Transient 3D heatflow analysisfor integrated circuit devices

using the transmissionline matrix methodon a quad tr eemesh

T. Smy�, D. Walkey, S.K. Dew

�Departmentof Electronics

CarletonUniversity

Ottawa,ON, CanadaK1S5B6�Departmentof ElectricalandComputerEngineering

Universityof Alberta,Edmonton,AB, CanadaT6G2G7

Abstract

Thispaperpresentsa3D transmissionline matrix implementation(TLM)for thesolutionof transient

heatflow in integratedsemiconductordevices. The implementationusesa rectangulardiscontinuous

meshto allow for localmeshrefinement.Thisapproachis basedonaquadtreemeshingtechniquewhich

canhave a complex geometryusingblocksof varyingsizes.Eachsuchblock canhave a maximumof

two adjacentblocksonany verticalsideanda maximumof four blockson thetopor bottom.

TheTLM implementationis basedon a physicalextractionof a resistanceandcapacitancenetwork

andthenthecreationof theappropriateTLM matrix. Theformulationallowsfor temperaturedependent

materialparametersandanon-uniformtimestepping.

The simulatoris first testedusinga 2D exampleof a heatsourcein a rectangularregion. Using

this examplethenumericalerror is determinedandfoundto belessthan0.4%. Next, nonlinearitiesare

included,anda numberof non-uniformtimesteppingalgorithmsaretested.Then,a 3D problemis also

comparedto ananalyticalsolutionandagaintheerroris verysmall.Finally, anexampleof afull solution

of heatflow in a realisticSi trenchdevice is presented.

Keywords:Simulation,Device,Thermal,Transient,TLM

�Correspondingauthor

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1 Intr oduction

Thesolutionof theheatdiffusionequationusingtransmissionline matrix (TLM) techniquesis well estab-

lished[1, 2]. A specificapplicationof TLM hasbeenthe solutionof steadystateand transientthermal

responsesof integratedcircuit (IC) devices[3, 4]. This problemis of substantialinterestdueto theneedto

know maximumoperatingtemperaturesfor reliability reasonsandto developtemperatureconsistentmodels

of thedevice.

Thebasicequationfor describingtheheatflow in IC devicesis[6, 7]:����� ��� ��������� �� � ���� � ����� �!�#"$� �#� (1)

where ��� ��is the temperature-dependent thermalconductivity, ��� thespecificheatof thematerial, � the

materialdensity, and� ����� �!�#"$� �#� describesthe local heatgeneration.A simplisticdescriptionof theprob-

lem to be solved is that of a small thin heatgenerationplateat the base/collectorjunction of the device

undersimulationwhich is placedin a large semiconductorsubstratecharacterizedby a constantthermal

conductivity, materialdensityandheatcapacity. Theboundaryconditionsaresomewhatproblematic,but a

fixedtemperatureon thebottomof thesemiconductorandtheassumptionof eitherdevice “tiling” (leading

to a reflective boundaryconditionat the edges)or of a semi-infinitesubstrateleadsto a problemthat can

be solved analyticallyusinga Green’s function approach.However, solving this equationfor realisticIC

devicesis complicatedby anumberof factors:

1. Theproblemmustbesolvedin 3D, becauseof thedominantroleof 3D heatspreadingfrom thesmall

sourceinto thesubstrate.

2. Theelectricaldevice equationleadsto non-uniformheatgenerationat thebase.

3. Thedevice metalizationmustbesimulated.This is particularlycrucialfor substrateswith poorther-

malconductivities suchasGaAs.

4. Temperature-dependentmaterialpropertiesmustbeincluded.BothGaAsandSi exhibit temperature-

dependentthermalconductivities andheatcapacities.ThismakesEq.1 anonlinearequation.

Accountingfor theeffectsdescribedabove necessitatestheuseof a3D numericalmodel.Someapplica-

tionssimply requirethesolutionof thesteadystateproblemandthentheextractionof aneffective thermal

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resistance.%'& � which is definedasthesteadystatetemperatureabove ambientdividedby thepower dissi-

patedin thedevice. However, many problemsrequirea full solutionof thetransientresponseof thedevice.

TLM hasshown itself highly appropriatefor solving time dependenttransientheatflow problemsandis

particularlyattractive for multimaterialandnonlinearproblemsasany temperatureor materialdependence

of thethermalparameterscanbelocally accommodated.

Anotherconstrainton this typeof problemrelatesto theefficiency of themethodin the faceof a large

spatialdynamicrange.Theheatflow problemfor IC devicesis characterizedby verylargethermalgradients

localizedaroundthe active componentswhich are relatively small regions ( ( 1 ) m) containinga power

source.In theseregions,ahighdensityof meshelementsis requiredtoobtainanaccuratesolution.However,

it is alsonecessaryto solve for relatively large volumesdueto substratethicknesseswhich are typically

50-500 ) m. Thesetwo requirementsmake the efficient solutionof the heatequationfor 3D IC devices

only possiblewith a highdegreeof meshrefinementaroundthedevice junctionsandmuchlargerelements

towardstheedgesof thesimulationregion.

The TLM methodis traditionally performedusinga continuousrectangularmesh. Clearly, a uniform

grid spacingwould be inefficient becausevery many cellswould be requiredif forcedto usea fine mesh

throughout.A nonuniformmeshcanbereadilyimplementedusingTLM, but this is usuallystill continuous

sothatthefinemeshrequiredneartheactive devicesmustalsoextendthroughat leastpartof thesubstrate,

therebystill requiringan unnecessarilylarge numberof cells. Ideally, a multigrid or quadtreeapproach

shouldbe usedsuchthat local meshrefinementneednot extendbeyond whereit is absolutelyrequired,

therebyallowing theminimumnumberof cellsin thecalculation.

In suchmeshes,two or moresmallcellsmaybeadjacentalonga singlesideof a largercell. While this

allows the necessarylocal meshrefinement,it leavesa problemof how to connectall of the transmission

lines (TL’s) of thesmall cells to thatof the large one. So far, threeapproachesto this problemhave been

made,generallytreatingthe2D case.[8, 9, 10] Of these,Pulko[10] andAit-Sadi[8] optedto connectall of

thesmallcell TL’s togetherat thecell boundary. ThesemergedTL’sarethenconnectedto asingleTL from

thelargecell. In effect, this imposesa constanttemperaturecriterionover mostof the largecell boundary

which is aqualitative changeto theproblembeingconsidered.Wong[9] avoidsthisproblemby connecting

only theTL of themiddlesmallcell to thatof thelargeone.Dependentvoltagesourcesarethenconnected

to eachof theTL’s of all of theothersmallcellsto linearly interpolatethetemperaturealongthe largecell

boundary. This appearsto correctsomeof theinaccuraciesintroducedby theearliermethods.[9] However,

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somepotentialproblemswith thisapproacharethatdependentsourcescanreducethenumericalstabilityof

themethodandalsothatsuchactive sourcesno longerexplicitly conserve energy. Theformersituationcan

leadto problemswhenutilizing dynamictime steppingto improve efficiency, andthe latter canintroduce

systematicerrorsinto thesolution. A preferableapproachwould strictly staywith passive componentsto

avoid thesedrawbacks.

This paperwill describethedevelopmentof a non-uniformmeshingtechniquebasedon a 3D quadtree

mesh(QTM) approachanda TLM implementationusingthis mesh.A distinct featureof this approachis

thatmultiple transmissionlinesareusedif necessaryto connectlargecells to eachof their smallerneigh-

bors. Suchan approachavoids imposingconstanttemperaturesalongthe cell boundaryandavoids using

unnecessarydependentsources.However, suchanapproachresultsin a variablenumberof TL’s for each

cell. Further, this TLM implementationwill beusedfor bothconstantandtemperature-dependent material

properties,andthetime stepwill ultimatelybeadjusteddynamicallyduringsimulationto reduceexecution

time. Resultsconfirmingtheaccuracy of this TLM approachwill bepresented,andfinally a full transient

simulationof a realisticSi-basedtrenchdevice will begiven.

2 Model Development

2.1 BasicTLM for Heat Flow

TLM modelingrepresentsa physicalmodelof heatflow asa sequenceof voltagepulsestravelling through

amatrixnetwork of transmissionlines.By computingthevoltage(temperature)at thenodesconnectingthe

transmissionlines,anexplicit, unconditionallystablesolutionto thetelegrapher’s equation canbedescribed

by asetof equationsthatareiteratedin astraightforwardmanner.

Thetelegrapher’s equationis givenby:

�+*-,.� %'/ � / � ,�� �10 / � / � * ,�� * (2)

where %�/ , 0 / and � / are respectively, the resistance,inductanceand capacitanceper unit length in the

transmissionline andV is thenodevoltage.Equations1 and2 have a similar structure.Replacing, with, setting %�/ � / � ����2 � , addinga currentsourceto the left handsideof Eq. 2 andrequiringthe term

involving thesecondtimederivative to benegligible resultsin acorrespondencebetweenthetwo equations.

Once %�/ and � / have beendeterminedby thephysicalproblem,a requirementthat thepulsestravel from

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nodeto nodein a time step 3 �resultsin the complex transmissionline impedance4 �65 0 / 27� / being

equalto 3 � 2 � � /83 � �where 3 � is thelocaldistancebetweennodes.Therequirementthatthesecondterm

in Eq.2 beneglectablethenbecomesaconditionthat 3 �besufficiently small.

With theseconditions,the transientheatequationcanbe solved in a TLM framework by iteratingthe

following steps:

Nodal Temperatures: At eachnode9 , thetemperatureat thetimestep: , ; � 9 �, is calculatedfrom thelo-

calsourcesandtheincidenttemperaturepulses,; ,=<> � 9 �. Thesubscript? indicateswhichtransmission

line thepulseis on.

Scattered Pulses: The scatteredtemperaturepulses; ,A@> � 9 �, arecalculatedfrom the nodal temperatures

andincidentpulses.

Incident Pulses: The incidentpulsefor the subsequentiteration, ;CB � , <> � 9 �, arecomputedfrom the scat-

teredpulsesfrom the previous time step,accountingfor any time stepchangesanddifferencesin

nodalseparation.

A substantialamountof work hasbeendoneonTLM solutionsof heatflow, includingthedevelopmentof

3D and2D modelsbasedon continuousrectangularmeshes[3]. Oneadvantageof TLM over conventional

finite differencetechniquesis greaterstability. In particular, this allows the time stepto be increasedas

the simulationprogressestowardssteadystate,therebyacceleratingthe computationsubstantially. Incor-

poratingthis into the TLM equationssimply requiressomeadditionalcomputationsfor scatteringeffects

at cell boundariesdueto changesin the time step[10], althoughcareis requiredto ensurethesecondtime

derivative termin Equation2 remainssmall. Finally, by updatingresistanceandcapacitancevaluesduring

thesimulation,nonlineartemperature-dependent materialpropertiesareeasilyincorporatedinto theTLM

framework.

2.2 TLM on a Quad TreeMesh

A full descriptionof the 3D modelbuilding codeis describedelsewhere[11]. This modelbuilding code

will, givenanIC technologydescriptionandlayoutinformation,automaticallygeneratea 3D modelof the

device,discretizethemodel,andsolve for thetemperaturedistribution in thedevice. Thebasisof thiscode

is aquadtreemeshingtechniquethatcreatesacomplex meshgeometryusingrectangularblocksof varying

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sizes.The modelis createdsuchthat a block andits neighborsform a relatively simplesetof topologies

wherebyeachblockcanhaveamaximumof two blocksonany verticalsideandamaximumof four blocks

on the top or bottom. Fromtheseblock topologies,a network of thermalresistancesandcapacitancescan

beextractedrepresentingthethermalequationto besolved.

Theessenceof themultigrid quadtreemesh(QTM) is shown in Fig. 1(a).Thebottomright cornershows

anareaof local refinement,anda reductionin meshdensityis obtainedby increasingtheblock sizeaswe

move away from it. Theprimaryrequirementof a 2D QTM is thattherecanbeat mosttwo smallerblocks

presentonany sideof agivenblock.Thisallowsfor averyquickexpansionof blocksize,but still minimizes

thenumberof localblock topologies.Theextensionof thisstyleof meshingto 3D canbedonein anumber

of ways,but perhapsthemostobviousis to allow thetopandbottomof ablock to haveatmostfour smaller

blocksadjacent.This resultsin block topologiessuchasshown in Fig. 1(b).

The fundamentalapproachto TLM modelingof the QTM is shown for a 2D casein Fig. 2(a). At the

centerof eachblock is a temperaturenode,anda transmissionline from this nodeto all adjacentnodesis

created.Eachtransmissionline consistsof two impedancesandtwo resistances,oneof eachassociatedwith

eachblock. This is shown in Fig. 2(b) for two nodesdenotedas 9 < and 9ED . Although the associationof%�/ � / with �$��2 � allows for a numberof choicesin choosingthevaluesof % and 4 of eachtransmission

line, a naturalchoiceis to usethe physicalvaluesof thermalresistanceandcapacitanceassociatedwith

nodalconnections.

Figure3(a) shows the basicgeometriesusedto calculatethe two resistorvaluesassociatedwith a link

betweenadjacentblocksin a 2D slab. Theresistance,% � , betweennode 9 � andthecommonboundaryis

simply given by % � 0 2 ���GFIH �where H is the thicknessof the slab,and F and

0arethe widths and

lengthsasshown in the figure. For cells which areof differentsizes,the sameformula holds; however,

multiple branchesareneededand the F usedin the calculationfor a given branchin the larger cell is

reducedto that of the correspondingsmallercell. This subdivision of the large cells resultsin several

parallelresistorswhoseequivalentresistanceis equalto thatwhichwouldhaveresultedhadtheentirewidth

of thecell beenusedin a singleresistor. Extendingthis to 3D just requirestreating H in thesamemanner

asis donefor F .

To calculatethethermalcapacitanceassociatedwith eachnode,thetotal capacitancein a block is deter-

minedby � &KJL& �M, &KJL& �$��� where , &KJN& is theblockvolume, � thematerialdensityand ��� thespecificheatof

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thematerial.As shown in Fig. 3(b),thiscapacitancecanbeapportionedto eachtransmissionline according

to thearea(or volume)it correspondsto. In 2D, this is given simply as � &KJL& 2PO divided by thenumberof

adjacentcellsalongthecorrespondingedge.

Althoughthecalculationof theline resistancesandcapacitanceswasdevelopedfor a2D mesh,verysimi-

lar expressionsaredeterminedfor the3D case.Theaboveapproachto theextractionof thetransmissionline

resistancescorrespondsmathematicallyto theapproachdescribedin [9] with theappropriatereductionfor

aquadtreemeshandanextensionto 3D. Oncethecapacitanceshave beendetermined,theline impedances

canbecalculatedusing 4RQ � 3 � 27� Q for agiventimestep.

Theuseof this formulationin 2D or 3D QTM resultsin a network of nodes(at theblock centers)con-

nectedby transmissionlines.Unlike previousmultigrid[8, 9, 10] approaches,connectionsonly occurat the

nodes.We denoteeachline asa “branch” with thenumberof branchesat eachnode 9 being S � 9 �. Each

TLM nodein the problemcanbe describedashaving a nodetemperature; � 9 �at time step : andasso-

ciatedwith eachbranch? of thenodeis a resistance% > � 9 �, an impedance4 > � 9 �

, a scatteredtemperature

pulse; , @> � 9 �anda incidentpulse; , <> � 9 �

. Also associatedwith thenodeis acurrentsourceT � 9 �if heatis

beinggeneratedin theblock representedby thatnode.A generalizedTLM nodeis shown in Fig. 4.

UsingthestandardTLM approachandtheabovenomenclature,weobtainfor thenodetemperatureat the

timestep : : ; � 9 � � UVNWYX Q[Z\ > ] ; , <> � 9 �% > � 9 �^� 4 > � 9 � � T � 9 �`_acbd (3)

with, d � WYX Q7Z\ > b% > � 9 ��� 4 > � 9 � (4)

thesummationsareoverall branchespresentat thatnode.

Thescatteredpulseoneachbranchis thengivenby:

; , @> � 9 � � ; � 9 � 4 > � 9 �% > � 9 ��� 4 > � 9 � � ; , <> � 9 � % > � 9 ��e 4 > � 9 �% > � 9 �^� 4 > � 9 � (5)

Theequationconnectingthescatteredpulsesat thetimestep: andtheincidentpulsesat : � b musttake

into accountboth thescatteringat thediscontinuityin 4 at theblock boundariesandthescatteringdueto

timestepchanges.Takingtheseinto accountweobtaina relationfor eachbranchof thenode:

;fB � , <> � 9 � � ; , @>hg � 9^i �Lj >hg � ; , @> � 9 � � > (6)

7

where j >hg � � b �lkY� 4 > � 9 �4 > � 9 �^� 4 >hg � 9 i � (7)� � 4 >hg � 9 i �Yemk 4 > � 9 �4 > � 9 �^� 4 > g � 9 i � (8)

andk � ;n3 � 2 ;Po � 3 �

is theratio of thecurrenttime stepto thepreviousone.For theabove expressions9 iand ? i denoterespectively thenodeandreversebranchat theendof thetransmissionline.

The TLM solutionof the heatdiffusion problemrequiresonly the repeatedsolutionof Eq. 3, 5 and6

with appropriateinitial andboundaryconditions. Two basicboundaryconditionsareneeded:1) a fixed

temperatureon a block faceor 2) a zero heatflow condition acrossa block face. Implementingthese

boundaryconditionsrequiresreplacingEq.6 for thebranchesconnectedto a boundary. Thefirst condition

requirestheimplementationof thefollowing equationfor thebranchconnectedto theface:

;fB � , <> � 9 � �M,!p e ; , @> � 9 �(9)

where ,!p is thefixedvoltageon theface.Thezeroflow conditionresultsin:

;fB � , <> � 9 � � ; , @> � 9 �(10)

for any branchwith a reflective boundaryconditionat theend.

Theinitial conditionusedwasto assumethedevice wasat a uniform initial temperature,,Eq . To impose

thiscondition,all initial incidentpulsesweresetto ,Eq 2 ] .

Theabove formulationassumedconstantmaterialparameters.It is, however, straightforward to update

temperature-dependent materialparametersat eachiteration. An additionalrequirementfor an accurate

solutionis thatthesecondtermin Eq.2 involving a secondtime derivative is negligible. This is essentially

a requirementfor asufficiently smalltimestep.A significantadvantageof theTLM methodis theability to

increasethetimestepsizeasthesimulationprogressestowardssteadystate.Theimplicationsof increasing

thetimesteponsolutionaccuracy will bediscussedbelow.

3 Results

In orderto evaluatetheaccuracy of theTLM algorithmdiscussedabove, theTLM solutiononbotha QTM

meshandauniformmeshwerecomparedtoananalyticalsolutionfor anumberof representativegeometries.

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3.1 2D slab

The first geometryconsideredwasa 2D slabwith a fixed temperatureon the left sideanda squareheat

generationregion in the centerof the slab. The otheredgesof theslabwereassumedto have a zeroheat

flow condition. The basicgeometryis shown for botha QTM anda uniform meshin Fig. 5(a). The slab

dimensionsare64 ) m by 64 ) m with a16 ) m squareuniformheatgenerationregion (100mF ). For these

initial simulations,the time stepwaskept uniform andwasat a valueof onetenththe smallestRC time

constantpresentin theslab. TheslabmaterialwasSi with a thermalconductivity of b[r ]ts bfu ov F 2 )^w - x ,

densityof ] rzy7y s bfu o � * � 2 )^w|{ andspecificheatof u�r~}�� 2 � - x wereused.

A time dependentanalyticalsolution for this problemcanbe developed[12] usinga Green’s function

approach.This leadsto thefollowing expressionfor thetemperature:

����� �!� �#� � �� � UV � * e � �� � �\Q[� � � * �������`� oE���P�� &�� ]R�C��� ��� Q � �� � � �#�K� ��� Q � * ��e �#�K� ��� Q � � �� Q � _a(11)UV �\� � � � { � ������� � oE�[� �� & � ]R�#�K� � k � � �� � � �C��� � k � � * �Ye �C��� � k � � � �k � � _a

where � and�

aretheslabdimensions,�

is thepower densityover theregion definedby thecornerpoints��� � � � � � and ��� * � � * � , � Q � 9�� 2P� ,k � � w|� 2 ] � and � ��� 27����� .

In Fig. 5, thebasicslabmeshgeometryis shown for boththeQTM anda uniform meshof 32 s 32. This

QTM meshshows an aggressive attemptat reducingthe numberof blocksandcontains220 blocks. The

characteristicQTM meshstructurecanbeseen.Theuniformmeshhas,of course,1024blocks.Also shown

in this figurearegrayscaledepictionsof the temperaturedistribution in theslabat timesof 0.1 ) s and15) s. As canbeseen,thetwo meshesproducevery similar distributions. In Fig. 6(a),a comparisonbetween

the two meshsolutionsandtheanalyticalsolutionis shown for the temperatureof a point at thecenterof

the slab. The temperatureis shown asa function of time andit canclearly be seenthat the solutionsfor

bothmeshescomparevery favorablywith theanalyticalsolution.However, theuniformmeshsolutiontook

considerablylongerto compute.

The residualerror (the absolutedifferencebetweenthe analyticalsolutionand the numericalsolution

normalizedto thefinal temperature)is alsoshown for anumberof casesin Fig. 6(b). Shown in thisplot are

threeTLM solutionsandtwo solutionsbasedon a simplefinite difference(FD) approach.(While bothFD

9

andTLM producevery similar resultsfor a given mesh,theTLM methodhasthepotentialto do sowith

considerablylesscalculationby stretchingthe time step. This is discussedfurther below.) The minimal

error is found for thehighestmeshdensityanda uniform grid of 1024elements.In this instance,theFD

solutioninitially shows a largererrorfor thetransientbut by 5 )�  , theresidualerrorsarevery similar. The

low densityQTM with 220elementsshows a larger steadystateerrorby a factorof about4, with theFD

solutionagainhaving a larger error initially. An intermediateQTM meshof 350 elementsusinga TLM

solutionreducesthe error to only around2.5 timesthat of the high densityresult. It is importantto note

that for all casestheresidualerror is lessthan0.4%of thenormalizedtransientandlessthan0.5 x for an

absoluteerror.

To enableaneasyevaluationof theeffect of thequadtreemeshon thespatialtemperaturedistribution,

thetemperaturehasbeenplottedin Fig. 7 in thex andy directionsthroughthecenterof theslabfrom one

edgeto theother. Thetemperatureprofile is plottedfor threedifferenttimesduringtheevolution to steady

state.As canbeseen,bothmeshesproducea reasonablyaccuratesolutionthroughouttheregion.

Theresultsdescribedabove indicatethata2D QTM meshessentiallyproducesasaccurateasolutionfor

thetemperatureasauniformhighdensitymesh,with areductionin thenumberof blocksby afactorof four.

3.2 Non-uniform time stepping

Oneof themainadvantagesof TLM over FD simulationmethodsis theability to reduceexecutiontimeby

optimizingthetimestepduringthesimulation.In this section,weexploresomesimplemethodsof altering

or controllingthetimestepduringthesimulation.To characterizetheerrorpresentin thesimulationduring

thetransient,anerrorparameterwasevaluated[5]. Theparameterusedwascalculatedusing:

� �¢¡¡¡¡ ] 0 / � / � ; � 9 ��e ] ;Po � � 9 �^� ;8o * � 9 �;[3 �^� ;8o � 3 � � ¡¡¡¡ (12)

which wasaveragedover theheatgenerationregion to calculate�-£¥¤¥¦ . This parameteris a directestimation

of thesizeof thesecondderivative errortermin Eq.2.

A variety of algorithmsfor stepsizegrowth was investigatedusingthe QTM of 220 blocksshown in

Fig. 5. The mostobvious methodof increasingthe stepsize is a uniform growth rategiven by ;P3 � �� b � T � ;Po � 3 �whereG is a fixed andtypically lessthan5%. Fig. 8(a) shows the transienttemperature

responseat thecenterof theblockandtheaverageerror, �-£#¤§¦ , for threedifferentgrowth rates.Thesmallest

growth rateof 0.5%follows theanalyticalcurveverywell. Increasingthegrowth rateto 5%causesanerror

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to develop in the steeppart of the transientwhich thendisappearsassteadystateis approached.Further

increasingthegrowth rateby a factorof 2 producesa largeovershootandthenoscillatorybehavior asthe

solutionmovestowardssteadystate.Theplot of averageerrorparameterclearlyshows the inaccuracy of

thesolutionathighergrowth rates.For T � u�rz¨ %, theerrorparameterstaysessentiallybelow bfu ov , andthe

solutionis well behaved. For the large growth rates,� £#¤¥¦ quickly becomesratherlarge introducingerrors

in the initial part of the transient.For the growth rateof 10%, the error parameteris large for the whole

simulation,rangingbetweenbfu o { and bfu o * .It was noticedthat for longer simulationtimes, the absolutesize of the time stepwould becometoo

large,producingerrorsandoscillationsin thesolution.An approachto minimizing bothsolutionerrorand

executiontime is to usea moderategrowth ratebut limit the time stepto a maximumvalue. In Fig. 8(b),

theresultsof this approacharepresented.A growth rateof 1% wasusedandthreesimulationsareshown

correspondingto amaximumtimestepof 20,50or 100timesthesmallestRC timeconstant(j � < Q ) present

in the slab. The plot of the temperaturetransientshows that whenlimited to 20 timesj � < Q the solution

closelyfollows theanalyticalresult.Using50or 100j � < Q producesincreasingerror;however, thesolution

doesstayrelatively closeto thecorrectanswerandapproachesthecorrectsteadystatevalue.Theeffect of

fixing thetimestepcanclearlybeseenin theplot of theaveragederrorparameterwith adampedoscillation

occurringin � £¥¤¥¦ from thepoint at which the time stepstopsgrowing. After this point, theerrorbecomes

progressively smaller, exceptin thecaseof 100j � < Q , indicatingthatthisfixedtimestepis too large.

Finally, a third approachto timestepcontrolwastriedwhich involvedmonitoringtheerrorparameter. In

thiscase,if �8£#¤¥¦ becamelargerthan ¨ s bfu ov , thegrowth ratewasheldconstantuntil it againdroppedbelow

this threshold,afterwhich it wasallowedto increaseagain.In Fig. 8(c), theresultsfrom this approachare

shown for four differentgrowth ratesof 1, 2.5, 5 and10%. As canbeseenfrom theplots, this methodis

quite successful.A high growth rateof even 2.5 or 5% canbeusedwith a goodresultobtained.A 10%

growth ratedoesproducesomeerror in the initial part of the transient,but even for this extremecasethe

erroris well behavedin thelatterpartof thetransient.

Of the above techniquesmonitoringthe errorparameterwasthe mostsuccessful.Of course,the point

of growing the time stepis to reducethenumberof iterationsusedto simulatethe transientresponseto a

particularvalueof�. In Fig. 9, the total numberof iterationsasa functionof

�is presentedfor four cases

which producean accuratesolution. Thesecasesare: a uniform time stepof u�rKb j � < Q , a uniform growth

rateof 0.5%,a growth rateof 1% anda maximumtime stepof ] u j � < Q , anda growth rateof 2.5% and

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a constraintof ¨ s bfu ov on the error parameter. As can be seen,the last caseproducedthe minimum

numberof iterationsand thereforeminimal executiontime; however, all the non-uniformtime stepping

algorithmsproduceaverydramaticdecreasein thesimulationtime. Theabove algorithmsareby nomeans

eitheroptimalor comprehensive, but do indicatethelargeadvantageof usingTLM with non-uniformtime

stepping.

3.3 Nonlinear materials

To investigatethe accuracy of the simulationwith respectto nonlinearmaterialproperties,the 2D slab

problemdescribedabove wassimulatedwith a temperature-dependent thermalconductivity givenby:

�©��� q � b � :�ª � � «e¬ q ��� :�ª * � ­e¬ q � * � (13)

where �Eq � b[r O ¨ s bfu ov F 2 )^w - x , : ª � � e O rz¨ s bfu o { x o �, : ª * � bfu oE® x o * and

q � y[u7u x for

Si. In Fig. 10(a),the transientresponsein theslabcenterfor constantandtemperature-dependent thermal

conductivity is shown. The importanceof including the nonlinearitycanclearly be seen. Unfortunately,

thereis no analyticsolutionto the nonlineartransientproblem,so that numericaltechniquesarerequired

for even thesimplestgeometries.Thesecondplot in Fig. 10 shows the time evolution of the temperature

profile in the � -directionthroughthe slabcenter. Although it is not possibleto get ananalyticalsolution

of the transientresponseof the non-lineartemperature-dependent problem,it is possibleto get a steady

statesolutionusinga Kirchoff transformation[13]. This analyticalresultis alsoshown in thesecondplot,

andit canbeseenthatthesimulationsdo smoothlyapproachthesteadystateanalyticalsolution,providing

confidencein thenon-linearTLM solution.

4 3D block

Theadvantageof theQTM is mostdramaticallyillustratedfor a3 dimensionalproblem.In Fig. 11,aQTM

for a3D blockof heightof 50 ) m anda width andlengthof 32 ) m is shown. Thebottomfaceof theblock

wasfixedat 300K, anda thin heatsourceregion wassituatednearthetop of theblock. Theheatsourceof

25 mW is centeredat theblock, is 8 s 8 ) m in its lateraldimensions,hasa thicknessof 0.2 ) m andis 0.2) m from thetop face.Theblockmaterialis Si.

As with the2D slab,ananalyticaltransientsolutionof this problemis available(assumingtemperature

independentmaterialparameters),resultingin a solutionsimilar to Eq. 11, but with a triple sum. Figure

12

11presentsthetemperaturedistribution of theblock usinga QTM modelwith only 1974blocks.However,

if a uniform � - � meshwereto be used,the numberof nodeswould be 122,880,assumingthe samenon-

uniform meshingin the " direction. A comparisonof the time responseof the modelandthe analytical

solutionis shown in Fig.12. As canbeseenfrom boththeplot of thetemperaturetransientresponseandthe

residualerror, theQTM TLM solutionis veryaccurateover theentireresponsetime. A spatialtemperature

distribution is shown in thesecondfigure. Both theanalyticalandQTM TLM solutionfor thetemperature

at thecenterof theblock down throughthedepthof theblock arepresented.The agreementis excellent,

lendingconfidenceto the3D extensionsof theQTM TLM approach.The situationanalyzedhereis very

similar to thefinal device simulationsto be investigated,consistingof a block of Si with a backsidefixed

temperatureandasmallthinheatingplateatthetopof theblockrepresentingtheheatingatthebase/collector

junctionof amicroelectronicdevice.

5 Full Si tr enchsimulation

Finally, a modelof a realisticSi trenchdevice is shown in Fig. 13(a). The figureshows 1/4 of thedevice

which hasa central1 ) m x 20 ) m emitterwith a collectorcontacton eachside. The technologyhasa

PtSi/W plug contactstructureandtwo aluminummetallayerswereincludedin the modelwith a W plug

via. Thesimulationregion is 64 ) m squareandthewaferthicknesswas100 ) m. Thebacksidetemperature

was300K andthepower generatedin thedevice was1.75mW, uniformly generatedin a region underthe

emitter. Theendsof theemitterandcollectormetalizationwerealsofixedat300K. Thetotalmodelconsists

of 15,000blocks.

A simulationof thetransientresponseof this Si device is shown in Fig. 13(b). Thetemperatureof point

at thecenterof thebase/collectorjunctionis presentedasa functionof time. TheFig. 14shows thechange

in thetemperatureprofileat four differenttimes.

6 Conclusions

This paperpresents2D and3D TLM implementationsof thesolutionfor heatflow, ultimately in realistic

integratedsemiconductordevices.Theimplementationis basedona quadtreemeshwhich is veryefficient

at localizedmeshrefinement. Model building codewill, given an IC technologydescriptionand layout

information,automaticallygeneratea 3D modelof the device anddiscretizethe model. The basisof the

13

quadtreemeshingtechniqueis to createa modelconsistingof differing sizeblocks,suchthata block and

its neighborsform a relatively simplesetof topologies,wherebyeachblock canhave a maximumof two

blocksonany verticalsideandamaximumof four blockson thetopor bottomof a block.

From this physicalmodel,a generalizedTLM implementationis formulatedby extractinga physical

resistanceandcapacitancenetwork. Theformulationallowsfor non-uniformtimesteppingandtemperature-

dependentmaterialparameters.

The TLM implementationwas first testedfor a simple 2D problemfor which an analyticalsolution

is obtainable. A direct comparisonbetweenthe QTM TLM solution,uniform meshsolutionsandfinite

differencesolutionsshowed the QTM meshTLM solutionworks very well. A numberof algorithmsfor

optimizingthetimesteppingwereinvestigatedwith thebestfoundto beastraightforwardmonitoringof an

errorparameter. It wasfoundthat increasingthe time stepduringsimulationcouldreduceexecutiontime

by up to 100times.This is on top of thereductionfactorresultingfrom usingtheQTM over a continuous

mesh.

A simple3D situationwasalsoanalyzedandit wasfound thata QTM of 1974blocks(versus122,880

blocksfor a uniform � - � mesh)produceda very accuratesolutionwith lessthan0.5%error. Finally, an

exampleof a full simulationof a Si trenchdevice using15,000blockswaspresented,displayingthe full

capabilitiesof thesimulator.

Acknowledgement

This work wassupportedby Nortel Networks, theNaturalSciencesandEngineeringResearchCouncilof

Canada(NSERC)andtheNetworksof Centersof Excellence(Micronet).

14

References

[1] PeterB. Johns,A simpleexplicit andunconditionallystablenumericalroutinefor the solutionof the

diffusionequation,Int. J.Numer. Eng. 11, 1407-1328,(1977).

[2] M.Y. Al-Zeben,A.H. SalehandM. A. Al-Omar, TLM modellingof diffusion,drift andrecombination

of chargecarriersin semiconductors,Int. J.Numer. Model.: Electron.Netw. DevicesFields,5, 219-225,

(1992).

[3] Xiang Gui, Paul W. Webb,andGuang-boGao,Useof three-dimensionalTLM methodin the thermal

simulationanddesignof semiconductordevices,IEEETrans.Electr. Dev., 39(6) 1295–1302(1992).

[4] Xiang Gui, Guang-boGao,HadisMorkoc, Theeffectsof surfaceMetallizationtheThermalBehavior

of GaAsMicrowave PowerDevices,IEEETrans.Microwave Th. Techniques,bf 42(2)342–344,1994.

[5] PaulW. WebbandIanA. D. Russell,Applicationof theTLM methodto transientthermalsimulationof

microwave power transistors,IEEETrans.Electr. Dev., 42(4) 624–631(1995).

[6] J.Higgens,ThermalPropertiesof PowerHBT’s, IEEETrans.Electr. Dev., 40(12) 2171-2177(1993).

[7] H. Carslaw, Conduction of heat in solids, Ch.1,Oxford,1959.

[8] R. Ait-Sadi, A.J. Lowery and B. Tuck, Combinedfine-coarsemeshtransmissionline modelling for

diffusionproblems,Int. J.Numer. Model.: Electron.Netw. DevicesFields,3, 111-126,(1990).

[9] C.C.WongandH. Xiao, A combinedfine-coarsemeshmethodfor the transmissionline modellingof

diffusion,Int. J.Numer. Model.: Electron.Netw. DevicesFields,9 405-415(1996).

[10] S. H. Pulko, I.A. HalleronandC.P. Phizacklea,Substructuringof spaceandtime in TLM diffusion

applications,Int. J.Numer. Model.: Electron.Netw. DevicesFields,3 207-214(1990).

[11] T. Smy, D. Walkey andS.Dew, A 3D thermalsimulationtool for integrateddevices– Atar, Accepted

by IEEETransonComputerAidedDesign(Sept.2000).

[12] M. NecatiOzisik,Heat Conduction, Ch.2-4,JohnWiley andSons,1980.

[13] W.B.Joyce, Thermal resistanceof heatsinks with temperature-dependent conductivity, Solid-State

Electronics,18321-322(1975).

15

List of Figures

1 (a) A 2D quadtreemeshshowing refinementand(b) a single3D elementwith 4 blocks

above,2 on theright and1 onall othersides. . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 (a)TLM transmissionline links in aquadtreeand(b) detailsof asinglelink in thequadtree

mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Geometryfor a 2D meshfor calculating(a) resistancesand(b) capacitances.. . . . . . . . . 18

4 A generalTLM nodefor aQTM mesh.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Meshgeometriesandtemperaturedistributionsfor a2D rectangularslab. . . . . . . . . . . 20

6 Transientresponseof the2D rectangularslabshowing (a) temperatureand(b) residualerror. 21

7 Spatial temperaturedistributions throughthe centerof the rectangularslab (a) in the x-

directionand(b) in they-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Transientresponseof theslabusinga QTM with differentstrategiesfor non-uniformtime

steppingsuchas(a) unconstrainedgrowth, (b) 1% growth constrainedto a maximumtime

stepand(c) growth constrainedby limiting �8£#¤¥¦ to ¨ s bfu ov . . . . . . . . . . . . . . . . . . 22

9 Totalnumberof iterationsasa functionof simulationtime. . . . . . . . . . . . . . . . . . . 23

10 Effect of a temperature-dependent heatconductivity on (a) transientresponsein the slab

centerand(b) temperatureprofile in thex-direction. . . . . . . . . . . . . . . . . . . . . . . 23

11 Grayscalerepresentationof thetemperaturedistribution at 100 ) s in a rectangular3D solid

usinga QTM mesh.Thedarkestregionscorrespondto a temperatureriseof 65 x . Dueto

thesymmetry, only onequadrantof theblock is shown. . . . . . . . . . . . . . . . . . . . . 24

12 Temperatureresponsewithin a 3D rectangularsolidshowing (a) transientresponseof max-

imumtemperatureand(b) temperatureprofileasa functionof depthnearthesteadystate.. . 24

13 (a)Detailsof aSi trenchdeviceshowing structureandmetalizationand(b) transientthermal

responseof thedevice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

14 Four temperaturedistributionsin a Si trenchdevice at increasingtimes. . . . . . . . . . . . 26

16

(a) (b)

Figure1: (a) A 2D quadtreemeshshowing refinementand(b) a single3D elementwith 4 blocksabove, 2on theright and1 onall othersides.

Ri

ZiZj

Rj

nj

in

(a) (b)

Figure2: (a)TLM transmissionline links in aquadtreeand(b) detailsof asinglelink in thequadtreemesh.

17

(a)

n1

n2

WL

T1 T2

T T2

T

1

3

n1

n2

n3

L

W

(b)

16totC

16totC

16totC

16totC

tot8

Ctot8

C

4C

tot8

C

tot8

C

tot

Figure3: Geometryfor a2D meshfor calculating(a) resistancesand(b) capacitances.

18

RjZi

Z

n

n

n

n

Gi

i

j+1

j

j

Ri

j+2

Figure4: A generalTLM nodefor aQTM mesh.

19

Uniform Mesh QTM Mesh

model

z

x

y

� � u�rKb )�  � £¥¯ � bfu x

� � b-¨ )�  � £#¯ � O ] x

Figure5: Meshgeometriesandtemperaturedistributionsfor a2D rectangularslab.

20

(a)0°

50°

100°

150±Time (µs)

0

10

20

30

40

Tem

pera

ture

Ris

e (K

)

Analytical²Grid Mesh TLM³Quad Tree Mesh TLM´

(b)0µ

50µ

100µ

150¶Time (µs)

0

0.001

0.002

0.003

0.004

Nor

mal

ized

Err

or· FD uniform mesh (1024)FD QTM (220)TLM uniform mesh (1024)¸TLM QTM (350)¸TLM QTM (220)¸

Figure6: Transientresponseof the2D rectangularslabshowing (a) temperatureand(b) residualerror.

(a)0¹

20¹

40¹

60¹

x (µº m)

0

10

20

30

40

Tem

pera

ture

Ris

e (K

) Analytical »Uniform mesh TLMQTM TLM¼

t=15 ½

µs

t=1.0 ½

µs

t=0.1 ½

µs

(b)0¹

20¹

40¹

60¹

y (µº m)

0

10

20

30

40

Tem

pera

ture

Ris

e (K

)

Analytical»Uniform mesh TLMQTM TLM¼

t=15 ½

µs

t=1.0 ½

µs

t=0.1 ½

µs

Figure7: Spatialtemperaturedistributionsthroughthecenterof therectangularslab(a) in thex-directionand(b) in they-direction.

21

(a)0¾

50¾

100¾

150¿Time (µs)

0

10

20

30

40

50

Tem

pera

ture

Ris

e (K

)

AnalyticalÀG = 0.5%ÁG = 5%ÁG = 10%Á

0Â

50Â

100Â

150ÃTime (µs)

10−7

10−6

10−5

10−4

10−3

10−2

10−1

eÄG = 0.5%ÅG = 5%ÅG = 10%Å

(b)0¾

50¾

100¾

150¿Time (µs)

0

10

20

30

40

50

Tem

pera

ture

Ris

e (K

)

AnalyticalÀMax Æ

∆t = 100τmin

Max Æ

∆t = 50τmin

Max Æ

∆t = 20τmin

0Ç

50Ç

100Ç

150ÈTime (µs)

10−7

10−6

10−5

10−4

10−3

10−2

eÉMax Ê

∆t at 100τmin

Max Ê

∆t at 50τmin

Max Ê

∆t at 20τmin

(c)0¾

50¾

100¾

150¿Time (µs)

0

10

20

30

40

50

Tem

pera

ture

Ris

e (K

)

AnalyticalÀ1%2.5%Ë5%¿10%

0Â

50Â

100Â

150ÃTime (µs)

10−7

10−6

10−5

10−4

10−3

10−2

eÄ1%Ì2.5%Í5%Ã10%Ì

TemperatureResponse AverageErrorParameter

Figure8: Transientresponseof theslabusinga QTM with differentstrategiesfor non-uniformtime step-ping suchas(a) unconstrainedgrowth, (b) 1% growth constrainedto a maximumtime stepand(c) growthconstrainedby limiting �8£#¤¥¦ to ¨ s bfu ov .

22

0Î

50Î

100Î

150ÏTime (µs)

0

2000

4000

6000

8000

10000

Num

ber

of It

erat

ions

G = 0.5%ÐG = 1%, Max Ð

∆t of 20τmin

G = 2.5%, e < 5x10Ð −4

Uniform time steps at 0.1τmin

Figure9: Totalnumberof iterationsasa functionof simulationtime.

(a)0Ñ

50Ñ

100Ñ

150ÒTime (µs)

0

10

20

30

40

50

Tem

pera

ture

Ris

e (K

)

Nonconstant κConstant Ó

κ

(b)0Ô

20Ô

40Ô

60Ô

x (µÕ m)

0

10

20

30

40

50

Tem

pera

ture

Ris

e (K

)

Steady StateÖTLM at 15 ×

µsTLM at 5 ×

µsTLM at 1 ×

µsTLM at 0.1 ×

µs

Figure10: Effect of a temperature-dependent heatconductivity on (a) transientresponsein theslabcenterand(b) temperatureprofile in thex-direction.

23

Figure11: Grayscalerepresentationof thetemperaturedistributionat100 ) sin arectangular3D solidusingaQTM mesh.Thedarkestregionscorrespondto atemperatureriseof 65 x . Dueto thesymmetry, only onequadrantof theblock is shown.

(a)0Ø

20Ø

40Ø

60Ø

80Ø

100Ø

Time (µs)

0

20

40

60

80

Tem

pera

ture

Ris

e (K

)

AnalyticalÙQuad Tree TLMÚ

0Ø0.002Ø0.004Ø0.006Ø0.008Ø

Nor

mal

ized

Err

or

(b)0Û

10Û

20Û

30Û

40Û

50Û

Depth (µm)

0

20

40

60

80

Tem

pera

ture

Ris

e (K

)

AnalyticalÜQuad Tree TLMÝ

Figure12: Temperatureresponsewithin a3D rectangularsolidshowing (a) transientresponseof maximumtemperatureand(b) temperatureprofileasa functionof depthnearthesteadystate.

24

(a) (b)0Þ

5ß

10Þ

15ß

Time (µs)

0

2

4

6

8T

empe

ratu

re R

ise

(K)

Figure13: (a) Detailsof a Si trenchdevice showing structureandmetalizationand(b) transientthermalresponseof thedevice.

25

t = 0.05 )�  , 3 � £#¯ � yàráu K t = 0.25 )�  , 3 � £#¯ � yàrzâ K

t = 5.0 )�  , 3 � £#¯ ��ã ráu�¨ K t = 15.0 )�  , 3 � £¥¯ ��ã r ] â K

Figure14: Four temperaturedistributionsin aSi trenchdevice at increasingtimes.

26