@@ -343,3 +343,184 @@ where $\tau_{srh}(N)$ is the Shockley-Read-Hall and $\tau_{a}(N)$ the Auger life
The latter is only taken into account for minority charge carriers.
This model can be selected in the configuration file via the parameter \parameter{recombination_model = "srh_auger"}.
\section{Trapping of Charge Carriers}
\label{sec:trapping}
\apsq provides the possibility to simulate the trapping of charge carriers as a consequence of radiation induced lattice defects.
Several models exist, that quantify the effective lifetime of electrons and holes, respectively, as a function of the fluence and, partially, the temperature.
The fluence needs to be provided to the corresponding propagation module, and is always interpreted as 1-MeVneutron equivalent fluence~\cite{niel}.
The decision on whether a charge carrier has been trapped during a step during the propagation process is calculated similarly to the recombination precesses, described in \ref{sec:recombination}.
It should be noted that the trapping of charge carriers is only one of several effects induced by radiation damage.
In \apsq, these effects are treated independently, i.e. defining the fluence for a propagation module will not affect any other process than trapping.
Until now, no models for de-trapping of charge carriers have been implemented.
In addition, for most modules, the parameters have been extracted under certain annealing conditions.
A dependency on annealing conditions has not been implemented here.
Please refer to the corresponding reference publications for further details.
The following models for trapping of charge carriers can be selected.
\subsection{Ljubljana}
\label{sec:trap:ljubljana}
In the Ljubljana (sometimes referred to as \textit{Kramberger}) model~\cite{kramberger}, the trapping time follows the relation
\begin{equation*}
\tau^{-1}(T) = \beta(T)\Phi_{eq} ,
\end{equation*}
where the temperature scaling of $\beta$ is given as
extracted at the reference temperature of $T_0=\SI{-10}{\celsius}$.
The parameters used in \apsq are
\begin{equation*}
\begin{split}
\beta_{e}(T_0) &= \SI{5.6e-16}{\cm^2 \per\ns}\\
\kappa_{e}&= -0.86 \\
\end{split}
\qquad
\begin{split}
\beta_{h}(T_0) &= \SI{7.7e-16}{\cm^2 \per\ns}\\
\kappa_{h}&= -1.52 \\
\end{split}.
\end{equation*}
While~\cite{kramberger} quotes different values for $\beta$ for irradiation with neutrons, pions and protons, the values for protons have been applied here.
The parameters arise from measurements of the were obtained evaluating current signals of irradiated sensors via light injection at fluences up to $\Phi_{eq}=\SI{2e14}{n_{eq}\per\cm^2}$.
This model can be selected in the configuration file via the parameter \parameter{trapping_model = "ljubljana"}.
\subsection{Dortmund}
The Dortmund (sometimes referred to as \textit{Krasel}) model~\cite{dortmundTrapping}, describes the effective trapping times as
\begin{equation*}
\tau^{-1} = \gamma\Phi_{eq} ,
\end{equation*}
with the parameters
\begin{equation*}
\begin{split}
\gamma_{e}&= \SI{5.13e-16}{\cm^2 \per\ns}\\
\end{split}
\qquad
\begin{split}
\gamma_{h}&= \SI{5.04e-16}{\cm^2 \per\ns}\\
\end{split}.
\end{equation*}
The values have been extracted evaluating current signals of irradiated sensors via light injection at fluences up to $\Phi_{eq}=\SI{8.9e14}{n_{eq}\per\cm^2}$, at a temperature of $\SI{0}{\celsius}$.
No temperature scaling is provided.
Values for neutron and proton irradiation have been evaluated in~\cite{dortmundTrapping}, \apsq makes use of the values for proton irradiation.
This model can be selected in the configuration file via the parameter \parameter{trapping_model = "dortmund"}.
\subsection{CMS Tracker}
This effective trapping model has been developed by the CMS Tracker Group.
It follows the results of~\cite{CMSTrackerTrapping}, with measurements at fluences of up to $\Phi_{eq}=\SI{3e15}{n_{eq}\per\cm^2}$, at a temperature of $\SI{-20}{\celsius}$ and an irradiation with protons.
The interpolation of the results follows the relation
\begin{equation*}
\tau^{-1} = {\beta\Phi_{eq}} + \tau_0^{-1} ,
\end{equation*}
\begin{equation*}
\begin{split}
\beta_{e}(T_0) &= \SI{1.71e-16}{\cm^2 \per\ns}\\
\tau_{0,e}^{-1}&= \SI{-0.114}{\per\ns}\\
\end{split}
\qquad
\begin{split}
\beta_{h}(T_0) &= \SI{2.79e-16}{\cm^2 \per\ns}\\
\tau_{0,h}^{-1}&= \SI{-0.093}{\per\ns}\\
\end{split}.
\end{equation*}
No temperature scaling is provided.
This model can be selected in the configuration file via the parameter \parameter{trapping_model = "cmstracker"}.
\subsection{Mandic}
The Mandi\'{c} model~\cite{Mandic} is an empirical model developed from measurements with high fluences ranging from $\Phi_{eq}=\SI{5e15}{n_{eq}\per\cm^2}$ to $\Phi_{eq}=\SI{1e17}{n_{eq}\per\cm^2}$ and describes the lifetime via
\begin{equation*}
\tau = c\Phi_{eq}^{\kappa}
\end{equation*}
with the parameters
\begin{equation*}
\begin{split}
c_e &= \SI{0.054}{\ns\per\cm^2}\\
\kappa_e &= -0.62 \\
\end{split}
\qquad
\begin{split}
c_h &= \SI{0.0427}{\ns\per\cm^2}\\
\kappa_h &= -0.62 \\
\end{split}.
\end{equation*}
The parameters for electrons are taken from~\cite{Mandic}, for measurements at a temperature of $\SI{-20}{\celsius}$, and the results extrapolated to $\SI{-30}{\celsius}$.
A scaling from electrons to holes was performed based on the default values in Weightfield2~\cite{Weightfield2}.
This model can be selected in the configuration file via the parameter \parameter{trapping_model = "mandic"}.
\subsection{Custom Trapping Model}
Similarly to the mobility models described above, \apsq provides the possibility to use fully custom trapping models.
In order to use a custom model, the parameter \parameter{trapping_model = "custom"} needs to be set in the configuration file.
Additionally, the following configuration keys have to be provided:
\begin{description}
\item[\parameter{trapping_function_electrons}] the formula describing the effective electron trapping time
\item[\parameter{trapping_function_holes}] the formula describing the effective hole trapping time.
\end{description}
The functions defined via these parameters can depend on the local electric field.
In order to use the electric field magnitude in the formula, an \parameter{x} has to be placed at the respective position.
Parameters of the functions can either be placed directly in the formulas in framework-internal units, or provided separately as arrays via the \parameter{trapping_parameters_electrons} and \parameter{trapping_parameters_electrons}.
Placeholders for parameters in the formula are denoted with squared brackets and a parameter number, for example \parameter{[0]} for the first parameter provided.
Parameters specified separately from the formula can contain units which will be interpreted automatically.
\begin{warning}
Note that both fluence and temperature are not inherently available in the custom trapping model, but need to be provided as additional parameters as described above.
\end{warning}
The following configuration parameters replicate the Ljubljana model described in Section~\ref{sec:trap:ljubljana} using a custom trapping model.
author={I. Mandi{\'{c}} and V. Cindro and A. Gori{\v{s}}ek and B. Hiti and G. Kramberger and M. Miku{\v{z}} and M. Zavrtanik and P. Skomina and S. Hidalgo and G. Pellegrini},
title={Measurements with silicon detectors at extreme neutron fluences},
journal={Journal of Instrumentation}
}
@online{weightfield2,
title={Weightfield2: a freeware 2D simulator for silicon and diamond detector},
