Signal modeling of high-purity Ge detectors with a small read-out electrode and application to neutrinoless double beta decay search in Ge-76

4 MB·English

by  M. Agostini

PreprinttypesetinJINSTstyle-HYPERVERSION
Signal modeling of high–purity Ge detectors with a
small read–out electrode and application to
neutrinoless double beta decay search in 76Ge
M. Agostini abcd∗, C. A. Ur b, D. Budjáš c, E. Bellotti e, R. Brugnera bd,
1
C.M.Cattadorie, A.diVacri f, A.Garfagninibd, L.Pandola f, andS.Schönertac
1
0 aPhysikdepartmentE15,TechnischenUniversitätMünchen,München,Germany
2
bINFNPadova,Padova,Italy

n cMax-Planck-InstitutfürKernphysik,Heidelberg,Germany
a
J d DipartimentodiFisicadell’UniversitàdiPadova,Padova,Italy
7 eINFNMilanoBicocca,Milano,Italy
1 f INFNLaboratoriNazionalidelGranSasso,Assergi,Italy

]
t E-mail: [email protected]
e
d
-
s
n ABSTRACT: The GERDA experiment searches for the neutrinoless double beta decay of 76Ge us-
i ing high–purity germanium detectors enriched in 76Ge. The analysis of the signal time structure
.
s
c provides a powerful tool to identify neutrinoless double beta decay events and to discriminate
i
s themfromgamma–rayinducedbackgrounds. Enhancedpulseshapediscriminationcapabilitiesof
y
Broad Energy Germanium detectors with a small read–out electrode have been recently reported.
h
p This paper describes the full simulation of the response of such a detector, including the Monte
[
Carlo modeling of radiation interaction and subsequent signal shape calculation. A pulse shape

2
discrimination method based on the ratio between the maximum current signal amplitude and the
v
0 eventenergyappliedtothesimulateddatashowsquantitativeagreementwiththeexperimentaldata
0
acquiredwithcalibrationsources. Thesimulationhasbeenusedtostudythesurvivalprobabilities
3
4 of the decays which occur inside the detector volume and are difficult to assess experimentally.
2. Suchinternaldecayeventsareproducedbythecosmogenicradio–isotopes68Geand60Coandthe
1 neutrinoless double beta decay of 76Ge. Fixing the experimental acceptance of the double escape
0
1 peak of the 2.614 MeV photon to 90%, the estimated survival probabilities at Q =2.039 MeV
ββ
v: are(86±3)%for76Geneutrinolessdoublebetadecays, (4.5±0.3)%forthe68Gedaughter68Ga,
Xi and(0.9+−00..42)%for60Codecays.
r
a
KEYWORDS: Detectormodelingandsimulations,Particleidentificationmethods,Gamma
detectors.
∗Correspondingauthor.
Contents
1. Introduction 1
2. Overviewofthesimulation 2
2.1 Fieldcalculation,chargecarriertransportandsignalgeneration 3
2.2 Read–outelectronicsresponseandnoise 5
3. ModelingofBEGedetectors 5
3.1 TheelectricfieldinsideBEGedetectors 6
3.2 Signaldevelopmentindependenceoftheinteractionposition 7
3.3 Discriminationbetweensingle–siteandmultiple–siteevents 11
4. Validationofthesimulation 12
4.1 Theexperimentalsetup 13
4.2 Pulseshapecomparisonwithlowenergygamma–raybeams 13
4.3 RisetimeandA/E distributionsstudies 15
5. Backgroundrejectionandsignalidentificationstudies 20
5.1 Comparisonofpulseshapediscriminationofsimulatedandexperimentaldata 20
5.2 SurvivalfractionsofdecaysinternaltheGecrystal 22
6. Conclusionsandoutlook 24
1. Introduction
Thesearchforneutrinolessdoublebetadecay(0νββ)istheonlypracticalmethodtoprobewhether
the neutrino is a Majorana particle, i.e. its own anti–particle. 0νββ decay violates lepton num-
berconservationandwouldestablishnewphysicsbeyondtheStandardModelofparticlephysics.
ProvidedthattheexchangeoflightMajorananeutrinosaregeneratingtheleadingtermin0νββ de-
cay,effectiveneutrinomassesdowntoafewtensofmeVwillbeprobedinthenextgenerationof
0νββ decayexperiments.
One promising isotope is 76Ge. High–Purity Germanium (HPGe) detectors can be produced
fromgermaniummaterialenrichedin76Ge,whichservessimultaneouslyasasourceandasdetector
medium. The advantages of the intrinsic radio–purity and excellent spectroscopic performance
(2−3 keV full width at half maximum resolution at Q =2039 keV) of HPGe detectors have
ββ
been recognized early [1], and until today, the most stringent experimental limits on 0νββ decay
havebeenachievedwithgermaniumcrystalsenrichedin76Ge[2]. Alsoonegroupreportedaclaim
ofevidence[3].
–1–
The research work presented here was carried out in the framework of the GERmanium
Detector Array (GERDA) experiment [4] which recently started its commissioning phase at the
Laboratori Nazionali del Gran Sasso (LNGS) of the INFN in Italy. An array of bare HPGe
detectors enriched to 86% in 76Ge will be immersed in a cryogenic liquid which serves simul-
taneously as a coolant for the germanium detectors and as a high–purity shield against exter-
nal radiation. The experiment pursues a staged implementation: the mentioned claim of evi-
dence will be scrutinized in its first phase with about 15 kg·years exposure and a background
index at Q of <10−2 counts/(keV·kg·year). The second phase of the experiment will ex-
ββ
plore half–lives up to 2·1026 years, with 100 kg·years of exposure and background index of
<10−3 counts/(keV·kg·year). Contingent on the results of the first two phases, a third phase
is conceived to probe half–lives > 1027 years. The corresponding effective neutrino mass of
(cid:38)10 meV is predicted based on the results of neutrino oscillation experiments assuming an in-
vertedmasshierarchy[5]. Toexplorethisparameterregime,anexposureofseveral1000kg·years
and backgrounds <10−4 counts/(keV·kg·year) are required. To reach such background levels,
which are about three orders of magnitude below the best current values, novel techniques are
requiredwhichexploitthedecaycharacteristicsandtopologyofbackgroundevents.
Inrecentworks,wecouldshowthatp–typeHPGedetectorsproducedbyCanberraSemicon-
ductor [6] with a small read–out electrode, referred to as thick–window Broad Energy Germa-
nium (BEGe) detectors, exhibit pulse shape discrimination performance [7, 8] superior to coaxial
HPGe detectors. This allows to efficiently distinguish between single–site events (0νββ–like)
andmulti–siteevents(gamma–raybackgrounds). InvestigationsofabareBEGedetectorinliquid
argon showed unaltered performance as compared to operation in a vacuum cryostat [9]. An ad-
ditional benefit of the small size of the read–out electrode is its low capacitance which results in
a lower noise and therefore a superior resolution and a lower threshold compared to other types
of large–volume HPGe detectors. Following a full production chain test [10] of BEGe detectors
withmodifiedisotopiccomposition(depletedin76Ge),the GERDA collaborationdecidedtoadopt
thick–windowBEGedetectorsastheirbaselinedesignforthesecondphaseofGERDA. R&Dwith
BEGedetectorsisalsocarriedoutbytheMajoranacollaboration[11,12].
Thescopeofthispaperistheab–initiomodelingofthesignalpulseshapesofBEGedetectors
createdbyparticleinteractionsandthecomparisonwithexperimentaldata. Theenergydepositions
and event topology were generated with the GEANT4 simulation package [13, 14] and the conse-
quentsignalpulseshapegeneratedbydriftingthechargecarrierinthecalculatedelectricfield. The
characteristicpulseshapesofthesimulatedpulsesarethenanalyzed. Firstwevalidatethesimula-
tion of pulse shapes and discrimination methods with experimental data using calibration sources
locatedexternallytothecrystal. Then,wecalculatetheexpectedpulseshapediscrimination(PSD)
cut efficiencies for 60Co, 68Ga and 0νββ decays internal to the detector. We also study the pulse
shapesofenergydepositionsclosetothesmallread–outelectrodeandinitsvicinity.
2. Overviewofthesimulation
The simulation presented here replicates the physical processes involved in the generation of the
Ge detector signals with application to the particular case of Broad Energy Germanium (BEGe)
detectors. The simulation can be divided into three logical blocks: 1) interaction of ionizing par-
–2–
experimentalgeometry
BlockI:
interactions
MonteCarlo
electricpotentialandfield
BlockII:
electronandholetrajectories
PulseShape
pulseshapes
libraryofpulses pulse
FE&DAQresponse
BlockIII:
Read–OutElectronics
experimentalnoise libraryofbaselines pulse+noise
Figure1.Dataflowofthesimulationorganizedintothreelogicalblocks.Thedataflowincludesafirstphase
(thinarrows)inwhichalibraryofpulsesandbaselinesiscreatedandasecondphase(thickarrows)inwhich
thelibrariesareusedtogeneratethedetectorsignals. InblockIII,FE&DAQreferstothepreamplifierand
thesamplingdevice.
ticles with the Ge crystal; 2) collection of the charge carriers produced at each interaction point
atthedetectorelectrodesandtimeevolutionoftheinducedsignals;3)signalshapingbyread–out
electronicdevices.
The first block consists of a Monte Carlo simulation, including all the physical processes
involved in the passage of gamma–rays or charged particles through matter. It provides the inter-
action points and the corresponding energy losses within the crystal. This part of the simulation
is performed within the MAGE framework [15], which is based on the GEANT4 simulation pack-
age[13,14].
The second block describes the dynamics of the charge carriers generated inside the detector
and provides the signals induced on the electrodes by the charge movement. It is calculated by
usingamodifiedversionoftheMultiGeometrySimulation(MGS)software[16].
Thelastblockofthesimulationtakesintoaccounttheeffectsoftheread–outelectronicdevices
andoftheelectronicnoise. Itgeneratessignalsthatcanbedirectlycomparedtothemeasuredones.
Figure1illustratestheimplementationdiagramoftheselogicalblocksinthesoftware. Inthe
following two subsections the second and the third block of the simulation are explained in more
detail.
2.1 Fieldcalculation,chargecarriertransportandsignalgeneration
ThepulsesimulationperformedwiththeMGS-basedsoftwareincludesthefollowingsteps: 1)com-
putationoftheelectricfieldinsidethedetectoratdefinedpotentialsoftheelectrodes; 2)transport
–3–
of charge carries toward the electrodes; 3) time evolution of the signal induced at the read–out
electrodebythemovingcharges.
Sinceasemiconductordetectorcanbeconsideredasanelectrostaticsystem,theelectricfield
canbecomputedbysolvingtheMaxwellequationsor, equivalently, bysolvingthePoissonequa-
tion with a set of boundary conditions for the potential. In p–type Ge detectors, the p–n junction
formsattheborderofthe∼1mmthickdonor–dopedsurfacelayer(n+layer). Atoperationalbias
voltagesthep–typevolumeisfullydepletedoffreecharges(activevolume),whilemostofthen+
layerretainselectronsintheconductionband(anode,formingadeadlayer). Inoursimulationwe
approximatethissituationbydefiningtheactivevolumewithanegativespacechargedistribution
ρ proportionaltothenetp–typeimpuritydistribution,adjoinedtothedeadlayeronwhichthebias
voltageisapplied1. Theboundaryconditionsarethenprovidedbythevalueofthepotentialonthe
conductivesurfaces:V andV . Thereforethesystemofequationstosolveis:
cathode anode

∇2φ((cid:126)r) = −ρ((cid:126)r)/(ε0εr)
φ| = V (2.1)
Scathode cathode

φ| = V
Sanode anode
where(cid:126)r is the position, φ((cid:126)r) is the electric potential, ρ((cid:126)r) is the charge density distribution, ε is
0
thevacuumelectricalpermittivity, ε ∼16istherelativepermittivityofGe, φ| isthepotentialat
r S
thesurfaceSsurroundingtheconsideredelectrode.
The movement of charge carriers within the active volume is computed by using two phe-
nomenological models [17, 18] which provide the drift velocity as a function of the electric field
magnitude and of the electric field direction relative to the crystallographic axes. We assume that
eachtransferofenergytotheGecrystallatticeresultsinthegenerationofacloudoffreeelectrons
and holes, which further drift as two independent clusters. These are approximated in our simu-
lation by two point-like charges with opposite sign. The trajectories are calculated with a fourth
orderRunge-Kuttamethodwith1nstimestep.
Using the simulated trajectories of charge carriers and the weighting potential2 distribution
inside the detector, the charge signal Q((cid:126)r(t)) induced on the electrodes is computed by using the
Shockley–Ramotheorem[19]:
Q((cid:126)r(t))=−q φ ((cid:126)r(t)) (2.2)
tot w
where(cid:126)r(t)isthepositionofthechargeclusteratthetimet,q thetotalchargeoftheclusterand
tot
φ ((cid:126)r(t))istheweightingpotential.
w
Thetotalsignalofasimulatedparticleeventcomposedofseveralinteractionsiscalculatedas
asumofpulsesfromeachhit,whosepositionandenergydepositionareprovidedbeforehandwith
the MAGE MonteCarlosimulation. Toreducethecomputationtime,thegenerationofthesignals
corresponding to the individual hits is performed by using a library of pre–calculated pulses. The
library is generated by dividing the detector active volume in 1 mm3 cubic cells, and simulating
1Sinceourfieldcalculationisperformedona0.5mmstepgrid,itcannotdescribethethinpartofthen+volumein
whichtheconduction–bandelectronsaredepleted(then–sideofthep–njunction,withpositivespacecharge).
2Theweightingpotentialisadimension–lessquantitydefinedastheelectricpotentialcalculatedwhentheconsidered
electrodeiskeptataunitpotential,allotherelectrodesaregroundedandallchargesinsidethedeviceareremoved.
–4–
a single normalized interaction at each corner. The library must be generated only once for each
simulateddetectorgeometryandbiasvoltagesetting. TheneachMonteCarlogeneratedinteraction
isassociatedtooneofthecubiccellsinthepulselibrary. Thesignaliscomputedastheweighted
averageoftheeightpulsesineachcornerofthecube,wheretheweightisgivenbytheinverseofthe
cubic euclidean distance between the interaction point and the considered corner. The amplitudes
oftheindividualinteractionpulsesarethennormalizedaccordingtotheenergydepositionsinthe
hitsandallthepulsesoftheeventareaddeduptoonecombinedsignal.
Theuseofapulselibrarydecreasessignificantlytheprocessingtimeofthesimulationinsuch
awaythatthepulsecomputationtimeiscomparablewiththatoftheMonteCarlosimulation.
2.2 Read–outelectronicsresponseandnoise
The read–out electronics includes the preamplifier and the digital sampling device. The response
functionwasdeterminedbyprovidingthepreamplifierinputwithanimpulsegeneratedbyahigh–
precisionpulser, andthenbydeconvolvingthedigitallysampledsignalwiththeinputsignal. The
simulated detector signals are consequently convolved with this response function. This is per-
formed for each pulse in the library mentioned in previous subsection, so that it needs to be done
onlyonceforeachsimulatedexperimentalsetup.
To reproduce the electronic noise present in the experimental data, samples are taken from a
library of experimentally recorded baselines. The amplitude of the noise is normalized according
totheexperimentalsignal–to–noiseratio,andthenoisesampleisthenaddedtothecalculatedfull
eventsignal. Theoutputofthesimulationisafileofsignalssimilartothoserecordedexperimen-
tallywithadigitaldataacquisitionsystem. Thereforeitispossibletoapplythesameanalysistools
tobothexperimentalandsimulateddata.
3. ModelingofBEGedetectors
Thedetectorusedforthepresentsimulationisamodifiedthick–windowversionofaBroadEnergy
Germanium (BEGe) detector [20], a standard p–type High–Purity Ge (HPGe) detector offered by
Canberra Semiconductor. These detectors have a cylindrical shape and a small B–implanted p+
electrode on one of the flat surfaces. The Li–diffused n+ electrode (between 0.4 mm and 0.8 mm
in the “thick window” modification) covers the rest of the outer surface and is separated from the
p+electrodebyacirculargroove. Thedetectorismountedina1.5mmthickcylindricalaluminum
housing. Thepositivehighvoltageisappliedtothen+electrodewhilethesignalisreadoutfrom
the p+ electrode where the holes are collected. For the purpose of the present work the specific
caseoftheBE3830/Smodelwasconsidered. TheBE3830/Smodelhasadiameterof71mmanda
thicknessof32mm. ThedetectorwasexperimentallycharacterizedinRefs.[21,22]. Aschematic
drawingofatypicalBEGedetectorisshowninFigure2.
InthissectionthesignalformationanddevelopmentintheBEGedetectorswillbediscussed
followingthestepsoutlinedinSection2.1. Moreover,theresultsofthesimulationwillbeusedto
show how it is possible to distinguish between single–site and multiple–site events by analyzing
thepulseshapes.
–5–
n+electrode
(highvoltagecontact)
p–typegermanium
p+electrode(signalread–outcontact)
Figure 2. Sketch of a BEGe detector. The signal read–out electrode and the groove are not to scale. The
plotshowsaverticalsectionofthedetectorpassingthroughthesymmetryaxis.
3.1 TheelectricfieldinsideBEGedetectors
The pulse shape discrimination properties of BEGe detectors have been attributed to the peculiar
internal electric field which is created by the small size of the read–out electrode. In Section 2.1
thespacechargedistributioninsidetheactivevolume(equivalenttothenetdistributionofacceptor
impurities)wasidentifiedalongwiththeelectrodepotentialsasasourceofelectricfieldinsidethe
detector. Tobetterunderstandthetwocontributions,thelinearsuperpositionprinciplecanbeused
toseparatethepotentialintothetwoindividualcomponents:
φ((cid:126)r)=φ ((cid:126)r)+φ ((cid:126)r) (3.1)
0 ρ
whereφ isthepotentialcalculatedconsideringonlytheelectrodepotentialsandnoimpuritycharge
0
(ρ((cid:126)r)=0 ∀(cid:126)r) and φ is the potential generated by the impurity charge distribution when ground-
ρ
ing all the electrodes. Therefore, we can solve the following two problems and then add up the
solutions:
 
∇2φ0((cid:126)r) = 0 ∇2φρ((cid:126)r) = −ρ((cid:126)r)/(ε0εr)
φ | = V φ | = 0 (3.2)
0 Scathode cathode ρ Scathode
 
φ | = V φ | = 0
0 Sanode anode ρ Sanode
Similarly, since the electric field is determined by the linear relation (cid:126)E = −∇φ, it can be also
separatedintotwocomponents:
(cid:126)E((cid:126)r)=(cid:126)E ((cid:126)r)+(cid:126)E ((cid:126)r) (3.3)
0 ρ
where(cid:126)E ((cid:126)r)=−∇φ ((cid:126)r)and(cid:126)E ((cid:126)r)=−∇φ ((cid:126)r).
0 0 ρ ρ
Figure3showstheelectricpotentialandtheelectricfieldstrengthofthetwocontributionsand
their sum for the BE3830/s detector operated in its nominal configuration, i.e. cathode grounded,
anodesetat3500Vandρ ∼ 1010 impurityatoms/cm3.
Figure 3.a and 3.b show the electric potential and field strength generated only by the elec-
trodes(φ ((cid:126)r)and||(cid:126)E ((cid:126)r)||). Thepotentialandthefieldshowasharpvariationintheregioncloseto
0 0
thesmall–sizep+electrode. Thefieldintherestofthevolumeissoweakthatthecollectiontime
wouldbelongerthanthecharacteristicrecombinationtimeandmostofthechargewouldbelost.
Figure3.cand 3.dshowtheelectricpotentialandfieldstrengthprovidedonlybytheimpurity
chargedistribution(φ ((cid:126)r)and||(cid:126)E ((cid:126)r)||). FortheconsideredBEGedetectorthechargeconcentra-
ρ ρ
tionisapproximatedbyauniformdistribution. Theresultingnegativeelectricpotentialreachesits
–6–
4.0e3 0.0e0 4.0e3
3.0e3 V] −5.0e2 V] 3.0e3 V]
[ [ [
2.0e3 ntial −1.0e3 ntial 2.0e3 ntial
e e e
1.0e3 pot −1.5e3 pot 1.0e3 pot
0.0e0 −2.0e3 0.0e0
(a) φ0((cid:126)r) (c) φρ((cid:126)r) (e) φ((cid:126)r)
1.2e6 3.0e5 1.6e6
m] 2.4e5 m] m]
V/ V/ 1.2e6 V/
8.0e5 h[ 1.8e5 h[ h[
gt gt 8.0e5 gt
n n n
e 1.2e5 e e
4.0e5 eldstr 6.0e4 eldstr 4.0e5 eldstr
fi fi fi
0.0e0 0.0e0 0.0e0
(b) ||(cid:126)E0((cid:126)r)|| (d) ||(cid:126)Eρ((cid:126)r)|| (f) ||(cid:126)E((cid:126)r)||
Figure 3. Simulated electric potential and electric field strength for different configurations of a BEGe
detector. In(a)and(b)theelectrodepotentialisconsidered,in(c)and(d)thechargedistribution,andin(e)
and (f) the sum of the two contributions. The plots show half of a vertical section of the detector passing
throughthesymmetryaxis. Thecathodeisdrawninredandtheanodeinblack.
peakvalueinthemiddleofthedetector. Thedirectionoftheelectricfieldintheregionclosetothe
small cathode is opposite to that of (cid:126)E , whereas in the rest of the detector the field helps to move
0
thechargesproducedclosetotheoutern+electrodetowardsthecentralsliceofthedetector.
Figure3.eand 3.fshowthetotalelectricpotentialandfieldstrength(φ((cid:126)r)and||(cid:126)E((cid:126)r)||). The
potentialclosetothesmallp+electrodeisdominatedbytheelectrodescontributionsmoothedout
bytheoppositecontributionoftheimpuritychargedistributionfield. Intherestofthevolumethe
dominant contribution is provided by E . The effect of the E field is to bring the holes in the
ρ ρ
centerofthedetectorwhilethefieldE subsequentlycollectsthemtotheread–outelectrode. This
0
peculiar way of charge transportation in BEGe detectors leads to the favorable signal shape as it
willbediscussedinthenextsections.
3.2 Signaldevelopmentindependenceoftheinteractionposition
As given by equation (2.2), the weighting potential determines the charge that a cluster of charge
carriersdriftinginsidethedetectorinducesontheread–outelectrode. Figure4showstheweighting
potential φ ((cid:126)r) and the strength of the weighting field ||(cid:126)E ((cid:126)r)||=||−∇φ ((cid:126)r)|| of the cathode in
w w w
a BEGe detector. The weighting potential (similarly to φ in Figure 3.a) has a sharp variation
0
in the region close to the small–size p+ electrode and it is very weak in the rest of the detector
volume (blue area in Figure 4). If an interaction occurs in the volume of weak φ , according to
w
equation (2.2), the signal induced by the holes drifting towards the small cathode remains small
until the charges arrive at about 1 cm away from the electrode and then rapidly grows until the
holesarecollected.
–7–
1 3
m]
0.8 2.5 1/c
ntial 2 gth[
00..46 htingpote 1.5 fieldstren
eig 1 ng
0.2 w hti
0.5 g
ei
w
1 0 0 1 0 0
[1/cm]
(a) φw((cid:126)r) (b) ||(cid:126)Ew((cid:126)r)||
Figure4. Weightingpotential(a)andweightingfieldstrength(b)ofthesmallread–outelectrode(cathode)
computed for half of a vertical section of the BEGe detector passing through the detector symmetry axis
(densitymaps)andforthesymmetryaxis(blacklinesinthesideplots). Thecathodeisdrawninredandthe
anodeinblack.
Ontheotherhand,inmostoftheactivevolumetheelectroncontributionwillbepresentonly
atthebeginningofthepulse,sinceelectronsarecollectedtotheoutern+electrodeandtheportion
of the detector volume that is close to the outer surface is much larger than the portion close to
thep+electrode. Thiseffectisfurtherenhancedduetothefactthatelectronsmovewithvelocities
roughlytwotimeshigherthantheholes. Moreover,thesignalinducedbytheelectrondriftingwill
berelativelysmall,asmostelectronclustersmoveintheregionofweakφ . Thus,thecontribution
w
oftheelectronstotheformationofthesignalsisexpectedtobenegligibleformostoftheinteraction
positions.
Figure5.ashowstheelectronandholetrajectoriesforthreeinteractionsinthe“bulk”detector
volume far from the p+ electrode. The holes follow the electric field (Figure 3.f) and are first
transported into the middle slice of the detector, then drift towards the center of the detector and
finally their trajectory bends towards the read–out electrode. It can be noticed that for all these
events, the last part of the hole collection happens along a common path which is independent of
the starting position. Accordingly, the last part of the induced signal is identical for the different
events. Sincethefirstpartofthesignal,inducedbytheholesandelectronsinweakφ regions,is
w
comparativelysmall,thesignalshapesareessentiallyindependentoftheinteractionposition. This
canbeclearlyseeninFigure5.b,whichshowsthechargeandcurrentpulsescorrespondingtothe
trajectories from Figure 5.a. The only visible difference is the time shift of the rising part, which
depends on the length of the charge carrier path to reach the strong φ region. It is important to
w
stress that this type of almost indistinguishable signals originates from interactions in most of the
detector volume, including corner regions. For further discussion we will refer to such signals as
“typeI”trajectories.
TheshapeofthecurrentsignalsinFigure5.bcanbeunderstoodbydifferentiatingthefunction
inequation(2.2). Thecurrentinducedatthecathodebyachargecarrieristhengivenby:
dQ((cid:126)r(t))
I((cid:126)r(t))= =q (cid:126)v((cid:126)r(t))·(cid:126)E ((cid:126)r(t)) (3.4)
tot w
dt
–8–
anode
cathode 1500
electrons
holes u.] 1200
interactionpoint [a. 900
e
arg 600
h
c 300
0
120
a.u.] 90
[
nt 60
e
curr 30
0
0 100 200 300 400 500 600 700 800
time[ns]
(a)Trajectories (b)Chargeandcurrentpulses
Figure 5. Simulated electron–hole trajectories (a) and corresponding charge and current pulses (b) for
three events occurring in different places in the bulk detector volume far from the p+ electrode (“type I”
trajectories). The small oscillation in the current signals after the peak originates from the experimentally
measuredFEandDAQresponsethepulsesareconvolvedwith.
FromthisequationwecanseethattheinducedcurrentI((cid:126)r(t))dependsonthevelocity(cid:126)v((cid:126)r(t))and
the weighting field (cid:126)E ((cid:126)r(t)) at the position(cid:126)r(t) of the charge cluster. The charge carrier velocity
w
can vary roughly between 5 cm/s and 10 cm/s [21], while E can increase by more than a factor
w
of 20 close to the read–out electrode (see Figure 4.b). It is evident that E has a dominant effect
w
onthecurrentsignal–thesignalsof“typeI”trajectoriesinFigure5.bfeatureasignificantcurrent
peakattheendoftheholecollection.
Two other types of less common trajectories can be identified when the interactions occur in
the close vicinity of the groove and the small p+ electrode. Examples of these kind of events are
displayedinFigure6. The“typeII”trajectoriesoriginateclosetothep+electrode(greenandblack
color). In these events the holes are directly and quickly collected at the cathode. Now also the
electronsdriftingintheoppositedirectionprovideasignificantcontributiontothesignal,sincethey
arenowmovinginaregionofstrongφ . Theclosertheinteractionoccurstothecathode,themore
w
important the signal induced by electrons becomes. The signal is fully dominated by the electron
contributionforinteractionswithin∼2mmfromthep+electrode(blackexampleinFigure6). The
inducedchargesignalrisesquicklyatthebeginningandthen,astheydriftawayfromthecathode
into the weaker φ regions, the signal growth slows down. The current peak appears at the very
w
beginning of the collection time. For events occurring few mm to ∼1 cm from the p+ electrode
(greenexampleinFigure6)neitherelectronsnorholestraversethefullthicknessofthestrongφ
w
region. Sothemainpartofthesignalisinducedinarelativelyshorttimeandtherisetimeisthus
faster. The current peak isamplified because contributions fromboth charge carrier types addup.
Thecurrentamplificationcanbefurtherenhancediftheinteractionhappensclosetotheinneredge
ofthegroove,becausehereE isstrongest(Figure4.b).
w
Startingpointsinazoneclosetotheanode,∼1.5cmfromthedetectorsymmetryaxis,resultin
–9–