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eqns.tex
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eqns.tex
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\documentclass[12pt,twoside]{report}
\usepackage{kjfsty}
\begin{document}
\begin{equation}
\tau_{therm}(T) = w \sqrt{\frac{2\pi m^*}{k_B T}} e^{\frac{\EE_{3c}}{k_B T}}
\end{equation}
\begin{equation}
g_{u\ell} = \frac{c \Gamma_{u\ell}}{N_p n_\textit{eff}} \frac{2 q^2 \EE_{u\ell} z_{u\ell}^2}{h c \epsilon_0 n_\textit{eff} \; \delta\!\EE_{u\ell} L_p}
\end{equation}
\begin{equation}
\tau_{ph,u\ell} = \frac{n_\textit{eff}}{c \left(-\frac{1}{L} \ln(R) + \alpha_{w,u\ell} \right)}
\end{equation}
\begin{equation}
\frac{1}{\tau_{u\ell}(T)} = \frac{1}{\tau_{u\ell}(T=0 \tn{K})} \left(1+\frac{2}{e^{\frac{\hslash\omega_{LO}}{k_B T}}-1}\right)
\end{equation}
\begin{equation}
\eta_{inj}(T) = \eta_{inj} \left(1+\frac{2}{e^{\frac{\hslash\omega_{LO}}{k_B T}}-1}\right)
\end{equation}
\begin{equation}
\begin{split}
\frac{d n_5}{d\!t} &=\left(1-\eta_\textit{inj}\right)\frac{J}{q}-n_5\left(\frac{1}{\tau_{54}}+\frac{1}{\tau_{53}}+\frac{1}{\tau_{52}}+\frac{1}{\tau_{51}}\right) -\frac{n_5}{\tau_{therm}} - S_{54} g_{54} \left(n_5-n_4\right)\\
\frac{d n_{4k}}{d\!t} &= \eta_\textit{inj} \frac{J}{q} + \frac{n_5}{\tau_5}-\frac{n_4}{\tau_{k4}}-S_{42} g_{42}\left(n_{4k}-n_{2k}\right)\\
\frac{d n_4}{d\!t} &= \frac{n_{4k}}{\tau_{4k}} - n_4 \left(\frac{1}{\tau_{43}}+\frac{1}{\tau_{42}}+\frac{1}{\tau_{41}}\right) + S_{54} g_{54} \left(n_5-n_4\right)\\
\frac{d n_3}{d\!t} &= \frac{n_5}{\tau_{53}}+\frac{n_4}{\tau_{43}} -n_3 \left(\frac{1}{\tau_{32}}+\frac{1}{\tau_{31}}\right)\\
\frac{d n_{2k}}{d\!t} &= \frac{n_5}{\tau_{52}}+\frac{n_4}{\tau_{42}}+\frac{n_3}{\tau_{32}}-\frac{n_{2k}}{\tau_{2k}} +S_{42} g_{42} \left(n_{4k}-n_{2k}\right)\\
\frac{d n_2}{d\!t}&= \frac{n_{2k}}{\tau_{2k}}-\frac{n_2}{\tau_{21}}+\frac{n_{inj} e^{-\frac{\Delta_{inj}+\EE_{21}}{k T}}}{\tau_{21}}\\
\frac{d S_{54}}{d\!t} &= \left(N_p g_{54}\left(n_5-n_4\right)-\frac{1}{\tau_{ph1}}\right) S_{54}\\
\frac{d S_{42}}{d\!t} &= \left(N_p g_{42}\left(n_{4k}-n_{2k}\right)-\frac{1}{\tau_{ph2}}\right) S_{42}
\end{split}
\end{equation}
\begin{table}[htbp]
\caption{Parameters used in the \emph{k}-space laser rate equation model.}
\begin{tabular}{ c c p{0.5in} c c }
\toprule
$R$ & 0.27 & & $L_p$ & 512~\AA\\
width & 12~\um & & $N_p$ & 40\\
$L$ & 1.5~mm & & $\Delta_\textit{inj}$ & 27.3~meV\\
$\hslash\omega_{LO}$ & 34~meV & & $n_{inj}$ & $1.6\times10^{11}$~cm\sup{-3}\\
$\EE_{21}$ & 67.4~meV & & $\EE_{3c}$ & 60~meV\\
$w$ & 197~\AA & & $m^*$ & $0.04 m_e$\\
\hline
$\Gamma_{54}$ & 0.60 & & $\Gamma_{42}$ & 0.67\\
$\alpha_{w54}$ & 7.4~cm\sup{-1} & & $\alpha_{w42}$ & 5.2~cm\sup{-1}\\
$\EE_{54}$ & 128~meV & & $\EE_{42}$ & 151~meV\\
$z_{54}$ & 23~\AA & & $z_{42}$ & 13~\AA\\
$\delta\!\EE_{54}$ & 24~meV & & $\delta\!\EE_{42}$ & 15~meV\\
\hline
\end{tabular}
\label{chpt3:tab1}
\end{table}
\begin{table}[htbp]
\caption{Calculated LO phonon lifetimes used in the rate equation model.}
\setstretch{1.2}
\begin{tabular}{c c | c c c c}
\toprule
\multicolumn{2}{c}{\multirow{2}{*}{\setstretch{0.9} \centering\parbox{0.2in}{$\tau_{u\ell}$\\(ps)}}} & \multicolumn{4}{c}{upper state $u$} \\
\multicolumn{2}{c|}{} & 5 & 4 & 3 & 2 \\
\cline{2-6}
\multirow{4}{*}{\rotatebox{90}{\mbox{lower state $\ell$}}} & 4 & 4.0 & & \\
& 3 & 4.2 & 1.2 & & \\
& 2 & 7.0 & 1.9 & 3.0 & \\
& 1 & 9.3 & 10.6 & 3.5 & 7.2\\
\hline
\end{tabular}
\label{chpt3:tab2}
\end{table}
\end{document}