# Intrinsic timing properties of ideal 3D-trench silicon sensor with fast
front-end electronics

Gian Matteo Cossu11footnotetext: Corresponding authors Davide Brundu  and
Adriano Lai

###### Abstract

This paper describes the fundamental timing properties of a single-pixel
sensor for charged particle detection based on the 3D-trench silicon
structure. We derive the results both analytically and numerically by
considering a simple ideal sensor and the corresponding fast front-end
electronics in two different case scenarios: ideal integrator and real fast
electronics (trans-impedance amplifier). The particular shape of the Time of
Arrival (TOA) distribution is examined and the relation between the time
resolution and the spread of intrinsic charge collection time is discussed, by
varying electronics parameters and discrimination thresholds. The results are
obtained with and without simulated electronics noise. We show that the
3D-trench sensors are characterized by a _synchronous region_ , i.e. a portion
of the active volume which leads to the same TOA values when charged particles
cross it. The synchronous region size is dependent on the front-end
electronics and discrimination threshold, and the phenomenon represents an
intrinsic physical effect that leads to the excellent time resolution of these
sensors. Moreover, we show that the TOA distribution is characterized by an
intrinsic asymmetry, due to the 3D geometry only, that becomes negligible in
case of significant electronics jitter.

##  1 Introduction

High-luminosity conditions expected for the next generation of particle
physics experiments require the use of vertex and tracking detectors with
extreme requirements in terms of radiation resistance, spatial and time
resolutions [1, 2, 3, 4]. The renewed interest within the scientific community
has led to several studies on different new technologies for future tracking
systems, investigating also the possibility to measure simultaneously time and
spatial coordinates (4D tracking) [5, 6, 7]. A comprehensive discussion on the
topic can be found in [8, 9].

In this work, we study silicon 3D sensors [6] with trench structure, which
have proven to be extremely radiation resistant and to have unprecedented time
resolution, with the drawback of higher capacitance and complex fabrication
[10, 11, 12]. Beyond the experimental and technological aspects, an analytical
study of their fundamental physical properties is not present in the
literature. On the contrary, in-depth studies have been carried out on the
physical properties of planar structures [13]. In this work we derive
analytically the fundamental and intrinsic timing properties of 3D silicon
sensors with trench technology in the case of ideal fast front-end
electronics, starting from first principles. In fact, it is well-known that,
for planar sensors, signal shape variations as a function of the hit position
in the pixel are caused by a non-uniform weighting field and non-saturated
drift velocity, while here we focus on the fact that for silicon sensors with
3D-trench structure another source of signal shape variations is due to the
different inducing times of charge carriers, which arise only from the
particular 3D geometry. Moreover, we discuss the results by using numerical
simulations with custom simulation packages [14], in the case of realistic
fast front-end electronics. All the results obtained both analytically and
numerically are first discussed without introducing electronics noise (Sec.
2), while in Sec. 3.4 we study the timing properties in the presence of noise.
These results are in agreement with already performed experimental
measurements on 3D-trench silicon pixel sensors, obtained at test beam [10,
15, 16] and with in-laboratory characterization with a laser setup [17].

##  2 Time resolution of ideal 3D-trench sensor with ideal fast electronics

In this section we discuss the transient currents in a 3D-trench silicon
sensor, starting from first principles and deduce the properties of the
voltage signal and the time of arrival (TOA) distribution by considering a
transfer function of an ideal fast electronics (i.e. an ideal integrator).

###  2.1 Properties of transient currents

We consider a 3D silicon sensor with a standard parallel electrode
configuration: two ohmic trenches are located at the two opposite sides of the
pixel, while a third trench, used as a readout electrode, is placed at the
pixel centre, parallel to the two ohmic trenches (Fig. 1). Without loss of
generality, we can study the transient currents in one half of the sensor, by
considering the region between the readout and one ohmic trench: the geometry
is then a simple parallel-plate sensor, as shown in Fig. 2.

In the case of pure vertical minimum ionizing particles (MIPs), the effect of
Landau fluctuation of energy loss influences only signal amplitudes, causing a
time walk effect on output signals. However, as we will discuss later, since
this effect can be corrected by the proper discrimination method (constant
fraction, use of time-over-threshold, etc.), we simplify the model by not
introducing Landau fluctuations. In the case of particles crossing the sensor
with a non-null tilt angle this approximation is, in principle, no more valid,
since also the shape of the signals is affected. Moreover, we will not
introduce higher-order effects like weighting field unevenness, non-saturated
drift velocity, charge diffusion, etc. in order to derive time resolution
contributions due to the 3D geometry only.

Figure 1: Silicon sensor with 3D geometry, with track and charge deposit
representation.

Figure 2: Ideal parallel-plate sensor with 3D geometry (left), example of
boxed current with induction duration $t_{e}$ for the electrons and $t_{h}$
for the holes (right)

We indicate with $d$ the distance between electrodes and with $x$ the position
of the vertical track passing through the sensor. By applying the Ramo theorem
[18] to the configuration just discussed, in the case of tracks parallel to
the electrodes, the resulting transient currents are simply boxed currents.
Since all the carriers with the same polarity have always the same coordinate
at every instant, all the electrons and holes currents can be summed up and
the current is equivalent to the sum of two contributions,

$I(t)=\frac{Q_{in}v_{e}}{d}\theta\Big{(}\frac{x}{v_{e}}-t\Big{)}+\frac{Q_{in}v_{h}}{d}\theta\Big{(}\frac{d-x}{v_{h}}-t\Big{)},$
(2.1)

where $Q_{\text{in}}$ is the overall charge deposited within the sensor,
$v_{e}$ and $v_{h}$ are the carriers velocities, $t$ is the time and $\theta$
is the Heaviside step function. The duration of the signal is given by the
carrier that has to cover the longer distance, depending on the $x$ position.
We can define the two times, $t_{e}$ and $t_{h}$,

$t_{e}=\frac{x}{v_{e}},\hskip 28.45274ptt_{h}=\frac{d-x}{v_{h}}.$ (2.2)

The duration of the signal will be then the maximum between these two values,

$t_{c}=\text{max}\\{t_{e},t_{h}\\}.$ (2.3)

The center of gravity time (COG), as defined in [13], is

${t_{\text{cog}}}(x)=\frac{1}{2d}\Big{(}\frac{x^{2}}{v_{e}}+\frac{(d-x)^{2}}{v_{h}}\Big{)}.$
(2.4)

For electronics with a time constant $\tau\gg t_{\text{cog}}$ the time
resolution can be estimated by considering the time displacement of the pulse
response due to signals with COG time equal to $t_{\text{cog}}$. Following
calculation in [19] the standard deviation of the COG time can be written as

$\sigma_{{t_{\text{cog}}}}=\sqrt{\overline{t_{\text{cog}}^{2}}-\overline{t_{\text{cog}}}^{2}}=\sqrt{\frac{4}{180}\frac{d^{2}}{v_{e}^{2}}-\frac{7}{180}\frac{d^{2}}{v_{e}v_{h}}+\frac{4}{180}\frac{d^{2}}{v_{h}^{2}}},$
(2.5)

where the over-line represents the average calculation. The resolution becomes
smaller if the distance $d$ is kept smaller. It is worth noting that, with a
planar detector without a gain mechanism, this leads to the fact that the
noise will become the dominant contribution to time resolution [13], not
allowing to improve further the performance. This is not the case with 3D
sensors, where the signal-to-noise ratio (SNR) can be kept high by choosing a
proper thickness of the detector, allowing it to have more charge and, at the
same time, maintaining the timing characteristics that depend only on the
electrode distance.

###  2.2 Properties of output signals and the synchronous region

In order to relate the intrinsic properties of currents with observed timing
properties of 3D-trench sensors we need to perform a convolution between the
transient currents in Eq. 2.1 and a transfer function describing a fast front-
end electronics. Let us consider a simple ideal integrator: the output voltage
signal, $V(t)$, is the one obtained across the capacitance $C_{D}$ of the
detector,

$V(t)=\frac{1}{C_{D}}\int^{t}I(t^{\prime})dt^{\prime}.$ (2.6)

Considering the current in Eq. 2.1, the voltage signal for every position $x$
can be written as

$V(t)=\frac{1}{C_{D}}\Bigg{(}\int_{0}^{t}\frac{Q_{in}v_{e}}{d}\theta\Big{(}\frac{x}{v_{e}}-t^{\prime}\Big{)}dt^{\prime}+\int_{0}^{t}\frac{Q_{in}v_{h}}{d}\theta\Big{(}\frac{d-x}{v_{h}}-t^{\prime}\Big{)}dt^{\prime}\Bigg{)}.$
(2.7)

Since the currents are boxed, with constant amplitude, $V(t)$ is a simple ramp
signal. It is important to note that the slope changes when one of the charge
carriers stops inducing, namely when it has arrived at the corresponding
electrode, as shown in Fig. 3.

Figure 3: a) Voltage signal for different boxed current, b) Different voltage
signal and fixed threshold $V_{th}$: the brown signal will cross the threshold
all at the same time $t_{s}^{min}$ while the blue signal will have different
TOA ranging from $t_{s}^{min}$ to $t_{s}^{max}$, c) Synchronous region inside
the detector (green region) with all events that produce the same time of
arrival $t_{s}^{min}$

Given the $V(t)$ signal, let us focus now on the discrimination properties: if
we consider a certain threshold, $V_{th}$, corresponding to a certain charge
$Q_{s}=V_{th}C_{D}$, the threshold is reached at a time $t_{s}$ such that

$V_{th}=V(t_{s})=\frac{1}{C_{D}}\Bigg{(}\int_{0}^{t_{s}}\frac{Q_{in}v_{e}}{d}\theta\Big{(}\frac{x}{v_{e}}-t^{\prime}\Big{)}dt^{\prime}+\int_{0}^{t_{s}}\frac{Q_{in}v_{h}}{d}\theta\Big{(}\frac{d-x}{v_{h}}-t^{\prime}\Big{)}dt^{\prime}\Bigg{)}.$
(2.8)

Neglecting the capacitance $C_{D}$ for the moment, since it is just a
multiplicative factor, the threshold charge $Q_{s}$ can be written as

$Q_{s}=\Bigg{(}\int_{0}^{t_{s}}\frac{Q_{in}v_{e}}{d}\theta\Big{(}\frac{x}{v_{e}}-t^{\prime}\Big{)}dt^{\prime}+\int_{0}^{t_{s}}\frac{Q_{in}v_{h}}{d}\theta\Big{(}\frac{d-x}{v_{h}}-t^{\prime}\Big{)}dt^{\prime}\Bigg{)},$
(2.9)

and by considering Eq. 2.2 the previous equation becomes

$Q_{s}=\frac{Q_{in}v_{e}}{d}\Big{[}\Big{(}t_{s}-t_{e}\Big{)}\theta\Big{(}t_{e}-t_{s}\Big{)}+t_{e}\Big{]}+\frac{Q_{in}v_{h}}{d}\Big{[}\Big{(}t_{s}-t_{h}\Big{)}\theta\Big{(}t_{h}-t_{s}\Big{)}+t_{h}\Big{]}.$
(2.10)

Depending on the impact point $x$ of the track, at the given threshold
$V_{th}$, the time $t_{s}$ ranges between a minimum $t_{s}^{min}$ and a
maximum $t_{s}^{max}$, as shown in Fig. 3 (b). It is interesting to examine
the case of $t_{s}^{min}$, which is obtained by requiring the following
conditions,

$t_{s}<t_{e}~{}~{}~{}\text{and}~{}~{}~{}t_{s}<t_{h},$ (2.11)

which means that we are taking into account only the signals where the charge
$Q_{s}$ is reached with both carriers still contributing to the induction, as
shown in Fig. 3 (b) and Fig. 4.

Figure 4: Integrals of the boxed current: for the red current the charge
$Q_{s}$ is reached with both carriers still contributing to the induction, for
the green current one carrier has arrived at the corresponding electrode and
the other has to induce the remaining charge to arrive at the value $Q_{s}$.

In this case, following Eq. 2.10, we obtain that both the functions
$\theta\big{(}t_{e}-t_{s}\big{)}$ and $\theta\big{(}t_{h}-t_{s}\big{)}$ are
equal to 1 and we can write

$Q_{s}=\frac{Q_{in}(v_{e}+v_{h})}{d}t_{s}^{min},$ (2.12)

and, correspondingly,

$t_{s}^{min}=\frac{Q_{s}d}{Q_{in}(v_{e}+v_{h})}.$ (2.13)

This is an important result because the time $t_{s}^{min}$ depends only on
constant values, having specified the threshold $Q_{s}$. Thus, it can be
concluded that all the tracks impacting the sensor in positions $x$ satisfying
the conditions in Eq. 2.11 produce transient currents whose output signals
reach the threshold at the same time $t_{s}^{min}$. This effect is peculiar of
3D-trench geometry, and the corresponding $x$ range defines a Synchronous
Region within the sensor. The events belonging to the synchronous region
satisfy

$x=\begin{cases}~{}~{}~{}~{}x\geq v_{e}t_{s}\hskip
42.67912pt~{}~{}~{}~{}\text{for electrons,}\\\ ~{}~{}~{}~{}x\leq
d-v_{h}t_{s}\hskip 34.14322pt\text{for holes.}\end{cases}$ (2.14)

For simplicity, we can now consider the case where the velocities $v_{e}$ and
$v_{h}$ are equal. The time becomes

$t_{s}^{min}=\frac{d}{2v}\frac{Q_{s}}{Q_{in}}=\frac{d}{2v}\beta,$ (2.15)

where $\beta$ indicates the fraction of the threshold with respect to
$Q_{in}$. The events satisfying Eq. 2.14 have positions $x\geq x_{s}$ and
$x\leq d-x_{s}$ with

$x_{s}=vt_{s}^{min},$ (2.16)

and stay in a region with width $L_{s}$ given by

$L_{s}=d-2x_{s}=d(1-\beta),$ (2.17)

which defines the width of the synchronous region, as shown in Fig. 3 (c). It
is worth noting that $L_{s}$ depends on the chosen threshold $Q_{s}$.

On the contrary, events with impact position outside the synchronous region,
$x<x_{s}$ or $x>d-x_{s}$, have a different time at threshold $t_{s}$, bigger
than $t_{s}^{min}$. Given the symmetry of the problem, considering a position
closer to the electrode where electrons induce and following Eq. 2.10 we
obtain

$Q_{s}=\frac{Q_{in}v_{e}}{d}t_{e}+\frac{Q_{in}v_{h}}{d}t_{s},$ (2.18)
$t_{s}=\frac{d}{v}\frac{Q_{s}}{Q_{in}}-t_{e},$ (2.19)
$t_{s}=2t_{s}^{min}-\frac{x}{v}.$ (2.20)

We can see now that the time $t_{s}$ depends on the impact point $x$, i.e. the
$x$ coordinates are defining a non-synchronous region. The maximum of $t_{s}$
is obtained for $x=0$, and peculiarly its value is twice $t_{s}^{min}$,

$t_{s}^{max}=2t_{s}^{min}.$ (2.21)

###  2.3 Time of arrival distribution and propagation coefficient

In order to study the time resolution of the ideal 3D-trench discussed above,
we consider vertical tracks uniformly distributed on the coordinates $x$. All
the tracks impacting the sensor within the synchronous region are
characterized by a peaking TOA distribution because all the events have the
same time of arrival $t_{s}^{min}$. A similar TOA distribution shape has been
already seen in the case of Germanium gamma-ray detectors, where the sharp
peak was due to events in the central region of the detector [20]. The events
populating the narrow peak correspond to a fraction $\alpha$ of the total,
given by

$\alpha=\frac{Q_{in}-Q_{s}}{Q_{in}}.$ (2.22)

The remaining fraction $\beta=(1-\alpha$) shows a uniform TOA distribution
(Fig. 5) with width $\Delta t_{s}=t_{s}^{max}-t_{s}^{min}=t_{s}^{min}$ and
standard deviation $\sigma_{\beta}=\frac{t_{s}^{min}}{\sqrt{12}}$.

Figure 5:  (Top) Distribution of time of arrival TOA with threshold $Q_{s}$:
it presents a delta distribution with TOA $t_{s}^{min}$ for a fraction
$\alpha$ of the events, and a uniform distribution due to fraction $\beta$ of
events outside the synchronous region. (Bottom) Single distribution of the two
regions: a delta distribution with $\sigma_{\alpha}=0$ and a uniform
distribution with average $\mu_{\beta}$ and standard deviation
$\sigma_{\beta}$ .

The synchronous events, having all the same TOA, have a standard deviation

$\sigma_{\alpha}=0.$ (2.23)

We remark in this context that this result is ideal, with the aim of
understanding the fundamental properties of 3D-trench geometry. Moreover, it
is obtained by considering an ideal fast electronics without noise. As we will
discuss later in the text, in the presence of noise the synchronous peak
becomes larger due to the noise convolution, and the overall distribution is
characterized by a narrow peak and a longer right tail, as observed
experimentally and found in simulations [10].

The total TOA distribution is characterized by a probability density function
(PDF) $p(t)$ as a mixture of two PDFs, indicated as $p_{\alpha}(t)$ and
$p_{\beta}(t)$, for the synchronous and not synchronous contribution
respectively. They are summed up with weights $\alpha$ and $\beta$ and are
characterized by averages $\mu_{\alpha}=t_{s}^{min}$ and
$\mu_{\beta}=\frac{3}{2}t_{s}^{min}$ and standard deviations $\sigma_{\beta}$
and $\sigma_{\alpha}$. The total PDF can be written as

$p(t)=\alpha p_{\alpha}(t)+\beta p_{\beta}(t).$ (2.24)

The overall standard deviation of the mixture distribution is given by

$\sigma_{t_{s}}^{2}=\alpha(\sigma_{\alpha}^{2}+\mu_{\alpha}^{2})+\beta(\sigma_{\beta}^{2}+\mu_{\beta}^{2})-\mu^{2},$
(2.25)

where $\mu=\alpha\mu_{\alpha}+\beta\mu_{\beta}$,

$\mu=(1-\beta)t_{s}^{min}+\beta\frac{3}{2}t_{s}^{min},$ (2.26)
$\mu=t_{s}^{min}\frac{(2+\beta)}{2}.$ (2.27)

Substituting the corresponding values we find

$\sigma_{t_{s}}^{2}=\frac{(t_{s}^{min})^{2}}{12}(4\beta-3\beta^{2}).$ (2.28)

Figure 6:  Propagation coefficient $\mathscr{P}(\beta)$ as a function of the
threshold $\beta$.

Since the minimum time $t_{s}^{min}$ is proportional to the chosen threshold
(Eq. 2.15), the total resolution can be also written as a function of the
carrier velocity $v$, the electrode distance $d$ and the fraction $\beta$ as

$\sigma_{t_{s}}=\frac{d}{2v}\frac{\beta\sqrt{4\beta-3\beta^{2}}}{\sqrt{12}},$
(2.29)

that is a monotonically increasing function with respect to the fraction
$\beta$. At this point, it is interesting to relate the time resolution
$\sigma_{t_{s}}$ of the output signal with the charge collection time (CCT)
distribution, which is an intrinsic property of the sensor, namely the
distribution populated with all the collection times $t_{c}$ as defined in Eq.
2.3. We find that for carriers with equal velocities this distribution is
symmetric, as shown in Fig. 7 (a), with limits given by

$t_{c}=\begin{cases}t_{c}^{min}=\frac{d}{2v}&\mbox{if }x=\frac{d}{2},\\\
t_{c}^{max}=\frac{d}{v}&\mbox{if }x=d\end{cases},$ (2.30)

and, correspondingly, the standard deviation is given by

$\sigma_{t_{c}}=\frac{t_{c}^{max}-t_{c}^{min}}{\sqrt{12}}=\frac{t_{c}^{min}}{\sqrt{12}}.$
(2.31)

Thus, the time resolution in Eq. 2.29 can be written as

$\sigma_{t_{s}}={\beta\sqrt{4\beta-3\beta^{2}}}\cdot\sigma_{t_{c}}\implies\sigma_{t_{s}}=\mathscr{P}(\beta)\cdot\sigma_{t_{c}},$
(2.32)

where we introduced the function of the threshold $\mathscr{P}(\beta)$ called
Timing Propagation Coefficient from the sensor to front-end electronics, as
defined in [21] and calculated analytically in this case for the ideal
integrator (Fig. 6),

$\mathscr{P}(\beta)={\beta\sqrt{4\beta-3\beta^{2}}}.$ (2.33)

The timing propagation coefficient measures the capability of fast electronics
to reduce the intrinsic time collection dispersion of the sensor transient
currents, as a function of the threshold chosen. In order to make a comparison
between slow and fast electronics, let us consider the resolution calculated
using Eq. 2.4 following [13], with equal carrier velocities $v$,

$\sigma_{t_{cog}}=\frac{d}{v}\frac{1}{\sqrt{180}},$ (2.34)

that can be written in terms of the standard deviation of the CCT
$\sigma_{t_{c}}$ as

$\sigma_{t_{cog}}=\frac{2}{\sqrt{15}}~{}\sigma_{t_{c}}.$ (2.35)

This calculation assumes electronics with time constant $\tau\gg t_{cog}$,
thus for slow electronics and equal velocities the propagation coefficient
$\mathscr{P}$ of an ideal 3D-trench silicon sensor is equal to

$\mathscr{P}=\frac{2}{\sqrt{15}}\sim 0.516.$ (2.36)

and does not depend on the threshold. In our calculation, we find that a fast
electronics, e.g an ideal integrator, reaches the same result if a threshold
$\beta\sim 0.46$ is chosen, while for lower thresholds the resolution improves
significantly (e.g. $\mathscr{P}=0.33$ for $\beta=0.33$ ).

Figure 7:  a) CCT distribution with same velocities $v$. b) CCT with different
velocities $v_{e}$ and $v_{h}$.

###  2.4 Asymmetry of the TOA distribution

Another important aspect of the TOA distribution is that the synchronous peak
causes the distribution to become asymmetric, even in the case of carriers
with the same saturation velocities $v$. Since we are studying an ideal
sensor, with a constant electric field and constant saturation velocities of
carriers over the entire active region of the sensor, this asymmetry cannot be
caused by unevenness or non-uniformity of the weighting field. Indeed, it is
an intrinsic characteristic of the 3D geometry and it is strictly related to
the different charge collection times, since the charged carriers have to
cover different distances depending on the position $x$. The intrinsic
asymmetry becomes stronger if a lower threshold $\beta$ is chosen (higher
$\alpha$) and the resolution improves (Fig. 8).

Figure 8:  a) TOA distribution for different thresholds $\beta$. doubling the
threshold, i.e $\beta_{2}=2\beta_{1}$ the distribution gets larger and
delayed; also the synchronous peak halves in amplitude. b) synchronous region
for $\alpha_{1}$. c) synchronous region for $\alpha_{2}=\alpha_{1}/2$

Indeed the size of the synchronous region becomes smaller with a higher
threshold $\beta$ (Fig. 8). We can generalize easily the results to the case
of different saturation velocities. The TOA distribution acquires another
source of asymmetry as shown in Fig. 9 (a). Following Eq. 2.18, we find two
values for the time $t_{s}^{max}$, in particular

$t_{s,e}^{max}=\frac{Q_{s}d}{Q_{in}v_{e}},~{}~{}~{}~{}~{}~{}~{}~{}t_{s,h}^{max}=\frac{Q_{s}d}{Q_{in}v_{h}}.$
(2.37)

The minimum value, $t_{s}^{min}$, is given by Eq. 2.13. In this case, the
synchronous region is no more symmetric with respect to the sensor geometry,
and it gets closer to the electrode at a lower potential, as shown in Fig. 9
(b). The size of the synchronous region is given by

$L_{s}=d-x_{s_{1}}-x_{s_{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{s_{1}}=v_{e}t_{s}^{min},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{s_{2}}=v_{h}t_{s}^{min}.$
(2.38)

Figure 9:  a) Time of arrival distribution in the case we have different
saturation velocities $v_{e}$ and $v_{h}$. b) Synchronous region with
different drift velocities for the carriers, that gets closer to the holes
electrode.

####  2.4.1 Propagation coefficient $\mathscr{P}$ with different saturation
velocities

The propagation coefficient $\mathscr{P}$ in case of different saturation
velocities can be obtained by considering the CCT distribution shown in Fig. 7
(b). The standard deviation of the total distribution can be calculated again
by considering the mixture of two distributions, finding

$\sigma_{t_{c}}^{2}=\frac{(t_{c}^{min})^{2}}{12}\frac{v_{e}^{4}+2v_{e}^{3}v_{h}-5v_{e}^{2}v_{h}^{2}+2v_{e}v_{h}^{3}+v_{h}^{4}}{v_{e}^{2}v_{h}^{2}}.$
(2.39)

If we introduce the parameter $k$ as the ratio of hole and electron
velocities, $v_{h}=k\cdot v_{e}$, we can express the standard deviation
$\sigma_{t_{cog}}$ in Eq. 2.5 in terms of $\sigma_{t_{c}}$,

$\sigma_{t_{cog}}^{2}=\frac{(4k^{2}-7k+4)(1+k)^{2}}{15(k^{4}+2k^{3}-5k^{2}+2k+1)}{\sigma_{t_{c}}^{2}}.$
(2.40)

Thus, the following function,

$\mathscr{P}(k)=\sqrt{\frac{(4k^{2}-7k+4)(1+k)^{2}}{15(k^{4}+2k^{3}-5k^{2}+2k+1)}},$
(2.41)

represents the Timing propagation coefficient of an ideal 3D-trench sensor
with slow front-end electronics as a function of the factor $k$. The function
is shown in Fig. 10 (a). For equal velocities (i.e $k=1$) we obtain the
propagation coefficient already discussed in Eq. 2.35, while for $k<1$ the
value decreases until reaches its minimum, given by

$\mathscr{P}\Big{(}k=\frac{3-\sqrt{5}}{2}\Big{)}=\frac{\sqrt{30}}{12}\sim
0.456.$ (2.42)

This justifies a criterion that, in order to estimate the achievable
resolution of a 3D silicon sensor that produces currents with different
collection times $t_{c}$, one can consider the CCT distribution and take half
the standard deviation $\sigma_{t_{c}}$.

It is instructive to consider an even simpler model of sensor, where the
currents are rectangular pulses with different duration $t_{c}$, uniform
collection time distribution with standard deviation $\sigma_{t_{c}}$,
following [21]. In this case, for slow front-end electronics, we find that
$\sigma_{t_{s}}$ is exactly half of $\sigma_{t_{c}}$, as the centroid
resolution would suggest.

Figure 10:  a) Timing propagation coefficient with slow electronics as a
function of the ratio of the saturation velocities $k$. For equal saturation
velocities, we get the value 2.35, for smaller $k$ (i.e smaller holes
velocities) the propagation coefficient gets smaller reaching a minimum where
$k=\frac{3-\sqrt{5}}{2}\sim 0.38$ (i.e $v_{e}\sim 2.63~{}v_{h}$). b) Timing
propagation coefficient for the ideal integrator as a function of the
threshold $\beta$ and the ratio $k$ of the velocities of the carriers.

The definition of the timing propagation coefficient becomes useful when we
consider faster electronics because, as shown in [21] and 2.33, this can be
expressed as a function of the threshold, to manifest the ability of a fast
electronics to reduce the intrinsic collection time dispersion. Considering
now the distribution Fig. 9, we can take the mixture of three distributions,
following the same method used in the previous sections, to find the
propagation coefficient for the ideal integrator with different saturation
velocities of carriers. In particular, we have

$\sigma_{t_{s}}^{2}=\frac{\beta^{3}\Big{(}4(v_{e}^{4}-v_{e}^{3}v_{h}+v_{e}^{2}v_{h}^{2}-v_{e}v_{h}^{3}+v_{h}^{4})-3\beta(v_{e}^{2}-v_{e}v_{h}+v_{h}^{2})^{2}\Big{)}}{(v_{e}^{4}+2v_{e}^{3}v_{h}-5v_{e}^{2}v_{h}^{2}+2v_{e}v_{h}^{3}+v_{h}^{4})}\sigma_{t_{c}}^{2},$
(2.43)

$\mathscr{P}(\beta,v_{e},v_{h})=\sqrt{\frac{\beta^{3}\Big{(}4(v_{e}^{4}-v_{e}^{3}v_{h}+v_{e}^{2}v_{h}^{2}-v_{e}v_{h}^{3}+v_{h}^{4})-3\beta(v_{e}^{2}-v_{e}v_{h}+v_{h}^{2})^{2}\Big{)}}{(v_{e}^{4}+2v_{e}^{3}v_{h}-5v_{e}^{2}v_{h}^{2}+2v_{e}v_{h}^{3}+v_{h}^{4})}}.$
(2.44)

For $k=1$ we find again the propagation coefficient given by 2.33. The
propagation coefficient for different drift velocities is shown in Fig.10 (b).

###  2.5 Correlation between time resolution and efficiency

The existence of the synchronous region has the effect to produce a
correlation between the resolution of the 3D detector and the efficiency. In
fact, once a certain threshold is chosen, if we are willing to sacrifice
efficiency and consider only the events crossing the synchronous region, we
would have a perfect detector with an ideal infinite resolution. This means
that with the ideal integrator and a threshold $Q_{s}$ at $\beta=30\%$, as an
example, the synchronous region would cover 70% of the sensor active volume
(i.e. not taking into account the trenches volume), thus obtaining a detector
with an efficiency $E_{\text{ff}}=70\%$ and resolution $\sigma_{t_{s}}=0\,ps$.
We remind that this result is derived by considering perpendicular tracks. The
calculation in the previous section can be extended to consider also the
efficiency, thus we can define a new Propagation Coefficient,
$\mathscr{P}(\beta,E_{\text{ff}})$, as

$\mathscr{P}(\beta,E_{\text{ff}})=\frac{\sqrt{(\beta+E_{\text{ff}}-1)^{3}(3+E_{\text{ff}}-3\beta)}}{E_{\text{ff}}},$
(2.45)

$\sigma_{t_{s}}=\mathscr{P}(\beta,E_{\text{ff}})\cdot\sigma_{t_{c}}.$ (2.46)

Figure 11:  a) TOA Distribution for a chosen $\beta$ with highlighted the
events for a reduced efficiency $E_{\text{ff}}$. b) Detector with a fraction
$E_{\text{ff}}$ of the detector and the corresponding synchronous region
$L_{s}$ inside. c) Propagation coefficient as a function of $\beta$ and
$E_{\text{ff}}$: the flat region is where the propagation coefficient
$\mathscr{P}(\beta,E_{\text{ff}})=0$ and the detector is a perfect detector.

In this case, if an efficiency $E_{\text{ff}}$ equal or smaller than $\alpha$
is chosen (also $E_{\text{ff}}\leq 1-\beta$), the resolution becomes zero
since we consider only events that are in the synchronous region (Fig. 11 and
Fig. 12 ). If we increase the efficiency the resolution has to gradually
worsen until the value in Eq. 2.32 is reached.

Figure 12:  a) Propagation coefficient $\mathscr{P}(\beta,E_{\text{ff}})$ for
different thresholds $\beta$. b) Resolution as a function of the efficiency
$E_{\text{ff}}$ for different thresholds $\beta$ of an ideal 3D detector with
$d=20~{}\mu m$ and saturation velocities $v=0.1\mu m/ps$.

##  3 Time resolution of ideal 3D-trench sensor with real fast electronics

In the previous section we discussed the properties of the voltage output
signals and the corresponding TOA distribution by considering a transfer
function of an ideal integrator. For a meaningful comparison with experimental
observations, a more realistic transfer function should be considered.
Unfortunately, the analytical calculations using such transfer functions are
no more straightforward, so we will derive the results by using numerical
simulations, with the possibility of interpreting the results in light of the
analytical description carried out for the simpler model. The simulation of
the front-end electronics has been performed by using the custom and open-
source software called TFBoost [14, 22], capable of carrying out precise and
very fast simulations of the entire read-out chain of 3D sensors [10].

###  3.1 Fast trans-impedance amplifier transfer function

The model of the ideal integrator is useful to understand what happens when
the electronics has a rise-time of the same order of magnitude as the current
pulse. A more realistic electronics response can be modelled as a second-order
trans-impedance amplifier (TIA), with DC trans-impedance $R_{m_{0}}$. The
linear system composed of sensor and electronics is characterized by a finite
bandwidth and a time constant $\tau$. The transfer function can be written as

$R_{m}(s)=\frac{R_{m_{0}}}{(1+s\tau)^{2}}.$ (3.1)

The output signals with this type of fast electronics, when the time constant
$\tau$ is of the same order of magnitude as the average current pulse
duration, can be obtained with TFBoost by making the convolution between the
transfer function in Eq. 3.1 and the boxed currents of the ideal 3D sensor
previously discussed (Fig. 2). Within TFBoost the transfer function is
described by using a small signal model, whose parameters are chosen in order
to have $\tau=160$ ps and $R_{m_{0}}=2.3~{}\text{k}\Omega$, representing a
realistic front-end electronics as described in [21].

Figure 13:  CCT distribution of the current used for the simulations with
TFBoost (left). Boxed current of the ideal 3D detector (right).

The parameters to obtain the transient currents (Fig. 13 (b)) used for the
simulations are chosen in order to represent a realistic 3D-trench silicon
sensor: electrode distance $d=20~{}\mu\text{m}$, uniform electric field, both
carriers in the condition of saturated drift velocity with
$v_{e}=0.107~{}\mu\text{m/ps}$ and $v_{h}=0.0837~{}\mu\text{m/ps}$. The
corresponding CCT distribution is shown in Fig. 13 (a). Using Eq. 2.39 we find
the standard deviation of the CCT distribution equal to
$\sigma_{t_{c}}=35.4~{}\text{ps}$, in agreement with the value found in the
simulation, and an average current duration of $~{}\overline{t_{c}}\sim
160~{}\text{ps}$. Since the ratio between the saturation velocities is equal
to $k=0.78$, looking at Fig. 10 we can estimate the time resolution obtainable
with slow electronics simply by considering half of $\sigma_{t_{c}}$, i.e.
$\sim 17.7~{}\text{ps}$. With fast electronics the time resolution will depend
on the chosen discrimination method and threshold used, thus a scan in the
$\beta$ value can be studied by applying a constant fraction discrimination to
the simulated output signals. The corresponding propagation coefficient is
shown in Fig. 14 (a).

Figure 14:  a) Comparison of the propagation coefficient relative to the ideal
integrator with that obtained with fast TIA electronics. b) Ballistic deficit
on the output signals with the fast TIA when $\tau=\overline{t_{c}}$.

As can be seen from Fig. 14 (b), the output signals with fast electronics are
characterized by different shapes, since they reflect the different transient
current shapes at electronics input. In contrast, in the case of slow
electronics the output signal shapes are all equal, being the impulsive
response of the electronics. Fig. 14 (b) shows the output signals obtained
with the convolution of the transfer function in Eq. 3.1 with the input
currents. Having chosen the time constant $\tau$ equal to the average current
duration $\overline{t_{c}}$, for some signals the input charge is not
completely integrated, which leads to the well-known phenomenon of ballistic
deficit. From the different rise-time values of the signals in Fig. 14 (b) it
is clear that the dispersion of the TOA is maximum when using higher
thresholds while it tends to zero as the threshold approaches the signal
baseline. Considering a threshold $\beta=0.2$ we obtain a propagation
coefficient $\mathscr{P}\sim 0.2$ and consequently an intrinsic temporal
resolution of $\sigma_{t_{s}}=\mathscr{P}(\beta)\sigma_{t_{c}}\sim 7ps.$

The real threshold that can be used will obviously depend on the amount of
noise present in the system. This will affect both the intrinsic resolution
obtainable and another fundamental contribution to the temporal resolution of
the system which is given by the electronic jitter $\sigma_{ej}$, which will
be discussed in Sec 3.4.

###  3.2 Intrinsic asymmetry and synchronous region

The output signals obtained with the fast TIA have been discriminated with a
CFD at different thresholds by using the TFBoost package. The TOA distribution
map in the case of two examples, with $\beta=0.35$ and $0.8$, are shown in Fig
15, where it is possible to see how the synchronous region is wider at a lower
threshold, covering almost half of the whole area of the sensor, while it is
tighter when the higher threshold is used. Moreover, in the case of
$\beta=0.8$, the TOA distribution is shifted towards higher time values, as
shown clearly in Fig 16.

Figure 15: TOA as a function of the impact point in the sensor, obtained by
using CFD with $\beta=0.35$ (left) and $\beta=0.8$ (right). Figure 16:
Intrinsic TOA distribution obtained by using CFD with $\beta=0.35$ (blue) and
$\beta=0.8$ (black).

This result is compatible with the analytical results obtained in the case of
the ideal integrator model, shown in Fig. 8. However, an interesting
difference to note concerns the shape of the synchronous peak, which is not
perfectly sharp as in the case of the ideal integrator, but has a smoother
behaviour for increasing time values, as shown in Fig. 16 where a log scale
for the vertical axis is used to emphasize this effect. Being the transition
between the two components smoother, also the edge of the synchronous region
is no longer abrupt, as can be seen also in Fig. 17, where the TOA as a
function of the particle impact point along the X axis is shown.

Figure 17: TOA as a function of the impact point in the sensor along the X
axis, obtained by using CFD with $\beta=0.35$ with the Ideal Integrator
(right) and with the Fast TIA (left).

It is important to point out that also with realistic electronics, such as the
fast TIA, we find the asymmetry introduced in Sec. 2.4 (see Fig. 16). Since we
are using transient currents from a simple 3D sensor model, where only the 3D
geometry is simulated, the asymmetry of the TOA distribution is due to the
intrinsic 3D geometrical properties only, and not due to higher-order effects.

###  3.3 Mixture of comprehensive TOA distributions

In the presence of electronics noise the intrinsic TOA distribution of the
ideal integrator will be distorted and enlarged (Fig. 18).

Figure 18: Effect of the electronic jitter on the TOA distribution of the
ideal integrator.

From a mathematical point of view the intrinsic TOA PDF, $p(t)$, is convoluted
with the jitter PDF, $J(t)$ with standard deviation $\sigma_{ej}$, leading to
the final PDF,

$P(t)=p(t)\otimes J(t).$ (3.2)

Recalling the mixture distribution in Eq. 2.24 we get

$P(t)=\alpha\Big{[}p_{\alpha}(t)\otimes
J(t)\Big{]}+\beta\Big{[}p_{\beta}(t)\otimes J(t)\Big{]}.$ (3.3)

In order to simplify the notation we indicate the two convoluted PDFs for the
synchronous and not synchronous regions as $P_{\alpha}(t)$ and $P_{\beta}(t)$
respectively, and their standard deviation as $\Sigma_{\alpha}$ and
$\Sigma_{\beta}$ respectively. Being the contributions from the two PDFs
uncorrelated, the standard deviation can be summed up in quadrature so that
$\Sigma_{\alpha}^{2}=\sigma_{\alpha}^{2}+\sigma_{ej}^{2}$ and
$\Sigma_{\beta}^{2}=\sigma_{\beta}^{2}+\sigma_{ej}^{2}$. The total time
resolution of the mixture will be then given by

$\sigma_{t}^{2}=\alpha(\Sigma_{\alpha}^{2}+\mu_{\alpha}^{2})+\beta(\Sigma_{\beta}^{2}+\mu_{\beta}^{2})-\mu^{2},$
(3.4)

where we made the assumption that the $J(t)$ has its mean $\mu_{J}=0$ so that
$\mu_{\alpha}$, $\mu_{\beta}$ and $\mu$ are the same mean values in Eq. 2.25 .
An example of the final PDF $P(t)$ is shown in Fig. 19.

Figure 19: PDF P(t) as a mixture of the distributions $P_{\alpha}(t)$ and
$P_{\beta}(t)$.

Since the PDF $p_{\alpha}(t)$ has $\sigma_{\alpha}=0$, we find that the TOA
distribution of the ideal integrator has the important characteristic that the
synchronous peak relative to the fraction of events $\alpha$, in the presence
of electronic jitter, represents the distribution of the jitter itself, being
the convolution of $J(t)$ PDF and the synchronous delta-function. However, in
the case of fast TIA we found that the synchronous peak, indicated here by
$p_{\alpha}(t)$, is no more a delta-function, as shown in Fig. 16. Thus, for a
real fast TIA the peak function $P_{\alpha}$ receives contributions from both
the jitter and the intrinsic sensor distribution.

The $P(t)$ function is able to describe experimental data in the case of TOA
measurements with 3D-trench sensors and realistic electronics, so it is
important to study its analytical properties in order to fit the data
accurately. For this purpose, the shape of $J(t)$ plays a crucial role. If the
voltage signal shapes and amplitude do not vary significantly event-by-event
and if the voltage noise is Gaussian, the electronics jitter $J(t)$ is itself
Gaussian, with a standard deviation $\sigma_{ej}$ that depends on the output
noise RMS, $\sigma_{V}$, of the amplification stage and on the slope
$V^{\prime}_{\text{thr}}$ of the signal at the chosen discrimination
threshold: $\sigma_{ej}=\sigma_{V}/V^{\prime}_{\text{thr}}$. The function
$J(t)$ starts to deviate from a Gaussian function if one of the mentioned
hypotheses is not true. In particular, as we observed in Fig. 14 (right), fast
front-end electronics can be sensitive to the different transient current
shapes and be affected by the ballistic deficit. Each different shapes
contribute to the jitter function with a different $V^{\prime}_{\text{thr}}$,
thus the final $J(t)$ is no more exactly Gaussian. Another example is when we
are in the presence of Landau fluctuations. The signals are characterized by a
wide range of amplitudes and, consequently, by a wide range of different
slopes $V^{\prime}_{\text{thr}}$. In order to study these effects we performed
simulations with the TFBoost package where we emulate realistic noise samples
starting from experimental measurements [10] (in particular we used a Gaussian
red noise with 0.985 correlation), and we introduced Landau fluctuations
considering the toy model 150 $\mu$m thick 3D sensor Fig. 20 (left).

Figure 20: Left: signal amplitude distribution considering a sensor thickness
of 150 $\mu$m obtained with the fast electronics. Right: electronics jitter
distribution, obtained as the difference between the TOA measured with and
without noise (event-by-event) in the simulation. The solid line represent a
fit using the mixture of two Gaussians while the dashed line is the result of
a fit with a single Gaussian function. Figure 21: TOA distributions of voltage
signals using the fast electronics, $\beta=0.35$ and SNR$\sim 20$. The blue
PDF is the mixture of the two Gaussians used to model the synchronous peak,
while the black PDF is a single Gaussian to model the not synchronous
contribution.

The corresponding $J(t)$ distribution is obtained by calculating the event-by-
event difference of TOA values, measured with and without noise, and it is
shown in Fig. 20 (right), where the deviation from Gaussianity is evident. In
this example the TOA are evaluated with a CFD with $\beta=0.35$, and the
threshold is kept fixed between the measurements with and without noise. The
$J(t)$ can be modelled as a mixture of Gaussian PDFs with different standard
deviation $\sigma_{ej}$, corresponding to the different slopes
$V^{\prime}_{\text{thr}}$. From a pure experimental point of view, it is
interesting to investigate how many Gaussian functions are needed within the
mixture model to fit realistic data distributions with acceptable accuracy. By
using one single Gaussian the systematic uncertainty on the $\sigma_{ej}$
evaluation is up to $\sim 10\%$, while by using two Gaussians it becomes $\sim
2\%$ and it is sufficient to properly model the jitter behaviour, as shown in
Fig. 20 (right). The comprehensive distribution, as described in Eq. 3.3, is
shown in Fig. 21, where the intrinsic asymmetry and effect of the electronic
jitter $J(t)$ on the shape of the final PDF can be seen. Given the nature of
the two original distributions (synchronous and not synchronous) the proper
model to fit the $P(t)$ has to be a mixture of PDFs.

From a pure experimental point of view it is essential to find a simple
mixture model. The one shown in Fig. 21 is composed of three Gaussians, two of
them with the same mean to describe the synchronous peak (which we expect is
mainly given by the electronic jitter of the system $J(t)$, as discussed
previously), while the third is used to model the not synchronous
contribution. The standard deviation of the overall mixture model is
consistent with the true standard deviation of the distribution.

###  3.4 Noise contribution to the symmetry of TOA distribution

In the previous Section 3.3, we have discussed how the jitter distribution
$J(t)$ can modify the shape of the final TOA distribution. This effect can be
more or less significant depending on the SNR or, equivalently, on the
relative value of the standard deviations $\sigma_{t_{s}}$ and $\sigma_{ej}$
of the two distributions, $p(t)$ and $J(t)$, respectively.

Figure 22: Intrinsic TOA distributions for different SNR: $\text{SNR}=\infty$
(top left), $\text{SNR}=40$ (top right), $\text{SNR}=20$ (bottom left),
$\text{SNR}=10$ (bottom right).

Fig 22 shows the TOA distribution for different SNR values. In the limit of
infinite SNR there is no jitter, and we find the intrinsic distribution of the
sensor already seen in Fig. 16, while for lower SNR, the TOA distribution
becomes more and more symmetrical. Depending on the amount of jitter present,
the mixture model for the final distribution must be adapted to the specific
case. For example, looking at Fig 22, for SNR lower than 20, a mixture model
with two Gaussian functions is accurate enough to properly describe the final
distribution, and allows to estimate the time resolution with an accuracy of
order $\mathcal{O}(1\%)$.

Since the intrinsic asymmetry is a peculiar feature of 3D sensors, it is
important to point out that the asymmetry of the comprehensive distribution,
which has been brought out with this simple 3D sensor model, experimentally
cannot be ignored just by taking into account the resolution of the peaking
structure, because this can lead to a severe underestimation of the time
resolution $\sigma_{t}$. Other aspects that affect the duration of induction
of the carriers, such as weighting field non-uniformity, will further modify
the TOA distribution, whose shape will be however characterised by intrinsic
asymmetry.

##  4 Conclusions

In this work the intrinsic timing properties of a single-pixel 3D-trench
sensor have been discussed in the case of fast front-end electronics. In order
to perform the study, the transient currents have been obtained analytically
by using a simple ideal model to take into account the 3D geometry of the
trench structure. The first result is that we can analytically relate the time
resolution, i.e. the standard deviation of TOA distribution, $\sigma_{t_{s}}$,
and the CCT standard deviation $\sigma_{t_{c}}$ through the timing propagation
coefficient $\mathscr{P}$, using an ideal integrator as fast electronics. In
this case, the propagation coefficient $\mathscr{P}$ depends on the
discrimination threshold, in contrast to the slow electronics case. Moreover,
we found that the TOA distribution of the single-pixel 3D-trench sensor is
composed of two contributions: a synchronous component ideally with
$\sigma_{t}=0$, corresponding to all the events given by the particles passing
through the synchronous region of the pixel active volume, and a non-
synchronous component. This property leads to a strong asymmetry of the TOA
distribution even considering non-ideal front-end electronics such as fast
TIA. This intrinsic asymmetry is clearly mitigated by the presence of noise,
but for SNR $>10$ it remains visible in the final TOA distribution, as already
observed in accurate simulations and experimental results [10, 16]. We remark
that the presence of the synchronous region and the corresponding TOA
intrinsic asymmetry arises from the 3D-trench geometry only and it represents
a fundamental feature of these sensors. Furthermore we proposed, as an
experimental procedure, the usage of a probabilistic mixture model in order to
properly describe the TOA contributions and determine correctly the final time
resolution. The usage of different models, in particular the ones neglecting
the asymmetric right tail of the distribution, could lead to a wrong
estimation of the time resolution.

## References

  * [1] CMS Collaboration.  A MIP Timing Detector for the CMS Phase-2 Upgrade. Technical Design Report.  CERN-LHCC-2019-003, CMS-TDR-020, 2019.
  * [2] ATLAS Collaboration.  Technical Proposal: a High-Granularity Timing Detector for the ATLAS Phase-II Upgrade.  CERN-LHCC-2018-023, LHCC-P-012, 2018.
  * [3] LHCb collaboration.  Physics case for an LHCb Upgrade II — Opportunities in flavour physics, and beyond, in the HL-LHC era.  CERN-LHCC-2018-027, LHCb-PUB-2018-009, 2018.
  * [4] FCC Collaboration.  The Future Circular Collider (FCC) Conceptual Design Report.  CERN-ACC-2018-0056, CERN-ACC-2018-0057, CERN-ACC-2018-0058, CERN-ACC-2018-0059, 2018.
  * [5] G. Pellegrini, P. Fernández-Martínez, M. Baselga, C. Fleta, D. Flores, V Greco, S. Hidalgo, I. Mandić, G. Kramberger, D. Quirion, and M. Ullan.  Technology developments and first measurements of low gain avalanche detectors (lgad) for high energy physics applications.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 765:12–16, 2014\.  HSTD-9 2013 - Proceedings of the 9th International "Hiroshima" Symposium on Development and Application of Semiconductor Tracking Detectors.
  * [6] Parker, S.I. and Kok A., and Kenney C. and Jarron P. and Hasi J. and Despeisse M. and Da Via C. and Anelli G.  Increased speed: 3d silicon sensors; fast current amplifiers.  IEEE Trans. Nucl. Sci., 58 (2), pp. 404-417, 2011.
  * [7] A. Lai, L. Anderlini, M. Aresti, A. Bizzeti, A. Cardini, G.-F. Dalla Betta, G.T. Forcolin, M. Garau, A. Lampis, A. Loi, C. Lucarelli, R. Mendicino, R. Mulargia, M. Obertino, E. Robutti, and S. Vecchi.  First results of the TIMESPOT project on developments on fast sensors for future vertex detectors.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 981:164491, 2020\.
  * [8] N. Cartiglia, R. Arcidiacono, M. Costa, M. Ferrero, G. Gioachin, M. Mandurrino, L. Menzio, F. Siviero, V. Sola, and M. Tornago.  4d tracking: present status and perspectives.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 1040:167228, 2022.
  * [9] Adriano Lai.  4d-tracking in the 10-ps range: A technological perspective.  Frontiers in Physics, 10, 2022.
  * [10] D. Brundu, A. Cardini, A. Contu, G.M. Cossu, G.-F. Dalla Betta, M. Garau, A. Lai, A. Lampis, A. Loi, M.M. Obertino, B.G. Siddi, and S. Vecchi.  Accurate modelling of 3d-trench silicon sensor with enhanced timing performance and comparison with test beam measurements.  Journal of Instrumentation, 16(09):P09028, Sep 2021.
  * [11] J. Lange, G. Giannini, S. Grinstein, M. Manna, G. Pellegrini, D. Quirion, S. Terzo, and D. Vázquez Furelos.  Radiation hardness of small-pitch 3d pixel sensors up to a fluence of $3\times 10^{16}$ ${\rm n}_{\rm eq}/{\rm cm}^{2}$.  Journal of Instrumentation, 13(09):P09009, 2018.
  * [12] R. Mendicino, G. T. Forcolin, M. Boscardin, F. Ficorella, A. Lai, A. Loi, S. Ronchin, S. Vecchi, and G.-F. Dalla Betta.  3D trenched-electrode sensors for charged particle tracking and timing.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 927:24–30, 2019\.
  * [13] Werner Riegler and Gianluca Aglieri Rinella.  Time resolution of silicon pixel sensors.  JINST, 12(11):P11017, 2017.
  * [14] D. Brundu, A. Contu, G. M. Cossu, and A. Loi.  Modeling of Solid State Detectors Using Advanced Multi-Threading: The TCoDe and TFBoost Simulation Packages.  Frontiers in Physics, 10, 2022.
  * [15] F. Borgato, D. Brundu, A. Cardini, G. M. Cossu, G. F. Dalla Betta, M. Garau, L. La Delfa, A. Lai, A. Lampis, A. Loi, M. M. Obertino, G. Simi, and S. Vecchi.  Charged-particle timing with 10 ps accuracy using timespot 3d trench-type silicon pixels.  Frontiers in Physics, 11, 2023.
  * [16] A. Lampis, F. Borgato, D. Brundu, A. Cardini, G.M. Cossu, G.-F. Dalla Betta, M. Garau, L. La Delfa, A. Lai, A. Loi, M. Obertino, G. Simi, and S. Vecchi.  10 ps timing with highly irradiated 3d trench silicon pixel sensors.  Journal of Instrumentation, 18(01):C01051, jan 2023.
  * [17] Mauro Aresti, Davide Brundu, Alessandro Cardini, Andrea Contu, Gian Matteo Cossu, Gian-Franco Dalla Betta, Michela Garau, Adriano Lai, Andrea Lampis, and Angelo Loi.  A complete in-laboratory characterization of 3d trench-type silicon pixel sensors.  In 2021 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), pages 1–5, 2021.
  * [18] Ramo, S.  Currents Induced by Electron Motion.  Proceedings of the IRE, vol. 27, no. 9, pp. 584-585, 1939, doi: 10.1109/JRPROC.1939.228757.
  * [19] Werner Riegler.  Time resolution of silicon sensors.  CERN Detector Seminar, url: https://indico.cern.ch/event/1083146/attachments/2328724/3975380/detector_seminar_oct_15_2021_timing_riegler_v2.pdf, 2021\.
  * [20] F.S Goulding.  Semiconductor detectors for nuclear spectrometry, i.  Nuclear Instruments and Methods, 43(1):1–54, 1966.  Proceedings of the Tenth Summer Meeting of Nuclear Physicists.
  * [21] Adriano Lai and Gian Matteo Cossu.  High-resolution timing electronics for fast pixel sensors.  arXiv:2008.09867, Aug 2020.
  * [22] D. Brundu and G. M. Cossu.  dbrundu/TFBoost: TFBoost v0.2-alpha.  November 2020, Zenodo. doi: 10.5281/zenodo.4291148.