\section{Introduction and motivation}
\label{sec:introduction}

In this paper, we present results from a test sensor
which was designed to investigate a novel approach
to electromagnetic calorimeters (ECAL).
In this approach, the
readout of the ECAL is binary, meaning that each channel
returns a yes/no result with no further amplitude 
information.

Although the potential applications for a binary ECAL are
wide, the sensor discussed in this paper was designed for
a specific application, namely the International Linear
Collider (ILC)~\cite{ref:introduction:ilc}. 
This is widely assumed to be
the next worldwide particle physics frontier accelerator
and it has a wide and rich physics 
programme~\cite{ref:introduction:ilcphysics}.
The work presented in this paper
has been done within the context of the
CALICE Collaboration~\cite{ref:introduction:calice},
which is studying calorimetry for the ILC.

Many of the ILC physics measurements require reconstruction of heavy
particles decaying 
into hadrons. To identify the decaying
particles through the invariant mass of the hadronic jets
requires excellent hadronic jet energy resolution. The
main techniques which should be able to provide this resolution are
based on Particle Flow Algorithms 
(PFA)~\cite{ref:introduction:pfa1, ref:introduction:pfa2},
which require
very high granularity calorimetry to allow the separation
of the showers into the individual particles in the jet.
For the ECALs, it is widely accepted that a sampling
calorimeter will provide the best PFA performance, due
to both transverse and longitudinal granularity. The
current state-of-the-art has silicon sensors as the
detecting layers and tungsten as the 
converter and this structure is being
considered by both the major ILC detector
concepts~\cite{ref:introduction:sid, ref:introduction:ild}. 


%\subsection{Electromagnetic shower properties}
An electron, positron or photon interacting in material produces
an electromagnetic (EM) shower containing a large
number of charged particles and photons.
A sampling EM calorimeter (ECAL) measures the charged
particle component of the shower at various positions
in the shower depth by having sensitive layers interspersed
with a dense material which efficiently produces the shower.
The sensitive layers operate
by measuring the
ionisation created by the charged particles in these
layers and normally the shower energy is estimated by
measuring the energy deposited through the ionisation.

In this paper, we present a new technique for a sampling
ECAL, in which an estimate of the number of particles in
the shower is used as the measure of the shower energy,
rather than the energy deposited. Clearly, these
quantities are strong correlated, as each
charged particle in the shower will deposit energy in
the sensitive layer. However, the energy deposited depends
on the speed and angle of the charge particle and also has
fluctuations around the average deposit, as described by
the Landau function. Hence, it would be expected that
the energy deposited would have a larger spread for any
given shower than the number of particles. This is
illustrated in figure~\ref{fig:introduction:mips}
which shows the energy deposited and the number
of charged particles in simulated photon-induced
EM showers using 
the GEANT4 program~\cite{ref:introduction:geant4}.
The ECAL geometry used for this example had 30 layers
of 500\,$\mu$m thick silicon sensors, with 0.6 radiation
lengths ($X_0$) of tungsten in front of each of the first 
20 layers and $1.2X_0$ in front of each of the last 10 layers.
This is typical of the designs being 
considered~\cite{ref:introduction:sid, ref:introduction:ild}. 
For all energy and resolution estimates below, 
the information from
the last 10 layers is weighted by a factor of two compared to
the first 20 layers.
No experimental effects have been included in this
simulation so the analogue energy deposits are assumed
to be measured perfectly, with no resolution 
effects.
The number of charged particles is determined from the
perfect simulation information as the number of separate 
particles
passing through the last 1\,$\mu$m of the silicon layer.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=6cm]{introduction/Ebulk.png}
\includegraphics[width=6cm]{introduction/Nepi.png}
\caption{\sl 
Comparison of the distributions of the quantity
measuring shower energy using (left) the energy deposited
in the sensitive layers and (right) the number of
charged particles crossing the sensitive layers.
Results for a range of incident photon energies
from 1 to 50\,GeV are shown.
}
\label{fig:introduction:mips}
\end{center}
\end{figure}

Figure~\ref{fig:introduction:compaE} shows the average
value of each measure as a function of photon energy and
shows both estimates demonstrate good linearity.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=12cm]{introduction/compaE.png}
\caption{\sl 
Comparison of the linearity as a function
of the incident photon energy, when
measuring shower energy using (left) the energy deposited
in the sensitive layers and (right) the number of
charged particles crossing the sensitive layers.
}
\label{fig:introduction:compaE}
\end{center}
\end{figure}
%
Figure~\ref{fig:introduction:ideal} shows the
root mean square (RMS) of each measure as an estimate
of the basic irreducible resolution in the two cases.
It is clear
the number of charged particles gives a very
significant improvement over the energy deposited as
an estimator of the incident photon energy. Fitting
these resolutions to a function of the form
\begin{displaymath}
\frac{\sigma_E}{E} = \frac{s}{\sqrt{E\ {\rm in\  GeV}}} \oplus c
\end{displaymath}
give values for the stochastic terms $s$ of 14.6\% for the
``analogue'' (energy deposited) case and 9.5\% for
the ``digital'' (number of charged particles) case,
and for the constant terms $c$ of 1.0\% and 0.8\%, respectively.
Hence, the analogue case is degraded by a factor of around 1.5
compared to the digital case. It is known that a realistic
analogue
ECAL with low noise electronics can approach the basic resolution
of the analogue case. The purpose of the study reported in this
paper is to estimate the resolution of a realistic binary ECAL
and compare it with this ideal case.
%
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=8cm]{introduction/resEsq.png}
\caption{\sl 
Comparison of the resolution dependence on 
the incident photon energy, when
measuring shower energy using (green) the energy deposited
in the sensitive layers and (red) the number of
charged particles crossing the sensitive layers.
The lines show the results of the fits to the
data points, as described in the text.
}
\label{fig:introduction:ideal}
\end{center}
\end{figure}

In the rest of this paper, Section~\ref{sec:requirements}
expands on
the ECAL and sensor requirements for the ILC application.
The design and fabrication of the test sensor are 
described in Section~\ref{sec:design}.
Section~\ref{sec:pixel} and Section~\ref{sec:sensor} describe
results from the test sensor for individual pixels
and sensor-wide measurements, respectively. 
Section~\ref{sec:physics}
gives the expected physics performance of a binary
ECAL based on the results from the test sensor.
