# First survey of centimeter-scale AC-LGAD strip sensors with a 120 GeV proton
beam

Christopher Madrid11footnotetext: Corresponding author. Ryan Heller  Claudio
San Martín  Shirsendu Nanda  Artur Apresyan  William K. Brooks  Wei Chen
Gabriele Giacomini,  Ohannes Kamer Köseyan  Sergey Los  Cristián Peña  René
Rios  Alessandro Tricoli  Si Xie  Zhenyu Ye

###### Abstract

We present the first beam test results with centimeter-scale AC-LGAD strip
sensors, using the Fermilab Test Beam Facility and sensors manufactured by the
Brookhaven National Laboratory. Sensors of this type are envisioned for
applications that require large-area precision 4D tracking coverage with
economical channel counts, including timing layers for the Electron Ion
Collider (EIC), and space-based particle experiments. A survey of sensor
designs is presented, with the aim of optimizing the electrode geometry for
spatial resolution and timing performance. Several design considerations are
discussed towards maintaining desirable signal characteristics with
increasingly larger electrodes. The resolutions obtained with several
prototypes are presented, reaching simultaneous $18\text{\,}\um$ and
$32\text{\,}\ps$ resolutions from strips of $1\text{\,}\mathrm{c}\m$ length
and $500\text{\,}\um$ pitch. With only slight modifications, these sensors
would be ideal candidates for a 4D timing layer at the EIC.

##  1 Introduction

The use of 4-dimensional (4D) trackers [1] that provide precise spatial and
timing information is critical for many proposed experiments ranging from high
energy collider physics to astroparticle experiments in space [2], and their
development represents a key technological challenge. Tracking detectors
capable of achieving 5–50 $\mathrm{ps}\text{/}$ timing resolution and 5–30
$\mathrm{\SIUnitSymbolMicro m}\text{/}$ position resolution are needed for
many proposed future colliders including the FCC-hh [3, 4], Muon colliders [5,
6], and the Electron–Ion Collider (EIC) [7]. A major breakthrough in 4D
tracking detector technology in recent years has been the development of
silicon based sensors such as AC-coupled Low Gain Avalanche Detectors (AC-
LGAD) [8, 9, 10, 11]. These sensors have been demonstrated to be capable of
achieving 30 $\mathrm{ps}\text{/}$ time resolution and 5-30
$\mathrm{\SIUnitSymbolMicro m}\text{/}$ position resolution [12, 13, 14, 15].

In this paper, we explore, for the first time, AC-LGADs of centimeter-scale
strip length and relatively coarse pitch, enabling coverage of large areas
with fewer channels. Longer strip sensors with sparse readout may offer better
cost and performance for applications where channel count or electrical power
density should be economized. Thanks to the excellent signal to noise ratio in
AC-LGADs, sparse readout can be exploited without significant degradation of
spatial or time resolution.

We also continue to explore the impact of the electrode metal size on the
overall performance. Studies of these key detector design parameters are of
particular importance for the development of detectors at EIC, which can
benefit from particle identification capability made possible by a 4D time-of-
flight layer with $30\text{\,}\mathrm{ps}\text{/}$ and tens of microns
resolution.

We reported first studies of the performance of AC-LGAD sensors exposed to
particle beams in previous publications [12, 13, 14, 15], focusing on fine
pitch sensors targeting ultra-fine spatial resolution on the order of
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. In this paper, we expand
those studies to include several new detector prototypes with larger area and
coarser pitch produced by Brookhaven National Laboratory (BNL) and tested at
the Fermilab (FNAL) Test Beam Facility (FTBF) [16], focusing on in-depth
studies of their performance. Section 2 presents a description of the AC-LGAD
sensors, and the experimental setup at the FTBF is described in Section 3.
Experimental results are presented in Sections 4–6. These results include
measurements of sensor signal properties, detection efficiency, and position
and time resolution. Conclusions and outlook are presented in Section 7.

##  2 The AC-LGAD sensors

In this campaign, we exposed approximately 15 different AC-LGAD sensors to the
FNAL proton beam. These were all produced in the same batch fabricated in a
class-100 clean room at BNL on 4 inch, p-type epitaxial wafers. The epitaxial
layer consists of high-resistivity silicon of thickness
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. The substrate is a low-
resistivity, $300\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ thick wafer,
serving as mechanical support, and its back is unpatterned. A photograph of a
wafer fabricated in this batch is shown in Figure 1. The wafer was populated
with AC-LGAD strip sensors of active area $5\times 5$, $5\times 10$, and
$5\times 25$ ${\mathrm{mm}}^{2}\text{/}$. The aluminum (about
$0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ thick) for the AC-coupled
strips is patterned on top of $\sim$ $150\text{\,}\mathrm{nm}\text{/}$ of
PECVD silicon oxide, which has been deposited over a uniform low-dose and high
resistivity implant of phosphorus. At the border of the resistive n-layer, a
junction-termination edge (JTE) is created by a deep phosphorus implantation.
Embedded into the JTE, which runs all around the device, a high dose
phosphorus implant is contacted by a metal frame, grounded during biasing of
the sensor. The gain layer, a deep low-dose boron implant, is ion-implanted
below the n-resistive layer and is $20\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ from the JTE. The termination region, including guard rings, is
the same for all devices. The wafers have been passivated with polyamide while
the full length of the metalized strip is exposed and available for wire-
bonding.

For each strip length, a few variations in the pitch and width of the metal
strips were produced:

  * •

$5\text{\,}\mathrm{mm}\text{/}$ long strips: pitch/gap 500/300
$\mathrm{\SIUnitSymbolMicro m}\text{/}$ or a multi-pitch structure of
pitch/gap 300/150 $\mathrm{\SIUnitSymbolMicro m}\text{/}$, 200/100
$\mathrm{\SIUnitSymbolMicro m}\text{/}$ and 100/50 $\mathrm{\SIUnitSymbolMicro
m}\text{/}$ into the same device;

  * •

$10\text{\,}\mathrm{mm}\text{/}$ long strips: pitch/gap 500/200
$\mathrm{\SIUnitSymbolMicro m}\text{/}$ or 500/300 $\mathrm{\SIUnitSymbolMicro
m}\text{/}$ or 500/400 $\mathrm{\SIUnitSymbolMicro m}\text{/}$, and the same
multi-pitch series as in the $5\text{\,}\mathrm{mm}\text{/}$ long strips;

  * •

$25\text{\,}\mathrm{mm}\text{/}$ long strips: pitch/gap 500/200 and the same
multi-pitch series as in the $5\text{\,}\mathrm{mm}\text{/}$ long strips.

We focused the detailed analysis on a set of five sensors with large collected
datasets that exhibit the characteristic behavior of this production. These
sensors were selected to span the full range of strip lengths and widths
produced, with the goal of extracting lessons for the geometric optimization
of the electrodes. They all feature $500\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ pitch, as envisioned for an EIC timing layer. The parameters for
these five sensors are shown in Table 1 along with the operating voltages
used. In the rest of this paper, we refer to them based on names that indicate
the strip length, in $\mathrm{mm}\text{/}$, and strip width, in
$\mathrm{\SIUnitSymbolMicro m}\text{/}$; for example, BNL 10-200 indicates the
sensor with $10\text{\,}\mathrm{mm}\text{/}$ length and
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ width.

Photographs of the three sensor length variations and wirebonding scheme are
shown in Figures 1 and 2.

Figure 1: Photograph of a 4 inch wafer fabricated with AC-LGAD strip sensors
of different lengths. The wafer has 24 devices with
$1\text{\,}\mathrm{cm}\text{/}$ long strips, 10 devices with
$2.5\text{\,}\mathrm{cm}\text{/}$ long strips, and 14 devices with
$0.5\text{\,}\mathrm{cm}\text{/}$ long strips.

Figure 2: The three strip length variations of BNL manufactured sensors tested at FNAL. BNL 5-200 (left), BNL 10-200 (center) and BNL 25-200 (right). The strips are read from alternating ends, to facilitate measuring of the longitudinal proton hit position (impact parameter) and compensating for signal propagation delays.  Table 1: A summary of the geometry and optimal operating voltage for the five sensors selected from the 15 sensor survey. Name | Pitch | Metal Width | Length | Thickness | Operating Voltage
---|---|---|---|---|---
Unit | $\mathrm{\SIUnitSymbolMicro m}\text{/}$ | $\mathrm{\SIUnitSymbolMicro m}\text{/}$ | $\mathrm{mm}\text{/}$ | $\mathrm{\SIUnitSymbolMicro m}\text{/}$ | V
BNL 5–200 | 500 | 200 | 5 | 50 | 245
BNL 10–100 | 500 | 100 | 10 | 50 | 220
BNL 10–200 | 500 | 200 | 10 | 50 | 255
BNL 10–300 | 500 | 300 | 10 | 50 | 240
BNL 25–200 | 500 | 200 | 25 | 50 | 215

##  3 The experimental setup at the FNAL Test Beam Facility

The results presented in this paper were collected at the Fermilab
$120\text{\,}\mathrm{GeV}\text{/}$ proton test beam facility, using the LGAD
characterization setup described in detail in previous results [12, 13]. This
setup is based on a silicon tracking telescope that measures the impact
position of each proton, and a fast microchannel plate detector (MCP-PMT),
which serves as a time reference with $10\text{\,}\mathrm{ps}\text{/}$
resolution. The AC-LGAD and MCP-PMT waveforms were recorded using an eight
channel Lecroy Waverunner 8208HD oscilloscope with a bandwidth of
$2\text{\,}\mathrm{GHz}\text{/}$ and a sampling rate of
$10\text{\,}\mathrm{G}\mathrm{S}\mathrm{/}\mathrm{s}$ per channel. The AC-LGAD
sensors were mounted on readout boards developed by Fermilab [17]. The readout
boards are optimized for LGAD sensors and are capable of accommodating 16
channels. The trigger signal is generated by a scintillator detector located
downstream from the AC-LGADs and distributed to the tracker and the
oscilloscope. Figure 3 shows a diagram of the setup along with an image of the
environmental chamber that houses the AC-LGADs inside the telescope.

Figure 3: Diagram of the AC-LGAD and reference instruments along the beamline
(top). The environmental chamber placed within the FTBF silicon telescope
(bottom). The telescope comprises five pairs of orthogonal strip layers and
two pairs of pixel layers, for a total of up to 14 hits per track.

The pixel detectors inside the FTBF telescope have recently been upgraded from
Phase 0 CMS pixel detectors with a pitch of $100\times
150~{}$$\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$ to Phase 2 CMS pixel
detectors with a pitch of $25\times
100~{}$$\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$. The new detectors use the
RD53A chip [18]. The new pixel detectors feature fast readout, which enables
reliable track-finding even at high beam intensity. The new pixel detector
modules also use significantly improved DAQ, yielding increased operational
robustness and enabling data-quality monitoring in real time. The active area
of each pixel layer is $13.6\times 9.6~{}$$\mathrm{m}\mathrm{m}^{2}$, and they
are aligned for an overlapping region of about $9.6\times
9.6~{}$$\mathrm{m}\mathrm{m}^{2}$ with four expected pixel hits per track.

Additionally, the procedure to align the sensors under test with the telescope
was also improved. Previous measurements relied on sensor alignment that only
considered variations in the sensor position along the beamline and rotation
around the beam axis. The new alignment procedure also considers the two
remaining rotational angles of the sensor, consequently improving the
telescope resolution to $5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. For
each AC-LGAD spatial and temporal resolution measurement presented in this
paper, the contributions from the tracking telescope
($5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$) and MCP–PMT time reference
($10\text{\,}\mathrm{ps}\text{/}$) have been subtracted in quadrature. Since
the sensors considered have resolutions greater than
$15\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ and
$30\text{\,}\mathrm{ps}\text{/}$, the reference contributions typically have a
very small impact.

##  4 Signal properties

As described in Section 3, the waveforms from up to seven channels of each AC-
LGAD are recorded by the oscilloscope. The waveforms are analyzed event-by-
event to extract the amplitude, risetime, slew rate, and other relevant signal
properties. The averaged pulse shapes from the three sensor length variants
can be seen in Figure 4, normalized by their integrals. The timing performance
is optimal for shorter sensors as the risetime tends to increase with the
length of the sensor.

Figure 4: Average waveforms, normalized by area, from sensors with varying
lengths (left) and varying metal widths (right).

Events considered in the analysis are required to have a high-quality track
and an MCP-PMT timestamp, ensuring reliable references for the proton position
and time of arrival. Furthermore, only events with a track pointing through
the active area of the AC-LGAD are considered. These events form the
denominator for all efficiency measurements. A minimum amplitude is required
to identify which channels have signals well-separated from noise. The
channels used in the multi-strip reconstruction methods, described in the
following sections, must have signal pulses with amplitude greater than
$15\text{\,}\mathrm{mV}\text{/}$. To avoid partially contained clusters at the
edge of the active sensor area, we only consider events for which the largest
amplitude signal is found in a channel that is not at the edge of the sensor.
The AC-LGADs considered in this paper typically have signals in one or two
neighboring strips in each event.

###  4.1 Gain uniformity

Gain uniformity across the full area of a sensor is a desirable feature for
AC-LGADs. We observed a degree of non-uniformity in the signal amplitude in
all sensors from the current production batch. A characteristic band pattern
is observed, as shown in Figure 5 and 6. These patterns arise most likely due
to non-uniformity in the gain implant. The ratio of amplitudes between high
and low gain regions is roughly two when the sensors are operated at bias
voltage close to the breakdown voltage. Because the highest gain regions enter
breakdown at lower voltages, they determine the maximum attainable bias
voltage for each sensor. As result, the lower gain regions remain underbiased,
and in some cases, these regions represent the majority of the surface because
the high gain structures can be highly localized. Many important performance
metrics are strongly dependent on gain, and therefore are observed to vary
across the surface.

To study the impact of design and geometry choices isolated from the effects
of the non-uniformity, we define three different regions within each sensor
that characterize the high-gain, low-gain, and gap regions. The gap region is
the space between adjacent metalized strips, where we also observe differences
in signal response. Examples of these regions of interest are overlaid in
Figure 5, and the corresponding average pulse shapes in each region are also
shown. We take the performance in the high-gain region as indicative of what
would be attained in a uniform sensor, where all regions would reach high gain
at similar bias voltage. Since uniform large-area LGADs are produced
routinely, it is expected that future large-area AC-LGADs productions by BNL
or alternative foundries will be able to achieve good uniformity.

In Subsection 4.2, we study a variety of signal shape metrics relevant for the
spatial and time resolutions as a function of the strip geometries. In order
to minimize the effect of the gain non-uniformity, we define the metrics in a
gain-independent manner by accounting for or dividing out the dependence on
the measured charge.

Figure 5: Signal MPV amplitude as a function of the telescope track $x$ and
$y$ position (left) for the BNL 10-200 sensor. The different colored boxes
represent the high-gain (red), low-gain (blue), and gap (black dotted)
regions. The averaged waveform (right) for the three different regions.

Figure 6: Signal MPV amplitude as a function of the telescope track $x$ and
$y$ position for the BNL 10-100 (left) and BNL 10-300 (right) sensors. The
different colored boxes represent the high-gain (red), low-gain (blue), and
gap (black dotted) regions.

###  4.2 Signal metrics

Achieving good time resolution requires signals with a steep rising edge well-
separated from the noise. More precisely, the jitter contribution to the time
resolution, representing the impact of the electronic noise, is given by:

$\sigma_{\rm{jitter}}=\frac{N}{dV/dt}\sim\frac{t_{\rm{rise}}}{S/N}$ (4.1)

where $N$ represents the noise amplitude, $dV/dt$ is the slew rate,
$t_{\rm{rise}}$ is the risetime, and $S$ is the signal amplitude [19].
Minimizing the jitter calls for sensors not only with large total charge or
gain, but also desirable pulse shapes that result in large $dV/dt$.

To understand the design elements affecting the jitter, the signal development
in AC-LGADs can be factorized into two stages as outlined in [14]. First, the
charge carriers drift within the bulk, are multiplied in the gain region, and
induce charges in the n+ surface layer. Then, the induced signal spreads
through the resistive layer, and induces signals in the AC-coupled electrodes
nearby. The first stage of charge generation and multiplication has little
design freedom among LGADs of $50\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ thickness with uniform gain. In absence of premature breakdown,
these sensors typically deliver the best performance at bias voltage roughly
$10\text{\,}\mathrm{V}\text{/}15\text{\,}\mathrm{V}\text{/}$ before breakdown,
before the onset of additional noise, and where the gain reaches approximately
a factor of 20–40. For modest variation of the gain implant doping, the
breakdown and operating voltages will shift, but the gain at optimal bias
voltage is more or less fixed [17]. With saturated drift velocities (bias
$>$200\text{\,}\mathrm{V}\text{/}$$ at room temperature), the signal risetime
for the initial charge development can be as fast as
$400\text{\,}\mathrm{ps}\text{/}500\text{\,}\mathrm{ps}\text{/}$.

Considering sensors with typical LGAD gain implants and good production
uniformity, then, most of the room for design optimization lies in tuning the
geometry to best preserve the signal as it is passed to the AC-electrodes in
the second stage of signal development. With the survey of varying geometries
in this campaign, significant sculpting of the signal due to the electrode
design has been observed. In particular, the largest electrodes are associated
with significantly slower risetimes, or alternatively, smaller $dV/dt$ for a
given input charge. Since many of the jitter contributions in Equation 4.1 are
correlated with the signal size, and the sensors considered in this survey
exhibit non-uniform gain, the various contributions cannot be directly
compared across sensors to deduce the influence of the electrode geometry.
Instead, we rely on several metrics that are deliberately gain-independent,
intended to help predict the performance in uniform sensors of the same
geometries. The first metric used is the risetime (from 10% to 90% of the
maximum amplitude), which naturally tends to be scale-invariant. Then, as the
amplitude and slew rate ($dV/dt$) are proportional to the gain, we consider
the ratios of these quantities to the charge, determined from the integral of
each pulse. These ratios serve as gain-independent metrics of the signal
shape. The final metric is an estimate of the expected jitter for an example
input charge of $10\text{\,}\mathrm{fC}\text{/}$ (gain of $\sim 15$), found by
multiplying $dV/dt/Q$ by $10\text{\,}\mathrm{fC}\text{/}$ and inserting into
Equation 4.1 with a characteristic noise of $2\text{\,}\mathrm{mV}\text{/}$
typical of the 16-channel board amplifiers.

These metrics are studied as a function of the strip length with a constant
electrode width of $200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ in
Figure 7. The variations observed between the three selected regions of
interest described in Section 4.1 form the uncertainty bands. As was visible
in the pulse shapes in Figure 4, increasing the strip length tends to slow the
risetime and yields smaller amplitudes and $dV/dt$ for a given input charge.
As result, the expected jitter at a given charge is also larger for the longer
strips. Similarly, we consider variations in strip width in Figure 8 at
constant $1\text{\,}\mathrm{cm}\text{/}$ length. In this case, the width does
not appear to have a significant impact on the signal metrics. Although all
five sensors were operated at slightly different bias voltage, specified in
Table 1, it was verified they were all within a range with approximately
saturated drift velocities and constant risetimes. As result, the difference
in bias voltage does not significantly affect the comparison between sensors.

Two distinct phenomena may contribute to the slowing of the signal observed in
the largest strips. First, the slowing may be due to the increased capacitive
load on the amplifiers. In this case, simply reducing the electrode width to
50 or $100\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ may be enough to
compensate for the length in the longest strips and maintain faster signals
shapes. Or, the amplifier design could be modified to better compensate for a
higher input capacitance. Alternatively, the slowing could be due to an
impedance mismatch at the strip-wirebond interface, causing part of the signal
to reflect back along the full length of the strip and finally be collected at
a small delay, leading to an apparent stretching of the rising edge. To
mitigate this effect, the transmission properties at the interface could be
improved, using shorter wirebonds as well as redundant bonds for each
connection. All of these potential solutions will be studied in upcoming beam
test campaigns, ideally leading to both improved understanding of the slowing
phenomenon, and improved timing performance in the longest strips. Although
the longest strips considered in this campaign do suffer from degraded
risetime, it seems possible to design AC-LGAD detector systems with electrode
lengths greater than $2\text{\,}\mathrm{cm}\text{/}$ that maintain the fast
signal shapes seen in smaller devices.

Figure 7: Risetime (top left), ratio of signal amplitude to charge (top
right), ratio of slew rate to charge (bottom left), and expected jitter at
$10\text{\,}\mathrm{fC}\text{/}$ (bottom right) for different length sensors
with $200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ electrode width. The
uncertainty bands represent variation observed across different regions of
each sensor.

Figure 8: Risetime (top left), ratio of signal amplitude to charge (top
right), ratio of slew rate to charge (bottom left), and expected jitter at
$10\text{\,}\mathrm{fC}\text{/}$ (bottom right) for different metal width
sensors with $10\text{\,}\mathrm{mm}\text{/}$ long electrode length. The
uncertainty bands represent variation observed across different regions of
each sensor.

##  5 Position reconstruction

In this section we discuss the reconstruction of the proton impact location,
highlighting aspects affected by the large size of these sensors relative to
previous measurements. By exploiting the signal sharing properties of the AC
contacts, the proton hit position (impact parameter) between strips, $x$, can
be inferred with resolution much finer than the pitch. This reconstruction is
presented in Sec. 5.1 and is similar to the previous analysis [13].

Because adjacent strips are read from alternating ends and the signals
propagate with a finite velocity, the time difference between adjacent
channels can be used to infer the proton hit position (impact parameter), $y$,
along the length of the strip with O($\mathrm{mm}\text{/}$) precision. This
analysis is described in Sec. 5.2.

###  5.1 X position reconstruction

The $x$ position reconstruction takes advantage of the fact that the signal is
generally shared between a pair of adjacent strips. The reconstruction
considers the two highest amplitude strips and interpolates between them to
determine the proton hit position (impact parameter). To achieve this, we
define the amplitude fraction, $f$, as $f=a_{1}/(a_{1}+a_{2})$, where $a_{1}$
and $a_{2}$ are the leading and subleading strip amplitudes. Using the tracker
reference, we find template histograms of the impact parameter as a function
of $f$, and fit them with a polynomial, denoted $h(f)$, unique to each sensor.
The polynomials $h(f)$ are shown for the different geometries in Figure 9. The
amplitude fraction varies linearly within the gap regions between strips, but
saturates upon reaching the boundary of the metalized contact.

Figure 9: Fit functions, $h(f)$, denoting the proton hit position (impact
parameter) as a function of the amplitude fraction, $f$, for varying metal
widths (left) and varying strip lengths (right). The fits are performed using
the full range of the dotted curves, but only the range of the solid curves
are used in the reconstruction. The dashed colored lines and grey bands
delineate the metalized region of each sensor.

In each event the amplitude fraction $f$ is found and the function $h(f)$ is
used to determine the impact parameter relative to the leading strip. This
method can be used in events that satisfy two conditions: at least two strips
must have an amplitude larger than the minimum threshold; and the amplitude
fraction must be less than some value, above which the $f$ no longer varies
with the proton impact position and typically occurs close to the metalized
strips. Events that fail either condition are instead classified as "one
strip" events and assigned an $x$ position at the center of the leading strip.

The resulting resolutions and efficiencies for the two reconstruction
categories for the three $1\text{\,}\mathrm{cm}\text{/}$ sensors are shown in
Figures 10, Figure 11 and Figure 12, and for all sensors in Table 2, after
subtracting a $5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ contribution
from the tracker reference measurement. The efficiency for a proton to produce
a signal in at least one strip reaches 100%.

Figure 10: Resolution (left) and efficiency (right) for the $x$ position
reconstruction as a function of the track $x$ position for the BNL 10-200
sensor. The left figure indicates the resolution that would be achieved with
binary readout, pitch $/\sqrt{12}$ (in red), as well as the expected and
observed resolution for the two-strip reconstruction (green), and the observed
exactly one strip resolution (blue). The efficiency is shown separately for
events in the single- and two-strip categories, as well as the union of both
categories. The event selection includes hits across the entire sensor
surface, including high- and low-gain regions.

Figure 11: Resolution (left) and efficiency (right) for the $x$ position
reconstruction as a function of the track $x$ position for the BNL 10-100
sensor. The left figure indicates the resolution that would be achieved with
binary readout, pitch $/\sqrt{12}$ (in red), as well as the expected and
observed resolution for the two-strip reconstruction (green), and the observed
exactly one strip resolution (blue). The efficiency is shown separately for
events in the single- and two-strip categories, as well as the union of both
categories. The event selection includes hits across the entire sensor
surface, including high- and low-gain regions.

Figure 12: Resolution (left) and efficiency (right) for the $x$ position
reconstruction as a function of the track $x$ position for the BNL 10-300
sensor. The left figure indicates the resolution that would be achieved with
binary readout, pitch $/\sqrt{12}$ (in red), as well as the expected and
observed resolution for the two-strip reconstruction (green), and the observed
exactly one strip resolution (blue). The efficiency is shown separately for
events in the single- and two-strip categories, as well as the union of both
categories. The event selection includes hits across the entire sensor
surface, including high- and low-gain regions.

When a proton strikes near the center of the gap region, the majority of
events fall in the two strip category, and achieves resolution of
$15\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}20\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ for all strip
widths ranging from 100 to $300\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$, significantly smaller than the
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ pitch size. We derived the
expected resolution for the two strip reconstruction by propagating the
uncertainty of the noise and signal amplitudes in the two leading strips,
$a_{1}$ and $a_{2}$, for the signal sharing polynomial $h(f)$:

$\begin{split}\sigma_{x}^{\rm{expected}}&=\mathrm{P}\left|\frac{dh}{df}\right|\frac{\sqrt{a_{1}^{2}+a_{2}^{2}}}{(a_{1}+a_{2})^{2}}N\end{split}$
(5.1)

where P is the sensor pitch and $N$ is the noise amplitude. The expected
resolution was evaluated as a function of $x$ and overlaid on Figures 10–12.
The expected uncertainties are generally quite close to the observed values,
but tend to be slightly smaller. This difference is attributed to the presence
of additional contributions to the observed resolution not captured in Eq.
5.1, including the effect of non-uniformity and channel-to-channel variations
that are neglected by using a single function $h(f)$ used to characterize the
signal sharing. Non-uniformity tracker telescope performance could also lead
to extra contributions to the observed resolution.

Signals from protons that strike the metalized strip mostly fall in the single
strip category because the majority of the signal is contained within that
single strip. In the absence of a second signal, we must assign its position
as the center of the strip, resulting in significantly degraded resolution
approaching the limit of the strip width / $\sqrt{12}$. The observed
resolutions shown in Figures 10–12 and Table 2 confirm this expectation,
though the limit is achieved only for the BNL 10-300 and BNL 5-200 sensors. In
the other cases, due to the gain non-uniformity, portions of the gap regions
with low gain also contribute single strip events. These events have larger
residuals and spoil the single strip resolution.

The low-gain regions of the prototype sensors compromise the overall
performance, since they contribute single-strip events in the gap regions.
However, sensors with uniform high gain would maintain two-strip efficiency in
the gaps and constrain the one-strip events to the metal regions. In this
case, the single strip resolution would approach the expected limit of metal
width / $\sqrt{12}$. As result, with sufficiently small metal width
($50\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}100\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$), the sensor
would yield $15\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}30\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ resolution
uniformly for both categories across the surface. Uniform DC-LGADs of similar
size and similar gain implant have already been demonstrated by HPK and FBK
for the CMS and ATLAS timing detector upgrades. Based on preliminary
understanding of the non-uniformity observed in the current production, we
expect that improved uniformity and consequently the desired performance will
be achieved in the next production from BNL as well.

From Equation 5.1, we would expect better two-strip resolution for a smaller
derivative of the proton hit position (impact parameter) with respect to
signal sharing fraction, $dh/df$. Based on the $h(f)$ shown in Figure 9, this
factor slightly favors the sensors with larger metal width (smaller gap). This
trend is observed in the data in Figures 10–12 and Table 2, where the
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ width strips achieve a few
micron better resolution than the sensors with smaller metal. This simply
reflects the fact that it is easier to interpolate within a smaller gap.
However, this effect is relatively small tradeoff given the substantial
advantages of increased 2-strip acceptance for sensors with reduced metal
width and larger gaps.

###  5.2 Y position reconstruction

Because signals from the strips were read out from alternating ends, the time
difference between adjacent channels for events in the two-strip category can
be used to reconstruct the $y$ position of the proton hit. The $y$ resolution
achievable using this method is of order O($\mathrm{mm}\text{/}$), which is
not sufficiently fine to be useful for track reconstruction but is sufficient
to correct for position dependent delays that impact the time measurement,
discussed in more detail in Section 6. This $y$ reconstruction enables both
standalone operation without a tracker as well as online timing measurements
made before the track reconstruction is available. This precision will also
assist in pattern recognition by avoiding mismatched hits along long strips.

We can estimate the relationship between the propagation time for signals in
two adjacent strips and the $y$ position reconstruction with some simple
approximations. First, we assume the paths the signal takes through the sensor
are first approximately aligned with the $x$ direction towards the nearest
strips, then along the $y$ direction following the electrode axis. Second, we
assume that the propagation velocity, $v_{x}$ ($v_{y}$), along the $x$ ($y$)
directions are constant. With these two assumptions, an analytic expression
for the difference between the $y$ position of the center of the sensor to the
proton hit $y$ position ($\Delta y$) can be derived as

$\begin{split}\Delta y&=\frac{v_{y}}{v_{x}}\left(\frac{P}{2}-\Delta
x\right)+\frac{1}{2}v_{y}\Delta t\end{split}$ (5.2)

where $P$ is the sensor pitch and $\Delta x$ is the $x$ impact parameter.
Based on this expression, the expected resolution for the $y$ position
estimate is given by:

$\begin{split}\sigma_{y}^{\rm{expected}}&=\sqrt{\frac{v_{y}^{2}}{v_{x}^{2}}\sigma^{2}_{x}+\frac{1}{2}v_{y}^{2}\sigma_{t}^{2}}\\\
&\approx\frac{1}{\sqrt{2}}v_{y}\sigma_{t},\end{split}$ (5.3)

in the limit that $\sigma_{x}$ is small compared to $\sigma_{y}$. Note this
expression does not depend directly on the sensor length. With observed
velocities $v_{y}$ on the order of
$50\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$, and assuming
$\sigma_{t}=$40\text{\,}\mathrm{ps}\text{/}$$, we expect a $y$ resolution of
about $1.4\text{\,}\mathrm{mm}\text{/}$, demonstrating the limit
$\sigma_{x}\ll\sigma_{y}$.

In these prototypes, the velocities $v_{x}$ and $v_{y}$ have nontrivial
variation across the surface, which prevents the use of the simple formula in
Equation 5.2. Instead, we derived an empirical lookup table relating the mean
$y$ position to the amplitude fraction, $f$, (a proxy for $\Delta x$), and
$\Delta t$, which is used to reconstruct the proton $y$ position in each
event. This procedure yields a resolution governed approximately by Equation
5.3.

The measured $y$ position resolutions for the BNL 10-200 and BNL 25-200
sensors are shown in Figure 13. The BNL 10-200 exhibits slightly better
performance, close to the expected value of $1.4\text{\,}\mathrm{mm}\text{/}$.
The longer BNL 25-200 sensor has coarser time and $x$ position resolution due
to its longer pulse shape and gain non-uniformity, which results in slightly
worse $y$ position as well. Since the $y$ resolution is not strongly dependent
on the strip length, the $y$ measurement would become increasingly more useful
for devices that have length larger than several $\mathrm{cm}\text{/}$, able
to achieve resolution several times smaller than the length / $\sqrt{12}$.

Figure 13: Residuals of the $y$ position reconstruction for the two longer
sensors, BNL 10-200 and 25-200, with Gaussian fit to extract the longitudinal
resolution.

##  6 Time measurements

The increased size of these sensors, particularly along the strip length,
introduces a position-dependent time delay, shown in Figure 14. If left
uncorrected, this position-dependent time delay significantly degrades the
resolution of the time measurement. Given the centimeter-scale length, the
variation in arrival time can be as large as $400\text{\,}\mathrm{ps}\text{/}$
and $1\text{\,}\mathrm{ns}\text{/}$ for the $10\text{\,}\mathrm{mm}\text{/}$
and $25\text{\,}\mathrm{mm}\text{/}$ sensors, respectively.

We demonstrate two strategies to correct for the position-dependent time
delay. The first strategy is to use the external tracker to determine the
proton hit position. The second strategy is to use the position reconstruction
procedure outlined in Section 5 to determine the proton hit position. In
general, the delay correction does not require extremely fine position
resolution to remove this contribution to the time resolution, and both
strategies perform adequately.

Figure 14: Variation in mean signal arrival time as a function of the proton
impact location along the strip for three length sensors (left). Map of the
arrival time for a single channel from the BNL 10-200 sensor (right). The mean
arrival time is defined to be zero for hits near the wirebond at the end of
the strip.

###  6.1 Time delay corrections

To correct the time delay, we first generate maps of the mean arrival time for
each channel on every sensor as a function of proton $x$ and $y$, as in Figure
14 (right). Then, in each event, we use the map and the reconstructed $x$ and
$y$ position of the proton hit to determine the appropriate correction for the
arrival time delay. The proton $x$ and $y$ position can be found using either
the external tracking information, or the AC-LGAD position reconstructions
described in Section 5.

The form of the delay correction could be further simplified if the delay
could be parametrized in terms of a simple function of the $x$ and $y$
positions in each event. However, the delays exhibits nontrivial position
dependence, as can be seen in the BNL 10-200 sensor in Figure 15. In general,
the sensors exhibit linearly varying delays across large regions, consistent
with a constant longitudinal propagation velocity. However, frequently these
regions are separated into two or more domains with different apparent
velocities. In some cases, as in BNL 10-200, the inflection points between two
domains is strongly correlated with features in the gain non-uniformity. In
Figure 15 (right), the delay domain boundary points are overlaid as black
points on the map of the signal amplitude variation, and it is clear that they
correspond with the locations of peak gain. Sensors with higher uniformity
tend to have less complex delay structures, but understanding this behavior in
detail is beyond the scope of this work. For this analysis, we do not use an
analytic parametrization of the delays and simply rely on the empirical delay
maps as in Figure 14. We note that within the non-metalized regions between
strips, significant variation in arrival time along the $x$ axis is observed,
with a delay accumulating with distance from the strip at roughly
$700\text{\,}\mathrm{ps}\text{/}\mathrm{mm}$. This variation would correspond
to an unphysically slow propagation velocity and its origin is not yet fully
understood.

Figure 15: Mean arrival time as a function of the track $y$ position for two
neighboring strips that are read from alternating ends (left). The transition
points near $y=-2$ between regions with different apparent velocities are
overlaid as black dots on the map of signal amplitude (right). The transitions
between domains occurs at the locations with highest gain, regardless of the
readout end.

###  6.2 Time resolution

The three main contributions to the AC-LGAD time resolution are the Landau
fluctuations in the ionization depth profile, the noise-induced jitter, and
the effect of the arrival time delays.

$\begin{split}\sigma_{t}^{2}&=\sigma_{\rm{Landau}}^{2}+\sigma_{\rm{jitter}}^{2}+\sigma_{\rm{delay}}^{2}\end{split}$
(6.1)

The Landau term grows with the active thickness of the sensor and sets an
approximately $30\text{\,}\mathrm{ps}\text{/}$ resolution floor for
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ sensors. The jitter term is
proportional to the ratio of the noise and the slewrate of the AC-LGAD signal.
The arrival time variation, left uncorrected, would contribute proportionally
to the dimensions of the sensor:

$\sigma_{\rm{delay_{y}}}\sim\frac{L/v_{y}}{\sqrt{12}},\quad\sigma_{\rm{delay_{x}}}\sim\frac{P/v_{x}}{\sqrt{12}}$
(6.2)

where $L$ and $P$ are the length and pitch of the sensor. If the delays are
instead corrected, the residual contribution to the time resolution due to
imprecision in the correction is given by:

$\begin{split}\sigma_{\rm{delay_{y}}}\sim\frac{\sigma_{y}}{v_{y}},\quad\sigma_{\rm{delay_{x}}}\sim\frac{\sigma_{x}}{v_{x}}.\end{split}$
(6.3)

Considering a typical value of
$v_{y}\sim$50\text{\,}\mathrm{m}\mathrm{m}\mathrm{/}\mathrm{n}\mathrm{s}$$,
the $y$ delay contribution can be brought to a negligible
$10\text{\,}\mathrm{ps}\text{/}$ with
$\sigma_{y}\sim$500\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$$. These
resolutions are easily achievable within a tracking system, and are nearly
achieved in standalone operation as well. As result, the time resolution even
with large sensors remains dominated by the Landau fluctuations and in some
cases, the jitter. Sensors with uniform, high gain would have improved
$\sigma_{\rm{jitter}}$, $\sigma_{y}$, and $\sigma_{x}$, and should achieve
resolutions at the Landau floor of $30\text{\,}\mathrm{ps}\text{/}$.

###  6.3 Time resolution results

Signal sharing between the strips can also be utilized to improve the time
measurement of the AC-LGAD sensor, as shown in previous results [13]. The
simplest approach we use to reconstruct the time for a given hit is to use the
timestamp measured in the channel with the largest signal amplitude, using a
constant fraction discriminator (CFD) algorithm at 50% level to the leading
edge of the signal. However, this approach can lead to non-uniform time
resolution across the sensor as the single-channel signal amplitudes are
smaller in the gaps between two channels.

An improved method calculates timestamps according to the amplitude squared
weighted average of multiple channels as follows:

$\begin{split}t_{\rm{reco}}&=\frac{a_{1}^{2}t_{1}+a_{2}^{2}t_{2}}{a_{1}^{2}+a_{2}^{2}}\end{split}$
(6.4)

where $a_{1}$ and $a_{2}$ are the amplitude of the leading and subleading
channels and $t_{1}$ and $t_{2}$ are the time measurements of leading and
subleading channels.

The time resolution is measured by comparing the AC-LGAD timestamps with those
from the Photek MCP-PMT time reference. The time resolution of the Photek MCP-
PMT has been measured to be 10 ps, and this value is subtracted in quadrature
from the time resolution measurements, as discussed in Section 3. The time
resolutions for the three $10\text{\,}\mathrm{mm}\text{/}$ sensors as a
function of the proton $x$ impact parameter are shown in Figures 16 and 17
using both the single and multi-channel timestamps and different forms of
delay correction, including events along the full length of the strips. The
purple curves show the single-channel timestamps with no position-dependent
delay correction. In this case, the uncorrected delays degrade the resolution
to $50\text{\,}\mathrm{ps}\text{/}70\text{\,}\mathrm{ps}\text{/}$. Adding the
tracked-based delay corrections (black curves) significantly improves the
resolution, achieving on average $40\text{\,}\mathrm{ps}\text{/}$ for direct
metal hits, and up to $55\text{\,}\mathrm{ps}\text{/}$ in the gaps. In the gap
regions, the signal is shared roughly equally between the two neighbor strips
and neither strip has a very large amplitude. However, combining both signals
into the multi-channel timestamp with delay correction (dark green) recovers
most of the performance in the gap regions, yielding average time resolutions
of $40\text{\,}\mathrm{ps}\text{/}45\text{\,}\mathrm{ps}\text{/}$ for all $x$
positions in all three sensors.

The absolute time resolution values shown in Figures 16 and 17 are dominated
by the sensor regions with low, non-uniform gain, which results in slightly
degraded time resolution with respect to the one shown in Figure 18 for the
high-gain regions only. As a result, these resolutions have a non-negligible
jitter contribution and do not reach the $30\text{\,}\mathrm{ps}\text{/}$
Landau limit. The impact of the gain non-uniformity can be seen in Figure 16
(right), which shows the multi-channel resolution across the surface of the
BNL 10-200 sensor. The spots with high gain (as visible in Figure 5) do
achieve $30\text{\,}\mathrm{ps}\text{/}$ resolution. The resolutions
considering high-gain regions alone are also shown in Table 2. Based on the
performance in these regions, we expect similar sensors with uniform, high
gain would achieve uniform
$30\text{\,}\mathrm{ps}\text{/}35\text{\,}\mathrm{ps}\text{/}$ resolution.

Additionally, we also consider the time resolution using delay corrections
based on the self-measured impact parameters in $x$ and $y$ described in
Section 5 rather than the external tracker (light blue curves in Figures 16
and 17). In the gap regions, where the position reconstruction performs well,
the resolution with AC-LGAD-based delay corrections is very close to what is
obtained with the tracker-based correction. In the metal strips, however, the
majority of events have a signal in only one strip, and the $y$ position
reconstruction is not possible. In this case, the time resolution degrades to
the uncorrected scenario.

Table 2 also summarizes the performance of the shorter and longer sensors. As
foreshadowed in Section 4, the slower risetimes in the
$25\text{\,}\mathrm{mm}\text{/}$ strips lead to significant degradation in the
time resolution, even considering only the high gain region. As described in
Section 4, we foresee several possible avenues to mitigate this effect and
deliver good timing performance even with very long strips.

Figure 16: Time resolution for various time definitions as a function of the
telescope track $x$ position for the BNL 10-200 sensor (left). The event
selection includes hits across the entire sensor surface, including high- and
low-gain regions. The four curves show different reconstruction and time delay
correction options. The time resolutions shown include single channel
reconstruction without a time delay correction, single channel reconstruction
with a telescope position dependent time delay correction, multi-channel
reconstruction with a telescope position dependent time delay, and multi-
channel reconstruction with a position dependence self-corrected using the AC-
LGAD $x$ and $y$ position reconstruction. The time resolution as a function of
the telescope track $x$ and $y$ position for the multi-channel reconstruction
with telescope tracker position time delay correction (right).

Figure 17: Time resolution for various time definitions as a function of the
telescope track $x$ position for the BNL 10-100 (left) and BNL 10-300 (right)
sensors. The event selection includes hits across the entire sensor surface,
including high- and low-gain regions. The four curves are based on a variety
of time reconstruction and time delay correction definitions, as described in
Figure 16.

##  7 Conclusions and Outlook

In this paper, we presented the results of the first comprehensive beam test
campaign studying centimeter-scale AC-LGAD strips. The sensors considered were
manufactured at the Brookhaven National Laboratory and have a coarse pitch of
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. They deliver precise 4D
measurements over a large area with relatively sparse readout and few
channels. These traits make these sensors particularly desirable for
applications including an Electron Ion Collider 4D tracking layer, or space-
based 4D trackers.

We performed a survey of approximately 15 sensors from a single BNL production
with varying strip geometries. We focused the complete analysis on a set of
five sensors spanning the full range of strip lengths and widths in order to
understand how to better optimize the electrode design. This production
suffered from non-uniform gain layer implantation, which degrades the time and
spatial resolutions across most of the surface of each sensor. However, with
careful analysis, we can still extract useful lessons for electrode
optimization and estimate the performance that would be obtained in devices
with uniform gain. The resolutions obtained with the 5 fully analyzed sensors
are summarized in Table 2. Despite the poor uniformity, most of the sensors
still deliver excellent time resolutions in the range of
$30\text{\,}\mathrm{ps}\text{/}50\text{\,}\mathrm{ps}\text{/}$ and spatial
resolutions from $10\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}80\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. This level of
performance as a function track $x$ position can also be seen for the
$5\text{\,}\mathrm{mm}\text{/}$ (BNL 5–200) and
$10\text{\,}\mathrm{mm}\text{/}$ (BNL 10-200) long sensors with
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ metal widths in Figure 18.

Considering the spatial reconstruction performance, we found in Section 5 that
events with clusters of two or more strips convey much more information than
single-strip events. The two-strip clusters allow precise interpolation for
the $x$ impact parameter, and to a lesser extent, even the longitudinal impact
parameter, $y$, along the strips. As particles that pass directly through a
metal strip tend to yield signals completely absorbed in that strip, this
presents a strong motivation to minimize the electrode width and maximize the
acceptance for two-strip clusters. In turn, reducing the electrode width
confines the remaining single-strip events to the small region within the
strips, reducing the spatial resolution of this category to approach that of
the two-strip category. Based on the performance observed, we believe strips
of $500\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ pitch and
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ electrode width and
uniform gain could attain uniform spatial resolution of
$15\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}20\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. We extrapolate
that strips with even narrower $50\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$ width may yield better performance, constraining the single-strip
population to a smaller region. Furthermore, the resistivity of the n+ layer
could also be tuned to ensure more uniform two-strip acceptance.

On the other hand, precision timing analysis with such large devices presents
interesting features not observed in shorter AC- or DC-LGADs considered in
previous studies. Signals propagating long distances across the sensor surface
are collected with varying delays at a scale at least as large as the desired
time resolution. Since these delays are determined by the impact position,
they can be easily corrected with a measurement of hit location with modest
resolution. Within a collider application, the tracking detector performance
is typically far in excess of what is needed for such a correction to make the
impact of the position-dependent delays negligible. Even for operation as a
standalone timing plane or in real-time operation at a collider before
tracking is performed, the 2D spatial reconstruction is sufficient to remove
the effect of the delays. This consideration, too, strongly favors thinner
electrodes and expanded two-strip cluster acceptance, which makes the
longitudinal reconstruction and standalone position-dependent delay correction
possible.

The other interesting challenge for the time resolution is related to the
impact of the strip length, as described in Section 4. The longest sensors
with strip lengths beyond $2\text{\,}\mathrm{cm}\text{/}$ exhibit slower
rising edges that increase the susceptibility to jitter in the timing
measurement. We outlined several potential mitigation techniques, and it is
likely possible to design $2.5\text{\,}\mathrm{cm}\text{/}$ strips with good
timing properties. In particular, reducing the electrode width, as already
motivated by the spatial reconstruction, would reduce the electrode
capacitance and may be enough to restore the fast rising edge even in the
longest strips. These options will be explored in the next beam campaign. The
strip length is less relevant for the spatial reconstruction. Although the
longest prototype analyzed in this campaign suffered from poor gain uniformity
and yielded poor performance, we do not find any indication that the strip
length itself should necessarily have a large impact on the spatial
resolution. Any modification of the pulse shape due to the length could be
mitigated using a measure of the pulse integral, rather than amplitude, for
the position reconstruction.

Taking all of the results together, we believe that large area, coarse pitch
sensors with uniform $30\text{\,}\mathrm{ps}\text{/}$ and
$15\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}20\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ resolutions are
well within reach. This performance can be achieved with a moderate
optimization of the layout and processing compared to the analyzed sensors,
including reducing the strip width and the surface resistivity. Improving the
gain uniformity is an important consideration as well, but uniformity at this
scale is routinely delivered by HPK and FBK for the CMS and ATLAS timing
detectors, and is expected to be achieved in future BNL productions, too.
Finally, although strips up to $1\text{\,}\mathrm{cm}\text{/}$ have already
demonstrated signal shapes adequate for $30\text{\,}\mathrm{ps}\text{/}$
resolution, maintaining this performance beyond
$2\text{\,}\mathrm{cm}\text{/}$ require further study and confirmation in the
next test beam campaign.

As shown in Table 2 and Figure 18, the BNL 10–200 sensor can achieve
simultaneous $32\text{\,}\mathrm{ps}\text{/}$ and
$18\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ resolutions considering the
high gain regions. As result, this class of sensors represents a strong
candidate for future 4D tracking detectors, providing excellent position and
time resolutions with small channel count compared to standard tracking
sensors.

Table 2: Summary of spatial and timing performance for the five fully analyzed sensors. The time resolution for the high gain regions is shown. The position resolutions and efficiencies are shown for both the one and two strip categories.  Name | Time resolution | Spatial resolution
---|---|---
| Exactly one strip | Two strip
High gain | Resolution | Eff. | Resolution | Eff.
Unit | $\mathrm{p}\mathrm{s}$ | $\mathrm{\SIUnitSymbolMicro m}\text{/}$ | - | $\mathrm{\SIUnitSymbolMicro m}\text{/}$ | -
BNL 5–200 | 30 $\pm$ 1 | 61 $\pm$ 1 | 35% | 12 $\pm$ 1 | 65%
BNL 10–100 | 35 $\pm$ 1 | 69 $\pm$ 1 | 23% | 19 $\pm$ 1 | 77%
BNL 10–200 | 32 $\pm$ 1 | 82 $\pm$ 1 | 43% | 18 $\pm$ 1 | 57%
BNL 10–300 | 36 $\pm$ 1 | 83 $\pm$ 1 | 51% | 16 $\pm$ 1 | 49%
BNL 25–200 | 51 $\pm$ 1 | 128 $\pm$ 1 | 82% | 31 $\pm$ 1 | 18%

Figure 18: Summary of the position and time resolutions as a function of the
track $x$ position for the BNL 5-200 sensor (left) and BNL 10-200 (right).
Both the position and time resolutions are measured in the sensor region with
high-gain only.

## Acknowledgments

We thank the Fermilab accelerator and FTBF personnel for the excellent
performance of the accelerator and support of the test beam facility, in
particular M. Kiburg, E. Niner, N. Pastika, E. Schmidt and T. Nebel. We also
thank the SiDet department, in particular M. Jonas and H. Gonzalez, for
preparing the readout boards by mounting and wirebonding the AC-LGAD sensors.
Finally, we thank L. Uplegger for developing and maintaining the telescope
tracking system, and R. Lipton for useful discussions.

This document was prepared using the resources of the Fermi National
Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of
Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance,
LLC (FRA), acting under Contract No. DE-AC02-07CH11359. This work was also
supported by the U.S. Department of Energy under grant DE-SC0012704; used
resources of the Center for Functional Nanomaterials, which is a U.S. DOE
Office of Science Facility, at Brookhaven National Laboratory under Contract
No. DE-SC0012704; supported by the Chilean ANID PIA/APOYO AFB180002 and ANID -
Millennium Science Initiative Program - ICN2019_044. This research was also
performed using the resources of the University of Illinois at Chicago and
supported by the U.S. Department of Energy under grant DE-FG02-94ER40865.

## References

  * [1] D. Berry, V. Cairo, A. Dragone, M. Centis-Vignali, G. Giacomini, R. Heller, S. Jindariani, A. Lai, L. Linssen, R. Lipton, C. Madrid, B. Markovic, S. Mazza, J. Ott, A. Schwartzman, H. Weber, and Z. Ye, 4-Dimensional Trackers, . https://arxiv.org/abs/2203.13900.
  * [2] M. Duranti, V. Vagelli, G. Ambrosi, M. Barbanera, B. Bertucci, E. Catanzani, F. Donnini, F. Faldi, V. Formato, M. Graziani, M. Ionica, L. Moriconi, A. Oliva, A. Serpolla, G. Silvestre, and L. Tosti, Advantages and Requirements in Time Resolving Tracking for Astroparticle Experiments in Space, Instruments 5 (2021) no. 2, . https://www.mdpi.com/2410-390X/5/2/20.
  * [3] E. Sicking, Detector requirements for future high-energy collider experiments, . https://indico.cern.ch/event/813597/contributions/3727952/attachments/1988376/3314100/EvaSicking_DetectorRequirements.pdf. TREDI 2020.
  * [4] C.-E. Wulz, Report from ECFA, the European Committee for Future Accelerators, EPJ Web of Conferences 95 (2015) 06003.
  * [5] K. M. Black et al., Muon Collider Forum Report, . https://arxiv.org/abs/2209.01318.
  * [6] N. Bartosik, A. Bertolin, L. Buonincontri, M. Casarsa, F. Collamati, A. Ferrari, A. Ferrari, A. Gianelle, D. Lucchesi, N. Mokhov, M. Palmer, N. Pastrone, P. Sala, L. Sestini, and S. Striganov, Detector and Physics Performance at a Muon Collider, Journal of Instrumentation 15 (2020) no. 05, P05001–P05001. https://doi.org/10.1088/1748-0221/15/05/p05001.
  * [7] R. A. K. et al, Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, arXiv:2103.05419 [physics.ins-det].
  * [8] G. Giacomini, W. Chen, G. D’Amen, and A. Tricoli, Fabrication and Performance of AC-coupled LGADs, JINST 14 (2019) P09004. https://doi.org/10.1088%2F1748-0221%2F14%2F09%2Fp09004.
  * [9] M. Mandurrino, R. Arcidiacono, M. Boscardin, N. Cartiglia, G. F. Dalla Betta, M. Ferrero, F. Ficorella, L. Pancheri, G. Paternoster, F. Siviero, and M. Tornago, Demonstration of 200-, 100-, and 50- $\mu$ m Pitch Resistive AC-Coupled Silicon Detectors (RSD) With 100$\%$ Fill-Factor for 4D Particle Tracking, IEEE Electron Device Letters 40 (2019) no. 11, .
  * [10] M. Mandurrino, R. Arcidiacono, M. Boscardin, N. Cartiglia, G. F. D. Betta, M. Ferrero, F. Ficorella, L. Pancheri, G. Paternoster, F. Siviero, V. Sola, A. Staiano, and A. Vignati, Analysis and numerical design of Resistive AC-Coupled Silicon Detectors (RSD) for 4D particle tracking, Nucl. Inst. Meth. A 959 (2020) 163479.
  * [11] N. Cartiglia, Issues in the design of Ultrafast silicon detectors, TREDI2015, Trento, Italy . https://indico.cern.ch/event/351695/contributions/828366/.
  * [12] A. Apresyan, W. Chen, G. D’Amen, K. F. Di Petrillo, G. Giacomini, R. Heller, H. Lee, S. Los, C. S. Moon, and A. Tricoli, Measurements of an AC-LGAD strip sensor with a 120 GeV proton beam, JINST 15 (2020) no. 09, P09038, arXiv:2006.01999 [physics.ins-det].
  * [13] R. Heller, C. Madrid, A. Apresyan, W. Brooks, W. Chen, G. D'Amen, G. Giacomini, I. Goya, K. Hara, S. Kita, S. Los, A. Molnar, K. Nakamura, C. Peña, C. S. Martín, A. Tricoli, T. Ueda, and S. Xie, Characterization of BNL and HPK AC-LGAD sensors with a 120 GeV proton beam, JINST 17 (2022) no. 05, P05001, arXiv:2201.07772 [physics.ins-det].
  * [14] M. Tornago, R. Arcidiacono, N. Cartiglia, M. Costa, M. Ferrero, M. Mandurrino, F. Siviero, V. Sola, A. Staiano, A. Apresyan, K. Di Petrillo, R. Heller, S. Los, G. Borghi, M. Boscardin, G.-F. Dalla Betta, F. Ficorella, L. Pancheri, G. Paternoster, H. Sadrozinski, and A. Seiden, Resistive AC-Coupled Silicon Detectors: Principles of operation and first results from a combined analysis of beam test and laser data, Nucl. Inst. Meth. A 1003 (2021) . https://www.sciencedirect.com/science/article/pii/S016890022100303X.
  * [15] J. Ott et al., Investigation of signal characteristics and charge sharing in AC-LGADs with laser and test beam measurements, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 1045 (2023) 167541. https://www.sciencedirect.com/science/article/pii/S0168900222008336.
  * [16] Fermilab Test Beam Facility, https://ftbf.fnal.gov.  https://ftbf.fnal.gov.
  * [17] R. Heller, A. Abreu, A. Apresyan, R. Arcidiacono, N. Cartiglia, K. DiPetrillo, M. Ferrero, M. Hussain, M. Lazarovitz, H. Lee, S. Los, C. Moon, C. Peña, F. Siviero, V. Sola, T. Wamorkar, and S. Xie, Combined analysis of HPK 3.1 LGADs using a proton beam, beta source, and probe station towards establishing high volume quality control, Nucl. Instrum. Meth. A 1018 (2021) 165828. https://www.sciencedirect.com/science/article/pii/S0168900221008135.
  * [18] RD53 Collaboration, M. Garcia-Sciveres, The RD53A Integrated Circuit, , CERN, Geneva, 2017.  https://cds.cern.ch/record/2287593.
  * [19] H. F.-W. Sadrozinski, S. Ely, V. Fadeyev, Z. Galloway, J. Ngo, C. Parker, B. Petersen, A. Seiden, A. Zatserklyaniy, N. Cartiglia, F. Marchetto, M. Bruzzi, R. Mori, M. Scaringella, A. Vinattieri, Ultra-fast silicon detectors, Nucl. Instrum. And Meth. in Phys. Res. A 730 (2013) 226–231.