\section{Pixel charge diffusion simulation}
\label{sec:simulation}

%\subsection{Expected signal characteristics}
%\label{sec:simulation:expected}

The signal from a minimum ionising particle (MIP)
arises as the particle passes through the sensor.
%A relativistic charged particle
%will create an average of 80 electron-hole pairs per
%micron in silicon, although there are fluctuations
%around this value, as described by the Landau 
%function.
Physically,
electron-hole pairs are created throughout the silicon
sensor. Within (or near) the epitaxial layer
the liberated electrons are
able to diffuse to the collecting diodes and
hence be seen as signal. 
Some electrons from the 
substrate close to the epitaxial layer can also
diffuse into the epitaxial layer and so be collected.
However, electrons in the deep P-well implant will
mostly be lost.
Overall, for a MIP
at normal incidence to a sensor with
a 12\,$\mu$m
thick epitaxial layer and deep P-well,
around 1100\,$e^-$ will be available
to contribute to the signal. Due to diffusion from
the substrate, a 5\,$\mu$m
thick epitaxial layer will yield around half this value.
Once liberated, the charge diffuses and may be absorbed by 
the collection
diodes of the neighbouring pixel. In addition, other N-well
structures can collect charge and hence charge will be
lost in terms of being signal. This last effect is the
reason for which the deep P-well implant was developed.

As described in Section~\ref{sec:testp:calibration},
for calibration,
the sensor was exposed to an intense $^{55}$Fe source.
This isotope produces gamma radiation including prominent
K$_\alpha$ lines with photon
energies close to 5.9\,keV. There is also a lower rate of
K$_\beta$ photons, with
energies close to 6.5\,keV.
Photons of these energies can interact in
the silicon, again causing ionisation, and will liberate
all the signal charge within a small O(1\,$\mu$m$^3$) volume.
In most cases when the photons convert in the epitaxial layer,
a significant fraction of the charge will diffuse 
out of the pixel and be absorbed in the neighbouring pixels,
being lost as signal to the hit pixel. Hence only a small
fraction of the signal will be collected by the diode and
a spread
of signal sizes will be observed, depending on the details
of the charge diffusion.
However, a
small number of the photons will interact
directly inside the signal diodes themselves.
In this case,
all the charge liberated, around 1620\,$e^-$,
will be collected by the diode and a fixed signal size
will be observed.

Another difference between an interaction in the diode and elsewhere
in the epitaxial layer is the speed of charge collection.
Charge deposited in the diode should be collected quickly,
of the order of 1\,ns, while the diffusion of charge from the
epitaxial layer takes of order 100\,ns. The diffusion time
is convoluted with the preamplifier response and 
signal shaping circuit, which results in a slower rise time and
lower peak for longer collection times.
Figure~\ref{fig:simulation:time} shows the simulated pre-shaper circuit
response resulting from the different signal time characteristics.
A diode interaction signal is found to be around 15\% larger 
than a more general epitaxial layer interaction, for the same
total charge collected.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=8cm]{simulation/pulse_exp_shaper3b.png}
\caption{\sl 
Simulated reponse of pre-shape circuit to pulses of the same magnitude
but with varying collection times. The pulse charge is injected at 13\,$\mu$s on
the time axis and
collected with an exponential rise, with time constants of 0\,ns (black, dashed),
25\,ns (green), 50\,ns (blue) and 75\,ns (red). Zero corresponds to
a diode interaction while a typical epitaxial interaction corresponds
to the 50 or 75\,ns curve.
}
\label{fig:simulation:time}
\end{center}
\end{figure}


\subsection{Detailed simulation}
\label{sec:simulation:sentaurus}

To estimate the signal spread quantitatively, a simulation of
the sensor pixel was performed using the 
Sentaurus~\cite{ref:simulation:sentaurus} package. The pixel
GDS design file was used as input to the simulation, 
ensuring the details of
the pixel were correctly simulated. 
Figure~\ref{fig:simulation:gds} shows the pre-sampler
pixel simulated. The pre-shaper diffusion is expected to
be very similar as the diodes are placed identically in
the two designs; this is particularly true for the deep P-well
sensors as very little charge is expected to be absorbed by
the circuit N-wells. Only modelling of the diffusion of MIP signals
was performed with this detailed simulation.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=7cm]{simulation/SentaurusNODPW.jpg}
\includegraphics[width=7cm]{simulation/SentaurusDPW.jpg}
\caption{\sl 
Diagram of the GDS simulated volume
for the pre-sampler pixel charge diffusion modelling;
without deep P-well (left) and with deep P-well (right).
%INVERT COORDINATES IN ONE PLOT TO MATCH.
}
\label{fig:simulation:gds}
\end{center}
\end{figure}

Charge equivalent to
that expected for a MIP passing perpendicularly through the epitaxial
layer was deposited at various locations
within the pixel and then its diffusion was simulated to investigate where
it was absorbed or lost. A $3\times 3$ array of pixels
was simulated to allow for neighbour pixel charge collection.
The simulation assumed a 15\,$\mu$m epitaxial layer and
simulated the volume
from the sensor surface down to 32\,$\mu$m 
below the surface ($150 \times 150 \times 32$\,$\mu$m$^3$ in total).
A correction is made for all results shown below to account for
the difference between the 15\,$\mu$m epitaxial thickness assumed
in this simulation and the 12\,$\mu$m eventually implemented in the
fabricated sensor.

The simulation was CPU-limited so only a small number of MIP
impact positions were modelled. Because of the approximate
eight-fold symmetry of the sensor, all positions
simulated were within a triangle with the pixel centre,
corner, and side centre as its three corners. Within this
triangle, the positions simulated were spaced 5\,$\mu$m
apart, resulting in 21 points in total. Using the 
approximate pixel symmetry,
these could be translated and rotated to give a 5\,$\mu$m
regular array across the whole of the central pixel.
Figure~\ref{fig:simulation:points} shows the layout of the
21 points simulated.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=6cm]{simulation/points2.png}
\caption{\sl 
Layout %and numbering 
of the 21 points (black circles) used for MIP charge deposits in
the Sentaurus simulation.
The red circles indicate the four diode positions.
}
\label{fig:simulation:points}
\end{center}
\end{figure}

The simulation
results are shown in figure~\ref{fig:simulation:simcharge}
for sensors with and without the deep P-well implant.
Here the fraction of the signal charge predicted in all nine pixels
of the $3 \times 3$ array
is shown. It is seen that the signal in all cases is predicted to 
be significantly larger with the deep P-well implant. In particular,
in the central pixel, the fraction of charge seen depends on the
position of the incident particle. The signal varies
between 20\% and 50\% with the deep P-well implant, with an average
over the pixel of 30\%. However, without the deep P-well implant,
the central pixel signal can be less than 1\% and only reaches a maximum
of 30\% when near the diode. The average for this case is less
than 10\%.
Similar large differences are seen for the neighbouring
pixels also.
%
\begin{figure}[ht!]
\begin{center}
%\includegraphics[width=7cm]{simulation/21point_GDSnoDPW_TCAD.png}
%\includegraphics[width=7cm]{simulation/21point_GDSDPW_TCAD.png}
\includegraphics[width=7cm]{simulation/gdsNodpwMip.png}
\includegraphics[width=7cm]{simulation/gdsDpwMip.png}
\caption{\sl 
Sentaurus simulation of %the fraction of 
the MIP charge (in unit of $e^-$) 
seen as signal
for no deep P-well (left)
and deep P-well (right), as a function of the MIP impact position
relative to the centre of the pixel.
The simulation assumed a 15\,$\mu$m thick epitaxial layer, but the
values shown are scaled to those expected for a 12\,$\mu$m layer.
%The colour code for each of
%the nine pixels (``cells'') is shown on the plots,
%where Cell 5 is the central pixel.
%TO DO: CLEAN UP AND MAKE Y AXES THE SAME
}
\label{fig:simulation:simcharge}
\end{center}
\end{figure}

\subsection{Diffusion simulation}
\label{sec:simulation:diffusion}

The Sentaurus simulation was too time-consuming to model a
large number of charge deposition points,
so a simplified model was used for studying bigger samples of
points and for modelling the $^{55}$Fe calibration response. 
A simple diffusion equation
\begin{displaymath}
\frac{\partial \rho}{\partial t} = - D \nabla^2 \rho
\end{displaymath}
was numerically solved, using an forward-time/centre-difference (FTCD)
method, to model the
charge density $\rho$ within the epitaxial layer.
The diffusion coefficient $D$,
and parameters to model the charge collection with and
without deep P-well implants at the surface, were adjusted
to agree with the results of the Sentaurus simulation for
the above 21 points. The simulation was extended to a
$5 \times 5$ array of pixels and both a 5 or 12\,$\mu$m epitaxial
depth using a 1\,$\mu$m grid. Thus a total of 
$250 \times 250 \times 5$ or $250 \times 250 \times 12$
volume elements were used.
The MIP deposits were again simulated within the same triangle
as shown in figure~\ref{fig:simulation:points} but with 1\,$\mu$m
spacing, giving a total of 351 points. 
Figure~\ref{fig:simulation:diffcharge} shows
the equivalent distributions to figure~\ref{fig:simulation:simcharge}
for MIP deposits resulting from
the simple diffusion model, showing the much finer spacing.
Only the central $3 \times 3$ pixels are shown, although the
simulation volume extended beyond this region.
Figure~\ref{fig:simulation:diffcharge05} shows the same but for
the 5\,$\mu$m epitaxial layer thickness.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=7cm]{simulation/nodpwMip.png}
\includegraphics[width=7cm]{simulation/dpwMip.png}
\caption{\sl 
Simple diffusion simulation of %the fraction of 
the amount of a MIP charge  (in unit of $e^-$) seen as signal
for no deep P-well (left)
and deep P-well (right), as a function of the MIP impact position
relative to the centre of the pixel. The epitaxial layer thickness
is 12\,$\mu$m in these cases.
}
\label{fig:simulation:diffcharge}
\end{center}
\end{figure}

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=7cm]{simulation/nodpw05Mip.png}
\includegraphics[width=7cm]{simulation/dpw05Mip.png}
\caption{\sl 
Simple diffusion simulation of %the fraction of 
the amount of a MIP charge  (in unit of $e^-$) seen as signal
for (left) no deep P-well
and (right) deep P-well, as a function of the MIP impact position
relative to the centre of the pixel. The epitaxial layer thickness
is 5\,$\mu$m in these cases.
}
\label{fig:simulation:diffcharge05}
\end{center}
\end{figure}

For modelling the response to the $^{55}$Fe calibration photons,
the charge deposits
need to be simulated not only over the area of the pixel but also
at various depths in the epitaxial layer.
The simple diffusion model was
used to estimate the spectrum of the charge collected for
randomly positioned charge deposits (as expected for $^{55}$Fe)
throughout the volume of the epitaxial layer. Examples
to illustrate the depth dependence are shown in
figure~\ref{fig:simulation:diffdepth}.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=4.5cm]{simulation/zDepth00.png}
\includegraphics[width=4.5cm]{simulation/zDepth06.png}
\includegraphics[width=4.5cm]{simulation/zDepth11.png}
\caption{\sl 
Simple diffusion simulation of the fraction of the charge seen as signal
for a deep P-well pixel, as a function of the $^{55}$Fe photon impact position
relative to the centre of the pixel at three different depths in the
epitaxial layer. Left: furthest from the diodes, centre: in the middle, right: nearest to the diodes.
The epitaxial layer thickness is 12\,$\mu$m in these cases.
}
\label{fig:simulation:diffdepth}
\end{center}
\end{figure}


As can be seen in figure~\ref{fig:simulation:diffdepth},
the simulation
indicates that for the deep P-well cases,
there is a broad ``plateau'' near the centre of
the pixel, effectively independent of epitaxial layer depth, 
where an approximately
constant fraction of the deposited charge is collected. This
spectrum of observed charge is shown in 
figures~\ref{fig:simulation:diffspectrum}
and \ref{fig:simulation:diffspectrum05},
which show the constant fraction is between 0.3 and 0.4,
meaning that a peak corresponding to around 
30-40\% of the $^{55}$Fe energy would be observed.
This fraction is only weakly dependent on epitaxial layer
depth.
(Note, the full energy peak resulting from hits within the signal
diodes is not due to diffusion and so is not seen with this
simple model.) For pixels with no deep P-well, then no such 
plateau peak
is seen in the simulation, so it is expected that only the
full energy peak should be visible in this case. 
Also, for no deep P-well,
the absolute rate of photons with a significant fraction of the full
energy is expected to be significantly lower.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=4.5cm]{simulation/nodpwSpectrum.png}
\includegraphics[width=4.5cm]{simulation/dpwSpectrum.png}
\includegraphics[width=4.5cm]{simulation/spectra12.png}
\caption{\sl 
Simple diffusion simulation of the fraction of the $^{55}$Fe
charge seen as signal
for a uniform distribution of photon hit positions 
throughout the epitaxial volume for no deep P-well (left) and
deep P-well (centre).
The $y$-axis represents the epitaxial layer interaction volume 
of a pixel which results in the given signal fraction.
The filled red histogram shows the fraction of
signal charge absorbed in the central pixel, while the white histogram
shows the spectrum of charge absorbed by the centre pixel as well as the
eight neighbouring pixels. Right: overlaid comparison of the total
distributions from the left and centre plots for
simulated deep P-well (solid, red)
and non-deep P-well (dashed, blue) spectra for an equal irradiation exposure.
The epitaxial layer thickness is 12\,$\mu$m in these cases.
}
\label{fig:simulation:diffspectrum}
\end{center}
\end{figure}

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=4.5cm]{simulation/nodpw05Spectrum.png}
\includegraphics[width=4.5cm]{simulation/dpw05Spectrum.png}
\includegraphics[width=4.5cm]{simulation/spectra05.png}
\caption{\sl 
Simple diffusion simulation of the fraction of the $^{55}$Fe
charge seen as signal
for a uniform distribution of photon hit positions 
throughout the epitaxial volume for no deep P-well (left)
and deep P-well (centre).
The $y$-axis represents the epitaxial layer interaction volume 
of a pixel which results in the given signal fraction.
The red histogram shows the fraction of
signal charge absorbed in the central pixel, while the white histogram
shows the spectrum of charge absorbed by the centre pixel as well as the
eight neighbouring pixels.
Right: overlaid comparison of the total
distributions from the left and centre plots for
simulated deep P-well (solid, red)
and non-deep P-well (dashed, blue) spectra for an equal irradiation exposure.
The epitaxial layer thickness
is 5\,$\mu$m in these cases.
}
\label{fig:simulation:diffspectrum05}
\end{center}
\end{figure}

The volume for photon interactions
which give the plateau constant value for the deep P-well case
is a significant fraction
of the epitaxial volume.
For a 12\,$\mu$m epitaxial layer thickness,
it is estimated from the simulation
to be around 30\% of the $50 \times 50 \times 12$\,$\mu$m$^3$
pixel volume, i.e. around $10,000$\,$\mu$m$^3$. 
For the 5\,$\mu$m epitaxial layer thickness case, it is around 25\%,
i.e. $3000$\,$\mu$m$^3$. 
The volume within the diodes
which would give a signal corresponding to the total $^{55}$Fe 
energy is estimated to be approximately a circle of diameter
5\,$\mu$m and depth 1\,$\mu$m, giving a total volume for the
four diodes of around 100\,$\mu$m$^3$. This is independent of
epitaxial layer thickness.
Hence, the plateau region is expected to be
much larger, by around 
two orders of magnitude, than the diodes for 
a 12\,$\mu$m epitaxial layer depth and somewhat smaller for
a 5\,$\mu$m epitaxial layer depth.
Therefore, two peaks are expected from the  $^{55}$Fe
exposure; a large one around 0.3 of the photon energy and 
a much smaller one at the
full photon energy.
