Study of Silicon Photomultipliers - Physikalisches Institut Heidelberg

Transcription

Joëlle Barral
Promotion X2001
Ecole Polytechnique 1 , France
Study of Silicon Photomultipliers
13th April-2nd July 2004
Supervisors : Masahiro Teshima 2 and Caroline Terquem 1 .
Max Planck Institut für Physik 2
Werner-Heisenberg Institut
Münich, Germany
Abstract
The registration of low-intensity light photons flux is one of the critical issues for experimental
physics. Silicon Photomultipliers (SiPM) are very promising new photodetectors. They are formed by an
area of many independent pixels (typically 576 for a 1 mm2 device). Each pixel is an avalanche photodiode (APD) working in Geiger-mode at a low bias voltage (56V) with a gain of 106 . The output signal
gives the number of pixels fired by photons, the detection efficiency for the actual devices is 20%.
In this study, the main properties of SiPMs (single photon counting, time resolution, recovery time,
dark noise, afterpulsing, cross-talk) are discussed. A comparison between measurement results and expected features - relying on previous studies, Monte-Carlo simulations and analytical calculations - is
presented. A control of the saturation of the dynamic range is obtained. Finally, SiPMs are demonstrated
to be good candidates for Positron Emission Tomographs as an alternative for APDs.
Résumé
La détection de flux de photons de faible intensité est un point clé de la physique expérimentale.
Les photomultiplicateurs à silicium (SiPM) sont de nouveaux photodétecteurs très prometteurs. Ils sont
formés d’une matrice de nombreux pixels indépendants (typiquement 576 pour un détecteur d’ 1 mm2 ).
Chaque pixel est une photodiode à avalanche fonctionnant en mode Geiger à faible tension de polarisation
(56V), dont l’amplification est de 106 . Le signal de sortie donne le nombre de pixels touchés, l’efficité de
détection est de 20% pour les détecteurs actuels.
Nous présentons les principales caractéristiques des SiPM (comptage de photons un à un, résolution
temporelle, temps mort, bruit noir, afterpulse, crosstalk). Nous comparons les résultats de mesure aux
résultats attendus, en nous référant aux études antérieures ainsi qu’aux simulations et aux calculs analytiques effectués. Nous montrons qu’un contrôle de la saturation de l’échelle dynamique est possible, et
que les SiPMs sont de bons candidats pour remplacer les APDs dans les tomographes par émissions de
positrons.
Acknowledgments1
I would like to thank Professor Masahiro Teshima for his invitation to work at the Max Planck Institute
for Physics and his welcome.
I am grateful to Nepomuk Otte, for his availability and his patience, and to the whole Magic/Euso
group, especially Florian, Daniel, Hendrik, Markus, Mustapha, beide Jürgen, Masaaki, Robert, David,
Keiichi, Satoko, Kenji, Nadia, Sybille and professors Wolfgang Wittek, Razmick Mirzoyan and Eckart
Lorenz.
1 I will also be really grateful to those who will send me the misprints or failures they would have noticed :
joelle.barral@polytechnique.org
1
Contents
Introduction
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Topology
1.1 Avalanche Photodiodes . . . . . . . .
1.1.1 Proportional & Geiger modes
1.1.2 Quenching . . . . . . . . . .
1.2 Silicon Photomultipliers . . . . . . .
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Energy features
2.1 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Calibration . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Single Photoelectron Counting . . . . . . . . . . . . .
2.2 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Dependence of the efficiency with the supplied voltage
2.3 Dynamic range . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Saturation . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Application . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Dispersion of the signal . . . . . . . . . . . . . . . .
2.4 Sensitivity to the environment . . . . . . . . . . . . . . . . .
2.4.1 Magnetic field . . . . . . . . . . . . . . . . . . . . .
2.4.2 Behavior with temperature . . . . . . . . . . . . . . .
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Time features
3.1 Randomness in physical mechanisms . . . . . . . . . . . . . .
3.2 Rise time & Time resolution . . . . . . . . . . . . . . . . . .
3.2.1 Rise time . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Time resolution . . . . . . . . . . . . . . . . . . . . .
3.3 Recovery time . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 First measurement . . . . . . . . . . . . . . . . . . .
3.3.2 Strange double afterpulses . . . . . . . . . . . . . . .
3.3.3 Recovery time depending on the number of pixels fired
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Noise
4.1 Dark-Counting Rate : Primary noise . . . . . . . .
4.2 Trapping phenomena - Afterpulsing . . . . . . . .
4.2.1 Definition . . . . . . . . . . . . . . . . . .
4.2.2 Method : Time Correlated Carrier Counting
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3
CONTENTS
4.3
5
4.2.3 Experiment & Results . . . . . . . . . . . . . . . .
Cross-talk . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Cross-talk reduction . . . . . . . . . . . . . . . . .
4.3.3 Measurement . . . . . . . . . . . . . . . . . . . . .
4.3.4 Cross-talk dependence on the number of pixels fired
Sodium Spectrum
5.1 Principle of Detection . . . . . . . . . . . . . . . . . . .
5.2 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Medical Applications . . . . . . . . . . . . . . . . . . .
5.4 Time Resolution . . . . . . . . . . . . . . . . . . . . . .
5.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Jitter in the LSO . . . . . . . . . . . . . . . . .
5.5.2 Time resolution for the coincidence measurement
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Conclusion
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A Leading Edge vs Constant Fraction Discriminator
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B Simulations : Scilab code
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C Dynamic range
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D Recovery time
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E PET
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F Temperature
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G Abbreviations used
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Introduction2
"Man was made to stand upright and contemplate the sky."
Immanuel Kant
Contemplating the sky has enabled mankind to better understand its role in the world. Moreover, this
contemplation has been the source of the most beautiful discoveries in physics, from the revolution of
planets to the cosmic microwave background. To pursue the human quest of investigating and understanding the cosmos, we continually need new instruments and better technology. The requirements of
astronomy and the extreme conditions prevalent in space force us to be more and more ingenious in finding new means of seeing. We need to develop detectors for the new messengers, such as neutrinos or
gravitational waves, that the cosmos is sending us. Astronomy proposes us a challenge. Instruments are
our means of scratching the veneer to discover what is hidden beyond.
EUSO (Extreme Universe Space Observatory) will investigate Extreme Energy Component of the Cosmic Radiation, E > 5.1019 , where the GZK effect is predicted to strongly attenuate cosmic ray background. This particle energy domain also approaches the Grand Unified Theory limits. EUSO is based
on the Extensive Air Showers induced in the atmosphere by the high energy primary cosmic radiation.
A highly beamed Cherenkov radiation is generated by the ultra-relativistic particles in the EAS and scattered by the atmosphere. The total energy caused by the charged secondary particles is converted into
fluorescence photons through the excitation of the air N2 molecules. EUSO will be looking downward,
using the Earth’s atmosphere as a giant detector. It will be sensitive in visible and near UV wavelength
region.
Historically, opening new energy windows has led to the discovery of totally unexpected phenomena...
Figure 1: Galilean telescope
2 Picture
on the cover : EUSO logo, Nuit étoilée, Vincent Van Gogh, June 1889
4
Chapter 1
Topology
1.1
Avalanche Photodiodes
An APD is basically a p-n junction diode operated at a large reverse bias voltage. The physical mechanism upon which avalanche gain depends is the impact ionization. It occurs when the electric field in the
depletion region is strong enough : an electron colliding with a bound valence electron transfers enough
energy to the electron to ionize it. This creates an additional electron-hole pair producing current gain.
The additional carriers, in turn, can gain sufficient energy from the electric field to cause further impact
ionization, creating an avalanche of carriers.
Figure 1.1: Basis structure of an APD
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CHAPTER 1. TOPOLOGY
6
Figure 1.2: Electric field distribution in epitaxy layer
Around an avalanche photodiode, a guard ring is sent to prevent breakdown and tunnelling (because
the diode is operating at large bias voltage) at the edge of the diode. Consequentially, the diode becomes
insensitive for photons of a relative large area at the circumference of the diode.
The reflection on the surface of top contact is reduced by an antireflection coating Si3 N4 .
1.1.1
Proportional & Geiger modes
In a proportional counter, each original electron leads to an avalanche which is basically independent
of all other avalanches formed from other electrons associated with the original ionizing event. The
collected charge remains proportional to the number of original electrons.
Figure 1.3: Impact ionization : Proportional mode
7
CHAPTER 1. TOPOLOGY
With an higher electric field, a situation is created, in which one avalanche can itself trigger a second
avalanche at a different position.
Figure 1.4: Impact ionization : Geiger mode
The difference between both modes relies on the holes : in Geiger mode they trigger avalanches,
whereas in proportional mode they have not enough energy to do so (because their ionization coefficient
is much lower than the ionization coefficient of electrons). From the critical value of the electric field
( corresponding to the breakdown voltage), a self propagating chain reaction occurs. In principle, an
exponentially growing number of avalanches could be created. As the quenching takes place to a particular level, i.e. when almost the same number of avalanches has been created, all pulses have the same
amplitude, regardless of the number of original ion pairs that initiated the process.
1.1.2
Quenching
• Passive quenching : The avalanche photodiode (i.e. each pixel for the silicon photomultiplier) is
connected to the power supply through a large series resistor Rs ( Rs ≈ 100 − 400kΩ according the manufacturer). If the current through the diode equals zero, the voltage across the diode
equals Ubias , which is larger than the breakdown voltage. If the diode breaks down, the series
resistor reduces the voltage across the APD, what quenches the avalanche. After the breakdown is
quenched, the diode is recharged through the resistor. A drawback of passive quenching is then the
slow recharge of APD.
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CHAPTER 1. TOPOLOGY
Figure 1.5: Passive quenching
• Active quenching : The resistor is replaced by an electronic circuit. This circuit has to detect the
breakdown and quench it by quickly reducing the voltage across the APD for a certain time. After
that, the circuit has to recharge the APD to the operating voltage. This active quenching is not used
with SiPM because an electronic circuit would be required for each pixel, what would be quite
expensive.
Figure 1.6: Active quenching
CHAPTER 1. TOPOLOGY
1.2
9
Silicon Photomultipliers
Figure 1.7: Silicon Photomultipliers MEPhI & P U LSARr
The SiPM are 1 mm * 1 mm, composed of 24*24=576 pixels. Each pixel is 20 µm wide (active
area). 3 mm * 3 mm SiPM are now considered but some improvements in the fabrication process
are required to make them efficient at room temperature.
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CHAPTER 1. TOPOLOGY
Figure 1.8: SiPM connections
The resistance of each pixel allows the passive quenching (1.1.2) and serves also as a decoupling
element between the individual pixels because Cpixel Rpixel ∼
= 10−8 s tdischarge ∼
= 1ns. Pixels
are electrically decoupled from each other by polysilicon resistors and are connected by common
Al strips, in order to readout the SiPM signal.
Each SiPM pixel operates as a binary device but SiPM in whole is an analogue detector. The
output signal allows us to determine the number of pixels fired. The wavelength of the incident light
must be known by another way, so that we can convert this number of pixels fired in an energy.
SiPM is intrinsically very fast due to the very small width of the depletion region and the extremely
short Geiger type discharge.
Chapter 2
Energy features
2.1
2.1.1
Gain
Calibration
When an electron is thermally generated in a pixel of the SiPM array, it triggers an avalanche,
exactly as if the pixel would have been fired. On this dark noise, some cross-talk is observed :
photons emitted by the pixel "fired" may migrate to other pixels and trigger avalanches (cf. §4.1
and §4.3.4). We calibrate on this dark noise.
Figure 2.1: Cross-talk : y-axis=number of counts, x-axis=area of the dark noise signal on the scope.
Ubias =57V.
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CHAPTER 2. ENERGY FEATURES
Figure 2.2: Calibration on the dark noise. Ubias =57V
The single pixel signal is determined by the total charge Qpixel collected during the Geiger
discharge of the single-pixel capacitor :
gainpixel =
(Ubias − Ubreakdown )
Qpixel
= Cpixel ∗
|e|
|e|
We measure the area on the oscilloscope :
Z
Z
U (t)dt =
signal
R ∗ I(t)dt = 50Ω ∗ Qdetector
(2.1)
(2.2)
signal
As the equation (2.1) indicates, the gain is expected to be proportional to the overvoltage Ubias −
Ubreakdown . The following plot confirms it and allows us to determine the capacitance of a pixel
Cpixel and the breakdown voltage Ubreakdown .
13
Figure 2.3: Gain dependence with the supplied voltage
We find Cpixel = 36f F and Ubreakdown =51.3 V, which is in good agreement with the value given
by the manufacturer (50fF, 50V).
The gain is around 106 : the signal from one pixel is high enough to make the electronic noise
(i.e. the pedestal width) negligible. It is a benefit compared to APDs used in proportional mode
(compulsory to have an analogue device), whose gain is about 200.
2.1.2
Single Photoelectron Counting
We fire the detector with the laser, through an optical fiber put about 20 cm far from the detector.
A preamplifier is used after the read-out of the SiPM, because without, the amplitude of a signal
given by one pixel fired is about 6mV (Ubias = 56V ), and the sensitivity of the oscilloscope cannot
go below 2mV per division). We vary the supplied voltage.
Figure 2.4: Single Photoelectron Counting : Ubias = 52V
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15
The precision is very good : the single photoelectron counting is achieved, as soon as the supplied
voltage is high enough above the breakdown voltage. The difference in the mean number of pixels
fired for each voltage (whereas the set-up remains the same) will be discussed in the next section.
Gaussian curves do not fit these plots, we tried to fit with Poisson curves, which is not better, maybe
because of the definition of the Poisson used by our software root.
2.2
Efficiency
2.2.1
Definition
The efficiency is given by the ratio of the total amount of charge produced per incident photon :
= QE ∗ geom ∗ Geiger .
– QE is the quantum efficiency, i.e. the probability of a firing photon to be absorbed. To estimate
this efficiency, we must take into account :
∗ The reflection on the surface of top contact
∗ the effective thickness of top contact
∗ the optical absorption as a function of the wavelength and of the effective thickness of
the depletion layer.
In Si, for λ ≈ 400nm, QE ≈ 70%.
– geom is the geometrical efficiency, i.e. the ratio of the active area per the total area of the
device. To improve the geometrical efficiency, bigger pixels are considered.
– Geiger is the Geiger efficiency, i.e. the probability for a photoelectron to trigger an avalanche.
16
N
QE ∗ geom = photoelectrons
.
Nphotons
The efficiency of actual SiPM is about 20%.
For a given material, the wavelength range detection is limited on one side (long-wavelength cutoff) by the energy gap of the detector and on the other side (short-wavelength cut-off) by the very
large value of optical absorption.
2.2.2
Dependence of the efficiency with the supplied voltage
As we mentioned in 2.1.2, the number of pixels fired seems to depend on the supplied voltage, for
a fixed set-up.
It may be due to a dependence of the quantum efficiency with the supplied voltage, enhanced by
the increase of the crosstalk with the overvoltage (cf. 4.3.4). Actually, the Geiger efficiency is
17
expected to increase with the overvoltage. Two different plots fit quite well the the data. The exact
dependence could be further investigated.
2.3
2.3.1
Dynamic range
Saturation
As the detector has a finite number of pixels, the dynamic range (i.e. the maximal number of
photons that can be simultaneously detected) is limited. The saturation of the signal with increasing light intensity can be calculated (see C for the details of the calculation), assuming that the
dispersion of each number of photons arriving on the detector is poissonian (we must take it into
account, because the tail of this Poisson distribution will be more affected by the saturation than its
head). The number of pixels fired depends on the number of photons arriving on the detector as :
Npixelsf ired = m ∗ (1 − e−
Nphotons ∗
m
)
(2.3)
where is the detection efficiency and m the total number of pixels of the detector.
Figure 2.7: Saturation in the dynamic range
In this plot, we assumed an efficiency of 100% (To take the efficiency into account is like to consider
for the plot the x-axis as the number of photoelectrons multiplied by the Geiger efficiency).
2.3.2
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Application
The increase of total pixels seems technologically possible up to ≈ 4000pixels/mm2 . In the
hadron calorimeter of the TESLA experiment, for which SiPM have already been considered as
possible detectors, the minimal signal is supposed to be about 20photons/mm2 and the maximal
one 5000photons/mm2
Figure 2.8: Saturation depending on the total number of pixels. 20 photons firing
Figure 2.9: Saturation depending on the total number of pixels. 5000 photons firing
In this case, we should take the efficiency into account : 15%∗5000 = 750, and with 750 "efficient"
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photons arriving on the detector, the saturation is well below. These plots show the worth effect of
the saturation, if the efficiency were 100%.
2.3.3
Dispersion of the signal
The knowledge of the saturation (formula (2.3)) enables to deduce the number of photons arrived
on the detector from the number of pixels fired. As the number are integers, the precision in the
number of photons deduced is given by the following curve :
Figure 2.10: Precision in the number of photons deduced
For more than 4056 "efficient" photons, all the pixels are fired.
To estimate the dispersion of the signal, we have to take into account :
p
– The Poisson statistics dispersion Nphotons
– The imprecision in the number of photons deduced, due to the saturation (previous plot)
– The statistical fluctuation around the average saturation.
We can simulate the saturation and compute its dispersion :
20
Figure 2.11: Simulation of the saturation
Figure 2.12: Fluctuation around the saturation (number of pixels)
We can now assess the ratio (total dispersion)/signal
and its dependence with the number of firing
√
photons . It can be compared to the ratio
Nphotons
signal
for a detector without saturation.
21
Figure 2.13: Dispersion/Signal ratio depending on the number of photons per pixel
It shows that the curves begin to diverge from 2500 firing photons and that until 3700 firing photons,
the ratio dispersion/signal is better than 10% (except for the first values, but this is due to the
Poisson dispersion).
2.4
2.4.1
Sensitivity to the environment
Magnetic field
An avalanche photodiode has a high immunity to magnetic fields, due to the short path traversed
by charge carriers. It is a great benefit compared to photomultipliers. For example, the red book of
EUSO requires that the detectors have "low sensitivity to magnetic field of the order of the gauss",
which is related to the Space mission (magnetic field on earth ≈ 0.5 Gauss). It is also a reason why
SiPM are considered for the hadron calorimeter of TESLA (magnetic field ≈ 4 T).
2.4.2
Behavior with temperature
The behavior with temperature has to be further investigated (cf. [6]). The set-up on which we
are about to work is presented in appendix F, and the calculation of the pressure required to avoid
condensation is given.
Chapter 3
Time features
3.1
Randomness in physical mechanisms
The avalanche process starts when the first impact ionization, by the photo-generated pair, takes
place. The depth of the depletion layer puts a distance between the place where the photon is
absorbed and the high field region where the impact ionization occurs. As a result, it creates a
statistical delay and the time jitter can be evaluated as vws if w, the depletion layer thickness is
La , the wavelength, and where vs is the saturated velocity of the carriers in the electric field.
The avalanche multiplication is a stochastic process whose time constant is about 10 ps, which
is much lower than the value of the time resolution attained with the fastest devices.
The time resolution is then limited by the mechanisms involved in the spreading of the avalanche
over the device area. The avalanche is triggered in a single seed point. The transverse propagation
of the avalanche activation depends on the device geometry. When the multiplication process
occurs over the whole active area, the avalanche current reaches its final steady-state value. The
closer the seed point is to the center of the junction area, the faster is the activation of the whole
device and thus the rise of the avalanche current. The time resolution increases by reducing the
illuminated area on the photodiode.
3.2
Rise time & Time resolution
A first important characteristic of a detector is its ability to determine with precision when an
event occurs. This relies on :
– the rise time : the time taken by the signal to go from 10% to 90 % of its maximal value.
– the time resolution : the precision with which the detector will integrate the signal, i.e. the
time dispersion due to the detector.
22
CHAPTER 3. TIME FEATURES
3.2.1
23
Rise time
As the two following screen captures show, the rise time is about 1ns (Ubias = 56V ), whatever the number of pixels fired might be. The APDs have also a rise time of about 1 ns,
whereas PMT can achieve a rise time of 200 ps.
Figure 3.1: Signal observed on the oscilloscope (Thermal noise ⇔ 1 pixel fired)
Figure 3.2: Signal observed on the oscilloscope (≈ 500 pixels fired)
3.2.2
Time resolution
Set-up
The purpose is to determine the time resolution σ of one detector.
24
We fire two detectors with a pulsed laser triggered by a frequency generator. The pulsed
laser gives very short pulses (FWHM=50ps). The frequency scale extends from 40 kHz to 40
MHz. We can vary the frequency and the intensity of the laser. We trigger on the laser pulse.
Figure 3.3: Set-up to determine the time resolution
√
The time resolution of two detectors (= 2σ) is given by the variance of ∆t, the difference
between the arrival times of the two signals received. We can measure σ directly : the oscilloscope can determine the arrival time of both signals and calculate the jitter. We can also use
the computer (coupled to the oscilloscope, using the software Labviewr ), that reads all the
differences of arrival times : we have then its distribution. We can also use a discriminator to
determine with a higher precision the arrival times.
Electronics noise
We need to take into account the electronics noise. It includes the jitter of the laser, the
time resolution of the oscilloscope, and the scattering in the wires. We use a constant fraction
discriminator and not a leading edge discriminator in order to reduce the jitter introduced by
the walk (cf. appendix A).
The electronic output of the laser is taken as trigger for our measurement. This electronic
synchronization presents a jitter relative to the optical signal, which is given by the manufacturer to be below 20 ps (Picosecond Pulsed Diode Laser PDL 800-B). The electronics
noise can be measured by splitting a signal in two parts, and by giving them to two different
channels. We measure the dispersion of the arrival time of both signals.
25
Figure 3.4: Jitter introduced by the electronics
It gives us σelectronics = 7ps. This measurement does not take into account the time jitter
introduced by the different rise times of the signals. This walk is guaranteed by the manufacturer of the CFD we used (Ortec Quad 934) to be below 50 ps.
Laser frequency
We check that our measurement doesn’t depend on the frequency of the pulses. Each time,
10000 events are counted.
frequency (kHz)
σ (ps)
43.4
77
99.9
78
994
75
10001
76
40000
86
Figure 3.5: Reliability of the set-up
The variations are explained by the fact that the time resolution depends on the number of
pixels fired,
√related to the number of photons arriving on the detector, number n whose dispersion is n. At very high frequency, the recovery time must be taken into account : the
number of pixels fired is lower, hence the increase of the time resolution (cf. next §).
26
Time resolution dependence with the number of pixels fired
Global plot The saturation seen when a lot of pixels are fired gives us the time resolution
of the laser and the scope. We find σelectronics = 27ps, which is in good agrement with
the
q evaluation made before. We can then deduct the value of the time resolution by taking
2
(σmeasured
− σelectronics2 ) (our signal is the sum of two gaussian signals).
Figure 3.6: Dependence of the time resolution with the number of pixels fired
This curve follows a Poisson law. Above 40 pixels fired, the electronics noise limits the time
resolution of the set SiPM + read-out. The Poisson law binds the time resolution with the
number of pixels fired, and not with the number of photoelectrons, because of the saturation
(cf. §2.3.3). However, we are only interested by a window below 40 pixels fired, where the
effects of the saturation are negligible.
Time resolution for one pixel fired We find F W HM = 570ps, σ = 242ps. The tail of
the curve can be explained by the carriers trapped and released inside the time resolution, or
by the drift of the carrier in the depletion region before the impact ionization occurs.
27
Figure 3.7: Time resolution when one pixel is fired
Best time resolution The best time resolution is achieved when a lot of pixels are fired. The
time resolution found is then of the same order of magnitude as the electronics jitter.
Figure 3.8: Time resolution for more than 40 pixels fired
28
Dependence of the time resolution with the supplied voltage
The period of the laser is 289.96 ns. As the number of pixels fired depends from the supplied
voltage, we made the plot as follows :
Figure 3.9: Time resolution dependence with the supplied voltage
If the time resolution depends on the supplied voltage, the variations are negligible (regarding
the precision with which we know the number of pixels fired).
3.3
3.3.1
Recovery time
First measurement
A dead time follows each pulse, due to the finite time taken to quench the avalanche and
then reset the diode voltage to its initial bias value (cf. §1.1.2). We fire all the pixels of the
detector with the laser at a low frequency, at which we are sure to let the detector enough
time to recover. If we increase the frequency, the gain is constant (all the pixels are fired and
each pixel acts as a binary device), until the pixels fired have no time to recover after a signal,
before the next one comes. The gain is then lower than expected.
The temporal stability (for the duration of a measurement) has been controlled : it is better
than 0, 6% in all our measurements. Furthermore, we can observe that this stability increases
with the frequency.
−t
In the diode model, the recharge is proportional to 1 − e RC . An order of magnitude of the
recovery time might then be given by Rpixel ∗ Cpixel = 400kΩ ∗ 36f F ≈ 15ns (Rpixel is
indicated by P ulsarr and Cpixel has been calculated in §2.1.1).
29
Figure 3.10: Recovery time when all the pixels are fired
−t
Expecting a curve in 1 − e τ , the recovery time τ is then defined by the time after which 63%
of the pixels fired have recovered. We find τ = 1.2µs, and this value is almost independent
from the supplied voltage.
Figure 3.11: Dependence of the recovery time with the supplied voltage
This long, not well defined dead time can be attributed to the passive quenching. (cf. [10])
30
3.3.2
Strange double afterpulses
Enigma When we fired the SiPM by the pulsed laser, we found some strange afterpulses
for a specific frequency window of the generator triggering the laser.
Figure 3.12: Two strange pulses
These two afterpulses were observed by both detectors if we used two of them side by side.
A further inquiry showed that these strange afterpulses also depended from the width of the
signal delivered by the triggering generator.
Figure 3.13: Signal delivered by the frequency generator
31
Explanation For those frequencies the laser was re-triggered because the pulse given by
the generator was too broad or bad-defined.
To avoid this phenomena of undesirable afterpulses, we put the trigger level of the laser at
+1V (the signal delivered by the generator is 4.5 V high, this level can be chosen between
-1V and +1V, it hadn’t been checked before).
This point shows us, that the SiPM is able to detect, after 10 ns, a second signal with an
amplitude of about the quarter of the normal one (the intensity of the laser is the same for the
three signals, even at this frequency). In this case, a naive interpretation of the recovery time
12
(cf. next §) of the detector gives us τ = 42ns (solving 0.25 = 1 − e− τ ).
3.3.3
Recovery time depending on the number of pixels fired
Actually, the previous set-up, in which all the pixels are fired, gives us the recovery time of
each pixel. If the pixels are not all fired each time, they can take over from each other, and the
recovery time of the whole detector is shorter. Furthermore, a more realistic situation consists
in taking that the detector is not fired all the time, but that "sometimes" two consecutive
signals appear.
Two consecutive signals of N photons
Let us assume than only two signals of the same number of photons Nphotons are firing the
detector, the second one coming a time t after the first one. If each fired pixel recharges as
1 − e−t/τ , then the calculation (of course taking the saturation into account, cf. appendix D)
gives us such a recharge for the whole detector :
t
1 N
1
1
1
) ) ∗ (1 − )N + m ∗ (1 − (1 − )N ) ∗ (1 − (1 − )N ) ∗ (1 − e− τ )
m
m
m
m
(3.1)
where m is the total number of pixels of the device (576 in our case). This formula allows
us to calculate the recovery time for each number of pixels fired. For example, for two firing
signals of 300 efficient photons (i.e. Nphotons ∗ = 300), we find τ300 = 119ns.
m ∗ (1 − (1 −
32
Figure 3.14: Recovery time for two 300 efficient photons firing signals (number of pixels fired depending
on time (ns))
For a number of photoelectrons above 264, we can set the level in the definition of the recovery
time to 63% (below, the number of pixels untouched is high enough to have "instantaneously"
more than 63% of the maximal signal). The recovery time is then given by :
τNphotons = τ1 ∗ (1 + ln(1 − (1 −
1 Nphotons
)
))
m
where Nphotons is the number of efficient photons firing.
Figure 3.15: Recovery time (ns) depending on the number of efficient photons firing
(3.2)
33
The plot (3.12) found when the laser is re-triggered can then be explained with 815 efficient
photons firing, 436 pixels fired, and a recovery time of 866 ns.
An infinity of consecutive signals of N photons
Let us now assume that an "infinity" of signals, with the same number of efficient photons
Nphotons , fires the detector, each signal coming a time t after the previous one. Actually, a
real situation would be between this case and the case studied in the previous paragraph.
The calculation (cf. appendix D) gives us for the recharge of the detector :
m ∗ (1 − e−
Nphotons
m
t
)∗(
1 − e− τ
t
1 − e− τ −
Nphotons
m
)
(3.3)
Figure 3.16: Recovery time (ns) depending on the number of efficient photons firing, for two consecutive
signals (red plot) and for an infinity of consecutive signals (black plot)
Chapter 4
Noise
4.1
Dark-Counting Rate : Primary noise
The main source of noise limiting the SiPM performances is the dark noise rate, which mainly
originates from the carriers created thermally in the depletion layer.
Each electron-hole recombination mechanism can be reverse leading to a carrier generation.
Recombination holds when excess carriers decay. Generation takes place when there is a
paucity of carriers. We can distinguish three ways of recombination.1
∗ Band-to-band recombination or radiative recombination : electron-hole pairs recombine
directly from band to band with the energy carried away by photons. An electron falls
from its state in the conduction band into the empty state in the valence band which is
associated with the hole. Its counterpart is the optical electron-hole pairs generation.
∗ Trap assisted recombination, also called Shockley Read Hall effect: electron-hole pairs
recombine through deep-level impurities. An electron falls into a "trap", an energy level
within the bandgap caused by the presence of a foreign atom or a structural defect. The
electron occupying the trap can in a second step fall into an empty state in the valence
band, thereby completing the recombination process. One can envision this process either as a two-step transition of an electron from the conduction band to the valence band
or also as the annihilation of the electron and the hole which meet each other in the trap.
The energy liberated during the recombination event is dissipated by lattice vibrations or
phonons. Its counterpart is thermal electron-hole pairs generation
∗ Auger recombination: an electron and a hole recombine in a band-to-band transition,
but the resulting energy is absorbed by a third carrier (another electron or hole). Its
counterpart is the impact ionization.
Usually, a recombination event needs a third partner to allow conservation of energy and crystal momentum. This third partner is often a lattice defect, most commonly an impurity atom,
with an energy state deep in the band gap, i.e. not close to the band edge. Recombination is
then determined by these deep states or deep levels (cf. [18]).
1 Recombination has been detailed rather than generation because the mechanisms are easier understood. Of course, dark noise
is due to generation processes.
34
CHAPTER 4. NOISE
35
Figure 4.1: Carrier recombination mechanisms in semiconductors
At room temperature, the dark counting rate (depending on the voltage applied) is about 1
MHz.
4.2
4.2.1
Trapping phenomena - Afterpulsing
Definition
Traps may result from the damage caused by an implantation in the fabrication process.
Deep levels trap some avalanche carriers and release them with a statistical delay. If the
delay is greater than the dead time after the previous avalanche pulse, a released carrier can
re-trigger an avalanche and cause a statistically correlated pulse. These carriers, captured by
trapping centers inside the depletion layer and released after the diode has been recharged,
form the afterpulse. During an avalanche, only a very small fraction of the trapping level is
filled : therefore, the trap population is always well below saturation and the carrier trapping
probability remains constant during all the avalanche pulse. The probability that an afterpulse
occurs increases with the amount of charge that flows through the diode during a Geiger
discharge. Thus, the afterpulsing probability increases with increasing bias voltage.
36
CHAPTER 4. NOISE
4.2.2
Method : Time Correlated Carrier Counting
Figure 4.2: Principle of Time Correlated Carrier Counting
The method consists in filling the deep levels with a pulsed stimulus, in measuring the
time interval from the filling pulse to the detection of a released carrier and in repeating the
procedure to collect an histogram of the carrier emission versus time.
The histogram is proportional to the emission probability of the first carrier and not to
the emission of a carrier. In case of multiple releases within the time range analyzed, only
the first one is detected. The deviation is supposed to be almost negligible if the total afterpulsing probability is low (< 5%) [16]. The probability density is given by P (t) =
P
− t
B ∗ e−θ∗t + j Aj e τj , where θ is the primary dark-counting rate (as we measure the first
signal detected, the time constant 1/θ of the first noise intervenes, which contradicts [17])
, and where the different exponential components are due to the different types of trapping
levels present in the device. Indeed, if we assume that each deep level has a constant probability of releasing a trapped carrier, then the population N(t) of the trapped carriers behaves
like dN
dt = −K ∗ N (t), where K is the probability for a trapped carrier at a certain depth to
be released, per time unit.
37
CHAPTER 4. NOISE
Figure 4.3: Experimental set-up
All the measurements have been made with a bias voltage of 56V.
4.2.3
Experiment & Results
As we need to get rid of the tail of the filling pulse - i.e. we need to begin our measurement
after it -, an hold-off time is required, which can be obtained by using a discriminator, possibly
coupled with a Dual Gate Generator. The hold-off time is also used to cover the first part of
an intense afterpulsing effect with fast decay, thus ensuring correct measurement conditions
also for the slowest decay components.
Long decay time We want to see what happens quite a long time after the pulse, that is in
a window of 5µs. Since the statistic is very low, we need to register a big number of events
(3.5 millions in our case). The period of the laser is 27µs. This time is sufficient to have the
trap population in equilibrium before each measurement (almost all the trapped carriers are
released), but limits also our investigation. A Dual Gate Pulse Generator delays the triggering
signal with 3.435µs.
We find the following curve, fitted with the sum of two exponentials and an additive constant.
CHAPTER 4. NOISE
38
Figure 4.4: Afterpulse : hold-off time = 3.435µs, fitted with two exponentials
Two time constants appear: 289ns and 141ns, which respective probabilities of 88% and
12%.
Shorter decay time In order to determine afterpulses with shorter decay time, we take the
Dual Gate Generator away, and use directly the features of the oscilloscope. The window
considered begins after the rise time of the filling pulse. The synchronized output of the laser
(start time) is delayed. The trigger level for the signal detection is set on negative slope,
absolute value of 3 mV (in order to count the afterpulses that could be taken for one-pixel
firing signals). We get rid of the 0.6µs tail of the filling pulse by cutting directly the histogram
obtained.
CHAPTER 4. NOISE
Figure 4.5: Afterpulse : Difficulties in the determination of time constants
39
40
CHAPTER 4. NOISE
As the plots show, the fits are quite difficult to establish, because we search a sum of
exponentials. Furthermore, the dark noise time constant (1/θ ≈ 1µs) is in the time window
of the constants found, and thus difficult to distinguish.
We can evaluate the afterpulsing probability for each measured trap level.
lifetime (ns)
probability (%)
499
1.15
532
1.23
167
5.17
36
8.34
275
4.6
107
9.8
Figure 4.6: Afterpulsing probability depending on the lifetime of deep levels
Upshots
The lifetime of a carrier is determined by the device parameters and the energy levels in the
band gap. Then, it depends of the features of each single device. In consequence, lifetime is
a very effective parameter to characterize the purity of a material or a device. It is one of the
very few parameters giving information about the low defect densities (1010 , 1011 cm−3 ).
We have characterized some deep levels, with lifetime constants between 30ns and 600ns,
what agrees with the values given by other studies [17]. The main important fact is that the
probability of the measured afterpulsing remains low. However, the precision of the dark
noise measurement and of the fits is not high enough and the set-up need to be improved.
41
CHAPTER 4. NOISE
4.3
Cross-talk
4.3.1
Definition
Hot carriers in avalanche p-n junction emit photons even in the visible range. Thus, during
the avalanche breakdown, a photodiode operating in Geiger mode may emit a few photons.
The physical origin of this radiation is still a debated issue [19]. The photons emitted will be
detected by neighboring pixels.
Figure 4.7: Three ways of cross-talk
1. Direct cross-talk
2. Inside the depletion layer
3. Through reflection
4.3.2
Cross-talk reduction
A solution to avoid the first way of crosstalk is to isolate pixels optically by trenches filled
with an opaque material.
Figure 4.8: Trenches to avoid direct cross-talk
42
CHAPTER 4. NOISE
4.3.3
Measurement
We can evaluate the proportion of crosstalk by measuring it on the dark noise (i.e. when a
thermal electron is generated in one pixel ⇔ one pixel fired).
Figure 4.9: Crosstalk. Ubias = 56V
The crosstalk increases with the supplied voltage (the statistics are not the same, but even if
they are lower in the plot below, we can distinguish until five-pixels crosstalk).
43
CHAPTER 4. NOISE
Figure 4.10: Crosstalk on the dark noise. Ubias = 57V
4.3.4
Cross-talk dependence on the number of pixels fired
A kind of saturation intervenes also here : an emitted photon arriving on a pixel already fired
will not induced cross-talk. Let us call Pi,N the probability of crosstalk on i pixels (i ≥ 1,
practically, i ≤ 5) for a total number of pixels fired N. Then,
P1,N =
X
i≥1
P2,N =
X
i≥2
i ∗ Pi,0 ∗ (1 −
N −1
N − 1 i−1
)∗(
)
m−1
m−1
(4.1)
N −1 2
N − 1 i−2
)) ∗ (
)
m−1
m−1
(4.2)
Ci2 ∗ Pi,0 ∗ (1 − (
and so on.
For example, with 300 pixels fired, the probability of one-pixel cross-talk is 12.2 %(m =
576). The value found on the dark noise are only maximal values.
Figure 4.11: One pixel cross-talk probability depending on the number of pixels fired.
Chapter 5
Sodium Spectrum
5.1
Principle of Detection
Na Radioactivity The sodium 22 N a decays by β + emission : p → e+ + n + νe . The
positron travels only a few millimeters before losing its kinetic energy. When its energy is
low, it combines with an electron in the absorbing material by a process of annihilation :
e+ + e− → 2γ. Each photon has an energy of 511keV , which is the rest mass of an electron
(= rest mass of a positron). These two oppositely directed photons form the annihilation
radiation. The radioactivity is about 400Bq, which means that the delay between two photons
arriving on the detectors is about 1ms (for each annihilation positron-electron, each detector
receives one photon since both are going in opposite directions).
Detection of γ Gamma-ray photon is uncharged and creates no direct ionization or excitation of the material through which it passes. A gamma-ray with a few MeV can go through
a few millimeters [21]. The size of the detector must take this fact into account. Three main
phenomenons can take place.
∗ The photoelectric effect : the direct absorption converts the energy of the photon into the
kinetic energy of the electron emitted, modulo the binding energy. To fulfill the place
let by the electron, an electronic rearrangement happens. It can give birth to an X-ray or
to a so-called Auger electron, whose energy is much lower than the energy of the first
electron.
44
CHAPTER 5. SODIUM SPECTRUM
45
Figure 5.1: Photoabsorption
∗ The Compton effect : the result of Compton scattering interaction is the creation of a
recoil electron and a scattering gamma ray photon. The maximum energy that can be
transferred to an electron in a single Compton interaction is given by : Ee− |θ=π =
2
0c )
, where θ is the angle of the direction of the electron created with respect
hν 2hν/(m
1+ 2hν
m0 c 2
to the direction of the incident gamma-ray.
Figure 5.2: Compton scattering
∗ The pair production : the creation of an electron-positron pair is possible when the energy
of the photon exceeds 1, 022M eV (it is not the case with Na decay). The positron is
unstable and will be annihilated : e+ + e− → 2γ. Two annihilation gamma-ray photons
can then escape from the detector.
The secondary gamma radiation is composed by the Compton scattered gammas and by the
annihilation photons formed at the end of the tracks of positrons created in pair production.
46
Figure 5.3: Principle of an inorganic scintillator
Scintillator The high Z-value of the constituents and high density of inorganic crystals favor
their choice for gamma-ray spectroscopy (rather than organic crystal) because heavy nucleus
accept better gammas than light nucleus. The scintillation mechanism in inorganic materials
depends on the energy states determined by the crystal lattice of the material. Absorption
of energy can result in the elevation of an electron from its normal position in the valence
band across the gap in the conduction band, leaving a hole in the normally filled valence
band. Impurities act as activators. There will be energy states created within the forbidden
gap through which the electron can de-excite back to the valence band. A charged particle
passing through the detection medium will form a large number of electron-hole pairs created
by the elevation of electrons from the valence to the conduction band. The positive hole will
quickly drift to the location of an activator and ionize it, because the ionization energy of
the impurity will be less than that of a typical lattice site. Meanwhile, the electron is free to
migrate through the crystal and will do so until it encounters such an ionized activator. At this
point, the electron can drop into the impurity site, creating a neutral impurity configuration
which can have its own set of excited energy states. If the activator state that is formed is
an excited configuration with an allowed transition to the ground state, its de-excitation will
occur very quickly and with high probability for the emission of a corresponding photon. The
migration time for the electron is more shorter than the drop-out time : therefore, the decay
time of these states determines the time characteristics of the emitted scintillation light. The
decay rate of the scintillation event establishes the time needed to process the event. It sets the
patient throughput or the quality of the images. The decay rate together with the light output
establishes the timing resolution of the event and therefore sets the minimum time for the
coincidence window. The coincidence window width gives the magnitude of random noise
events for a given single event rate. In order to fully utilize the scintillation light, the spectrum
should fall near the wavelength region of maximum sensitivity for the device used to detect
the light.
Light collection Two effects limit the light collection :
47
∗ The optical self-absorption (The medium should be transparent to the wavelength of its
own emission.)
∗ The losses at the scintillator surfaces
We can expect such a spectrum, depending on the size of the detector.
Figure 5.4: Expected spectrum of Sodium
5.2
Set-up
We used two different crystals : a LSO crystal (2∗2∗15mm from CT I r , who has exclusive
rights for this crystal) and a LYSO crystal (2 ∗ 2 ∗ 10mm from Saint − Gobainr ). Each
crystal had all faces but one wrapped in a reflective foil and a teflon tape, and the non-covered
face is fixed with optical grease1 to the SiPM (cf. appendix E). This coupling agent prevent
the internal trapping of the scintillation light (LSO index of refraction = 1.82, LYSO index of
refraction = 1.81). However, we have to take care that the optical grease does not creep along
the foil, creating a light guide also trapping many photons.
1 BC-630 Silicone Optical Grease is a clear, colorless, silicone, optical coupling compound that features excellent light transmission and low evaporation and bleed at 25◦ C. It has specific gravity of 1.06 and an index of refraction of 1.465.
Saint-Gobain Crystals & Detectors
48
Figure 5.5: Set-up for the coincidence measurement
5.3
Medical Applications
This kind of detection is used for medical applications. Positron Emission Tomography
(PET) is an in vivo technique for imaging biological processes. The availability of short lived
positron-emitting isotopes of carbon, nitrogen, oxygen and fluorine allows most compounds
of biological interest to be labelled in trace amounts and introduced into the body for imaging with PET. The patient gets a biologically active compound (for example Glucose), that
is marked with a radioactive nucleus (positron-emitter, for example 18 F , 11 C, 13 N , 15 O).
Tumors need a lot of energy. The biggest concentration of the tracer will then be in the tumor,
what allows us to detect it. The problem with usual PM is their high magnetic sensitivity,
when we want to combine PET with MRT (Magnetic Resonant Tomography), to have a well
defined anatomy image.
Figure 5.6: Coincidence measurement
49
Figure 5.7: Philipsr , Allegror , Positron Emission Tomography
Figure 5.8: Philipsr , Allegror , Clinical image
5.4
Time Resolution
In the current PETs, APDs are used. In order to know if SiPMs could also be used, we
have to compare both time resolutions. Time resolution of PETs is between σ = 2.04ns (4*4
APDs coupled to 2*2*10mm LSO arrays, [23]) and σ = 1.4ns (2*8 LSO-APD matrix, [22]).
We register the spectrum, to be able to discriminate and take only events whose energies
fit the photo-peak. To set the energy window around the 511 keV photo-peak allows us to
reduce the chance of fortuit coincidence.
50
Figure 5.9: Spectrum of Sodium before cut
A capacitor of 50pF has been added to our SiPM, to integrate the signal. Preamplifiers and
a Constant Fraction Discriminators are used. With the LSO crystal, and a 5ns delay for the
discriminator, we find σ = 1.286ns (statistic=413 events). With the LYSO crystal, we just
made the first measurements. We found σ = 4ns. We would expect a better light collection,
and all the same a slight increase of the time resolution [27], but not such a deterioration, so
the set-up has to be improved (value surely due to a bad optical coupling between the LYSO
crystals and the SiPMs).
The conditions of these measurements have to be compared to the real conditions of a
PET measurement. The decay is here much lower than the decay of tracers in PET, and the
statistics are then low (if we prolong the measurement, an electronic drift false the value, and
the time resolution found get worth). However, this shows us that SiPM should be considered
as an alternative to APDs in PETs.
Figure 5.10: Time resolution, cuts on the 511 keV photopeaks
51
5.5
52
Simulation
Our purpose is to determine which time resolution should be expected.
We use the acception-rejection algorithm to simulate the distribution of light after the scintillator. This one is suppose to have a 5ns rise time and a 40ns decay time. As explained in §5.1,
typical half time for excited states in the processus of light emission are more longer than the
migration time for the electrons. Thus, it is the decay time of these states that determines the
time characteristics of the emitted scintillation light. We then take into account the one pixel
time resolution of the SiPM found in §3.2.2.
5.5.1
Jitter in the LSO
Here is a preliminary simulation of the jitter due to the reflections in the scintillator. We
took a 2D-model,whose size is 2mm*10mm. As the gamma emission from Na is isotropic,
all the possible incident angles are equiprobable. If the source is in front of the middle of the
detector (which is not the best solution, but a realistic one), we find such a distribution :
Figure 5.11: Dispersion due to reflections, x-axe scale=ps
FWHM is about 200 ps, which is negligible in comparison with the 40ns of decay time (what
is not the case with PM, because its decay time is much shorter).
5.5.2
Time resolution for the coincidence measurement
Simulation of the signal
The repartition function of the probability distribution taken cannot be inverted, hence the use
of the acception-rejection algorithm. We used two different rejection functions (an exponential, and a Lorentz function), whose convergences look very similar. The signal was simulated
for 200 photons firing (signal read on the oscilloscope).
53
Figure 5.12: Signal simulated : 200 photons firing, one pixel width = 10ns
Figure 5.13: Signal simulated : 200 photons firing, one pixel width = 15ns
Simulation of the time resolution
Both were simulated, without or with a constant fraction discriminator, in order to determine
how wide one-pixel signal should be, i.e. which capacitor should be used. We find the
following time resolution depending on the one-pixel width (delay of the discriminator=5ns,
fraction=1, which occurred to be the best values with our experimental set-up).
54
Figure 5.14: Without constant fraction discriminator
Figure 5.15: With constant fraction discriminator
The simulation gives us a best time resolution of 1.19 ns, achieved with a single pixel width
of 15 ns. The study has not been exhaustive because of the good agreements of these values
with the time resolution found. However, this simulation could be further exploited.
Conclusion
Let us summarize the main features found :
Rise time
Time resolution
Recovery time
Gain
Single Photon Resolution
Dark noise
Crosstalk
Afterpulse
Efficiency
Low sensitivity to magnetic field
Low sensitivity to electric field
1 ns
30 ps (more than 40 photons) - 170 ps (1 photon)
1.2 µs when saturation - 120 ns for 300 photons firing
106
OK
1 MHz @ room temperature
< 15% for 300 pixels fired
< 10 %
15-20 %
OK
OK
All these features make SiPMs very promising detectors. As most of them depends on the
number of photons forming the signal received, the mean expected value of this one in a
particular experiment has to be known to assess if SiPMs are conceivable detectors. The
irradiance at EUSO, produced in clear sky conditions by a typical EAS induced by a primary
proton with an energy of 1020 eV at 45◦ zenith angle and falling at half of full Field of View,
is estimated at 550 photons per squared meters. An appropriate lens system is used to focus
the light but this value is low enough to consider SiPM as a suitable detector candidate.
55
Appendix A
Leading Edge vs Constant
Fraction Discriminator
Fast timing discriminators aim at counting narrow pulses at very high counting rates and
precisely marking the arrival time of these same pulses. Rise time refers to the time taken
to make the transition from 10 % to 90 % of the pulse amplitude on the leading edge of the
pulse, and fall time specifies the transition time from 90% to 10 % of the amplitude on the
trailing edge of the pulse.
Leading Edge Discriminator A leading-edge discriminator incorporates a voltage comparator with its threshold set to the desired voltage. When the leading edge of the analog
pulse crosses this threshold, the comparator generates a logic pulse. The logic pulse ends
when the trailing edge of the analog pulse crosses the threshold in the opposite direction. If
enoise is the voltage amplitude of the noise superimposed on the analog pulse, and dV
dt is the
slope of the signal when its leading edge crosses the discriminator threshold, then the timing
jitter is given by enoise
. If the noise cannot be reduced, the minimum timing jitter is obtained
dV
dt
by setting the discriminator threshold for the point of maximum slope on the analog pulse.
The main problem with leading edge discriminators is that signal with different amplitudes
induce walk, because the trigger point is fixed.
56
APPENDIX A. LEADING EDGE VS CONSTANT FRACTION DISCRIMINATOR
57
Figure A.1: Walk induced by different signal amplitudes ( leading edge discriminator)
Constant Fraction Discriminator The input signal is split into two parts. One part is
attenuated to a fraction of the original amplitude, and the other part is delayed and inverted.
These two signals are then added : the zero-crossing corresponds to the original point of
optimum fraction on the delayed signal. The time of zero-crossing is independent of pulse
amplitude, this kind of walk is eliminated. However, rise time may also induce walk, which
has to be adjusted.
Appendix B
Simulations : Scilab code
Pixels saturation
function plot_nb_pixels_fired_stat(p,q)
// p=number of statistics for each number of photons firing
// q= maximal number of photons firing
stacksize(20000000)
xxx=zeros(1,q)
pixels=zeros(1,q)
for k=1:1:q
pixels(1,k)=k; // number of pixels fired
xxx(k)=k;
// (x-scale)
end
for n=1:1:q // number of photons firing
pixel_temp=zeros(1,p)
//number of pixels fired for each statistical round
for tt=1:1:p
area=zeros(576,576) //the array is empty
for i=1:1:n
a=int((grand(1,1,’def’)*24))+1; // 24*24 pixels
b=int((grand(1,1,’def’)*24))+1;
if (area(a,b)==0)
area(a,b)=1; // a pixel not already fired is fired
else pixels(1,n)=pixels(1,n)-1;
end
end
pixel_temp(1,tt)=pixels(1,n);
pixels(1,n)=n; // next statistical round
end
pixels(1,n)=mean(pixel_temp); //statistical mean
end
plot2d(xxx,pixels);
endfunction
58
APPENDIX B. SIMULATIONS : SCILAB CODE
59
Reflections in the LSO crystal
function plot_scintill2d(n)
stacksize(100000000)
tab=zeros(1,n)
r=1
lar=2 //width in mm
long=10 //length in mm
eloi=10 //distance from Na to LSO in mm
Na_lar=1 // width of Sodium in mm
ang_min=atan(2*eloi/(long-Na_lar)) // minimal incident angle
while(r<=n)
ang=ang_min+grand(1,1,’def’)*(pi/2-ang_min)
// absolute value of the incident angle
cote=(1-grand(1,1,’def’))
// to know if the photon is going to the right or to the left
if cote>=0.5
imp=long/2+eloi/tan(ang)+(grand(1,1,’def’)-0.5)*Na_lar;
// arrival point on the LSO, O on the left, SiPM on the right
end
if cote<0.5
imp=long/2-eloi/tan(ang)+(grand(1,1,’def’)-0.5)*Na_lar;
end
x=imp
y=0
if cote>=0.5
// the photon is going to the right
b=%T;
i=0;
while b
x=x+lar/tan(ang);
i=i+1;
b=(x<long);
end
temps=((eloi/sin(ang)+i*lar/sin(ang)-(x-long)/cos(ang))*1.82*0.001/3)*10000;
// scale in ps
end
if cote<0.5
// the photon is going to the left
bb=%T;
ii=0;
60
while bb
x=x-lar/tan(ang);
ii=ii+1;
bb=(x>0);
end
x=-x;//to compensate
b=%T;
while b
x=x+lar/tan(ang);
ii=ii+1;
b=(x<10);
end
temps=((eloi/sin(ang)+ii*lar/sin(ang)-(x-long)/cos(ang))*1.82*0.001/3)*10000;
end
if temps<2000 // cut to eliminate extrem values
tab(1,r)=temps;
r=r+1;
end
end
histplot(200,tab)
endfunction
61
LSO+SiPM time resolution
function LSO_SIPM(n,tau1,tau2,c0,t0,a0,p,largeur)
//c0=0.019 t0=10.986125 a0=35 tau1=5 tau2=40 n=200
//p=number of statistics wanted
//largeur= one-pixel-width
stacksize(10000000)
fwhm=zeros(1,p)
for (ff=1:1:p)
coll=zeros(1,n)
i=1
while(i<=n)
U1=grand(1,1,’def’)*1.1871692;
X=a0*(tan(U1/(a0*c0)+atan(-t0/a0)))+t0;
U2=1-grand(1,1,’def’);
if
U2<=(tau1+tau2)/(tau2*tau2)*(1-exp(-X/tau1))*exp(-X/tau2)/c0*(1+(X-t0)^2/a0^2)
coll(1,i)=round(X+rand(1,1,’normal’)*0.171);
// SiPM time resolution for one pixel
i=i+1;
end
end
mon_plot=zeros(1,400) // cut above 400ns
xxx=zeros(1,400)
for j=1:1:400
xxx(1,j)=j
end
for i=1:1:n
if round(coll(1,i))<=400 & round(coll(1,i))<>0
ecart=0;
while (exp(-ecart^2/(2*(largeur/2.35)^2))>=0.05)
if round(coll(1,i)+ecart)<=400
mon_plot(round(coll(1,i)+ecart))=
mon_plot(round(coll(1,i)+ecart))+exp(-ecart^2/(2*(largeur/2.35)^2))
end
if ecart <>0 &
round(coll(1,i)-ecart)>=1&round(coll(1,i)-ecart)<=400
mon_plot(round(coll(1,i)-ecart))=
mon_plot(round(coll(1,i)-ecart))+exp(-ecart^2/(2*(largeur/2.35)^2));
end
ecart=ecart+1;
end
end
end
//search for the maximum of counts
mi_haut=max(mon_plot)/2
//first time that this maximum is got
t1=0
k=1
while (t1==0)
if mon_plot(1,k)>=mi_haut
t1=k;
end
k=k+1;
end
fwhm(1,ff)=t1;
end
disp(st_deviation(fwhm))
endfunction
LSO+SiPM+CFD time resolution
function LSO_SIPM_CFD(n,tau1,tau2,c0,t0,a0,p,delay,fraction)
//tau1=5 tau2=40
// c0=0.019 t0=10.986125 a0=35 (Lorentz function parameters)
//n=200 (can vary)
//p=number of statistics wanted
//delay=5ns (can vary)
//fraction=1 (can vary)
stacksize(10000000)
tab_largeur=zeros(1,25)
for largeur=2:1:50
fwhm=zeros(1,p)
for (fff=1:1:p)
mon_plot=zeros(1,400) // cut above 400ns
coll=zeros(1,n)
i=1
U1=0
X=0
U2=0
while(i<=n)
U1=grand(1,1,’def’)*1.1871692;
X=a0*(tan(U1/(a0*c0)+atan(-t0/a0)))+t0;
62
63
U2=1-grand(1,1,’def’);
if U2<=(tau1+tau2)/(tau2*tau2)*(1-exp(-X/tau1))
*exp(-X/tau2)/c0*(1+(X-t0)^2/a0^2)
coll(1,i)=round(X+rand(1,1,’normal’)*0.171);
i=i+1;
end
end
ff=zeros(1,800)
xxx=zeros(1,800-delay)
mon_plot_retar=zeros(1,800-delay)
// signal inverted and delayed
mon_plot_redui=zeros(1,800-delay)
// signal divided
mon_plot_total=zeros(1,800-delay)
// sum of both
mon_plot_neg=zeros(1,400)
mon_plot_bis=zeros(1,800)
for ii=1:1:n
ecart=0;
if round(coll(1,ii))<=400 & round(coll(1,ii))<>0
while (exp(-ecart^2/(2*(largeur/2.35)^2))>=0.05),
if ii>=1 & ii<=200 & (round(coll(1,ii)+ecart)<=400)
then mon_plot(1,round(coll(1,ii))+ecart)=
mon_plot(1,round(coll(1,ii))+ecart)
+0.93751/largeur*exp(-ecart^2/(2*(largeur/2.35)^2))
end
if ecart <>0 &
round(coll(1,ii)-ecart)>=1 & round(coll(1,ii)-ecart)<=400
then mon_plot(1,round(coll(1,ii)-ecart))=
mon_plot(1,round(coll(1,ii)-ecart))
elseif ecart <>0 &
round(coll(1,ii)-ecart)<1 & round(coll(1,ii)-ecart)<=400
then mon_plot_neg(1,1-round(coll(1,ii)-ecart))=
mon_plot_neg(1,1-round(coll(1,ii)-ecart))
end
ecart=ecart+1;
end
end
end
for j=1:1:800
ff(1,j)=j
64
if j<=400
mon_plot_bis(1,j)=mon_plot_neg(1,401-j)
end
if j>400
mon_plot_bis(1,j)=mon_plot(1,j-400)
end
end
for jj=1:1:800-delay
xxx(1,jj)=jj
mon_plot_retar(1,jj)=-mon_plot_bis(1,jj+delay)
mon_plot_redui(1,jj)=mon_plot_bis(1,jj)/fraction
mon_plot_total(1,jj)=mon_plot_bis(1,jj)+mon_plot_redui(1,jj)
end
for jjj=1:1:800
ff(1,jjj)=jjj
end
// we search for the first zero-crossing (at the beginning, it’s <0)
toto0=0
k=1
while (toto0==0)&(k<800)
if mon_plot_total(1,k)>0
toto0=k;
end
k=k+1;
end
mi_haut=mon_plot_bis(1,toto0)/2
//we search for the first time we get the half of the first maximum
toto1=0
k=1
while (toto1==0)&(k<800)
if mon_plot_bis(1,k)>=mi_haut
toto1=k;
end
k=k+1;
end
fwhm(1,fff)=toto1;
end
disp(st_deviation(fwhm))
tab_largeur(1,largeur/2)=st_deviation(fwhm)
end
plot2d(2:1:50,tab_largeur)
endfunction
Appendix C
Dynamic range
We want to determine the number of pixels fired depending on the number of photons firing.
Let us call f(q) the number of pixels fired when q photons have already come, and m the
total number of pixels of the device. P(event) indicates the probability of the event. Then
: f(q+1)=f(q)+ P(the new incoming photon fires the active area)*P(the pixel has not already
been fired)*P(the photon is absorbed)*P(the photoelectron triggers an avalanche).
f (q + 1) = f (q) + geom ∗ (1 −
f (q)
) ∗ QE ∗ Geiger
m
(C.1)
Hence,
q+1
) )
(C.2)
m
We now have to consider the poissonian fluctuations around the mean value N of photons
x
arriving. The distribution around N is given by πN (x) = e−N ∗ Nx! with x ∈ N . Then, the
expected value of f(N) is given by :
X
E(f (N )) =
(πN (x) ∗ m ∗ (1 − (1 − )x ))
(C.3)
m
x
f (q) = m ∗ (1 − (1 −
E(f (N )) = m ∗ (1 − e−
65
∗N
m
)
(C.4)
Appendix D
Recovery time
Two consecutive signals of N photons
Let us call h(q) the amplitude (normalized to one) of the signal due to the q th photon arriving.
m is the total number of pixels of the detector. N is the total number of incoming photons. t is
the lapse of time between both signals, τ is the recovery time for one pixel. P(event) indicates
the probability of the event. To simplify, the efficiency is not written in the intermediate
calculations (it is easier to consider q as the number of efficient photons). f(q) is the number
of pixels fired, when q photons have come on the detector (cf. C).
h(q+1)=P(the pixel has not already been fired while this shot)*P(the pixel has not been fired
while the previous shot)*1 + P(the pixel has not already been fired while this shot)*P(the
t
pixel has been fired while the previous shot)*(1 − e− τ ).
t
f (q)
f (N )
f (q)
f (N )
) ∗ (1 −
) ∗ 1 + (1 −
)∗
∗ (1 − e− τ )
(D.1)
m
m
m
m
PN
and the total signal S is given by q=1 h(q).
Hence, (we don’t take into account the Poisson distribution, the previous calculation shows
that, considering the value of m, the difference is negligible) :
h(q + 1) = (1 −
S(N ) = m ∗ (1 − (1 −
t
1
1
1
1 N
) ) ∗ (1 − )N + m ∗ (1 − (1 − )N ) ∗ (1 − (1 − )N ) ∗ (1 − e− τ )
m
m
m
m
(D.2)
An infinity of consecutive signals of N photons
We use the same notation as in the previous section. k indicates the number of shots since the
pixel considered was fired for the last time.
∞
h(q + 1) = (1 −
k∗t
f (q) X f (N )
f (N ) k−1
)∗
∗ (1 −
)
∗ (1 − e− τ )
m
m
m
(D.3)
k=1
To simplify the expression, we make the approximation (1 −
S(N ) = m ∗ (1 − e−
Nphe
m
1 N
m)
t
)∗(
1 − e− τ
t
1 − e− τ −
66
N
≈ e− m . Then
Nphe
m
)
(D.4)
Appendix E
PET
Folien
Figure E.1: Reflective foil around the LYSO crystal
LSO-LYSO characteristics
density
hygroscopy 1
relative light output (photons/keV)
relative rad2 hardness
wavelength of maximum emission
index of refraction
primary decay time
afterglow
7.4
no
27
high
420
1.82
40
<0.1% @6ms
7.1
no
32
high
420
1.81
40
<0.1% @6ms
1 The property of materials such as paper and acetate to absorb or release moisture and, in so doing, to expand or contract their
dimensions.
2 rad : radiation absorbed dose = 0.01 Joule/kg (1 gray=100 rads)
67
Appendix F
Temperature
Set-up
Figure F.1: Set-up for measurements at low temperatures
This set-up is put in a vacuum chamber to avoid condensation on the SiPM when the
temperature is lowered. The low temperatures are achieved thanks to the Peltier element.
Peltier devices, also known as thermoelectric modules, are small semi-conductor
components that function as heat pumps. A "typical" unit is a few millimeters thick by a few
millimeters to a few centimeters square. When a DC current is applied, heat is moved from
one side of the device to the other (Peltier effect, discovered in 1834)- where it must be
removed with a heatsink. The "cold" side is commonly used to cool an electronic device
such as a microprocessor or a photodetector. If the current is reversed the device makes an
68
69
APPENDIX F. TEMPERATURE
excellent heater. (http://www.peltier-info.com)
The heat draining by the Copper unit is not enough : a water cooling should be put in.
Pressure required
If we consider that the vapor is an ideal gas, and that the latent heat of evaporation does not
vary with the temperature in the window considered, then the Clapeyron formula gives us
ln(
Psat
M ∗ Lv
1
1
)=
∗( − )
P0
R
T0
T
where (water parameters in brackets):
∗
∗
∗
∗
∗
∗
T=temperature in K
Psat =saturation vapor pressure in mbar
T0 =ebullition temperature at P0 in K (373K for 1013 mbars)
M=molar mass in kg/mol (0.018 kg/mol)
Lv =latent heat of evaporation in J/kg (2.26 ∗ 106 J/kg)
R=ideal gas constant = 8.31 J/K/mol
(F.1)
APPENDIX F. TEMPERATURE
Figure F.2: Saturation pressure vapour
Figure F.3: Saturation pressure vapour
70
Appendix G
Abbreviations used
APD
CFD
EAS
EUSO
FWHM
GZK
LSO
LYSO
MRT
PDL
phe
PM
PMT
SiPM
QE
UV
Avalanche Photodiode
Constant Fraction Discriminator
Extensive Air Shower
Extreme Universe Space Observatory
Full Width Half Maximum
Greisen-Zatsepin-Kuzmin
Lutetium OxyorthoSilicate
Lutetium Yttrium OxyorthoSilicate
Magnetic Resonant Tomography
Pulsed Diode Laser
photoelectron
Photomultiplier
Photomultiplier tube
Silicon Photomultiplier
Quantum Efficiency
ultraviolet
71
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Scintillator" IEEE Transactions on Nuclear Science, NS-46, p. 474-478, 1999
74

Study of Silicon Photomultipliers - Physikalisches Institut Heidelberg

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