Simulation of Heavily Irradiated Silicon Pixel Detectors

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SimulationofHeavilyIrradiatedSiliconPixelDetectors

M.Swartz,D.Kim

JohnsHopkinsUniversity,Baltimore,MD21218,USA

V.Chiochia,Y.Allkofer,C.Amsler,C.Regenfus,T.Speer

¨Z¨PhysikInstitutderUniversitaturich-Irchel,8057Z¨urich,Switzerland

A.Dorokhov,C.H¨ormann,K.Proko ev

arXiv:physics/0605215v1 [physics.ins-det] 24 May 2006

¨Z¨PhysikInstitutderUniversitaturich-Irchel,8057Z¨urich,Switzerlandand

PaulScherrerInstitut,5232VilligenPSI,Switzerland

D.Kotlinski,T.Rohe

PaulScherrerInstitut,5232VilligenPSI,Switzerland

D.Bortoletto,S.Son

PurdueUniversity,WestLafayette,IN47907,USA

S.Cucciarelli,M.Konecki

¨Basel,4056Basel,SwitzerlandInstitutf¨urPhysikderUniversitat

L.Cremaldi,D.A.Sanders

UniversityofMississippi,University,MS38677,USA

Weshowthatdoublypeakedelectric eldsarenecessarytodescribegrazing-anglechargecollectionmeasure-mentsofirradiatedsiliconpixelsensors.Amodelofirradiatedsiliconbasedupontwodefectlevelswithopposite

chargestatesandthetrappingofchargecarrierscanbetunedtoproduceagooddescriptionofthemeasuredchargecollectionpro lesinthe uencerangefrom0.5×1014neq/cm2to5.9×1014neq/cm2.Themodelcor-rectlypredictsthevariationinthepro lesasthetemperatureischangedfrom 10 Cto 25 C.Themeasuredchargecollectionpro lesareinconsistentwiththelinearly-varyingelectric eldspredictedbytheusualdescrip-tionbaseduponauniforme ectivedopingdensity.Thisobservationcallsintoquestionthepracticeofusinge ectivedopingdensitiestocharacterizeirradiatedsilicon.ThemodelisnowbeingusedtocalibratepixelhitreconstructionalgorithmsforCMS.

1.Introduction

Asiliconpixeldetectoriscurrentlybeingde-velopedfortheCMSexperimentattheCERNLargeHadronCollider(LHC).Thedetectorwillbeakeycomponentinthereconstructionofprimaryandsec-ondaryverticesintheparticularlyharshLHCenviron-mentcharacterizedbylargetrackmultiplicitiesandhighradiationbackgrounds.Theinnermostlayer,lo-catedatonly4cmfromthebeamline,isexpectedtobeexposedtoa1MeVneutronequivalent uenceof3×1014neq/cm2peryearatfullluminosity.

Theresponseofthesiliconsensorsduringthede-tectoroperationisofgreatconcern.Itiswellunder-stoodthattheintra-diodeelectric eldsinthesedetec-torsvarylinearlyindepthreachingamaximumvalueatthep-njunction.Thelinearbehaviorisaconse-quenceofauniformspacechargedensity,Ne ,causedbythermallyionizedimpuritiesinthebulkmaterial.Itiswellknownthatthedetectorcharacteristicsarea ectedbyradiationexposure,butitisgenerallyas-sumedthatthesamepictureisvalidafterirradiation.Infact,itiscommontocharacterizethee ectsofirra-diationintermsofavaryinge ectiveuniformchargedensity.In[2]wehaveprovedthatthispicturedoesnotprovideagooddescriptionofirradiatedsiliconpixelsensors.Inaddition,itwasshownthatitispossibletoadequatelydescribethechargecollectioncharacteristicsofaheavilyirradiatedsilicondetectorintermsofatuneddoublejunctionmodelwhichpro-

ducesadoublypeakedelectric eldpro leacrossthesensor.Themodelingissupportedbytheevidenceofdoublypeakedelectric eldsobtaineddirectlyfrombeamtestmeasurementsandpresentedinThedependenceofthemodeledtrapconcentrationsupon uencewaspresentedin[4]andthetemperaturede-pendenceofthemodelwaspresentedin[5].Wesum-marizetheseresultsinthisdocument.

Thispaperisorganizedasfollows:Section2de-scribestheexperimentaldetails,Section3describesthecarriertransportsimulationusedtointerpretthedata.ThetuningofthedoublejunctionmodelanditsresultingpredictionsarediscussedinSection4.ThetemperaturedependenceofthedataandmodelaresummarizedinSectionTheconclusionsaregiveninSection2.ExperimentalDetails

ThemeasurementswereperformedintheH2beamlineoftheCERNSPSin2003/04using150-225GeVpions.Thebeamtestapparatusisdescribedin[6].Asiliconbeamtelescopeconsistedoffourmoduleseachcontainingtwo300µmthicksingle-sidedsilicondetectorswithastrippitchof25µmandreadoutpitchof50µm.Thetwodetectorsineachmodulewereori-entedtomeasurehorizontalandverticalimpactco-ordinates.Apixelhybriddetectorwasmountedbe-tweenthesecondandthirdtelescopemodulesona

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SNICSymposium,Stanford,California-3-6April,2006

cooledrotatingstage.AtriggersignalwasgeneratedbyasiliconPINdiode.TheanalogsignalsfromalldetectorsweredigitizedinaVME-basedreadoutsys-tembytwoCAEN(V550)ADCsandonecustom-built ashADC.Theentireassemblywaslocatedinanopen-geometry3THelmholtzmagnetthatproducedamagnetic eldeitherparallelororthogonaltothebeam.ThetemperatureofthetestedsensorswascontrolledwithaPeltiercoolerthatwascapableofoperatingdownto-30 C.Thetelescopeinformationwasusedtoreconstructthetrajectoriesofindividualbeamparticlesandtoachieveaprecisedeterminationoftheparticlehitpositioninthepixeldetector.Theresultingintrinsicresolutionofthebeamtelescopewasabout1µm.

Theprototypepixelsensorsareso-called“n-in-n”devices:theyaredesignedtocollectchargefromn+structuresimplantedinton–bulksiliconusingp-sprayisolation.Alltestdeviceswere22×32arraysof125×125µm2pixelsthatwerefabricatedbyCiS.Thesubstrate,producedbyWacker,was285µmthick,n-doped,di usively-oxygenated oatzonesiliconoforientation 111 ,resistivity3.7k ·cmandoxygenconcentrationintheorderof1017cm 3.Individualsensorsweredicedfromfullyprocessedwafersafterthedepositionofunder-bumpmetalizationandin-diumbumps.AnumberofsensorswereirradiatedattheCERNPSwith24GeVprotons.Theirradia-tionwasperformedwithoutcoolingorbias.Thede-liveredproton uencesscaledto1MeVneutronsbythehardnessfactor0.62[8]were0.5×1014neq/cm2,2×1014neq/cm2and5.9×1014neq/cm2.Allsampleswereannealedforthreedaysat30 C.Inorderto avoidreverseannealing,thesensorswerestoredat-20Caf-terirradiationandkeptatroomtemperatureonlyfortransportandbumpbonding.AllsensorswerebumpbondedtoPSI30/AC30readoutchips[9]whichallowanalogreadoutofall704pixelcellswithoutzerosup-pression.ThePSI30settingswereadjustedtoprovidealinearresponsetoinputsignalsrangingfromzerotomorethan30,000electrons.

3.Sensorsimulation

Theinterpretationofthetestbeamdatareliesuponadetailedsensorsimulationthatincludesthemodel-ingofirradiatione ectsinsilicon.Thesimulation,pixelav[2,10,11],incorporatesthefollowingele-ments:anaccuratemodelofchargedepositionbyprimaryhadronictracks(inparticulartomodeldeltarays);arealistic3-Dintra-pixelelectric eldmap;anestablishedmodelofchargedriftphysicsincludingmobilities,HallE ect,and3-Ddi usion;asimula-tionofchargetrappingandthesignalinducedfromtrappedcharge;andasimulationofelectronicnoise,response,andthresholde ects.Theintra-pixelelec-tric eldmapwasgeneratedusingtcad9.0[12]to

simultaneouslysolvePoisson’sEquation,thecarriercontinuityequations,andvariouschargetransportmodels.A nalsimulationstepreformattedthesim-ulateddataintotestbeamformatsothatitcouldbeprocessedbythetestbeamanalysissoftware.

Thesimulationwascheckedbycomparingsimu-lateddatawithmeasureddatafromanunirradiatedsensor.Aplotofthechargemeasuredinasinglepixelasafunctionofthehorizontalandverticaltrackim-pactpositionfornormallyincidenttracksisshowninFig.1.Thesimulationisshownasthesolidhis-togramandthetestbeamdataareshownassolidpoints.Notethatthesensorsimulationdoesnot+in-cludethe“punch-through”structureonthenim-plantswhichisusedtoprovideahighresistancecon-nectiontogroundandtoprovidethepossibilityofon-waferIVmeasurements.Thereisreducedchargecol-lectionfromthisportionoftheimplantandthedatashowsreducedsignalinbothprojectionsatthebiasdot.Anothercheck,showninTableI,isthecompar-isonoftheaverageLorentzanglemeasuredatseveralbiasvoltages[6].Inbothcases,reasonableagreementisobservedbetweenmeasuredandsimulateddata.

Figure1:Collectedchargemeasuredinasinglepixelasafunctionofthehorizontal(left)andvertical(right)trackimpactpositionfortracksthatarenormallyincidentonanunirradiatedsensor.Thesimulationisshownasasolidhistogramandthetestbeamdataareshownassoliddots.

TableIMeasuredandsimulatedvaluesofaverage

LorentzangleθLversusbiasvoltageforanunirradiatedsensor.

BiasVoltageMeasuredθL[deg]SimulatedθL[deg]

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SNICSymposium,Stanford,California-3-6April,2006

oneproposedin

[13],isbasedontheShockley-Read-Hall(SRH)statisticsandproducesane ectivespacechargedensityρe fromthetrappingoffreecarriersintheleakagecurrent.Thee ectivechargedensityisrelatedtotheoccupanciesanddensitiesoftrapsasfollows,

ρe =e[NDfD NAfA]+ρdopants

(1)

where:NDandNAarethedensitiesofdonorandac-ceptortrappingstates,respectively;fDandfAaretheoccupiedfractionsofthedonorandacceptorstates,respectively,andρdopantsisthechargedensityduetoionizeddopants(describestheresistivityofthemate-rialbeforeirradiation).Thedonorandacceptoroccu-panciesarerelatedtothetrapparametersbystandardSRHexpressionsfvvD=

hσDhp+eσDenieED/kT

veσAe(n+nieEA/kT)+vhσAh(p+ni

e EA/kT)where:veandvharethethermalspeedsofandholes,respectively;σDe,σD

electrons

haretheelectronholecapturecrosssectionsforthedonortrap;σAand

e,σA

aretheelectronandholecapturecrosssectionsforhtheacceptortrap;n,parethedensitiesoffreeelec-tronsandholes,respectively;niistheintrinsicden-sityofcarriers;ED,EAaretheactivationenergies(relativetothemid-gapenergy)ofthedonorandac-ceptorstates,respectively.Notethatthesingledonorandacceptorstatesmodelthee ectsofmanyphysicaldonorandacceptorstatesmakingthetwo-trapmodelan“e ectivetheory”.

ThephysicsofthemodelisillustratedinFig.2.Thespacechargedensityandelectric eldareplot-tedasfunctionsofdepthzforamodeltunedtore-producetheΦ=5.9×1014neqcm 2chargecollec-tiondataat150Vbias.TheSRHprocessproduceselectron-holepairsmoreorlessuniformlyacrossthethicknessofthesensor.Astheelectronsdrifttothen+implant,thetotalelectroncurrentincreasesaszdecreases.Theholecurrentsimilarlyincreaseswithincreasingz.Trappingofthemobilecarriersproducesanetpositivespacechargedensitynearthep+back-planeandanetnegativespacechargedensitynearthen+implant.Sincepositivespacechargedensitycor-respondston-typedopingandnegativespacechargecorrespondstop-typedoping,therearep-njunctionsatbothsidesofthedetector.Theelectric eldinthesensorfollowsfromasimultaneoussolutionofPois-son’sequationandthecontinuityequations.There-sultingz-componentoftheelectric eldvarieswithanapproximatelyquadraticdependenceuponzhavingaminimumatthezeroofthespacechargedensityandmaximaatbothimplants.Amoredetaileddescrip-0014

tionofthedoublejunctionmodelanditsimplemen-tationcanbefoundin[2].

Figure2:Thespacechargedensity(solidline)and

electric eld(dashedline)atT= 10 Casfunctionsofdepthinatwo-trapdoublejunctionmodeltunedtoreproducetheΦ=5.9×1014neqcm 2chargecollectiondataat150Vbias.

4.Modeltuningandresults

Chargecollectionacrossthesensorbulkwasmea-suredusingthe“grazingangletechnique”[14].AsisshowninFig.3,thesurfaceofthetestsensorisori-entedbyasmallangle(15 )withrespecttothepionbeam.Severalsamplesofdatawerecollectedwithzeromagnetic eldandattemperatureof 10 Cand 25 C.Thechargemeasuredbyeachpixelalongtheydirectionsamplesadi erentdepthzinthesen-sor.Preciseentrypointinformationfromthebeamtelescopeisusedtoproduce nelybinnedchargecol-lectionpro les.

Figure3:Thegrazingangletechniquefordeterminingchargecollectionpro les.Thechargemeasuredbyeachpixelalongtheydirectionsamplesadi erentdepthzinthesensor.

Thechargecollectionpro lesforasensorirradi-atedtoa uenceofΦ=5.9×1014 neq/cm2andop-eratedatatemperatureof 10Candbiasvoltagesof150Vand300VarepresentedinFig4.Themea-suredpro lesareshownassoliddotsandthesimu-latedpro lesareshownashistograms.Inordertoinvestigatetheapplicabilityofthetraditionalpictureoftype-invertedsiliconafterirradiation,thesimulated

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SNICSymposium,Stanford,California-3-6April,2006

pro lesweregeneratedwithelectric eldmapscorre-spondingtotwodi erente ectivedensitiesofaccep-torimpurities.Thefullhistogramsarethesimulatedpro leforNe =4.5×1012cm 3.Notethatthe300Vsimulationreasonablyagreeswiththemeasuredpro- lebutthe150Vsimulationisfartoobroad.ThedashedhistogramsshowtheresultofincreasingNe to24×1012cm 3.Atthise ectivedopingdensity,thewidthofthesimulatedpeakinthe150Vdistribu-tionisclosetocorrectbutitdoesnotreproducethe“tail”observedinthedataatlargey.The300Vsim-ulateddistributionisfartoonarrowandthepredictedchargeislowerthanthedata(notethatthepro lesareabsolutelynormalized).Itisclearthatasimu-lationbaseduponthestandardpictureofaconstantdensityofionizedacceptorimpuritiescannotrepro-ducethemeasuredpro les.

Figure4:Themeasuredandsimulated chargecollectionpro lesforasensoratT= 10Cirradiatedtoa uenceofΦ=5.9×1014neq/cm2.Thepro lesmeasuredatbiasvoltagesof150Vand300Vareshownassoliddots.Thefullhistogramsarethesimulatedpro lesforaconstante ectivedopingNe =4.5×1012cm 3ofacceptorimpurities.Thedashedhistogramsarethesimulatedpro lesforaconstante ectivedopingNe =24×1012cm 3.

Thesamemeasuredpro lesandthosefrombiasvoltagesof200Vand450VareshowninFig.5.Theyarecomparedwithsimulationsbasedupontheelec-tric eldproducedbythetwotrapmodel.Themodel

hassixfreeparameters(ND,NA,σDe,σDh,σAe,σA

h)thatcanbeadjusted.Theactivationenergiesarekept xedtothevaluesof[13]:ED=EV+0.48eV,EA=EC 0.525eVwhereEVandECaretheen-ergiesofthevalenceandconductionbandedges.Theelectric eldmapproducedbyeachtcadrunisin-putintopixelav.Theelectronandholetrappingrates,ΓeandΓh,arealsoinputstopixelavandaretreatedasconstrainedparameters.Althoughtheyhavebeenmeasured[15],theyareuncertainatthe20%levelduetothe uenceuncertaintyandpossi-bleannealingofthesensors.Theyarethereforeal-lowedtovarybyasmuchas±20%fromtheirnom-0014

inalvalues.Thedonorconcentrationofthestartingmaterialissetto1.2×1012cm 3correspondingtoafulldepletionvoltageofabout70Vforanunirradi-ateddevice.Becauseeachmodeliterationtookap-proximatelytwodays,itwasnotpossibletousestan-dardstatistical ttingtechniques.Theparametersofthedoublejunctionmodelweresystematicallyvariedandtheagreementbetweenmeasuredandsimulatedchargecollectionpro leswasjudgedsubjectively.The“best ts”showninthispaperareprobablynottruelikelihoodminimaandthecalculationofeightparam-etererrormatricesisbeyondavailablecomputationalresources.Adequateagreementwasachievedbyset-tingtheratioofthecommonholeandelectroncrosssectionsσh/σeto0.25andtheratiooftheacceptoranddonordensitiesNA/NDto0.40.ThereisarangeofparametersintheND-σespacethatproducesrea-sonableagreementwiththemeasuredpro les.TherangeisshowninFig.6aasthesolidlineintheloga-rithmicspace.Ifthedonordensitybecomestoosmall(ND<20×1014cm 3),the150Vsimulationproducestoomuchsignalatlargez.Ifthedonordensitybe-comestoolarge(ND>50×1017cm 3),the300Vsim-ulationproducesinsu cientsignalatlargez.SincethesimulatedleakagecurrentvariesasIleak∝σeND,di erentpointsontheallowedsolidcontourcorre-spondtodi erentleakagecurrent.Contoursofcon-stantleakagecurrentareshownasdashedcurvesandarelabeledintermsofthecorrespondingdamagepa-rameterαwhereα0=4×10 17A/cmistheexpectedleakagecurrent[16].Itisclearthatthesimulationcanaccommodatetheexpectedleakagecurrentwhichissmallerthanthemeasuredcurrentbyafactorofthree.

Theelectronandholetrapsinthemodelshouldalsocontributetothetrappingofsignalcarriers.Thecontributionsofthesestatestothee ectivetrappingratesofelectronsandholesaregivenbythefollowingexpressions

Γe=ve σA

e

NA(1 fA)+σD

eNDfD veσAeNAΓh=vh σD

hND(1 fD)+σA

hNAfA (3)

vhσDhNDwhereithasbeenassumedthatthetrapoccupancies

aresmall.BecauseNA/NDisassumedtobeconstant,contoursofconstantelectrontrappingrateareparal-leltocontoursofconstantleakagecurrentinND-σespace.Thebest“ t”ofthesimulationtothemea-suredpro lesreducedΓeto85%oftheun-annealedtrappingrateΓ0forthenominal uence[15].ThesecontoursarecomparedwiththeallowedcontourinFig.6b.Itisclearthatthesimulationcanaccommo-datethemeasuredtrappingrateinthesameregionofparameterspacethatmaximizestheleakagecurrent.Figure6balsosuggestsasolutiontothepuzzlethatthetrappingrateshavebeenshowntobeuna ectedbythepresenceofoxygeninthedetectorbulk[15]

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SNICSymposium,Stanford,California-3-6April,2006

Figure5:Themeasuredchargecollectionpro lesatatemperatureof 10 Candbiasvoltagesof150V,200V,300V,and450Vareshownassoliddotsfora uenceof5.9×1014neq/cm2.Thetwo-trapdoublejunction

simulationisshownasthesolidhistogramineachplot.

Figure6:TheallowedregionintheND-σespaceforthebest t5.9×1014neq/cm2modelisshownasthesolidlinein(a)and(b).Contoursofconstantleakagecurrentareshownasdashedcurvesin(a)andarelabeledintermsofthecorresponding17damageparameterαwhereα0=4×10 A/cmistheexpectedleakagecurrent[16].Contoursofconstantelectrontrappingrateare

shownasdashedcurvesin(b)andarelabeledintermsofthestandard-annealedtrappingrateΓ0forthenominal uence[15].

whereasitiswell-establishedthatthespacechargeef-fectsarequitesensitivetothepresenceofoxygeninthematerial[17].ItisclearfromFig6bthatsmall-cross-sectiontrappingstatescanplayalargeroleinthee ectivechargedensitybutasmalloneinthee ectivetrappingrates:everypointonthedj44line0014

produces100%ofthee ectivechargedensitybutonlythelargercrosssectionpointscontributesubstantiallytothetrappingrate.Iftheformationoftheadditionalsmall-cross-sectionstatesweresuppressedbyoxygen,thenρe couldbesensitivetooxygenationwhereasΓe/hwouldbeinsensitivetooxygenation.Thisisan-otherconsequenceoftheobservationthattheoccu-panciesfD/Aofthetrappingstatesareindependentofthescaleofthecrosssectionsinthesteadystate(seeeq.3).Thetrappingofsignalcarriersisnotasteady-statephenomenonandissensitivetothescaleofthetrappingcrosssections.

Thesimulationdescribesthemeasuredchargecol-lectionpro leswellbothinshapeandnormalization.The“wiggle”observedatlowbiasvoltagesisasig-natureofthedoublypeakedelectric eldshowninFig.2.Therelativesignalminimumneary=700µm(seeFig.5)correspondstotheminimumoftheelectric eldz-component,Ez,wherebothelectronsandholestravelonlyshortdistancesbeforetrapping.Thissmallseparationinducesonlyasmallsignalonthen+sideofthedetector.Atlargervaluesofy,Ezincreasescaus-ingtheelectronsdriftbackintotheminimumwheretheyarelikelytobetrapped.However,theholesdriftintothehigher eldregionnearthep+implantandaremorelikelytobecollected.Thenetinducedsignalonthen+sideofthedetectorthereforeincreasesandcreatesthelocalmaximumseenneary=900µm.Thechargecollectionpro lesatT= 10 Cforsen-sorsirradiatedto uencesofΦ=0.5×1014neq/cm2andΦ=2×1014neq/cm2andoperatedatseveralbiasvoltagesarepresentedinFig.7(a-c)andFig.7(d-g),respectively.Themeasuredpro les,shownassoliddots,arecomparedtothesimulatedpro les,shownashistograms.Notethatthe“wiggle”ispresentatlowbiasevenatΦ=0.5×1014neq/cm2whichisjustabovethe“type-inversion” uence.Thissuggeststhatadoublypeaked eldispresentevenatrathersmall uences.

Thedoublejunctionmodelcanprovideareasonabledescriptionofthelower uencechargecollectionpro- lesusingtheparametersobtainedwiththe ttingprocedureshowninTableII.Weobservethatthedonortrapconcentrationincreasesmorerapidlywith uencethandoestheacceptortrapconcentration.Theratiobetweenacceptoranddonortrapconcen-trationsis0.76atthelowest uenceanddecreasesto0.40at5.9×1014neq/cm2.Inaddition,the tsexcludealineardependenceofthetrapconcentrationswiththeirradiation uence.AtΦ=5.9×1014neq/cm2thecrosssectionratioσh/σeissetto0.25forbothdonorandacceptortrapswhileat ndσAdonorh/σAe=0.25andσDD

lower uenceswe

traps,respectively.h/σe=1fortheacceptorandThesimulatedleakagecurrentisapproximatelylinearin uence,butthera-tioNA/NDisclearlynotconstant.Thismaybeaconsequenceofthequadratic uencescalingofoneormoredi-vacancystatesoritmayre ectthefact

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SNICSymposium,Stanford,California-3-6April,2006

Figure7:Measured(fulldots)andsimulated(histogram)chargecollectionpro lesforsensorsirradiatedto uencesof0.5×1014neq/cm2(a-c)and2×1014neq/cm2(d-g),atT= 10 Candseveralbias

voltages.

Figure8:Thez-componentofthesimulatedelectric eldatT= 10 Cresultingfromthemodelbest tisshownasafunctionofzforasensorirradiatedto uencesof0.5×1014neq/cm2(a)and2×1014neq/cm2(b).

thatthetwotrapmodelwiththeparticularchoiceofactivationenergiesdoesnotaccuratelymodelthede-pendenceofthetrapoccupanciesonleakagecurrent.TheallowedND-σeparameterspacesforthelower u-encemodelsaremuchmoreconstrainedthanintheΦ=5.9×1014neq/cm2caseandpredicttheexpectedleakagecurrent.Theelectronandholetrappingrates,ΓeandΓharefoundtoscalemoreorlesslinearlywith uence.

TableIIDoubletrapmodelparametersextractedfromthe ttothedata.

Φ[1014neqcm 2]2.0

14 3

NA[10cm]6.8

2.540

A/D

[10 15cm2]σe6.60

1.651.65

D 152σh[106.60cm]

2.728.

2 2

Γh[10ns]13.

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SNICSymposium,Stanford,California-3-6April,2006

5.Temperaturedependence

Thetemperaturedependenceofthechargecollec-tionpro leswasstudiedbyaccumulatingdataatT= 25 C.Thepixelavsimulationincludestem-peraturedependentmobilities,di usion,andtrappingrates.Thetcadcalculationoftheelectric eldmapisalsobasedupontemperaturedependentquantitiesincludingthebandgapenergyandSRHlifetimes.TheT= 25 Ccharge2collectionpro lesfortheΦ=2.0×1014neq/cmandΦ=5.9×1014n

eq/cm2sensorsarecomparedwiththesimulationinFig.10.Itisclearthatthesimulationcorrectlytracksthetemperature-dependentvariationsinthemeasuredpro les.

(a)(b)

(c)(d)

(e)(f)(g)

Figure10:Measured(fulldots)andsimulated

(histogram)chargecollectionpro lesatT= 25 Candseveralbiasvoltagesforsensorsirradiatedto uencesof2.0×1014neq/cm2(a-d)andof5.9×1014neq/cm2(e-g).

Thee ectoftemperatureonthez-componentofthesimulatedelectric eldatΦ=5.9×1014neq/cm2isshowninFig.11forbiasvoltagesof150Vand300V.Itisclearthatdecreasingthetemperaturealsodecreasesthe eldsonthep+sideofthesensorandincreasesthemonthen+side.

6.Conclusions

Inthispaperwehaveshownthatdoublypeakedelectric eldsarenecessarytodescribegrazing-angle0014

Figure11:Thesimulatedz-componentoftheelectric eldasafunctionofthezcoordinateatthe

5.9×1014neq/cm2 uencefortemperaturesT= 10 CandT= 25 C.The eldpro lesareshownforbiasvoltagesof150Vand300V.

chargecollectionmeasurementsofirradiatedsiliconpixelsensors.Amodelofirradiatedsiliconbasedupontwodefectlevelswithoppositechargestatesandthetrappingofchargecarrierscanbetunedtoproduceagooddescriptionofthemeasuredchargecollectionpro lesinthe uencerangefrom0.5×1014neq/cm2to5.9×1014neq/cm2.Themodelcorrectlypredictsthevariationinthepro lesasthetemperatureischangedfrom 10 Cto 25 C.

Thedoublypeakedelectric eldpro leshavemax-imaneartheimplantsandminimanearthedetectormidplane.Thiscorrespondstonegativespacechargedensitynearthen+implantandandpositivespacechargedensitynearthep+backplane.We ndthatitisnecessarytodecreasetheratioofacceptorcon-centrationtodonorconcentrationasthe uencein-creases.Thiscausestheelectric eldpro letobecomemoresymmetricasthe uenceincreases.Thee ectofdecreasingthetemperaturehastheoppositee ectofsuppressingthe eldsonthep+sideofthesensorandincreasingthemonthen+side.

Themeasuredchargecollectionpro lesofirradi-atedsensorsareinconsistentwiththelinearly-varyingelectric eldspredictedbytheusualdescriptionbaseduponauniforme ectivedopingdensity.Thissuggeststhatthecorrectnessandthephysicalsigni canceofef-fectivedopingdensitiesdeterminedfromcapacitance-voltagemeasurementsarequiteunclear.Inaddition,weremarkthatthenotionofpartlydepletedsiliconsensorsafterirradiationisinconsistentwiththemea-suredchargecollectionpro lesandwiththeobserveddoublypeakedelectric elds.

Thecharge-sharingbehaviorandresolutionfunc-tionsofmanydetectorsaresensitivetothedetailsoftheinternalelectric eld.Aknownresponsefunc-

We show that doubly peaked electric fields are necessary to describe grazing-angle charge collection measurements of irradiated silicon pixel sensors. A model of irradiated silicon based upon two defect levels with opposite charge states and the trapping o

SNICSymposium,Stanford,California-3-6April,2006

tionisakeyelementofanoptimalreconstructionprocedure.Thee ectivemodeldescribedinthispa-perisbeingusedtocalculatedetailedresponsefunc-tionsthatarebeingincorporatedintoanewhitrecon-structionalgorithmfortheCMSpixel

trackingsystem[18,19].Thiswillpermitthe“calibration”ofthere-constructionalgorithmtobetrackedasthepixelsareirradiatedduringLHCoperation.

Acknowledgments

WegratefullyacknowledgeSilvanStreulifromETHZurichandFredyGlausfromPSIfortheirimmensee ortwiththebumpbonding,FedericoRavotti,Mau-riceGlaserandMichaelMollfromCERNforcarry-ingouttheirradiation,KurtB¨osigerfromtheZ¨urichUniversityworkshopforthemechanicalconstruction,Gy¨orgyBenczeandPascalPetiotfromCERNfortheH2beamlinesupportand, nally,thewholeCERN-SPSteam.ThisworkwassupportedinpartbyNSFgrantPHY-0457374.

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