Journal of Computational Physics 152,zhao/publication/mypapers/pdf/void.pdfState University,...

Journal�

of ComputationalPhysics152,281–304(1999)

Article�

ID jcph.1999.6249,availableonlineathttp://www.idealibrary.comon

Zhilin Li,��

1 HongkaiZhao,�� 2 and� HuajianGao

CenterFor Research in ScientificComputationandDepartmentof Mathematics,NorthCarolinaState�

University, Raleigh,NorthCarolina 27695-8205;� Departmentof Mathematics,StanfordUniversity, Stanford, California 94305;and Department

�of MechanicalEngineering,

Stanfor�

d University, Stanford, California 94305-4040E-mail:� �

[email protected],� [email protected],and� [email protected]

Received October23,1998;revisedFebruary24,1999

A numericalmethodfor studyingmigration of voids driven by surfacediffu-sionandelectriccurrentin a metalconductingline is developed.Themathematicalmodelinvolvesmoving boundariesgovernedby a fourthordernonlinearpartialdif-ferential equationwhich containsa nonlocalterm correspondingto the electricalfield anda nonlineartermcorrespondingto thecurvature.Numericalchallengesin-cludeefficient computationof theelectricalfield with sufficient accuracy to affordfourthorderdifferentiationalongthevoid boundaryandto capturesingularitiesaris-ing in topologicalchanges.We usethe modifiedimmersedinterfacemethodwitha fixed Cartesiangrid to solve for the electricalfield, and the fast local level setmethodto updatethepositionof moving voids.Numericalexamplesareperformedto demonstratethe physical mechanismsby which voids interact under electro-migration. c�� 1999AcademicPress

1. INTRODUCTION

In many physicalproblems,masstransportalong interfacessuchassurfacediffusionand� grainboundarydiffusionbecomesincreasinglyimportantasthecharacteristiclengthscale� is reduced.Thediffusionalmasstransportisgovernedbyarelevantchemicalpotentialalong� the interface.Converging or diverging atomicfluxescausemotion or relocationofboundaries.�

Thedynamicsof theseprocessesis of greatinterestto materialscientistsand

1 Theauthoris partially supportedby theNSFunderGrantDMS-96-26703andNorth CarolinaStateFR&PDFund.

2 The�

authoris partially supportedby theNSFunderGrantDMS-97-06566.

281

0021-9991/99�

$30.00Cop�

yright c�� 1999by AcademicPressAll rightsof reproductionin any form reserved.

282 LI, ZHAO, AND GAO

biologists.�

There is a large literatureon this topic. We refer the readersto two recentsurv� ey articlesby Mullins [14] andCahnandTaylor [5] andthe referencestherein.Theproblem� consideredin this paperinvolvesthe evolution of voids underelectro-migrationin�

a conductingmetal line wherethe driving forcesfor diffusion are the gradientof thecurv� atureandthe electricpotentialalongthe void boundary. The normalvelocity of thev� oid surfaceis given by thepartialdifferentialequation(PDE)

Un !#" s$ % C1 &(' C2) *,+.- (1.1)

/where0 1

s$ is�

thesurfaceLaplacian,2 is�

thepotentialfunctionassociatedwith anappliedelectric3 field, and 4 is themeancurvaturealongtheboundary;for acircle, thecurvatureisa� positiveconstant.ThecoefficientsC1 5 C2

) are� relatedto thephysicalconstants

C1 6 eD7 s$ 8 19 3: Z; <k=

BTk> ? C2

) @ DA

s$ B 4C D

3: E

s$k=

BTk> F (1.2)

/

where0 e7 is the charge of an electron,G is the atomicvolume,Z H is a phenomenologicalconstant� relatedto theeffective valenceof anatom,k

=BI is�

Boltzmann’s constant,TJ

k> is�

thetemperature,K L

s$ is thesurfaceenergy. TheconstantDs$ is definedas

Ds$ M D Ns$ O s$k=

BI TJ

k> e7 P Qs Q k> BTk

R S(1.3)/

where0 Ts$ is thethicknessof thediffusionlayer, D Us$ e7 V Qs W k> BTk

Ris thesurfacediffusioncoeffi-

cient,� andQX

s$ is�

theactivationenergy for surfacediffusion.TheelectricpotentialY satisfies�theK

LaplaceequationZ\[^] 0,_

with no-fluxboundarycondition `ba` n c 0_

on thevoidboundaryas� well asotherappropriateboundaryconditionson thecomputationalboundary.

For a void boundedby a closedsurface,it canbeshown from thedivergencetheoremthatK

thevoid conservesits volume(or area)duringsurfacediffusion.In Eq. (1.1), thefirstterm,K d

s$ e ,f is a nonlocaldriving forcewhich tendsto drift thevoid alongwith theelectriccurrent.� Thesecondterm, g s$ h ,f isa fourthordernonlineartermwhichonly dependsonthelocal geometry. The boundaryevolution governedby the surfaceLaplacianof the meancurv� ature,asstudiedby numerousauthors[3–5, 14], canbe regardedasa gradientflowwith0 H i 1 innerproductfor thesurfaceareagiven by theequation

d Ajdtj k

Sl Un m ds

j nSl oqp s$ r ds

j s#tSl u v s$ wqx 2) ds

j y(1.4)/

Suchz

aprocesstendsto minimizethesurfaceareawhile conservingthevolume.Anisotropycan� be includedin both the freeenergy andtheabove innerproductin a variationalform[5]. Anothertypeof gradientflow which minimizesthesurfaceareawhile conservingthev� olume with L2

)inner product is {}| ¯~ ,f where ¯� is the averageof the meancurvature

along� the interface.Thesetwo typesof gradientflows correspondto two limiting casesof� combinedsurfacediffusion and growth process.The connectionbetweensharpanddif�

fuseinterfacemotion laws via a gradientflow canbefoundin [19]. In general,surfacedif�

fusionproblemsadmit few analyticalresults.Dueto theintrinsic nonlinearityandlackof� a maximumprinciple, smoothsolutionsonly exist locally in time while topologicalsingularities� occur in finite time. Surfaces(curves) can merge or pinch-off in both twoand� threedimensions.A linearstability analysis[15, 18] indicatesRayleighinstability at

ELECTRO-MIGRATION 283

longwavelengthperturbations.It hasbeenshown usingaperturbationanalysis[3] thattheonly� stableequilibria aresurfaceswith constantmeancurvaturewhich locally minimizetheK

surfacearea.The dynamicsand stability of self-similar pinch-off were also studiedin�

[3]. For a closedplanecurve, suppose� x� ��q� t� �.� y� �� t� �b� is�

someparametrizationof thecurv� e. Definethemetric g� ��q� t� �� x� 2�� y� 2� and� angle �� tan

K 1 ¡ y� ¢¤£ x� ¥§¦ . Following thederi�

vationgiven in [16], wehave

g� t¨ ©�ªq« t� ¬q¯®±° Un ² ³ t¨ ´�µq¶ t� ·q¸º¹ 1

g�»Un ¼�½¿¾

where0 Un is�

thenormalvelocity of thecurve. Usingthefact ÀÂÁÄÃ s$ ÅÇÆ±È¤É g� and� ÊÊ s$ Ë 1gÌ ÍÍbÎ ,f

where0 sÏ is thearclength,wegetthetimeevolutionequationbelow for thecurvatureÐ with0respectÑ to thearclengthsÏ ,f

Òt¨ Ó#Ô}Õ 2Un Ö

sÏ 2 ×ÙØ 2Un ÚIf Un ÛÇÜ ss$ ,f then

Ýt¨ Þºßáà 2

) âss$ ãÙä ssss$ å (1.5)

/Wæ

e canseefrom theevolution equationthat thefirst term, ç�è 2 éss$ ,f is a secondorderand

nonlinearê termcorrespondingto a backwardheatequationwhich causesinstability, whiletheK

secondterm, ë�ì ssss$ ,f isa linearfourthorderstabilizingterm.Throughsimplelinearizedanalysis� wecanseethelongwaveinstabilities.If wemultiply theequationby í and� integrateit with respectto sÏ ,f afterintegrationby partswehave theenergy equation

djdtj L

î0� 1

2ï 2 dsj ð

3ñ L

î0� ò 2 ó 2

s$ dsj ô L

î0� õ 2

ss$ dsj ö

where0 L is thetotal lengthof thecurve.A similar typeof equation,discussedin [7], showsthatK

aclosedcurvewill convergetoacircleonly if its initial shapedoesnotdeviatetoomuchfrom÷

a circle; otherwisesingularitiesmay occur in finite time. This phenomenonis alsodiscussed�

througha modifiedKuramoto–Sivashinsky equationfor nearlyplanarinterfacemotionø for phasetransitions[2].

In thispaperweshalladdresssomedifficultiesin thenumericalimplementationof mov-ing�

boundaryproblemsarisingin electro-migrationvoiding.Theseinclude(a)constructingan� efficient andaccuratePoissonsolver for the equationsdefinedon arbitrarydomains,(b)/

trackingtopologicalchanges(breaking,merging) alonga moving interface,(c) evalu-ating� thesurfaceLaplacianoperatoralongtheinterface.Wewish to explorethepossibilityof� simulatingmoving interfacesonafixed Cartesiangrid withouthaving to re-meshastheinterfacesmigrate.For thispurpose,weintroducethemodifiedimmersed-interface-method(IIM)/

developedin [10,11] tosolvethePoissonequationfor theelectricpotential.WeshowthatK

thesystemcanbepreconditionedsothattheconvergenceis almostindependentof themeshø size.Weusethelocal level setmethodto updatetheinterfaceaccordingto Eq.(1.1).Wæ

ewill deriveageneralformulafor thesurfaceLaplacianoperatorin Cartesiancoordinatesalong� the interfacerepresentedimplicitly by a level setfunction.As discussedin [5], thelevel setfunctionfor a geometricPDEcannotbearbitrarily chosen.SinceeachcontouroftheK

level setfunctionmoves accordingto its own curvaturevariation,andbecauseof the


lackof amaximumprinciple,differentcontoursmaycrosseachother. Weadoptamodifiedleù

vel set function method[22] which allows us to get aroundthis problemby (i) usingtheK

local level setmethodso thatonly thoselevel setsthatarevery closeto the interfaceare� involved;(ii) stickingto thesigneddistancefunctionasour uniquechoiceof level setfunction;and(iii) extendingthequantitiesat theinterface,if necessary, to otherlevel setsso� that they arenormalto the interface.It hasbeenshown [22] that this extensionof thenormalvelocitymaintainsthenormaldistancebetweendifferentcontours.After thesemod-ifications,�

thelevel setmethodcanbeusedefficiently to capturemoving interfacesfor theeletro-migration3 problemsdiscussedhere.An alternativeapproachwasproposedin [9, 20]usingú the finite-elementformulationwith adaptive re-meshing.The numericalalgorithmthatK

weproposein thispaperhastheadvantageof simulatingamoving interfaceonafixedCartesianû

grid, which demandsa relatively smallamountof computationalresourcesandmaybedesirableasapracticalanalysistool for many applications.

2.ü

OUTLINE OF THE NUMERICAL ALGORITHM

Wæ

e considera moving boundarywith the normal velocity given by surfacediffusionunderú a linearcombinationof electricalpotentialandsurfacetension,

Un ýºþ s$ ÿ� 2)�sÏ 2) � C1 �� C2

) �� (2.6)/

where0 � is theelectricalpotentialwhichis asolutionof theLaplaceequationonthedomainoutside� of voids (seeFig. 1), � is

�the curvaturealong the boundaryof voids which are

immersedin afixed Cartesiangrid over thecomputationaldomain.Belo

w is an outline of our numericalalgorithm along with referenceto the specificsections� wheremoredetailsaregiven.

Outline�

of theMethod

� From�

the level set function at a time level t� n ,f find necessaryinterfaceinformationsuch� asthe normalandtangentialdirections,the projectionsof irregular grid points,thecurv� atures,etc.� Use

�theIIM to solve theelectricpotentialfunction;seeSection3.� Find

�thecomponentof thenormalvelocityof thesurfacedueto theelectricpotential;

see� Section4.1.� Find�

thecomponentof thenormalvelocityof thesurfacedueto thecurvaturevariation;see� Section4.2.� Find

�thenormalvelocityof thelevel setfunction.� Update

�thelevel setfunction.� Reinitialize

�thelevel setfunction.� Go

�to thenext time level.

3.�

A FAST ALGORITHM FOR COMPUTING THE ELECTRICAL

POTENTIAL ON IRREGULAR DOMAINS

Wæ

eneedto solve thePoissonequation

�� f! "

x� # y� $% & x� ' y� (*),+ 1 - (3.7)/


FIG..

1. A�

sketchof a computationaldomain.The regionsof / 2 arevoids.We want to track themotionoftheir0

boundaries.

1321n4 576

2

8:9<; sÏ => (3.8)/

where0 sÏ is�

thearclengthof theinterface?A@ 2) . Thesolutionis definedonamulti-connected

domain�

with given boundaryconditionson BAC 1 (see/

Fig. 1). Thereis alsoa compatibilitycondition� if theboundaryconditionsareof Neumanntype.In ourelectro-migrationproblem,DFE sÏ GAH 0.

_InI

recentyears,thelevel setmethod,first proposedin [16], hasbecomea powerful toolfor computingmoving boundary/interfaceproblems,especiallyfor problemsthat involvetopologicalK

changesand/orin threedimensions.Weintendtopresentasimple,secondorderdiscretization�

methodfor solving(3.7)with theinterfacedescribedby a level setfunction.ThereJ

areavarietyof methodsbasedonfinitedifferencefor thiskindof problems,for exam-ple,� thecapacitancematrixmethodandfastmethodsbasedonintegralequations.However,manø y of thesemethodsrequireexplicit informationabouttheboundaryof exclusions,suchas� splineinterpolationsor Fourierexpansionsfor closedregions.It is not clearhow to im-plement� thesemethodsif theboundaryis describedby atwo dimensionallevel setfunctionbecause�

theinformationis given only atgrid points.Ourdiscretizationmethodis basedontheK

immersedinterfacemethod[12, 21], andis designedfor treatingNeumannboundaryconditions� underthelevel setformulationwith anefficientpreconditioningtechnique.

Totakeadvantageof afastPoissonsolver, weextendtheequationto theentirerectangularreÑ gion

K�LNMf! O

x� P y� QR S x� T y� U*V,W 1 XZY 2) [ (3.9)

/\A]_^`n4 acb

2

dfeFg sÏ hi (3.10)/

where0 f! j

x� k y� l is definedaszeroin m 2. We usea n sign� to indicatethelimit of a functionwhen0 approachingfrom theexterior of o 2

) . Thesolution p inside� q

2) depends�

on how the


jumpr

conditionsalongtheinterfacearespecified.If a singlelayer is introduced,thereis ajumpr

in thenormalderivative. If adoublelayeris introduced,thefunctionis discontinuousacross� theinterface.Usually, wecannotrequirethefunctionandits normalderivative to becontinuous� acrosstheinterfaceat thesametime.Following theideaintroducedin [12, 21],we0 eitherintroduceanunknown jump in thesolutionandrequirethenormalderivative tobe�

continuousor vice versa.However, to force thesolutionto becontinuousmay leadtorapidchangesin thefunction,andanill-conditionedsystemfor theunknown jump in thederi�

vative.Nos

w considerthefunctional tvu g� w ,fx�y{z

f! |

x� } y� ~� � x� � y� �*�,� 1 �� 2) �

(3.11)/

[ � ]� � �7�

2 � sÏ �� g� � sÏ �� [ � n ]� � �c� 2 � 0_ �

where0 thejumpacross�3� 2) is�

thedifferencebetweenthelimiting valuefrom theexterioroftheK

voidsandthatfrom theinterior. Sincethejumpconditionsalwaysreferto theinterface�A�2) ,f wewill omit theexplicit referenceto A¡ 2

) from÷

now on.Thesolution¢¤£ g� ¥ of� thesystemabo� ve is a functionalof the jump g� ¦ sÏ § with0 [u¨ ]� © g� ª sÏ « . We areinterestedin theparticularg� ¬ sÏ such� that ®A¯v° g� ±7²´³ n4 µ:¶F· sÏ ¸ . Thiscorrespondsto adoublelayerin thepotentialtheory.

3.1.ñ

DiscrA

etizationof theInterface

In order to constructa fast and convenientnumericalalgorithm for moving interfaceproblems,� we adopta uniform Cartesiangrid with a fastPoissonsolver andusethe levelset� methodto updatetheinterfaceateachtimestep.

TheJ

gridpointsaredividedinto twocategories:r¹ egulargridº pointsarethoselocatedawayfrom theinterfaceandirr

»egular ones� arethoselocatedwheretheinterfacecutsthroughthe

standard� fivepoint stencil.Ourattentionis focusedon theirregulargrid points.A two dimensionallevel setfunction ¼¾½ x� ¿ y� À ,f where Á¾Â x� Ã y� ÄAÅ 0

_describesthe location

of� theinterface,is introduced.In thelevel setformulation,theinterfaceinformationis onlygiº ven at grid points.To solve for theunknown jump functiong� Æ sÏ Ç so� that thethesolutionsatisfies� theNeumannboundarycondition,weneedtodiscretizeg� È sÏ É as� well. In otherwords,we0 needto find valuesof g� Ê sÏ Ë at� discretepointson the interface.Too many pointsoftenleadù

to a largeandill-conditionedsystemwhich alsomeansmorestorage.Too few pointsoften� leadto lossof accuracy. Ourstrategy is to find theprojectionsof irregulargrid pointson� the interface.Let X

Ì ÍÏÎx� i Ð y� jÑ Ò be�

an irregular grid point. Its projectionsXÌ ÓÕÔ×Ö

x� ØÚÙ y� ÛÝÜcan� befoundusingthefollowing procedure(seealsoFig. 2):

1. Find theunit steepestascentdirectionpÞ at� X:

pÞ ß àâáãåäâæ¾ç<è (3.12)/

2. Locatetheprojectionof X on� theinterfacealongthedirection pé ,fXÌ ê¾ë

XÌ ìîí

pÞ ï (3.13)/

where0 ð is determinedfrom thequadraticequation

ñ¾ò XÌ óõô÷öåøâù¾úÝûýü 12 þ pÞ T He

ÿ ��pÞ �� 2

) �0_

(3.14)


FIG..

2. Finding

thecontrolpoint X �

from anirregulargrid point � x i � y� j� � , �� x i � y� j

� �� 0. It canbechosenasthe0

projectionof thegrid pointon theinterface.

and� He�� is theHessianmatrixof � ,f

Heÿ �� "! # xx$ % xy$&

yx' ( yy' ) (3.15)/

e3 valuatedatX.

Wæ

eonlydefineg� * sÏ + on� thoseprojectionsfromaparticularside,for example,,.- x� / y� 021 0._

Wæ

ewill call themSet3

I4 . Theprojectionsfrom theothersideof theinterfacearecalledSet

3II4 .

Thevaluesof g� 5 sÏ 6 on� thoseprojectionsin Set3

II can� bedefinedvia interpolationusingthev� aluesdefinedat thosein Set

3I4 . Theinterpolationschemewill beexplainedlater. Notethat

we0 do not needto or7 der theK

projectionswherethe unknown jumps are defined,a veryimportant�

featureof thelevel setformulationcomparedwith theLagrangianformulation.

3.2.ñ

DiscrA

etizationof thePoissonEquationwith Jumps

Gi�

ven jumpconditionsacrosstheinterface8:9 2) ,f

[ ; ]� <

g� = sÏ >@? [ A n ]� B 0_ C

[ f!

]� D

f! EGF

sÏ H (3.16)/

at� all projections,we usethe immersedinterfacemethodto solve the Poissonequation.Theessenceof thismethod[10–12]consistsof afinite differenceschemewith thestandarddiscrete�

Laplacianplus a correctionterm at the right handsideassociatedwith irregulargridº points, leadingto a discretesystemwhich can be solved by a fast Poissonsolver.TheJ

detailscanbefoundin [10–12].Theinterfaceinformationsuchasthetangentialand


FIG..

3. A�

diagramof thelocalcoordinates.TheinterfacecanbeexpressedasIKJ�L�M with NPO 0Q�R 0andS�T U 0V�W 0.

normalê derivatives andthecurvatureat projectionsis obtainedfrom thevaluesof thelevelset� functionat grid pointsplusa bilinear interpolation;see[12] for details.A key part ise3 valuatingthefirst andsecondderivatives of thejump [ X ]

�alongtheinterface.

EvaluationY

of thederivativesof thejumpfunction[ Z ]�.[ TheJ

ideais to usetheweightedleastsquaresinterpolation[12] toapproximatethederivativesof thejumpalongtheinterfaceat� agivenprojectionX

Ì \^]`_x� acb y� dPe . Weusethelocalcoordinatesat f x� g�h y� iPj in

�thetangential

and� normaldirections(seeFig. 3),

kmlonx� p x� qPr cos� sutwv y� x y� yPz sin� {�|

(3.17)/}m~o�� x� � x� �P� sin� �u�� y� � y� �� cos� ��

where0 � is�

the anglebetweenthe x� -axis andthe normaldirection,pointing to the regionof� �

1.InI

orderto usetheweightedleastsquaresinterpolation,wetakeasmallcircle � centered�at� � x� �� y� �P� with0 aradius� ; usually� isbetween1 � 5� h

�and� 4h

�so� thatatleastthreeprojections

from÷

Set3

I4 are� enclosed.If we canfind thesignedarclengthstartingfrom � x� �� y� P¡ , wf ecan

useú anappropriateinterpolationschemetoapproximatethederivativesof thejumpfunctionwith0 respectto thearclength.Near ¢ x� £�¤ y� ¥P¦ ,f theinterfacehastheform

§^¨�©:ª¬«C 2 ® D

A ¯ 3: °

O� ±�² 4 ³µ´ (3.18)

/

Let X ¶1 ·`¸ x� ¹1 º y� »1 ¼ be�

a projectionin Set3

I b�ut differentfrom ½ x� ¾�¿ y� ÀPÁ . We candetermine

theK

constantsC and� DA

usingú theinterfaceinformation.Denote Â n4 ÃÅÄ n4 Æ�Ç as� theunit normal


direction�

of theinterface;wehave

2C È 1 É 3ñ

D Ê 21

1 Ë 2C Ì 1 Í 3ñ

D Î 2)1

2) ÏÑÐ n4 ÒcÓ

(3.19)/

1

1 Ô 2C Õ 1 Ö 3ñ

D × 2)1

2) Ø n4 ÙcÚ

where0 Û�Ü1 ÝPÞ 1 ß are� thecoordinatesunderthetransformation(3.17).Wearriveatthefollowing

linearù

systemof equationsfor C and� DA

:

C à 2)1 á D

A â 3:1 ã�ä 1 å

(3.20)/

2C æ 1 ç 3ñ

D è 21 éÑê n4 ë

n4 ì2íIn otherwords,thecurveisapproximatedbyaHermitesplineinterpolationbetweenX î and�XÌ ï

1. Oncewe have solvedfor C and� DA

,f we have an analyticexpressionfor approximatingtheK

interface.ThearclengthbetweenX ð and� X ñ1 is determinedfrom

òsÏ 1 óõô

ö1

0� 1 ÷�ø 2ù C úuû 3

ñDA ü 2) ý

2)

dj þ2ÿ

(3.21)/

This definiteintegral canbeapproximatedby theSimpsonrule or a Gaussianquadratureformula÷

using the approximateanalyticexpressionof the interface.In this way, the arclengthis evaluatedto third orderaccuracy, which is necessaryfor thesecondorderschemefor÷

thePoissonequation.Notethatthedistancebetweentwopointsontheinterfaceisonlyasecond� orderapproximationtothearclengthandisnotaccurateenoughfor ourrequirement.Finally�

, we needto determinethesignof thearc lengthaccordingto therelative positionbetween�

X � and� X�1,f

sÏ 1 ��sÏ 1 � if � X �1 � X �� t �� 0

_�� sÏ 1 � otherwise� � (3.22)

/

where0 t �

is thetangentialvectoratX � .Wæ

euseuppercaselettersto expressdiscretequantities.For example,G� �

sÏ � is�

thediscreteform of g� � sÏ � . Oncewe have thesignedarc lengthbetweenX � and� otherprojectionsfromSet3

I4 ,f it is easyto interpolateG

� �sÏ � from÷

Set3

I4 toK

obtainG� "!

sÏ #%$ and� G� & &"'

sÏ (%) at� any projectionon� the interface,either in Set

3I or� in Set

3II ,f using the following weightedleast squares

interpolations,

G� *

sÏ +%,�-X .k / Set0

I 1324

k> G� k> 5 XÌ 6k> 798 (3.23)

/

G� :";

sÏ <%=�>X ?k @ Set0

I A3BC

k> G� k> D XÌ Ek> F9G (3.24)

/

G� H H"I

sÏ J%K�LX Mk N Set0

I O3PQ

k> G� k> R XÌ Sk> T9U (3.25)

/


where0 Vk> ,f W k

> ,f and X k> are� thecoefficientsof theinterpolationwhich is determinedfrom the

system�k> dj

k>

k> dj

k> sÏ k>

k> dj

k> s$ 2

k2)

[ Y[Z]\_^a` ]� b 1

0_0_ or�

0_10_ or�

0_0_1

c (3.26)/

where0 dj

k> is�

the distancebetweenXÌ d

k> and� X

Ì eand� f ,f g ,f and h are� the vectorswhose

components� arethecoefficients i k> ,f j k

> ,f and k k> . Theseareunder-determinedsystemswith

theK

samecoefficient matrix of full row-rank. We usethe leastsquaressolutionfrom thepseudo-in� verse.ThesolutionminimizestheEuclideannormamongall possiblesolutions.Wæ

ith theweightedleastsquaresinterpolation,theerrorsin theinterpolationvarysmoothlyand� thosepointscloserto thecenteraregiven moreweightthantheotherpoints.

Wæ

ith the schemediscussedabove for evaluatingthe jump [ l ]�

andits first andsecondderi�

vatives,therestof theimplementationof theIIM methodisstraightforward.Theprocessof� thediscretizationcanbewritten in amatrix–vectorform:

Am npo

BGq r

Fs t

(3.27)/

Hereÿ

Am

is�

thediscreteLaplacianusingthestandardfive point stencil,Bq

is�

thedeviation oftheK

differenceschemedueto the jump in the solution,and F is the sourceterm plus thecorrection� termsat irregulargrid points.

3.3.ñ

DiscrA

etizationof theResidual

Wæ

ewantto find g� u sÏ v such� that wyx g� z satisfies� theNeumannboundarycondition

{}|n ~�� sÏ �9� (3.28)

/F�or an arbitrary[ � ],

�thesolution �� x� � y� � usuallyú doesnot satisfytheequationabove. The

discrete�

differenceof the two sidesis the residual.An iterative schemeis neededuntil(3.28)/

is satisfiedto a givenaccuracy. Taking into accountall of � i j enclosed3 in a circlesurrounding� a projection,the residual,onceagain,canbe computedusing the weightedleastù

squaresinterpolation,

��n � G� � sÏ ��

i j

�i j � i j � C � sÏ �9� (3.29)

/

where0 C � sÏ � is�

avectorof thecorrectiontermsat theprojections.Wereferthereaderto [12]for thedetails.

TheJ

matrix–vector form of suchdiscretizationandthe interpolationcanbe written intermsK

of � and� G�

as�A BmE D

�G� F

sV ¡ (3.30)

/

TheSchurcomplementfor G�

thenK

is

¢DA £

E AY ¤ 1B

q ¥G� ¦

V § EAY ¨ 1F

s(3.31)/

def© ª

F̄ «


Wæ

e usethe GMRESmethodto solve the muchsmallerSchurcomplementsystem.Eachiteration�

involvesonefastPoissonsolverandaninterpolationschemeto evaluatetheresid-ual.ú A preconditioningstrategy is usedto acceleratetheconvergence.Theinitial guesshaslittleù

effect on thenumberof iterations.Sowe simply take theinitial guessaszerovector.Notes

that if the interfaceis a circle, andthedomainis infinite, thenthesolutionis indeedzero¬ (seethediscussionsin thefollowing section).

3.4.ñ

A PreconditioningStrategy

Sincez

the homogeneousNeumanncondition is prescribedon the boundary ¯® 2,f thesolution� in theinteriorof thevoidsis definedonly upto anarbitraryconstant.However, theconditition� numberof theSchurcomplementdependson theconstantsolutionin ° 2. Thissituation� is remediedby settingthesolutionin ± 2

) toK

betheaveragevalueof thesolutionatall� irregulargrid pointsfrom theothersideof theinterfaceateachiteration.

TheJ

preconditioningtechniquesdiscussedabove can be justified using the theory ofintegral equations.Thesolutionof thePoissonequationcanbewritten asthedistributionof� sourcesanddipolesalongtheboundaries,

²�³x́�µ 1

2ù ¶ ·�¸

1

¹1 º sÏ » log ¼ x ½ X ¾ sÏ ¿9À ds

j Á 1

2ù Â ÃÅÄ

2

Æ2) Ç sÏ È�ÉÉ nÊ log Ë x Ì X Í sÏ ÎÐÏ ds

j Ñ(3.32)/

TheJ

sourcestrengthsÒ 1 Ó sÏ Ô and� Õ2) Ö sÏ × are� determinedfrom the boundaryconditionsonØÚÙ

1 and� ÛÚÜ2. SeveralfastPoissonsolversfor problemson irregulardomainsarebasedon

solving� the integral equations.However, for a boundeddomain,thenumberof unknownsusuallyú is morethandoublethat in theapproachdescribedherebecauseof anadditionalunknoú wn sourcestrengthalong ÝÚÞ 1. Note that the sourcestrengthß 2

) à sÏ á corresponds� totheK

jumpof thesolutionacrosstheinterfaceâ¯ã 2. Sincethesolutioninsidethevoidscanbean� y arbitraryconstant,we needto have anadditionalconditionto uniquelydeterminethesolution.� A convenientconditionis to definetheconstantastheaverageof thelimit solutionfrom ä 1 side� approachingto åÚæ 2,f thatis,

C ç è�é2

u¨ êìë X í dsj î

(3.33)/

ThisJ

condition minimizes the jump in the solution, which is equivalent to minimizingtheK

conditionnumberfor theSchurcomplement.If ï¯ð 2 is a singlecircle andtheoverallrectangularÑ domainis very large,thenthesolutionis closeto beingaxi-symmetric,andwewill0 obtaintheconvergencein acoupleof iterations.

Ourñ

numericalexamplesindicatethat the numberof iterationsfor solving the Schurcomplement� is smallandis independentof thegrid size.It turnsout this is alsotruefor thecases� whereV is

�not zero;seetheexamplein Section3.5.

In practice,thematricesandvectorsin (3.30)and(3.31)arenever formedexplicitly. Thealgorithm� is outlinedbelow:

Outline�

of thealgorithmto evaluatetheelectricalpotential

1. Initialization: Settingup thegrid andtheboundaryconditionson ò¯ó 1; indexing thegridº pointsasregularor irregular;obtainingprojectionsof irregulargrid points;evaluatingtheK

normaldirectionandthecurvatureinformationof the interfaceôÚõ 2 at� theprojectionsof� irregulargrid points.It is affordableto storethesedatasincethenumberof irregulargridpoints� is roughlythesquarerootof thetotal numberof grid points.


2. Findingtheright handsideof theSchurcomplementsystem(3.31).Thiscanbedoneby�

settingG� ö

0_

in (3.27)to obtainthesolutionof thePoissonequationandcalculatingtheresidualof (3.31).

3.ñ

Applying theGMRESiterationwith an initial guessof G�

. Themainprocessof theGMRES�

methodrequiresonly matrix–vectormultiplications.This involvesessentiallytwosteps� in our algorithm.Thefirst stepis to solve thePoissonequation,which correspondstoK

thematrix–vectormultiplication in theGMRESmethod.Themaincostof this stageisobtaining� thecorrectiontermateachirregulargridpointdueto thejumpin thesolution.ThefastPoissonsolverthatweemployedis themodifiedHWSCRT routinefromFISHPackage.TheJ

secondstepis to interpolatethesolutionof thePoissonequationto obtaintheresidual.

3.5.ñ

Numerical÷

Examplesfor theEvaluationof thePoissonEquation

Wæ

eprovideoneexamplewith differentboundaryconditionsto show theefficiency of thealgorithm� proposedabove. Thisis themostcostlypartin thesimulationof electro-migrationv� oids.Wewantto verify secondorderaccuracy of thesolutionprocedure,andmoreimpor-tant,K

alsoto verify theassumptionthatthenumberof iterationsis nearlyindependentof themeshsizeexceptfor a factorof logh

�.

Wæ

econstructanexactsolution:

ø�ùx� ú y� ûìüþý 1

2) logù

r¹ ÿ r¹ 2 � r¹ � x� 2) �

y� 2) �

(3.34)/

f! �

x� � y� �� 4

TheJ

boundary� � 1 is�

the unit rectangle:� 1 � x� � y� � 1. The interior boundary� � 2) is�

anellipse3

x� 2

a� 2 � y� 2

b�

2 � 1 �Case 1. TheDirichlet boundaryconditionon �� 1 and� thenormalderivativeboundary

condition� on � � 2) are� given usingtheexactsolution.Thefirst partof TableI showsthegrid

refinementanalysisandotherinformationwith a� � 0_ �

5�

andb�

0_ !

4.

InI

TableI, n4 is�

thenumberof grid lines in the x� - andy� -directions;e7 is�

theerrorof thecomputed� solutionin themaximumnorm;n4 1 is the total numberof irregulargrid points;n4 2 is�

the numberof irregular grid pointson oneside,which is alsothe dimensionof theSchurz

complement;k=

is thenumberof iterationsof theGMRESmethod,which is alsothenumberê of the fastPoissonsolver called.We observe a secondorderrateof convergencein themaximumnorm.Thenumberof iterationsfor theGMRESiterationis proportionaltoK

logh�

. Notethattheerrormaynot bereducedexactly by a factorof four but rathermayfluctuate,asexplainedin [12].

Case 2. Thenormalderivative " #%$&" n4 is prescribedon '�( 1 usingú theexactsolution.InI

this case,thesolutionis not uniqueandthecompatibilityconditionmustbeimposedinorder� to obtaina reasonablyaccuratesolution.To obtaina uniquesolution,we specifythesolution� at onecornerusingthe exact solution.In this way, we canmeasurethe error oftheK

computedsolution.TableII shows theresultsof thegrid refinementanalysis.We haveresultsÑ similar to thoseanalyzedabove.


TABLE I

The Grid RefinementAnalysiswith a) = 0.* 5,+

b,

= 0.* 4, and b,

= 0.* 15

n a- b e r n1 n- 2 k.

40 0.5 0.4 5.7116/ 100 4 100 52 1680 0.5 0.4 1.45911 102 4 3.9146 204 104 17

160 0.5 0.4 3.52363 104 5 4.1408 412 208 19320 0.5 0.4 8.16385 106 6 4.3161 820 412 21

n a- b e r n1 n- 2 k.

40 0.5 0.15 4.48207 108 3 68 36 1380 0.5 0.15 1.14669 10: 3 3.9089 132 68 15

160 0.5 0.15 2.6159; 10< 4 4.3832=

68 136 17320 0.5 0.15 6.7733> 10? 5 3.8621 68 268 20

Note@

. Dirichlet boundaryconditionsareprescribedon ACB 1, andNeumannboundaryconditionsareprescribedD on ECF 2. Secondorderconvergenceisobserved.Thenumberof iterationsisalmostindependentof themeshsize.

TABLE II

TheG

Grid RefinementAnalysis for PureNeumannTypeBoundary Conditions

n a- b e r n1 n- 2 k.

40=

0.5 0.15 4.5064H 10I 3 84 44 1480 0.5 0.15 1.2529J 10K 3 3.5967 164 84 17

160 0.5 0.15 3.3597L 10M 4 3.7292 332 168 19320 0.5 0.15 7.9409N 10O 5 4.2309

=668 336 21

4. EVALUATION OF THE NORMAL VELOCITY OF THE SURFACE

Calculatingû

the surfaceLaplacianoperatoralong a curve in two dimensionsis fairlystraightforw� ard.In higherdimensions,however, aniceparameterizationof thehypersurfacecan� beverydifficult tomaintainin aLagrangianformulationandhencethesurfaceLaplacianoperator� canbeproblematicin numericalcomputation.Sinceweareusingafixed Cartesiangridº wewill usea formulafor thesurfaceLaplacianoperatorin Cartesiancoordinates.Weare� usingthelevel setmethodtocapturethemoving interfaceandthegeometryisembeddedin thelevel setfunction.OursurfaceLaplacianoperatorthusmustalsobeexpressedin termsof� this level setfunction.We startwith thetwo dimensionalcaseandthenextendit to anynumberê of dimensions.Let P be

�the tangentialdirectionof thecurve andnQ be

�its normal

direction.�

AssumesÏ is the arc length parameter. If f!

is a function that is definedin aneighborhoodê of thecurve, thenthesurfaceLaplacianof f

!is�

Rs$ f! S d

j 2)

f!

dsj

2 T djdsj UWVYX[Z f

! \^]`_nQ acb f

! dfe T Heÿ g

f! hjilk`mon f

!pn4 qsr f

! tvu 2)

f!

wn4 2 x

where0 y is�

curvatureandHez f! { is�

theHessianof f!

. Thereforewehave

|f! }�~ T He� f! �j�� nT He� f! � n �

� 2 f!

� � 2 �� 2 f!

�n4 2 �


In general,let � be�

an n4 � 1 dimensionalhyper-surfacein Rn and� f!

be�

a function de-fined�

in a neighborhoodof � . Let e� 1 � e� 2) �� e� n be

�a local coordinateframeof R

� n ,f wheree� 1 � e� 2 �� e� n � 1 are� in thetangentplaneof � and� e� n is thenormalof .ThesurfaceLaplacianof� f!

on� ¡ is�

¢s$ f! £¥¤§¦ f

!¨e7 n ©

i ª 1« n ¬ 1

2®

f!

¯e7 2®i

°¥±§² f!³e7 n´µ·¶

f! ¸v¹ 2

®f!

ºe» 2®n´ ¼ (4.35)

½

where¾ ¿ is themeancurvature(sumof theprincipalcurvatures)of thesurface.In thelevelsetÀ formulation,with thesurfacedefinedby thezerolevel setof Á ,ÂÃ§Ä¥ÅsÆ ÅÈÇÉ ÊÈËoÌÎÍ e� n´ Ï

ÐÈÑÒ ÓÈÔoÕÎÖ

×fØÙe» n´ Ú

ÛÈÜÝ ÞÈßoàâá[ã f

Ø ä å 2 fØ

æe» 2

n´ çèÈé Tê ëÈìoí Heî fØ ïñðÈòó ôÈõoöÎ÷

Ifø

fØ ùûúýüûþsÿ�� is

themeancurvatureof thelevel setfunctionwe obtaina fourth order

nonlinearPDEfor thelevel setformulationof surfacediffusionof themeancurvatureflow.There

is verylittle analysisfor thisPDEasfar asweknow. If thelevel setsof � are� parallel,i.e., �� constant,� then

� 2 fØ

�e» 2

n´ � e� n´ �� e� n´ � � fØ !#"%$�&' (�)�*,+ -

.�/0 1�2�354�6 f

Ø 7

Ifø

fØ

is

constantin thenormaldirection,i.e., 8 fØ 9

e� n´ : 0,;

wehave < fØ =?>

e» n´ @ 0.;

Numericallyif we canmake A a� distancefunction(throughthere-initializationprocess)andextend f

ØsuchÀ that B f

Ø C�D�E�F0;

(see[17]), thenwehave

GsH fØ IKJ

fØ L

which¾ is veryeasyto computeona Cartesiangrid. For accuracy we preferto use(4.35)inourM numericalcomputation.

4.1. Evaluationof theSurfaceVelocitydueto theElectricalPotential

The

electric potentialfunction is only definedoutsideof the voids. However, for thelevel setformulation,we needthe velocity field in a thin tubesurroundingthe interface.The

difficulty is that we do not have any informationinsidethe voids andthe computedpotentialN is only secondorder accuratenearor on the interfacewherethe error of thepotentialN is notsmooth.As aresult,oscillationsin thevelocitywill bepresentaftersecondorderM differentiation.Suchoscillationswill alsoaffect thechoiceof thetime stepandareaconserv� ation(seeSection4.3aswell). Our strategy is to usea secondorderextrapolationtoO

extendthe informationoutsideof thevoids to the irregulargrid pointsinside,andthentoO

computethederivativealongthetangentialdirectionto obtainthenormalvelocityof thesurfÀ acedueto the variationin the electricpotential.Therearea variety of choicesastowhen¾ andhow to do theextrapolationalongthenormaldirectionanddifferentiatingalongtheO

tangentialdirection.Theschemeusedin oursimulationis thefollowing:P Use

Qthestandardcentralschemeto compute

R�SRsT UWV

XZYX

x[ n\ y] ^_�`_

ya n\ xb (4.36)½

at� rc egular gridd pointswhich lie inside»

theO

computationaltubebut outsidee theO

voids.


f UseQ

theweightedleastsquaresinterpolationto obtain g�hg sH at� all irr»

egular gridd points,bothi

insideandoutsidethevoidsby extrapolationandinterpolation,respectively.j Extend kmlk sH from theinterfacealongthenormaldirectionn toO

all grid pointsaswell astoO

thosepointswhich arelocatedinsidethevoid within thecomputationaltubeusingtheformula

n�onsT tp q

rts�us sT v

wyxz {y|�} signÀ ~�� 0

; �(4.37)½

Heresign�� is thesignfunctionof � . It switchesthedirectionof informationpropagationalong� characteristicsat theinterface.For moreinformationon velocity extension,seealso[6, 13,17,22].� Finally

�, usethe standardcentralschemeto computethe part of the surfacevelocity

due�

to theelectricpotential

��sT

�Z��sT �K�

��

x[��sT n\ y] �

��

ya�Z��sT n\ xb � (4.38)

½

F�or two dimensionalsurfacediffusionproblems,the total internalareaof voidsshould

remainunchanged,by theconsideration

��u� dx�

dy � Vn´ ds� ¡ 2

®¢sT 2® £ C1 ¤�¥ C2

® ¦¨§ ds� ©

0; ª

In ourmethod,thesurfacevelocity is proportionalto thetangentialderivativeof afunction.Theoretically«

andnumericallythisapproachpreservestheareabetterthantheapproachthate¬ xtendsthevelocityafterit is computedator neartheinterface.

4.2. Evaluationof theSurfaceVelocitydueto theCurvature

Inø

two dimensions,it is relatively straightforwardto evaluatethesurfacevelocityduetotheO

variationsin curvaturesincethecurvatureis definedatall grid points.It canbedonebyfinding

theseconddirectionalderivative of thecurvature;see(4.36)and(4.38).Sincethecomputational� processinvolvesfourthorderderivativesof thelevel set,onemaybetemptedtoO

usea fourthorderschemeto computethefirst orderderivatives,a third orderschemetocompute� thesecondorderderivatives,andsoforth.Whilethisapproachissoundin principle,it

resultsin largererrorscomparedwith thestandardcentraldifferenceschemeandrequiresa� largerwidth for thecomputationaltubeusedin thelocal level setmethod.Thetime stepsizeÀ alsoneedsto becut to keepstability. Furthermore,with widersupportandfourthorderderi�

vatives,specialcaremust be taken to handlethe boundaryconditionsof the voids.Numerically®

, ahighordermethoddoesnotgivebetterresultsin ourcasebecauseerrorscanbei

amplifiedandoscillationscanoccurat or nearthe interface.We find that thestandardcentral� differencescheme,also usedin [1] for etchingand depositionproblems,worksbesti

in our numericalexperiment.In fact,it canbeshown that,if a functionis sufficientlysmooth,À thecomputedfourthorderderivativesusingthestandardcentraldifferenceschemeare� secondorderaccurate.In [1], atwo dimensionalmodelinvolving only surfacediffusiontermsO

is considered.Thereareno topologicalchangesin thegeometry.After we computedthe normal velocity of the boundariesof the voids, we usedthe

weighted¾ ENO (essentiallynon-oscillatory)schemeto updatethe level set function and


performN there-initializationprocess.TheWENO schemeis morestableandaccurateandtheO

errorsaresmootherthanthoseof theENOscheme(see[8]).

4.3. ChoosingAdaptiveTimeStepSteps

TheCFL conditionfor thesurfacediffusionis

¯t° 1 ± h

² 4³ ´

C2® µ (4.39)

½

TheCFL conditionfor theHamilton–Jacobianequationin updatingthevelocity is

¶t° 2® · h

² ¸Umax¹ º (4.40)

½

where¾ Umax is thelargestmagnitudeof thenormalvelocityin thecomputationaltube.SincetheO

electricalpotentialis theinversionof theLaplacianoperator, theCFL conditionfor itssurfÀ acediffusionis

»t° 3¼ ½ h

² 2® ¾

C1 ¿ (4.41)½

TheÀ

adaptive time stepis chosenfrom the smallestof the threefor the explicit method.UsuallyQ

C2 is relatively small comparedwith C1,Â and the time stepis tolerablefor thesimulationsÀ on mostworkstations.However, it is known thatthedesiredlevel setfunctionis thesigneddistancefunction.Sucha level setfunctionhaskink at somepointsor alonga� curvewhere Á ÂyÃÅÄ is notwell defined(seeFig. 4). WhenthecomputationaltubecontainsthoseO

points,the calculationsarenot correctat thosepointssincethe derivatives do note¬ xist. Thenormalvelocity canhave very largemagnitudesat thesepoints,leadingto verysmallÀ time stepsizesandsometimesinstabilities.Anothermathematicalconcernis, aswepointedN out in the Introduction,that if eachlevel setfunctionmoves accordingto its owngeometryd andmotionlaw independentlythesetswill run into eachotherdueto thelackofa� maximumprinciple in this problem.Our numericalsolutionto this problemis to useacut-of� f function ÆÈÇ x[ É ,Â

ÊÈË x[ Ì#Í 1 if x[ Î 0Ï

Ðx[ Ñ 1Ò 4

³ Ó20Ô

x[ 3¼ Õ

10x[ 2® Ö

4×x[ Ø 1Ù if

0 Ú x[ Û 1

0 iÏ

f 1 Ü x[ Ý (4.42)½

TheÀ

new velocity is definedas

UneÞ wn´ ßáà

â ã�ä�åçæ0èé Un´ ê (4.43)

½

Inø

averythin tubecontainingtheinterface,ë ì�íïîñð 0è ,Â thenormalvelocity is unchanged.It is

graduallyò loweredto zeroto takecareof thetroublesomepointswithoutasignificanteffectonM accuracy. Wechooseó,ô 3

õh²

and� ö0è is

between2h²

and� 4h²

in

ournumericaltests.NoticethatO

weusedacut-off functionwith up to third ordercontinuousderivatives,in aneffort toreduce÷ theoscillationsintroducedby thecut-off neartheboundary. Numericallyspeaking,theO

level setfunctionprovidesbodyfittedparallelframes(level sets)neartheinterfacewhichcan� captureboththegeometryandthemotionof theinterfaceon a Euclideanframework.


FIG.ø

4. Grid pointsshown asopencirclesindicatepointsthatareeitheronor closeto thekink of thesigneddistancefunction;thesoliddotsareprojectionsof irregulargrid pointson theinterface.

Physicallyandmathematically, what really mattersis just the motion of level setsveryclose� to the interface.Due to a numericalCFL conditionandcorrectscalingof the levelsetÀ function (distancefunction) the level setformulationis guaranteedto be troublefree.The reasonwhy we usea cut-off function insteadof extendingthe normalvelocity is toenforce¬ areaconservation,aswementionedbefore.Ourapproacheffectively eliminatesthepossibilityN of having overlargenormalvelocitiesnearthekink of thelevel setfunctionneartheO

interface.

Remark�

4.1.×

Whentopologicalchangestake place,we canno longercut off thekink.The calculationmay lose someaccuracy at thosepoints. On the other hand,we needsomeÀ mathematicalinsightsbeforewe can fix the problemnumerically. Somedifferentapproaches� combiningperturbationtechniques,the particlemethod,and/orthe boundaryinte

gralmethodwith thelevel setformulationareunderinvestigation.

5. NUMERICAL EXPERIMENTS

Wù

ehaveperformedaseriesof numericalexperimentson theinteractionof voidsduringtheO

electro-migrationprocessin an attemptto draw somephysical insights from thesee¬ xperiments.For all thetestsshown here,weusedNeumannboundaryconditions(flux) attheO

left andright edgesof therectangles,andhomogeneousNeumannboundaryconditionsat� thetopandbottomof therectangles.


FIG.ø

5. Aú

void evolutionwith largechemicalpotential.Themotionis relatively stable.

In thefirst two experiments,wetry to reproducetheresultsfrom [20]. Thecomputationaldomain�

is therectangleû 2Ô ü

5ý þ

xÿ � 2Ô �

5,ý �

0Ï �

5ý �

y� � 0Ï �

5.ý

Theinitial void is acirclecenteredat� ( � 1.5,0) with radiusr 0

Ï 25.In thefirst case,thecoefficientsof thepotentialsaretaken

toO

beC1 � 0Ï �

625

andC2® � 0

Ï �625 �

6 �

25Ô �

10� 3¼,Â correspondingto the parameterschosen

in [20]. The computationalgrid is 300 � 60,

or h� �

1� 60.

Figure5 shows the evolutionofM thevoid with time. For this testproblem,thesurfacetensionis relatively largeandtheconfiguration� is stable.Therelativeareachangefor this caseis lessthan0.0619%.

In thesecondcasesee(Fig. 6), thecoefficient of thechemicalpotential,C2® � 1 � 875

� �2Ô �

08333Ï �

10� 4,Â is small comparedto thatof theelectricalpotential,C1 � 1 875,�

andthemotion is lessstablethanthat in the first case.The grid spacingis h

� !1" 100. The area

loss#

is about6%justbeforethevoid breaksup.OursimulationhasgonebeyondtheresultsobtainedM in [20]. Theevolution processseemsto agreewith theresultsin thatpaperup tosomeÀ timebycomparingthegraphsthere.NotethatweusedNeumannboundaryconditionsat� the left andright edgesof therectanglewhile Dirichlet boundaryconditionswereusedin

[20]. Usuallyit is harderto solvePoissonequationswith NeumannboundaryconditionsthanO

with thoseof Dirichlet type.As

$discussedin [20], thestabilityof thevoidshapedependsontheratiobetweenthedriv-

ing forceassociatedwith thesurfaceenergy andthatassociatedwith theelectro-migration.F%or fixed line geometryandvoid size,this ratio is proportionalto C1 & C2

® . Ourfirst two testcases,� whichessentiallyreproducethecorrespondingexamplesgiven in [20], demonstratethatO

a nearlycircular void shapeis stablewhenC2® ' C1 ( 6

)25Ô *

10+ 3¼

and� unstableat asmallerÀ ratio whenC2 , C1 - 2 . 08333

Ï /100 4. In the first testcase,andfor all caseswith

lar#

gerC2® 1 C1,Â thenearlycircularvoid shapeis stablebecausethedominatingdriving force


FIG.ø

6. Aú

void evolutionwith smallchemicalpotential.Themotionis lessstable.

is thesurfaceenergy, which tendsto resistdeviation from thecircularshape.In thesecondcase,� andfor all caseswith smallerC2

® 2 C1,Â electro-migrationforcesbecomedominantandcause� drasticdeviationsfrom thecircularvoid shape.Figure7 shows anevolutionprocesswith¾ even smallerC2

® 3 C1. The calculationwas donewith a coarsegrid, h� 4

15 50.ý

Whilewe¾ seemorelossin the area,we seea similar patternin the evolution process.The lossin

theareais dueto thepoor resolutionandthe largecurvaturesthatappearimmediatelyfollo

6wing topologicalchange.

Thealgorithmdevelopedherecanbeusedto studytheinteractionsof multiple voidsinan� electro-migrationline. Figure8 shows theevolution processof two voids.Thefirst oneis acirclecenteredat( 7 2.1,0) with radius0.15.Thesecondcircle is centeredat( 8 1.55,0)with¾ radius0.3.Thecoefficient of thechemicalpotentialis chosenasC2

® 9 3õ :

9062; <

10= 5>

and� thecoefficient of theelectricalpotential1.875,with theratio C2 ? C1 @ 2 A 08Ï B

10C 5>

intheO

rangeof instability. Theelectricalpotentialis dominantandthemotionis towardamoreunstableD patternwith thecreationof anumberof smallvoidsfrom thelargervoidsthroughshapeÀ instabilities.In thiscase,h

� E0Ï F

01Ï

andtheareachangeis lessthan2.7%.Mostof thelossin theareaoccursaftersmallvoidsareproduced.Thelargecurvatureof thesmallvoidscompensates� for thesmallchemicalpotentialcoefficientC2

® relati÷ veto theelectro-migrationcoef� ficient C1. This is not unexpectedbecausea small C2 G C1 indicatesshapeinstabilityofM large voids, which tend to breakup to into smallervoids in order to seeka balancebetweeni

thedriving forceassociatedwith thesurfaceenergy andthatassociatedwith theelectro-migration.¬ Thesameargumentwould indicatethatconditionsof smallC2

® H C1 tendO

toO

preventmergerof voids into largerones,which is consistantwith our observations.InthisO

simulation,weseealsothatsmallvoidsmove fasterthanlargeones.

300õ

LI, ZHAO, AND GAO

FIG.ø

7. Aú

void evolutionwith verysmallchemicalpotentialonacoarsegrid.

It appearsthattheinteractionof voidsevolvestowardanenergeticbalancebetweenthesurfÀ acetensionandelectro-migration.Wehavealreadyseenthatlargervoidstendtobreakupintosmallerones.If thesurfacetensionforceissufficiently large,twoor moresmallervoidsmayI mergeto form largervoids,asis demonstratedin Fig. 9, wherewe take C2

® J 0Ï K

0391Ï

and� C1 L 1 M 875.�

In thesimulation,h� N

1O 60.

Initially thevoidsaretwo ellipses:ThefirstoneM is centeredat (0.35,0.5)with aP Q 0

Ï R2Ô

andb� S

0Ï T

14,whereaP is

themajorandb�

is

the

FIG. 8. An evolutionprocessof two voidswith smallchemicalpotentials.

ELECTRO-MIGRATION 301õ

FIG.ø

9. Anú

evolutionprocessof two voidswith largechemicalpotentials.

minorI axis.Theellipseto theright is centeredat(0.65,0.5)with aP U 0Ï V

24Ô

andb� W

0Ï X

12.Thearea� changeis about10%,which is muchworsethanthevaluesin our previouscases.Webeliei

vetherearetwo factorswhichaffectthelossof thearea:Themajorfactoris thelargeh�

,Âwhich,¾ of course,meanslackof resolutionin thesimulation.For problemsinvolving fourthorderM derivatives,we find thatreasonableresolutionis necessaryto preserve thearea.ThesecondÀ factoris the singularitybeforeor after the topologicalchanges,i.e., the merging,where¾ somepartsof thelevel sethavevery largecurvatures.

Figure%

10 shows the interactionof two voidsundera relatively largeC2 Y C1 ratio.÷ Thefirst oneis anellipsecenteredat ( Z 2.1,0) with themajoraxis0 [ 3õ andtheminor axis0.2.TheÀ

forcecoefficientsareC1 \ 1 ] 875�

andC2 ^ 0Ï _

007324.Ï

Thevoid to theright is a circlecentered� at ( ` 1.5,0) with radius0 a 2. Theelectricfield becomesconcentratedbetweentheboundaryi

of theconductingline andthe largervoid. Due to this concentration,theupperand� lower tips of thelargervoid aredriven fasterthanthemediumportionandthesmallervb oid is shieldedfrom the electricfield by the presenceof the large void. As a result,thelargevoid wrapsaroundthesmall void without breakingup into smallervoids.Thecasedemonstrates�

thestrongshieldingeffectof largervoidswhenthey arerelatively stable.As a final example,we investigatetheeffect of initial shapeof a void. Figure11 shows

theO

evolution processof an ellipsecenteredat ( c 1.8, 0) with the major axis 0.4 andtheminor axis 0.2. The coefficientsareC1 d 1 e 875

�andC2 f 0

Ï g3õ h

90625; i

10j 3¼. The surface

tensionO

forcetendsto smooththevoid shapeinto acircle.Theeffectof electricalpotential,onM theotherhand,causesa protrusionof thevoid with a relatively largecurvatureat thefrontal6

tip of thevoid. Thissharptip moves rapidlyundertheelectro-migrationforcesanddrags�

the restof the void along.Sucha worm-like motion of void demonstratesanotherdelicate�

balancebetweensurfacetensionandelectro-migrationforces.

302õ

LI, ZHAO, AND GAO

FIG.ø

10. Anú


FIG.ø

11. Anú


ELECTRO-MIGRATION 303õ

6.k

CONCLUSION

In thispaper,wehavedescribedanumericalmethodfor thesimulationof movinginterfaceproblemsN whicharisein electro-migrationvoidingin integratedcircuits.In comparisonwitha� conventionalfiniteelement-oraboundaryelement-basednumericalmethodwith adaptivemeshing,I our methodhasthe advantageof employing a fixed Cartesiangrid without theneedof re-meshingat eachtime step.A fastiterative methodusingtheGMRESiterationhasl

beenadoptedto solve theLaplaceequationsassociatedwith electricpotentialon thee¬ xteriorof voids.Theevolutionof void boundariesis describedby atwo dimensionallevelsetÀ function which is updatedat eachtime stepaccordingto surfacediffusion equationsgoò vernedby combineddriving forcesassociatedwith surfacetensionandelectricpotential.TheÀ

level setmethodis usedto efficiently updatethe motion of voids andis capableofcapturing� complex topologicalchangesof voids suchas void merging and void break-up.D Our numericalexampleshave reproducedand gonebeyond sometest casesin theliteratureusingafinite element-basedsimulationmethod.WehavealsoperformedaseriesofM numericalexperimentsto elucidatethe mechanismsby which electro-migrationvoidsinteract.It appearsthattheinteractionof multiple voidstendsto evolve theshapeandsizeofM the voids into a statein which the surfacetensionis betterbalancedwith the electro-migrationI force.This is demonstratedin exampleswheresmallvoidsdominatedby surfacetensionO

tendto merge into larger voids while large voids dominatedby electromigrationforces6

tendto undergo an instability andbreakup into smallervoids.Also, themotionofsmallÀ voids could be significantlyretardedby large voids in their vicinity asthe electricfield

at small voids is shieldedby the large voids,especiallywhenthe sizesof the voidsare� ontheorderof theconductingline width. Themethodthatweproposedoesnotrequirea� sophisticatedmeshingtechniqueandmakesa relatively low demandon computationalresources.All thesimulationsinvolveinteractionof multiplevoidsandcanbedonerapidlyonM commonworkstations.It is alsorelatively straightforward to generalizethemethodtothreeO

dimensionalsimulationswherethecomputationalsavingsover re-meshingisexpectedtoO

beevenmoresignificant.It is true that for fourth orderdifferentialequations,the time steplimitation is still a

concern� for explicit algorithms.An implicit or semi-implicit level set methodis underinvestigation.Adaptive Cartesiangrid refinementtechniqueswill be consideredalso infurther6

developmentof themethoddiscussedin thispaper.

ACKNOWLEDGMENT

Them

authorsthanktherefereesfor their usefulcommentsandsuggestions.

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