From 357568e1fb77afed9dfa203e62da237bf7ce51b3 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 9 Apr 2018 00:24:18 -0500
Subject: 2018-04-09 00:24

---
 PbSe_global_analysis/SI.tex | 279 ++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 279 insertions(+)
 create mode 100644 PbSe_global_analysis/SI.tex

(limited to 'PbSe_global_analysis/SI.tex')

diff --git a/PbSe_global_analysis/SI.tex b/PbSe_global_analysis/SI.tex
new file mode 100644
index 0000000..d3cb783
--- /dev/null
+++ b/PbSe_global_analysis/SI.tex
@@ -0,0 +1,279 @@
+% document
+\documentclass[11pt, full]{article}
+\usepackage[letterpaper, portrait, margin=0.75in]{geometry}
+\usepackage{setspace}
+\usepackage{color}
+
+% text
+\usepackage[utf8]{inputenc}
+\setlength\parindent{0pt}
+\setlength{\parskip}{1em}
+\usepackage{enumitem}
+\renewcommand{\familydefault}{\sfdefault}
+\newcommand{\RomanNumeral}[1]{\textrm{\uppercase\expandafter{\romannumeral #1\relax}}}
+
+% math
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage[cm]{sfmath}
+\usepackage{commath}
+\usepackage{multirow}
+\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}
+
+% graphics
+\usepackage{graphics}
+\usepackage{graphicx}
+\usepackage{epsfig}
+\usepackage{epstopdf}
+\usepackage{xpatch}
+\graphicspath{{./figures/}}
+
+% "S" prefix
+\renewcommand{\theequation}{S\arabic{equation}}
+\renewcommand{\thefigure}{S\arabic{figure}}
+\renewcommand{\thetable}{S\arabic{table}}
+
+% bibliography
+\usepackage[backend=biber, natbib=true, url=false, sorting=none, maxbibnames=99]{biblatex}
+\bibliography{mybib}
+
+% hyperref
+\usepackage[colorlinks=true, linkcolor=black, urlcolor=blue, citecolor=black, anchorcolor=black]{hyperref}
+\usepackage[all]{hypcap}  % helps hyperref work properly
+
+\begin{document}
+\pagenumbering{gobble}
+
+\begin{center}
+	\LARGE
+	
+	Supplementary Information
+
+	Global Analysis of Transient Grating and Transient Absorption \\ of PbSe Quantum Dots
+	
+	\normalsize
+	
+	\textit{Daniel D. Kohler, Blaise J. Thompson, John C. Wright*}
+
+	Department of Chemistry, University of Wisconsin--Madison\\
+	1101 University Ave., Madison, Wisconsin 53706
+\end{center}
+
+\vspace{\fill}
+
+*Corresponding Author \\
+\hspace*{2ex} email: wright@chem.wisc.edu \\
+\hspace*{2ex} phone: (608) 262-0351 \\
+\hspace*{2ex} fax: (608) 262-0381
+
+\pagebreak
+\renewcommand{\baselinestretch}{0.75}\normalsize
+\tableofcontents
+\renewcommand{\baselinestretch}{1.0}\normalsize
+
+\pagebreak
+\setcounter{page}{1}
+\pagenumbering{arabic}
+
+\section{Absorbance}  % ---------------------------------------------------------------------------
+
+\autoref{figure:absorbance} displays the absorbance spectra of the two batches considered in this
+work.
+The lower spectra are plotted relative to each batches 1S peak center,
+emphasizing the peak-shape differences around the 1S.
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{absorbance}
+  \label{figure:absorbance}
+  \caption{Normalized absorbance spectra of the two baches considered in this
+    work. In the upper plot, the spectra are plotted directly against energy. In
+  the lower plot the spectra are plotted relative to the 1S peak center.}
+\end{figure}
+
+To extract peak parameters from the rising continuum absorption, the data was fitted on the second
+derivative level, as described in the supplementary information of \textcite{Czech2015}.
+The script used to accomplish this fit, full parameter output, and additional figures showing the
+separate excitonic features and fit remainder are contained in the supplementary repository, as
+described in \autoref{section:repository}.
+
+Note that the aliquots used for each of the two Batch A experiments were at
+slightly different concentrations, a crucial detail for m-factor corrections
+(see \autoref{section:m-factors}). The two Batch B experiments were done using
+the same aliquot. The absorbance spectrum of each sample is kept in an
+associated ``cal'' directory in the supplementary repository (see \autoref{section:repository}).
+
+
+\pagebreak
+\section{Artifact correction}  % ------------------------------------------------------------------
+
+\subsection{Spectral delay correction}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{spectral_delay_correction}
+  \label{figure:spectral_delay_correction}
+  \caption{Frequency dependent delay calibration using CCl$_4$. (a) Measurement
+    of the pulse overlap position in $\tau_{21}$ space with respect to
+    $\omega_1$ ($\omega_2$ = 7500 cm$^{-1}$). The thick black line shows the
+    center of the temporal profile, as determined by Gaussian fits. (b) Same as
+    (a), but now $\omega_1$ is kept static while $\omega_2$ is scanned. (c) Same
+    as (a), but now active spectral delay corrections have been applied. (d)
+    Two-dimensional frequency-frequecy scan of CCl$_4$ with spectral delay
+    correction applied.}
+\end{figure}
+
+\pagebreak
+\subsection{Power factors}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{power_factors}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+\pagebreak
+\subsection{m factors} \label{section:m-factors}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{m_factors}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+\pagebreak
+\subsection{Processing}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{TG_artifacts}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{TA_artifacts}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+
+\pagebreak
+\section{Auger recombination dynamics}  % ---------------------------------------------------------
+
+%\begin{figure}[!htb]
+%	\centering
+%	\includegraphics[scale=0.5]{"fsb19-3"}
+%	\label{fig:matrix_flow_diagram}
+%	\caption{$S_{\mathsf{TG}}$ measured before and after multiexciton relaxation dynamics.}
+%\end{figure}
+
+Using a Poisson distribution to model the effects here: keep in mind that
+Poisson is only valid when excitation probability is "low". m
+Scholes thinks an equations of motion approach might be more fitting.
+Others have approached this by truncating the Poisson model so that dots are effectively "off" at
+high fluence (this is when pumping the continuum, so no SE contributions from the pump).
+
+According to the Poisson distribution, initial population created by pump is given by
+
+\begin{equation}
+P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}.
+\end{equation}
+
+Assumes all absorption events have equal probability.
+The absorption of the pumped sample will be proportional to
+
+\begin{eqnarray}
+a_{\mathsf{NL}} &=& a_0 \left(1-e^{-\lambda}\sum_{k=1}\frac{\lambda^k}{k!}\right)
++ e^{-\lambda}\sum_{k=1}a_k\frac{\lambda^k}{k!} \\
+&=& a_0 - e^{-\lambda}\sum_{k=1} (a_0 - a_k)\frac{\lambda^k}{k!}.
+\end{eqnarray}
+
+So the difference in the absorption is
+
+\begin{equation}
+S(T=0) = a_{\mathsf{NL}} - a_0 = -e^{-\lambda}\sum_{k=1}(a_0-a_k)\frac{\lambda^k}{k!}.
+\end{equation}
+
+We will assume that absorption is proportional to the number of ground state excitons: $a_k = ck$
+for all $k$.
+Then
+
+\begin{eqnarray}
+S(T=0) &=& ce^{-\lambda}\sum_{k=1}k\frac{\lambda^k}{k!} \\
+&=& c\lambda e^{-\lambda}\sum_{k=0}\frac{\lambda^k}{k!} \\
+&=& c \lambda,
+\end{eqnarray}
+
+and the mean value corresponds to the response (as we expect when the relationship between
+occupation and signal is linear i.e. $<ck> = c\lambda$).
+
+After Auger recombination, the excited state distribution has homogenized to $k=1$.
+Signal is thus given by
+
+\begin{eqnarray}
+S &=& ce^{-\lambda}(a_0-a_1)\sum_{k=1}\frac{\lambda^k}{k!} \\
+&=& ce^{-\lambda}(e^\lambda-1) \\
+&=& c(1-e^{-\lambda}).
+\end{eqnarray}
+
+Previous work has analyzed this.
+
+Comparing the distribution theory with our results.
+
+The mean number of excitations should be proportional to our fluence: $\lambda = mI$.
+This predicts the linear scaling of intensity close to zero delay, and it also predicts the
+exponential saturation observed at longer delays.
+Both observations qualitatively agree with our results.
+Quantitatively, however, our two delays suggest different scaling constants with respect to pump
+fluence: the long-time $m$-value is roughly 40\% larger than the short time scaling.
+This means that our initial scaling underestimates how quickly the band saturates.
+
+Philosophically, there are two problems with this distribution: (1) I should use the equations of
+motion for degenerate pumping, and (2) the pump is filtered by $k$-vector conservation (two pumps).
+My strategy: come up with an expression for the distribution using coupled equations of motion.
+Assume the driven limit, so that a steady state is reached.
+We can account for these issues by utilizing the more general Conway-Maxwell-Poisson distribution.
+
+Estimate spot size as 300 um: 1 um ~ 1 mJ per cm squared.
+
+I think I should revisit the scaling of my exciton signal---I do not expect it to be the same as a
+Poisson distribution because of the stimulated emission channels.
+
+\begin{eqnarray}
+\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
+\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
+\frac{d \rho_{00}}{dt} &=& \cdots
+\end{eqnarray}
+
+\pagebreak
+\section{Supplementary repository} \label{section:repository}  % ----------------------------------
+
+All scripts and raw data used in this work have been uploaded to the Open Science Framework (OSF),
+a project of the Center for Open Science.
+These can be found at DOI: \href{http://dx.doi.org/10.17605/OSF.IO/N9CDP}{10.17605/OSF.IO/N9CDP}.
+
+To download the contents of this repository from your command line...  % TODO
+
+To completely reproduce this work, simply execute \texttt{./run.sh all} from your terminal.
+You will require the following:
+
+\begin{enumerate}
+  \item python 3.6
+  \item WrightTools VERSION TODO (and dependencies)
+  \item latex
+\end{enumerate}
+
+You can replace \texttt{all} with one of \texttt{data}, \texttt{simulations}, \texttt{figures},
+or \texttt{documents}.
+
+Otherwise, the OSF repository attempts to be generally self-explanatory.
+README files and comments are used to explain what was done.
+
+\pagebreak
+\printbibliography
+
+\end{document}
-- 
cgit v1.2.3

