This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 129.173.72.87

This content was downloaded on 13/11/2014 at 13:05

Please note that terms and conditions apply.

Computational modelling of pixelated CdZnTe detectors for x- and γ- ray imaging applications

View the table of contents for this issue, or go to the journal homepage for more

2012 JINST 7 P03004

(http://iopscience.iop.org/1748-0221/7/03/P03004)

Home Search Collections Journals About Contact us My IOPscience

2012 JINST 7 P03004

PUBLISHED BY IOP PUBLISHING FOR SISSA MEDIALAB

RECEIVED: December 20, 2011REVISED: February 3, 2012

ACCEPTED: February 14, 2012PUBLISHED: March 12, 2012

Computational modelling of pixelated CdZnTedetectors for x- and γ- ray imaging applications

M.E. Myronakis,a,1 M. Zvelebilb and D.G. Darambaraa

aJoint Department of Physics, Division of Radiotherapy and ImagingThe Institute of Cancer Research and Royal Marsden Foundation NHS Trust,Fulham Road, London, SW3 6JJ, U.K.

bBreast Breakthrough Centre, The Institute of Cancer Research,237 Fulham Road, SW3 6JB, London, U.K.

E-mail: marios.myronakis@icr.ac.uk

ABSTRACT: Cadmium Zinc Telluride (CdZnTe) detectors are currently used in medical imag-ing systems employingγ-ray photons. As new imaging techniques such as photon-counting andenergy-weighted x-ray imaging are gaining research interest, CdZnTe is seen under a new light forpotential use in computed tomography, tomosynthesis and other x-ray imaging applications. How-ever, being relatively expensive, CdZnTe could be favouredby advanced computational modellingto assist in detector and imaging system optimisation. In this work, pixelated CdZnTe detectorsare computationally modelled using an integrated framework that combines the Finite Element andMonte Carlo numerical methods to obtain realistic detectormodels.Various detector thickness andpixel sizes are designed and their performance is investigated in terms of charge induction effi-ciency, detection efficiency and energy resolution. Detection efficiency and energy resolution areassessed for monoenegergetic photon beams within the energy range used in medical x-ray imag-ing applications such as mammography and computed tomography. Some of the capabilities of theframework are demonstrated. Small pixel sizes, below 100µm are prone to charge transport effectssuch as diffusion, especially in larger thickness (> 0.5 mm) and may have limited use in pixelatedgeometries. Detection efficiency is affected by fluorescence and photon escape as thickness andpixel size decrease. Energy resolution is affected by beam geometry and can vary from∼3% to11% depending on the beam width. The framework provides a generic platform and a powerfultool that can be used in the design and optimisation of semiconductor detectors made from anysemiconductor material, imaging systems and signal correction techniques.

KEYWORDS: Solid state detectors; X-ray detectors; Detector modelling and simulations II (elec-tric fields, charge transport, multiplication and induction, pulse formation, electron emission, etc);Detector modelling and simulations I (interaction of radiation with matter, interaction of photonswith matter, interaction of hadrons with matter, etc)

1Corresponding author.

c© 2012 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1748-0221/7/03/P03004

2012 JINST 7 P03004

Contents

1 Introduction 1

2 Methods 22.1 Modelling framework 2

2.1.1 General description 22.1.2 Features 32.1.3 Monte Carlo Model 42.1.4 Finite Element model 4

2.2 Detector performance calculations 42.2.1 Charge induction efficiency 52.2.2 Detection efficiency 52.2.3 Energy resolution 5

3 Results 63.1 FEM results 63.2 MC results 6

4 Discussion 9

5 Conclusion 14

1 Introduction

The advent of digital detectors in x-ray imaging, particularly in medical applications, promoted fur-ther research towards image quality improvement and dose reduction to the patient [1–4]. Photon-counting and energy-weighting techniques for image acquisition are actively investigated and newdetector materials such as the semiconductor Cadmium Zinc Telluride (CdZnTe) are assessed as po-tential candidates for x-ray imaging [5, 6]. In this context, computational models of x-ray systemsand detectors are used to assist in design optimisation, performance improvement and assessmentof new detector materials and imaging systems.

Realistic semiconductor detector modelling requires the integration of several layers of modelsof physical processes to obtain meaningful and valid results. In semiconductors, signal is generatedfrom electron-hole pairs (e-h pairs) that drift along the detector volume towards opposite electrodes.The electrodes are held under constant bias potential that creates an electric field within the detectormoving the pairs in opposite directions. The e-h pairs (charge carriers) are produced through theinelastic interaction of electrons generated by x-ray photon interacting within the detector material.Electron inelastic collisions may also produce characteristic x-rays or Auger electrons which canescape the detector or the primary pixel of interaction (when pixelated electrodes are utilised) and

– 1 –

2012 JINST 7 P03004

degrade the detector spatial resolution and detection efficiency. In addition, e-h pair diffusion to ad-jacent pixels, charge trapping (in energy levels due to material defects and impurities) and chargesharing between pixels reduce the induced signal in the collecting electrode. Hence, the combi-nation of all the aforementioned physical processes is essential to accurately model the pixelatedCdZnTe detector.

An early attempt to create computational models of CdZnTe strip detectors was made by Her-man et al. [7] and Tousignant et al. [8] using analytical solutions of the continuity equations andMonte Carlo simulation. Although the results obtained werequalitatively good there was a lack ofproper modelling of the response of various parameters, forexample the signal readout [7] or thelack of three-dimensional information [8]. Later work by Mathy et al. [9] used the finite elementmethod to obtain the solution to the continuity equation in pixelated CdZnTe, the main focus ofthis work being the biparametric spectra analysis. More recent works include those from Guerra etal [10], who used simplified analytical description of the charge transport with Monte Carlo simu-lations, and Benoit & Hammel [11], who used numerical and analytical solutions with Monte Carloto model strip detectors. Although most models produce reasonably good qualitative results, thereis a lack of a generic and one-includes-all framework available. Several omissions such as properelectronic modelling, electric field polarisation or even acomplete one-to-one match between themodel and the real detector properties limit the capabilities of most models. There is also somelag regarding the integration of modelling levels that willallow for compatibility between differentcomputational methods and require minimum effort to construct the model.

In the current work, a comprehensive integrated computational modelling framework of pixe-lated CdZnTe is described based on a previous work from our team [12]. It can be used to modeland assess the performance of x- andγ- ray imaging semiconductor detectors and systems. Theframework is flexible and can be effectively applied to modelany semiconductor detector materialand imaging system. It combines the Finite Element numerical Method (FEM) to model chargetransport effects and Monte Carlo numerical computationalexperiments to obtain information ofphoton interactions and energy deposition. The method has been previously used forγ-ray imag-ing detector models [12]. In this work, the framework is used to investigate the performance ofpixelated CdZnTe detectors under x-ray irradiation for various geometrical specifications (i.e. pixelsize and thickness) and photon energies.

2 Methods

2.1 Modelling framework

Detector modelling is based on the integration of Monte Carlo (MC) method and Finite ElementMethod (FEM) using the equations and implementation described in a previous study [12]. TheGeant4 Application for Tomographic Emission (GATE) [13, 14] was used for the MC simulationsand the COMSOL Multiphysics

TMcommercial applications for FEM models.

2.1.1 General description

The general concept of the integrated framework is to combine output data obtained by each com-putational method to achieve a realistic description of thesignal generation process in CdZnTe

– 2 –

2012 JINST 7 P03004

Table 1: CdZnTe material properties

Property Value Unit

Dielectric constant 10.9Pair generation energy 4.64×10−6 MeV

Electron mobility 10−1 m2/V ·sHole mobility 5×10−3 m2/V ·s

Electron lifetime 3×10−6 sHole lifetime 1×10−6 s

detectors. Photon interaction position and energy deposition information are obtained from theMC output and combined with the solution of continuity equations, obtained from FEM, that de-scribe electron and hole flow (i.e signal induction) within the detector. Previously [12], the datawere post-processed externally, using MatlabR© [15] to load and combine the output of both meth-ods. This technique required massive computational resources and was difficult to utilize when thenumber of incident photons was high

(

> 108)

.

Currently, solution data obtained from the FEM models are implemented in GATE through anew GATE extension, theGate Solid State Detector Module (GateSSDmodule)developed withinour team. The new module is specifically developed for pixelated geometries and can load solutionmatrices of any size into GATE (depending on computational resources). Trilinear interpolation inthree-dimensional space is used to obtain values not included in the implemented solution matrix.The interpolation points coincide with energy deposition locations within the detector, obtained“on-the-fly” during MC simulation. The accuracy of the method depends on the amount of datapresent in the solution matrix.

2.1.2 Features

The module has the capability toa) extract signal from the anodes and the cathode of the detector,b) adjust the shaping time and observe the outcome, e.g on the energy spectrum,c) obtain differentCIE per pixel, i.e. different mobility-lifetime product per pixel, d) set different noise level per pixel,e) model bad pixels,f) provide energy resolution information depending on material propertiesand detector geometry rather than user input as is the case inGeant4 and GATE, andg) it canpractically be used to model any solid state detector by simply changing the material propertiesand the solution map imported to the module. Implementationof the partial differential equationsused with the finite element method was described in [12] based on the methods initially describedby Barrett et al. [16] and Prettyman [17].

The physical processes considered in the modelling framework can be separated to those de-scribed in MC models and those in FEM models. The first includeparticle interactions, energydepositions in the detector, charge-carrier generation and the readout scheme. The latter includecharge-carrier drift, diffusion and trapping, and signal induction at the semiconductor detector elec-trodes. Separate models for each method are constructed according to the demands of our study asdescribed in the following subsections.

– 3 –

2012 JINST 7 P03004

(a) Monte Carlo model of the detector. (b) Finite Elelement model of the detector inCOMSOL Multiphysics

TM.

Figure 1: (a) Detector model used in Monte Carlo simulations. The pixel size varied from 40 to400µm and the thickness between 0.2 and 3 mm. The mono-energetic photon beams directed tothe central pixel and had energies from 10 to 140 keV. (b) Detector model used in Finite ElementAnalysis computations. The pixel size varied in accordancewith the MC model. Due to symmetryonly 9 pixels (3x3 area) were created to save computational resources.

2.1.3 Monte Carlo Model

The detector was modelled as a solid block of 100×100 pixels (figure1a). The pixel size variedfrom 40 to 400µm with 10µm step for pixel sizes up to 100µm and 50µm for pixels sizes between100 and 400µm. Detector thickness is varied from 0.2 to 3 mm (0.2, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0mm).

Each geometrical configuration (i.e specific pixel size and thickness) is irradiated with mono-energetic photon beams, focused at the central pixel of the detector, and photon energies between10 and 140 keV with 5 keV step size. The total number of computational experiments resultingfrom the combination of these specifications reached 4212.

The Photoelectric effect, Compton and Rayleigh scatter as well as x-ray fluorescence wereincluded in the physics interactions modelling.

2.1.4 Finite Element model

Finite Element models are computationally more demanding than MC models. Hence, due tosymmetry, the detector geometry was confined to 9 pixels (3× 3 pixel area) with pixel size andthickness corresponding to the MC models (figure1b). Smaller geometries require fewer meshelements and computational resources. The geometrical model included ohmic contacts (pixelatedanodes and single cathode). The contact separation was 50µm.

The bias potential applied at the electrodes was determinedfor each thickness by increasingthe potential up to the point that electron trapping was negligible (i.e. fast electron drift time).Further increase of the potential did not produce any difference in energy spectrum.

2.2 Detector performance calculations

Detectors were compared in terms of charge induction and photon detection using the charge in-duction efficiency (CIE), photon detection efficiency and photo-peak efficiency as figures of merit.

– 4 –

2012 JINST 7 P03004

In addition, the energy resolution for detectors with 1 mm thickness is calculated to demonstratethe framework ability to provide energy resolution depending on detector properties rather thanuser adjustment.

2.2.1 Charge induction efficiency

Charge induction efficiency is defined as the ratio of the charge induced at the collecting anode ofthe pixelated geometry to the charge deposited by a photon interaction. This is directly providedin the solution of the partial differential equations from COMSOL Multiphysics. It is affected bythe detector geometry through the small pixel effect [16] that renders the contribution of positivecharges (holes) to the induced signal negligible. It is alsoaffected by material properties throughdefects and impurities acting as energy traps for drifting charges and prevent their collection, thusreducing charge induction. The induced signal is further reduced through charge diffusion andrepulsion to neighbouring pixels [18]. These effects (excluding repulsion) were implemented inthe FEM models of the detector as described in [12] and the CIE is given by the solution of theadjoint continuity equation (for electrons) [17]:

∂n∗

∂ t= µe ·∇φ ·∇φk +Dn ·∇2n∗−

n∗

τe+ µe ·∇φ∇n∗ (2.1)

wheren∗ is the adjoint electron concentration,φ is the bias potential applied to the electrodes,φk

is the weighting potential [19, 20], µe is the electron mobility andτe is the electron lifetime. ThetermDn is the diffusion coefficient for electrons in CdZnTe.

An additional set of FEM computations is acquired with varying mobility-lifetime product(µτ) to calculate CIE for material properties that are not constant across different pixels in realdetectors. The calculations are obtained for 1mm thick detector and 100µm pixel size. Theelectron mobility (µe) varied from 0.01 to 0.4 m2/Vs and the lifetime (τe) from 10−7 to 3·10−6 s.The bias potential is constant and equal to -200 V. Material properties used in the models are shownin table 1.

2.2.2 Detection efficiency

In the current work the detection efficiency in the irradiated primary pixel, around the photopeakand in other pixels that surround the primary one, is calculated. Pixel detection efficiency (εpix) iscalculated as the ratio of the number of photons detected in the primary pixel of interaction to thenumber of photons incident. Photopeak detection efficiency(εphot) is calculated as the ratio of thephotons detected within an extremely narrow energy window around the incoming photon energy inthe primary pixel to the number of the incident photons. The energy window completely containedthe photpeak. Detection efficiency in other pixels (εoth) is calculated as the number of photonsdetected in all pixels except the primary to the number of photons incident to the primary pixel.

2.2.3 Energy resolution

Energy resolution (Eres) was calculated asEres = 2.355·σ , where the standard deviation (σ ) wasobtained by Gaussian fitting of the energy spectrum photopeak.

– 5 –

2012 JINST 7 P03004

3 Results

Results are separated into two sections. In the first section, COMSOL MultiphysicsTM

results arepresented. These are related with the operating characteristics and material properties of CdZnTeand provide information regarding charge induction at the electrodes, charge diffusion to adjacentlateral pixels and the effect of mobility and lifetime on charge induction.

In the second section, preliminary results from 4212 GATE computations are also presented.This is the first set of the computations where Monte Carlo is not combined with the FEM data.The second set which is under way provides the integrated output. Results from GATE are relatedto photon interactions (i.e. photoelectric effect, Compton and Rayleigh scattering), the generationof photoelectrons and fluorescence x-rays. The method provides information about interactionposition, energy deposition and photon escape. Preliminary energy resolution results (only for 1mm thickness and 100µm pixel size) obtained through integration of FEM and MC models arealso presented in this section.

3.1 FEM results

The change of CIE across the pixel thickness is presented in figure2 for 40 to 100µm pixel sizeand 0.2 to 3.0 mm thickness. The horizontal axis (thickness)corresponds to energy depositionlocations along the pixel thickness. Similar curves were obtained for 150 to 400µm pixel sizeand 1.5 and 2.5 mm thickness. As the ratio of pixel size to thickness decreased, the CIE near thecathode decreased as well. This is particularly observed inthe 40µm pixel size where there isa dramatic decrease in CIE of more than 70% from 0.2 to 3.0 mm thickness. The bias potentialapplied to the electrodes was -40, -100, -200, -400 and - 600 Volts for 0.2, 0.5, 1.0, 2.0 and 3.0 mmdetector thickness respectively.

A similar decrease in CIE is also seen across the lateral dimension of the pixel in figure3where the extent of lateral diffusion is observed. The CIE along a line that laterally traversed thepixel in the middle of the pixel thickness is presented for the same geometrical configuration as infigure 2. The zero on the x- axis of the figure denotes the middle of the pixel size in the lateraldimension. Again, as the ratio of pixel size to thickness decreased the CIE decreased and chargediffusion to adjacent pixels increased.

Basic material properties such as electron mobility and lifetime can affect CIE. The modellingframework is able to provide information regarding the effect of small -or large- variations ofthese properties on the CIE. In figure4, the variation of CIE with various electron mobility andlifetime values is depicted for a pixel size equal to 100µm and 1 mm thickness. Three electron-mobility values were used (µe = 0.01, 0.1, 0.4 m2/Vs) and five different lifetime values (τe =

0.1, 0.825, 1.55, 2.27, 3·10−6) s) for each mobility value. There is an electron mobility-lifetimeproduct value (µτe) above which the CIE seemed to saturate. From figure4b the saturation valuecan be deducted asµτe ≈ 8.25×10−8 m2/V.

3.2 MC results

In figure 5 the pixel detection efficiency (εpix) is depicted against the incoming photon energy atpixel size equal to 40, 100, 200 and 400µm and thickness equal to 0.2, 0.5, 1.0, 1.5, 2.0, 2.5 and3.0 mm. For low energy photon the absorption was almost 100% for all thickness and pixel sizes.

– 6 –

2012 JINST 7 P03004

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.2

0.4

0.6

0.8

1

1.2

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(a) 0.2 mm thickness

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

1.2

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(b) 0.5 mm thickness

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(c) 1.0 mm thickness

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(d) 2.0 mm thickness

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 30

0.2

0.4

0.6

0.8

1

1.2

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(e) 3.0 mm thickness

Figure 2: Charge induction efficiency across thickness of the pixel at the middle of the pixel for40, 50, 60, 70, 80, 90 and 100µm pixel size and thickness equal to a) 0.2, b) 0.5, c) 1.0, d) 2.0, e)3.0 mm.

However, there is an acute drop inεpix between 20-30 keV close to the fluorescence x-ray of Cd(∼

23.1 keV) and Te (∼ 27.5 keV), and is more evident in thinner detectors and smaller pixels. Infigure 5d, for 400µm size and above 1 mm thickness there is not an evident drop around 20 to30 keV. In addition, as the energy increases, photons becomemore penetrating and thus there arefewer photons detected producing a smooth exponential dropin εpix.

Detection efficiency in other pixels (εoth) is depicted in figure6 using the same geometricalproperties as in figure5. There is a steep increase inεoth just after 25 keV energy which indicatesphoton escape. This peak is observed even when the thicknessis increased and hence provides ahint that there is lateral photon escape from the primary pixel and detection in other pixels. This isalso supported by the decrease inεoth as the pixel size increases. In figure6d (400µm pixel size),εoth is less than 20% even for 140 keV.

In figure7, the photopeak detection efficiency (εphot) is shown. As in figure5, there is a steepdrop in εphot around 20 to 30 keV energy attributed to x-ray fluorescence escape. However,εphot

accounts for the energy of the incoming photons, and as shownin figure 7 only a portion of thedetected photons actually deposit all their energy in the primary pixel. In figure8, εphot is depictedagainst pixel size for specific energies (i.e. 25, 40, 80 and 140 keV). Larger pixels and thickerdetectors improveεphot.

Additional information regarding the energy spectrum and energy resolution is provided infigure9. Figure9adepicts the energy spectrum obtained after irradiation of adetector with 100µmpixel size and 1 mm thickness with a mono-energetic beam of 50keV photons with width equal to

– 7 –

2012 JINST 7 P03004

−150 −120 −90 −60 −30 0 30 60 90 120 150−0.2

0

0.2

0.4

0.6

0.8

1

Lateral position (µm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(a) 0.2 mm thickness

−150 −120 −90 −60 −30 0 30 60 90 120 150−0.2

0

0.2

0.4

0.6

0.8

1

Lateral position (µm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(b) 0.5 mm thickness

−150 −120 −90 −60 −30 0 30 60 90 120 150−0.2

0

0.2

0.4

0.6

0.8

1

Lateral position (µm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(c) 1.0 mm thickness

−150 −120 −90 −60 −30 0 30 60 90 120 150−0.2

0

0.2

0.4

0.6

0.8

1

Lateral position (µm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(d) 2.0 mm thickness

−150 −120 −90 −60 −30 0 30 60 90 120 150−0.2

0

0.2

0.4

0.6

0.8

1

Lateral position (µm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

40 µm

50 µm

60 µm

70 µm

80 µm

90 µm

100 µm

(e) 3.0 mm thickness

Figure 3: Charge induction efficiency across lateral position of thepixel at the middle of thedetector thickness for 40, 50, 60, 70, 80, 90 and 100µm pixel size and various thickness.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

τe=1.00e−07 s

τe=8.25e−07 s

τe=1.55e−06 s

τe=2.27e−06 s

τe=3.00e−06 s

(a)µe = 0.01 m2/(Vs)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

τe=1.00e−07 s

τe=8.25e−07 s

τe=1.55e−06 s

τe=2.27e−06 s

τe=3.00e−06 s

(b) µe = 0.1 m2/(Vs)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Thickness (mm)

Ch

arg

e In

du

ctio

n E

ffic

ien

cy

τe=1.00e−07 s

τe=8.25e−07 s

τe=1.55e−06 s

τe=2.27e−06 s

τe=3.00e−06 s

(c) µe = 0.4 m2/(Vs)

Figure 4: Charge induction efficiency across the detector thicknessat the middle of the pixelfor different electron mobility (µe) and lifetime (τe) values, a)µe = 0.01 m2/(Vs), b) µe =

0.1 m2/(Vs), c) µe = 0.4 m2/(Vs). The bias potential is equal to -200 V.

half the pixel size (i.e. 50µm). The photopeak and other spectrum features (such as fluorescencepeaks) are clearly visible and the energy resolution is 2.78%. In figure9b the beam width is equalto the pixel size. The photopeak is still visible , however the energy resolution is 11.30%. Theenergy resolution obtained using the wider beam is shown in figure9c against the photon beamenergy. Energy resolution data are fitted(R2 = 0.846) using the power functiony = 23.09·x−0.16,wherex is the photon energy andy the energy resolution. There is a gradual improvement in energyresolution as the photon beam energy increases, although there are local discrepancies. The best

– 8 –

2012 JINST 7 P03004

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 40 µm

Photon energy (keV)

Pix

el d

ete

ctio

n e

ffic

ien

cy

(εp

ix)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(a)

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 100 µm

Photon energy (keV)

Pix

el d

ete

ctio

n e

ffic

ien

cy

(εp

ix)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(b)

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 200 µm

Photon energy (keV)

Pix

el d

ete

ctio

n e

ffic

ien

cy

(εp

ix)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(c)

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 400 µm

Photon energy (keV)

Pix

el d

ete

ctio

n e

ffic

ien

cy

(εp

ix)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(d)

Figure 5: Detection efficiency in primary pixel against photon energy for a) 40, b) 100, c) 200 andd) 400µm and thickness between 0.2 and 3 mm.

energy resolution (9.69%) is computed at 140keV and the worst (16.24%) at 10 keV.

4 Discussion

In figures2 and3, CIE curves against depth and lateral position of interaction are shown. Increaseddetector thickness has a detrimental effect on the CIE whichis more evident in smaller pixel sizes(figures2c, 2d, 2eand3c, 3d, 3e). As the pixel thickness increased, the distance that electrons hadto cover until reaching the collecting electrode was increased. Hence, they remained in the detectorfor a longer time which resulted in an increase of the diffusion extent and the probability to becaptured in deep energy levels within the detector. In pixelsizes less than 70µm and thicknessabove 1 mm, charge diffusion extended to the whole pixel. Thediffusion extent was large evenfor 90 and 100µm pixel size when the thickness was greater than 2 mm. Imagingapplicationsusually employ isotropic emission of photons that can interact at any point within the detector.Thus, a pixel severely affected by diffusion would induce, at any location, much less signal than

– 9 –

2012 JINST 7 P03004

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 40 µm

Photon energy (keV)

De

t. e

ff.

in o

the

r p

ixe

ls (

εo

th )

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(a)

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 100 µm

Photon energy (keV)

De

t. e

ff.

in o

the

r p

ixe

ls (

εo

th )

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(b)

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 200 µm

Photon energy (keV)

De

t. e

ff.

in o

the

r p

ixe

ls (

εo

th )

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(c)

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 400 µm

Photon energy (keV)

De

t. e

ff.

in o

the

r p

ixe

ls (

εo

th )

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(d)

Figure 6: Detection efficiency in other pixels against photon energyfor a) 40, b) 100, c) 200 andd) 400µm and thickness between 0.2 and 3 mm.

what is expected from the actual deposited energy, and therefore has limited usability. In smallerthickness, i.e. 0.2 and 0.5 mm (figures2a, 2b, 3a, 3b), the CIE of smaller pixel sizes (40-80µm) wasnot greatly affected by diffusion or trapping. However, thelimited thickness may only be usefulin applications that require photon energies less than 30 keV to avoid the fluorescence effectsdiscussed in one of the following paragraphs. Varying the geometrical properties of the detector,allows to determine different geometrical constraints suitable for the needs of the application inquestion, avoiding costly experimental work.

Charge induction efficiency depends on material properties, such as carrier mobilityµ andlifetime τ. It was possible through COMSOL Multiphysics to investigate the effect of those twoproperties on CIE by varying their values as shown in figure4. Higher carrier lifetime values de-creased the probability of trapping within the detector andhence the CIE curve is more levelledfrom anode to cathode. Higher mobility values increased thecarriers velocity within the detectorand hence were collected faster, reducing the probability of trapping and the extent of diffusion.The capability to vary such material properties can be amplyexploited to match real detector char-

– 10 –

2012 JINST 7 P03004

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 40 µm

Photon energy (keV)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(a)

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 100 µm

Photon energy (keV)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(b)

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 200 µm

Photon energy (keV)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(c)

0 15 30 45 60 75 90 105 120 135 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Pixel size = 400 µm

Photon energy (keV)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(d)

Figure 7: Photopeak detection efficiency in primary pixel with photon energy for a) 40, b) 100, c)200 and d) 400µm and thickness between 0.2 and 3 mm.

acteristics and performance as obtained, for example, fromexperimental work in the lab or inimaging systems.

Imaging applications require good detection efficiency to form an acceptable image whileminimising the delivered dose. In figures5 to 7 the pixel detection efficiency, detection in otherpixels and the photopeak detection efficiency were presented for selected, representative pixelsizes against the incoming photon energy. These were results obtained from GATE MC com-putations only.

The pixel detection efficiency (figure5) increases with thickness and decreases with energy,as expected. However, larger thicknesses (greater than 1 mm) retained good detection efficiency(> 90%) for up to 90 keV photon energy. Increasing the pixel sizeresulted in a slight increase inεpix

and is evident by theεpix increase at 25 keV for thick pixels (> 1 mm). The steep drop observedat 25 keV, specifically in 0.2 mm thickness is due to x-ray fluorescence photons from Cadmium(∼ 23.17 keV) that escape from the pixel volume. Increasing thepixel size results in an increasein the available pixel volume for the photon or the photoelectron to move within. Intuitively, thisdecreased the probability that the photon or the electron crosses the boundaries of the pixel and

– 11 –

2012 JINST 7 P03004

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Energy = 25.0 keV

Pixel size (µm)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(a)

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Energy = 40.0 keV

Pixel size (µm)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(b)

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Energy = 80.0 keV

Pixel size (µm)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(c)

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Energy = 140.0 keV

Pixel size (µm)

Ph

oto

pe

ak

de

tec

tio

n e

ffic

ien

cy

(εp

ho

t)

0.2 mm

0.5 mm

1.0 mm

1.5 mm

2.0 mm

2.5 mm

3.0 mm

(d)

Figure 8: Detection efficiency with pixel size for a) 25, b) 40, c) 80 and d) 140 keV photon energyand thickness between 0.2 and 3 mm. Detection efficiency obtained within a narrow energy windowaround the incoming photon energy.

deposits its energy to adjacent pixels. Hence, the probability the photon to escape decreases and aminimal improvement inεpix is observed.

Photons that escape the primary pixel of interaction may be detected by other pixels surround-ing the primary pixel. In figure6, εoth is depicted and shows a steep increase inεoth at 30 keV,attributed to x-ray fluorescence and Compton-scattered photon escape from the primary pixel ofinteraction. Increasing the pixel size results in a decrease in εoth as photons have more volumein the primary pixel to be reabsorbed. As the photon energy increases,εoth gradually dropped,although with different trends depending on detector thickness. In thinner detectors (0.2, 0.5 mm)there is an almost immediate drop whereas in thicker there isan initial increase up to 90 keVfollowed by a gradual drop.

Detecting photons in other pixels whereasεpix remained high indicates that there were photonsthat interacted within the primary pixel but escaped and deposited part of their energy to anotherpixel. This is confirmed by obtainingεphot against photon energy as shown in figure7 and against

– 12 –

2012 JINST 7 P03004

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pixel size 100 µm

Energy (keV)

No

rma

lise

d c

ou

nts

(a) Beam width equal to half pixel size

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pixel size 100 µm

Energy (keV)

No

rma

lise

d c

ou

nts

(b) Beam width equal to pixel size

0 25 50 75 100 125 1509

10

11

12

13

14

15

16

17

Energy (keV)

En

erg

y r

eso

lutio

n (

%)

y = 23.09⋅ x−0.16

R2 = 0.846

(c) Energy resolution obtained using beam width equalto pixel size

Figure 9: Energy spectrum for 50 keV photons for two different irradiations conditions. a) thebeam width is half the pixel size and b) the beam width is equalto the the pixel size. The geometryof the dector is the same (i.e 100x100x1000µm). c) Energy resolution against photon energyobtained after irradiation of the detector with the wide beam. Fitting is depicted with the reddashed line.

pixel size for selected photon energies depicted in figure8. There is a steep drop inεphot after25 keV meaning that photons do not deposit all their energy inthe primary pixel. Fluorescencephotons and Compton scattered photons escaping the primarypixel contribute to the observeddrop. Photopeak detection efficiency increases with pixel size (figure8) and thickness, in orderto achieve values ofεphot greater than 50% up to 100 keV at least 2 mm thickness and morethan 200µm pixel size is required. This choice however is subject to the imaging applicationscope. For example, an x-ray system employing up to 30 keV photons (i.e mammography) wouldobtain reasonableεphot andεpix even with 0.5 mm detector thickness . The thinner detector (0.2mm) would be difficult to employ in imaging systems using energies above 20 keV as it had theworst performance in terms of detection efficiency, largelyunderachieving in comparison to otherdetector thicknesses.

– 13 –

2012 JINST 7 P03004

The photopeak detection efficinecy (εphot) results would be more accurate after the integrationof the FEM results in GATE. Realistic spectra would be obtained and thus for the same energywindow the number of photons detected at the photopeak will be different. Preliminary resultsfrom current computational experiments that integrate thetwo main components of the modellingframework are presented in figure9. It is evident from the comparison of figures9aand9bthat theirradiating photon beam geometry can have a significant impact on the form of the photopeak andthe spectrum in general. Currently, results for 100µm pixel size and 1 mm thickness have beenobtained and processed for two beam widths that are directedat the central pixel of the detector.The spectrum in figure9b is affected by carrier diffusion which is more intense at theperipheryof the pixel. Since the beam width is equal to the pixel size and the photon flux is homogeneous,the CIE at the centre and the periphery are equally affectingthe generated signal. In real imagingsystems the photon interactions occur in arbitrary locations within the pixel and an “ideal” spectrumsuch that of figure9awould be difficult to obtain. Hence, energy resolution(figure 9c) is computedusing a beam width equal to the pixel size. Discrepancies observed in the figure attributed mainlyto the manual Gaussian fitting where the coefficient of determination varied from 0.872 to 0.973between photopeaks. Energy resolution depends on the Fano factor and the number of chargecarriers generated by the energy deposition [21]. Signal induction in semiconductors is subjectto the amount of charge deposited which depends on the energyof the incoming photon. Higherenergy photons deposit more charge and hence the statisticsof charge induction are better, leadingto improved energy resolution. The power fit in figure9c is included as an indicator of the energyresolution improving trend with photon energy and for future reference to compare results frommore computational experiments and experimental work. Once the full set of computations isfinished, it will be possible to assess the relation of energyresolution with pixel-to-thickness ratio.

5 Conclusion

A comprehensive, integrated modelling framework for semiconductor detectors and imaging sys-tems was presented. COMSOL Multiphysics and GATE software applications were integratedthrough the implementation of a new module developed by our team. Results from a large numberof computational experiments with CdZnTe detector models were obtained to demonstrate the ca-pabilities and flexibility of this framework. The applications can be used either separately to obtainisolated information regarding the CIE, the interactions and the energy deposition of the detectoror integrated to obtain realistic results about detection efficiency and energy resolution.

This work provided insight as to what the geometric specifications could be for specific x-rayimaging applications, although a definite answer regardinga universally optimal CdZnTe detectorgeometry could not be drawn. Very small pixel sizes are proneto diffusion effects and may requireelectronic corrections to be used, if they are to be used at all. A more feasible implementation forphoton-counting systems for example, for energies up to 100keV would be a 100µm pixel size and1 mm thickness. In addition, for energy-weighting techniques, the photopeak detection efficiencyand the energy spectrum obtained from each detector configuration has to be considered. However,more accurate results are expected from our current, on-going simulations that fully exploit themodelling framework using the GATE-FEM integration feature.

– 14 –

2012 JINST 7 P03004

Currently, the framework is also able to provide features not used in this work such as pixelnoise, adjusting shaping time, different CIE for each pixeland cathode signal. Future implemen-tations will include polarization effects to study and investigate optimal photon fluxes and thecapability to import CIE or mobility-lifetime product mapsobtained experimentally to achieve a100 % match between the model and the real detector. Future computational experiments cur-rently planned include the performance assessment of complete imaging systems for both x-rayand gamma-ray imaging based on detector geometrical configurations decided by the current work.Overall, the modelling framework developed by our team is currently the most advanced, modernand accurate method that combines state-of-the-art software to computationally investigate the per-formance of any semiconductor detector.

References

[1] M.J. Yaffe and J.A. Rowlands,X-ray detectors for digital radiography, Phys. Med. Biol.42 (1997) 1.

[2] E.D. Pisano and M.J. Yaffe,Digital mammography, Radiology234(2005) 353.

[3] W. Kalender and Y. Kyriakou,Flat-detector computed tomography (FD-CT),Eur. Radiol.17 (2007) 2767.

[4] J.M. Park, E. Franken, M. Garg, L.L. Fajardo and L.T. Niklason,Breast tomosynthesis: presentconsiderations and future applications, Radiographics27 (2007) S231.

[5] S. Yin et al.,Direct conversion CdZnTe and CdTe detectors for digital mammography,IEEE Trans. Nucl. Sci.49 (2002) 176.

[6] C. Herrmann, K-J. Engel and J. Wiegert,Performance simulation of an x-ray detector for spectral CTwith combined Si and Cd[Zn]Te detection layers, Phys. Med. Biol.55 (2010) 7697.

[7] M. Mayer et al.,Performance and simulation of CdZnTe strip detectors as sub-millimeter resolutionimaging gamma radiation spectrometers, IEEE Trans. Nucl. Sci.44 (1997) 922.

[8] O. Tousignant et al.,Transport properties and performance of CdZnTe strip detectors,IEEE Trans. Nucl. Sci.45 (1998) 413.

[9] F. Mathy, J. Bonnefoy, A. Gliere, C. Mestais and L. Verger, Simulation and measurement of pulseheight and rise-time for electron signals in CZT detectors:influence of material and electronicsparameters,Nucl. Instrum. Meth.A 458 (2003) 484.

[10] P. Guerra, A. Santos and D.G. Darambara,Development of a simplified simulation model forperformance characterization of a pixelated CdZnTe multimodality imagingsystem,Phys. Med. Biol.53 (2008) 1099.

[11] M. Benoit and L.A. Hamel,Simulation of charge collection processes in semiconductor CdZnTegamma-ray detectors,Nucl. Instrum. Meth.A 606 (2009) 508.

[12] M.E. Myronakis and D.G. Darambara,Monte Carlo investigation of charge-transport effects onenergy resolution and detection efficiency of pixelated CZTdetectors for SPECT/PET applications,Med. Phys.38 (2011) 455.

[13] S. Agostinelli et al.,Geant4-a simulation toolkit, Nucl. Instrum. Meth.A 506 (2003) 250.

[14] S. Jan et al.,GATE: a simulation toolkit for PET and SPECT, Phys. Med. Biol.49 (2004) 4543.

[15] MATLAB version 7.9.0.The Language Of Technical ComputingNatick, The MathWorks Inc.,Massachusetts (2009).

– 15 –

2012 JINST 7 P03004

[16] H.H. Barrett, J.D. Eskin and H.B. Barber,Charge Transport in Arrays of Semiconductor Gamma-RayDetectors, Phys. Rev. Lett.75 (1995) 156.

[17] T. Prettyman,Method for mapping charge pulses in semiconductor radiation detectors,Nucl. Instrum. Meth.A 422 (1999) 232.

[18] A. Bolotnikov, G. Camarda, G. Wright and R. JamesFactors limiting the performance of CdZnTedetectors, IEEE Trans. Nucl. Sci.52 (2005) 589.

[19] W. Shockley,Currents to Conductors Induced by a Moving Point Charge, J. Appl. Phys.9 (1938) 635.

[20] S. Ramo,Currents induced by electron motion, Proc. I.R.E.(1939) 584.

[21] G.F. Knoll,Radiation detection and measurements, John Wiley and Sons, Inc., New York (2000).

– 16 –