\chapter{PEDOT:PSS}

\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{dfigure}
  \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{dfigure}
\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{dfigure}
	\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{dfigure}

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

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

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

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

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

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

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{dtable}
 \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{dtable}

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

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{Frequency-domain transient grating spectroscopy}  % --------------------------------------

This section describes preliminary, unpublished work accomplished on PEDOT:PSS.  %