USF Libraries
USF Digital Collections

Sequential quantum dot cellular automata design and analysis using Dynamic Bayesian Networks

MISSING IMAGE

Material Information

Title:
Sequential quantum dot cellular automata design and analysis using Dynamic Bayesian Networks
Physical Description:
Book
Language:
English
Creator:
Venkataramani, Praveen
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla
Publication Date:

Subjects

Subjects / Keywords:
Probabilisitic modelling
QCA
Bayesian networks
Quantum-dot cellular automata
Dissertations, Academic -- Electrical Engineering -- Masters -- USF   ( lcsh )
Genre:
non-fiction   ( marcgt )

Notes

Summary:
ABSTRACT: The increasing need for low power and stunningly fast devices in Complementary Metal Oxide Semiconductor Very large Scale Integration (CMOS VLSI) circuits, directs the stream towards scaling of the same. However scaling at sub-micro level and nano level pose quantum mechanical effects and thereby limits further scaling of CMOS circuits. Researchers look into new aspects in nano regime that could effectively resolve this quandary. One such technology that looks promising at nano-level is the quantum dot cellular automata (QCA). The basic operation of QCA is based on transfer of charge rather than the electrons itself. The wave nature of these electrons and their uncertainty in device operation demands a probabilistic approach to study their operation. The data is assigned to a QCA cell by positioning two electrons into four quantum dots. However the site in which the electrons settles is uncertain and depends on various factors.In an ideal state, the electrons position themselves diagonal to each other, through columbic repulsion, to a low energy state. The quantum cell is said to be polarized to +1 or -1, based on the alignment of the electrons. In this thesis, we put forth a probabilistic model to design sequential QCA in Bayesian networks. The timing constraints inherent in sequential circuits due to the feedback path, makes it difficult to assign clock zones in a way that the outputs arrive at the same time instant. Hence designing circuits that have many sequential elements is time consuming. The model presented in this paper is fast and could be used to design sequential QCA circuits without the need to align the clock zones. One of the major advantages of our model lies in its ability to accurately capture the polarization of each cell of the sequential QCA circuits.We discuss the architecture of some of the basic sequential circuits such as J-K flip flop (FF), RAM memory cell and s27 benchmark circuit designed in QCADesigner. We analyze the circuits using a state-of-art Dynamic Bayesian Networks (DBN). To our knowledge this is the first time sequential circuits are analyzed using DBN. For the first time, Estimated Posterior Importance Sampling Algorithm (EPIS) is used to determine the probabilistic values, to study the effect due to variations in physical dimension and operating temperature on output polarization in QCA circuits.
Thesis:
Thesis (M.S.E.E.)--University of South Florida, 2008.
Bibliography:
Includes bibliographical references.
System Details:
Mode of access: World Wide Web.
System Details:
System requirements: World Wide Web browser and PDF reader.
Statement of Responsibility:
by Praveen Venkataramani.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 58 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 002059583
oclc - 503563189
usfldc doi - E14-SFE0002787
usfldc handle - e14.2787
System ID:
SFS0027104:00001


This item is only available as the following downloads:


Full Text

PAGE 1

Sequential Quantum-Dot Cellular Automata Design And Analysis Using Dynamic Bayesian Networks by Praveen Venkataramani A thesis submitted in partial fulfillment of the requirements for the degree of Master of Electrical Engineering Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Sann jukta Bhanja, Ph.D. Paris Wiley, Ph.D. Wilfrido A. Moreno, Ph.D Date of Approval: October 29, 2008 Keywords: probabilisitic modelling, qca, baye sian networks, quant um-dot cellular automata Copyright 2008 Praveen Venkataramani

PAGE 2

DEDICATION Tomyparents.

PAGE 3

ACKNOWLEDGEMENTS IwouldliketotakethisopportunitytothankmyprofessorDr.SanjuktaBhanja,whohad guidedmethroughoutmyGraduatestudies.Shehadhelpedmeinlearningdifferentaspects inresearchandtothinkasaresearcher.Hervaluableadvicehadhelpedmethroughoutmy researchwork. IthankDr.ParisH.WileyandDr.WilfredoA.Morenoforservinginmycommittee. Ithankmyfriends,familyandcolleagueswhohadgivenmesupportandhadboostedmy condencewhileitwaslow.

PAGE 4

TABLEOFCONTENTS LISTOFTABLESiii LISTOFFIGURESiv ABSTRACTvii CHAPTER1INTRODUCTION1 1.1Motivation2 1.2NoveltyofthisWork2 1.3ContributionofthisWork4 1.3.1ReliabilityAnalysis4 1.4OrganizationofthisWork5 CHAPTER2QUANTUM-DOTCELLULARAUTOMATA6 2.1History6 2.2ComputingusingQCA8 2.2.1ElementsofQCA9 2.2.1.1QuantumDots9 2.2.2QCACells10 2.3MechanicsofQCA12 2.4LogicDevicesinQCA16 2.4.1MajorityLogicSynthesis17 2.5ModelingQCADesigns22 CHAPTER3DYNAMICBAYESIANMODEL25 3.1Introduction25 3.2QuantumMechanicalProbabilities25 3.3OverviewofBayesianModeling27 3.4DynamicBayesianModel30 3.4.1SequentialDesignofJKFFinQCAusingDBN33 3.4.2DesignofaSingleMemoryCellinQCAusingDBN34 3.4.3Designofs27SequentialBenchmarkCircuitinQCAusingDBN36 3.5EstimatedPosteriorImportanceSampling36 CHAPTER4VALIDATIONOFDBNMODEL41 i

PAGE 5

CHAPTER5ANALYSISOFSEQUENTIALQCACIRCUITS44 5.1Introduction44 5.2TemperatureCharacterization45 5.3CharacterizationUsingCellDimensions50 CHAPTER6CONCLUSIONANDFUTUREWORKS55 REFERENCES56 ii

PAGE 6

LISTOFTABLES Table4.1.ComparisonoftheResultsobtainedfromQCADesginerandDBNModel, forJKFFat2.0Kusing18nmCells42 Table4.2.ComparisonoftheResultsobtainedfromQCADesginerandDBNModel, forRAMat2.0Kusing18nmCells42 Table4.3.ComparisonoftheResultsobtainedfromQCADesginerandDBNModel, forS27at2.0Kusing18nmCells43 iii

PAGE 7

LISTOFFIGURES Figure1.1.TableofEmergingLogicDevices3 Figure2.1.UnpolarizedQCACell11 Figure2.2.QCACellanditsOrientations11 Figure2.3.BinaryWireinQCA11 Figure2.4.DataTransferinQCA12 Figure2.5.InniteWell14 Figure2.6.PlotoftheEnergySplittingbetweentheGroundStateandtheFirstExcited StateofanElectronundergoingAdiabaticSwitching[1].15 Figure2.7.TypicalAdiabaticClockingOperationinQCA[2].16 Figure2.8.MajorityGate17 Figure2.9.MajorityComputations17 Figure2.10.InverterDesigninQCA18 Figure2.11.ANDGateusingMajorityFunctioninQCA18 Figure2.12.ORGateusingMajorityFunctioninQCA18 Figure2.13.NANDGateusingMajorityFunctioninQCA19 Figure2.14.FullAdderDesigninQCA21 Figure2.15.MajoritySimplicationofJKFF22 Figure2.16.JKSchematicfromg.2.15.22 Figure2.17.JKFFDesignusingQCADesigner23 Figure3.1.ASmallBayesianNetwork27 Figure3.2.JKFFDesignedinQCADesignerusing18nmCells32 Figure3.3.JKFFUnraveledfortwoTimeSlicesusingDBN33 iv

PAGE 8

Figure3.4.SchematicofRAM34 Figure3.5.RAMProposedinciteWalus0335 Figure3.6.RAMDesignedusingDBN35 Figure3.7.Schematicofs27SequentialBenchmarkCircuit36 Figure3.8.s27SequentialBenchmarkCircuit37 Figure3.9.s27unraveledfortwoTimeSlicesusingDBN38 Figure5.1.ResultsforOutputPolarizationVersusTimetatDifferentTemperatures forJKFFCircuit.45 Figure5.2.ResultsforLoopPolarizationVersusTimetatDifferentTemperaturesfor RAM[3]CircuitDuringWriteMode(R/W=0).46 Figure5.3.ResultsforLoopPolarizationversusTimet,whenInput=0andtheValue intheFeedback=1atDifferentTemperaturesforRAM[3]CircuitDuring ReadMode(R/W=1).47 Figure5.4.ResultsforLoopPolarizationversusTimet,whenInput=1andtheValue intheFeedback=0atDifferentTemperaturesforRAM[3]CircuitDuring ReadMode(R/W=1).47 Figure5.5.ResultsForOutputPolarizationVersusTimetfortheInputsIN0=1IN1=1 IN2=1IN3=1atDifferentTemperaturesforS27BenchmarkCircuit49 Figure5.6.ResultsForOutputPolarizationVersusTimetfortheInputsIN0=1IN1=0 IN2=0IN3=1atDifferentTemperaturesforS27BenchmarkCircuit49 Figure5.7.ResultswithOutputPolarizationversusTimetatDifferentCellDimensionsforJKFFCircuit50 Figure5.8.ResultswithLoopPolarizationversusTimetatDifferentCellDimensions forRAMCircuitDuringWriteMode(R/W=0).51 Figure5.9.ResultswithLoopPolarizationversusTimet,whenInput=0andtheValue intheFeedback=0AtDifferentCellDimensionsForRAMCircuitDuring ReadMode(R/W=1).52 Figure5.10.ResultswithLoopPolarizationversusTimet,whenInput=1,Valueinthe Feedback=0atDifferentCellDimensionsforRAMCircuitDuringRead Mode(R/W=1).52 Figure5.11.ResultswithOutputPolarizationversusTimet,fortheInputsIN0=1IN1=1 IN2=1IN3=1,atDifferentCellDimensionsForS27BenchmarkCircuit53 v

PAGE 9

Figure5.12.ResultswithOutputPolarizationversusTimet,fortheInputsIN0=1IN1=0 IN2=0IN3=1,atDifferentCellDimensionsforS27BenchmarkCircuit53 vi

PAGE 10

SEQUENTIALQUANTUMDOTCELLULARAUTOMATADESIGNANDANALYSIS USINGDYNAMICBAYESIANNETWORKS PraveenVenkataramani ABSTRACT TheincreasingneedforlowpowerandstunninglyfastdevicesinComplementaryMetal OxideSemiconductorVerylargeScaleIntegration(CMOSVLSI)circuits,directsthestream towardsscalingofthesame.Howeverscalingatsub-microlevelandnanolevelposequantum mechanicaleffectsandtherebylimitsfurtherscalingofCMOScircuits.Researcherslookinto newaspectsinnanoregimethatcouldeffectivelyresolvethisquandary.Onesuchtechnology thatlookspromisingatnano-levelisthequantumdotcellularautomata(QCA).ThebasicoperationofQCAisbasedontransferofchargeratherthantheelectronsitself.Thewavenature oftheseelectronsandtheiruncertaintyindeviceoperationdemandsaprobabilisticapproachto studytheiroperation. ThedataisassignedtoaQCAcellbypositioningtwoelectronsintofourquantumdots. Howeverthesiteinwhichtheelectronssettlesisuncertainanddependsonvariousfactors.Inan idealstate,theelectronspositionthemselvesdiagonaltoeachother,throughcolumbicrepulsion, toalowenergystate.Thequantumcellissaidtobe polarized to+1or-1,basedonthealignment oftheelectrons. Inthisthesis,weputforthaprobabilisticmodeltodesignsequentialQCAinBayesian networks.Thetimingconstraintsinherentinsequentialcircuitsduetothefeedbackpath,makes itdifculttoassignclockzonesinawaythattheoutputsarriveatthesametimeinstant.Hence designingcircuitsthathavemanysequentialelementsistimeconsuming.Themodelpresented inthispaperisfastandcouldbeusedtodesignsequentialQCAcircuitswithouttheneedto vii

PAGE 11

aligntheclockzones.Oneofthemajoradvantagesofourmodelliesinitsabilitytoaccurately capturethepolarizationofeachcellofthesequentialQCAcircuits. WediscussthearchitectureofsomeofthebasicsequentialcircuitssuchasJ-Kipop (FF),RAMmemorycellands27benchmarkcircuitdesignedinQCADesigner.Weanalyze thecircuitsusingastate-of-artDynamicBayesianNetworks(DBN).Toourknowledgethisis thersttimesequentialcircuitsareanalyzedusingDBN.Forthersttime,EstimatedPosterior ImportanceSamplingAlgorithm(EPIS)isusedtodeterminetheprobabilisticvalues,tostudythe effectduetovariationsinphysicaldimensionandoperatingtemperatureonoutputpolarization inQCAcircuits. viii

PAGE 12

CHAPTER1 INTRODUCTION Transistorshavebeenthefundamentaldeviceinalmostalloftoday'stechnologies.Gordon Moore,co-founderofIntelCorporation,observedinearly1960'sthat,thenumberoftransistors thatcouldbeinexpensivelyembeddedinaspecicareaonachipwouldincreaseexponentially anddoubleevertwoyears.Thishasbeenthechallengeintransistortechnologyeversince.Industrieshavelinkedeverymeasureofdevicecapabilitieswiththislaw.Transistorshaveevolved throughmanylevelsoftechnologytothecurrentandmostwidelyusedcomplementarymetal oxidesemiconductortechnology(CMOS).Semiconductorindustrieshaveusedthistechnology tocrowddeviceslikemicroprocessorswithbillionsoftransistorstoday.Eventhoughtheyhave beenabletokeepupwithMoore'slawbyshrinkingthesizeoftransistorsfrommicronstosubmicronsandnowtonanolevels,researcheshaveshownalimitationinthedeviceoperationat suchsmallsizes.Thisismainlyduetotheinterventionofquantummechanicalpropertiesof electronsatsuchminisculesizes.Itproducedmanyundesiredeffectsindeviceoperation,such aselectrontunnelingandpowerdissipation,whichhinderedfurtherscalingofintegratedcircuits. HoweverindustriesstillstrivetokeepMoore'slawtruebyreducingdevicesizestonanometer levelwhilealsolookingintoalternativesthatcouldaidthehighspeedoperationwithoutthe problemsexperiencedatsubnanolevelCMOStechnologies.Nowresearchersacrosstheworld havebeenproposingnovelmethodstosatisfytheevergrowingneedforhighspeeddevicesand havestartedtolookbeyondCMOS,intothepropertiesoftheelectrons. 1

PAGE 13

1.1Motivation Itisclearfromthepacingtechnologicalgrowththatnano-scaledevicesarealreadyinproduction.ThisisevidentfromthenewIntelprocessorsthatuse32nmlogictechnologytodesign SRAMwith1.9billiontransistors[4].HoweverdesigningandfabricatingCMOSatnanolevel comeswithacost.Issuesrelatedtooxidethickness,powerdissipation,andthermalreliability arisestoquestion.Soindustriesareinsearchfornewmaterialsanddesignsthatcouldaidthe scalingofCMOS,buttheyarealsowellawarethatCMOStechnologycouldbecontinuedonly foradecade[5].Someoftheemergingalternativesincludecarbonnanotubes,spintronics,spinvalvesetc[5].Figure1.1.showtheemerginglogicdevicesandalongwiththeirperformance parameters. Thesecircuitshavedevicesizesnearatomicandmoleculardimensions.Atsuchsmallsizes thequantummechanicaleffectsaremoredominant.Quantum-dotcellularautomation(QCA)is onesuchtechnologythatpromisesthefutureofnano-technologies.QCAexploitsthequandary ofdevice-deviceinteractionthatexistsinCMOSscalingatnano-level,byusingdeviceinteractionsfordatapropagation.Designingsuchcircuitsrequireanewdesignmethodologythat couldhelptocontrolthedirectionofpropagationofdata.TheupperhandofQCAdevicesover conventionalCMOScircuits,apartfromtheoneputforthaboveare,theabsenceofmetallic interconnectswhichisthemainsourceofIRlosses,theabsenceofelectronowforcharge transferandofcourseitsextremelowpowerconsumption. 1.2NoveltyofthisWork Researchershavedevelopedseveralmodelsfordefectcharacterizationanddesigntominimizetheuncertaintyofpropercircuitoperation.Thisuncertaintyofthecircuitaidsindeveloping aprobabilisticmodelforanalysis.OnesuchmodelistheBayesiannetwork(BN)modeling[6], whichexploitsthecausalrelationshipsinclockedQCAcircuitstoobtainamodelwithlow complexity.Itisbasedondensitymatrixformulationsandalsotakesthedependenciesinduced 2

PAGE 14

Figure1.1.TableofEmergingLogicDevices byclockingofcells.OneofthemanyinterestingfeaturesofBNisthatitnotonlycaptures thedependenciesexistingbetweentwoQCAcells,butcanalsobeusedtoconductsteadystate operationswithouttheneedfortemporalcomputationofquantummechanicalequations. Manycircuitshavebeendesignedandanalyzedusingthismodel.Mostlythecircuitsare combinationalinnature,andwereanalyzedfordefectsandthermalrobustnesstonameafew[7]. TheinfeasibilityinusingtheBNmodelforsequentialdesignistheacyclicnatureofthemodel. Inthiswork,weuseanextendedmodeltodesignandanalyzetheeffectsonsequentialcircuitsbytimecouplingtheBN.Themodelcanaccuratelycapturethedevicecharacteristicsand 3

PAGE 15

provideresultsfasterthanthetraditionalmethodsthatusetimeconsumingquantummechanicaliterations.WeusethedynamictimecoupledBayesianmodeltoanalyzethephysicaland thermalreliabilityasameasureofpolarization.Tothebestofourknowledgethisistherst evermodelthatprovidesarealisticworkforreliabilityanalysisonsequentialQCAcircuits.In thisworkwehaveusedanewsamplingalgorithmforbeliefpropagationknownasEstimated PosteriorImportanceSampling(EPIS)algorithm.Itestimatestheprobabilityofposteriornodes withbothaccuracyandspeed.Themainadvantageofthisalgorithm,overthecommonlyused clusteringalgorithm,isthatitcanbeusedforanycircuitindependentofthecircuitsizewithout anycompromiseonthespeed. 1.3ContributionofthisWork Asanemergingtechnology,manyworksarebeingperformedinQCA.Amongtheother interestingareasforexplorationwechosetoexplorethesequentialdesigninQCAcircuitsdue toitsdynamicnature. 1.3.1ReliabilityAnalysis Whiletherehasbeenanalysisofcombinationalcircuitsasameasureofthermalrobustness earlierin[7],andmanydesignsinQCAusecombinationallogictobuildALU,microprocessors andFPGA[8]wheretheeffectofpolarizationisnotamajorfactor.Memorycircuits[9,10,11] havegreatereffectonpolarizationduetothesequentialnatureandrequiresindepthanalysisto determineanoptimalconditionthatwillaidtheefciencyofthesystem.SimilarstudiesinQCA includesdefectcharacterizationandthermaleffects[12],[13]and[2].Inthiswork,weanalyze thereliabilityofsequentialcircuitswithrespecttoelectronpolarizationundervariousthermal andphysicalconditions. 4

PAGE 16

1.4OrganizationofthisWork Inchapter2weexplainthefundamentalsbehindQCA.Wealsolookintofewlogicdevices builtinQCAandtechniquesusedtobuildthem.Inchapter3,welookintoanoverviewofBN andthedetailsofDBNwithexamples.Inchapter4,wevalidatethemodelwithagroundtruth simulatedinQCADesignerandcomparetheresultswithdetailedexplanations.Inchapter5, weanalyzesequentialcircuitssuchasJKFF,RAM,ands27benchmarkcircuits,undervarying operatingtemperatureandphysicaldimensions,fortheeffectonoutputpolarization. 5

PAGE 17

CHAPTER2 QUANTUM-DOTCELLULARAUTOMATA 2.1History TheconceptofnanotechnologywasrstintroducedbyRichard.P.Feynmanthroughhis famouslectureonnano-technologytitled There’sPlentyofRoomattheBottom in1959.Since thenresearchershaveconstantlyworkedonmethodstoimplementcomputationatnano-level. Theterm QuantumCellularAutomata wasrstcoinedbytheresearchersGerhardGrossingand AntonZeilingertorepresenttheirmodelin1988[14].Nonethelesstheirconceptsofmodern computingwasonlyremotelyrelatedtotheconceptsintroducedbyDavidDeutsch[15]in1985 andhencefailedtodevelopintoamodelofcomputation.TherstexhaustiveresearchinQCA wasconductedbyJohnWatrous[16]. Therstproposalforimplementationofthecellularautomatatechniquesusing quantumdots wassuggestedbyCraigS.LentandPaulTougaw[17]underthename QuantumCellular Automata ,asthesuccessortothecurrentCMOStechnologybasedcomputations.Toclassifythe modelsforcomputationmethodsdesignedusingthisconcept.Eversince Quantum-dotCellular Automation hasbecometheterminologyusedtoclassifythemodelsforcomputationmethods designedthisconcept. TheQCAconceptinvolvesinkeepingthebinaryswitchoperationintroducedbyKonrad Zuse,butreplacingtheswitchwithacellhavingbi-stablechargeconguration,whereone congurationofchargerepresents0andtheother1,forthe OFF and ON statesrespectively. Withtheaidofclockingschemetomodulatetheeffectivebarrierbetweenthetwostates,we couldestablishsufcientsupportforgeneral-purposecomputing.WhileQCAdeviceshave beendemonstratedtoworkundercryogenictemperatures,workisunderwaytoimplementsuch 6

PAGE 18

devicesatroomtemperaturesusingmoleculesaschargecontainersorbyusingmagneticdots thatcouldalignitselfaspertheinputdata. Thereareanumberofresearchgroupsinleadingresearchlabsaroundtheworldworking onQCA.TheresearchgroupatUniversityofNotredamehasbeenspearheadingQCAresearch formorethanadecade.AnothergroupcreditedwiththeadvancementofQCAresearchisfrom UniversityofPisa,Italy.ThisgroupleadbyMassimoMacucciconductedananalyticalresearch inQCAinvolvingseveralinstitutionsallovertheworldundertheQUADRANTproject.Some oftheleadingresearchgroupscurrentlyinvolvedindifferentareasofQCAresearchis; 1.C.Lentet.al.,J.Timleret.alof UniversityofNotreDame -DeviceandFabricationlevel 2.D.Tougawet.alof Valparasio,IN -DeviceandLogiclevel 3.M.Macucciet.alof UniversityofPisa -DeviceandFabricationlevel 4.K.Waluset.al.of UniversityofCalgary -Logiclevel 5.P.Koggeet.al.andNiemieret.alof UniversityofNotreDame -Architectureandtesting 6.F.Lombardiet.al.andTahooriet.alof NorthEasternUniversity -Architectureandtesting 7.K.Wanget.al.of UniversityofCalifornia-LosAngels -Architectureandtesting 8.J.Abrahamet.al.of UnivofTexas,Austin -Architectureandtesting 9.N.Jhaet.al.of PrincetonUniversity -Architectureandtesting 10.R.Karriet.al.of PolytechnicUniversityBrooklyn -Architectureandtesting 11.K.Wuet.al.of UniversityofIncheon,Korea -Architectureandtesting 12.K.Kimet.al.of UniversityofIllinois,Chicago -Architectureandtesting 13.E.Peskinet.alof RIT -Architectureandtesting 14.A.Orailogluet.alof UniversityofCalifornia-SanDiego -Architectureandtesting 7

PAGE 19

15.NASAJetPropulsionLab-Architectureandtesting 16.M.Liebermanet.al.,T.Felhneret.al.,G.Bernsteinet.al,G.Snideret.al.,W.Porodof Univ ofNotreDame -Fabricationlevel 17.A.Dzuraket.al.of UniversityofNewSouthWales -Fabricationlevel 18.D.Jamiesonet.al.of UniversityofMelbourne -Fabricationlevel MostoftheresearchgroupsareeitherinvolvedinQCAtestingandotherarchitecturalissues orinthefabricationofQCA.Atthelogiclevel,QCAresearchreceivedagreatboostfromthe workdoneattheUniversityofCalgary,underKonradWalus.Thisgroupintroducedtherst eversimulatorknownasQCADesigner[9].EventodayQCADesignerisamongsttheleading QCAdesignandsimulationtoolusedallovertheworld. 2.2ComputingusingQCA ItisinterestingandsimpletocomeupinvariousmethodstorepresentbinaryvaluesinQCA physicallywithonebasicidea-useofchargecongurationofacell.Thesechoicesareofthe following; 1.Electronicchargestate. 2.Electronicchargeconguration(QCA). 3.Electronspinstate. 4.Nuclearspinstate. 5.Nuclearpositions. 6.Collectivemagneticmoment. 8

PAGE 20

7.Coherentelectronicquantumstate. 8.Superconductinggroundstate. Atthemacroscopiclevel(1)couldbeseenasaCMOSmodel.Ithasseensuccessinmemory application.Itisalsopossibletousemolecularchargecentersintheplaceofgateofthetransistorsandencodeinformationinitschargedstate.Approach(4),(7)havefounditsapplication inCoherentquantumcomputinginthegasphaseandsolidstateproposals,while(3)hasfound itselfinsolidstateCoherentquantumcomputing.Howevertheweaknessofthesemethodsis decoherence.SuperconductorsbasedQCAdeviceshaveusedacombinationof(7)and(8),but limitedtotemperature.Thebasisofconventionalmemorysystemsusetheapproach(8).Most ofthelogicsystemcouldberealizedusingthisapproachandhaveanadvantageofhavinghigh magneticcouplingenergies.Approach(2)providesfeaturesforhighspeedandmorerobust general-purposecomputingapplications. 2.2.1ElementsofQCA 2.2.1.1QuantumDots AbasicQCAcellisshowningure2.1.(a).Itisasquarecellconsistingof4-dots.These dotsareknowas quantumdotsorqdots Thefunctionofthisdotistolocalizethechargeofthe electron.Thedotisessentiallyaregionofspacewithpotentialbarrierssurroundingitwhich aresufcientlyhighandwidesothatthechargewithinitisquantizedtoamultipleelementary charge.Thisbarrier,however,mustbecontrolledsuchthattheelectronscouldtunnelthrough thematthetimeofswitching.Thebi-stabilityofQCAisnothingbutthequantizationofcharge andhenceitisimportanttoknowtherelationshipbetweentheenergylevelsofasingleparticleandtheenergylevelsofthedot.Quantumdotscanbeeithermetallic,ormolecular,ora semiconductor. Themetallicimplementationofquantumdotsconsistsofmetalislandsonaninsulating substrate.Inthistype,asingledotconsistsofbillionsoffreeelectrons.ButtheCoulomb 9

PAGE 21

costofoneelectron,tunnelingthedotwouldbelarge.Thechargingofthedotisestablished bythiselectrostaticeffect.Thesingleparticleenergyofthesedotsexistsveryclosetogetherin energyandisinsignicantduringtunneling. Themolecularimplantationofquantumdotsaresimplybasedonredoxcenters,areasin amoleculethataccepts(reduced)ordonates(oxidize)anelectronwithoutbreakingthebonds thatholdthemoleculetogether,withinthemolecule.Thesetypesofdotshaveverylargesingle particleenergylevelspacingunlikemetallicdots,andahighCoulombcostforaddingadditional chargeandhencebotheffectsarestrong. Thesemiconductorbaseddotsareformedbyelectrostaticallydepletinga2-dimensional electrongas.Themetallicpatternsonthesurfacearetypicallyusedtoshapetheconning potential.Dotscanalsobeformedfromselfassembledstructuressuchaspyramids.Dotsizes andseparationareintheorderoftensofnanometers.Thesingleparticleenergyleveldistances couldbevariedbyconstrictingtheconningpotential.Howevertheyareextremelysensitiveto variationsingeometry. 2.2.2QCACells Asdescribedintheprevioussection,thesimplestformofaQCAcellisasquarecellconsistingoffourquantumdots.Theessentialfeatureofthecellthatprovidesthebi-stablenature isthat,itpossessesanelectricquadrupolethatcouldhavetwostableorientations.Thesetwo orientationsareusedtorepresentthe0and1duringwhich,thetwostatestheelectronsoccupy antipodaldots.Thebitinformationisstoredintheformofthein-planequadrupolemoments. Intheabsenceofanyenvironmentalconditions,thetwoorientationshavethesameelectrostatic energy.Theorientationoftheneighboringcellscausesoneoftheorientationstobethedesired low-energycongurations. Theorientationoftheelectronsinthedots,onaneventofaninput,isbasedonthetunneling oftheelectronsfromonedottotheother.Theeffectivepolarizationofthecellisthecharge 10

PAGE 22

Figure2.1.UnpolarizedQCACell (a)P=+1 (b)P=-1Figure2.2.QCACellanditsOrientations distributionamongthefourdotsandisdenedas; P = ( 1+ 3) Š ( 2+ 4) 1+ 2+ 3+ 4(2.1) Where iistheelectronicchargeineachofthefourdotsofthecell.Oncepolarized,the QCAcellcouldbeinoneofthetwoorientationsasshowningure2.2.(a)and2.2.(b).Thebistableorientationsarecausedduetothecoulumbicrepulsionofelectronsandcouldbedenoted as+1and-1for1and0respectively.Howeverthesestatesarethe"mostlikely"statesbutare nottheonlystates.Thereisanegligiblepossibilityofhavinganerroneousstate. DatatransferinQCAarchitectureisestablishedbythemutualinteractionbetweenneighboringcellsduetocolumbicforces.Thusbyalteringthedrivercell,alsoknownastheinput cell,thedatacouldbealteredorpropagatedthroughtheneighboringcells.Figure2.3.(a)shows asimplebinarywiremadeofQCAcellsplacedadjacenttoeachother.Figure2.4.(a)showsa simpleQCAdatatransferoperation.Consideringtheinitialcongurationofthecellstoin+1,if thedrivercellischangedto-1,thechangeinpolarizationaffectstheimmediateneighborwhich Figure2.3.BinaryWireinQCA 11

PAGE 23

Figure2.4.DataTransferinQCA orientsitselfparalleltothedrivercell.Thisorientationaffectsitsneighborandsoon,leadingto alineartransferofdatafromthedrivercell. Asportrayedinthegurethedataistransferredinalinearfashioninabinarywire.This typeofarrangementisutilizedinbuildinginterconnectsbetweenlogicdevicesexplainedlater inthischapter.Thepolarizationofelectronsinthecelldependsonvariousconditionssuchas temperature,kinkenergy,clockenergy,andquantumrelaxationtime[18]. 2.3MechanicsofQCA Forbetterunderstandingoftheoperationofasimple4-dotQCAcell,itisusefultostudythe behaviorofelectronatthequantumlevel.Abriefoverviewofsomeofthepostulatesofquantum mechanicsisgivenbelow; 1.Thephysicaluniverseisnotdeterministic,i.e.atsub-atomiclevelwecanonlyobtain probabilitiesofoutcomesofasystembutneveractuallypredictitscertainty. 2.Bothlightanmattershowwave-likeandparticlelikecharacteristics, 12

PAGE 24

3.Undercertainconditionssomephysicalquantitiesarequantized,i.e.theycanonlytake certaindiscretevalues. Thusfromthepostulatesitisclearthatelectronscouldposesawavelikeandaparticlelike natureatquantumlevel.TheSchr odingerwaveequationoftheelectronby; Š h2 2 m 2 + V = i h t (2.2) where = ( x y z t ) and 2 = { 2 ( x y z t ) x2+ ... + 2 ( x y z t ) z2} (2.3) Soconsideringthemotionofparticleinanyonedirectionforsimplicity,equation2.2becomes Š h2 2 m 2 ( x t ) x2+ V ( x t ) ( x t )= i h ( x t ) t (2.4) Š h2 2 m 2 ( x t ) x2+ V ( x t ) ( x t )= E ( x t ) t (2.5) WhereVisthepotentialactingontheparticle, E = i h theenergyoftheparticleandmisthe mass. Theelectroninitswavenatureisconsideredtoexistinstatesnamely 1.BoundState:Theparticlemovesinanitestate. 2.UnboundState:Theparticlecouldescapetoinnity. Forthisworkweconsidertheparticletoexistinaboundstate,i.e.inan innitepotentialwell gure2.5. Apotentialwellisabarrierwithinniteenergysurroundingtheelectronsthuspreventing electronsfromtunneling.Whiletheelectronexitsinthisbarrier,thewavefunctionoftheelectronisgivenby ( x y z ) ,whichistheprobabilityofndingtheelectroninthatwell.This probabilityisproportionalto | ( x y z )2| .where ( x y z )2= ( istheconjugateof 13

PAGE 25

Figure2.5.InniteWell ).Figure2.5.showstheinnitewellwhichholdstheelectronwithcharge eŠ.Thesystem couldbedescribedwiththeboundaryconditionsas; V(x)= 00 x a 0 x a (2.6) d2 dx + 2 m h ( E Š V ( x )) = 0(2.7) ThesolutiontotheSchr odingerequationforafreeelectron(V(x)=0)isgivenas, d2 dx + 2 m h ( E ) = 0(2.8) Using k2= 2 mE / h2thisreducesto d2 dx + k2 = 0(2.9) SolutionofSchr odinger'sequationforthiswavefunctionisasin/cosfunctionanditalsogives thevalueoftheenergyofanelectronwithinapotentialwell.Theelectroncanonlyhavecertain discreteenergies( En)matchingtheallowedwavefunctions.Alower(higher)energyelectron 14

PAGE 26

Figure2.6.PlotoftheEnergySplittingbetweentheGroundStateandtheFirstExcitedStateof anElectronundergoingAdiabaticSwitching[1]. willhaveasmaller(larger)valueof k (wavevector)andalarger(smaller)wavelengthrespectively. Sincetheboundaryconditionsdemandthewavefunctiontobezeroatthewallsofthewell, thewavevectorcanonlytakediscretequantitiesandhencetheelectroncanonlyexistinquantizedenergylevels.Thespacingbetweenadjacentenergylevelsdependsonthewidthofthe potentialwell.Ifweconsidertheheightofthepotentialwelltobenite,thenthereisapossibilityfortheelectronstotunneloutofthepotentialwell. Nowwithregardstoquantumdots,eachdotinaQCAcellbehavesasapotentialbarrierfor theelectron.Incaseofaninnitepotentialwell,theelectronsarepreventedfromtunnelingdue tothehighbarrierpotential.Howeverifthebarriercouldbelowered,thentheprobabilityofthe electrontotunnelitincreases.Ifthepotentialbarrierisloweredwellenough,theelectronscould tunnelfreelybetweenthequantumdots.Oncethetunnelingisestablishedandtheelectrons congurethemselvesbycolumbicrepulsion,thebarrierpotentialofthedotsisraisedagain therebytrappingtheelectronsintheirrespectivedots.Thuswenoticedthatorientation,and hencethepolarization,ofanelectroncouldbealteredbyraisingorloweringofpotentialbarrier. 15

PAGE 27

Figure2.7.TypicalAdiabaticClockingOperationinQCA[2]. PotentialbarrierisgiventotheQCAcellsintheformofclockenergy[1].Theworkdone inraisingandloweringoftunnelingbarrierscontrolledbytheclockenergycanbetermedas leakagepowerdissipationasthiswilltakeplaceeveniftheQCAcelldoesnotswitchstate.In asimilarway,aclockcontrolsthetunnelingbarriersina4-dotQCAcellusedinthiswork. Howeversinceaninnitebarrierisnotfeasibleinrealworld,thereisalwaysanitepossibility ofsomeelectronchargetoescapethebarrierwhenheldforalongdurationoftime.Inthiswork, wehaveneglectedanylossofcharge.Electronsathigherenergyhavemoreprobabilitytotunnel thewellthananelectronatlowerenergyorgroundstate.Thermalerrorsoccurwhenelectrons settleathigherenergylevelsandaremorelikelytotunnelsthewell.Furtherinthisdocument wewillprovethatthepolarizationoftheelectronreduceswithincreaseintemperature,i.e.the electronsaremorelikelytosettleathigherenergylevelratherthanatgroundstate.Figure2.6. showstheplotoftheenergysplittingbetweenthegroundstateandtherstexcitedstateofan electronundergoingadiabaticswitching[1]and2.7.showsthetypicaloperationoftheclocking giveninQCA[2]. 2.4LogicDevicesinQCA Likewementionedearlier,anidealizedQCAcouldbedescribedasasetof4chargedcontainercalled dots .Thecellcontainstwomobileelectronswhichquantummechanicallytunnel intooneofthefourdotseach.Howeverthedesignissuchthattheelectronstunnelbetween cells.Againasmentionedinsection2.2.1.1,quantumdotscouldberealizedbyformingquan16

PAGE 28

Figure2.8.MajorityGate Figure2.9.MajorityComputations tumdotselectro-staticallyinasemiconductor,smallmetallicislandorbyusingredoxcentersin molecules.Thebarrierofthedotiskepthighenoughsuchthattheelectronscouldnottunnel easily. ALineararrangementofQCAcellsusedasabinarywire(gure2.3.(a))couldcarryeither +1or-1aroundthelayout.TheunderstandingoflogicdevicesbuiltforcomputationinQCA couldbewellunderstoodbyunderstandingthefundamentalsbehinddesigningit. 2.4.1MajorityLogicSynthesis ComputationusingQCAisachievedbyusingthemajoritylogicformulation.Itcouldbe explainedas; MajorityM ( A B C )= A B + B C + C A (2.10) Usingequation2.10circuitsinconventionaltechnologycouldberealizedinQCA.Figure2.8. depictsasimplemajoritygatewhichisobtainedfromtheequation2.10andistheunderlying gaterepresentationforanygaterealizedinQCA.Asseeninthegure,asimplemajoritygate consistsof3inputsandoneoutput.Thecentercellisthe majorityvoter becausetheelectronsin thatcellcongurethemselvesbasedonthemajorityvalueofthethreeinputs.Theoutputthen 17

PAGE 29

Figure2.10.InverterDesigninQCA Figure2.11.ANDGateusingMajorityFunctioninQCA followsthisvalue.Figure2.9.shows3differentcomputations.Asweseefromthegurethe outputisthemajorityofthethreeinputsandhencethename. Aninvertercouldbeconstructedasshowningure2.10..Theinversionoftheinputvalue isobtainedbycolumbicrepulsionofelectronsbetweenthecellsinthenormalplaneandthe cellsdiagonaltoit.ANDandORfunctionscouldbeconstructedbyhavingoneoftheinputs tothemajoritygateas-1and1respectivelyasshowningures2.11.and2.12..Thiscouldbe easilyrealizedfromtheequation2.10bysubstitutingwithervalues.ANANDfunctioncouldbe obtainedbyusinganinverterinfrontofanANDgateasshowningure2.13.. Eventhoughattimes,obtainingmajoritylogicforacircuitcouldsometimesbedirect,while designingbigcircuitsitisquitecomplicated.Insuchcases,wecaneitherreducetheoutput Figure2.12.ORGateusingMajorityFunctioninQCA 18

PAGE 30

Figure2.13.NANDGateusingMajorityFunctioninQCA functionofthecircuitorbyreducingusingKarnaughmapsintosetsofmajorityequations. Eachofthereductiontechniquesarebrieyexplainedwithanexample. ForanexampleofconstructingQCAcircuitsbyreducingtheoutputfunction,wemakeuse ofthealgorithmandtheaddercircuitproposedin[19]. Aone-bitfulladderisreducedas; Sums = a b cin+ a b cin+ a b c in+ a b c in. Carrycout= a b + b cin+ a cin. Byusingthemajorityfunction2.10andthefunctionforcarryin2.11weget cout= a b + b cin+ a cin= m ( a b cin) c out= a b+ b c in+ a c in= m ( a, b, c in) Thenthesumcouldberewrittenas s = a b + a b) cin +( a b c in+ a b c in) =[( a b+ a c in+ b c in)+( a b + a c in+ b c in)] cin+( a b c in+ a b c in) =( a b+ a c in+ b c in) cin+( a b + a c in+ b c in) cin+( a b c in+ a b c in) 19

PAGE 31

=( a b+ a c in+ b c in) cin+( a b + a c in+ b c in) cin+( a c in+ b c in)+( a c in+ b c in) =( a b+ a c in+ b c in) cin+( a b + a c in+ b c in) cin+( a b+ a c in+ b c in)+ ( a b + a c in+ b c in) = m ( a, b, c) cin+ m ( a b c) cin+ m ( a, b, c) cin m ( a b c) cin= m ( m ( a, b, c) cin, m ( a b c)) = m ( c out, cin, m ( a b c)) Figure2.14.showstheaddercircuitbuiltusingtheequationsderivedabove.ThemajorityreductiontechniqueusingKarnaughmapisexplainedusingaJKipop(FF).Weobtain themajorityequationsfromtheKarnaughmap(K-map)usingmajorityreductiontechnique,by followingoneofthreebasicrulesputforthin[20],toreducetheKarnaughmaps(K-maps). Figure2.15.showsthereductionofthebasicK-mapofaJKFFusingthistechnique. HerewerstcreateaKarnaughmapusingtheJKFFtruthtable.Wemakeuseof3primitives andtrytoderiveamajorityfunction.Theconditionliesthatthemajorityofthethreeprimitives shouldresultintheoriginalK-map.Asshowning.2.15.therstsetofprimitivesisobtained byplacing1'sinappropriateplacessuchthatcollectivelyitendsuptotheoriginalK-map.If foranyinstancethecombinedvaluedoesnotendup,theprimitivethatpresentstheproblem isreducedagainasshowning.2.15.andtheprocessisrepeateduntiltheoverallcondition issatised.Wenowgrouptheone'stoobtainanequationasineq.2.10.Forexample,ifwe considertherstK-mapwegettheequation M = J K+ K QP+ J QP= > M = Maj ( J K, QP) (2.11) Figure2.17.showstheJKFFdesignedusingtheequationderivedabove. 20

PAGE 32

Figure2.14.FullAdderDesigninQCA 21

PAGE 33

Figure2.15.MajoritySimplicationofJKFF Figure2.16.JKSchematicfromg.2.15. 2.5ModelingQCADesigns Thereareseveralapproximatesimulatorsavailableatthelayoutlevel,suchasthebistable simulationengineandthenonlinearapproximationmethods.Thesemethodsareiterativeand donotproducesteadystatepolarizationestimates.Inotherwords,theyestimatejuststateassignmentsandnottheprobabilitiesofbeinginthesestates.Thecoherencevectorbasedmethod doesexplicitlyestimatethepolarizations,butitisappropriatewhenoneneedsfulltemporaldynamicssimulation(Blochequation),andhenceisextremelyslow.Perhaps,theonlyapproach thatcanestimatepolarizationforQCAcells,withoutfullquantum-mechanicalsimulationisthe 22

PAGE 34

Figure2.17.JKFFDesignusingQCADesigner 23

PAGE 35

thermodynamicmodelproposedin[21],butitisbasedonsemi-classicalIsingapproximation.In thenextchapter,wedemonstratehowwecanuseaBayesianprobabilisticcomputingmodelto exploittheinducedcausalityofclockinginaQCAdesigntoarriveatamodelwiththeminimum possiblecomplexity. 24

PAGE 36

CHAPTER3 DYNAMICBAYESIANMODEL 3.1Introduction InthischapterwedescribethemodeldevelopedfortheanalysisofsequentialQCAdevices. ThemodelextendstheBayesianmodel[6]tocapturethetemporaldependenciesthatexitsin sequentialdevices.Weusethedensitymatrixformulationtoobtainthesteadystateprobabilities forcellpolarizations.Themodelisnon-iterativeandallowsquickestimationandcomparison ofquantummechanicalquantitiesforsequentialQCAcircuitssuchastheirdependenceontemperatureandanyparameterthatcouldresultinanerroneousoutput.Thishelpsinanalyzingthe circuitandtheirtemporalbehaviourwithoutexhaustivesimulations. 3.2QuantumMechanicalProbabilities FollowingTougawandLent[22]andothersubsequentworksonQCA,weusethetwo-state approximatemodelofasingleQCAcell.Wedenotethetwopossible,orthogonal,Eigenstates ofacellby | 1 and | 0 .Thestateattime t ,whichisreferredtoasthewave-functionanddenoted by | ( t ) ,isalinearcombinationofthesetwostates,i.e. | ( t ) = c1( t ) | 1 + c2( t ) | 0 .Note thatthecoefcientsarefunctionoftime.Theexpectedvalueofanyobservable, A ( t ) ,canbe expressedintermsofthewavefunctionas A = ( t ) | A ( t ) | ( t ) orequivalentlyasTr [ A ( t ) | ( t ) ( t ) | ] ,whereTrdenotesthetraceoperation,Tr [ ]= 1 || 1 + 0 || 0 .Theterm | ( t ) ( t ) | isknownasthedensityoperator, ( t ) .Expectedvalueofanyobservableofaquantumsystemcanbecomputedif ( t ) isknown. 25

PAGE 37

A2by2matrixrepresentationofthedensityoperator,inwhichentriesdenotedby ij( t ) canbearrivedatbyconsideringtheprojectionsonthetwoEigenstatesofthecell,i.e. ij( t )= i | ( t ) | j .Thiscanbesimpliedfurther. ij( t )= i | ( t ) | j = i | ( t ) ( t ) | j =( i | ( t ) )( j | ( t ) )= ci( t ) c j( t ) (3.1) ThedensityoperatorisafunctionoftimeandusingLoiuvilleequationswecancapturethe temporalevaluationof ( t ) inEq.3.2. i h t ( t )= H ( t ) Š ( t ) H (3.2) whereHisa2by2matrixrepresentingtheHamiltonianofthecellandusingHartreeapproximation.ExpressionofHamiltonianisshowninEq.3.3[22]. H = Š1 2iEkPifiŠ Š 1 2iEkPifi = Š1 2Ek P Š Š 1 2Ek P (3.3) wherethesumsareoverthecellsinthelocalneighborhood. Ekisthe"kinkenergy"orthe energycostoftwoneighboringcellshavingoppositepolarizations. fiisthegeometricfactor capturingelectrostaticfalloffwithdistancebetweencells. Piisthepolarizationofthe i -thcell. And, isthetunnelingenergybetweentwocellstates,whichiscontrolledbytheclocking mechanism.Thenotationcanbefurthersimpliedbyusing P todenotetheweightedsumofthe neighborhoodpolarizations iPifi.UsingthisHamiltonianthesteadystatepolarizationisgiven by Pss= Š ss 3= ss 11Š ss 00= Ek P E2 k P2+ 4 2tanh ( E2 k P2/ 4 + 2 kT ) (3.4) 26

PAGE 38

Figure3.1.ASmallBayesianNetwork Eq.3.4canbewrittenas Pss= E tanh ( ) (3.5) where E = 0 5 iEkPifi,totalkinkenergyandRabifrequency = E2 k P2/ 4 + 2and = kTis thethermalratio.Wewillusetheaboveequationtoarriveattheprobabilitiesofobserving(upon makingameasurement)thesystemineachofthetwostates.Specically, ss 11= 0 5 ( 1 + Pss) and ss 00= 0 5 ( 1 Š Pss) ,wherewemadeuseofthefactthat ss 00+ ss 11= 1. 3.3OverviewofBayesianModeling WeproposeaBayesianNetworkbasedmodelingandinferencefortheQCAcellpolarization. ABayesiannetworkisaDirectedAcyclicGraph(DAG)inwhichthenodesofthenetwork representrandomvariablesandasetofdirectedlinksconnectpairsofnodes.Thelinksrepresent causaldependenciesamongthevariables.Eachnodehasaconditionalprobabilitytable(CPT) excepttherootnodes.Eachrootnodehasapriorprobabilitytable.TheCPTquantiestheeffect theparentshaveonthenode.Bayesiannetworkscomputethejointprobabilitydistributionover allthevariablesinthenetwork,basedontheconditionalprobabilitiesandtheobservedevidence aboutasetofnodes. Fig.3.1.illustratesasmallBayesiannetworkthatisasubsetofaBayesianNetworkfora majoritylogic.Ingeneral, xidenotessomevalueofthevariable XiandintheQCAcontext, each Xiistherandomvariablerepresentinganeventthatthecellisatsteady-statelogic"1"orat 27

PAGE 39

steadystatelogic"0".Theexactjointprobabilitydistributionoverthevariablesinthisnetwork isgivenbyEq.3.6. P ( x5, x4, x3, x2, x1)= P ( x5| x4, x3, x2, x1) P ( x4| x3, x2, x1) P ( x3| x2, x1) P ( x2| x1) P ( x1) (3.6) InthisBN,therandomvariable, X5isindependentof X1,giventhestateofitsparents X4This conditionalindependence canbeexpressedbyEq.3.7. P ( x5| x4, x3, x2, x1)= P ( x5| x4) (3.7) Mathematically,thisisdenotedas I ( X5, { X4} { X1, X2, X3} ) .Ingeneral,inaBayesiannetwork, giventheparentsofanode n n anditsdescendentsareindependentofallothernodesinthe network.Let U bethesetofallrandomvariablesinanetwork.UsingtheconditionalindependenciesinEq.3.7,wecanarriveattheminimalfactoredrepresentationshowninEq.3.8. P ( x5, x4, x3, x2, x1)= P ( x5| x4) P ( x4| x3, x2, x1) P ( x3) P ( x2) P ( x1) (3.8) Ingeneral,if xidenotessomevalueofthevariable Xiand pa ( xi) denotessomesetofvalues for Xi'sparents,theminimalfactoredrepresentationofexactjointprobabilitydistributionover m randomvariablescanbeexpressedasinEq.3.9. P ( X )=mk = 1P ( xk| pa ( xk)) (3.9) Notethat,BayesianNetworksareproventobeminimalrepresentationthatcanmodelallthe independenciesintheprobabilisticmodel.Also,thegraphicalrepresentationinFig.3.1.and probabilisticmodelmatchintermsoftheconditionalindependencies.SinceBayesianNetworks 28

PAGE 40

usesdirectionalpropertyitisdirectlyrelatedtoinferenceundercausality.InaclocklessQCA circuit,causeandeffectbetweencellsarehardtodetermineasthecellswillaffectoneanother irrespectiveoftheowofpolarization.ClockedQCAcircuitshoweverhaveinnateordering senseinthem.Partoftheorderingisimposedbytheclockingzones.Cellsinthepreviousclock zonearethedriversorthecausesofthechangeinpolarizationofthecurrentcell.Withineach clockingzone,orderingisdeterminedbythedirectionofpropagationofthewavefunction[22]. Let Ne ( X ) denotethesetofallneighboringcellsthatcaneffectacell, X .Itconsistsofall cellswithinapre-speciedradius.Let C ( X ) denotetheclockingzoneofcell X .Weassumethat wehavephasedclockingzones,ashasbeenproposedforQCAs.Let T ( X ) denotethetimeit takesforthewavefunctiontopropagatefromthenodesnearesttothepreviousclockzoneor fromtheinputs,if X sharestheclockwiththeinputs.Notethatonlytherelativevaluesof T ( X ) areimportanttodecideuponthecausalorderingofthecells.Thus,givenasetofcells,wecan exactlypredict(dependentontheeffectiveradiusofinuenceassumed)theparentsofeverycell andallthenon-parentneighbors.Inthiswork,weassumetouse four clockzones.Wedenote thisparentsetby Pa ( X ) .Thisparentsetislogicallyspeciedasfollows. Pa ( X )= { Y | Y Ne ( X ) ( C ( Y )
PAGE 41

chosenchildrenstatesare ch( X )= argmaxch ( X ) = argmaxch ( X )i ( Pa ( X ) Ch ( X ))Ek P (3.11) HoweverBayesiannetworkisadirectedacyclicgraph(DAG),soitisinfeasibletodesignany circuitthatformsaloopwhichincludesanysequentialcircuit.Thuswecouldonlyportray thespatialredundancyexistinginthecircuitataparticulartimeinstancebutnotthe temporal redundanciesthatexistsbetweenvarioustimeinstances. 3.4DynamicBayesianModel Theapproachwetaketodesignasequentialcircuitisbygivingadynamicnaturetothe Bayesianmodel.Sequentialcircuitscouldbeviewedasacombinationalcircuitatvarioustime instances,whilelinkingtheoutputofonetimeslicetotheinputofthenexttimeslice,thisprocess isknownas unraveling .ThuswerepresentthesequentialcircuitsdynamictimecoupledBNof thecombinationpart.ThistypeofmodeliscalledDynamicBayesianmodel(DBN).Similar techniquehasbeenusedtodesignsequentialcircuitinCMOStechnology[23],butneverhave beenimplementedtodesignasequentialQCAcircuit. MuchlikethespecialcaseformalismssuchashiddenMarkovmodelsandlineardynamic systems,DBNhandlesdependenciesbetweenvarioustimesliceswithoutdisturbingtheinternal dependenciesusingrandomsetofvariables.If tirepresentthetimesliceat ithinstanceandthe underlyingdependenciesforthecombinationalpartisrepresentedasafunctionofnodes Vtiand links Etiattimeslice tias Gti=( Vti, Eti) ,thenthenodesoftheDBNcouldberepresentedasa unionofallnodesforeachtimeslice. V =ni = 1Vti(3.12) 30

PAGE 42

HoweverthelinksofaDBN E includesboththeunionofthelinksforonetimesliceandthe temporaledges(linksconnectingtwotimeslices) Eti, ti + 1,denedas Eti, ti + 1= { ( Xi ti, Xj ti + 1) | Xi ti Vti, Xj ti + 1 Vti + 1} (3.13) Where Xi ti, Xj tiarethe i -thand j -thnodeoftheDAGfortimeslice ti.Eveninageneralized structure,wherethetemporaledgescanbebetweenanynodefromthetimeslice titoanynode oftimeslice ti + 1,theoverallstructuremustrepresenttheminimalidentitymapoftheunderlying model.Thecompletesetofedges E is E = Et1ni = 2( E ( ti)+ Eti Š 1, ti) (3.14) Thesteadystatedensitymatrixdiagonalentries(Eq.3.5withthesechildrenstateassignmentsareusedtodecideupontheconditionalprobabilitiesintheBayesiannetwork(BN). P ( X = 0 | pa ( X ))= ss 00( pa ( X ) ch( X )) P ( X = 1 | pa ( X ))= ss 11( pa ( X ) ch( X )) (3.15) Oncewecomputealltheconditionalprobabilities,weprovidepriorprobabilitiesfortheinputs.WecantheninfertheBayesianNetworkstoobtainthesteadystateprobabilityofobserving allthecellsincludingtheoutputsat1or0 31

PAGE 43

Figure3.2.JKFFDesignedinQCADesignerusing18nmCells 32

PAGE 44

Figure3.3.JKFFUnraveledfortwoTimeSlicesusingDBN 3.4.1SequentialDesignofJKFFinQCAusingDBN TheJKFFisdesignedusingthereductionmethodexplainedinsection2.4.Theschematic ofthesameisshowning.2.16.andtheQCArepresentationofthecircuitisshowning.3.2.. Itconsistsof421cellsandoutputreachesafter3clockcycles.Weuseeachofthecellsofthe QCAcircuitandrepresentthemasnodesoftheDBNeachnodeisthenassignedtobeaparent orchildbasedonthecellcharacteristics.Linksarethendirectedfromtheparenttoitschildren. Thenumberofchildrenaparentcouldhaveatanypointoftimedependsonthepredenedarea ofinuence.Ifthecellhappenstobeacrosswireweagthecellsuchthatitdoesnotaffectthe normalwire.Thenetworkisthenreplicatedforoverthetimeinstancesrequirefortheanalyses. Theconditionalnaturebetweendifferenttimeslicesisthenestablishedbyconnectingtheoutput ofonetimeslicetotheinputoftheother,thisisshowning.3.3.. 33

PAGE 45

Figure3.4.SchematicofRAM 3.4.2DesignofaSingleMemoryCellinQCAusingDBN TheRAMusedforthisanalysisisproposedin[3].TheoperationoftheRAMisasfollows; gure3.4.isamemorycellwhichisselectedbysettingthe rowselectRS inputto+1.Theread andwritemodecanbechosebygivingan+1or-1signalrespectivelyatthe R/W input.During thereadmode,i.e.whenthesignalislogical0intheR/Winput,thevalueoftheinputisretained intheloopandtheoutputremainszero.Duringthewritemode,i.e.whenwehavealogical1in theR/W,thevalueintheloopisfedtotheoutput.Figure3.5.showsthesameasinglememory celloftheRAMproposedin[3]. Figure3.6.showstheRAMobtainedfromtheDBNmodel.Forunravelingpurposes,we modifythememoryloopbygivingaarbitraryvalueintotheinputoftheORgatefedfromthe memoryloopandfeedingtheoutputoftheANDgateinthememorylooptoaseparateORgate similartotheoneinthecircuit.TheORgateandtheANDgateinthememoryloopisthen replicatedaccordingtothenumberoftimeinstancesrequiredforanalyses.Thisapproachis normallynotrequiredbutinthecaseoftheRAMtheconditionalnatureofthememoryloopdue totheexistenceoftheANDgatecallsforsuchanaction. 34

PAGE 46

Figure3.5.RAMProposedinciteWalus03 Figure3.6.RAMDesignedusingDBN 35

PAGE 47

Figure3.7.Schematicofs27SequentialBenchmarkCircuit 3.4.3Designofs27SequentialBenchmarkCircuitinQCAusingDBN Weconstructthes27benchmarkcircuitbyreducingthespicemodelfortheCMOStechnologyofthesame,intosetsofmajorityequations.Figure3.7.showstheschematicrepresentation ofs27sequentialbenchmarkcircuitwhichhas4inputsandoneoutput.Thecircuitconsistsof 2D-typeFFsthatisrepresentedasawireintheQCAequivalent(gure3.8.).TheNOR,AND andORlogicdevicesarebuiltbytheircorrespondingmajorityfunctions.TheDBNnetworkfor thisdesignisconstructedby unraveling thethreeFF'sandtimeslicingthecircuit.Figure3.9. showstheDBNformofthes27benchmarkcircuit. 3.5EstimatedPosteriorImportanceSampling Inthissection,wewouldlookintothebasicsofestimatedposteriorsamplingalgorithm (EPIS)[24]usedinthisworkforbeliefpropagation. AfterbuildingtheBayesiannetworkitisimportanttoupdatetheconditionalprobability table(CPT)ofeachnodebasedonthevalueassignedtoit.Thisassignmentofvaluestoa nodeisknownasevidence.Themethodofpropagating,inotherwordsupdatingtheCPT, isknownasbeliefpropagatingorbeliefupdating.Thereareseveralalgorithmsproposedto 36

PAGE 48

Figure3.8.s27SequentialBenchmarkCircuit updatethebeliefs.Thesealgorithmsfallundertwobasiccatagoriesnamely, exactalgorithm and approximatealgorithm Thedifferencebetweenthetwoisthat,inexactalgorithmthenetworkcomputesthebelief overallthenodesconsideringthemostlikelyinstantiationvaluegivenparticularevidence.Exactalgorithmsarethepreferredalgorithmasitgivesexactlikelihoodvalues.Someoftheexact algorithmsareclusteringalgorithm,polytreealgorithm,variableeliminationetc.Clusteringalgorithmisthemostcommonalgorithmusedforbeliefpropagation.Inclusteringalgorithm,the directedgraphisbrokenintosetsofjunctiontreesandthentheprobabilityisupdatedinthis junctiontree.Howeverthecomplexityofcomputationinclusteringalgorithm,oranyexactal37

PAGE 49

Figure3.9.s27unraveledfortwoTimeSlicesusingDBN gorithmforthatmatter,liesinthecircuitsize.Thisindicatesthat,forlargecircuitthecomplexity increasesandexactinferencebecomesinfeasible.Forthisreasonweuseapproximatemethods toobtaintheprobabilityvalues. Approximatealgorithmsarenotaspreciseasexactalgorithms;however,severalalgorithms havebeenproposedtoobtainnearexactvalues.Variationalsampling,probabilisticpartialevaluation,andstochasticsamplingaresomeoftheapproximatealgorithms.Amongtheseveral approximatealgorithms,stochasticsamplingisproventoproducealmostexactvalues.However theaccuracyoftheprobabilisticvaluedependsonthenumberofsamplesusedtocomputethe belief,whichmeansthatforalargenumberofsamples,theprobabilisticvaluecomputedusing stochasticsampling,wouldconvergetoanexactvalue.Severalalgorithmsthatfallunderthe familyofstochasticsamplingmethodsare,probabilisticlogicsampling(PLS),adaptiveimpor38

PAGE 50

tancesamplingforBayesiannetwork(AIS-BN),andestimatedposteriorimportancesampling (EPIS-BN). Importancesamplingstatesthat,thevalueofarandomvariableVinadomain Rnwith afunctionf(x),where Rnistheregionunderthecurve ,canbepredictedbyintroducinga importancefunction,whichisanon-zeroprobabilitydensityfunctionforanyvalueofX assumingthat 1.f(x) probabilitydensityfunctiondenedon 2. { Xi} isasequenceofindependentandidenticallydistributedrandomvariable(i.i.d). 3.I(X)includes 4.Vexistsandisnite. Thevalueoftherandomvariableforabondedregion isgivenbytheequation V ( X )=f ( X ) I ( X ) dx I ( X ) = 1 NNif ( Xi) I ( Xi) (3.16) Theadaptiveimportancesamplingparameterizestheimportancefunctionusingasetofparametersandupdatesthebelieftablebasedonthecurrentdistributionofthegradient.TheAISBNlearnstheimportancefunctionmodiesthepriorvalueintwosteps.Firstitinitializesthe probabilitydistributionsoftheparentoftheevidencenodestotheuniformdistribution.Then itadjustssmallprobabilitiesintheCPTcomposingtheimportancefunctiontohighervalues. FinallytheAIS-BNcomputestheimportancefunctionthatapproachestheoptimalimportance function.EPISfollowssimilarmethodindeterminingtheimportancefunction.Howeverituses aloopybeliefpropagationtocomputeanapproximationofoptimalimportancefunctionand thenapplya -cutoffheuristicstocutoffsmallprobabilitiesintheimportancefunction. 39

PAGE 51

ThealgorithmofEPIS,asgivenin[24],isasitemized; 1.Thenodesareorderedaccordingtothenetworktopology. 2.Thenodesareinitializedwiththeparameterssuchasthenumberofsamplesm,thethresholdvalue ,andthepropagationlengthd. 3.Itthencomputestheimportancefunctionusingthevalueofevidenceobtainedoverall nodes. 4.AppliesthethresholdvaluetotheimportanceCPT(ICPT)toenhanceimportancefunction. 5.Calculatestheoptimalimportancefunctionbasedonthenumberofsamplesm. TheimportancefunctionforEPIS-BNisgivenbytheequation ( X \ E )=niP ( Xi| PA ( Xi) E ) (3.17) Previousstudies[25,26]showthatEPIS-BNisthebetterthanAIS-BNinbothspeedand accuracyofthenodeprobabilities.Henceinthiswork,weimplementtheEPISalgorithmusing theBayesiannetworktoolGeNIe[27],topropagatethebelief. 40

PAGE 52

CHAPTER4 VALIDATIONOFDBNMODEL Inthissection,wepresentthesimulationresultsobtainedfromtheDBNmodeldescribedin theprevioussection.Wethencompareandanalyzetheresultsobtainedfromtheproposedmodel againstQCADesignerforaccuracy.Wedesignedthecircuitsusing18nmcellsandsimulated usingcoherentvectorengineunderatemperatureof2.0K.ThedesignedJKFFcircuitcontains 431cellsandtheentiresimulationtook22minutesinQCADesignerand6secondsusingour model.TheRAMdesignedinQCADesignerconsistsof175cellsandthes27consistsof344 cells.ThesimulationswereperformedinIntelcore2duo1.4GHzPCwith4GBRAM. TheDBNisexecutedinGeNIedevelopmentenvironment[27].Sinceinexactalgorithms, suchasclusteralgorithm,itisinfeasibletorunlargecircuitsweuseastochasticalgorithm knownasEstimatedPosteriorImportanceSampling(EPIS)algorithm.TheEPISalgorithmuses loopybeliefpropagationtocomputeanestimateoftheposteriorprobabilityoverallnodesof thenetworkandthenusesimportancesamplingtorenethisestimate[24]. Thecircuitdiagramsforthecorrespondingcircuitsareshowningures3.2.,3.5.,and3.8.. Tables4.1.,4.2.and4.3.showtheresultsobtainedfromQCADesignerandDBNmodel.From theresultsweobservethefollowingbasicdetails; 1.ThesimulationresultsobtainedfromQCADesignershowsbetterpolarizationthantheresultsobtainedfromDBN.Thiscouldbeexplainedwiththereasonthatthe quantumboost providedinQCADesigner,isnotconsideredinourmodel. 2.ThePolarizationoftheoutputdependsonthecurrentinputstateandthepreviousoutput. FromtheresultsobtainedforJKFF,weseethatthepolarizationvaluedeteriorateswhenthe circuitholdsthevalueduringtheinputsJ=-1,K=-1.Thiseffectoccursbecausethereisdrop 41

PAGE 53

Table4.1.ComparisonoftheResultsobtainedfromQCADesginerandDBNModel,forJKFF at2.0Kusing18nmCells QCADesigner DBNModel J K Qn Qn -1 -1 1 0.944 1 -1 1 0.962 -1 -1 1 0.908 -1 1 -1 -1.000 -1 -1 -1 -0.984 1 1 1 0.966 inpolarizationduringdatapropagationofthepreviousoutputvalue,hencethegatesthatare dependentonthevalueofthepreviousstateoftheoutputproduceaweaklypolarizedoutput. Whenoneoftheinputsishigh,thepolarizationoftheoutputimprovesduetotheinuenceof thenewvalueasseenfromthetable. Table4.2.ComparisonoftheResultsobtainedfromQCADesginerandDBNModel,forRAM at2.0Kusing18nmCells R/S=1 QCADesigner DBNModel R / W I / P O / P FBP FBN 1 O / P -1 -1 -1 -1 X -0.994 1 1 -1 -1 0.996 -1.000 -1 -1 1 -1 X 0.974 1 -1 -1 1 0.976 -1.000 -1 -1 1 1 X 0.914 FortheRAMcircuit,weseefromthetablethattheinternalpolarizationisnotobservedin QCADesignerhoweverusingourmodelitispossibletoseethepolarizationofthevalueinthe memoryloop.Intable4.2.(b),wehaveshownthesameforonetimeinstanceunder FBN 1and wecanseethatthevaluehasaprettygoodpolarizationinsidetheloop.The don’tcares indicate thatthememorylooppolarizationvaluedoesnotrevolveinthememoryloopduringwritemode. Theresultsfors27benchmarkcircuitsimulatedinQCADesignerisshownintable4.3.(a), theanalysisofthesamecircuitusingDBNmodelisshownintable4.3.(b).Forvalidation purposewehaveshownonlyoneoftheinputcombinations.Fromthetableitisobservedthat 42

PAGE 54

Table4.3.ComparisonoftheResultsobtainedfromQCADesginerandDBNModel,forS27at 2.0Kusing18nmCells QCADesigner DBNModel IN0 IN1 IN2 IN3 OUT FBP 1 FBP 2 OUT -1 -1 -1 1 1 -1.000 -1.000 0.920 1 1 1 1 1 -0.980 -0.916 0.988 -1 1 -1 -1 1 -1.000 -0.986 0.922 1 1 1 1 1 -0.986 -0.926 0.988 -1 -1 1 -1 1 -0.994 -0.988 0.916 1 1 1 1 1 0.996 -0.906 0.986 1 1 -1 1 1 -0.994 -0.986 0.982 1 1 1 1 1 -0.988 -0.978 0.988 itispossibletoobtainthepolarizationvaluesatthefeedbackloopsusingtheDBNmodel.As wecanseefromtable4.3.(b),theoutput FBP 1ofthetwoDFF(gure3.7.)hasamuchbetter polarizationthantheoutput FBP 2bottomDFFwhichistheinvertedvalueoftheoutput. 43

PAGE 55

CHAPTER5 ANALYSISOFSEQUENTIALQCACIRCUITS 5.1Introduction AsthedesignofQCAcircuitsincreases,thecircuitsizeanditscomplexityincreasesrelatively.Designingsequentialcircuitsposedifcultyasalltheoutputsmustarriveatthesame timeinstancetoavoid raceconditions .Duetothenatureofthefeedbacks,theclockinginsequentialQCAcircuitsshouldbecarefullylaidout.Thisisoftendifcultinlargecircuits,where itisnecessarytotakeallthecircuitsusingthatvalueintoaccount.Severalalgorithmshavebeen developedtoresolvethiscomplexityindesigningsequentialQCAcircuitsbutcannotbeused toanalyzethecircuit.Inthispaper,wepresentanovelprobabilisticmodeltodesignsequential QCAcircuitsusingitsdynamicnature.Themodelcouldalsobeusedtoexplorethecircuitfor variousdefectstudies[28]. Duetotheapparentsmallsizeandoperation,QCAcouldachieveveryhighprocessing speeds,evenwithlargecircuits.Thereliableoperationofsuchcircuitsunderdifferentconditionsisimportanttodeterminethecircuit'sresponsewhileoperatinginrealtime.Bayesian network(BN)provestobeaneffectivemethodtoanalyzespatialdependenciesthatexistsbetweentwocells[6]. Whilecombinationalcircuitsareanalyzedforspatialrelationshipsbetweenthecells,sequentialcircuitmustbeanalyzedforbothspatialandtemporaldependenciesduetothepresenceof feedbacks.AsBNaredirectedacyclicgraphs(DAG),thereexistsasetbackwhilerepresenting sequentialcircuitsinBNmodel.Toresolvethisproblemweutilizethemodelpresentedinthis paperwhichcapturesnotonlythespatialdependenciesbutalsothetemporalrelationshipthat existsinallsequentialcircuits.Theconditionalprobabilityexistingbetweentheinputandthe 44

PAGE 56

outputthroughthefeedbackismodeledbyviewingthesequentialcircuitasaseriesofcombinationalcircuitsatdifferenttimeinstances.Similarmodelhasbeenusedtoanalyzesequential CMOScircuits[23]buthasnotyetbeenperformedinQCAdevices.Soweusethemodelput forthinthispapertoanalyzethesequentialQCAcircuitsundervaryingoperatingtemperatures andphysicaldimensions.BNprovidesaccurateandslightlypessimisticresultthatwhichhelps designertobuildmorereliablecircuits. -1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000 0246810 Time tOutput Polarization 1.5K 2K 2.5K 3K 01 11 00 00 10 10 00 11 10 In p ut VectorsFigure5.1.ResultsforOutputPolarizationVersusTimetatDifferentTemperaturesforJKFF Circuit. 5.2TemperatureCharacterization Inthissectionwepresentthesimulationresultsobtainedusingdynamictime-coupledBayesian networkofsequentialQCAcircuitswithregardstothetemperaturevariations.Thesimulation isperformedtoseethereliabilityofsequentialcircuitsacrosscertaintimeduration,underdifferenttemperatures.Theincreaseintemperaturehasdifferenteffectsondifferentcircuits,based onthesizeofthecircuitandthenumberofcomputationsmade.Figures5.1.,5.2.,5.3.and5.4., 45

PAGE 57

and5.5.and5.6.showstheestimatesplottedwithoutputpolarizationversustimeforthecircuits usedinthiswork.Asweareconcernedaboutthevaluebeingheldinthememory,werealize theeffectonthefeedbackintermsofthevalueattheoutputduringcaseswhentheoutputdirectlydependsonthepreviousstatevalueinthefeedback,suchasinJKFFands27benchmark circuits.HoweverinthecaseofRAMtheoutputdoesnotdependonthememoryvaluebut ratherisconditionaltotheR/Wvalue,i.e.theoutputdependsonthefeedbackonlywhenthe R/Wmodeis0.Henceitisnecessarytoconcentrateonthefeedbackdirectlyratherthanlooking attheoutputduringreadmode.Duringwritemode,astheoutputdependsonthevalueinthe feedback,thepolarizationoftheoutputisseenwithrespecttothefeedbackvalue. Wesimulatethecircuitsbyconsideringallpossibleinputcombinations.Thisgivesusawide viewofhowthedeviceoperatesatallpossiblestatesandunderdifferentoperatingtemperature. -1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000 0123 Time tLoop Polarization 1.5K 2K 2.5K 3K Figure5.2.ResultsforLoopPolarizationVersusTimetatDifferentTemperaturesforRAM[3] CircuitDuringWriteMode(R/W=0). Wesimulatethecircuitunderfourbasictemperatures,1.5K,2.0K,2.5K,and3.0K.Fromthe resultsweobservethefollowing: 46

PAGE 58

-1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000 00.511.522.533.5 Time tLoop Polarization 1.5K 2K 2.5K 3K Figure5.3.ResultsforLoopPolarizationversusTimet,whenInput=0andtheValueinthe Feedback=1atDifferentTemperaturesforRAM[3]CircuitDuringReadMode(R/W=1). 0.000 0.200 0.400 0.600 0.800 1.000 00.511.522.533.5 Time tLoop Polarization 1.5K 2K 2.5K 3K Figure5.4.ResultsforLoopPolarizationversusTimet,whenInput=1andtheValueinthe Feedback=0atDifferentTemperaturesforRAM[3]CircuitDuringReadMode(R/W=1). 47

PAGE 59

1.Polarizationofthecellsiswidelyaffectedbytemperature,independentofthetypeofthe circuit,andthetimeperiodforwhichthecircuitholdsthevalue. 2.Rateofdecayinpolarizationisnon-linearwithtemperature. 3.LossinpolarizationoccursinJKwhenthecircuitholdsthevalue. 4.PolarizationofthememoryvalueinRAMduringREADmodeisbetterthanduringthe WRITEmode. InJKFFweobservethatthepolarizationreducesasthecircuitholdsthevalueduringthe inputsJ=0K=0asshowningure5.1..Asseenfromthegurethecircuitholdsthevalue prettygoodatlowtemperaturesevenforalongdurationoftime,butathightemperaturesthe polarizationoftheoutputreducesdependinguponhowlongthevalueisstoredinthecircuit.As theriseintemperaturehasanon-lineareffectonthedropinpolarization,asthepreviousoutput propagates,itlosesafractionofitspolarizationateachcell.Thisdropmitigatesatthemajority gateswheretheotherinputalsohasalowpolarization,therebyresultinginaweaklypolarized output.Henceitisnecessarytooperatethecircuitatanominaltemperatureoruseasmaller designathighertemperaturesuchthatthelossinpolarizationateachcelldoesnotimpactmuch ontheeffectiveoutputpolarization. InRAM,thevalueinthememorylooplosesitspolarizationattheANDgatethatfeeds backthevaluetotheORgateinthememoryloopshowningure3.4..Duringthewritemode theinputtotheANDgateistheinvertedvalueoftheRead/Writesignal,whichis+1.Careful examinationrevealsthatwhenthepolarizationofthisvalueandthevalueoftheotherinputis weak,aconictoccursbetweenthetwoweaklypolarizedvalueof+1(fromthetwoinputs)and thestrongpolarizationof-1,thatisusedtogiveanANDfunctiontothemajoritygate.Dueto thisconicttheoutputsettlestoaweaklypolarizedvalue.Butduringreadmode,astheinverted valueis-1theoutputoftheANDgatealwaysremains-1irrespectiveofthevalueatitsother inputandhencethemajorityoftheoutputdependssolelyonthevalueoftheinputunlikeincase ofthewritemode. 48

PAGE 60

0.000 0.200 0.400 0.600 0.800 1.000 01234567 Time tOutput Polarization 1.5K 2K 2.5K 3K 0001 1111 0100 1111 0010 1111 1101 1111 Input VectorsFigure5.5.ResultsForOutputPolarizationVersusTimetfortheInputsIN0=1IN1=1IN2=1 IN3=1atDifferentTemperaturesforS27BenchmarkCircuit 0.000 0.200 0.400 0.600 0.800 1.000 01234567 Time tOutput Polarization 1.5K 2K 2.5K 3K 0001 1001 0100 1001 0010 1001 1101 1001 In p ut VectorsFigure5.6.ResultsForOutputPolarizationVersusTimetfortheInputsIN0=1IN1=0IN2=0 IN3=1atDifferentTemperaturesforS27BenchmarkCircuit 49

PAGE 61

-1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000 0246810 Time tOutput Polarization 9nm 15nm 18nm 01 11 00 00 10 10 00 11 Input VectorsFigure5.7.ResultswithOutputPolarizationversusTimetatDifferentCellDimensionsforJK FFCircuit Ins27benchmarkcircuit,weobservethattheoutputpolarizationdependsontheinputs. Fromgure5.5.and5.6.,wecouldseethatatthe5thtimeslicetheoutputisbetterwhenthe inputswitchesfrom0010to1111,thanwhenitswitchesfrom0010to1001.Thisisbecause, asmostofthegatesinthecircuitisanORgate(inQCAweuseanORandinvertertoobtaina NOR)whenalltheinputstothegateare1thepolarizationisbetter.Thereasonissimilartothe oneexplainedabovewithANDgate. Fromthesimulationresults,itisevidentthatsincethetimeperiodforwhichthecircuit holdsthevalueisnon-deterministic,itwillbehighlydependentontheoperationaltemperature, toachievearelativelyhighandstablevalueforoutputpolarization. 5.3CharacterizationUsingCellDimensions Inthissectionweobservetheeffectonpolarizationduetochangesincelldimensions.Cell dimensionisanimportantfeature,whichwhenreducedhelpstoembedmanydevicesovera 50

PAGE 62

-1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000 0123 Time t Loop Polarization 9nm 15nm 18nm Figure5.8.ResultswithLoopPolarizationversusTimetatDifferentCellDimensionsforRAM CircuitDuringWriteMode(R/W=0). smallarea.Reducingthefeaturesizehascertaindifcultieswithrespecttofabrication,asitrequiresmoreprecisionandadvancetechnologiestoobtainsuchsmallfeaturesize.Aseverything endsupintermsofcosteventually,havingsmallfeaturesizenarrowsdowntheallowableerror boundary.Thismakesitimportantfortheresearcherstostudythevariouseffectsthatcould occurbyreducingthesizeandmaximumextentitcouldbedonewithinthaterrorboundary.In short,wehavetoknowhowreliablethecellis,atsuchsmallfeaturesize.Figures5.7.,5.8.,5.9. and5.10.,and5.11.and5.12.,plotstheoutputpolarizationversustimetforthecircuitsJK, RAM(duringreadandwritemode),ands27respectively. Fromthesimulationresultsweobservethat; 1.Ataconstanttemperature,thepolarizationisnothighlyaffectedbythecelldimensions. 2.Thedropinpolarizationattheoutput,forlargecelldimensions,isduetothedropinkink energybetweenthecells.Thekinkenergylargelydependonthecelldimensions,hence smallerthecell,betterthekinkenergy. 51

PAGE 63

-1.000 -0.800 -0.600 -0.400 -0.200 0.000 0.200 0.400 0.600 0.800 1.000 0123 Time tOutput Polarization Size 09nm Size 15nm Size 18nm Figure5.9.ResultswithLoopPolarizationversusTimet,whenInput=0andtheValueinthe Feedback=0AtDifferentCellDimensionsForRAMCircuitDuringReadMode(R/W=1). 0.000 0.200 0.400 0.600 0.800 1.000 0123 Time tOutput Polarization Size 09nm Size 15nm Size 18nm Figure5.10.ResultswithLoopPolarizationversusTimet,whenInput=1,ValueintheFeedback=0atDifferentCellDimensionsforRAMCircuitDuringReadMode(R/W=1). 52

PAGE 64

0.000 0.200 0.400 0.600 0.800 1.000 01234567 Time tOutput Polarization 9nm 15nm 18nm 0001 1111 0100 1111 0010 1111 1101 1111 In p ut VectorsFigure5.11.ResultswithOutputPolarizationversusTimet,fortheInputsIN0=1IN1=1IN2=1 IN3=1,atDifferentCellDimensionsForS27BenchmarkCircuit 0.000 0.200 0.400 0.600 0.800 1.000 01234567 Time tOutput Polarization 9nm 15nm 18nm 0001 1001 0100 1001 0010 1001 1101 1001 Input VectorsFigure5.12.ResultswithOutputPolarizationversusTimet,fortheInputsIN0=1IN1=0IN2=0 IN3=1,atDifferentCellDimensionsforS27BenchmarkCircuit 53

PAGE 65

3.ThestabilityoftheRAMisunaffectedwiththechangeincelldimensions 4.Ins27benchmarkcircuittheinuenceoftheinputvaluescouldbeseenatcelldimension=18nm. InJKFF,thevalueisheldgoodevenatlargecelldimensionsandevenbetteratsmaller dimensions.ThewritemodeinRAMappearstoprovideaverygoodpolarizedvalueatthe outputtoo.Thereadmodeproducesidealresultswithoutasignicantdecreaseinpolarization andtheeffectobservedinthesection5.2isnotobservedhere.Ins27,weobservenoticeable decreaseinpolarizationatlargercelldimensions,whichismoreduetothesizeofthecircuitand thenumberofcomputationsmaderatherthanthevariationsincelldimensions.Wecanseethat thecellpolarizationimproveswithreductionincelldimensions.Hencetheoutputpolarization forthecircuitswithcelldimensionsbetween9nm-18nmdoesnotvarymuch. 54

PAGE 66

CHAPTER6 CONCLUSIONANDFUTUREWORKS Inthisthesis,wehaveputforthcertaindesignrulesthatcanbefollowedtodesignefcient sequentialQCAcircuits.Wehavedesignedfewsequentialcircuitsincludingonesequential benchmarkcircuit.Wehaveusedthestate-of-artDynamicBayesianmodelthatovercomesthe limitationsofthedirectacyclicnatureofBNmodel,toanalyzesequentialQCAcircuits.We haveshownthatitcanbeusedforcomprehensiveanalysisofsequentialQCAcircuitswithout theneedtoalignclockzones.Wehavevalidatedthemodelwithagroundtruth,simulatedusing QCADesigner.Wehavealsoshowntheadvantagesofourmodelinexhaustiveanalysisofthe temporalbehaviorofthecircuit.Wehaveusedthemodeltoanalyzetheeffectsonpolarizationduringvariationintemperatureandcelldimensions.Wehavediscussedtheobservations fromtheresults.Inourpreviousworks[28,6],wehaveshowntheefciencyofBNmodelin analyzingcombinationcircuits.Inthiswork,wehavedevelopedamodeltoanalyzesequential circuits.Forthersttimewehaveusedanapproximatealgorithmusingstochasticsampling method,knownasestimatedposteriorimportantsampling(EPIS),toobtaintheprobabilistic valuesforQCAcircuits. DynamicBayesianmodelwouldbeusefulinanalyzingbothspatialandtemporalbehavior ofsequentialQCAcircuitsfordefectsandreliableoperationunderphysicalandenvironmental variations.Themodeldevelopedinthisworkcanaidincharacterizingthesequentialoperation ofmemoryandregistercircuitsinQCA,underdifferentconditionsthatpossiblywillinuence thenormaloperationofthecircuitinrealtime. 55

PAGE 67

REFERENCES [1]C.S.LentandP.D.Tougaw,"AdeviceArchitectureforcomputingwithQuantumdots," IEEEProceedings ,vol.85,April1997. [2]A.O.Orlov,R.K.Kummamuru,R.Ramasubramaniam,C.S.Lent,G.H.Bernstein,and G.L.Snider,"ClockedQuantumdotCellularAutomataShiftRegister," SurfaceScience vol.532-535,pp.11931198,June2003. [3]K.Walus,A.Vetteth,G.Jullien,andV.Dimitrov,"RAMDesignUsingQuantum-Dot CellularAutomata," NanoTechnologyConference ,vol.2,pp.160163,2003. [4]"www.intel.com/technology/architecture-silicon/32nm." [5]"Internationalroadmapforsemiconductors,"2007. [6]S.BhanjaandS.Sarkar,"ProbabilisticModellingofQCACircuitsUsingBayesianNetworks," IEEETransanctionsonNanotechnology ,vol.5,November2006. [7]S.Bhanja,M.Ottavi,S.Pontarelli,andF.Lombardi,"QCACircuitsforRobustCoplanar Crossing," JournalofElectronicTesting:TheoryandApplications ,vol.23,pp.193210, 2007. [8]M.Niemier,M.Kontz,andP.Kogge,"ADesignofandDesignToolsforaNovelQuantumDotBasedMicroprocessor," 37thDesignAutomationConf.(DAC) ,pp.227232,June 2000,aCMPressLosAngelesC.A. [9]K.Walus,T.J.Dysart,G.A.Jullien,andA.R.Budiman,"QCADesigner:ARapidDesign andSimulationToolforQuantum-DotCellularAutomata," IEEETransactiononNanoTechnology ,vol.3,March2004. [10]B.TaskinandB.Hong,"Dual-PhaseLine-BasedQCAMemoryDesign," IEEEConference onNanotechnology ,vol.1,pp.302305,June2006. [11]P.M.Kogge,T.Sunaga,andE.Retter,"CombinedDRAMandLogicChipforMassively ParellelApplications," 16thIEEEConferenceonAdvancedResearchinVLSI,Raliegh,NC pp.416,March1995,iEEEComputerSocietyPressNo.PR070747. [12]M.Momenzadeh,J.Huang,andF.Lombardi,"DefectcharacterizationofQCASequential DevicesandCircuits," IEEEInternationalsymposiumonDefectandFaultTolerancein VLSISystems ,pp.199207,October2005. 56

PAGE 68

[13]R.K.Kummamuru,A.O.Orlov,R.Ramasubramaniam,C.S.Lent,G.H.Bernstein,and G.L.Snider,"OperationofQuantumDotcellularAutomataShiftRegisterandAnalysisof Errors," IEEETransactionsonElectronDevices ,vol.50,pp.19061913,September2003. [14]G.Gr ossingandA.Zeilinger,"Quantumcellularautomata," ComplexSyst. ,vol.2,no.2, pp.197208,1988. [15]D.Deutsch,"QuantumTheory,theChurch-TuringPrincipleandtheUniversalQuantum Computer," ProceedingsoftheRoyalSocietyofLondon.SeriesA,MathematicalandPhysicalSciences ,vol.400,no.1818,pp.97117,1985. [16]J.Watrous,"Onone-dimensionalquantumcellularautomata," FoundationsofComputer Science,AnnualIEEESymposiumon ,vol.0,p.528,1995. [17]P.TougawandC.Lent,"Logicaldevicesimplementedusingquantumcellularautomata," JournalofAppliedPhysics ,vol.75,pp.18181825,feb1994. [18]S.Srivastava,S.Sarkar,andS.Bhanja,"PowerDissipationBoundsandModelsforQuantumDotCellularAutomataCircuits," IEEEConferenceonNanotechnology ,vol.1,pp. 375378,June2006. [19]W.Wang,K.Walus,andG.Jullien,"Quantum-dotcellularautomataadders," Nanotechnology,2003.IEEE-NANO2003.2003ThirdIEEEConferenceon ,vol.1,pp.461464 vol.2,Aug.2003. [20]K.Walus,G.Schulhof,G.Jullien,R.Zhang,andW.Wang,"CircuitDesignBasedonMajorityGateforApplicationswithQuantum-DotCellularAutomata," Signals,Systemsand Computers,2004.ConferenceRecordofthe38AsilomarConference ,vol.2,pp.1354 1357,November2007. [21]Y.WangandM.Lieberman,"Thermodynamicbehaviorofmolecular-scalequantum-dot cellularautomata(qca)wiresandlogicdevices," Nanotechnology,IEEETransactionson vol.3,no.3,pp.368376,Sept.2004. [22]P.TougawandC.Lent,"DynamicBehaviourofQuantumCellularAutomata," Applied Physics ,vol.80,pp.47224736,October1996. [23]S.Bhanja,K.Lingasubramanian,andN.Ranganathan,"AStimulusFreeGraphicalProbabilisticSwitchingModelforSequentialCircuitsusingDynamicBayesianNetworks," ACM TransactiononDesignAutomationandElectronicSystem ,2006. [24]C.YuanandM.J.Druzdzel,"Theoreticalanalysisandpracticalinsightsonimportancesamplinginbayesiannetworks," Int.J.Approx.Reasoning ,vol.46,no.2,pp.320333,2007. [25]S.S.RamaniandS.Bhanja,"Any-timeprobabilisticswitchingmodelusingbayesiannetworks,"in ISLPED’04:Proceedingsofthe2004internationalsymposiumonLowpower electronicsanddesign .NewYork,NY,USA:ACM,2004,pp.8689. 57

PAGE 69

[26]S.Bhanja,K.Lingasubramanian,andN.Ranganathan,"Estimationofswitchingactivity insequentialcircuitsusingdynamicbayesiannetworks," VLSIDesign,2005.18thInternationalConferenceon ,pp.586591,Jan.2005. [27]"www.genie.sis.pitt.edu." [28]S.SrivastavaandS.Bhanja,"HierarchicalProbabilisticMacromodelingforQCACircuits," Computers,IEEETransactionson ,vol.56,no.2,pp.174190,Feb.2007. 58


xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam 2200373Ka 4500
controlfield tag 001 002059583
005 20100205121855.0
007 cr mnu|||uuuuu
008 100205s2008 flu s 000 0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0002787
035
(OCoLC)503563189
040
FHM
c FHM
049
FHMM
090
TK145 (Online)
1 100
Venkataramani, Praveen.
0 245
Sequential quantum dot cellular automata design and analysis using Dynamic Bayesian Networks
h [electronic resource] /
by Praveen Venkataramani.
260
[Tampa, Fla] :
b University of South Florida,
2008.
500
Title from PDF of title page.
Document formatted into pages; contains 58 pages.
502
Thesis (M.S.E.E.)--University of South Florida, 2008.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
520
ABSTRACT: The increasing need for low power and stunningly fast devices in Complementary Metal Oxide Semiconductor Very large Scale Integration (CMOS VLSI) circuits, directs the stream towards scaling of the same. However scaling at sub-micro level and nano level pose quantum mechanical effects and thereby limits further scaling of CMOS circuits. Researchers look into new aspects in nano regime that could effectively resolve this quandary. One such technology that looks promising at nano-level is the quantum dot cellular automata (QCA). The basic operation of QCA is based on transfer of charge rather than the electrons itself. The wave nature of these electrons and their uncertainty in device operation demands a probabilistic approach to study their operation. The data is assigned to a QCA cell by positioning two electrons into four quantum dots. However the site in which the electrons settles is uncertain and depends on various factors.In an ideal state, the electrons position themselves diagonal to each other, through columbic repulsion, to a low energy state. The quantum cell is said to be polarized to +1 or -1, based on the alignment of the electrons. In this thesis, we put forth a probabilistic model to design sequential QCA in Bayesian networks. The timing constraints inherent in sequential circuits due to the feedback path, makes it difficult to assign clock zones in a way that the outputs arrive at the same time instant. Hence designing circuits that have many sequential elements is time consuming. The model presented in this paper is fast and could be used to design sequential QCA circuits without the need to align the clock zones. One of the major advantages of our model lies in its ability to accurately capture the polarization of each cell of the sequential QCA circuits.We discuss the architecture of some of the basic sequential circuits such as J-K flip flop (FF), RAM memory cell and s27 benchmark circuit designed in QCADesigner. We analyze the circuits using a state-of-art Dynamic Bayesian Networks (DBN). To our knowledge this is the first time sequential circuits are analyzed using DBN. For the first time, Estimated Posterior Importance Sampling Algorithm (EPIS) is used to determine the probabilistic values, to study the effect due to variations in physical dimension and operating temperature on output polarization in QCA circuits.
538
Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
590
Advisor: Sanjukta Bhanja, Ph.D.
653
Probabilisitic modelling
QCA
Bayesian networks
Quantum-dot cellular automata
690
Dissertations, Academic
z USF
x Electrical Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.2787