1

DefectDiagnosisusinga Curr ent Ratio basedQuiescentSignalAnalysisModelfor Commercial Power Grids

Chintan Patel, Ernesto Staroswiecki, Smita Pawar, Dhruva Acharyya and Jim PlusquellicDepartment of CSEE, University of Maryland, Baltimore County

for JETTA fr om a special issue guest edited by Dr. Rafic Makki

Contact Author

Jim Plusquellic

UMBC, CSEE, ECS 212

1000 Hilltop Circle, Baltimore, MD - 21250

Email: plusquel@cs.umbc.edu

Phone: 410-455-1349

Fax: 410-455-3969

DefectDiagnosisusinga Curr ent Ratio basedQuiescentSignalAnalysisModelfor Commercial Power Grids

Authors:

Chintan Patel, cpatel2@cs.umbc.edu

Ernesto Staroswiecki, estaro1@umbc.edu

Smita Pawar, psmita1@cs.umbc.edu

Dhruva Acharyya, adhruva1@cs.umbc.edu

Jim Plusquellic, plusquel@cs.umbc.edu

Addr ess:Department of CSEE, University of Maryland Baltimore County

1000 Hilltop Circle,Baltimore,

MD - 21250.

Abstract

QuiescentSignalAnalysis(QSA)is a novel electrical-test-baseddiagnostictechniquethat usesIDDQ measure-

mentsmadeat multiple chip supplypadsas a meansof locating shortingdefectsin the layout.Theuseof multiple

supplypadsreducestheadverseeffectsof leakagecurrentbyscalingthetotal leakagecurrentovermultiplemeasure-

ments.In previous work, a resistancemodelfor QSAwas developedand demonstratedon a small circuit. In this

paper, theweaknessesof theoriginal QSAmodelare identified,in thecontext of a productionpowergrid (PPG)and

probecard model,anda new modelis described.Thenew QSAalgorithmis developedfromtheanalysisof IDDQ con-

tour plots. A “family” of hyperbolacurvesis shownto be a goodfit to the contourcurves.Theparameters to the

hyperbolaequationsare derivedwith thehelpof insertedcalibration transistors.Simulationexperimentsare usedto

demonstrate the prediction accuracy of the method on a PPG.

Keywords:

IDDQ, IDDT, quiescent signal analysis, test, power grids

Abstract

QuiescentSignalAnalysis(QSA)is a novel electrical-test-baseddiagnostictechniquethat usesIDDQ measurements

madeat multiplechip supplypadsasa meansof locatingshortingdefectsin the layout.Theuseof multiplesupply

padsreducestheadverseeffectsof leakage currentby scalingthetotal leakage currentover multiplemeasurements.

In previouswork, a resistancemodelfor QSAwasdevelopedanddemonstratedon a smallcircuit. In this paper, the

weaknessesof theoriginal QSAmodelare identified,in thecontext of a productionpowergrid (PPG)andprobecard

model,anda new modelis described.Thenew QSAalgorithmis developedfromtheanalysisof IDDQ contourplots.A

“family” of hyperbolacurvesis shownto bea goodfit to thecontourcurves.Theparameters to thehyperbolaequa-

tionsarederivedwith thehelpof insertedcalibration transistors.Simulationexperimentsareusedto demonstratethe

prediction accuracy of the method on a PPG.

1.0 Introduction

IDDQ hasbeenamain-streamsupplementaltestingmethodfor defectdetectionfor morethanadecadewith many

companies.With theadventof thedeepsubmicrontechnologies,theuseof single-thresholdIDDQ techniqueresultsin

unacceptableyield loss.Settinganabsolutepass/fail thresholdfor IDDQ testinghasbecomeincreasinglydifficult due

to the increasingsubthresholdleakagecurrents[1]. Currentsignatures[2], delta-IDDQ [3] andratio-IDDQ [4] have

beenproposedasameansfor calibratingfor thesehighsubthresholdleakages.Thesetechniquesrely onaself-relative

or differentialanalysis,in which the averageIDDQ of eachdevice is factoredinto the pass/fail threshold.However,

these proposed forms of calibration are expected to become less effective over successive technology generations.

An alternative calibrationstrategy thatmayhave betterscalingpropertiesis to distributethetotal leakagecurrent

acrossa setof measurements.This is accomplishedby introducingprobinghardwarethatallows themeasurementof

IDDQ at eachof the supply ports.The methodproposedin this work, called QuiescentSignal Analysis (QSA), is

designedto exploit this typeof leakagecalibrationfor defectdetectionandasameansof providing informationabout

DefectDiagnosisusinga Curr ent Ratio basedQuiescentSignalAnalysisModelfor Commercial Power Grids

Chintan Patel, Ernesto Staroswiecki, Smita Pawar, Dhruva Acharyya and Jim Plusquellic

Department of CSEE, University of Maryland, Baltimore County

This work is supported by a Faculty Partnership Award from IBM’s Austin Center for Advanced Studies(ACAS) Program and by an NSF grant, award number 0098300.

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the defect’s location in the layout [5][6]. This latter diagnosticattribute of QSA may provide an alternative to

image-basedphysical failure analysisproceduresthat arechallengedby the increasingnumberof metal layersand

flip chip technology.

A resistance-baseddiagnosticmodelfor QSAwasdevelopedin previousworksandsimulationexperimentswere

usedto demonstratethe diagnosticcapabilitiesof the QSA methodon a small circuit [5][6]. In this paper, several

weaknessesof theresistance-basedmodelareuncoveredfrom simulationsof a productionpower grid (PPG).A cur-

rent-ratio-basedmodel is proposedand demonstratedto improve on defect localizationaccuracy of the original

method[7]. Thenew methodrequirestheinsertionof calibrationtransistors(CT), oneundereachof thesupplypads

in thedesign,thatpermit theshortingof thepower andgroundsupplyrails at pointscloseto thesubstrate.Thestate

of theCTsarecontrolledby scanchainflip-flops.TheIDDQs obtainedwhenoneof theCTsis turnedon areusedto

calibratethe IDDQs measuredundera failing IDDQ pattern.The calibrationtechniqueis shown to addressseveral

weaknessesof thepreviousmodelincludingnon-zeroprobecardresistanceandirregularsupplygrid topologies.Cur-

rent ratios,asopposedto absolutecurrents,areproposedasa meansof reducingthedependenceof the localization

algorithmon thevalueof thedefectcurrent.SPICEsimulationexperimentsdemonstratethat themaximumpredic-

tion error is 650 units in a 30,000 by 30,000 unit area.

It is notpossibleto evaluatetheQSA algorithmon theentire80,000by 80,000unit areaof thePPGusingSPICE

dueto the largesizeof theR model.Instead,a specializedpower grid simulationenginecalledALSIM is used[8].

The anomaliesin the grid’s structurein this larger areaincreasethe maximum prediction error to 1,340 units.

Although the predictionaccuracy is goodfor mostcases,an alternative “lookup table” approach(in contrastto the

hyperbola-basedapproach)is likely to be moreaccuratefor irregular grid regionsor configurations.The enhanced

simulationcapabilitiesof ALSIM enablethis strategy, aloneor in combinationwith the hyperbola-basedapproach

described in this paper.

Theremainderof this paperis organizedasfollows.Section2.0describesrelatedwork. Section3.0givesa brief

descriptionof theoriginal resistance-basedQSAtechnique,identifiesits weaknessesanddescribesthebasisof anew

model.Section4.0presentsthedetailsof thecurrent-ratio-basedQSA model.Section5.0givesexperimentalresults.

Section 6.0 gives our conclusions and areas of future research.

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2.0 Background

Severaldiagnosticmethodshavebeenproposedbasedon IDDQ measurements.In general,thesemethodsproduce

a list of candidatefaultsfrom a setof observedtestfailuresusinga fault dictionary. Thelikelihoodof eachcandidate

fault canbe determinedby several statisticalalgorithms.For example,signatureanalysisusesthe Dempster-Shafer

theory, which is basedon Bayesianstatisticsof subjective probability [9]. Delta-IDDQ makesuseof theconceptsof

differential currentprobabilisticsignaturesand maximumlikelihood estimation[10]. Although thesemethodsare

designedto improve theselectionof fault candidates,in many cases,they areunableto generatea singlecandidate.

Otherdifficultiesof thesemethodsincludetheeffort involvedin building thefault dictionaryandthetime requiredto

generate the fault candidates from the large fault dictionary using tester data.

TheQSA procedurecanhelpprunethecandidatelist producedby IDDQ andothervoltagebaseddiagnosticalgo-

rithms.Thephysical layout informationgeneratedby our methodcanbeusedwith informationthatmapsthelogical

faultsin thecandidatelists to positionsin thelayout.In addition,it maybepossibleto usethe(x,y) locationinforma-

tion provided by QSA asa meansof reducingthe searchspacefor likely candidatesin the original fault dictionary

procedure. This can reduce the processing time and space requirements significantly.

3.0 QSA Models

QSAanalyzesasetof IDDQ measurements,eachobtainedfrom individualsupplypads,to predictthelocationof a

shortingdefect.Theresistive elementof thepower grid causesthecurrentdrawn by thedefectto benon-uniformly

distributedto eachof the supplypads.In particular, the defectdraws the largestfraction of its currentfrom supply

padstopologically “nearby”. The sameis true of the leakagecurrents.However, only the leakagecurrentsin the

vicinity of the defectcontribute to the measuredcurrentin thesepads.The smallerbackgroundleakagecomponent

improvestheaccuracy of thedefectcurrentmeasurement.As describedin previousworks,QSAalsoproposestheuse

of regression analysis as a means of eliminating the remaining leakage component from the measured values [5][6].

3.1 The Resistance-based QSA Model

The fraction of the defectcurrentprovided by eachof the padsin the region of the defectis proportionalto the

equivalent resistancebetweenthe defectsite andeachof the pads.The differencesin thesevaluescanbe usedto

localizethedefectusinga methodbasedon triangulation.For example,Figure1 shows a shortingdefectin anequiv-

alentresistancemodelof a simplepower grid. Here,Req0 throughReq3 representtheequivalentresistancesbetween

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eachof the supplypads,Padi, andthe defectsite shown in the centerof the figure.The following setof equations

describetherelationshipbetweenthepower supplybranchcurrents,I0 throughI3, andVdef, thevoltageat thedefect

site.

In Eq.1, theIsarethemeasuredIDDQs.TheRps representtheprobecard’s resistances,whichweassumearevery

smallwith respectto theReqandcanbeignored(thisassumptionis addressedbelow). This leavestheReqandVdefas

unknowns.Withoutadditionalinformation,it is notpossibleto solve theseequationssincethereare4 equationsand5

unknowns.However, for thepurposeof diagnosis,only therelationshipsbetweentheReqareneeded.Relative equiv-

alent resistances can be computed with respect to a reference equivalent resistance, Reqj, as given by Eq. 2.

Under the condition that Rp << Req (otherwisethe modelshown if Figure1 is not complete),it is possibleto

obtainanaccuratepredictionof thedefect’s locationby solving thecircle expressionsgiven in Eq. 3 for a common

point of intersection given byx andy.

Theparametershi andki representthex andy coordinatesof thecenterof the ith circle.Thethreecircleequations

arerelatedto correspondingequationsfrom thesetdescribedby Eq.2 throughtheReq. Here,Reqj is assumedto be1.0

andReqaandReqbarecomputedfrom Eq.2 usingtheIDDQ measurements.Parameterm is usedto maptheresistances

given on the left in Eq. 3 to distances in the layout.

Thechoiceof thesupplypadsto beusedin thetriangulationprocedureis basedon two criteria.First, thesupply

I i Reqi Rpi+( )× VDD V–def

= for i = 0,1,2,3 (1)

I i Reqi Rpi+( )× I j Reqj Rpj+( )×=

Reqi

I jI i---- Reqj×

I jI i---- Rpj× Rpi–+=

with

(2)

i j≠

solving for Reqi in terms of Reqj gives

m Reqj× x h j–( )2 y k j–( )2+=

m Reqa× x ha–( )2 y ka–( )2+=

m Reqb× x hb–( )2 y kb–( )2+=

(3)

Figure 1. Equivalent resistance model of the power grid with a shorting defect.

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padsaresortedaccordingto themagnitudeof their correspondingIDDQ. Thesupplypad,j, with the largestIDDQ is

selectedfollowed by two orthogonallyadjacentsupplypads,a andb, to pad j sourcingthe next two largestvalues.

Notethatthismodelis basedon two simplifying assumptions:auniformresistance-to-distancemappingfunctionand

negligible valuesfor Rp. A uniform resistance-to-distancemappingfunctionis usedto describepower gridsin which

the equivalent resistance and Euclidean distance between any two points on the grid are proportional.

An exampleapplicationof this triangulation-basedmethodis shown in Figure2. Threedottedcirclesareshown

whosecentersaredefinedby thepositionsof thePad1, Pad2 andPad3. Theradii arelabeledwith theappropriateReq

valuesasgiven in Eq. 3. For example,Pad3 definesthecenterof thecircle with smallestradius,i.e., it is thesupply

padwith thelargestIDDQ. Its radiusis labeledwith Reqj in thefigure.Theinitial radii of thethreecirclesarethenmul-

tiplied by a commonfactor, m, to a commonpoint of intersection.This point is labeledas“PredictedDefectLoca-

tion” in the figure to contrast it with the “Actual Defect Location”.

3.2 Weaknesses of the Resistance-based Model

Unfortunately, theassumptionsof theresistance-basedmodelarenotvalid in many situations.Here,it is assumed

thattheRpsaresmallrelative to theReq. Underthisassumption,themeasuredIDDQsarerelatedto theReqasgivenby

Eq. 4 (derived from Eq. 2).

Therefore,theresistance-basedQSA modelassumesthatthecurrentratiosareinverselyproportionalto theresis-

tanceratios.If thevaluesof Rps aresimilar to or largerthantheReq, thentherelationshipgivenby Eq.4 is weakened

and the accuracy of the triangulation approach is correspondingly reduced.

In thenext section,we presenta morecompleteequivalentresistancemodelof theCUT thatbetterrepresentsan

actualprobecardmodelin which theRps aresignificant.Thenew modelrequiresadditionalinformationin orderto

solve for unknownssuchastheReq andRp. A new QSA methodis proposedthatobtainsthis informationfrom cali-

Reqi

I jI i---- Reqj×≅ or

ReqiReqj-----------

I jI i----≅

If these terms are negligible then

(4)

Reqi

I jI i---- Reqj×

I jI i---- Rpj× Rpi–+=

Figure 2. Triangulation under r esistance model.

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bration transistorsmeasurements.However, it shouldbe notedat this point that large valuesof Rp will adversely

affect the precisionrequiredin the measurementof the IDDQs underany proposedstrategy. This follows from a

numericalanalysisof Eq. 2, that shows the convergenceof all currentratiosto the ratiosdefinedby the Rps asthe

magnitude of theRps are increased to and above theReq.

Anotherweaknessof the resistance-basedQSA modelis with regard to theuniform resistance-to-distancemap-

ping function. Most supply topologiesarepoorly modeledasuniform. In previous work, we proposeda mapping

function basedon resistancecontoursto dealwith complicatedirregular topologies[5]. In this work, we proposea

secondstrategy basedon theuseof a currentratio lookup-table.Both techniquesrequireresistanceandcurrentpro-

files of thegrid to bederived in advancethroughsimulations,andshouldbeavoided,if possible,in casesinvolving

more regular topologies.

The topologyof the PPGunderinvestigation in this work fits betweenthe totally regular and totally irregular

extremes.The mappingfunction is not strictly uniform but, becausethe physical structureof the grid is regular in

many places,it is possibleto modeltheresistanceperunit distancebetweeneachpairingof supplypadsusinga con-

stant.Thenew hyperbola-basedQSA methoddescribedin this paperis ableto calibratefor this typeof power grid

resistance-to-distanceprofile usingmeasureddataonly. Therefore,it providesa simpleralternative to a lookup-table

approach.

3.3 The PPG’s Physical Characteristics

Figure3 shows the80,000by 80,000unit layoutof thePPG.ThePPGinterfacesto a setof externalpower sup-

pliesthroughanareaarrayof VDD andGND C4 pads.A C4 padis a solderbumpfor anareaarrayI/O scheme.The

PPGhas64 VDD C4sand210GND C4s(not shown in Figure3). The64 VDD C4sdivide thePPGinto 49 different

regionscalledQuads.ALSIM simulationsexperimentswererun on theentirePPG.However, dueto spaceandtime

constraints,it wasnotpossibleto runSPICEsimulationsontheentirePPG.Rather, aportionof thePPGconsistingof

9 quadswassimulatedusingSPICE.Thisportionconsistsof thelower left 9 Quadsasshown in Figure3, andis sub-

sequently referred to as the Q9. The Q9 occupies a 30,000 by 30,000 unit area.

Figure 4. Layout details of the PPG.

Figure 3. Layout of the PPG.

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In orderto deriveanelectricalmodelof thePPG,wefocusedouranalysisontheportionshown in thelower left of

Figure3 identifiedastheQuad.Figure4(a)expandson this view by showing a moredetaileddiagramof this 10,000

by 10,000unit region. This is again expandedin Figure4(b) which shows a stacked four metallayerconfiguration,

with m1 andm3 runningvertically andm2 andm4 runninghorizontally. TheC4sareconnectedto wide runnersof

verticalm5, shown in the top portionof Figure4(a),thatare,in turn, connectedto them1-m4grid. In eachlayerof

metal,theVDD andGND railsalternate.In theverticaldirection,eachm1rail is separatedby adistanceof 432units.

The alternatingvertical VDD and GND rails are connectedtogetherusing alternatinghorizontal metal runners.

Stackedcontactsareplacedat theappropriatecrossingsof thehorizontalandvertical rails. Thegrid is fairly regular

exceptin the region labeled“irregular region” in theupperright cornerof Figure4(a).Them1 in this region of the

layout varies from the regular pattern shown in Figure 4(b).

TheR modelof theQuadwasobtainedfrom anextractionscriptwhichusesprocessparametersfrom theTSMC’s

0.25µm process[11]. 1Ω resistanceswereinsertedbetweenthepower suppliesandtheR modelof thegrid to model

thetesterpower supply(s)andprobecardcontactresistancesto thechip.Althoughour simulationmodeluses1Ω for

all probecardresistances,theanalyticalmodelthatwederivebelow accommodatesamorerealisticprobecardmodel

in which probe card resistanceis different from one pad to another. The combinedresistancenetwork contains

approximately 27,000 resistors.

3.4 The Quad’s Electrical Characteristics

Figure4(b) alsoshows a setof currentsourcesthatwereinsertedindividually in a sequenceof simulationsasa

meansof evaluatingtheelectricalbehavior of theresistancemodelat theVDD C4s.Thecurrentsources,whichmodel

thepresenceof a shortingdefect,wereplacedat regularintervalsbetweenm1 VDD andGND runners.An equivalent

resistancemodelof theQuadis shown in Figure5 with oneof thecurrentsourcesinserted.Thefour grid equivalent

resistances,Req, in theuppercenterportionof thefigurearethesourceof resistancevariationasseenfrom thepower

supplies,asthecurrentsourceis movedin the layout.Thestrengthof thecorrespondenceof theseresistancesto the

positionof thedefectdeterminestheaccuracy of the triangulationprocedureusedin QSA. It is thereforeprudentto

evaluate this relationship for the Quad.

Thereareseveral significantdifferencesbetweenthis modelandthe modelshown in Figure1. First, underthe

assumptionthatthevaluesof theRp arenon-zero,thegrid resistancesbetweentheC4s,e.g.R01 shown on thetop left

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of Figure5, areneededin any completeequivalentresistanceexpressionsuchasthatgivenby Eq.2. Second,theReq

areactuallythreedimensionalin natureandcanbemodeledasRz andRxy asshown on theright sideof Figure5. Rz

adversely impacts the accuracy of the triangulation procedure for the same reasons given earlier forRp.

A third limitation in our original resistance-basedmodelis relatedto the resistanceprofile thatcharacterizesthe

PPGunderinvestigation in this research.Our analysisrevealsthat the variationin equivalentresistanceover small

verticalintervalsof theQuad,e.g.,alongtheinterval betweentwo contactpointsin m1, is onorderwith thevariation

acrossthe entireQuad.For example,the segmentlengthgiven betweenpointsA andB in Figure4(b) is approxi-

mately630units.Usingthem1 resistanceparameterfor TSMC’s 0.25µm processyieldsa valueof 5.6Ω. Therefore,

in m1alone,theresistancevariesfrom 0Ω at thecontactto 5.6||5.6= 2.8Ω in thecenter. On theotherhand,theaver-

ageresistancefrom thecenterof theQuad(shown in Figure4(a))to any of theVDDs (distanceof ~7,000units)is less

than6Ω. The increasingwidth of the metal runnersfrom m1 to m5 is responsiblefor theseresistanceto distance

anomalies.

In orderto gain insightinto otheralternativediagnosticstrategies,wefirst derivedtheprofilesof thenetwork vari-

ablesincludingReq, Vdef (thevoltageat thedefectsite)andtheIDDQsat theVDD C4s.Theprofileswerederivedfrom

the resultsof 2,600SPICEsimulationexperimentsof the Quad.In eachsimulation,a 20mA currentsourcewas

placedbetweenm1 VDD andGND rails at differentlocationsin thelayout.Figure6 shows thecurvesfor Req0andI0

(at C40) andVdef (thecurrentsource’s terminalvoltageat theconnectionpoint on them1 VDD rail) for a setof simu-

lationsrunalongthelinesidentifiedasx-sliceandy-slicein Figure4(a).TheReq0valueswerecomputedusingEq.5.

It is clearfrom thesegraphsthatthevariationsin Req0andVdefalongthey dimensionaresignificantlylargerthan

Req0

VDD Vdef–( )I 0

-----------------------------------=

VDD = 2.5VVdef = voltage at the defect siteI0 = current through VDD0

(5)

where,

Figure 5. Equivalent resistance model of the Quad.

Figure 6. Network variable plots for sources along x-slice (top) and y-slice (bottom) lines of Figure 4(a).

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thosealongthex dimension.In contrast,thecurrentsarewell behavedalongeitherdimension.Thestaggeredarrange-

mentof VDD andGND grids,asshown in Figure4(b), causesthe total resistancebetweenVDD andGND to change

slowly acrossthegrid, throughtheexchangeof nearlyequalresistancefragmentsbetweentheVDD andGND grids.

This keepsthecurrentswell behavedwhile theresistancesto, andvoltagesat, thedefectsiteoscillateinverselywith

each other.

3.5 Contour Profiles of the Quad

Anotherusefulview of thebehavior of thesenetwork variablesis throughcontourplots.A line within a contour

plot is definedastheparametervaluesover which thevalueof the function remainsconstant.Contoursareparticu-

larly usefulwhendatais to befit to a function.Figures7 and8 show theequivalentresistanceandcurrentcontoursof

theQuadfor VDD0 (only every 3rd contourcurve is shown.) Thex andy axescorrespondto the(x,y) coordinatesof

theQuadasshown in Figure4(a).Thejaggednatureof thecurvesasshown in Figure8 modelsabandwhosewidth is

definedby thevertical line segmentsin thecurves.It is clearthat theequivalentresistancecontourplot is difficult to

make useof. Thesameis trueof theVdef contourplot (not shown). In contrast,thecurrentcontoursareelliptical in

shape,(exceptfor a region in theupperright handcorner, identifiedas“irregularregion” givenearlierin referenceto

Figure 4(a)). Similar patterns are present in the current contour plots of the other VDDs.

Therefore,a diagnosticmethodbasedon currentsis likely to yield the bestresults.However, unlike equivalent

resistance,thedisadvantageof usingthecurrentsdirectly is thedependency that is createdbetweenthecontoursand

themagnitudeof thedefect’s shortingcurrent.Currentratiosareanalternative thatreducethis dependency sincedif-

ferent values of defect current are reflected as the same ratio in the C4 IDDQs.

Thecontourplot for I0/I1 is shown in Figure9. Like theI0 contourplot, thecontourlinesarewell behaved.How-

ever, theelliptical curvescharacterizingthe I0 plot now appearashyperbolacurves,particularlyin theregion to the

left of x=5000.Thesetor “f amily” of hyperbolasis centeredat themidpointbetweenthepositionof C40 (lower left)

andC41 (upperleft). Thecontourcurvethatpassesthroughthismidpoint(y=5000)onthey axisis nearlylinearalong

Figure 7. Req0contours of Quad.

Figure 8. I0 contours of Quad.

Figure 9. I0/I1 contours of Quad.

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aline to thecenterof theQuad(shown by the‘dot’ in thecenterof thefigure).Thiscurvedefinesthepointsin thelay-

out thatareexpectedto produceanI0/I1 currentratio closelyapproximatedby 1.0.TheI0/I1 increaseto a maximum

in the lower left corner. ThemaximumI0/I1 currentratio is largely determinedby theRz componentof resistanceat

C40 andRp0. As anexample,theI0/I1 maximumfor theQuadis 1.55andtheI0/I2 maximumis 1.84.Thesemaximum

currentratioscanbedeterminedexperimentallyusinga simpletestcircuit. We describethis testcircuit andits other

benefits after we derive the analytical model for the new QSA procedure.

4.0 The Current Ratio Model for QSA

Thedensityof thecontourcurvesin thelower left quarterof Figure9, i.e. theregionwith x andy coordinatesless

than5000,is higherthanthedensityin otherregionsof theQuad.For example,thenumberof contourcurvesbelow

they=5000is 10 while thenumberabove this point is 6. Therefore,the I0/I1 andI0/I2 currentratiosareexpectedto

provide thebestresolutionfor defectsthatoccurin this region.UndertheassumptionthattheC4swith largestIDDQs

areclosestto thedefectsite,it is straightforwardto identify therelevantregionandto computetheappropriatecurrent

ratios from the measureddata.(This assumptionis later removed.) The morechallengingproblemis to determine

how to use these ratios to identify the location of the defect in the selected region.

Themoststraightforwardmethodis to usethemeasuredratiosto selecttwo contourcurves.For example,Figures

10and11show theI0/I1 andI0/I2 contourcurvesobtainedfrom simulationsperformedonthelower left quarterof the

Quad(only every othercurve is shown). Thepoint of intersectionof two curves,onefrom eachfigure,identifiesthe

positionof the defectin the layout.This is the generalideabehindthe lookup-tablemethodreferredto above. The

drawbackof this methodis the large numberof simulationsthat areneeded(onefor eachcandidatepositionin the

layout)to build thetable.Powergrid simulatorssuchasALSIM makethispracticalandweexpectthisapproachto be

useful for irregular grid topologies. However, a simpler method is possible in many situations.

An alternative strategy is to derive a functionthatapproximatesthecontourcurvesusingthemeasuredquantities,

i.e. thecurrentratios,asparameters.As notedabove, thecurrentratiocontourcurvesaresimilar in shapeto hyperbo-

las.Figures10 and11 show a setof curvesderivedfrom hyperbolassuperimposedon thecontourcurvesfor illustra-

Figure 10. I0/I1 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.

Figure 11. I0/I2 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.

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tion. In orderto realizethis mapping,it is necessaryto derive expressionsfor the hyperbolaparameters.Eq. 6 and

Figure12 defineandillustrate“horizontally-oriented”hyperbolas,suchasthoseshown in Figure11. Thearrows on

the right of Figure 11 illustrate the region in which these curves are represented in Figure 12.

Figure12portraystheroleof thea andb parametersin agraphanddefinesanadditionalparameter, c, thatis used

to definetherelationshipamongthesetsor “f amilies” of hyperbolasin Figures10 and11.A family of hyperbolasis

definedasasetthatshareacommoncenterandfocus.Theh andk parametersin Eq.6 definethecenterof thehyper-

bolas.Thecentersof thehyperbolacurvesshown in Figures10 and11 areidentifiedat (h,k)=(0,5000)and(5000,0),

respectively, andrepresentthemidpointbetweenthefoci. Thefoci of thehyperbolasaregivenby F1 andF2 in Figure

12. These points represent the (x,y) coordinates of the C4 VDDs.

Thea andb parametersof thehyperbolasneedto bedefinedin termsof thecurrentratios.Fortunately, thenature

of thecontoursdefinedby thegrid allow analternative formulationof Eq. 6 asgivenby Eq. 7. Here,b2 is replaced

with (c2 - a2). Sincec is fixedfor all hyperbolasin thefamily asthedistancebetweentheir center, (h,k), andthecoor-

dinates of the C4 supply pad, this makesb dependent ona. Therefore, onlya needs to be defined.

Fromthediagramshown in Figure12,a definesthepointof intersectionof theI0/I2 hyperbolawith thehorizontal

line definedbetweenthecenter(h,k) = (5000,0)andC40 (F1 in thefigure).Therefore,a variesfrom 0, at thecenter, to

L/2 atC40, whereL is definedasthedistancebetweenC40 andC42 (10,000for theQuad).Thecurrentratiosatpoints

alongthis line increasefrom 1.0 at the centerto the maximumcurrentratio, e.g.1.84 for I0/I2 in the Quad.If this

maximumcurrentratio is known, thenthefunctionthatdefinesa canbederivedfrom thelumpedR modelshown in

Figure 13 as follows.

x h–( )2

a2

------------------- y k–( )2

b2

-------------------– 1= (6)

x h–( )2

a2

------------------- y k–( )2

c2

a2

–-------------------– 1= (7)

Figure 12. Definitions of the hyperbola parameters

Figure 13. Lumped R model for the hyperbola a parameter.

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Eqs.8 and9 give theexpressionsfor thecurrentratio β02 anda without proof. ReqT (total resistance)is equalto

thesumof theequivalentresistances,i.e. Req0+ Req2, betweenthedefectsiteandeachof the two C4s.Rp0 andRp2

are the probe card resistances at C40 and C42, respectively.

4.1 Calibration Transistors

As pointedoutearlier, Req0andReq2, andthereforeReqT, cannotbeobtainedin thedefectivechip.However, under

thespecialcasewherethedefectshown in Figure13 is positionedon a line betweenC40 andC42, we canobtaina

closeapproximationof ReqT experimentally. This is accomplishedby insertinga calibrationtransistor(CT0) under

C40, asshown in Figure14. Thesourceanddrainof theCT0 connectto VDD andGND in m1 andprovide a way to

conditionallyshortthesenodestogether. By positioningtheCT0 directly underC40 (at thelowestresistanceposition

from m1 to C40), the maximumcurrentratio, β02(CT0)= I0(CT0)/I2(CT0), canbe obtained.This is accomplishedby

placingthechip into astatethatdoesnotprovoke thedefectandturningonCT0 usingthescanchainflip-flop driving

its gate.

Themeasuredvaluesof I0(CT0)andI2(CT0) resolveseveralissuesrelatedto theapplicationof this technique.First,

β02(CT0)allowsEq.8 to besolvedundertheboundaryconditionm=0. If thesameprocessis repeatedusingacalibra-

tion transistorCT2, positionedunderC42, thenEq. 8 canbesolvedundera secondboundarycondition,m=L, using

β02(CT2). With threeequations,ReqT, Rp0, andRp2 in Eq. 8 cannow be eliminated,allowing a to be expressedasa

functionof themeasuredcurrentratios,β02, β02(CT0)andβ02(CT2). This is possiblebecausethevaluesof ReqT, Rp0,

(8)

β02 =I 0I 2-----

ReqT L m–( )× L Rp2×+

m ReqT× L R× p0+----------------------------------------------------------------=

and aL2--- m–=

aL2---

LReqT-------------

Rp2 β02 Rp1×( )– ReqT+

1 β02+( )---------------------------------------------------------------

–=

Substituting and solving fora yields

(9)

Figure 14. Calibration Transistor and controlling scan-chain FF.

15

andRp2 arenearlyinvariantacrossthethreetests.Eq.10and11givestheexpressionsfor a andb in termsof thecur-

rent ratios derived from C40 and C42 CT tests.

Thus,a nicefeatureof this calibrationtechniqueis that it is independentof theRp, which arelikely to vary from

touch-down to touch-down of the probe card.

A secondproblemaddressedby the CTs is relatedto the proceduredescribedin Section3.1. Pad selectionis

accomplishedby sortingtheIDDQ valuesandidentifying thepadwith the largestIDDQ asthe“primary” pad(padj).

Two (of thefour) orthogonallyadjacentsupplypadsto padj arethenselectedfrom thetopof thesortedlist. Unfortu-

nately, this algorithmfails to selectthepadssurroundingthedefectundercertainconditions.For example,Figure15

shows a portionof thesupplygrid with 9 C4s.Thedefectis locatedin theupperleft Quadandtherefore,thealgo-

rithm shouldselectC43 asthe“primary” padandC41 andC47 astheorthogonallyadjacentpads.However, if Reqa>

Reqb, thesortedlist placesC45 above C41 andthealgorithmincorrectlyselectsC45. This typeof resistanceanomaly

can occur, for example, if the power grid mesh is denser between C43 and C45 than it is between C43 and C41.

TheCT datacanbeusedto instrumentamorerobustpadselectionalgorithm.Thecurrentratioβ31(CT3)= I3(CT3)/

I1(CT3)obtainedby turningontheCT underC43 givestheupperboundonthecurrentratiobetweenC43 andC41. The

currentratiocomputedunderthecircuit statewith thedefectprovoked,β31, is necessarilylessthantheβ31(CT3), since

β31(CT3)is themaximumratio.Therefore,animprovedalgorithmselectsthecorrectsecondarypads,e.g.C41 instead

mL β02 CT0( ) 1 β+ 02 CT2( )( ) β02– 1 β+ 02 CT2( )( )( )

1 β+ 02( ) β02 CT0( ) β02 CT2( )–( )--------------------------------------------------------------------------------------------------------------------------=

with,β02= current ratio I0/I2 at state with defect provokedβ02(CT0)= current ratio I0(CT0)/I2(CT0) with CT0 on

L= distance between two adjacent C4 VDDs

aL2--- m–= (10)

β02(CT2)= current ratio I0(CT2)/I2(CT2)with CT2 on

b c2

a2

–L2---

2a

2–= =

(11)

Figure 15. Anomalies in complex grids.

16

of C45, by using CT ratio data.

It is alsopossiblein somegrid configurationsthatthelargestIDDQ is not drawn from thepadthatis closestto the

defectsite.In this case,theexisting algorithmdoesnot selectthecorrect“primary” pad.We arecurrentlyinvestigat-

ing the use of CT data to solve this problem, and hope to describe a solution in a future work.

A third problemaddressedby theCTs is relatedto theassumptionthat theunity currentratio line (the1-line or

centerfor thehyperbolas)is positionedmidwaybetweentheC4s.This is only truefor simplegrids(suchastheQuad

shown in Figure4) if theRp areequal.If theRp arenot equal,Eq.10 canbeusedto derive theoffset,c’, of the1-line

by settingβ02 to 1 and simplifying.

A similar shift occursin morecomplex grids, suchas that shown in Figure15, but for a reasonrelatedto the

degreeof symmetryin theC4ssurroundinga region. For example,thebottomportionof thegrid in Figure15 con-

tainsa row of threeC4s,C40, C42 andC44. The1-line in the lower left Quadis shown skewedto theright from the

midpointgivenby L/2. Theasymmetryin theC4ssurroundingthis region,e.g.C4s0, 1 and6 ontheleft andC4s2-5,

7 and8 on theright, areresponsiblefor this shift. We arecurrentlyevaluatingmorecomplex circuit modelssuchas

the oneshown in Figure5 asa meansof formulatingan expressionthat accountsfor this shift. Experimentally, we

determined that Eq. 12 yields a good approximation of the offset,c’, of 1-lines for Quads within the PPG.

4.2 Leakage Current

Oneelementthat we haven’t addressedis the impactof leakagecurrent.A secondcalibrationmethodwaspro-

posedin previouswork to dealwith leakage[5]. Althoughcalibratingfor leakageis clearlyanimportantissue,wedo

not focus on it in this work becauseof spacelimitations. The limited numberof experimentsconductedthus far

involving leakageindicatethat it hasonly a small impacton the accuracy of the predictions.The sameis true for

experimentsconductedusingdifferentvaluesof defectcurrent.Currentratiosarenaturallyrobust to thesevariables

but a quantitative analysis of their impact remains to be determined and will be addressed in a future work.

4.3 The QSA Procedure

Theprocedureto localizeadefectfollows from thediscussiongivenin theprevioussection.Onceachip is identi-

c′ L2--- β02 CT0( )β02 CT2( )( )= (12)

17

fied asdefective,e.g.from a Stuck-Ator IDDQ go-nogotest,thefollowing testsareperformedundertheQSA proce-

dure.

• First, thechip is setto a statethatprovokesthedefectandtheindividual IDDQ valuesaremeasured.TheC4 pad,

j, sourcingthelargestIDDQ andtwo orthogonallyadjacentC4pads,x andy, areidentifiedasdescribedin Section

4.1. The current ratiosβjx andβjy are computed.

• Thechip is thenput into a statethatdoesn’t provoke thedefect.TheCT for the jth padis turnedon andthecur-

rent ratiosβjx(CTj) andβjy(CTj) are computed.

• Similarly, the currentratios βjx(CTx) and βjy(CTy) are computedfrom measurementsmadewith CTx and CTy

turnedon. Eq. 12 givestheoffsetsneededto derive the two centersof thehyperbolas,(h’,k)x and(h,k’)y, along

the x- and y-dimension, respectively, from padj.

• Eq. 10 is then used to deriveax anday parameters using Lx = 2*c’x and Ly = 2*c’y for L.

• Thebx andby parameters are computed using Eq. 11.

• Thesetwo pairsof a andb parametersdefineboththepositionandshapeof onehyperbolafrom eachof thetwo

families,e.g.asillustratedin Figure16 usingthehyperbolacurvesfrom Figures10 and11. The intersectionof

these two hyperbolae gives the predicted location of the defect.

The algorithm,asstated,requiresa changein the stateof the CUT after the first setof IDDQ measurementsare

made.Therefore,the contribution of leakageto the currentsmeasuredwith the CTs turnedon is different thanthe

contribution underthe statewith the shortingdefectprovoked. The vector-to-vector leakagevariation is likely to

adverselyaffect theaccuracy of thepredictions.An alternative testprocedurethatdoesnot changetheCUT’s stateis

to performtheCT testswith thedefectprovoked.ThecurrentsmeasuredundertheCT testscanbe“adjusted”by sub-

tractingthe currentsmeasuredunderthe defectprovoking test.Even thoughthe presenceof the defect’s currentis

likely to changetheequivalentresistancesof theCUT undertheCT tests,we expecttheerrorintroducedby this type

of procedureto besmallerthantheerror introducedundertestscenariosin which thevector-to-vectorleakagevaria-

tion is large.

Figure 16. Example prediction using the hyperbola curves from Figures 10 and 11.

18

5.0 Experimental Results

This algorithmwasappliedto thedataobtainedfrom 200SPICEsimulationsof the30,000by 30,000unit region

of thePPGreferredto asQ9 in Figure3. A threedimensionalerrormapplotting thepredictionerroragainstthe(x,y)

coordinateof theinserteddefect(modeledusingacurrentsource)is shown in Figure17.Thepredictionerroris com-

putedasthedirecteddistancebetweenthepredictedlocationandtheactuallocationof thedefect.Theaverageand

worst case prediction errors are 215 and 650 units, respectively.

Thesizeof thesimulationmodelfor theentirePPGshown in Figure3 madeit impossibleto performSPICEsim-

ulationson it. Instead,thePPGwassimulatedusinga specializedpower grid simulationenginecalledALSIM. The

predictionerror map from 500 ALSIM simulationsis shown in Figure18. The averageandworst caseprediction

errorsare410 and1,340units, respectively. The increasein predictionerror is largely dueto the moresignificant

anomalies in the grid’s structure over the larger region defined by the entire PPG.

6.0 Conclusions

The weaknessesof our previously derived resistance-basedQuiescentSignalAnalysismodelareaddressedin a

new current-ratio-basedtechnique.Calibrationtransistorsareproposedto reducethe adverseeffectsof probecard

resistancevariationsonthepredictionaccuracy of thenew QSAtechnique.Thecalibrationtransistordatais alsoused

to accountfor power grid resistancevariationsfrom oneregion to the next andasymmetricalor irregular arrange-

ments in the positions of the power supply pads.

Thecurrentratio contoursderived throughSPICEsimulationsof a commercialpower grid areshown to bewell

approximatedby “f amilies” of hyperbolacurves.An analyticalframework is derivedthatallows themeasuredIDDQ

data to be translated to physical (x,y) layout coordinates, that represent the position of the defect.

Although the analyticalmodel that we presentin this work accountsfor testerenvironmentvariablessuchas

probecardresistancevariations,thesimulationdatawasderivedfrom a simplermodel.For example,theprobecard

resistancewasheldconstantat1Ω ateverysupplypad,20mAwasusedfor defectcurrents,andleakagecurrentswere

not included.As pointedout,currentratiosarenaturallyrobustto variationsin defectcurrentandthecalibrationtran-

sistorsin combinationwith regressionanalysisare expectedto be effective in dealingwith leakages.Simulation

Figure 18. Prediction error map for the entire PPG.

Figure 17. Prediction error map of the Q9.

19

experiments are currently underway to verify these hypotheses.

Thelast issuethat remainsto beexploredis theeffectivenessof this techniqueon othertypesof grid topologies.

For significantly irregular grids, we expecta lookup-tableapproachto be moreaccuratethan the hyperbola-based

technique.We areinvestigating theuseof power grid simulatorssuchasALSIM asa meansof makingthis typeof

approach practical.

Acknowledgments

We thank Dr. Anne Gattiker and Dr. Sani Nassif at IBM Austin Research Lab for their support of this research.

References

[1] T.W.Williams, R.H.Dennard,R.Kapur,M.R.Mercer& W.Maly, “I DDQ test:SensitivityAnalysisof Scaling”, ITC, 1996,

pp.786-792.

[2] A.E.Gattiker and W.Maly, “Current Signatures”, VTS, 1996, pp.112-117.

[3] C. Thibeault, “On the Comparison of Delta IDDQ and IDDQ Test”, VTS, 1999, pp. 143-150.

[4] PeterMaxwell, PeteO’Neill, RobAitken, RolandDudley,NealJaarsma,Minh Quach,Don Wiseman,"CurrentRatios:A

self-Scaling Technique for Production IDDQ Testing", ITC, 1999, pp.738-746.

[5] Jim Plusquellic,“IC DiagnosisUsing Multiple SupplyPadIDDQs”, DesignandTest,Vol. 18, No. 1, Jan/Feb2001,pp.

50-61.

[6] ChintanPatelandJim Plusquellic,“A ProcessandTechnology-TolerantIDDQ Methodfor IC Diagnosis”,VTS, 2001,pp.

145-150.

[7] C.Patel,E.Staroswiecki,D. Acharyya,S.PawarandJ.Plusquellic,”A CurrentRatioModel for DefectDiagnosisusingQui-

escent Signal Analysis”, Workshop on Defect Based Testing, VTS, 2002.

[8] S. Nassif and J. Kozhaya,”Fast Power Grid Simulation”, DAC, 2000, pp. 156-161.

[9] ChristopherL.Hendersonand Jerry M.Soden,“SignatureAnalysis for IC Diagnosisand Failure Analysis”, ITC, 1997,

pp.310-318.

[10] C. Thibeault,L. Boisvert, “Diagnosismethodbasedon delta IDDQ probabilisticsignatures:Experimentalresults”, ITC,

1998, pp.1019-1026.

[11] MOSIS at http://www.mosis.edu/Technical/Testdata/tsmc-025-prm.html.

[12] ChintanPatel,ErnestoStaroswiecki,DhurvaAcharyya,SmitaPawar,andJimPlusquellic,“DiagnosisusingQuiescentSignal

AnalysisonaCommercialPowerGrid”, Acceptedfor publicationin InternationalSymposiumfor TestandFailureAnalysis,

2002.

Figure Captions

Figure 1: Equivalent resistance model of the power grid with a shorting defect.

Figure 2: Triangulation under r esistance model.

Figure 3: Layout of the PPG.

Figure 4: Layout details of the PPG.

Figure 5: Equivalent resistance model of the Quad.

Figure 6: Network variable plots for sources along x-slice (top) and y-slice (bottom) lines of Figure 4(a).

Figure 7: Req0contours of Quad.

Figure 8: I0 contours of Quad.

Figure 9: I0/I1 contours of Quad.

Figure 10: I0/I1 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.

Figure 11: I0/I2 contours (jagged) and hyperbolas (smooth curves) for lower left quadrant of Quad.

Figure 12: Definitions of the hyperbola parameters.

Figure 13: Lumped R model for the hyperbola a parameter.

Figure 14: Calibration Transistor and controlling scan-chain FF.

Figure 15: Anomalies in complex grids.

Figure 16: Example prediction using the hyperbola curves from Figures 10 and 11.

Figure 17: Prediction error map of the Q9.

Figure 18: Prediction error map for the entire PPG.

21

defect

Pad1

Req0

Req1Req3

Req2

Rp3

Rp2

Rp1

VDD

Supply Ring

Rdef

I1

I0

I3

I2

Vdeflocation

Rp0

on probe cardPad3

Pad0 Pad2

Figure 1

22

Pad1

ActualDefectLocation

DefectLocation

Predicted

scaled by m

Pad3

Pad2Pad0

Reqa

Reqb

Reqj

Figure 2

23

Quad Q9 C4 VDD Pads

Figure 3

24

Figure 4

VDD1

C4Pads

y slice

x slice

irregular region

(a)

M5

M5

VDD0 VDD2

VDD3Gnd2 Gnd5

Gnd1

Gnd0

Gnd4

Gnd3

Vdd

Gnd Rails

M1

M2

M3

M4

Contacts

(b)

B

A

Rails

Currentsources

+

+

+

+

+

+

+

++

+

+

+

25

Figure 5

VDD1

VDD0 VDD2

VDD3

Defect Current

GND3

GND5GND2

GND1

GND0

VDD Grid

GND Grid

Rp3

Rp2

Rp0

Rp1

Rpg5

Rpg4

Rpg3Rpg0

Rpg2

GND4

Rz

Rxy

+- +

-

+-

+-

+-

+-

+-

+-

+-

+-

Req0

Req1 Req3

Req2

Rpg1

C4s

Source: 20mA

R01

26

Figure 6

I0

Req0

10

00

0

50

00

Oh

ms

mA

mV

2.0

9.0

6.6

4.22.3

Vdef

0

2.4

10

00

0

50

00

Oh

ms

mA

mV

2.5

8.5

6.5

4.72.4

0

2.4

I0

Req0

Vdef

x-coordinate

y-coordinate

27

Figure 7

Mon Aug 5 01:04:20 2002

1900.0 2840.0 3780.0 4720.0 5660.0 6600.0 7540.0 8480.0 9420.0 10360.0 11300.0 12240.0 13180.0 14120.0 15060.0 16000.0 16940.0 17880.0 18820.0 19760.0 20700.01.50

2.24

2.98

3.72

4.46

5.20

5.94

6.68

7.42

8.16

8.90

9.64

10.38

11.12

11.86

12.60

Kili13.34

14.08

14.82

15.56

16.30

17.04

17.78

18.52

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20.74

21.48

22.22

22.96Kili

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Figure 8

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4.46

5.20

5.94

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7.42

8.16

8.90

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Kili13.34

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22.96Kili

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irregularregion

29

Figure 9

Mon Aug 5 01:13:52 2002

1900.0 2840.0 3780.0 4720.0 5660.0 6600.0 7540.0 8480.0 9420.0 10360.0 11300.0 12240.0 13180.0 14120.0 15060.0 16000.0 16940.0 17880.0 18820.0 19760.0 20700.01.50

2.24

2.98

3.72

4.46

5.20

5.94

6.68

7.42

8.16

8.90

9.64

10.38

11.12

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12.60

Kili13.34

14.08

14.82

15.56

16.30

17.04

17.78

18.52

19.26

20.00

20.74

21.48

22.22

22.96Kili

23.70

24.44

25.18

25.92

26.66

27.40

28.14

28.88

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30.36

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C40 C42

C41 C43irregularregion

30

Figure 10

Mon Aug 19 14:44:08 2002

2016.4 2507.6 2998.8 3490.1 3981.3 4472.5 4963.7 5454.9 5946.1 6437.3 6928.5 7419.7 7910.9 8402.1 8893.3 9384.5 9875.7 10366.9 10858.1 11349.3 11840.51687.09

2054.41

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31

Figure 11

Tue Aug 13 19:49:26 2002

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32

Figure 12

a

bF1 F2

cC40

C42

Horizontal Hyperbolas

50000 10000

center

(h, k)

33

Figure 13

Rp0 I0

VDD0+-

Req0

+- Vdef

+-

Req2

I2

VDD2

m

L

where,

L/2

a

C42

ReqT = Req0 + Req2

C40

Rp2

34

Figure 14

CalibrationGate controlledusing a scan-flop

C4

Transistor

VDDGND

35

Figure 15

C41

Defect’s

C40

C46

C43

C42

C47

C45

C44

C48

position

Reqa Reqb

L/2

1-line

36

Figure 16

Mon Aug 19 19:34:20 2002

2100.0 2520.0 2940.0 3360.0 3780.0 4200.0 4620.0 5040.0 5460.0 5880.0 6300.0 6720.0 7140.0 7560.0 7980.0 8400.0 8820.0 9240.0 9660.0 10080.0 10500.01900.00

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Figure 17

X

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dic

tion

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or

Y

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Figure 18

Pre

dic

tion

Err

or

X

Y