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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} |