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\begin{document}

%Title of paper
\title{Semiconductor saturable absorber model with circuitry}

\author{ede}
% \affiliation{Humboldt-Universit\"at zu Berlin, Institut f\"ur Physik, Newtonstrasse 15, D-12489 Berlin, Germany}

\date{\today}


\maketitle
%

\section{Problem description}
%----------------------------

A short semiconductor section negatively biased with $U_0<0$ is often used as a saturable absorber. A pulse train $P(t)$ is injected causing an ac photocurrent. In order to model the behavior of the absorber under these conditions, its  series resistivity $R$, wire inductivity $L$, and parasitic capacity $C$ have to be considered. 

\section{Model components}
%---------------------------

I use the following circuit schema  of semiconductor lasers, as given in the Petermann-Book. \\

~ \hfill \fig{0.3}{schaltbild} \hfill ~ \\

The basic voltage-current relations of a resistivity, inductivity, and capacity are
%
\begin{eqnarray} \label{rlc}
U_R = RI_R, ~~~
U_L = L \dot{I}_L, ~~~ \text{and} ~~~ 
C \dot{U}_C = I_C,
\end{eqnarray}
%
respectively. The intrinsic voltage-current relation of the semiconductor absorber can be modeled by the Fermi voltage
%
\begin{eqnarray}
U_F(N)=U_g+U_T \ln\left(\frac{N}{n_t}\right), ~~~
eU_g=E_g, ~~~
eU_T=kT,
\end{eqnarray}
%
with temperature $T$, Boltzmann constant $k$, band gap $E_g$ of the active layer, transparency concentration $n_t$. This simple logarithmic Boltzmann law is valid as long as $N$ does not exceed $n_t$, which holds in the absorber regime of operation (but no more when operating the device as a laser). Furthermore, it has been used the reasonable assumption that $U_F=U_g$ holds at transparency. 

The Fermi voltage depends on the excess carrier density, which in turn obeys the rate equation
%
\begin{eqnarray} \label{rate}
\dot{N}=\frac{I}{eV}-r(N)-G(N)S(t).
\end{eqnarray}
%
$I$: injection current (positive in forward direction), $V$: volume of the active layer, $r(N)$: nonstimulated recombination rate, $G(N)$: gain function (per unit time), $S(t)\sim P(t)$: average photon density in the active layer. 
The recombination rate can be modeled by
%
\begin{eqnarray}
r(N)=\frac{N-n_i}{\tau }, ~~~ \text{with} ~ n_i=n_t \exp\left(-\frac{U_g}{U_T}\right)
\end{eqnarray}
%
being the intrinsic concentration of the active material. The $n_i$-contribution to $r$ is usually neglected in lasers, because it is extremely small. When operating with reverse voltage, however, $N$ can become even smaller, in which case this contribution gives the saturation value of the reverse dark current
%
\begin{eqnarray}
I_s = \frac{eVn_i}{\tau }.
\end{eqnarray}
%
For the gain function, the logarithmic model
%
\begin{eqnarray}
G(N)=G'n_t \ln\left(\frac{N}{n_t}\right) \sim U_F(N)-U_g
\end{eqnarray}
seems to be more reasonable than the linear one, because it provides the very high absorption values at reverse voltages $U_F<0$. 
%

\section{Full model}
%--------------------

Applying Kirchhoff's fundamental node and mash laws to the given quasistationary\footnote{radiation losses neglected} electrical circuit, all the given model components can be combined. When using those quantities as dynamical variables whose derivatives appear in Eqs.~(\ref{rlc}) and (\ref{rate}), one arrives at
%
\begin{eqnarray}
\dot{N}   &=& \frac{I}{eV}-r(N)-G(N)S(t), ~~~~~~ I=\frac{U_C-U_F(N)}{R}\\
\dot{U}_C &=& \frac{I_L-I}{C},\\
\dot{I}_L &=& \frac{U_0-U_C}{L}.
\end{eqnarray}
%
In my opinion, this is the appropriate model for a short semiconductor saturable absorber. Before a further general discussion or application, I need typical values for the parameters $C,L$. I know only $R\approx 2 \Omega$mm/length. 

\end{document}


\section{Stationary case}
%------------------------

Setting $S(t)=S=$const. and all temporal derivates to zero, one gets $U_C=U_0, I_L=I$, and
%
\begin{eqnarray}
\dot{N}=\frac{U_0}{eVR}-\left[\frac{U_F(N)}{eVR}+r(N)\right]-G(N)S=0
\end{eqnarray}
%
