphics[width=\linewidth]{"PEDOT_PSS/raw"}
	\caption[PEDOT:PSS 3PE raw data.]{
    CAPTION TODO
  }
	\label{fig:PEDOTPSS_raw}
\end{figure}

\subsection{Assignment of zero delay}  % ----------------------------------------------------------

The absolute position of complete temporal overlap of the excitation pulses (zero delay) is a
crucial step in determining the magnitude of th epeak shift and therefore the total rephasing
ability of the material.  %
The strategy for assigning zero delay relies upon the intrinsic symmetry of the two-dimensional
delay space.  %
\autoref{fig:PEDOTPSS_delay_space} labels the six time-orderings (TOs) of the three pulses that are
possible with two delays.  %
The TO labeling scheme follow from a convention first defined my Meyer, Wright and Thompson.
[CITE]  %
[CITE] first discussed how these TOs relate to traditional 3PE experiments.  %
Briefly, spectral peak shifts into the rephasing TOs \RomanNumeral{3} and \RomanNumeral{5} when
inhomogeneous broadening creates a photon echo in the \RomanNumeral{3} and \RomanNumeral{5}
rephasing pathways colored orange in \autoref{fig:PEDOTPSS_delay_space}.  %
For both phase-matching conditions, there are two separate 3PE peak shift traces (represented as
black arrows in \autoref{fig:PEDOTPSS_delay_space}), yielding four different measurements of the
photon echo.  %
Since both 3PE and 3PE* were measured using the same alignment on the same day, the zero delay
position is identical for the four photon echo measurements.  %
We focus on this signature when assigning zero delay---zero is correct only when all four peak
shifts agree.  %
Conceptually, this is the two-dimensional analogue to the traditional strategy of placing zero such
that the two conjugate peak shifts (3PE and 3PE*) agree. [CITE]  %

We found that the 3PEPS traces agree best when the data in \autoref{fig:PEDOTPSS_raw} is offset by
19 fs in $\tau_{22^\prime}$ and 4 fs in $\tau_{21}$.  %
\autoref{fig:PEDOTPSS_processed} shows the 3PEPS traces after correcting for the zero delay
value.  %
The entire 3PEPS trace ($\tau$ vs $T$) is show for regions \RomanNumeral{1}, \RomanNumeral{3}
(purple and light green traces) and \RomanNumeral{5}, \RomanNumeral{6} (yellow and light blue
traces) for the [PHASE MATCHING EQUATIONS] phase matching conditions, respectively.  %
Peak-shift magnitudes were found with Gaussian figs on the intensity level, in accordance with
3PEPS convention. [CITE]
The bottom subplot of \autoref{fig:PEDOTPSS_overtraces} shows the agreement between the four traces
for $T > 50$ fs where pulse-overlap effects become negligible.  %
These pulse-overlap effects cause the 3PEPS at small $T$ even without inhomogeneous broadening.
[CITE]  %
At long $T$, the average static 3PEPS is 2.5 fs.  %

\begin{figure}
	\includegraphics[width=\linewidth]{"PEDOT_PSS/delay space"}
	\caption[PEDOT:PSS 3PE delay space.]{
    CAPTION TODO
  }
	\label{fig:PEDOTPSS_delay_space}
\end{figure}

\begin{figure}
	\includegraphics[width=\linewidth]{"PEDOT_PSS/processed"}
	\caption[PEDOT:PSS 3PE processed data.]{
    CAPTION TODO
  }
	\label{fig:PEDOTPSS_processed}
\end{figure}

\begin{figure}
	\includegraphics[width=\linewidth]{"PEDOT_PSS/overtraces"}
	\caption[PEDOT:PSS 3PE traces.]{
    CAPTION TODO
  }
	\label{fig:PEDOTPSS_overtraces}
\end{figure}

\begin{figure}
	\includegraphics[width=\linewidth]{"PEDOT_PSS/traces"}
	\caption[PEDOT:PSS 3PE traces.]{
    CAPTION TODO
  }
	\label{fig:PEDOTPSS_traces}
\end{figure}

There is a deviation of the TO \RomanNumeral{1}-\RomanNumeral{3} 3PEPS* trace (green line) from the
other traces.  %
It is attributed to a combination of excitation pulse distortions and line shape differences
between OPA1 and OPA2 (see \autoref{fig:PEDOTPSS_linear}) and small errors in the zero delay
correction.  %
\autoref{fig:PEDOTPSS_traces} shows what the four 3PEPS traces would llike like for different
choices of zero-delay.  %
The inset numbers in each subplot denote the offset (from chosen zero) in each delay axis.  %

\subsubsection{Numerical model}  % ----------------------------------------------------------------

We simulated the 3PEPS response of PEDOT:PSS through numerical integration of the Liouville-von
Neumann Equation.  %
Integration was performed on a homogeneous, three-level system with coherent dynamics described by

\begin{equation}
  \frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_2^*},
\end{equation}

where $T_2$, $T_1$ and $T_2^*$ are the net dephasing, population relaxation, and pure dephasing
rates, respectively.  %
A three-level system was used because a two-level system cannot explain the population relaxation
observed at long populations times, $T$.  %
This slow delcay may be the same as the slowly decaying optical nonlinearities in PEDOT:PSS.
[CITE]  %
Inhomogeneity was incorporated by convolving the homogeneous repsonse with a Gaussian distribution
function of width $\Delta_{\mathsf{inhom}}$ and allowing the resultant polarization to interfere on
the amplitude level.  %
This strategy captures rephasing peak shifts and ensemble dephasing.  %

It is difficult to determine the coherence dephasing and the inhomogeneous broadening using 3PE if
both factors are large.  %
To extract $T_2^*$ and $\Delta_{\mathsf{inhom}}$, we focused on two key components of the dataset,
coherence duration and peak shift at large $T$.  %
Since dephasing is very fast in PEDOT:PSS, we cannot directly respove an exponential free induction
decay (FID).  %
Instead, our model focuses on the FWHM of the $\tau$ trace to determine the coherence duration.  %
At $T > 50$ fs, the transient has a FWHM of $\sim$ 80 fs (intensity level).  %
For comparison, our instrumental response is estimated to be 70-90 fs, depending on the exact value
of our puse duration $\Delta_t$ (35-45 fs FWHM, intensity level).  %
An experimental peak shift of 2.5 fs was extracted using the strategy described above.  %
Taken together, it is clear that both pure dephasing and ensemble dephasing influence FWHM and peak
shift so it is important to find valuse of $T_2^*$ and $\Delta_{\mathsf{inhom}}$ that uniquely
constrain the measured response.  %

\begin{figure}
	\includegraphics[width=\linewidth]{"PEDOT_PSS/parametric"}
	\caption[PEDOT:PSS 3PE traces.]{
    CAPTION TODO
  }
	\label{fig:PEDOTPSS_parametric}
\end{figure}

We simulated the $\tau$ trance for a variety of $\Delta_{\mathsf{inhom}}$ and $T_2$ values.  %
The results for $\Delta_t = 40$ fs are summarized in \autoref{fig:PEDOTPSS_parametric}.  %
The lines of constant $T_2$ span from $\Delta_{\mathsf{inhom}} = 0$ (green left ends of curves) to
the limit $\Delta_{\mathsf{inhom}} \rightarrow \infty$ (blue right ends of curves).  %
The lines of constant $T_2$ demonstrate that ensemble dephasing reduces the transient duration and
introduces a peak shift.  %
The influence of inhomogeneity on the observables vanishes as $T_2 \rightarrow \infty$. % 

We preformed simulations analogus to those in \autoref{fig:PEDOTPSS_parametric} for pulse durations
longer and smaller than $\Delta_t = 40$ fs.  %
Longer pulse durations create solutions that do not intersect our experimental point (see
right-most subplot of \autoref{fig:PEDOTPSS_parametric}), but shorter pulse durations do.  %
[TABLE] summarizes the coherence dephasing time and inomogeneous broadening values that best
matches the experimental FWHM and inhomogeneous broadening value for $\Delta_t = 35, 40$ and 45
fs.  %
Clearly, there is no upper limit that can provide an upper limit for the inhomogeneous
broadening.  %

\begin{table}
 \begin{tabular}{ c | c c c }
  $\Delta_t$ (fs) & $T_2$ (fs) & $\hbar T_2^{-1}$ (meV) & $\Delta_{\mathsf{inhom}}$ (meV) \\ \hline
  45 & --- & --- & --- \\
  40 & 10 & 66 & $\infty$ \\
 \end{tabular}
 \caption[]{
   CAPTION TODO
 }
 \label{tab:PEDOTPSS_table}
\end{table}

\begin{figure}
	\includegraphics[width=\linewidth]{"PEDOT_PSS/agreement"}
	\caption[PEDOT:PSS 3PE traces.]{
    CAPTION TODO
  }
	\label{fig:PEDOTPSS_agreement}
\end{figure}

Our model system does ans excellent job of reproducing the entire 2D transient within measurement
error (\autoref{fig:PEDOTPSS_agreement}).  %
The most dramatic disagreement is in the upper right, where the experiment decays much slower than
the simulation.  %
Our system description does not account for signal contributions in TOs \RomanNumeral{2} and
\RomanNumeral{4}, where double quantum coherence resonances are important.  %
In additon, excitation pulse shapes may cause such distortions.  %
Regardless, these contributions do not affect our analysis.  %

Extremely fast (single fs) carrier scattering time constants have also been observed for PEDOT-base
conductive films. [CITES]

\section{Conclusions}  % ==========================================================================

% TODO