Universit de Montral

Characterization Studies of a CdTe Pixelated Timepix Detector for Applications inParticle Physics and Medical Physics.

parConstantine Papadatos

Dpartement de physiqueFacult des arts et des sciences

Mmoire prsent la Facult des tudes suprieuresen vue de lobtention du grade de Matre s sciences (M.Sc.)

en physique

Aot, 2016

c Constantine Papadatos, 2016.

Universit de MontralFacult des tudes suprieures

Ce mmoire intitul:

Characterization Studies of a CdTe Pixelated Timepix Detector for Applications inParticle Physics and Medical Physics.

prsent par:

Constantine Papadatos

a t valu par un jury compos des personnes suivantes:

Pierre Bastien, prsident-rapporteurClaude Leroy, directeur de rechercheHugo Bouchard, membre du jury

Mmoire accept le: . . . . . . . . . . . . . . . . . . . . . . . . . .

RSUM

Un dtecteur pixels Timepix a t caractris en vue dapplications dans la physique

des particules et en imagerie mdicale. Ce Timepix est fait dune couche sensible de

CdTe dune paisseur de 1000 m divise en (256 256) pixels avec des contactsohmiques. Chaque pixel a une aire de (55 55) m2.

Leffet de polarisation dans le dtecteur Timepix-CdTe (TPX-CdTe) a t tudi dans

le but de dterminer si un dtecteur aux contacts ohmiques dmontre cet effet. Il a t

observ que le biais effectif du dtecteur a diminu pendant une priode de mesure de

38 heures, ce qui signifie que le dtecteur est devenu polaris.

Le dtecteur a t expos des photons de 59,5 keV et de 662 keV mis par des

sources d241Am et 137Cs respectivement et les nergies mesures avec le TPX-CdTe

taient prcises 1,18% et 0,03% respectivement. Un talonnage global du dtecteur

a t dvelopp qui a t compar un talonnage conventionel pixel par pixel sou-

vent utilis. Les nergies (0,8 10 MeV) de protons ont t mesures et les rsolutions

en nergie obtenues par les deux talonnages ont t compares. Ltalonnage global a

montr une amlioration dans la rsolution den moyenne 8,6% pour des protons.

La prsence de piges dans le matriau du CdTe et leur influence sur la collection

de charge ont t tudies. La valeur du produit de la mobilit et du temps de vie des

porteurs de charge a t mesure pour des protons de 1,4 MeV partir de lquation de

Hecht.

Mots cls: Timepix, CdTe, polarisation, mobilit des porteurs de charge, temps

de vie des porteurs de charge, rsolution en nergie, quation de Hecht.

ABSTRACT

A Timepix pixel detector has been characterized for the purpose of applications in par-

ticle physics and medical imaging. This Timepix detector consists of a segmented CdTe

sensor layer with a thickness of 1,000 m divided into 256 256 pixels and manufac-tured with Ohmic contacts. Each pixel has an area of 55 55 m2.

The polarization effect in the Timepix-CdTe (TPX-CdTe) was studied with the aim

of determining whether a detector possessing Ohmic contacts exhibits the said effect.

It has been observed that the effective bias of the detector decreased over a 38 hour

measurement period. This indicates that the detector became polarized.

The detector was exposed to photons with energies of 59.5 keV and 662 keV emitted

from 241Am and 137Cs sources, respectively. The energies measured with the TPX-CdTe

were accurate to within 1.18% and 0.03%, respectively. A global calibration of the

detector was implemented and compared to a conventionally used per-pixel calibration.

Proton energies (from 0.8 up to 10 MeV) have been measured and the energy resolutions

obtained by both calibrations have been compared. The global calibration improves the

energy resolution by 8.6% for protons.

The presence of traps in CdTe and their influence on charge collection have been

investigated. The mobility lifetime product of charge carriers has been measured for1.4 MeV protons based on the Hecht equation.

Keywords: Timepix, CdTe, polarization, carrier mobility, carrier lifetime, en-

ergy resolution, Hecht equation.

CONTENTS

RSUM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . xii

NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . 1

1.1 The Project Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3

CHAPTER 2: INTERACTION OF PARTICLES WITH MATTER . . . . 4

2.1 Direct Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 The Bethe-Bloch Equation . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Minimum Ionizing Particles (MIPs) . . . . . . . . . . . . . . . 7

2.1.3 Bragg Curve and Range . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Electrons and Positrons . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Indirect Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . 14

vi

2.2.2 Compton Scattering . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4 Photon Attenuation and Total Absorption . . . . . . . . . . . . 22

2.3 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Thermal Neutrons . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Fast Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CHAPTER 3: SEMICONDUCTORS . . . . . . . . . . . . . . . . . . . . 26

3.1 Classification of Semiconductors and Band Structure . . . . . . . . . . 26

3.2 Electrical Conductivity and Mobility . . . . . . . . . . . . . . . . . . . 29

3.3 Properties of CdTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Fundamental Absorption . . . . . . . . . . . . . . . . . . . . . 31

3.3.2 Applications of CdTe . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 The TIMEPIX Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CHAPTER 4: POLARIZATION EFFECTS IN COMPOUND SEMICON-

DUCTORS . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Polarization in CdTe Detectors . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Polarization Reduction Techniques . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Schottky and Ohmic Contacts . . . . . . . . . . . . . . . . . . 40

4.3 Polarization Studies with the Timepix-CdTe Detector . . . . . . . . . . 41

CHAPTER 5: CHARACTERIZATION OF THE TIMEPIX-CDTE DETEC-

TOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1 Energy Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.1.1 Per-Pixel Energy Calibration . . . . . . . . . . . . . . . . . . . 48

5.1.2 Global Energy Calibration . . . . . . . . . . . . . . . . . . . . 51

5.2 Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

CHAPTER 6: CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . 61

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

LIST OF TABLES

2.I The interaction cross sections for photoelectric absorption and for

Compton scattering in Si and in CdTe, taken from NIST XCOM.[15] 17

3.I Mobility values of common semiconductors at 300 K, expressed

in cm2V1s1.[21] . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.I The measured proton energies Em with the per-pixel calibration

compared to the expected energies after Rutherford backscattering

with their percent deviation %Dev. The FWHM and the resolu-

tions R are provided for each energy. . . . . . . . . . . . . . . . . 52

5.II The energies Em of photons emitted from 241Am (59.5 keV) and137Cs (662 keV). The energy of 59.5 keV is measured using the

per-pixel calibration while the 662 keV photons were measured

with the global calibration. %Dev gives the percent deviation of

each measured energy from its expected value. The FWHM and

the resolution R are given for each. . . . . . . . . . . . . . . . . . 53

5.III Les nergies de protons mesures Em avec ltalonnage global

compares aux nergies attendues aprs la rtrodiffusion de Ruther-

ford avec leur diffrence en pourcentage %Dev. Les FWHM et les

rsolutions R sont donns pour chacune des nergies. . . . . . . . 57

5.IV The product of the TPX-CdTe detector for 1.4 MeV protons

with the corresponding range in m[9] and the quality of the fit (R2). 58

5.V The ee values found in the literature. The errors on these values

are not given for these references. . . . . . . . . . . . . . . . . . 60

LIST OF FIGURES

2.1 The electronic (black) and nuclear (red) stopping powers for hy-

drogen ions in CdTe. The contribution of dE/dxnucl is small com-

pared to dE/dxelec. The plot is graphed based on data from the

SRIM program.[9] . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The stopping power of muons as a function of their energy in cop-

per. The Bethe-Bloch equation describes the stopping power in the

region of 0.08 - 800 MeV with the minimum ionization energy at

3.5 MeV.[12]. . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 The rate of average energy loss in various materials. The energy of

minimum ionization of each aligns roughly at the same value (3.5

MeV). Different particles (muon, pion and proton) become MIPs

at different momenta.[12] . . . . . . . . . . . . . . . . . . . . . . 9

2.4 The energy deposited as a function of the average range in water

for a 250 MeV proton is described by the Bragg curve (red). Note

that the variation of the deposited energy as a function of the depth

for photons (10 MeV) is described by an attenuation curve (see

2.2.4).[13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 The difference between the trajectories of heavy and light charged

particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 The energy loss per radiation length, X0 (see Eq. (2.8)) in lead as

a function of the electron or positron energy.[12] . . . . . . . . . 13

2.7 The cross sections of the three principle interactions of photons

with matter.[14] . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.8 The photoelectric process. . . . . . . . . . . . . . . . . . . . . . 15

2.9 The photoelectric interaction cross section in lead is given as a

function of the photon energy. The edges correspond to the L and

K shells, indicated on the graph.[10] . . . . . . . . . . . . . . . . 15

2.10 The Compton scattering process. . . . . . . . . . . . . . . . . . . 18

ix

2.11 The Compton edges corresponding to different energies.[10] . . . 19

2.12 The interaction cross section for Compton scattering.[10] . . . . . 20

2.13 The pair production process. . . . . . . . . . . . . . . . . . . . . 20

2.14 The pair production interaction cross section in lead. The thresh-

old effect at low energy is shown at 1.022 MeV.[10] . . . . . . . . 21

2.15 a) A Timepix detector (ATLAS-TPX) covered by a mosaic of con-

verters: (1) LiF (5 mg/cm2), (2) Polyethylene (PE) of 1.3 mm, (3)

PE (1.3 mm) + Al (100 m, (4) Al (100 m), (5) Al (150 m),

(6)Uncovered Si. b) An X-ray radiogram of the converter layers.[16] 25

3.1 A unit cell showing a diamond lattice structure.[19] . . . . . . . . 27

3.2 A unit cell showing a cubic (zincblende) crystal structure which is

distinguishable from the diamond structure due to the presence of

a second type of atom.[20] . . . . . . . . . . . . . . . . . . . . . 27

3.3 The band structure of Silicon, showing the conduction band with

the valence band below it. The energy is graphed as a function of

the wave-vector~k.[23] . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 A comparison between the position of the energy bands of a metal,

a semiconductor and an insulator with respect to the Fermi energy.

[24]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 The Fermi-Dirac distribution for a few values of kBT/EF . . . . . 30

3.6 a) A direct band gap, like for CdTe. b) An indirect band gap, like

for Si. The photon absorption process is illustrated for each case

with an arrow indicating the transition of an electron from the top

of the valence band to the bottom of the conduction band.[25] . . 32

3.7 A Timepix detector equipped with a FITPix interface.[27] . . . . 33

3.8 A Timepix detector. (A) is the semiconductor sensor layor. (B)

shows the readout chip and the electronic bump bonds.[27] . . . . 34

3.9 The electronics scheme of a single pixel in a Timpepix detector.[26] 34

x

3.10 A frame showing the tracks of 1.4 MeV protons in the TPX-CdTe

detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1 Band bending model near the semiconductor-metal interface, used

to calculate the depth of the depletion layer and the electric field.

l is the depletion layers thickness. W is the function describing

the band bending and whose value corresponds to the separation

between the deep acceptor energy (E) and the Fermi energy. V is

the applied potential, is the contact potential, is the region in

which the charges are accumulated and = l .[8] . . . . . . . 374.2 Schematic plot of the electric field as a function of the position

in the situation shown in Figure 4.1. The slope of the electric

field changes abrubtly where the energy band intersects the Fermi

energy.[8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 A 38 hour measurement of the deposited energy by particles

from an 241Am source at 150 V. . . . . . . . . . . . . . . . . . . 43

4.4 Measurements of the change in the cluster size as a function of

the applied voltage for particles from an 241Am source. A lin-

ear fit around 150 V (120-180 V) gives the correspondance CS =

0.110860V +44.8359. The R2 value of this fit is 0.994426. . . . 434.5 The influence of the bias on the radial diffusion of charges created

by 3.5 MeV protons at an incidence of 0 in a Si Timepix.[30] . . 44

4.6 A measurement in time of the cluster size of particles over 38

hours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 The behaviour of the effective bias over the course of the 38 hour

measurement period with an initial applied voltage of 150 V. . . . 47

5.1 The dependence of the TOT on the X-ray energy for each pixel,

expressed by the calibration function f . The cxt term takes into

account the calibrations non-linearity near the threshold.[31] . . . 49

xi

5.2 The photon energy spectrum of an 241Am source with a peak at

58.8 keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 The energy spectrum of photons coming from a 137Cs source (662

keV photons) with a peak at 711,1 keV. . . . . . . . . . . . . . . 50

5.4 The calibration function for the heavy ionizing particles (3, 5 et 9

MeV protons and 20 MeV 6Li ions). . . . . . . . . . . . . . . . . 53

5.5 The energy spectrum for 662 keV photons emitted from a 137Cs

source processed with a global calibration. The measured energy

is 661.8 keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6 A comparison between the energy spectra for 10 MeV protons ob-

tained with the per-pixel (black) and global (red) calibrations. . . 56

5.7 The resolutions of each energy for the per-pixel (black) and global

(red) calibrations. Protons with energies from 0.8 to 10 MeV are

shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.8 The calculation of the coefficient of determination. . . . . . . . . 59

5.9 The measured energy as a function of the bias for 1.4 MeV protons

with statistical error bars. The fit function (red) has an R2 value of

0.99914. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.1 A) A Timepix-CdTe irradiated by -particles in 2008 with a large

quantity of localized trapping centres. B) A second Timepix-CdTe

irradiated by -particles in 2015 displaying a more homogeneous

structure. The vertical lines correspond to columns of dead pixels

resulting from the implantation of the bump bonds during fabri-

cation. The scales indicate the number of -particles registered

in each pixel. A) and B) comme from measurements taken at the

Tandem accelerator at the University of Montreal. . . . . . . . . . 62

LIST OF ABBREVIATIONS

ATLAS A Toroidal LHC Apparatus

CERN Conseil europen pour la recherche nuclaire

CMS Compact Muon Solenoid

CS Cluster Size

FWHM Full-Width at Half-Maximum

HPGe High-Purity Germanium

IEAP Institute of Experimental and Applied Physics

LHC Large Hadron Collider

MAFalda Medipix Analysis Framework

MIP Minimum Ionizing Particle

NIST National Institute of Standards and Technology

PET Positron Emission Tomography

SPECT Single Photon Emission Computed Tomography

SRIM Stopping and Range of Ions in Matter

ToA Time of Arrival

TOT Time Over Threshold

TPX Timepix

USB Universal Serial Bus

XRF X-Ray Fluorescence

NOTATION

Al Aluminium241Am Americium-241

C Carbon

c Speed of light in vacuum. c = 3.00 108 m/s2.CdTe Cadmium Telluride137Cs Cesium-137

eV electronvolt

fC femtocoulomb

GaAs Gallium Arsenide

GaP Gallium Phosphide

Ge Germanium

h Planck constant. h = 4.14 1015 eV s.He Helium

InSb Indium Antimonide

keV kiloelectronvolt6Li Lithium-6

6LiF Lithium Fluoride

MeV Megaelectronvolt

m micrometre

Si Silicon

Sn Tin99mTc Technetium-99m

ZnS Zinc Sulfide

o o o

ACKNOWLEDGMENTS

I would like to thank, first, my research supervisor, Claude Leroy, for having given

me the opportunity to work on this project. Your patience and advice were greatly ap-

preciated. Thanks to my research and office colleague, Thomas, for having shared his

knowledge, his company, and innumerable pints with me over the course of this mas-

ters project. I wish to also thank all my colleagues in the bunker, in the Groupe de

Physique des Particules, and at the IEAP in Prague for enriching these past two years

with discussions on physics and distractions from it.

Thanks to Chen for always having a solution to my computer problems. Thanks to

Louis Godbout for managing the Tandem, providing the particle beams and helping with

any technical problem encountered during our experiments.

Finally, I would like to thank my friends and family. In one way or another, each of

you has supported and encouraged me. I love you all.

PREFACE

My contributions to this masters project are centred on the collection and analysis

of data. I participated in the experimental component of the spectroscopic characteriza-

tion of the Timepix-CdTe detector through the use of protons produced by the Tandem

accelerator at the University of Montreal and the Van der Graaf accelerator in Prague, of

-rays coming from 241Am and 137Cs sources and of -particles emitted by the 241Am

source. In addition, I conducted a study on the polarization effect in the Timepix-CdTe

detector.

I wrote supplementary algorithms to the ones already included in the MAFalda anal-

ysis software which permitted the analysis of the measured data. Two calibrations of the

detector were performed. The per-pixel calibration was completed by Thomas Billoud

at the IEAP in Prague, while the global calibration was developed and implemented by

myself using protons and 6Li ions. The sum of my research activities and my results can

be found in chapters 3 and 4 of this thesis.

Additional research activity: I also participated in the luminosity analysis of data

from the ATLAS experiment at the LHC at CERN as part of the ATLAS-TPX collabo-

ration. I worked on reconstruction software validation for proton-proton collision events

in the context of "class 2 shifts" as a member of the ATLAS collaboration.

CHAPTER 1

INTRODUCTION

CdTe detectors are used in medical imaging and in particle physics in general because

their -ray detection efficiency is relatively high owing to their large atomic number

(ZCd = 48, ZTe = 52). Applications in SPECT and in PET are thus possible. The ability

to measure precisely the energies of photons from a few tens to hundreds of keV allows

the measurement of 140 keV photons emitted by 99mTc in SPECT imaging devices and

511 keV photons in PET imaging devices. CdTe is also a good candidate as a photon

detection material in particle physics, in accelerator experiments and in space, where its

high density gives it good radiation hardness.[1][2]

The development of CdTe as a particle detector had been hampered in favour of Si,

which has become the most well-studied material, historically. Nonetheless, the past 40

years have seen an increase in the research of CdTe as a photon detector.

It is important that CdTe detectors have a short electronic response time and can

therefore distinguish the signal from noise. Additionally, they must permit the charac-

terization and identification of -rays coming from unknown sources.

The energy resolution of a -ray detector is quantified by the full-width at half-

maximum (FWHM) of the energy peak divided by the mean measured energy (the

barycentre). The resolution is expressed as a percentage. High purity semiconductor

detectors, like HPGe (high purity Germanium) at 77 K, can have resolutions as good as

1% for 662 keV photons.[3]

When an ionizing particle interacts with the semiconductor, a charge cloud is created

made up of free charges possessing either negative (electrons) or positive (holes) charge.

A bias voltage applied across the electrodes provokes the drift of these two types of

charges in opposite directions. The speed with which each charge drifts towards the

electrode depends on the bias voltage and on the structure of the crystal lattice of the

material.[4]

These charge carriers can become trapped by impurities in the lattice. They can also

2

recombine with each other. In both cases, the result is a loss in the collected charge. The

performance of a CdTe detector depends on the number and distribution of these traps

in the material. The quantity of traps can be inferred by measuring the product of the

mobility and the lifetime of charge carriers (). A large amount of traps results in a

shorter lifetime and, therefore, a smaller value.[5]

To reduce the leakage current, Schottky-type CdTe detectors have been made, the

result of which was a good energy resolution for -rays. For 59.5 keV and 662 keV

photons, the obtained resolutions were 2.5% and 1.5% respectively [1]. However, these

Schottky-type detectors displayed time-instability problems under an external bias volt-

age. In practice, time-stability is of great importance to the functioning of a detector.

This instability phenomenon is called "polarization". Polarization is characterized by a

decrease in the energy resolution as well as a decrease in charge collection with time,

shifting the measured photopeaks toward lower energies after applying an external bias.

The term polarization is used to describe any temporal instability in compound semi-

conductors. Previous research has shown that this phenomenon is the result of an accu-

mulation of negative charges at a deep acceptor level during the application of the bias

voltage. This results in the variation of the electric field inside the CdTe with time.[1]

To eliminate the polarization effect, detectors with Ohmic contacts were used, the

compromise being that the energy resolution is inferior. It is nonetheless possible that

an Ohmic-type detector may become polarized under certain conditions.

1.1 Objectives of this Project

The research activity presented in this thesis has been made possible by the use of a

Timepix-CdTe (TPX-CdTe) detector possessing Ohmic contacts. Data from 0.8 MeV to

10 MeV protons and 20 MeV 6Li ions have been taken at the Tandem accelerator of the

University of Montreal and 1 MeV protons at the Van der Graaf accelerator of the IEAP

in Prague. 137Cs and 241Am sources have also been used for photon measurements and

the same 241Am source has been used for measurements of -particles. The prepara-

tion and analysis of the data was done using the Pixelman[6] and MAFalda[7] analysis

3

softwares.

The characterization of the TPX-CdTe detector is separated in three parts. First,

an investigation into polarization was conducted in order to show that an Ohmic-type

detector can become polarized. This would prove that the electric field in the sensor layer

of the TPX-CdTe can become unstable with time under the application of a bias voltage.

Next, a study of the energy resolution of the detector has been completed comparing the

reliability of two calibrations (a per pixel calibration and a global calibration). Lastly,

the mobility-lifetime product was measured in order to determine the contribution of

traps within the detector to its charge collection.

1.2 Structure of the Thesis

In the first chapter, an overview of the interaction of particles with matter is pre-

sented. A particular emphasis is placed on the interaction of photons with matter via

the photoelectric effect and Compton scattering. The second chapter is dedicated to the

description of CdTe semiconductors and to the operation of the Timepix detector. The

polarization effect in CdTe detectors is described in chapter 3, extending the method in-

troduced by [8] for non-pixelated CdTe detectors. An investigation of this effect on the

TPX-CdTe is included here. Chapter 4 presents the spectroscopic properties of the TPX-

CdTe detector and its charge collection capabilities with a description of its calibration.

This chapter also reports the measurement of the mobility lifetime product of the saiddetector.

CHAPTER 2

INTERACTION OF PARTICLES WITH MATTER

2.1 Direct Ionization

Particles such as electrons, protons and -particles (alphas) directly ionize the matter

through which they pass. They lose their kinetic energy over the course of their path

through collisions or radiative processes. The energy loss of these particles over the

distance travelled is quantified as the stopping power dE/dx,

dEdx

=

(dEdx

)col

+

(dEdx

)rad

=

(dEdx

)elec

+

(dEdx

)nucl

+

(dEdx

)rad

(2.1)

The collision stopping power, (dE/dx)col , is composed of two parts. The first partdescribes collisions between the particle and the atomic electrons, (dE/dx)elec. Thesecond part describes nuclear collisions, (dE/dx)nucl . The interaction cross-sectionwith electrons being much greater than that for nuclear interactions at high energy (E >

100 keV),

(dE

dx

)elec(dE

dx

)nucl

, (2.2)

as is shown in Figure 2.1.

For heavy particles, such as protons and alphas, the energy loss by collisions is sub-

stantially greater than that by radiation. For light particles (electrons, positrons), radia-

tive losses become increasingly important at energies greater than 10 MeV. The energy

loss by collisions represents losses from excitation and ionization. Excitation is the pro-

cess that moves an electron from a given shell to a more energetic shell. Consequently,

an incident particle can lose energy without removing an electron from an atom. Ion-

ization removes one or more electrons from the atom. It creates electron-hole pairs in

a semiconductor and electron-ion+ pairs in a gas. The number of pairs created is given

5

Figure 2.1 The electronic (black) and nuclear (red) stopping powers for hydrogen ionsin CdTe. The contribution of dE/dxnucl is small compared to dE/dxelec. The plot isgraphed based on data from the SRIM program.[9]

6

by the ratio (E/) of the energy, E, deposited by the particle in the material and , the

energy required to create a pair. varies from one material to another (Si = 3.62 eV et

Argas = 26.4 eV)[10].

2.1.1 The Bethe-Bloch Equation

The average rate of energy loss per unit lenght of a heavy charged particle is given

by the Bethe-Bloch equation which assumes i) that the mass of the incident particle is

greater than that of an atomic electron and ii) that the motion of the atomic electron is

negligible. If Tmax is defined to be the maximum energy transferred to an atomic electron

by a massive charged particle,

Tmax = 2mec2 22(

1+2meM

+m2eM2

)1(2.3)

2 =(1 2

)1 is the Lorentz factor, me is the mass of the electron, M is the massof the incident particle and = v/c (i.e., the speed of the particle in units of the speed

of light). When the mass of the incident particle is much greater than the mass of the

electron, M me and when is not very large, (2.3) becomes

Tmax 2mec2 22 (2.4)

The Bethe-Bloch equation gives the average energy loss per unit length:

dEdx

= 4NAr2emec2 Z

A

z2

2B (2.5)

where the factor B is given by

B ln(

2mec2 22TmaxI2

) 2

2C

Z. (2.6)

Z is the atomic number of the target atom, A is its atomic mass, NA is Avogadros

number, is the atomic density and z is the electric charge of the incident particle. The

classical electron radius is defined as re = e2/mec2 = 2.82 1013cm. I is the average

7

ionization energy. The factor takes into account the shielding of atomic electrons

through density effects in the material. This is due to polarization of the atomic electrons

by the electric field created by the incident particle. Inner shell electrons are not sensitive

to this field and, as a consequence, contribute less to the energy loss. The C/Z term is

associated to the non-participation of inner shell electrons (primarily coming from the K-

shell) in the collision process at low energy[11]. This shielding effect is produced when

the incident particle has a speed that is, at its maximum, comparable to the orbital speed

of atomic electrons. It is at these speeds that the assumption of a stationary electron with

respect to the incident particle becomes invalid. On the whole, the contribution of this

correction is relatively small. As an example for muons (m = 105 MeV/c2), the region

of energy over which the Bethe-Bloch equation is valid is represented in Figure 2.2.

2.1.2 Minimum Ionizing Particles (MIPs)

Energy loss is not strongly dependent on the mass of the incident particle unless

its mass is much greater than that of an electron. The stopping power (dE/dx) isproportional to z2/ 2 where z is the particles charge. The stopping power depends on

the absorber, according to Z/A. The quantity dE/dx is at a minimum when 3.5, asshown in Figure 2.3. Particles that undergo a minimal energy loss are called minimum

ionizing particles, or MIPs. The rate of average energy loss of particles in different

materials is shown in Figure 2.3.

For example, the proton, whose mass is close to the GeV scale (938 MeV/c2), must

have a kinetic energy of many GeV in order to become a MIP. For an electron, whose

mass is 0.511 MeV/c2, only a few MeV ( 2 MeV) are necessary. The stopping powersvalue at minimum ionization is roughly the same for all particles possessing the same

charge. However, each particle attains minimum ionization with a different momentum,

as is seen in Figure 2.3, which allows for its identification. A particle with a charge

z 2 (-particle, z = 2) has a rate of loss that is larger because of the proportionality ofdE/dx to the square of the charge. Such a particle can become a MIP for much higherenergies, as is the case with cosmic rays or in relativistic energy ion accelerators.

8

Figure 2.2 The stopping power of muons as a function of their energy in copper. TheBethe-Bloch equation describes the stopping power in the region of 0.08 - 800 MeV withthe minimum ionization energy at 3.5 MeV.[12].

9

Figure 2.3 The rate of average energy loss in various materials. The energy of mini-mum ionization of each aligns roughly at the same value (3.5 MeV). Different particles(muon, pion and proton) become MIPs at different momenta.[12]

10

2.1.3 Bragg Curve and Range

Due to the dependence of dE/dx on 1/ 2 for heavy charged particles, these particles

lose the majority of their energy immediately before being completely stopped in the

medium. This is illustrated by the Bragg peak, represented in Figure 2.4. The range is

the distance travelled by a particle of a given energy in a given material. It is obtained

experimentally by sending a beam of energetically coherent particles across layers of

identical materials of different thickness. The current of particles transmitted across the

layer is compared to the initial intensity of the beam. Figure 2.5 illustrates the difference

between the ranges of heavy and light charged particles. Heavy particles are not subject

to strong deviations from their trajectories because only a small fraction of kinetic energy

is lost with each collision of the particle with an electron.

In practice, the statistical nature of the energy loss causes fluctuations in the value of

the range of the particle across a material. These fluctuations manifest themselves as lat-

eral and longitudinal straggling. Consequently, two identical particles having the same

initial energy and travelling through the same material will not lose the same amount

of energy along their trajectory and will therefore not have the same range values. The

range is expressed as a Gaussian distribution. The straggling effect diminishes with an

increase of the particles mass.

The CSDA 1 range is calculated by integrating the inverse of the Bethe-Bloch for-

mula.

R(E0) = 0

E0

1dEdx

dE (2.7)

2.1.4 Electrons and Positrons

For electrons and positrons, whose mass is much smaller than that of a proton, the

radiative losses (bremsstrahlung) are much more significant at energies 10 MeV. 2 Inaddition, (dE/dx)col , given by Bethe-Bloch is also modified because the incident particle

1. Continuous Slow Down Approximation2. Bremsstrahlung is the radiation created by the acceleration of negative electric charges.

11

Figure 2.4 The energy deposited as a function of the average range in water for a250 MeV proton is described by the Bragg curve (red). Note that the variation of thedeposited energy as a function of the depth for photons (10 MeV) is described by anattenuation curve (see 2.2.4).[13]

12

Figure 2.5 The difference between the trajectories of heavy and light charged particles.

possesses the same mass as the atomic electron. It can then no longer be assumed that

the motion of the atomic electron is negligible compared to the incident electron. The

effects, including bremsstrahlung, that contribute to the total energy loss per radiation

length (X0) for electrons and positrons are presented in Figure 2.6. For electrons, the

radiation length is defined as,

X0 =A

4NAZ2r2elog183

Z1/3, (2.8)

where is the fine structure constant of the electromagnetic force.

2.2 Indirect Ionization

Photons interact indirectly with matter according to three principle interaction pro-

cesses. These interactions are photoelectric absorption, Compton scattering and pair

production. The dependence on the energy and on the atomic number of the absorber

dictates which process dominates. This is illustrated in Figure 2.7.

13

Figure 2.6 The energy loss per radiation length, X0 (see Eq. (2.8)) in lead as a functionof the electron or positron energy.[12]

Figure 2.7 The cross sections of the three principle interactions of photons withmatter.[14]

14

2.2.1 The Photoelectric Effect

Photoelectric absorption occurs when an incident photon of energy E interacts with

an atomic electron. The photon is converted into a photoelectron with kinetic energy

KE = E BEi, where BEi is the binding energy of the electron to the i shell. Anillustration of this is given in Figure 2.8. This is the dominant effect for low energy

photons, typically less than 50 keV. The resulting photoelectrons also possess lowenergies. The binding energy depends on the atomic number Z and on the electronic

shell. The binding energies of the K, L and M shells are given by,

BEK = Ry(Z1)2eV (2.9)

BEL =Ry4(Z5)2eV (2.10)

BEM =Ry9(Z13)2eV (2.11)

where Ry = 13.62 eV is the Rydberg constant.

When the photon energy increases, the photon can penetrate the electron cloud more

profoundly, which permits it to attain an electron of an internal shell. The photoelectron

is emitted, creating a vacancy that can be occupied by the transition of an electron on an

external shell. This transition is accompanied by the emission of an X-ray.

A photon with a lower energy can only attain the external shells, producing an edge.

An edge corresponds to the minimum photon energy necessary to eject an electron in a

given shell. This threshold effect is characteristic to each atom. As an example, Figure

2.9 presents the photoelectric interaction cross section in lead as a function of the photon

energy, indicating the edge effect.

The photoelectric interaction cross section of low energy photons for the K edge is

given by

15

Figure 2.8 The photoelectric process.

Figure 2.9 The photoelectric interaction cross section in lead is given as a functionof the photon energy. The edges correspond to the L and K shells, indicated on thegraph.[10]

16

Kpe =83

[2(

mec2

E

)7]1/24r2e

4Zn (2.12)

where n is a number that varies between 4 and 5. For n = 5, the photoelectric cross

section displays a general dependence of

Kpe Z5

E7/2(2.13)

For E mec2 (so E/mec2 1),

Kpe =(

mec2

E

)4r2e

4Z5 (2.14)

showing the following dependence:

Kpe Z5

E(2.15)

Table 2.I compares the interaction cross sections of the photoelectric effect and of

Compton scattering for photons from 20 keV to 2 MeV in Si and CdTe.

2.2.2 Compton Scattering

Compton scattering dominates for photons with energies greater than a few hun-

dred keV to a few MeV. The incident photon transfers a part of its energy to an atomic

electron. The resulting products of the Compton interaction are a recoil electron and a

scattered photon. This is illustrated in Figure 2.10. In the laboratory reference frame,

for an incident photon with an energy E , the scattered photon has an energy E ,

E = hc

=E

1+ Emec2 (1 cos)(2.16)

The energy of the recoil electron is

17

Table 2.I The interaction cross sections for photoelectric absorption and for Comptonscattering in Si and in CdTe, taken from NIST XCOM.[15]

18

Figure 2.10 The Compton scattering process.

ER = E E = EE

mec2(1 cos)

1+ Emec2 (1 cos)(2.17)

For a forward collision, = 0, E = E and ER = 0. For a backward collision, =

and,

E =E

1+2 Emec2(2.18)

ER = E

2Emec2

1+ 2Emec2(2.19)

which corresponds to a Compton edge, illustrated in Figure 2.11.

The interaction cross section as a function of the atomic number of the absorber and

the energy of the photon is

C = s +a ZE

, (2.20)

where s and a are the scattering and absorption cross sections, respectively. These

two contributions are represented in Figure 2.12 as a function of the photon energies.

19

Figure 2.11 The Compton edges corresponding to different energies.[10]

The cross sections of Si and CdTe are given in Table 2.I.

2.2.3 Pair Production

If the photon energy is E 2mec2 = 1.022MeV , and approaches the atomic nucleusor an electron, the photon can be converted into an electron-positron pair. This process

is shown in Figure 2.13. The created positron is annihilated immediately with the nearby

electron, which causes the emission of two photons with equal energies (0.511 MeV).

The dependence of the pair production interaction cross section on the atomic num-

ber of the absorber and on the photon energy is

PP Z2 logE (2.21)

and is graphically represented in Figure 2.14 where the threshold effect can be ob-

served (E = 1.022 MeV).

20

Figure 2.12 The interaction cross section for Compton scattering.[10]

Figure 2.13 The pair production process.

21

Figure 2.14 The pair production interaction cross section in lead. The threshold effectat low energy is shown at 1.022 MeV.[10]

The photon energy is

Emin = 2mec2(

1+2mec2

2Mc2

)(2.22)

If the mass M of the target particle is much greater than me, for example if it is an

atomic nucleus, this enables pair production to occur. If the target is an electron, M =me,

and a triplet is produced. The kinetic energy available is

T = h1,022MeV (2.23)

for pair production (2mec2 = 1,022 MeV). It is

T = h2,044MeV (2.24)

for triplet production (4mec2 = 2.044 MeV).

22

2.2.4 Photon Attenuation and Total Absorption

When photons travel through matter, their numbers decrease exponentially with the

distance. This gives

I (x) = I0e x, (2.25)

where is the density of the material through which the photon travels, x is the

distance travelled and is the total absorption coefficient. This quantity is proportional

to the inverse of the mean free path of photons in that material.

The total interaction probability of a photon in the material is the sum of all possible

interactions: photoelectric, Compton, pair production, Rayleigh scattering and photonu-

clear interactions. 3 The total cross section, , multiplied by the density of atoms, N,

gives the interaction probability per unit of length, with the total absorption coefficient,

, given by

= N = NAA, (2.26)

where A is the molar mass and = pe +C +PP. Thus, the probability that a

photon interacts with the material is given by

P = 1 e x. (2.27)

2.3 Neutrons

Neutrons interact with matter via elastic and inelastic scattering and nuclear reac-

tions. These interactions allow for their detection. The neutron interaction probability in

matter depends on the neutron energy.

3. Rayleigh scattering and photonuclear interactions dont play a big role in the context of this study.

23

2.3.1 Thermal Neutrons

Thermal neutrons undergo certain nuclear reactions which cause the emission of

charged particles which are used for their detection.

A practical application that is used by the ATLAS-TPX silicon pixel detectors oper-

ated in the ATLAS experiment is to cover a portion of the detectors chip with a layer

of 6LiF that acts as a thermal neutron converter [16]. The resulting neuclear reaction

caused by the thermal neutrons on the 6LiF is the following:

n+ 63Li + 31H. (2.28)

The interaction cross section of this reaction is 940 barns. The -particles and the tri-

tium emitted have energies of 2.73 MeV and 2.05 MeV respectively and can be detected

in Si.

2.3.2 Fast Neutrons

Fast neutrons are those whose kinetic energy is typically greater than 100 keV. These

neutrons are detected thanks to nuclei that recoil from a converter. The recoil energy of

a nucleus in an elastic collision with a fast neutron is

EA = En4A

(A+1)2cos2 , (2.29)

where En is the energy of the incident neutron, A is the atomic mass of the nucleus

and is the scattering angle in the laboratory reference frame. The maximum recoil

energy of the nucleus is obtained for = 0.

EAEn

=4A

(A+1)2. (2.30)

This relation makes evident the fact that these materials possessing small atomic

numbers are more favourable for the detection of fast neutrons. Consequently, polyethy-

lene (CH2) is used for this purpose.

24

The mosaic of neutron converters used on the ATLAS-TPX detectors in the ATLAS

experiment is shown in Figure 2.15.

25

Figure 2.15 a) A Timepix detector (ATLAS-TPX) covered by a mosaic of converters:(1) LiF (5 mg/cm2), (2) Polyethylene (PE) of 1.3 mm, (3) PE (1.3 mm) + Al (100 m,(4) Al (100 m), (5) Al (150 m), (6)Uncovered Si. b) An X-ray radiogram of theconverter layers.[16]

CHAPTER 3

SEMICONDUCTORS

The applications of semiconductors as particle detection instruments are varied. In

medical physics, the use of semiconductors, in particular CdTe, whose atomic number is

high, is generally associated with imaging via SPECT 1 and PET 2 applications [1] [17].

They are primarily used in particle physics as dosimeters and trackers, as in the ATLAS

and CMS experiments at the Large Hadron Collider (LHC) at CERN [16].

3.1 Classification of Semiconductors and Band Structure

Semiconductors can be classified into three large families corresponding to their po-

sition on the periodic table. Group IV materials display a tetragonal binding structure,

with four valence electrons. These are referred to as elemental semiconductors [C, Si,

Ge, -Sn] and their crystal lattice is characterized by a diamond structure.

The second family of materials occupies the III-V group and possesses a cubic crystal

(zincblende) structure, again with tetragonal bonds. Being composites, the bonds of

this group of semiconductors are covalent and ionic, which induces polarity within the

material (i.e., possessing an intrinsic electric polarity). They are made up of one trivalent

atom (3 valence electrons) and one pentavalent atom (5 valence electrons). The members

of this group are GaAs, InSb, GaP, etc. Finally, there is also the II-VI group, of which

CdTe and ZnS are members. These semiconductors also possess cubic (zincblende)

structures. The bonds of these materials are also covalent and ionic, but are even more

ionic than those of group III-V [18].

At temperature T=0, a semiconductor contains energy bands that are either empty

or full. The electrons occupy the valence band completely while the conduction band

remains empty. Between the valence and conduction bands exists a band gap.

Figure 3.4 illustrates the differences between metals, semiconductors and insulators

1. Single Photon Emission Computed Tomography2. Positron Emission Tomography

27

Figure 3.1 A unit cell showing a diamond lattice structure.[19]

Figure 3.2 A unit cell showing a cubic (zincblende) crystal structure which is distin-guishable from the diamond structure due to the presence of a second type of atom.[20]

28

Figure 3.3 The band structure of Silicon, showing the conduction band with the valenceband below it. The energy is graphed as a function of the wave-vector~k.[23]

in the context of energy bands. The conduction and valence bands are indicated in blue

and red respectively. The position of these bands with respect to the Fermi level dictates

how charges are conducted within the material. For conduction to occur, electrons are

displaced from the valence band to the conduction band. For metals, the two bands

overlap, thus the energy required to excite an electron is small. For these materials,

conduction can result simply by thermal excitation. In the case of insulators, the large

separation between the bands prohibits electrons from jumping from the valence band to

the conduction band.

The bands of a semiconductor are neither overlapping, as they are for metals, nor are

they as far as they are for insulators. Their band gap is on the order of a few eV, which

significantly limits conduction by thermal excitation but also allows for conduction by

virtue of the energy transferred by radiation entering the material.

It is important to note that the density of charge carriers in a semiconductor is much

lower than it is for metals and is highly temperature dependent. Furthermore, the bands

below the valence band and above the conduction band do not play a role in the con-

ductivity of the material [18]. The number of charge carriers varies as a function of the

29

temperature according to the Fermi-Dirac distribution [21].

f (E) =1

eEEF

kBT +1(3.1)

3.2 Electrical Conductivity and Mobility

The current density of a semiconductor is given by [18] [21]

j = |~E| (3.2)

The electric field causes charge carriers to drift with a velocity

~vde,h = e,h~E (3.3)

with a mobility, expressed in cm2V1s1,

e,h =qe,he,h

me,h, (3.4)

me,h and e,h are the mass and the lifetime of the electron or the hole. The lifetime

represents the elapsed time prior to which the charge carrier is captured by a trap in the

crystal lattice [4]. The product of the mobility and the lifetime is a parameter that char-

acterizes the spectroscopic quality of a detector. It is the only parameter of the material

which describes the dependence of the charge collection efficiency on an applied bias.

This product will be evaluated in section 5.2 of this study.

A few mobility values for common materials are presented in Table 3.I.

Material Electrons HolesSi 1350 480

CdTe 1100 100GaAs 8000 300

Ge 3600 1800

Table 3.I Mobility values of common semiconductors at 300 K, expressed incm2V1s1.[21]

30

Figure 3.4 A comparison between the position of the energy bands of a metal, a semi-conductor and an insulator with respect to the Fermi energy. [24].

Figure 3.5 The Fermi-Dirac distribution for a few values of kBT/EF .

31

3.3 Properties of CdTe

3.3.1 Fundamental Absorption

CdTe possesses a direct band gap, meaning that the trough of the conduction band

lies directly above the peak of the valence band. The band width is the difference in

energy between these two points. Figure 3.6 compares direct and indirect band gaps.

The property of having a direct band gap grants CdTe with the ability to be used as

an optoelectronic instrument [18]. In fundamental absorption, an electron absorbs an

incident photon and jumps from the valence band to the conduction band. This photon

must have an energy greater or equal to the energy of the band gap. Its frequency must

therefore be

Egh

(3.5)

where Eg is the energy of the band gap (Eg = 1.52 eV for CdTe)[21]. For a material

possessing an indirect band gap, like Si (Eg = 1.12 eV)[21], the electron cannot make

the transition between bands in the same way. In this case, the process takes two steps.

The electron absorbs a photon and a phonon 3 simultaneously. The photon provides the

energy necessary to make the jump while the phonon provides the momentum required

to transition to the bottom of the conduction band.

3.3.2 Applications of CdTe

CdTe has a high atomic number (ZCd = 48. ZTe = 52), a high density ( = 5.85

g/cm3) and a wide band gap (Eg = 1.52 eV). Its electron-hole pair creation energy ()

is 4.43 eV and its resistivity is 109 cm[22]. The band gap of CdTe is relatively large

compared to other semiconductors (Eg = 1.12 eV for Si), which allows it to be oper-

ated at relatively high temperatures (up to 100C) [5]. Beyond 100, the value of Eg

decreases significantly with the temperature. Its high atomic number endows it with a

3. A phonon is a quasi-particle corresponding to an excitation of the vibrational energy of the crystallattice

32

Figure 3.6 a) A direct band gap, like for CdTe. b) An indirect band gap, like for Si.The photon absorption process is illustrated for each case with an arrow indicating thetransition of an electron from the top of the valence band to the bottom of the conductionband.[25]

high detection efficiency for -rays due to their interactions with matter through pho-

toelectric absorption (PE Z45), Compton scattering (C Z) and pair production(PP Z2). In medical imaging, this high sensitivity to -rays is crucial for SPECTand PET [1] [17]. In particle physics experiments, CdTes high density gives it good

radiation hardness[2] and its high photon detection efficiency allows it to be used as a

-ray detector in high radiation fields produced by particle collisions where photons are

present in large quantities.

The performance of CdTe is limited by the existence of traps within the material.

These traps capture charge carriers and are a result of impurities in the crystal lattice or

crystal fabrication defects.

3.4 The TIMEPIX Detector

The Timepix (TPX) detector [26], illustrated in Figure 3.7, is a pixelated semicon-

ductor detector. In this study, the detector used is made up of a sensor layer of 1 mm

which is connected by bump-bonding to a readout chip. The CdTe chip is equipped with

a common backside electrode and with a matrix of electrodes (256 256 pixels squared,each with an area of 55 55 m2) on the front side. Figure 3.8 shows the different

33

components of the TPX. Each pixel contains its own amplification circuit, composed of

a preamp, a discriminator and a counter. The Timepix electronics scheme is shown in

Figure 3.9.

The device can be connected with a USB FITPix interface (for Silicon) or a Canpix

interface (for the TPX-CdTe)[28], which is then connected to a computer via USB cable.

The data acquisition and device control is done using the Pixelman[6] software.

The charges produced from electron-hole pair creation migrate towards the elec-

trodes under the influence of an electric field induced by the bias voltage applied across

the detector. At the cathode of each pixel, the charge is collected and the signal is ampli-

fied by the current amplifier. This signal is then compared to a threshold. If it is superior

to the threshold, the counter increments by 1, otherwise it stays at 0. A frame is produced

indicating the state of each pixel during the data acquisition period (Figure 3.10). The

adjacent pixels that also have counter values greater than 0 are collectively called a clus-

ter or a track, whose shape depends upon the type of interaction of the incident particle

with the detectors active layer. The shape varies according the the energy, the incidence

and the species of the particle. It is by studying these shapes that particle identification

is possible.

The measurement of events can be accomplished using one of three possible modes

of operation: 1) tracking and counting mode, 2) TOT (Time-over-Threshold) mode and

3) ToA (Time of Arrival) mode. In TOT mode, an internal clock counts the time during

which the signal is above the threshold. This clock operates at a defined frequency of

9.6, 24, 48 or 96 MHz for FITPix interfaces. Thus, the TOT values obtained are simply

Figure 3.7 A Timepix detector equipped with a FITPix interface.[27]

34

Figure 3.8 A Timepix detector. (A) is the semiconductor sensor layor. (B) shows thereadout chip and the electronic bump bonds.[27]

Figure 3.9 The electronics scheme of a single pixel in a Timpepix detector.[26]

35

Figure 3.10 A frame showing the tracks of 1.4 MeV protons in the TPX-CdTe detector.

the clock frequency multiplied by the measured time. It is therefore, possible to measure

energy values of incident particles using an appropriate TOT-energy calibration. ToA

mode allows one perform coincidence measurements with the aim of measuring time of

flight (ToF).

CHAPTER 4

POLARIZATION EFFECTS IN COMPOUND SEMICONDUCTORS

The so-called "polarization effect" contributes towards the reduction of the charge

collection efficiency of the studied CdTe detector. Polarization in the TPX-CdTe detector

has been observed and studied in this chapter.

4.1 Polarization in CdTe Detectors

When an external bias of the order of a few tens to a few hundreds of Volts is applied

on a detector, an electric field is produced under whose influence charge carriers drift

towards the electrodes, where they are collected. It has been observed that, at room

temperature (T = 25C), the collected charges decrease as a function of time after the

application of an initial bias. This has led to the consensus that there is a reduction in the

electric field intensity across the detector thickness, implying a reduction in the effective

internal voltage[1]. The result of this is the partial depletion of the detector. The depleted

zone is partially lost near the electrodes and is considered to be polarized, leading to the

reduction of the applied voltage.

An accumulation of charges is produced at the interface of the electrode and of the

material which provokes the diminished performance of the detector. A model explain-

ing this effect considers the energy band bending in the region of the electrode contacts.

In addition to this bending, hole detrapping by deep acceptor levels creating a charge

accumulation is proposed as the physical phenomenon responsible for polarization[8].

Such deep acceptor levels can cross the Fermi level when the bands are bent at a close

proximity to the electrode contacts, as is shown in Figure 4.1. This bending translates to

the behaviour of the electric field observed in Figure 4.2.

The temporal dependence of this behaviour is a result of the high activation energy

associated with the ionization of deep acceptor levels [5]. These deep acceptors are

ionized in a time I , on the order of

37

Figure 4.1 Band bending model near the semiconductor-metal interface, used to cal-culate the depth of the depletion layer and the electric field. l is the depletion layersthickness. W is the function describing the band bending and whose value correspondsto the separation between the deep acceptor energy (E) and the Fermi energy. V is theapplied potential, is the contact potential, is the region in which the charges areaccumulated and = l .[8]

Figure 4.2 Schematic plot of the electric field as a function of the position in thesituation shown in Figure 4.1. The slope of the electric field changes abrubtly where theenergy band intersects the Fermi energy.[8]

38

I =1

N0veE/kT (4.1)

where N0 is the density of states of deep acceptors, is the acceptor capture cross-

section, v is the average thermal velocity of charge carriers and E is the depth, inenergy, of the traps. It is immediately seen that, since E kT , I is large. In the case ofdetrapping, the time becomes shorter and is given by

D =1

NV v(4.2)

where NV is the electronic density of states at the top of the valence band.

Solving the Poisson equation for the potential V as a function of the position x gives

the form of the band bending at equilibrium (without an applied bias). The space is

characterized by two regions:

d2Vdx2

=(p+N) e, 0 < x < , (4.3)

d2Vdx2

=p e, < x < l, (4.4)

where p is the equilibrium hole concentration, is the dielectric constant, e is the unit

of electronic charge, l is the total depletion thickness of the detector and N is the number

of ionized deep acceptors. The region is the region in which charges are accumulated.

Solving (4.3) and (4.4), an expression describing the region can be obtained[8].

=[

2e(N + p)

] 12(

V0N

N + pW) 1

2

pN + p

, (4.5)

where

= l =[

2Wep

] 12

(4.6)

The potential V0 is the sum of the applied potential V and the contact potential 1

1. The contact potential is the difference between the electrostatic potentials of two metals in contact

39

and W is the separation between the deep acceptor and the Fermi level. These quantities

are all indicated in Figure 4.1.

At t = 0, N = 0. In general, we can assume

N = N0[1 et/I

](4.7)

When N0 p and W V0, we get

pN0

[1 et/I

]=

(0

)21, (4.8)

where 0 is the final value (t ) of :

0 =[

2e(N0 + p)

] 12(

V0N0

N0 + pW) 1

2

pN0 + p

. (4.9)

Then, for short time periods, t I , we obtain:[(0

)21

]=

(tI

)(p

N0

), (4.10)

so when t I , (4.8) becomes[(0

)21

]=

(p

N0

). (4.11)

4.2 Polarization Reduction Techniques

There have been many different methods traditionally used to eliminate or, at the

very least, strongly reduce the effects of polarization in CdTe detectors.

One method reinitializes the applied bias to a value of 0 V, thus dissipating the spa-

tial charge that is collected at the electrodes. The procedure consists of periodically

removing the bias during the measurement to induce depolarization. This is done over

the course of a charge collection measurement for a long period of time. This technique

with each other. It is determined by the difference in the work functions of each metal.

40

exploits the fact that I > D. In practice, for example when a detector is used in dosime-

try, its detection efficiency is reduced whenever the applied bias is reset to 0 V. Even if

the overall stability of the detector is improved, periodically, a fraction of the particles

penetrating the material is not detected.

A more effective technique is to use electrodes with high work functions. In doing

so, the contact potential at the electrode-semiconductor interface is modified to prevent

the deep acceptor level from crossing the Fermi level. This is accomplished by using a

p-type metal that behaves like an Ohmic contact. Gold and platinum have been conven-

tionally used as contact metals.

4.2.1 Schottky and Ohmic Contacts

It has been mentioned that the type of contact used in CdTe devices influences the de-

gree of polarization of the semiconductor. A description of Schottky and Ohmic contacts

is therefore necessary.

A Schottky contact is a metal-semiconductor contact whose contact potential is large

and whose dopant concentration is small (i.e., a concentration which is less than the

electronic density of states in the valence or in the conduction band). In this scenario,

the conduction band is bent away from the Fermi energy and the semiconductor becomes

intrinsic in this region. There are now fewer mobile charge carriers and the depletion

layer is produced.

An Ohmic contact is a metal-semiconductor contact that is characterized by a negli-

gible contact resistance with respect to that of CdTe. In the case of the TPX-CdTe, the

contact is made of Pl [28]. There exists an unimpeded majority carrier transfer going

from the semiconductor to the contacts. This means that the contact does not limit the

current [29]. In this case, the conduction band energy is less than the Fermi energy close

to the interface. The semiconductor behaves electrically like a metal in this region, the

result being the formation of an Ohmic contact.

41

4.3 Polarization Studies with the Timepix-CdTe Detector

A study of the possible effects of polarization in the TPX-CdTe detector were under-

taken at the University of Montreal. Previous studies on this topic primarily used diodes

and their results were principally taken from charge or deposited energy measurements

from photons in the active layer. For detectors possessing Schottky contacts, this type of

analysis based on collected charge measurements was adequate for such studies. How-

ever, when the same procedures were applied to Ohmic-type detectors, the polarization

effect was sufficiently reduced, the result being that the charge collection within the

detector was reliable for long periods of time.

This does however raise the question of whether the electric field inside the detector

is truly unchanged after a long measurement period. The advantage of using a Timepix

detector is that it allows for the measurement of the deposited energy or charge as well as

the position of particles, permitting the measurement of charge sharing between pixels.

By using the charge sharing effect, a small weakening of the electric field can be deduced

from an increase in the size of certain clusters.

The TPX-CdTe detector has been exposed to an 241Am source during a 38 hour

period inside a vacuum chamber set at 5 107 Torr with an external bias of 150 Vwhich was fixed for the duration of the experiment. The 241Am source is placed at

a distance of 1.5 cm from the TPX-CdTe. The particles reached the detector at an

average angle of incidence of 0. The energy of the emitted particles was measured

in increments of an hour and is represented as a function of time in Figure 4.3. As is

expected from a detector with Ohmic contacts, the results show no significant reduction

in the measured energy, indicating the absence of polarization. The average particle

energy measured over the 38 hour period was 4920 keV, as is shown in Figure 4.3.

It is important to note that this energy is less than that expected from an 241Am

source, 5.48 MeV. This is due to the CdTe thickness being only partially depleted at 150

V. Consequently, a fraction of the energy is not converted into charge carriers and, thus,

does not contribute to the signal. The measured energy being practically unchanged over

38 hours, a method using the cluster size as a variable was developed. It is for this reason

42

that the value of 150 V was chosen as the reference bias voltage. This assures that the

variation in the size of clusters is sufficiently large to observe the weak variations in the

bias. To illustrate this, Figure 4.4 shows the variation of the cluster size as a function of

the bias. It is clear that operating at a reference bias of 300 V or 400 V would not reveal

a significant increase in the cluster size after a long measurement period if the reduction

in the voltage is weak. This then makes the task of determining any variation in effective

voltage with time difficult.

The behaviour seen in Figure 4.4 can be explained in the following manner. An

incident particle creates a column of charges (electrons and holes) so these charges drift

towards their respective electrodes under the influence of the electric field created by the

applied voltage across the detector thickness. At the same time, there exists a lateral or

radial drift, caused in part by the diffusion due to the density gradient which leads this

charge column to grow radially before each carrier is collected at the electrode. If the

voltage increases, the intensity of the field increases as well. This accelerates the drift

towards the electrodes and subsequently reduces the lateral drift. The result of this is that

the cluster size becomes smaller, which is illustrated in Figure 4.5 for 3.5 MeV protons

at an angle of incidence normal to a Silicon Timepix detector, where the radial diffusion

at 40 V and 7 V are compared.

For the same measurement period, the average cluster size of particles has been

plotted in the same way as for the energy, showing the evolution of this variable in

time. Its behaviour is shown in Figure 4.6. In the absence of polarization caused by

an electric field, there would not be any change in the size of clusters over the course

of the measurement. Figure 4.6 shows that this is not the case. There is an increase

in the cluster size as of the beginning of the measurement until the end, indicating a

weakening of the electric field across the layer of CdTe. Such a weakening slows the

longitudinal drift of carriers, increasing their drift time. This enables a longer radial drift

and, therefore, a larger cluster size.

This behaviour can then be compared to the values of the cluster size obtained during

an analysis at multiple biases, as in Figure 4.4. We can then assume that, for small

variation in bias, a linear relation between the bias and the cluster size is valid. By

43

Figure 4.3 A 38 hour measurement of the deposited energy by particles from an241Am source at 150 V.

Figure 4.4 Measurements of the change in the cluster size as a function of the ap-plied voltage for particles from an 241Am source. A linear fit around 150 V (120-180V) gives the correspondance CS = 0.110860V + 44.8359. The R2 value of this fit is0.994426.

44

Figure 4.5 The influence of the bias on the radial diffusion of charges created by 3.5MeV protons at an incidence of 0 in a Si Timepix.[30]

45

Figure 4.6 A measurement in time of the cluster size of particles over 38 hours.

assuming that the change in cluster size is linear around the 150 V point (Figure 4.4), a

linear approximation over the 120-180 V interval is taken and a fit function is produced:

CS =0.110860V +44.8359, (4.12)

where CS is the cluster size.

Evaluating the fit function for the cluster size at the end of the measurement, ex-

tracted from Figure 4.6, the effective bias can be calculated. In fact, the behaviour of

the bias and, as a consequence, that of the electric field can be obtained for the entire

measurement period. An expression describing the behaviour of the effective bias for

small deviation from 150 V can be obtained.

V =9.02039CS+404.437 (4.13)

From (4.13), the effective bias is calculated for each point in Figure 4.6. Each cal-

culated voltage corresponds to an hour during the measurement. Together, they show

the behaviour of the effective bias as a function of time during the 38 hour period. This

is illustrated in Figure 4.7. During the 38 hours, the effective bias does not vary by

46

more than 15 V. Furthermore, the majority of its variation occurs in the first 17 hours,

after which, the bias seems to stabilize for a time. As of the 29 hour mark, a weakly

decreasing trend resumes.

This proves that an Ohmic-type detector can become weakly polarized. The reason

for this is that the resistivity of the Pl contact is larger than that of the CdTe, which

causes a very small deviation from Ohms law and, therefore, a small quantity of deep

acceptors can be produced near the electrodes.

47

Figure 4.7 The behaviour of the effective bias over the course of the 38 hour measure-ment period with an initial applied voltage of 150 V.

CHAPTER 5

CHARACTERIZATION OF THE TIMEPIX-CDTE DETECTOR

5.1 Energy Resolution

A study of TPX-CdTes energy resolution has been completed using data from pro-

tons and photons. The measured energies were obtained from a per-pixel calibration

conducted at the IEAP in Prague and are compared to those obtained with a global cali-

bration using protons and 6Li ions performed in Montreal.

5.1.1 Per-Pixel Energy Calibration

A calibration by X-ray fluorescence (XRF)[31] has been performed for each pixel in

the TPX-CdTe detector. The detector has been irradiated by monoenergetic X-rays in

order to limit to a single pixel the propagation of charge carriers created by an incident

photon. The necessary cluster size for this calibration is one pixel. Events producing

larger cluster sizes are rejected. Such events can be caused by a photon that interacts

with the detector at the boundary of multiple pixels or by charges that diffuse from one

pixel to its neighbour. Each pixel registers an energy spectrum whose peak is fitted with

a Gaussian function. This step is repeated for at least three other X-ray energies in order

to produce an adequate calibration function of the form

f (x) = ax+b cx t

(5.1)

where the cxt term takes into account the non-linear TOT response at low energy.

The a, b, c, and t parameters are calculated for the function f and at least five least

squares fit tests are performed for each pixel. The function f is shown in Figure 5.1,

where seven calibration points have been used.

This calibration method is applicable for per-pixel energies less than 500 keV. Todemonstrate this, the TPX-CdTe has been exposed to a 59.5 keV photon 241Am source.

A 3.2 mm Al plate was positioned between the source and the detector in order to stop

49

Figure 5.1 The dependence of the TOT on the X-ray energy for each pixel, expressed bythe calibration function f . The cxt term takes into account the calibrations non-linearitynear the threshold.[31]

the 5.48 MeV particles also emitted. The source has been placed on a mounted support

at a distance of 1.5 cm from the detector, at an angle of 0 with respect to its surface.

The detector was operated at a bias of 400 V. Figure 5.2 shows the peak at 58.8 keV in

the spectrum produced by the X-rays.

The energy difference between the observed peak and the expected energy of 59.5

keV is 1.18%.

If this calibration is applied to higher energy photons, the same precision is not repro-

duced. Figure 5.3 illustrates the energy spectrum of a 137Cs source of 662 keV photons.

Once again, a 3.2 mm Al plate was placed between the source and the detector to stop

the emitted electrons 1. The distance between the source and the detector was 1.2 cm.

The measured energy of these photons is 711.1 keV, a difference of 7.4% from its

expected value of 662 keV. It is clear that the measurement of higher energy photons and

particles would be poorly served by this calibration.

Energy measurements of 0.796 - 9.949 MeV protons have been taken at the Van

der Graaf accelerator at the University of Montreal and the measurement of 0.995 MeV

1. Electrons of 514 keV emitted from 137Cs are stopped in 0.84 mm of Al.

50

Figure 5.2 The photon energy spectrum of an 241Am source with a peak at 58.8 keV.

Figure 5.3 The energy spectrum of photons coming from a 137Cs source (662 keVphotons) with a peak at 711,1 keV.

51

protons have been taken at the IEAP in Prague with the goal of measuring the energy

resolution of the TPX-CdTe detector. We describe here the experimental setup used

in Montreal which makes use of Rutherford backscattering. The TPX-CdTe is placed

inside a vacuum chamber under a pressure of 107 Torr. A proton beam (0.8 up to 10MeV) has been backscattered off a gold foil with a thickness of 0.12 m. The detector

is bombarded by backscattered protons. It is placed at a scattering angle of 90 after the

gold foil. The gold foil is placed in an adjacent chamber, at 45 with respect to the beam

direction.

Table 5.I gives the resolutions for each energy with percent differences between the

measured and expected energies.

The resolution is calculated and expressed as a percentage according to

R =FWHM

Em100, (5.2)

where FWHM is the full-width at half-maximum of the Gaussian distribution fitted

to the spectrum for each energy and Em is the measured energy.

The accuracy of the calibration is quantified in terms of the percent deviation %Dev

of the measured and expected energy of the backscattered proton from the gold foil. It is

observed that the calibration is accurate for energies between 3 to 7 MeV but becomes

less so outside of this energy interval.

5.1.2 Global Energy Calibration

Due to the per-pixel calibrations limitations, it was necessary to calibrate the TPX-

CdTe detector using some other method in order to measure the energy resolution. The

method employed uses protons of different energies as calibration points. Henceforth,

this method will be called "global calibration".

The detector was operated at a bias of 300 V and was exposed to a flux of heavy

ionizing particles of known energies. 3, 5, and 9 MeV protons have been used as well

as 20 MeV 6Li ions. The mean TOT value measured for each spectrum is illustrated

as a function of the backscattered energy in Figure 5.4. A curve has been fitted on the

52

E (keV) Em (keV) FWHM (keV) R (%) % Dev795.95 430.898 73.095 16.96 45.86895.44 585.320 84.712 14.47 34.63994.94 718.528 135.200 18.82 28.13

1,094.43 873.908 112.458 12.87 20.151,193.92 981.923 132.357 13.48 17.761,293.42 1,116.610 134.320 12.03 13.671,392.91 1,239.910 155.245 12.52 10.981,492.41 1,411.620 74.230 12.36 5.411,989.87 1,919.320 112.352 13.76 3.552,984.81 3,021.590 151.194 11.76 1.233,979.75 3,997.160 206.310 12.13 0.444,974.68 4,954.210 253.156 12.01 0.415,969.62 5,966.670 261.112 10.28 0.056,964.56 6,952.300 298.755 10.10 0.188,954.43 8,502.680 395.129 10.92 5.049,949.37 9,055.420 450.306 11.69 8.98

Table 5.I The measured proton energies Em with the per-pixel calibration compared tothe expected energies after Rutherford backscattering with their percent deviation %Dev.The FWHM and the resolutions R are provided for each energy.

four points producing a quadratic function. Inverting this function gives the calibration

function,

E =1,3848.7(0.000144418(TOT +8.6129)+6.013242.45219

). (5.3)

Reprocessing the 662 keV proton data shown in Figure 5.3 with the global calibration

gives the spectrum illustrated in Figure 5.5. The accuracy in energy is visibly improved.

The photopeak is now located at 661.8 keV. Table 5.II provides the measured photon

energies.

Figure 5.6 compares two energy spectra measured with both calibrations for 10 MeV

protons. The expected energy after backscattering is 9,949 keV. The per-pixel calibra-

tion gives a value of 9,055 keV, which corresponds to a difference of 8.98% from the

expected value. The measured energy using the global calibration is 9,737 keV, corre-

53

Figure 5.4 The calibration function for heavy ionizing particles (3, 5 et 9 MeV protonsand 20 MeV 6Li ions).

E (keV) Em (keV) FWHM (keV) R (%) % Dev59.5 58.8 14.994 25.50 1.18662 661.8 85.515 13.22 0.03

Table 5.II The energies Em of photons emitted from 241Am (59.5 keV) and 137Cs (662keV). The energy of 59.5 keV is measured using the per-pixel calibration while the 662keV photons were measured with the global calibration. %Dev gives the percent devia-tion of each measured energy from its expected value. The FWHM and the resolution Rare given for each.

54

sponding to a difference of 2.13% from the expected value. The shift in the peak of the

global calibration shows how it can correct for the energy loss observed by the per-pixel

calibration, caused by saturation effects in the pixels that measure energies greater than

500 keV.

The values presented in Table 5.I are calculated once again in Table 5.III using the

global calibration. On average, the global calibration improves the resolution by 8.6%

compared to that of the per-pixel calibration. This improvement is shown in Figure 5.7.

It is also observed that the accuracy of the measured energy decreases for low ener-

gies, quantified by %Dev. Alternatively, by increasing the proton energy, the quantity of

charge carriers increases as well. This means that the relative fraction of trapped carriers

decreases with increasing energy.

There are, however, limitations to the global calibration. It can only be applied to

data taken at the bias used during calibration measurements.

5.2 Studies

The CdTe material contains impurities and defects which are caused by fabrication

limitations and by the presence of two types of atoms within the crystal lattice[4]. These

imperfections manifest themselves in the form of traps that prevent charge carriers from

freely moving through the material.

One way of determining how the TPX-CdTe detector is affected by these traps is to

measure the product of the charge carriers mobility and their lifetime (). In section

3.2, the importance of this parameter was mentioned in relation to the spectroscopic

characterization of a detector. The lifetime varies depending on the trap concentration in

the detector. The more traps there are, the higher is the possibility that a charge carrier

can become captured and, thus, becomes smaller.

To calculate the product for charge carriers in the detector, measurements of

deposited energy in the TPX-CdTe detector at multiple bias voltages have been taken.

A curve was fitted to these points. By assuming that the electric field is uniform and

by neglecting detrapping, one obtains an expression for the charge collection efficiency,

55

Figure 5.5 The energy spectrum for 662 keV photons emitted from a 137Cs sourceprocessed with a global calibration. The measured energy is 661.8 keV.

CCE, given by the Hecht Equation[4],

CCE =QQ0

=

(eeE

d

)[1 e

(dx0)eeE

]+

(hhE

d

)[1 e

x0hhE

]. (5.4)

where Q is the measured charge, Q0 is the charge created near the cathode, d is the

detector thickness and x0 is the distance between the cathode and the charge carrier. The

pixels, which are located on the anode side, collect electrons. In an Ohmic detector, the

signal is induced only by electrons. If the penetration of an incident particle is small

enough, i.e., corresponding to a small range compared to the thickness of the detector

(1,000 m), the x0 term becomes negligible (x0 0) and the Hecht Equation becomes

Q(V ) = Q0

(ee(V V0)

d2

)[1 e

d2ee(VV0)

], (5.5)

where V0 takes into account the imprecision of the bias while the Canpix interface

applies a bias of 0 V.

56

Figure 5.6 A comparison between the energy spectra for 10 MeV protons obtainedwith the per-pixel (black) and global (red) calibrations.

Figure 5.7 The resolutions of each energy for the per-pixel (black) and global (red)calibrations. Protons with energies from 0.8 to 10 MeV are shown.

57

E (keV) Em (keV) FWHM (keV) R (%) % Dev795.95 464.10 71.33 6.54 41.69895.44 615.56 79.91 5.52 31.26994.94 789.69 98.07 5.28 20.63

1,094.43 898.01 94.14 4.46 17.951,193.92 1,001.42 104.31 4.43 16.121,293.42 1,136.33 109.49 4.10 12.151,392.91 1,254.65 121.53 4.12 9.931,492.41 1,422.860 128.50 3.83 4.661,989.87 1,912.720 198.50 4.42 3.882,984.81 3,014.020 277.75 3.92 0.983,979.75 4,001.590 352.70 3.75 0.554,974.68 5,023.260 474.21 4.02 0.985,969.62 6,100.430 463.00 3.23 2.196,964.56 7,216.970 564.18 3.33 3.628,954.43 9,078.090 786.33 3.69 1.389,949.37 9,737.610 897.28 3.92 2.13

Table 5.III The measured proton energies Em with the global calibration compared tothe expected energies after Rutherford backscattering with their percent deviation %Dev.The FWHM and the resolutions R are provided for each energy.

The ee, Q0 and V0 factors are treated as free parameters and are extracted from the

fit. The values of Q and Q0 are obtained in fC by dividing the measured energies with the

per-pixel calibration by the mean energy required for electron-hole pair creation in CdTe

(4.43 eV). The measurements were made at the Tandem accelerator at the University

of Montreal. The incident particles were 1.4 MeV protons to reduce the penetration in

the TPX-CdTe as well as to limit the energy lost inside the platinum contact material.

Figure 5.9 shows the Hecht fit calculated using the Minuit2 software package. The result

is summarized in Table 5.IV 2 with the average range of protons in CdTe, giving an idea

as to how precise the approximation is with respect to the 1,000 m thickness of the

TPX-CdTe. Due to the e factor, the electron lifetime, a low ee value would indicate

2. The coefficient of determination is given by

R2 = 1ni=1(yi fi)2

ni=1(yi y)2, (5.6)

where yi are measured energy values, fi are the predicted values based on the fit and y is the average ofthe measurements. The code that was used to calculate this value is shown in Figure 5.8.

58

that trapping contributes significantly to the decrease of the charge collection efficiency.

It should be noted that the charge collected (Q0 = 45.37 fC) is 11% less than the

expected charge, Q, given by

Q =1.39291106eV

4.43eV1.6021019C = 50.37 fC, (5.7)

which means that there is a significant quantity of charges that are trapped in the

material.

This ee value is in agreement with those measured by other researchers. A list of

values taken from the literature can be found in Table 5.V.

Proton Energy Range (m) (103 cm3V

)R2

1,393 keV 14.21 1.32 0.43 0.99914

Table 5.IV The product of the TPX-CdTe detector for 1.4 MeV protons with thecorresponding range in m[9] and the quality of the fit (R2).

59

Figure 5.8 The calculation of the coefficient of determination.

Figure 5.9 The measured energy as a function of the bias for 1.4 MeV