Voronoi diagrams for VLSI manufacturing: Robustness and Implementation

VoronoidiagramsforVLSImanufacturing:
RobustnessandImplementation
EvanthiaPapadopoulou
IBMT.J.WatsonResearchCenter
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20021
TalkOverview

VLSILayout

VLSIData



:simplerVoronoidiagramforVLSIapplications(smalldegree)

CriticalAreaComputationProblem
ImportantprobleminVLSIYieldPrediction

Criticalarea

variationsof


VoDs

Planesweepconstructionof


VoD

Voronoi-basedCADtoolforCriticalAreaExtraction:tobeusedbyIBM
manufacturingin’03
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20022
VLSILayout

Layersofdifferentmaterials

Layer:shapesrealizingdevices(e.g.transistors,interconnect)
Gate:PC(polysilicon)

RX(diffusion)
Interconnect:M1,M2,M3.Contacts:CA,V1

VLSILayout:Compacthierarchicalformfollowinglogical(notphysical)hierarchy

Librarycell:groupofshapes(e.g.inverter).Mayappearinhundredsofplaces

ASICchip:thousandsofinterconnectedcells(Manyrepeatingcells)

HierarchicalLayout:compact,cellrepetitionusingtransistions.Logical,not
physicalhierarchy

FlatLayoutprintedinMask:millionsofshapes.Notavailable
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20023
VLSIData

Vastmajorityofsegmentsareaxisparallel




orientationsverycommon.Otherorientationspossible.Constantorientationscanbe
assumed.

Coordinatesonintegergrid
(verylargeintegers)

Degenerateconfigurationsaretherule

Noperturbationtechniquestoavoiddegeneracies:destroyaxisparallelproperty

Datavolumeofflatdata:orderofmillions(evenbillions)
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20024
VLSIProximityProblems

Width-spacinginteractionsessential.E.gnoise,yield,polishing

DesignRuleCheck:Checkwidth-dependentspacingrules

Lithography:Printingofshapesdependsonwidth/neighborinteractions

YieldPrediction:CriticalAreaestimation

Medialaxis/Voronoidiagram:
Addressrobustnessissue.AdapttoVLSIhierarchy

OurVoronoitoolforCriticalArea
–Robustnessissue:use


metric
–Constructionnothierarchical:Basedonplanesweep.
NeverkeepVoDinmemory:onlyactiveportionnearscanline
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20025
CriticalAreaProblem
VLSIYield:Percentageofmanufacturedchipsthatareworkingoverallchips
manufactured
Defects:Dustparticles,Contaminantsonmaterials
HighYield

Profit
LowYield

Loss
YieldPrediction:ControlCostofManufacturing


 

 


 

 
ave.#defectsperunitarea,


aclusteringparameter,



thecriticalareaoflayout
CriticalArea:measurereflectingsensitivityoflayouttodefects
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20026
DefectTypes
Twotypesofdefects:

Extramaterial

Shorts

Missingmaterial

Opencircuits
MissingMaterialdefects:Breaks,Via
Blocks
Defectofsize

=circle/squareofradius

E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20027
CriticalArea
CriticalArea:


















:
areaofcriticalregionfordefectsize

Criticalregion:
locusofpointswhereifadefectofradius

iscenteredcausesacircuitfailure






 

:
densityfunctionofthedefectsize
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,20028
CriticalRegion(
CriticalRegion(
CriticalRegion(
CriticalAreaviaVoronoidiagrams
Shorts



2ndorder


Voronoidiagramofpolys
Papadopoulou&Lee99
Opens



(weightd)


VordgrmofMAsegmts
Papadopoulou01
ViaBlocks





min-Max(Hausdorff)Voronoidgrmofpolys
Combinatorialstructureofindependentinterest
Papadopoulou01





termsderivedfromVoronoiedges

Bound:


 








metric

squaredefectmodel
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200212
Why

?
Algorithmicdegree
Liotta,Preparata,Tamassia96
Testcomputationseval.multivariatepolynomialsofarithmeticdegree

.Testcomputationsbitprecision:

 


(input

-bitintegers)


in-circletest(segments):
degree



v
(Burnikel96)


in-circletest(segments):
degree


l
i
l
k
l
query
j
(Papadopoulou&Lee99,01)
VLSIshapes:slopesaresmallconstants

degree1
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200213

Quadrants
p
north,southquadrant:


dist=verticaldist
eastwestquadrant:



horizontaldistance
Non-orthogonalsegments:
Considersix


rays
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200214

Bisector
q
p
q
p
q
p
Degeneracy:Collinearaxisparallelpoints
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200215

Bisectors:Straight-linesegments
L
p
L
p
Bisectorofapointandalinein


Consistsof


parts,oneforeachquadrantof

Theunboundedportionsarealways


rays
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200216

VoronoiDiagram
Straight-lineskeleton
Ortho45polygons:8orientations:


  

 
 


-slope
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200217

Voronoidiagramofsegments

Straight-lineskeleton

Maintainsproximityinformationin


(goodenoughforpracticalapplications)

VoronoiverticesonRationalcoordinates
(assuminginputonrationalcoordinates)
Ortho-45shapes:integercoordinatesaftermultiplyingby12

In-circletest,degree5
(


:degree40)
VLSIshapes:slopessmallconstants

degree1


 




-timeplanesweepalgorithm
(basedonwavefrontparadigmofDehneandKlein97andoriginalplanesweepofFortune87
Algorithmicdegree:7
Papadopoulou&Lee99,01
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200218
PlaneSweep
L
t


PlaneSweep:
Sweep-line

sweepingfromlefttoright
Papadopoulou&Lee99,01
basedonFortune87,Dehne&Klein97


:
polygons,portionsofpolygonstotheleftof

Maintain:Voronoi


 


Wavefront:
boundaryofVoronoicellof

SpikeBisectors:
bisectorsincidenttowavefront
Boundary:
treatedasspikebisector
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200219
Data-Structures



















scanline face

PlanarSubdivisionDataStructure(PSD):Half-edgedatastructure
(const.degree)

Wavefront:Red-Blacktree
Parameterizespikeendpoints



 




ascenterofimpliedsquare.Key:




.TiesresolvedbyPSDorder

Eventlist:Priorityqueue
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200220
Events

SiteEvents:endpointsofsegments
Priority:



SpikeEvents:intersectionsofneighboringspikebisectors
Priority:rightsideofimpliedsquare
HandlingofEvents:

PSDOperations:CreateFace,SplitFace,SplitEdge,ContractEdge

AssignmentofVoronoi/GeometricData:slope,owner,spikeequation
constantorientations

lookuptables

Wavefrontsearchesby


-degrays
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200221
ExampleEventHandling
b
a



















Searchwavefrontfor

,


FixPSDverticesbetween

,

using


 





UpdatePSD:createfournewfaces

AssignVoronoidata:slopes,owners,newspikeequations
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200222
ExampleofSiteEvent
F1
F3
F2
F2
F3
F1

UpdatePSD

AssignVoronoidata:newslope,spikeequation
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200223
VoronoiCriticalAreaTool
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200224
SomeExperimentalResults
Machine:RS6000/PPCbi-processormachine,processorspeed200MHz,memory
8GBs(single-processorrun)
Window:








micronsabout



chip
ScanLineProcessing
Level
M1M2
Time(min:sec)








Peakmemory(MBs)

 






E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200225
SomeExperimentalResults(continued)
FirstOrderVoronoi(nofacedeletions)
Level
M1M2
Time(min:sec)








Peakmemory(MBs)








 
N.VoronoiFaces












TotalN.Edges











ShortsVoronoi
Level
M1M2
Time(min:sec)









Peakmemory(MBs)








 

N.2ndOrderVoronoiFaces




 
 




N.Polygons






 




TotalN.Edges












E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200226
Voronoidiagramforshorts
2ndorderVoronoidiagramofpolygons:everyregionhasauniqueownerwhichis
responsibleforshortswithintheregion









,


2ndorderVoronoicell
(Papadopoulou&Lee99,01)
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200227
CriticalAreaIntegral












 
















 


























 








k
r
k
r
j
r
e
j
r
e


lengthofverticalside,



maxcriticalradius,



mincriticalradius
Addupformulas

internalterms




,






cancelout
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200228
CriticalArea

SummationofVoronoiedges
+
r
i
r
j
k
r
e
-
Criticalareawithin

:







































 










 


ExactCriticalAreaintegrationinoneLayoutpass

  




Papadopoulou&Lee99,01
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200229
VoronoidiagramforOpens
Papadopoulou01
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200230
VoronoidiagramforVia-Blocks


Min-MaxVDofrectangles
Papadopoulou01
Voronoidiagramunder







 




 




VoronoidiagramofpolygonsunderHausdorffdistance
Papadopoulou&Lee02
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200231
Conclusion

CriticalAreaExtractionTool
–Basedon


Voronoidiagramofsegments
–PlaneSweep:unfoldslayouthierarchyonfly
–DataVolumeofChip:breakintosmallwindows

Advantageof


metricforVLSIapplications

Current/FutureWork
–Usestatisticsonmanysmallwindows
–TakeadvantageofLayoutrepetition
E.Papadopoulou,IBMT.J.WatsonResearchCenterDIMACSWorkshopDec.4-6,200232