From c15660a15d2ac72ad8b385ac9d54fbc0e257af65 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Sun, 15 Apr 2018 14:12:07 -0500
Subject: 2018-04-15 14:12

---
 PEDOT_PSS/chapter.tex | 295 +++++++++++++++++++++++++++++---------------------
 1 file changed, 169 insertions(+), 126 deletions(-)

(limited to 'PEDOT_PSS/chapter.tex')

diff --git a/PEDOT_PSS/chapter.tex b/PEDOT_PSS/chapter.tex
index 10a7472..29b411c 100644
--- a/PEDOT_PSS/chapter.tex
+++ b/PEDOT_PSS/chapter.tex
@@ -1,4 +1,5 @@
-\chapter{PEDOT:PSS}  \label{cha:pps}
+\chapter{Measurement of ultrafast dynamics within PEDOT:PSS using three-pulse photon echo
+  spectroscopy}  \label{cha:pps}
 
 \textit{This Chapter presents content first published in \textcite{HorakErikH2018a}.
   The authors are:
@@ -32,32 +33,32 @@ As a polymer, PEDOT:PSS implicitly contains a large amount of structural inhomog
 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)
+even more structural inhomogeneity.
+In summary, PEDOT:PSS has a complex, nested microstructure.  %
+From smallest to largest:
+\begin{ditemize}
+  \item PEDOT oligomers (6---18-mers) [CITE 14]
   \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
+    films to as many as 13---14 oligomers for more conductive solvent treated films [CITE 15]
+  \item nanocrystallites then arrange into globular conductive particles in a pancake-like shape
+    [CITE 16, 17]
   \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.  %
+    nanofibril geometry akin to a string of pearls [CITE 6, 21]
+  \item nanofibrils interweave to form thin films, with PSS capping layer at surface [CITE 19, 22]
+\end{ditemize}
+
+In order to be conductive, PEDOT:PSS needs to have good spatial and energetic overlap between
+electronic states throughout the thin film.  %
+The exact energetics and dynamics of these electronic states, then, is a crucial piece of
+information needed to understand the mechanism of conductivity in PEDOT:PSS.  %
+The electronic states responsible for conductivity have very broad and featureless transitions in
+the mid infrared.  %
+Bulk, linear spectroscopy cannot tease out the relative contribution of homogeneous and
+inhomogeneous broadening in the breadth of these transitions.  %
+Multidimensional spectroscopy is able to tease these two broadening mechanisms apart.  %
+
+In this chapter, I report on my usage of three pulse echo (3PE) spectroscopy to constrain
+homogeneous and inhomgeneous linewidths in PEDOT:PSS.  %
 
 \section{Methods}  % ==============================================================================
 
@@ -86,62 +87,74 @@ Signal was detected using an InSb photodiode (Teledyne Judson J10D-M204-R01M-3C-
 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*).  %
+\autoref{pps:fig:mask} diagrams the phase matching mask used in this set of experiments.  %
 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]  %
+Comparing both pathways, rephasing-induced peak shifts can be extracted as in 3PE.
+\cite{WeinerAM1985a}  %
 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.]{
+	\caption[Phase matching mask for 3PE, 3PE*.]{
     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}
+	\label{pps:fig:mask}
 \end{figure}
 
-\autoref{fig:PEDOTPSS_raw} shows the ten raw 2D delay-delay scans that comprise the primary dataset
+\section{Results}  % ==============================================================================
+
+\autoref{pps:fig:linear} shows the transmission, reflectance, and extinction spectrum of the
+thin film used in this work.  %
+The region under investigation is shaded green.  %
+Reflectance is remarkably low across the visible and near infrared, and transmission does not
+change much at all over the region under investigation.  %
+
+\autoref{pps:fig: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.  %
+The data is presented on the intensity level, as raw as possible.  %
+The five repetitions of each experiment are truly remarkably similar, showing that no damage was
+being done during the experiment.  %
+Each row is normalized to the same factor, showing the remarkable \emph{quantitative} agreement
+between scans.  %
+In total, these 10 scans comprise roughly eight hours of continuous illumination for this
+sample.  %
+
+\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{pps:fig:linear}
+\end{figure}
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{"PEDOT_PSS/raw"}
-	\caption[PEDOT:PSS 3PE raw data.]{
-    CAPTION TODO
+	\caption[Raw 3PE data.]{
+    Raw ultrafast data.
+    Unprocessed two-dimensional delay-delay plots.
+    Each discrete acquisition is plotted as a single colored pixel.
+    Grey pixels correspond to negative results, which appear in the no-signal regions due to random
+    noise.
   }
-	\label{fig:PEDOTPSS_raw}
+	\label{pps:fig:raw}
 \end{figure}
 
+\section{Discussion}  % ===========================================================================
+
 \subsection{Assignment of zero delay}  % ----------------------------------------------------------
 
 The absolute position of complete temporal overlap of the excitation pulses (zero delay) is a
@@ -149,81 +162,88 @@ crucial step in determining the magnitude of th epeak shift and therefore the to
 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
+\autoref{pps:fig: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.  %
+\cite{MeyerKentA2004a}  %
+\textcite{KohlerDanielDavid2014a} 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}.  %
+rephasing pathways colored orange in \autoref{pps:fig: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
+black arrows in \autoref{pps:fig: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]  %
+that the two conjugate peak shifts (3PE and 3PE*) agree. \cite{WeinerAM1985a}  %
 
-We found that the 3PEPS traces agree best when the data in \autoref{fig:PEDOTPSS_raw} is offset by
+We found that the 3PEPS traces agree best when the data in \autoref{pps:fig: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
+\autoref{pps:fig: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}
+The entire 3PEPS trace ($\tau$ vs $T$) is shown 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.  %
+traces) for the $\vec{k_1} - \vec{k_2} + \vec{k_{2^\prime}}$ and $\vec{k_1} + \vec{k_2} -
+\vec{k_{2^\prime}}$ 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
+3PEPS convention. \cite{WeinerAM1985a}  %
+The bottom subplot of \autoref{pps:fig: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]  %
+\cite{WeinerAM1985a}  %
 At long $T$, the average static 3PEPS is 2.5 fs.  %
 
+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{pps:fig:linear}) and small errors in the zero delay
+correction.  %
+\autoref{pps:fig: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.  %
+
 \begin{figure}
 	\includegraphics[width=\linewidth]{"PEDOT_PSS/delay space"}
-	\caption[PEDOT:PSS 3PE delay space.]{
-    CAPTION TODO
+	\caption[3PE, 3PE* delay space.]{
+    Representation of 2D delay space.
+    Representation of symmetry between the two phase-matched experiments performed in this work.
+    In each two-dimensional delay space, the six TOs are labeled.
+    Pathways III and V are rephasing (orange), all other pathways are non-rephasing (blue).
+    Thick black arrows are drawn along the $\tau$ trace for constant T = 125 fs, with arrowheads
+    pointing in the direction of shift for positively correlated systems.
+    The region with signal above 10\% (processed dataset, amplitude level) is shaded to guide the
+    eye.
   }
-	\label{fig:PEDOTPSS_delay_space}
+	\label{pps:fig:delay_space}
 \end{figure}
 
 \begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT_PSS/processed"}
-	\caption[PEDOT:PSS 3PE processed data.]{
-    CAPTION TODO
+	\includegraphics[width=\linewidth]{"PEDOT_PSS/traces"}
+	\caption[Delay offsets.]{
+    Delay offsets.
+    Comparison between 3PEPS traces at different delay offsets.
+    Inset is D1, D2 offset in fs.
   }
-	\label{fig:PEDOTPSS_processed}
+	\label{pps:fig:traces}
 \end{figure}
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{"PEDOT_PSS/overtraces"}
-	\caption[PEDOT:PSS 3PE traces.]{
-    CAPTION TODO
+	\caption[Peak shift traces drawn in delay space.]{
+    3PEPS traces.
+    Fully processed 2D delay scans (upper) and 3PEPS traces for both rephasing pathways and both
+    phase matching conditions.
+    The 3PEPS traces are shown mapped onto the 2D space (upper) and overlaid for comparison
+    (lower).
   }
-	\label{fig:PEDOTPSS_overtraces}
+	\label{pps:fig: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}  % ----------------------------------------------------------------
+\subsection{Numerical model}  % -------------------------------------------------------------------
 
 We simulated the 3PEPS response of PEDOT:PSS through numerical integration of the Liouville-von
 Neumann Equation.  %
@@ -238,7 +258,7 @@ 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]  %
+\cite{MeskersStefanCJ2003a}  %
 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.  %
@@ -259,64 +279,87 @@ Taken together, it is clear that both pure dephasing and ensemble dephasing infl
 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 results for $\Delta_t = 40$ fs are summarized in \autoref{pps:fig: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
+We preformed simulations analogus to those in \autoref{pps:fig: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.  %
+right-most subplot of \autoref{pps:fig: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.  %
 
+Our model system does ans excellent job of reproducing the entire 2D transient within measurement
+error (\autoref{pps:fig: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.  %
+
+\begin{figure}
+	\includegraphics[width=\linewidth]{"PEDOT_PSS/parametric"}
+	\caption[3PEPS parameter space.]{
+    3PEPS parameter space.
+    Interplay of pure and ensemble dephasing on the coherent transient duration and the peak shift
+    value for the three pulse-widths considered in \autoref{pps:tab:table}.
+    Red lines represent the parameters for constant values of $T_2$ and varying amounts of
+    $\Delta_{\text{inhom}}$.
+    The domain of possible observables is bounded (blue hash for $\Delta_{\text{inhom}} \rightarrow
+    \inf$, green hash for $\Delta_{\text{inhom}} = 0$).
+    Also shown is the measured FWHM for the PEDOT:PSS thin film (star).
+  }
+	\label{pps:fig:parametric}
+\end{figure}
+
 \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
+ \caption[Fitted parameters.]{
+   Fitted parameters for the coherent transient.
+   The FWHM of the homogeneous line shape is $\hbar T_2^{-1}$.
  }
- \label{tab:PEDOTPSS_table}
+ \label{pps:tab:table}
 \end{table}
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{"PEDOT_PSS/agreement"}
-	\caption[PEDOT:PSS 3PE traces.]{
-    CAPTION TODO
+	\caption[Agreement between simulation and experiment.]{
+    Agreement between simulation and experiment.
+    Experiment and simulation in the full 2D representation (left) and transient grating slices
+    (right), for both phase matching conditions (top and bottom).
+    The identity of each slice can be inferred from its color.
+    In this case the displayed simulation is for $\Delta_t=35$ fs, with the appropriate $T_2$ and
+    $\Delta_{\text{inhom}}$ as seen in \autoref{pps:tab:table}.
+    Simulatinos for other pulse-widths look very similar.
   }
-	\label{fig:PEDOTPSS_agreement}
+	\label{pps:fig: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
\ No newline at end of file
+To asses homogeneous and inhomogeneous linewidth, we performed ultrafast four wave mixing
+spectroscopy on a drop-cast PEDOT:PSS thin film.  %
+Under Redfield theory, the homogeneous linewidth of any transition is determined by pure dephasing
+and population relaxation \cite{SkinnerJL1988a}, although ensemble dephasing can become relevant
+for very inhomogeneously broadened systems.  %
+Three-pulse photon echo (3PE) analysis can distinguish between homogeneous and inhomogeneous
+broadening. \cite{WeinerAM1985a}  %
+We collecte the transient grating population relaxation trace and 3PE traces and find that the net
+dephasing and population relaxation are both fast, comparable to our pulse width.  %
+Through numerical modeling, we extract a population time of 80 fs, a homogeneous dephasing time of
+$<18$ fs ($>73$ meV), and an inhomogeneous broadening factor of $>43$ meV.  %
+Extremely fast (single fs) carrier scattering time constants have also been observed for PEDOT-base
+conductive films. \cite{ChangYunhee1999a, ChoShinuk2005a, ZhuoJingMei2008a}  %
\ No newline at end of file
-- 
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