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Probabilistic modeling of quantum-dot cellular automata

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Probabilistic modeling of quantum-dot cellular automata
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Srivastava, Saket
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University of South Florida
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Probabilistic model
QCA
Power dissipation
Bayesian networks
Quantum-Dot Cellular Automata
Dissertations, Academic -- Electrical Engineering -- Doctoral -- USF   ( lcsh )
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bibliography   ( marcgt )
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Abstract:
ABSTRACT: As CMOS scaling faces a technological barrier in the near future, novel design paradigms are being proposed to keep up with the ever growing need for computation power and speed. Most of these novel technologies have device sizes comparable to atomic and molecular scales. At these levels the quantum mechanical effects play a dominant role in device performance, thus inducing uncertainty. The wave nature of particle matter and the uncertainty associated with device operation make a case for probabilistic modeling of the device. As the dimensions go down to a molecular scale, functioning of a nano-device will be governed primarily by the atomic level device physics. Modeling a device at such a small scale will require taking into account the quantum mechanical phenomenon inherent to the device. In this dissertation, we studied one such nano-device: Quantum-Dot Cellular Automata (QCA).^ We used probabilistic modeling to perform a fast approximation based method to estimate error, power and reliability in large QCA circuits. First, we associate the quantum mechanical probabilities associated with each QCA cell to design and build a probabilistic Bayesian network. Our proposed modeling is derived from density matrix-based quantum modeling, and it takes into account dependency patterns induced by clocking. Our modeling scheme is orders of magnitude faster than the coherent vector simulation method that uses quantum mechanical simulations. Furthermore, our output node polarization values match those obtained from the state of the art simulations. Second, we use this model to approximate power dissipated in a QCA circuit during a non-adiabatic switching event and also to isolate the thermal hotspots in a design. Third, we also use a hierarchical probabilistic macromodeling scheme to model QCA designs at circuit level to isolate weak spots early in the design process.^ It can also be used to compare two functionally equivalent logic designs without performing the expensive quantum mechanical simulations. Finally, we perform optimization studies on different QCA layouts by analyzing the designs for error and power over a range of kink energies.To the best of our knowledge the non-adiabatic power model presented in this dissertation is the first work that uses abrupt clocking scheme to estimate realistic power dissipation. All prior works used quasi-adiabatic power dissipation models. The hierarchical macromodel design is also the first work in QCA design that uses circuit level modeling and is faithful to the underlying layout level design. The effect of kink energy to study power-error tradeoffs will be of great use to circuit designers and fabrication scientists in choosing the most suitable design parameters such as cell size and grid spacing.
Thesis:
Dissertation (Ph.D.)--University of South Florida, 2007.
Bibliography:
Includes bibliographical references.
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Statement of Responsibility:
by Saket Srivastava.
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Title from PDF of title page.
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Document formatted into pages; contains 129 pages.
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Includes vita.

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ABSTRACT: As CMOS scaling faces a technological barrier in the near future, novel design paradigms are being proposed to keep up with the ever growing need for computation power and speed. Most of these novel technologies have device sizes comparable to atomic and molecular scales. At these levels the quantum mechanical effects play a dominant role in device performance, thus inducing uncertainty. The wave nature of particle matter and the uncertainty associated with device operation make a case for probabilistic modeling of the device. As the dimensions go down to a molecular scale, functioning of a nano-device will be governed primarily by the atomic level device physics. Modeling a device at such a small scale will require taking into account the quantum mechanical phenomenon inherent to the device. In this dissertation, we studied one such nano-device: Quantum-Dot Cellular Automata (QCA).^ We used probabilistic modeling to perform a fast approximation based method to estimate error, power and reliability in large QCA circuits. First, we associate the quantum mechanical probabilities associated with each QCA cell to design and build a probabilistic Bayesian network. Our proposed modeling is derived from density matrix-based quantum modeling, and it takes into account dependency patterns induced by clocking. Our modeling scheme is orders of magnitude faster than the coherent vector simulation method that uses quantum mechanical simulations. Furthermore, our output node polarization values match those obtained from the state of the art simulations. Second, we use this model to approximate power dissipated in a QCA circuit during a non-adiabatic switching event and also to isolate the thermal hotspots in a design. Third, we also use a hierarchical probabilistic macromodeling scheme to model QCA designs at circuit level to isolate weak spots early in the design process.^ It can also be used to compare two functionally equivalent logic designs without performing the expensive quantum mechanical simulations. Finally, we perform optimization studies on different QCA layouts by analyzing the designs for error and power over a range of kink energies.To the best of our knowledge the non-adiabatic power model presented in this dissertation is the first work that uses abrupt clocking scheme to estimate realistic power dissipation. All prior works used quasi-adiabatic power dissipation models. The hierarchical macromodel design is also the first work in QCA design that uses circuit level modeling and is faithful to the underlying layout level design. The effect of kink energy to study power-error tradeoffs will be of great use to circuit designers and fabrication scientists in choosing the most suitable design parameters such as cell size and grid spacing.
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Probabilistic Modeling of Quantum-Dot Cellular Automata by Saket Srivastava A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Sanjukta Bhanja, Ph.D. Nagarajan Ranganathan, Ph.D. Wilfrido A. Moreno, Ph.D. Rudy Schlaf, Ph.D. Natasha Jonoska, Ph.D. Date of Approval: December 14, 2007 Keywords: probabilistic model, qca, power dissipation, baye sian networks, quantum-dot cellular automata Copyright 2008 Saket Srivastava

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DEDICATION Tomylovingwifeandmyparents.

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ACKNOWLEDGEMENTS IwouldliketotakethisgoldenopportunitytothankmymajorprofessorDr.SanjuktaBhanja.Withoutherguidancethisworkwouldn'thavebeenpossible.Shegave mecompletefreedominresearchprospective.Shehashelpedmealottonurturemyself asaresearcher.Shehastrainedmeineveryaspectofresearchlikereading,writingetc. Moreovershehasalsobeenagoodfriendtome. MysincerethankstoDr.NagarajanRanganathan,Dr.RudySchlaf,Dr.Natasha JonoskaandDr.WilfredoA.Morenoforservinginmycommittee. IwouldalsoliketothankDr.SudeepSarkarforhishelpandguidanceinmyresearch. IamreallyverygratefulfortheinvaluablesupportandmotivationthatIrecievedfrom myfamilyandfriends.

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TABLEOFCONTENTS LISTOFTABLESiii LISTOFFIGURESv ABSTRACTxi CHAPTER1INTRODUCTION1 1.1Motivation2 1.2NoveltyofthisWork6 1.3ContributionofthisDissertation7 1.3.1PowerDissipationModel8 1.3.2HierarchicalCircuitDesign8 1.3.3StudyofKinkEnergyVariationinQCADesign10 1.4Organization11 CHAPTER2QUANTUM-DOTCELLULARAUTOMATA12 2.1QCABasics14 2.2PhysicsofQCADeviceOperation17 2.3ImplementationofaQCACell21 2.3.1MetalIsland21 2.3.2Semiconductor22 2.3.3MolecularQCA25 2.3.4MagneticQCA27 2.4LogicalDevicesinQCA29 2.5ClockinginQCA32 2.6QCAArchitecture36 2.7DefectTolerance39 2.8ModelingQCADesigns41 CHAPTER3PROBABILISTICBAYESIANNETWORKMODELING43 3.1Introduction43 3.2QuantumMechanicalProbabilities44 3.3BayesianModeling45 3.4ExperimentalResults49 i

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CHAPTER4POWERDISSIPATIONINQCA53 4.1Introduction53 4.2QuantumMechanicalPower56 4.3UpperBoundforPowerDissipation60 4.4EnergyDissipatedperClockCycleinaQCACircuit63 4.5Results65 4.5.1EnergyDissipationperClockCycleinaSingleQCACell66 4.5.2EnergyDissipationperClockCycleinBasicQCACircuits68 4.5.3EnergyDissipationperClockCycleinQCAAdderCircuits70 4.5.4EnergyDissipationperClockCycleinLargeQCACircuits72 CHAPTER5HIERARCHICALDESIGNINQCAUSINGPROBABILISTIC MACROMODELING77 5.1Introduction77 5.2ModelingTheory81 5.2.1QuantumMechanicalProbabilities84 5.2.2LayoutLevelModelofCellArrangements86 5.2.3Macromodel87 5.2.4CircuitLevelModeling91 5.3ErrorComputation93 5.4Results95 5.4.1Polarization95 5.4.2ErrorModes97 5.4.3DesignSpaceExploration98 5.4.4ComputationalAdvantage106 CHAPTER6EFFECTOFKINKENERGYINQCADESIGN109 6.1KinkEnergy110 6.2Results111 6.2.1NodePolarizationError111 6.2.2SwitchingPower113 CHAPTER7CONCLUSIONANDFUTUREWORKS116 REFERENCES119 ABOUTTHEAUTHOREndPage ii

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LISTOFTABLES Table4.1.BlochHamiltonianbeforeandafterachangeinclockortheneighboringpolarization63 Table4.2.Thermallayoutvisualizingtheenergydissipatedateachcellaveraged overdifferentinputtransitionsforsomebasicQCAlogicelements. Darkerthecolor,morethedissipation.69 Table4.3.Thermallayoutvisualizingtheenergydissipationateachcell,averagedoverallpossibleinputcombinationsfortwoQCAadderdesigns.71 Table4.4.Statisticsoftheenergydissipationpercellfora4x1MUXandasinglebitALUoverallpossibleinputcombinationsandfordifferent possibleclockenergies.Weshowtheaverage,maximum,andminimumenergypercelloverallinputcombinations.74 Table5.1.Macromodeldesignblocks88 Table5.2.Macromodeldesignblocks89 Table5.3.AbbreviationsusedforMacromodelBlocksfordesigningQCAarchitecturesofFullAddersandMultiplier93 Table5.4.Layoutandmacromodeltime( Tc)andspace( Ts)complexities.Please seetextforanexplanation Cmax, n ,and p .107 Table5.5.Comparisonbetweensimulationtiming(inseconds)ofaFullAdder andMultipliercircuitsinQCADesigner(QD)andGenieBayesian Network(BN)ToolforFullLayoutandMacromodelLayout108 Table6.1.DifferenttypesofQCAcellsandgridspacingusedinthisstudy111 Table6.2.OutputnodepolarizationofasimplemajoritygatefordifferentKink Energies112 Table6.3.OutputnodepolarizationofaQCAInverterfordifferentKinkEnergies112 iii

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Table6.4.PowerdissipationinQCAmajoritygatefordifferentKinkEnergies113 Table6.5.PowerdissipationinQCAInverterfordifferentKinkEnergies113 Table6.6.OutputnodepolarizationatSUMoutputnodeofAdder-1andAdder2QCAdesigns114 Table6.7.Non-AdiabaticEnergydissipationinAdder-1andAdder-2QCAdesigns114 iv

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LISTOFFIGURES Figure1.1.Candidatesforthenewnano-switch3 Figure1.2.PerformanceEvaluationforEmergingLogicDeviceTechnologies. Imagefrom[6]5 Figure1.3.Parallelbetweendesignmethodologiesfor(a)CMOSandthatproposedfor(b)QCA(Hendersonet.al.[26]).9 Figure2.1.VariousGroupsinvolvedindifferentareasofQCAresearch13 Figure2.2.Ansimple4-dotunpolarizedQCAcell15 Figure2.3.Twopolarizedstatesofa4-dotQCAcell15 Figure2.4.TransferofpolarizationbetweenadjacentQCAcellswhenthepolarizationofthedrivercellischangedfro mP=+1toP=-116 Figure2.5.TemperaturedependenceonthepolarizationofaQCAcellwithrespecttothechangeinpoalizationofthedrivercell.16 Figure2.6.Thevariousquantizedenergystatesofanelectroninaonedimensionalinnitepotentialwell.Foreachquantizedstatethepossiblewavefunctionsandprobabilitydistributionsfortheelectronare shown.ImageredrawnfromPrinciplesofElectronicMaterialsand Devices[43].18 Figure2.7.Ananexampleofanelectrontunnelingacrossanitepotentialwell. ImageredrawnfromPrinciplesofElectronicMaterialsandDevices [43].20 Figure2.8.Controllingthetunnelingbarrierbyvariationofclockenergy.Asthe clockenergysuppliedtoaQCAcellincreases,thetunnelingbarriers lower,makingitpossibleforelectrontotunnelacrosstotheother side.21 Figure2.9.(a)SEMimageofaMetallicDotQCAand(b)Schematicdiagram. Imagefrom:Orlovet.al.[44]22 v

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Figure2.10.ExamplequantumdotpyramidcreatedwithInAs/GaAs.Imagefrom: UniversityofNewcastle:CondensedMatterGroup[49]23 Figure2.11.(a)ElectronmicrographofaGaAs/AlGaAsQCAcell.(b)Simplied circuitequivalentofthefour-dotcell.Imagefrom:Perez-Martinez et.al.[53]24 Figure2.12.TwoviewsofMolecule1asaQCAcell.Imagefrom:Lentet.al.[57]24 Figure2.13.DifferentpossiblestatesofMolecule1(a)showsa+1state(b)anonidealstatethatisaunwantedstateand(c)showsa-1state.Image from:Lentet.al.[57]26 Figure2.14.(a)SEMimageofaroomtemperatureMQCAnetworkshownin (Cowburnet.al.[74])(b)Majoritygatesdesignedfortestingallinputcombinationsofthemajority-logicoperation.Thearrowsdrawn superimposedontheSEMimagesillustratetheresultingmagnetizationdirectionduetoahorizontallyappliedexternalclock-eld(Imre et.al.[79])27 Figure2.15.AQCAmajoritygate30 Figure2.16.QCAmajoritygatelogicwithdifferentinputs30 Figure2.17.AQCAinverter31 Figure2.18.ANDandORgaterepresentationofamajoritygatebyxingoneof theinputsasP=-1orP=+1respectively31 Figure2.19.QCANANDgateusinganANDgateandaninverter32 Figure2.20.AsinglebitQCAadderdesign33 Figure2.21.FourstagesinaQCAclock(1)Tunnelingbarriersstarttorise(2) Hightunnelingbarrierspreventelectronsfromtunneling(3)Tunnelingbarriersbegintolower(4)Electronsarefreetotunnel.34 Figure2.22.FourQCAclocksphaseshiftedby90degrees.34 Figure2.23.FlowofinformationinQCAlinecontrolledbyclockpropagation35 Figure2.24.LandauerandBennettclockingofQCAcircuits.(Lentet.al.[90])37 Figure2.25.AQCAMemoryCell(Waluset.al.[42])39 Figure2.26.DifferentcongurationsofdisplacedQCAcellsinamajoritygate. Conguration(a)isfaultfree(Tahooriet.al.[30])40 vi

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Figure3.1.AsmallBayesiannetwork46 Figure3.2.ClockedQCAmajoritygatelayout49 Figure3.3.BayesiannetdependencystructurecorrespondingtotheQCAmajoritygatewithnodescorrespondingtotheindividualcellsandlinks denotingdirectdependencies.50 Figure3.4.ExplodedviewoftheBayesiannetstructure,layingbarethedirected linkstructureandthenodeinformation.51 Figure3.5.Dependenceofprobabilityofcorrectoutputofthemajoritygatewith temperature and inputs.Notethedependenceoninputs.51 Figure3.6.ValidationoftheBayesiannetworkmodelingofQCAcircuitswith HartreeFockapproximationbasedcoherencevectorbasedquantum mechanicalsimulation.Probabilitiesofcorrectoutputarecompared forbasiccircuitelements52 Figure4.1.Polarizationchange(topplot)andpowerloss(bottomplot)inasingle cellwhenitspolarizationchangesfrom(a)-1to1(or0to1logic) and(b)-1to-1(remainsatstate0)duringaquasi-adiabaticclocking scheme.60 Figure4.2.Polarizationchange(topplot)andpowerloss(bottomplot)inasingle cellwhenitspolarizationchangesfrom(a)-1to1(or0to1logic)and (b)-1to-1(nochangeinstate)duringnon-adiabaticclockingscheme61 Figure4.3.Variationof(a)switchingpowerand(b)leakagepowerdissipated inasinglecellwithdifferentamountofclocksmoothingfordifferentclockenergy levels.Adiabaticityoftheswitchingprocessis controlledbysmoothnessoftheclocktransition.Thehorizontalline plotstheupperboundsforeachcaseascomputedusingthederived expressions.66 Figure4.4.Dependenceofenergydissipated(upperbound)inacellwithclock energyfordifferentclocktransitions.(a)00(b)01(c)10and (d)11.Notethattheplotsforcases(a)and(d)overlapcompletely andsodoestheplotsforcases(b)and(c).67 Figure4.5.EnergydissipationboundspercellfordifferentQCAlogicelements, averagedoverdifferentinputcombinations.Thenumberofcellsfor eachcircuitreferstothenumberofcellsthatdissipateenergyduring aswitchingevent.Thegraphshownhereisfor EK=0.5.Notethat thecolormappingscaleforeachcircuitisdifferent.70 vii

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Figure4.6.ThermalLayoutforaverageenergydissipatedineachcellofa4x1 MUXcircuit.Thedarkspotsaretheonesthatdissipatelargeramount ofenergyonanaverage.Thelayoutwasobtainedbysimulatingover allpossibleinputswitchingcombinationsfrom000000111111for EK=0.5.Theenergydissipationscaleforeachcellisintermsof 103eV.73 Figure4.7.ThermalLayoutforaverageenergydissipatedineachcellofasinglebitALUcircuit.Thedarkspotsaretheonesthatdissipatelarger amountofenergyonanaverage.Thelayoutwasobtainedbysimulatingoverallpossibleinputswitchingcombinationsfrom00000001111111for EK=0.5.Theenergydissipationscaleforeachcellis intermsof103eV.73 Figure4.8.ThermalLayoutforenergydissipatedineachcellofanALUcircuit for(a)Maximumenergydissipatinginputcombinationand(b)for leastenergydissipatinginputcombination.Energydissipationscale isinmultiplesof103.75 Figure4.9.GraphsshowingenergydissipatedinaQCAALUcircuit(a)Shows thevariationofleakageandswitchingcomponentsofenergydissipatedforvariousvaluesof EK(b)Showsthevariationinmaximum andminimumenergydissipatedforvariousvaluesof EK75 Figure5.1.ANANDlogicgate(a)QCAlayout(b)BayesianmodelofQCAlayout(c)Macromodelblockdiagram(d)Bayesiannetworkofmacromodelblockdiagram.82 Figure5.2.Majoritylogic(a)QCAcelllayout(b)Bayesiannetworkmodel(c) Macromodel(d)Probabilityofthe correct outputvaluefora5cell majoritygateatdifferenttemperaturesandfordifferentinputs.86 Figure5.3.Afulladdercircuit(Adder-1)(a)QCAcelllayout(b)Layoutlevel Bayesiannetworkrepresentation.(c)Circuitlevelrepresentation.(d) CircuitlevelBayesiannetworkmacromodel.Note:Nodeelements aregeneric.92 Figure5.4.ProbabilityofcorrectoutputforsumandcarryofAdder-1basedon thelayout-levelBayesiannetmodelandthecircuitlevelmacromodel, atdifferenttemperatures,fordifferentinputs(a)(0,0,0)(b)(0,0,1)(c) (0,1,0)(d)(0,1,1).96 Figure5.5.AQCAFullAddercircuit(Adder-2)(a)QCAFulladdercelllayout (b)Macromodelrepresentation(c)MacromodelBayesiannetwork. Note:Nodeelementsaregeneric.96 viii

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Figure5.6.ProbabilityofcorrectoutputforsumandcarryofAdder-2basedon thelayout-levelBayesiannetmodelandthecircuitlevelmacromodel, atdifferenttemperatures,fordifferentinputs(a)(0,0,0)(b)(0,0,1)(c) (0,1,0)(d)(0,1,1).96 Figure5.7.AQCA2x2Multipliercircuit(a)QCAmultipliercelllayout(b)Macromodelrepresentation99 Figure5.8.MacromodelBayesiannetworkofaQCA2x2Multipliercircuit.Note: Nodeelementsaregeneric.100 Figure5.9.Probabilityofcorrectoutputatthefouroutputnodesof2x2Multipliercircuitbasedonthelayout-levelBayesiannetmodelandthecircuitlevelmacromodel,atdifferenttemperatures,fordifferentinputs (a)(0,0),(0,1)(b)(0,0),(1,1)(c)(0,1),(0,1)(d)(0,1),(1,1)100 Figure5.10.Probabilityofcorrectoutputatthefouroutputnodesof2x2Multipliercircuitbasedonthelayout-levelBayesiannetmodelandthecircuitlevelmacromodel,atdifferenttemperatures,fordifferentinputs (a)(1,0),(0,1)(b)(1,0),(1,1)(c)(1,1),(0,1)(d)(1,1),(1,1).101 Figure5.11.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-1 CircuitanditsMacromodeldesign.Itcanbeseenthattheerroneous nodesinthelayoutareeffectivelymappedinthemacromodeldesign. Inputvectorsetfor(a)and(b)is(0,0,0)andthatfor(c)and(d)is (1,0,0).Note:Nodeelementsaregeneric.102 Figure5.12.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-1 CircuitanditsMacromodeldesign.Itcanbeseenthattheerroneous nodesinthelayoutareeffectivelymappedinthemacromodeldesign. Inputvectorsetfor(a)and(b)is(0,1,0)andthatfor(c)and(d)is (1,1,0).Note:Nodeelementsaregeneric.103 Figure5.13.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-2 CircuitanditsMacromodeldesign.Itcanbeseenthattheerroneous nodesinthelayoutareeffectivelymappedinthemacromodeldesign. Inputvectorsetfor(a)and(b)is(0,0,0)andthatfor(c)and(d)is (1,0,0).Note:Nodeelementsaregeneric.104 Figure5.14.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-2 CircuitanditsMacromodeldesign.Itcanbeseenthattheerroneous nodesinthelayoutareeffectivelymappedinthemacromodeldesign. Inputvectorsetfor(a)and(b)is(0,1,0)andthatfor(c)and(d)is (1,1,0).Note:Nodeelementsaregeneric.105 ix

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Figure6.1.KinkenergybetweentwoneighboringQCAcells110 x

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PROBABILISTICMODELINGOFQUANTUM-DOTCELLULARAUTOMATA SaketSrivastava ABSTRACT AsCMOSscalingfacesatechnologicalbarrierinthenearfuture,noveldesignparadigms arebeingproposedtokeepupwiththeevergrowingneedforcomputationpowerandspeed. Mostofthesenoveltechnologieshavedevicesizescomparabletoatomicandmolecular scales.Attheselevelsthequantummechanicaleffectsplayadominantroleindeviceperformance,thusinducinguncertainty.Thewavenatureofparticlematterandtheuncertainty associatedwithdeviceoperationmakeacaseforprobabilisticmodelingofthedevice.As thedimensionsgodowntoamolecularscale,functioningofanano-devicewillbegovernedprimarilybytheatomicleveldevicephysics.Modelingadeviceatsuchasmall scalewillrequiretakingintoaccountthequantummechanicalphenomenoninherenttothe device. Inthisdissertation,westudiedonesuchnano-device:Quantum-DotCellularAutomata (QCA).Weusedprobabilisticmodelingtoperformafastapproximationbasedmethodto estimateerror,powerandreliabilityinlargeQCAcircuits.First,weassociatethequantum mechanicalprobabilitiesassociatedwitheachQCAcelltodesignandbuildaprobabilistic Bayesiannetwork.Ourproposedmodelingisderivedfromdensitymatrix-basedquantum modeling,andittakesintoaccountdependencypatternsinducedbyclocking.Ourmodelingschemeisordersofmagnitudefasterthanthecoherentvectorsimulationmethodthat usesquantummechanicalsimulations.Furthermore,ouroutputnodepolarizationvalues xi

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matchthoseobtainedfromthestateoftheartsimulations.Second,weusethismodelto approximatepowerdissipatedinaQCAcircuitduringanon-adiabaticswitchingeventand alsotoisolatethethermalhotspotsinadesign.Third,wealsouseahierarchicalprobabilisticmacromodelingschemetomodelQCAdesignsatcircuitleveltoisolateweakspots earlyinthedesignprocess.Itcanalsobeusedtocomparetwofunctionallyequivalent logicdesignswithoutperformingtheexpensivequantummechanicalsimulations.Finally, weperformoptimizationstudiesondifferentQCAlayoutsbyanalyzingthedesignsfor errorandpoweroverarangeofkinkenergies. Tothebestofourknowledgethenon-adiabaticpowermodelpresentedinthisdissertationistherstworkthatusesabruptclockingschemetoestimaterealisticpowerdissipation.Allpriorworksusedquasi-adiabaticpowerdissipationmodels.Thehierarchical macromodeldesignisalsotherstworkinQCAdesignthatusescircuitlevelmodeling andisfaithfultotheunderlyinglayoutleveldesign.Theeffectofkinkenergytostudy power-errortradeoffswillbeofgreatusetocircuitdesignersandfabricationscientistsin choosingthemostsuitabledesignparameterssuchascellsizeandgridspacing. xii

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CHAPTER1 INTRODUCTION Inlate1960'sGordonMoore,ofIntelCorporation,predictedthatthetransistordensity orthenumberoftransistorsonanICchipwillgrowexponentiallyovertime[1].This trendofsemiconductorscalinghasbeenthebenchmarkforthegrowthofresearchand developmentactivityallovertheworld.Untilrecently,semiconductorindustryhasbeen abletokeepupwithMoore'slawovertheyears,packingmoreandmorecomputational powerintoourmicroprocessors.Withtransistorsizeshrinkingtonanometerscales,ithas beenahardbattleintherecentyearsfortheindustrytokeepupwiththescalingprocess. Thesmallesttransistorsinproductiontodayoperatedespitequantumeffects.Inthenear future,theoperationoftransistorswillbedominatedbythephysicsofquantumworld. Physicallimitationsofconventionaltransistorsincludingpowerdissipation,interconnects andfabricationarebecomingincreasinglydifculttosurmountwitheachtechnologygeneration[2,3].Therewillbeanurgentneedinthenearfuturetoreplacethecurrentdevice, theCMOStransistor,byonethatembracesthesequantumeffectsandtakesadvantageof thenanoscalephysics.Keepingthisinmind,noveldesignparadigmsarebeingproposedto keepupwiththeevergrowingneedforcomputationpowerandspeed.Thereneedstobe achangeinperspectivefromthedesignersandfabricationscientistsaliketolookbeyond CMOS. 1

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1.1Motivation WhileeventhecurrentgenerationCMOStechnologyisnanoscale,theissuesrelated todopingregions,oxidethickness,stickinglayers,diffusionbarriers,powerdissipation, andleakagecurrentsetc.haveputabigquestiononthefeasibilityofpursuingwithCMOS technologyinfuture[4,5].Thebigquestionfornanotechnologyis,whathappensafterthat andwhatkindofnanotechnologywillbeusedtoreplacethestandardCMOStransistors? Thequestionhastwoanswers.Inordertostayontheroadmapwithitsinexorable progress,nanotechnologyisalreadyrequired.ItisclearthatCMOSscalingwillcontinue foratleast10moreyearstillaround2018[6].Thiswillrequiretheincreasinguseof nanotechnologyinnewmaterialsfordielectrics,gates,interconnects,andchannels.There willbenewprocesses,materials,andstructuresthatrequireengineeringatthenanoscale. Thus,foratleast10years,nanotechnologywillextendandenhancestandardCMOSVLSI technology.Whileinadecadeormoremuchofthestandardapproachwillbenanoscale, thoughitwillnotbearevolutionbutlikeanrapidevolution,itwillstillbeacontinuation ofwhathasgoneonbefore. Sothequestionagainarises,whathappensafterthat?Whichdevicewillbethenew switch?CandidatesincludeQuantumCellularAutomata[7],CarbonNanotubeTransistors[8,9],siliconnanowires[10],spintransistors[11],superconductingelectronics[12, 13],molecularelectronics[14,15],SingleElectronTransistors[16,17],ResonantTunnelingDevices[18]andTunnelingPhaseLogic[19]asportrayedinFig1.1.TheInternational TechnologyRoadmapforSemiconductors(ITRS)describeshowthesetechnologieswork anddiscusssomeofthechallengesinimplementingthem.TheITRSroadmap[6]presents the"bestcurrentestimate"basedonanindustry-wideconsensusofitsR&Dneedsforthe next15years.Itisconsideredtobeanunbiaseddocumentthatisusedbyindustryand 2

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The new Nano-switch? CNT QCA SET Silicon Nanowire Spin Transistors RTD TPL Molecular Electronics Superconducting Electronics Figure1.1.Candidatesforthenewnano-switch 3

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researchgroupsallaroundtheworldtokeepthemselvesupdatedaboutthecurrentlevelof researchinseveraltechnologies. ThetableshowninFig1.2.showstheresultsofthecriticalreviewassessmentofemerginglogictechnologieshighlightedin[6].Itisclearthatanumberofdevicesshowagreat promiseforfutureresearch.Eachofthemcanbeconsiderednovelandsomeevenrevolutionary.Sometechnologiesoffertremendousscopeforcomputationwithhighpacking densities,whileothersofferextremelylowpowerdissipation.Atpresentitisrequiredto continueresearchfordevicesthatwillbemostsuited[20]tosupporttheevergrowing computationneedsbyofferingsmallsizesandhighpackingdensitiesandatthesametime providingtremendoussavinginpowerdissipation.Thiswilleliminatethebottleneckthat existsinthescalingofCMOSdevices. Mostoftheproposednoveltechnologiesdiscussedabovehavedevicesizescomparabletoatomicandmolecularscale.Attheselevelsthequantummechanicaleffectsplay adominantroleindeviceperformanceandinduceuncertainty.Quantum-dotCellularAutomata(QCA)isonesuchemergingnanotechnologythatoffersarevolutionaryapproachto computingatnano-level.QCAtechnologytriestoexploittheinevitablenano-levelissues, suchasdevicetodeviceinteraction,toperformcomputing.Inthecurrenttechnologiesthis devicetodeviceinteractionatnano-levelisoneofthebiggestroadblocksinfurtherscaling ofCMOSdevices.OtheradvantagesofQCAinclude:thelackofinterconnects,potential forimplementationinmetal,andusingmolecules.SinceQCAconceptdoesnotinvolve transferofelectrons,ithasapotentialforextremelylow-powercomputing,evenbelowthe traditional kBT [21].MagneticandmolecularimplementationsofQCAhaveapotential forroomtemperatureoperation. 4

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Figure1.2.PerformanceEvaluationforEmergingLogicDeviceTechnologies.Imagefrom [6] 5

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1.2NoveltyofthisWork Theunderlyinguncertaintyinnanoscaledeviceoperationmakesacaseforprobabilistic modelingofthesetechnologies.Inthisworkwedevelopafast,Bayesianprobabilistic computingmodel[22,23]thatexploitstheinducedcausalityofaclockedQCAcircuit toarriveatamodelwiththeminimumpossiblecomplexity.Theprobabilitiesdirectly modelthequantum-mechanicalsteady-stateprobabilities(densitymatrix)orequivalently, thecellpolarizations.Theattractivefeatureofthismodelisthatnotonlydoesitmodel thestrongdependenciesamongthecells,butitcanbeusedtocomputethesteadystatecell polarizations,withoutiterationsortheneedfortemporalsimulationofquantummechanical equations.Theimpactofourproposedmodelingisthatitisbasedondensitymatrixbasedquantummodeling,takesintoaccountdependencypatternsinducedbyclocking, andisnon-iterative.Itallowsforquickestimationandcomparisonofquantum-mechanical quantitiesforaQCAcircuit,suchasQCA-stateoccupancyprobabilitiesorpolarizations atanycell,theirdependenceontemperature,oranyparameterthatdependsonthem.This willenableonetoquicklycompare,contrastandnetuneclockedQCAcircuitsdesigns, beforeperformingcostlyfullquantum-mechanicalsimulationofthetemporaldynamics. In[24,25],itwasshownthatlayout-levelQCAcellprobabilitiescanbemodeledusing Bayesianprobabilisticnetworks. Inotherwords,wemakeuseofafastBayesiancomputingmodelinwhicheachQCA cellisdenedbythequantummechanicalprobabilitiesassociatedwiththecellandits neighborsandthecausalityofthedesignisderivedfromthedirectionofpropagationof clocksignal.Thisprobabilisticmodelhasbeenshowntoaccuratelycapturethedevice characteristicsandprovideresultsthatareordersofmagnitudefasterthanthetraditional methodsinvolvingtimeconsumingquantummechanicalsimulations.Theresearchpresentedinthiscanbebroadlycategorizedinthefollowingareas: 6

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WeusetheBayesianprobabilisticmodeltoestimatepowerdissipatedinaQCA circuitduringanonadiabaticswitchingevent.Tothebestofourknowledge,this workistherstworkthatprovidesarealisticestimateofpowerdissipationusinga non-adiabaticclockingscheme.Wehavetermedthispowerdissipationasworstcase power.ThehierarchicalcircuitdesignschemepresentedinthisworkmakesuseofprobabilisticmacromodelstodesignaQCAcircuit.Thisnotonlyreducesthecomplexity ofcircuitdesignbyordersofmagnitude,ithasshowntobemuchmoretimeefcient withresultscomparabletothelayoutleveldesign.Tothebestofourknowledge,this istherstworkinQCAthatisusedtoisolateweakspotsinadesignatearlyonina designprocess.Itcanalsobeusedtoperformdesignspaceexplorationtocompare twofunctionallyequivalentcircuitswithouthavingtodesignitatlayoutlevel.Deviceparametervariationtoperformtradeoffstudieshavebeenthehallmarkof researchinCMOSandothertechnologies.Itisnaturaltoperformsuchstudiesto evaluatetheoptimumdesignparametersforaQCAcircuit.Inthisworkweundertake onesuchstudyinQCAdesignbyvaryingthemaximumkinkenergyofaQCA design. 1.3ContributionofthisDissertation Inthissectionweprovideamoredetaileddescriptionofthenovelcontributionsoutlinedintheprevioussection.Asadevelopingtechnology,thereareanumberofresearch areasinQCA.Wetargettedthefollowingthreeareasinthiswork. 7

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1.3.1PowerDissipationModel SinceQCAisaeld-coupledcomputingparadigm,statesofacellchangeduetomutualinteractionsofeitherelectrostaticormagneticelds.Duetotheirsmallsizes,power isanimportantdesignparameter.Wederiveanupperboundforpowerlosstoestimate powerdissipatedinlargeQCAcircuits.WecategorizepowerlossinclockedQCAcircuitsintotwowellknowngroups:switchingpowerandleakagepower.Leakagepower lossisindependentofinputstatesandoccurswhentheclockenergyisraisedorlowered todepolarizeorpolarizeacell.Switchingpowerisdependentoninputcombinationsand occurswhenthecellactuallychangesstate.Totalpowerlosscanbemadeverysmallby controllingtherateofchangeoftheclock,i.e.adiabaticclocking.Ourmodelprovidesa realisticestimateofpowerlossinaQCAcircuitundernon-adiabaticclockingscheme.We deriveexpressionsforupperboundsofswitchingandleakagepowerthatareeasytocompute.Upperboundsobviouslyarepessimisticestimates,butarenecessarytodesignrobust circuits,leavingroomformanufacturingvariability.Giventhatthermalissuesarecritical toQCAdesigns,weshowhowourmodelcanbevaluableforQCAdesignautomationin multipleways.ItcanbeusedtoquicklylocatepotentialthermalhotspotsinaQCAcircuit. Themodelcanalsobeusedtocorrelatepowerlosswithdifferentinputvectorswitching; powerlossisdependentontheinputvector.Wecanstudythetrade-offbetweenswitching andleakagepowerinQCAcircuits.And,wecanusethemodeltovetdifferentdesignsof thesamelogic,whichwedemonstrateforthefulladder. 1.3.2HierarchicalCircuitDesign ToadvancedesignwithQCA,itisnecessarytolookbeyondthelayoutlevel.Hierarchicaldesignatmultiplelevelsofabstraction,suchasarchitectural,circuit,layout,and devicelevels,hasbeenasuccessfulparadigmforthedesignofcomplexCMOScircuits.It 8

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Conceptual Design Create and verify Behavioral model Create and verify structural logic model Create and verify Structural Transistor model Individual Transistor Layout SPICE simulation of Layout More Detail Less Detail Conceptual Design Create and verify Behavioral model Create and verify structural logic model Create and verify Structural QCA model Individual QCA Cell Layout Quantum Mechanical Simulation of Layout More Detail Less Detail (a)(b) Figure1.3.Parallelbetweendesignmethodologiesfor(a)CMOSandthatproposedfor(b) QCA(Hendersonet.al.[26]). isonlynaturaltoseektobuildasimilardesignstructureforemergingtechnology.Hendersonetal.[26]proposedahierarchicalCMOS-liketop-downapproachforQCAblocksthat areanalyzedwithrespecttotheoutputlogicstates;thisissomewhatsimilartofunctional logicvericationperformedinCMOS(Fig1.3.).Wealsoadvocatebuildingahierarchical designmethodologyforQCAcircuits.However,suchahierarchyshouldbebuiltbasedon notjustthefunctionalityofthecircuit,butitshouldalsoallowtheabstractionofimportant nano-deviceparameters[27,28]. RecognizingthatthebasicoperationofQCAisprobabilisticinnature,wepropose probabilisticmacromodelsforstandardQCAcircuitelementsbasedonconditionalprobabilitycharacterization,denedovertheoutputstatesgiventheinputstates.Anycircuit modelisconstructedbychainingtogethertheindividuallogicelementmacromodels,formingaBayesiannetwork,deningajointprobabilitydistributionoverthewholecircuit. 9

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Wedemonstratethreeusesforthesemacromodelbasedcircuits.First,theprobabilistic macromodelsallowustomodelthelogicalfunctionofQCAcircuitsatanabstractlevel; the'circuit-level'abovethecurrentpracticeoflayoutlevelinatimeandspaceefcient manner.Weshowthatthecircuitlevelmodelisordersofmagnitudefasterandrequires lessspacethanlayoutlevelmodels,makingthedesignandtestingoflargeQCAcircuits efcientandrelegatingthecostlyfullquantum-mechanicalsimulationofthetemporaldynamicstoalaterstageinthedesignprocess.Second,theprobabilisticmacromodelsabstractcrucialdevicelevelcharacteristicssuchaspolarizationandlow-energyerrorstate congurationsatthecircuitlevel.Wedemonstratehowthismacromodelbasedcircuitlevel representationcanbeusedtoinferthegroundstateprobabilities,i.e.cellpolarizations,a crucialQCAparameter.ThisallowsustostudythethermalbehaviorofQCAcircuitsat ahigherlevelofabstraction.Third,wedemonstratetheuseofthesemacromodelsforerroranalysis.Weshowthatthatlow-energystatecongurationofthemacromodelcircuit matchesthoseofthelayoutlevel,thusallowingustoisolateweakpointsincircuitsdesign atthecircuitlevelitself. 1.3.3StudyofKinkEnergyVariationinQCADesign Whiletherehavebeenexperimentalstudiesrelatedtodefectandfaulttalerancein QCA[29,30,31],notmuchworkhasbeendonetostudytheeffectsofvariationdevice parametersonerrorandpowerinQCAdesign.Similarstudiesofhiskindhavebeenthe hallmarkofCMOSresearchovertheyearsthatcontributedsignicantlyinthedevelopmentofCMOStechnology.Itisnaturaltoperformsuchastudywithrespecttoparameter variationsinQCA.WeperformastudyoferrorandpowerdissipationinaclockedQCA designbyvaryingoneofthemostcrucialparameterinQCAdesign;thekinkenergy.Kink energyistheenergycostofkeepingtwoadjacentcellsinoppositepolarizationandvaries withthesizeofaQCAcellandthegridspacing.Weanalyzetheeffectsofkinkenergy 10

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withadesignperspectivetohelpdesignersandfabricationscientiststochoosethemost optimumsizeofQCAcellandspacingbetweenadjacentQCAcells. 1.4Organization Thisdissertationisorganizedasfollows:Chapter2providesanoverviewofQCA technologyandthelogiclogicasscociatedwithit.Italsoincludesabriefdiscussionof varioustypesofQCAimplementation,currentlyunderresearch.Chapter3describesthe probabilisticmodelofQCAdevelopedinthiswork.Itelaboratesthederivationofquantum mechanicalprobabilitiesassociatedwitheachQCAcelltakingintoaccountthedependancy patternsinducedbyclocking.Theseprobabilitiesarethenusedtoderiveanoveralljoint probabilitydistributionfunctionofaQCAcircuitrepresentedasaBayesiannetwork.In chapter4,weusethismodeltoapproximatepowerdissipatedinaQCAcircuitduring anon-adiabaticswitchingevent.Inchapter5,wemakeuseofhierarchicalprobabilistic macromodelingschemetomodelQCAdesignsatcirsuitlevel.Weshowtheuseofthis hierarchicaldesignschemetoisolateweakspotsearlyoninthedesignprocess.Finally, inchapter6,weshowasetofstudiesrelatedtoerror-powertradeoffsinQCAdesign. Concludingremarksarelistedinchapter7. 11

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CHAPTER2 QUANTUM-DOTCELLULARAUTOMATA Theconceptofacellularautomatonoperatingonquantummechanicalprinciplesdates backtoRichardFeynman[32],whosuggestedaninitialapproachtoquantizingamodel ofcellularautomata.GerhardGrssingandAntonZeilingerintroducedtheterm"quantum cellularautomata"torefertoamodeltheydenedin1988[33]however,theirmodelhas verylittleincommonwiththeconceptsdevelopedinquantumcomputationafterDavid Deutsch'sformaldevelopmentofthatsubjectin1989[34]andsohasnotbeendeveloped signicantlyasamodelofcomputation. Aproposalforimplementingclassicalcellularautomatabysystemsdesignedwith quantumdotshasbeenproposedunderthename"QuantumCellularAutomata"byPaul TougawandCraigLent,asareplacementforclassicalcomputationusingCMOStechnology[35,36].Inordertobetterdifferentiatebetweenthisproposalandmodelsofcellular automatawhichperformquantumcomputation,manyauthorsworkingonthissubjectnow refertothisasaQuantum-dotCellularAutomata. QCAoffersarevolutionaryapproachtocomputingatnano-level[37,38].Ittriesto exploit,ratherthantreatasnuisanceproperties,theinevitablenano-levelissues,suchas devicetodeviceinteraction,toperformcomputing.Otheradvantagesincludethelackof interconnects,potentialforimplementationinmetal[39],andusingmolecules[40,41]. MagneticandmolecularimplementationsofQCAhavepotentialforroomtemperatureoperations. 12

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Architecture/Testing Logic FabricationP. Koggeet.al. Niemier/Bernstein et.al.Univ. of Notre DameF. Lombardi/Tahooriet. al.NE Univ.K. Wang et.al.UCLAJ. Abraham et.al.Univof Texas, AustinN. Jhaet.al. Princeton Univ. R. Karri et.al.Poly. Univ. BrooklynK. Wu et.al.Univ. Incheon, KoreaD. Tougaw/J.Willet.al. Valparaiso,IN K Waluset.al. Univof Calgary THIS WORK M. Lieberman et.al. T. Felhneret.al. C. Lent et.al. G. Bernstein et.al G. Snider et.al. W.PorodUnivof Notre Dame.M. Macucciet. al.Univ. of PisaA. Dzuraket.al.Univ. of New South WalesD. Jamieson et.al.Univ. of MelbourneDeviceC. Lent et.al. J. Timleret.alUnivof Notre Dame.D. Tougawet.al.Valparaiso,INM. Macucciet.al. Univ. of PisaK. Kim et.al.Univ. Ill., ChicagoE. Peskinet.alRITA. Orailogluet.alUCSDNASA Jet Propulsion Lab. QCA Figure2.1.VariousGroupsinvolvedindifferentareasofQCAresearch 13

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ThereareanumberofresearchgroupsinleadingresearchlabsaroundtheworldworkingonQCA.TheresearchgroupatUniversityofNotredamehasbeenspearheadingQCA researchformorethanadecade.AnothergroupcreditedwiththeadvancementofQCA researchisfromUniversityofPisa,Italy.ThisgroupleadbyDr.M.Macucciconducted ainvestigativeresearchinQCAinvolvingseveralinstitutionsallovertheworldunderthe QUADRANTproject.Fig2.1.showssomeoftheleadingresearchgroupscurrentlyinvolvedindifferentareasofQCAresearch.Aswecanseefromthegure,mostofthe researchgroupsareeitherinvolvedinQCAtestingandotherarchitecturalissuesorinthe fabricationofQCA.Atthelogiclevel,QCAresearchreceivedagreatboostfromthework doneattheUniversityofCalgary,underDr.KonradWalus.Thisgroupintroducedthe rsteversimulatorknownasQCADesigner[42].EventodayQCADesignerisamongst theleadingQCAdesignandsimulationtoolusedallovertheworld. 2.1QCABasics InaQCACell,twoelectronsoccupydiagonallyoppositedotsinthecellduetomutual repulsionoflikecharges.AsimpleunpolarizedQCAcellconsistsoffourquantumdots arrangedinasquare,showninFig2.2..Dotsaresimplyplaceswhereachargecanbe localized.Therearetwoextraelectronsinthecellthatarefreetomovebetweenthefour dots.Tunnelinginoroutofacellissuppressed.Thenumberingofthedotsinthecellgoes clockwisestartingfromthedotonthetopright.ApolarizationPinacell,whichmeasures theextenttowhichtheelectronicchargeisdistributedamongthefourdots,istherefore denedas: P 13 24 1234(2.1) 14

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Unpolarized QCA Cell Figure2.2.Ansimple4-dotunpolarizedQCAcell P = +1 P = -1 Figure2.3.Twopolarizedstatesofa4-dotQCAcell Where iistheelectronicchargeineachdotofafourdotQCAcell.Oncepolarized, aQCAcellcanbeinanyoneofthetwopossiblestatesdependingonthepolarizationof chargesinthecell.Becauseofcoulumbicrepulision,thetwomostlikelypolarizationstates ofQCAcanbedenotedasP=+1andP=-1asshowninFig2.3.Thetwostatesdepicted herearecalled"mostlikely"andnottheonlytwopolarizationstatesisbecauseofthesmall (almostnegligible)likelihoodofexistanceofanerroneousstate. InQCAarchitectureinformationistransferredbetweenneighboringcellsbymutual interactionfromcelltocell.Hence,ifwechangethepolarizationofthedrivercell(left mostcellalsoknowasinputcell),rstitsnearestneighborchangesitspolarization,thenthe nextneighborandsoon.Fig2.4.depictsthetransferofpolarizationbetweenneighboring QCAcells.Whenthedrivercell(input)isP=-1(orP=+1),alineartransferofinformation amongstitsneighboringcellsleadstoallofthembeingpolarizedtoP=-1(o rP=+1). 15

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P = +1 P = -1 P = -1 P = -1 P = -1 Figure2.4.TransferofpolarizationbetweenadjacentQCAcellswhenthepolarizationof thedrivercellischangedfro mP=+1toP=-1 1 0.5 0 0.5 1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Driver PolarizationDriven Polarization Temp = 0.0K Temp = 1.0K Temp = 3.0K Temp = 5.0K Figure2.5.TemperaturedependenceonthepolarizationofaQCAcellwithrespecttothe changeinpoalizationofthedrivercell. 16

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Aswecansee,achangeinpolarizationofthedrivercellpromptsalltheneighboring cellstochangepolarizationinordertoattainthemoststableconguration.Theabove exampleshowshowinformationcanbetransferredinalinearfashionovera"line"ofQCA cells.SuchalineofcellsisusedasinterconnectsbetweenvariousQCAlogiccomponents thatwewillseeinthefollowingsection.ThespeedofchangeinpolarizationofaQCA celldependsonanumberoffactorssuchastemperature,kinkenergy,clockenergyandthe quantumrelaxationtime.Fig2.5.showsthethermaldependanceofthepolarizationofa QCAcellwithrespecttothepolarizationofthedrivercell. 2.2PhysicsofQCADeviceOperation Inordertounderstandtheoperationofasimple4-dotQCAcellwerststudythe motionofanelectroninaninnitepotentialwell.Thewallsofthispotentialwellprevent electrontotunnelbetweenadjacentdots.Electronsinaninnitepotentialwellexistas awavefunction xyzthatgivesustheprobabilityofndinganelectronwithinthat potentialwell.Thisprobabilityisproportionaltoxyz 2.SolutiontotheSchrodinger's waveequationforafreeelectron(V=0)isgivenby: d2 dx2 m hEV0(2.2) WhereVisthepotentialactingonthepaerticle,Eistheenergyoftheparticleandmis themass.TakingV=0forfreeelectronweget: d2 dx2 m hE0(2.3) Using k22 m h2thisreducesto 17

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Figure2.6.Thevariousquantizedenergystatesofanelectroninaonedimensionalinnite potentialwell.Foreachquantizedstatethepossiblewavefunctionsandprobabilitydistributionsfortheelectronareshown.ImageredrawnfromPrinciplesofElectronicMaterials andDevices[43]. d2 dxk2)Tj/T1_0 1 Tf11.9552 0 0 11.9552 356.3998 330.3 Tm(0 (2.4) SolutionofSchrodingersequationforthiswavefunctionisasin/cosfunctionandit alsogivesthevalueoftheenergyofanelectronwithinapotentialwell.Theelectroncan onlyhavecertaindiscreteenergies(En)matchingtheallowedwavefunctions.Alower (higher)energyelectronwillhaveasmaller(larger)valueof k (wavevector)andalarger (smaller)wavelength. Sincetheboundaryconditionsdemandthewavefunctiontobezeroatthewallsofthe well,thewavevectorcanonlytakediscretequantitiesandhencetheelectroncanonlyexist inquantizedenergylevels.Thespacingbetweenadjacentenergylevelsdependsonthe 18

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widthofthepotentialwell.Ifweconsidertheheightofthepotentialwellasnite,thereis apossibilityofelectronstunnelingoutofthepotentialwell.Fig2.7.showsanexampleof anelectrontunnelingacrossanitepotentialwell.Thepotentialenergy(PE)ofpointAis lessthanthatofpointD.HenceacarreleasedfrompointAcanatmostmakeittoCbut notE.WhenthecarisatthebottomofthehillitsenergyistotallyKineticEnergy(KE). Theenergybarrier(betweenCandD)preventsthecarfrommakingittoE.Inquantum theory,ontheotherhand,thereisachancethatthecarcouldtunnelthrough(leak)the energybarrierbetweenCandEandemergeontheothersideofthehillatE.Fig2.7.(b) showsthethewavefunctionoftheelectronwhenitisincidentonaPEbarrier( Vo).The interferenceoftheincidentandreectedwavesgive yIx.Thereisnoreectedwavein regionIII.InregionIIthewavefunctiondecayswith x because EVo. SolvingtheSchrodingerequationforthenitebarrierregion(II)yieldsanexponential decayfunction.Thisisthemaindifferencetotheouterregionsoftheinnitewell,where thewavefunctionmustbezero.SolutionsforIandIIIarethesameasfortheinnite potentialwell.However,boundaryconditionsnowdemandthatthewavefunctionmatch theexponentialfunctioninregionII,causingnon-zeroamplitudeinregionIII.Sincethe probabilityofndinganelectronisproportionaltothesquareoftheamplitude,therefore, thereisanon-zeroprobabilitytondtheelectronontheoutside,i.e.itcanescapefrom regionI. TakingthisintoaccountwenowlookatasimpleQCAcellwithtwoelectronsplacedin neighboringpotentialwells(calleddots).Incaseofaninnitepotentialbarrierbetweenthe dots,electronsarenotallowedtotunnelwithinthedots.Asthepotentialbarrierdecreases, thepossibilityofanelectrontotunnelacrossthepotentialbarrierincreases.Whenthe potentialbarriersareverylow,electronscantunnelfreelyacrossthetwoquantumdots. InQCAtechnology,clockenergyisprovidedasameanstolowerorraisethetunneling barriersaswewillseeinSection2.5.Fig2.8.showshowthetunnelingbarriersbetween 19

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A B C D E VoE < VoI(x)II(x)III(x) V(x) A1A2Incident Reflected Transmitted x = 0 x = a x I II III(a) (b) Figure2.7.Ananexampleofanelectrontunnelingacrossanitepotentialwell.Image redrawnfromPrinciplesofElectronicMaterialsandDevices[43]. twodotsarelowered(raised)whentheclockenergysuppliedtotheQCAcellisraised (lowered). Theworkdoneinraisingandloweringoftunnelingbarrierscontrolledbytheclock energycanbetermedasleakagepowerdissipationasthiswilltakeplaceeveniftheQCA celldoesnotswitchstate.Inasimilarwayaclockcontrolsthetunnelingbarriersina4-dot QCAcellusedinthiswork. Sinceinpracticeitisnotpossibletoimplementaninnitepotentialwelltoprevent theelectronsfromtunnelingacross,thereisalwaysanitepossibilityofsomeelectronic chargeescapingtheQCAcelloveralongperiodoftime.However,inthisworkwehave neglectedanylossofcharge.Electronsinhigherenergystateswithinapotentialwellare morepronetotunnelacrossifthetunnelingpotentialisofniteheight.Thermalerrorsare causedwhentheelectronstosettleinhigherenergyorbitsandaremorelikelytotunnel acrossthebarriersascomparedtowhentheyareingroundstate.Wewillseeinlater chaptershowtheoutputnodepolarizationprobabilityfallswiththeriseintemperature. 20

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Tunneling Barrier Clock EnergyInfinite Potential Barrier No Tunneling High Potential Barrier Insignificant Tunneling Low Potential Barrier Electrons can Tunnel Figure2.8.Controllingthetunnelingbarrierbyvariationofclockenergy.Astheclock energysuppliedtoaQCAcellincreases,thetunnelingbarrierslower,makingitpossible forelectrontotunnelacrosstotheotherside. 2.3ImplementationofaQCACell ThebasicelementforQCAcomputationisabistablecellcapableofinteractingwithits localneighbors.Thecellisnotrequiredtoremainquantum-mechanicallycoherentatall times;therefore,manynon-quantum-mechanicalimplementationsofQCAhaveemerged. Generallyspeaking,therearefourdifferentclassesofQCAimplementations:Metal-Island, Semiconductor,MolecularandMagnetic.Inthissection,abriefdescriptionofeachimplementationisprovidedwithitsadvantagesanddisadvantages. 2.3.1MetalIsland TheMetal-Islandimplementation[44,39,45]wastherstfabricationtechnologycreatedtodemonstratetheconceptofQCA.Itwasnotoriginallyintendedtocompetewith currenttechnologyinthesenseofspeedandpracticality,asitsstructuralpropertiesare notsuitableforscalabledesigns.Themethodconsistsofbuildingquantumdotsusing aluminumislands.Earlierexperimentswereimplementedwithmetalislandsasbigas1 21

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(a)(b) Figure2.9.(a)SEMimageofaMetallicDotQCAand(b)Schematicdiagram.Imagefrom: Orlovet.al.[44] micrometerindimension.Becauseoftherelativelylarge-sizedislands,Metal-Islanddeviceshadtobekeptatextremelylowtemperaturesforquantumeffects(electronswitching) tobeobservable.Again,thismethodonlyservedasmeanstoprovethattheconceptisattainableinpractice[46,47].ASPICEmodeldevelopmentmethodologyforQCAcellswas proposedin[48]. 2.3.2Semiconductor Semiconductor(orsolidstate)QCAimplementations[50]couldpotentiallybeusedto implementQCAdeviceswiththesamehighlyadvancedsemiconductorfabricationprocessesusedtoimplementCMOSdevices[51].SemiconductorquantumdotsarenanostructurescreatedfromstandardsemiconductivematerialssuchasInAs/GaAs[52]and GaAs/AlGaAs[53,54].Thesestructurescanbemodeledas3-dimensionalquantumwells. Asaresult,theyexhibitenergyquantizationeffectsevenatdistancesseveralhundredtimes 22

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Figure2.10.ExamplequantumdotpyramidcreatedwithInAs/GaAs.Imagefrom:UniversityofNewcastle:CondensedMatterGroup[49] largerthanthematerialsystemlatticeconstant.Cellpolarizationisencodedaschargeposition,andquantum-dotinteractionsrelyonelectrostaticcoupling[55].Today,mostQCA prototypingexperimentsaredoneusingthisimplementationtechnology[56]. Advantages:Easiertointegrateinthefabricationprocessbecauseofthesuccessofsemiconductorsinmicroelectronicsforwhichmanytoolsandtechniqueshavebeendeveloped.EasiertouseexistingfacilitiesandmethodstocreateaviableQCAsolution. Disadvantages:Currentsemiconductorprocesseshavenotyetreachedapointwheremassproduction ofdeviceswithsuchsmallfeatures(20nanometers)ispossible.Seriallithographicmethods,however,makeQCAsolidstateimplementationachievable,butbynomeanspractical.Seriallithographyisslow,expensiveandunsuitable formass-productionofsolid-stateQCAdevices. 23

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(a) (b) Figure2.11.(a)ElectronmicrographofaGaAs/AlGaAsQCAcell.(b)Simpliedcircuit equivalentofthefour-dotcell.Imagefrom:Perez-Martinezet.al.[53] Figure2.12.TwoviewsofMolecule1asaQCAcell.Imagefrom:Lentet.al.[57] 24

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2.3.3MolecularQCA MolecularQCA[58,41,57,59]conceptconsistsofbuildingQCAdevicesoutofsingle molecules.MajorityoftheworksofarhasbeenpresentedbytheresearchgroupatNotre Dame. ThebasiccellforMolecularQCAin[57]consistsofapairofidenticalmolecules (Fig2.12.).ThemoleculeofourdiscussionisMolecule1whichisformallyknownas1,4diallylbutaneradicalcation.Thismoleculeiscomprisedoftwoallylsconnectedtobutyl bridgeononeendandtwomoreallylsconnectedtothesamebridgeontheotherend.This particularmoleculeisneutralononeendwhiletheotherendbehavesasacation. Thesemoleculeshaveanextraelectronorholethatcantunnelfromonesideofthe moleculetotheother.Byplacinganelectriceldnearonesideofthemoleculeoncan forcetheholetoeitherbeattractedorrepelled,thiscanbecalledthedrivercell.Ithasbeen calculatedthatMolecule1hasnonlinearswitchingcharacteristicsmakingitanidealswitch. Ifthemoleculescanbeplacedveryclosetoeachother,about7angstroms,theelectrostatic interactionswillcausetheholestobeatoppositeends.Thisallowspropagationofthea statetoothercells.TheFig2.13.showsthedifferentpossiblestatesthemoleculecanbe in(a)showsa+1(c)showsa-1and(b)anon-idealstatethatisaunwantedstate. Whilefabricationmethodsarecurrentlybeingresearched,noonemethodhasbeenpredominate.EffortsareontofabricatemolecularQCAcircuitsusingselfassemblymonolayermethods[60,61,62,63].Themoleculesthemselvesareproducedbystandardchemicalprocedure[64,65,66,67,68]. Advantages:RoomtemperatureoperationUltrasmalldevices:densityofdevicecanbeveryhigh.FastSwitching:deviceshouldbeabletooperateintheGigahertzrange 25

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(a) (b) (c) Figure2.13.DifferentpossiblestatesofMolecule1(a)showsa+1state(b)anon-idealstate thatisaunwantedstateand(c)showsa-1state.Imagefrom:Lentet.al.[57]LowPowerconsumption:Theonlypowerneededistodrivetheinputandforsome typeofclockingmechanism.LowPowerloss:Calculationsstatethatthereshouldbelowpowerlossduetoheat producedfromswitching. Disadvantages:SingleMoleculesareinexistenceandhavebeenproducedbuttheplacementofthese moleculesinaregularpatternorfashionhasnotbeendone.Becauseofthesizeoftheofthedevices,todrivetheinputtoacertainstateand sensingtheoutputscanbeverydifcult.Onedoesnotwanttoinuenceothercells besidestheintendeddrivercellandtheoutputcells.ClockingMethod:Thoughtheoriticalclockingmethodshavebeenproposed[69],no practicalclockingmethodhasbeendemonstrated. 26

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(a)(b) Figure2.14.(a)SEMimageofaroomtemperatureMQCAnetworkshownin(Cowburn et.al.[74])(b)Majoritygatesdesignedfortestingallinputcombinationsofthemajoritylogicoperation.ThearrowsdrawnsuperimposedontheSEMimagesillustratetheresulting magnetizationdirectionduetoahorizontallyappliedexternalclock-eld(Imreet.al.[79]) 2.3.4MagneticQCA AbasiccellinMagneticQCAisanano-magnet[70,71].Thesenano-magnetsarearrangedinvariousgrid-likefashionstoaccomplishcomputing[72].CellsinMagneticQCA areenumeratedbasedontheirsingledomainmagneticdipolemomentsandareinherently energyminimums[73].ThereareseveralpopularschemesofMagneticQCAthathave beenproposed:CowburnandWellandsnanodotQCAAutomata[74],ParishandForshaws Bi-stableMagneticQCA[75,76],andCsabaetal.,FieldCoupleNanomagnets[77,78]. CowburnandWellandshavefabricatedtheMagneticQCAmodelthathasbeendescribed here. Anano-magnetconsistsofasinglecircularnanodot.Thesenanodotsweremadeof amagneticSuperMalloy(mainlyNi).Thenanodotsare110nmindiameterandhada thicknessof10nm.Inordertohaveasingledomaininthenanodotsitwasfoundthatthe nanodotsmusthaveasizeofabout100nmandbelow.Singledomainswereimportantwhen 27

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analyzingthehysteresisloopsandretaininginformationwithnoexternalenergypresent. Thenanodotswerefabricatedusinghighresolutionelectronbeamlithography. Thebasiccellconsistsofasinglenanodot.Thesenanodotswereplaceabout20nm apartinastraightlineasshowninFig2.14.(a)Anelongateddotwasplacedatthebeginning ofthechain,thiswasusedastheinputdot.Therewasanoscillatingeldappliedtodots whichwas+25and-25Oe.Therewasalsoa-10Oebiasalongthechainofdots.Forthis experimentifthedotswerepointingtotherightitwasalogical1,totheleftwasconsidered a0.Whentheinputswheresetto1aresponsewasfoundwhereaswhenthedotswhere setto0noresponsewaspresent. Therehavebeensomeprogressrecentlyinfabricatingamajoritylogicgateusingnanomagnets[79].Fig2.14.(b)showstheimplementationofamajoritygateusingnanodotsby Imreetal.TheirapproachtoMQCAissimilartoCowburnet.al.,buttheyuseanadditional shape-inducedanisotropycomponenttoseparatethedirectionsformagneticinformation representationandinformationpropagationinthearray. Advantages:Roomtemperatureoperation:BecauseMagneticQCAusemagnetostaticforcesinsteadofcolumbic,thisschemecanoperateatroomtemperatureLessstringentfabrequirementsthanotherQCA:Sinceitisnotnecessarytohave featuresizeslike5-10nmandspacingof2nmforoperationHighDensitywhencomparedtoCMOS:Futureoutlookhavebeenpredictedtohave devicedensityashighas250,000millionpersquaredcmHighlyresearchedarea:Otherdevicessuchasharddrivesandmemoryusemagnetismastheirprimarymechanism. 28

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Lowpowerloss:Oncetheinputissetthenanodotscansettleintoalocalenergy minimumneedingnoexternalenergytokeeptheirstate.HigherthermalrobustnessthanotherQCA:CurrentlyotherQCAimplementations requirecryogenictemperaturestooperate.Thisschemestates40 kBT willkeepthermalerrorsbelow1peryear.3DArchitecturesPossible:Sincethemagneticforcesfromthenanodotsareinplane itispossibletohavestackedmultipleplanes Disadvantages:Individualsensingofdots:Thesensingofstatesisdonebymeasuringthetotalresultantmagneticeldoftheentirewireofdotsandnotasignalpointonthewire.Frequency:Estimatedspeedofoperationofmageticdevicesislowerthancurrent technologies[75].Eventhoughthisisnotveryfastitwouldhaveanicenichefor devices.Unknownunderthe20nmsizes:Currentpredictionsstatethatnanodotsmightbecomeunstableunder20nm. 2.4LogicalDevicesinQCA AsweknowthatpresentdaylogicarchitecturesarebasedonBooleanalgebra.Inorder toperformlogicaloperationusingQCA,wedenotethetwopolarizationstatesinterms ofBooleanlogic.Wetak eP=-1 congurationas"HIGH"andP=+1congurationas "LOW".InBooleanlogictheANDgate,theORgateandInverterformthemostbasic logiccomponents.InQCAarchitecture,themajoritygateandinverterformthemostbasic logiccomponents[80].Therehasalsobeenalotofresearchrecentlytopresentnovellogic designsusingQCA[81,82,83]. 29

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Figure2.15.AQCAmajoritygate P= +1 P= +1 P= +1 P= +1 P= +1 P= -1 P= -1 P= -1 P= -1 P= +1 P= -1 P= -1 Figure2.16.QCAmajoritygatelogicwithdifferentinputs Athreeinputmajoritygateconsistsofthreeinputsandoneoutput.Theoutputissaid tobeHIGH(LOW)ifatleasttwooutofthethreeinputsareHIGH(LOW)andviceversa. HereHIGH(LOW)referstothepolarizationstateP=+1(P=-1). AMajoritygateofthreeinputsA,BandCisdenotedas: Maj(A,B,C)=A.B+B.C+C.A Fig2.15.showsasimplethreeinputmajoritygateinQCA.DifferentinputcongurationsofaQCAmajoritygateisshowninFig2.16.. AQCAinverterdesignisshowninFig2.17..Thepurposeofaninverteristotakeone inputandproduceitsinverse.SoiftheinputisHIGH(LOW)theoutputisLOW(HIGH). Usingmajoritygatesandinverterswecanbuildlogiccircuitshavingsimilarfunctionalityasthatoffunctionsimplementedusingtheconventionallogicgates(suchasAND, ORandNOT).AnANDgateandanORgatecanbeeasilybuiltusingaMajoritygateby 30

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Figure2.17.AQCAinverter QCA OR Gate QCA AND Gate Figure2.18.ANDandORgaterepresentationofamajoritygatebyxingoneoftheinputs asP=-1orP=+1respectively settingoneofitsinputstoeitherP=-1orP=+1(LOWorHIGH)respectively.Oncewe xthepolarizationofoneoftheinputs,themajoritygateactsasasimpletwoinputAND gateoranORgate(Fig2.18.). NowthatwehaveshownhowwecanimplementanANDgate,ORgateandanInverter, wecanbuildanyBooleanlogiccircuitinQCA.WeshowasmallexampleofaNANDgate inQCAusingaANDgateandInverterinFig.2.19..Infact,itispossibletoimplementmost oftheBooleanlogiccircuitsinQCAusingonlythreeinputmajoritygatesandinverters[84, 85]. Thereareseveralsynthesisalgorithms[86,87]availabletoconvertaBooleanlogic functiontoaMajoritygatelogicfunction.Asingle-bitaddercircuitimplementedusing onlyMajoritygatesandinvertersisshowninFig2.20.. 31

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Figure2.19.QCANANDgateusinganANDgateandaninverter 2.5ClockinginQCA AQCAcellcanonlybeinoneoftwostatesandtheconditionalchangeofstateinacell isdictatedbythestateofitsadjacentneighbors.However,amethodtocontroldataowis necessarytodenethedirectioninwhichstatetransitionoccursinQCAcells.Theclocks ofaQCAsystemservetwopurposes:providingpowertothecircuit,andcontrollingdata owdirection.Likestatedbefore,QCArequiresverysmallamountsofpower.Thisis duetothefactthatcellsdonotrequireexternalpowerapartfromtheclocks.Theseclocks areareasofconductivematerialundertheQCAlattice,modulatingtheelectrontunneling barriersintheQCAcellsaboveit. AQCAclock[88]inducesfourstagesinthetunnelingbarriersofthecellsaboveit.In therststage,thetunnelingbarriersstarttorise(clocksignalgoeslow).Thesecondstage isreachedwhenthetunnelingbarriersarehighenoughtopreventelectronsfromtunneling. Thethirdstageoccurswhenthehighbarrierstartstolower(clockbeginstoriseagain).And nally,inthefourthstage,thetunnelingbarriersallowelectronstofreelytunnelagain.In simplewords,whentheclocksignalishigh,electronsarefreetotunnel.Whentheclock signalislow,thecellbecomeslatched.Fig2.21.showsaclocksignalwithitsfourstages. AtypicalQCAcircuitrequiresfourclocks,eachofwhichiscyclically90degreesoutof phasewiththepriorclockasshowninFig2.22..Ifaverticalwireconsistedofsay,8cells 32

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Sum Carry Out Figure2.20.AsinglebitQCAadderdesign 33

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SwitchHold Release RelaxTime Electric Field Barrier Figure2.21.FourstagesinaQCAclock(1)Tunnelingbarriersstarttorise(2)Hightunnelingbarrierspreventelectronsfromtunneling(3)Tunnelingbarriersbegintolower(4) Electronsarefreetotunnel. Clock 1 Clock 2 Clock 3 Clock 4 Figure2.22.FourQCAclocksphaseshiftedby90degrees. 34

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Clock zone 0 Clock zone 1 Clock zone 2 Clock zone 3Signal Propagation Time Figure2.23.FlowofinformationinQCAlinecontrolledbyclockpropagation andeachconsecutivepair,startingfromthetopweretobeconnectedtoeachconsecutive clock,datawouldnaturallyowfromtoptobottom.Therstpairofcellswillstaylatched untilthesecondpairofcellsgetslatchedandsoforth.Inthisway,dataowdirectionis controllablethroughclockzones[89].ThisprocessisdepictedinFig2.23.. In[90],Lentet.al.examinetheefcacyof Landauer-Bennett clockingapproachin molecularQCAcircuits.Landauerclockinginvolvestheadiabatictransitionofamolecularcellfromthenullstatetoanactivestatecarryingdatathatcanresultinpowerdissipation lesssthan kBTln2.Landauershowedthatforlogicallyreversiblecomputationthereisno necessaryminimumenergydissipationassociatedwithreadingabit,butratherwitherasing information.BennettextendedtheLandauerresultbyshowingthatinprincipleanycomputationcouldbeembeddedinalogicallyreversibleoperation.ThismethodsuggestsQCAas 35

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apracticalmeanstoimplementreversiblecomputing.Bennettclockingschemecanreduce thepowerdissipatedtomuchlessthan kBTln2withoutchangingcircuitcomplexity. Fig2.24.showsLandauerandBennettclockingofQCAcircuits.Eachgurerepresentsasnapshotintimeastheclockingeldsmoveinformationacrossthecircuit.The leftcolumn(L1)-(L5)representsLandauerclocking.Awaveofactivitysweepsacrossthe circuitastheclockingeldcausesdifferentcellstoswitchfromnulltoactive.Thecircuit shownincludesashiftregisterontopandathree-inputmajoritygateonthebottom.The rightcolumn(B1)-(B7)representsBennettclockingforacomputationalblock.Hereasthe computationaledgemovesacrossthecircuitintermediateresultsareheldinplace.When thecomputationiscomplete(B4),theactivitysweepsbackwards,undoingtheeffectofthe computation.Thisapproachresultsinminimumenergydissipation. TherehavebeenexperimentalstudiestodemonstratetheclockedQCAshiftregisters [91,92].SinglewalledCNTshavebeenproposedasapossiblemechanismtoprovide clockinginQCAcircuits[93].NotonlydoesaQCAclockclockprovideameansto controldataowdirectionandpowertothecircuit,ithasbeenalsoseenthatclockenergy alsoplaysasignicantroleintheoverallpowerdissipatedbyacircuit.Thiseffectisless prominentifthetheclockingschemeisadiabatic[94]however,ifanon-adiabaticclocking schemeisappliedtothecircuit[95,96],itcontributessignicantlytotheoverallpower dissipatedinthecircuit,aswewillseeinthelaterpartofthisdissertation. 2.6QCAArchitecture Foranytechnology,architecturaldesignisoneofthemostimportantparametersof itssuccess.Anytechnologycanprovetobereliableandefcientintermsofscalingand powerrequirements,butitisoflittleuseunlessonecanimplementarchitecturaldesignsin it.QCAisconsideredtobeoneofthemostcompleteemergingtechnologieswhenitcomes 36

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Figure2.24.LandauerandBennettclockingofQCAcircuits.(Lentet.al.[90]) 37

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toarchitecture.Severalnovelarchitectureshavebeenproposedbothinlogicandmemory design.ThemajoritylogicdesigninvolvedinQCAarchitecturepresentedaninteresting challengeforresearcherstodesigncomplexarchitectures[97]thatuseddifferentlogic comparedwithBooleanlogic(NAND/NOR)usedinCMOS. IntitiallogicgatesandadderdesignswereproposedbyLentet.al.in[80,98,99]. WhilemostoftherecentworkinQCAarchitecturehasbeenperformedunderDr.Kogge's groupatUniversityofNotredame,thereareseveralothergroupsthathavecontributed signicantlyinthedevelopmentofQCAbaseddesignsovertheyears.Anumberof combinational[100,101]andsequentialdesigns[102]havebeenproposedandsimulated.MostprominentlogicdesignthathasbeenimplementedinQCAisasinglebit adder[103,104,105].Severalothercomplexarchitecturaldesignshavebeenproposedin QCAsuchasasinglebitALU[106]andasimple12microprocessordesign[107,108]. OtherlogicaldesignsproposedinQCAincludemultiplexers,decodersandshiftregisters. ManysimpleandcomplexQCAarchitecturaldesignsarepresentedinlaterchapters. QCAreliesonnoveldesignconceptssuchas"memoryinmotion"and"processingin wire"toimplementuniqueparadigms[109,110,111].Lackofinterconnectsandpotential implementationoflogicinwiremakesQCAaveryattractivetechnologyformemorydesign[112].AsimplememoryelementisshowninFig2.25.AsQCAisimplementedona gridbasedarchitecture,somepromisingFPGAandapplicationspecicarchitectureshave alsobeenproposed[113,114,115]. Anumberofgroupsalsoworkinmajoritysynthesisalgorithms[87,116]thatprovidea majoritylogicsolutionofcommonBooleanexpressions.Oncethelogicalexpressionsare synthesizedtheycanbeusedfordesigncircuitsinQCA. 38

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1.00 -1.00 -1.00 -1.00 -1.00 -1.00 o u RowSelect Input R/W out Figure2.25.AQCAMemoryCell(Waluset.al.[42]) 2.7DefectTolerance Recentlytherehasbeenalotofworkdoneinnano-levelissuesinQCAdesignrelating todesignissuessuchascellsize,polarizationanddefecttolerance[117,118,119].Defect tolerancestudiesarecrucialforanynanoscaletechnologybecausedefectsinlayoutare unavoidable.TheQCAapproachisinherentlyrobustandcanbemadeevenmoresoby simplyusingwide(3-or5-cell)wirestobuildinredundancyateverystage.OtherdefecttolerantstrategiesinQCAareunderinvestigation. In[30]presentsastudyofdefectcharacterizationinQCAdesigns.Effectsofdefects areinvestigatedatthelogiclevel.TestingofQCAisalsocomparedwithtestingofconventionalCMOSimplementationsoftheselogicdevices.Fig2.26.showsthedifferent congurationsofadefectivemajoritygate.Anotherworkin[120]explorestheuseof enlarged linesandmajoritygatestostudydefectsincoplanarcrossingsinQCAdesign. 39

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Figure2.26.DifferentcongurationsofdisplacedQCAcellsinamajoritygate.Conguration(a)isfaultfree(Tahooriet.al.[30]) 40

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2.8ModelingQCADesigns Thereareseveralapproximatesimulatorsavailableatthelayoutlevel,suchasthe bistablesimulationengineandthenonlinearapproximationmethods.Thesemethodsare iterativeanddonotproducesteadystatepolarizationestimates.Inotherwords,theyestimatejuststateassignmentsandnottheprobabilitiesofbeinginthesestates.Thecoherence vectorbasedmethoddoesexplicitlyestimatethepolarizations,butitisappropriatewhen oneneedsfulltemporaldynamicssimulation(Blochequation),andhenceisextremely slow.Perhaps,theonlyapproachthatcanestimatepolarizationforQCAcells,without fullquantum-mechanicalsimulationisthethermodynamicmodelproposedin[121],butit isbasedonsemi-classicalIsingapproximation.Inthenextchapterwedemonstratehow wecanuseaBayesianprobabilisticcomputingmodeltoexploittheinducedcausalityof clockinginaQCAdesigntoarriveatamodelwiththeminimumpossiblecomplexity. OneofthemajoradvantagesofQCAlogicdesignisthatitiscapableofextremely lowpowercomputation.Lentetal.proposedamodeltoestimatepowerdissipationduringquasi-adiabaticswitchingeventinaQCAshiftregister.Perhapsthemostpurepower modelisthequantum-mechanicalmodelofthetemporaldynamicsofpowerderivedby TimlerandLent.Theyidentiedthreecomponentsofpower:clockpower,celltocell powergain,andpowerdissipation.Whilethismodelgivesusphysicallycloseestimates, itiscomputationallyexpensivetoestimate.WhendesigningQCAcircuits,wewouldlike estimatepowerquicklyinordertochooseamongmanydifferentalternativesandparameters.Theneedforfullblownquantum-mechanicalestimationwillberelegatedtothevery endofthedesignprocess.Tothisend,somestudiespresentlowerboundsofpowerdissipatedthatareeasytocompute.However,fromadesignautomationpointofviewitis importanttodesignfortheworstcase,leavinguswithmarginforerrorsduetoprocess 41

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variability. Hereworstcasereferstothepowerdissipatedduringanon-adiabaticclocking scheme .Forworstcaseconsiderations,theupperboundforpowerismorerelevant.Inthis dissertation,wepresenttheresultsofthisnon-adiabaticpowermodelinchapter4. Asresearchprogresses,itisnaturaltolookbeyonddevicelevelissuesinQCAdesigns andexplorecircuitlevelissuessoastoscopeoutthetypesofcircuitsthatcanbebuilt. However,QCAmodelingtoolsavailableforsuchdesignshavebeenatthelayoutlevel.The operationsofnanoscaledevicesaredominatedbyquantummechanics,makingitdifcult tomodelvariousissues,suchaserrororpowerdissipationwithdeterministicstatemodels. Thishasimplicationsinthestructureofthedesignmethodologytobeapplied.Hierarchical designatmultiplelevelsofabstraction,suchasarchitectural,circuit,layout,anddevice levels,isprobablystillpossible.However,thenatureofcouplingoftheissuesbetween levelswouldbedifferentandstronger.Forthis,weneedcomputingmodelsathigherlevels ofabstractionthatarestronglydeterminedbylayout-levelquantum-mechanicalmodels. Wepresentahierarchicaldesignschemethatusesprobabilisticmacromodelstoimplement circuitlevelQCAarchitectureinchapter5.Theprobabilisticmacromodelsusedascircuit blocksinthisdesignschemearederivedfromthelayoutlevelgraphicalmodelsthatare presentedinthenextchapter. 42

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CHAPTER3 PROBABILISTICBAYESIANNETWORKMODELING 3.1Introduction Inthischapter[24,25]wedevelopafast,BayesianProbabilisticComputingmodel thatexploitstheinducedcausalityofclockingtoarriveatamodelwiththeminimumpossiblecomplexity.Theprobabilitiesdirectlymodelthequantum-mechanicalsteady-state probabilities(densitymatrix)orequivalently,thecellpolarizations.Theattractivefeature ofthismodelisthatnotonlydoesitmodelthestrongdependenciesamongthecells,but itcanbeusedtocomputethesteadystatecellpolarizations,withoutiterationsortheneed fortemporalsimulationofquantummechanicalequations. Ourproposedmodelingisbasedondensitymatrix-basedquantummodeling,which takesintoaccountdependencypatternsinducedbyclocking,andisnon-iterative.Itallows forquickestimationandcomparisonofquantum-mechanicalquantitiesforaQCAcircuit, suchasQCA-stateoccupancyprobabilitiesorpolarizationsatanycell,theirdependence ontemperature,oranyparameterthatdependsonthem.Thiswillenableonetoquickly compare,contrastandnetuneclockedQCAcircuitsdesigns,beforeperformingcostly fullquantum-mechanicalsimulationofthetemporaldynamics. Wevalidateourmodelingwithcoherencevectorbasedtemporalsimulationforvarious QCAsystems(Fig.3.5.).Wealsoshow,usingthe clocked majoritygate,howthemodel canusedtostudydependencieswithrespecttotemperatureandinputs(Fig.3.5.). 43

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3.2QuantumMechanicalProbabilities FollowingTougawandLent[36]andothersubsequentworksonQCA,weusethetwostateapproximatemodelofasingleQCAcell.Wedenotethetwopossible,orthogonal, eigenstatesofacellby1and0.Thestateattime t ,whichisreferredtoasthewavefunctionanddenotedbyt ,isalinearcombinationofthesetwostates,i.e.t c1t 1 c2t 0.Notethatthecoefcientsarefunctionoftime.Theexpectedvalueof anyobservable, At ,canbeexpressedintermsofthewavefunctionas A t At t orequivalentlyasTr At t t ,whereTrdenotesthetrace operation,Tr 11 00.Thetermt t isknownasthedensity operator, t.Expectedvalueofanyobservableofaquantumsystemcanbecomputedif tisknown. A2by2matrixrepresentationofthedensityoperator,inwhichentriesdenotedby ijtcanbearrivedatbyconsideringtheprojectionsonthetwoeigenstatesofthecell,i.e. ijt i t j.Thiscanbesimpliedfurther. ijt i t j it t j it jt citcjt(3.1) ThedensityoperatorisafunctionoftimeandusingLoiuvilleequationswecancapturethe temporalevaluationof tinEq.3.2. i h ttH t tH (3.2) 44

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whereHisa2by2matrixrepresentingtheHamiltonianofthecellandusingHartree approximation.ExpressionofHamiltonianisshowninEq.3.3[36]. H 1 2iEkPifi1 2iEkPifi 1 2Ek P1 2Ek P (3.3) wherethesumsareoverthecellsinthelocalneighborhood. Ekisthe"kinkenergy"or theenergycostoftwoneighboringcellshavingoppositepolarizations. fiisthegeometric factorcapturingelectrostaticfalloffwithdistancebetweencells. Piisthepolarizationof the i -thcell.And, isthetunnelingenergybetweentwocellstates,whichiscontrolledby theclockingmechanism.Thenotationcanbefurthersimpliedbyusing P todenotethe weightedsumoftheneighborhoodpolarizations iPifi.UsingthisHamiltonianthesteady statepolarizationisgivenby Pss ss 3ss 11ss 00Ek P E2 k P24 2tanh E2 k P242 kT(3.4) Eq.3.4canbewrittenas PssE tanh(3.5) where E05 iEkPifi,totalkinkenergyandRabifrequency E2 k P242and kTisthethermalratio.Wewillusetheaboveequationtoarriveattheprobabilitiesof observing(uponmakingameasurement)thesystemineachofthetwostates.Specically, ss 11051Pssand ss 00051Pss,wherewemadeuseofthefactthat ss 00ss 111. 3.3BayesianModeling WeproposeaBayesianNetworkbasedmodelingandinferencefortheQCAcellpolarization.ABayesiannetwork[122,22,23]isaDirectedAcyclicGraph(DAG)inwhich 45

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X2 X4 X3 X5 X1 Figure3.1.AsmallBayesiannetwork thenodesofthenetworkrepresentrandomvariablesandasetofdirectedlinksconnect pairsofnodes.Thelinksrepresentcausaldependenciesamongthevariables.Eachnode hasaconditionalprobabilitytable(CPT)excepttherootnodes.Eachrootnodehasaprior probabilitytable.TheCPTquantiestheeffecttheparentshaveonthenode.Bayesian networkscomputethejointprobabilitydistributionoverallthevariablesinthenetwork, basedontheconditionalprobabilitiesandtheobservedevidenceaboutasetofnodes. Fig.3.1.illustratesasmallBayesiannetworkthatisasubsetofaBayesianNetwork foramajoritylogic.Ingeneral, xidenotessomevalueofthevariable XiandintheQCA context,each Xiistherandomvariablerepresentinganeventthatthecellisatsteady-state logic"1"oratsteadystatelogic"0".Theexactjointprobabilitydistributionoverthe variablesinthisnetworkisgivenbyEq.3.6. Px5x4x3x2x1 Px5x4x3x2x1Px4x3x2x1Px3x2x1Px2x1Px1 (3.6) 46

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InthisBN,therandomvariable, X5isindependentof X1,giventhestateofitsparents X4This conditionalindependence canbeexpressedbyEq.3.7. Px5x4x3x2x1 Px5x4(3.7) Mathematically,thisisdenotedas IX5 X4 X1X2X3 .Ingeneral,inaBayesiannetwork,giventheparentsofanode n n anditsdescendentsareindependentofallothernodes inthenetwork.Let U bethesetofallrandomvariablesinanetwork.Usingtheconditional independenciesinEq.3.7,wecanarriveattheminimalfactoredrepresentationshownin Eq.3.8. Px5x4x3x2x1 Px5x4Px4x3x2x1Px3Px2Px1 (3.8) Ingeneral,if xidenotessomevalueofthevariable Xiand paxidenotessomeset ofvaluesfor Xi'sparents,theminimalfactoredrepresentationofexactjointprobability distributionover m randomvariablescanbeexpressedasinEq.3.9. PXmk1Pxkpaxk(3.9) Notethat,BayesianNetworksareproventobeminimalrepresentationthatcanmodel alltheindependenciesintheprobabilisticmodel.Also,thegraphicalrepresentationin Fig.3.1.andprobabilisticmodelmatchintermsoftheconditionalindependencies.Since BayesianNetworksusesdirectionalpropertyitisdirectlyrelatedtoinferenceundercausality.InaclocklessQCAcircuit,causeandeffectbetweencellsarehardtodetermineasthe cellswillaffectoneanotherirrespectiveoftheowofpolarization.ClockedQCAcircuits howeverhaveinnateorderingsenseinthem.Partoftheorderingisimposedbytheclockingzones.Cellsinthepreviousclockzonearethedriversorthecausesofthechangein 47

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polarizationofthecurrentcell.Withineachclockingzone,orderingisdeterminedbythe directionofpropagationofthewavefunction[36]. Let NeXdenotethesetofallneighboringcellsthatcaneffectacell, X .Itconsists ofallcellswithinapre-speciedradius.Let CXdenotetheclockingzoneofcell X .We assumethatwehavephasedclockingzones,ashasbeenproposedforQCAs.Let TXdenotethetimeittakesforthewavefunctiontopropagatefromthenodesnearesttothe previousclockzoneorfromtheinputs,if X sharestheclockwiththeinputs.Notethat onlytherelativevaluesof TXareimportanttodecideuponthecausalorderingofthe cells.Thus,givenasetofcells,wecanexactlypredict(dependentontheeffectiveradius ofinuenceassumed)theparentsofeverycellandallthenon-parentneighbors.Inthis work,weassumetouse four clockzones.Wedenotethisparentsetby PaX.Thisparent setislogicallyspeciedasfollows. PaX YYNeX CY mod 4CX TY TX (3.10) The causes ,andhencetheparents,of X arethecellsinthepreviousclockingzoneandthe cellsarenearertothepreviousclockingzonethan X .Thechildrenset, ChX,ofanode, X ,willbetheneighbornodesthatarenotparents,i.e. ChXNeX PaX. ThenextimportantpartofaBayesiannetworkspecicationinvolvestheconditional probabilities PxpaX,where paXrepresentsthevaluestakenonbytheparentset, PaX. Wechoosethechildrenstates(orpolarization)soastomaximize E2 k P242, whichwouldminimizethegroundstateenergyoverallpossiblegroundstatesofthecell. Thus,thechosenchildrenstatesare ch XargmaxchXargmaxchXi PaX ChXEk P (3.11) 48

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B Figure3.2.ClockedQCAmajoritygatelayout Thesteadystatedensitymatrixdiagonalentries(Eq.3.5withthesechildrenstateassignmentsareusedtodecideupontheconditionalprobabilitiesintheBayesiannetwork(BN). PX0paXss 00paX ch XPX1paXss 11paX ch X(3.12) Oncewecomputealltheconditionalprobabilities,weprovidepriorprobabilitiesfor theinputs.WecantheninfertheBayesianNetworkstoobtainthesteadystateprobability ofobservingallthecellsincludingtheoutputsat"1"or"0". 3.4ExperimentalResults Inthissection,wediscusstheresultsofourmodelwithasmallexampleofthreeinput majoritygateasthecelllayoutofforQCAcanbeeffectivelydrawnwithsynthesisusing inverterandmajoritygates.Fig.3.4showsthecelllayoutofaclockedmajoritygate.The 49

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Figure3.3.BayesiannetdependencystructurecorrespondingtotheQCAmajoritygate withnodescorrespondingtotheindividualcellsandlinksdenotingdirectdependencies. BayesianNetworkstructureisshowninFig3.3.Notethatweobtainthestructurebased onthecausalowofthewavefunctionandtheinformationregardingtheclockzone.We use"Genie"[123]softwaretoolforBayesianinference.Wepresenttheextendedviewof theBayesianNetworkshowninFig.3.4.withthepolarizationofeachcellshownfora particularinputset. InFig3.5.,wereportthesteadystateprobabilitiesofthecorrectoutputsw.r.ttemperatureandweshowthattheprobabilityofcorrectoutputvarywiththeinputspace.Aswe canseethatthetemperatureplaysakeyroleinobtainingcorrectsignalbehavior.More effectoftemperatureislessforsomeinputssay011than001.Also,theinputset001and010showsdifferentsensitivity.Hencelayoutplaysanimportantrolein theerrorbehaviorofQCA.WevalidatedourmodelwithrespecttotheQCADesigner(Fig 3.6.)andreceivedthesameaccuracyusingthetemporalsimulation.However,thetimefor thesimulationisanorderofmagnitudefaster. 50

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Figure3.4.ExplodedviewoftheBayesiannetstructure,layingbarethedirectedlinkstructureandthenodeinformation. Majority Gate0 0.2 0.4 0.6 0.8 1 1.20 .0 5 0 .3 5 0 .6 5 0 .9 5 1 .2 5 1 .5 5 1 .8 5 2 .1 5 2 .4 5 2 .7 5 3 .0 5 3 .3 5 3 .6 5 3 .9 5 4 .2 5 4 .5 5 4 .8 5TemperatureProbability of Correct Output 0,0,0 1,0,0 0,1,0 1,1,0 0,0,1 1,0,1 0,1,1 1,1,1 Figure3.5.Dependenceofprobabilityofcorrectoutputofthemajoritygatewithtemperature and inputs.Notethedependenceoninputs. 51

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0 0.2 0.4 0.6 0.8 1C orne r 1 C orne r 2 C r o ssbar I n v e rt ertap(y) I n v e rt ertap(z) Li ne (B ) Maj ori t y1(0 0,0) Maj ori t y1(0 0,1) Maj ori t y1(0 1,0) Maj ori t y1(0 1,1) Maj ori t y1(1 0,0) Maj ori t y1(1 0,1) Maj ori t y1(1 1,0) Maj ori t y1(1 1,1) W i ret a pProbability of Correct Output HFSCA BN Figure3.6.ValidationoftheBayesiannetworkmodelingofQCAcircuitswithHartreeFock approximationbasedcoherencevectorbasedquantummechanicalsimulation.Probabilities ofcorrectoutputarecomparedforbasiccircuitelements 52

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CHAPTER4 POWERDISSIPATIONINQCA 4.1Introduction Forcomputationmechanismsthatinvolvethetransferorowofelectrons,suchas CMOSgates,ithasbeenshownthatwithcontinuedscalingifsingleelectronsareinvolved inthecomputationthepowerpergatewouldapproachthe kT limitbutthepowerdensity wouldbeextremelyhigh[124,125].Unlikecomputationmechanismsthatinvolvethe transferorowofelectrons,suchasCMOSgates,QCAcomputationdoesnotinvolve electrontransferbetweenadjacentQCAcells.Sinceonlyfewelectronsareinvolvedin QCAcomputations,itissusceptibletothermalissues.Thereforeitisimportanttomodel andtoconsiderpowerasanimportantparameterduringtheQCAdesignprocessatmultiple levelsofdesignabstraction. WhileworkondefectandfaultsinQCAcircuits,whichareotherimportantissues, havestarted[117,118,119],powerissueshavenotbeenconsideredextensively.Perhaps themostpurepowermodelisthequantum-mechanicalmodelofthetemporaldynamicsof powerderivedbyTimlerandLent[126,127].Theyidentiedthreecomponentsofpower: clockpower,celltocellpowergain,andpowerdissipation.Whilethismodelgivesus physicallycloseestimates,itiscomputationallyexpensivetoestimate.Whendesigning QCAcircuits,wewouldlikeestimatepowerquicklyinordertochooseamongmanydifferentalternativesandparameters.Theneedforfullblownquantum-mechanicalestimation willberelegatedtotheveryendofthedesignprocess.Tothisend,somestudies[128,90] 53

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presentlowerboundsofpowerdissipatedthatareeasytocompute.However,fromadesignautomationpointofviewitisimportanttodesignfortheworstcase,leavinguswith marginforerrorsduetoprocessvariability. Hereworstcasereferstothepowerdissipated duringanon-adiabaticclockingscheme .Forworstcaseconsiderations,theupperbound forpowerismorerelevant.Wederivesuchanupperboundandshowhowitcanberelevant forQCAdesignautomation.Someotherrelevantpowerrelatedworksincludetheenergy vs.speedtrade-offstudyin[129]fordifferentclockingschemes,however,inthecontextof reversiblecomputing.In[130],aRCmodelforaclockedQCAchainisusedtoinvestigate powerdissipationunderadiabaticclockingscheme. UndertheHartree-Fockquantummechanicalapproximation,whichhasbeenfoundto beadequate,thedynamicsofacollectionofQCAcellscanbeexpressedintermsofthe dynamicsofindividualcells.Asaresult,thepowerdissipationforaQCAcircuitcan expressedasthesumofpowerestimatescomputedonaper-cellbasis.EachcellinaQCA circuitseesthreetypesofevents:(i)clockgoingfromlowtohighsoasto"depolarize" acell,(ii)inputorcellsinpreviousclockzoneswitchingstates,and(iii)clockchanging fromhightolow,latchingandholdingthecellstatetothenewstate.Eachoftheseevents areassociatedwithpowerloss.Aninterestingpointisthatthepowerdissipatedduring therstandthethirdtransitionsisduetotheclockchangingandoccursevenifthestate ofacelldoesnotchange.Thisisanalogousto"leakage"powerinCMOScircuits.The powerlossduetothesecondeventcanbetermedasthe"switching"powersinceitis dependentonthecellsactuallychangingstate.Clockenergyneedstobehightodrive thecellintoanintermediate,depolarizedstate.Inafullydepolarizedstate,thechangein driverpolarizationhasnoeffectonthedrivencell,hencethe"switching"poweriszero. Thisistheidealcase.However,toachievethistheclockingenergyneedstobehighand, consequently,theassociated"leakage"powerwouldbehigh.Thus,thesetwocomponents ofpowerareinverselyrelated.Theupperboundsderivedinthispaperwillhelpusquantify 54

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thisrelationship.However,arealclockimplementationwillalsoaddtopowerlossinthe clockingcircuititself.ThiswilladdtotheoverallpowerdissipationinaQCAcircuit.In thisstudywedonotaccountforthispowerloss. Ashortenedversionofthisboundderivationispresentedinin[95],however,using verysmallQCAlogicelements.Thetheoreticalcontributionsofthisworkare(i)the computationofupperboundofpowerdissipatedinaQCAcellrepresentingtheworst caseinputswitchingvectorsetand(ii)thecharacterizationofpowerintotwocomponents: leakageandswitching.Thisupperbound,whichiseasytocompute,canbeusedinthe QCAcircuitdesignprocess.Further,wedemonstratehowtheseestimatescanbeused(iii) tocharacterizethepowerdissipationinbasicQCAelementsliketheinverter,majoritygate, andcrossbar,(iv)tocomparetwofunctionallyequivalentaddercircuitsintermsofpower dissipatedduringanyswitchingevent,(v)tocomputepowerinlargeQCAcircuitlikea4x1 MultiplexerandasinglebitALU,(vi)tostudyvariationofpowerexpendedwithdifferent inputstates,and(vii)tolocatethethermalweakspotsinadesign. Theorganizationofthischapterisasfollows.Section4.2summarizesthequantum formulationofthepowerdissipationinQCAsaspresentedin[126].Usingthisexpression, wederiveinSection4.3theupperboundforthepowerdissipatedinaQCAcellduring eachclockcycle.Wethenusethispercellboundtoestimatethepowerinthewholecircuit asdescribedinSection4.4.InSection4.5weshowsimulationresultsusingthisupper bound.Werstvalidatetheboundusingquantummechanicalsimulationsandshowthat theboundholds.Wethendemonstratepowertheestimationprocessandstudythepower dissipationinanumberoflogicelementssuchasmajoritygates,inverter,singlebitadders andalsoforlargeQCAcircuitssuchasa4x1multiplexerandasinglebitALU[131]. 55

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4.2QuantumMechanicalPower Wedenotetheeigenstatesofacell,correspondingtothetwogroundstates,ofeach cellby0and1.Thiscouldrepresentthegroundstateofacellwithoneelectron intwodots,asinmolecular-QCA,oracellwithtwoelectronswithfourdots.Anarrayofcellscanbemodeledfairlywellbyconsideringcell-levelquantumentanglement ofthesetwostatesandjustCoulombicinteractionswithnearbycells,usingtheHarteeFock(HF)approximation[36,132].Thisallowsonetocharacterizetheevolutionof theindividualwavefunctions.Thestateofacellattime t ,whichisreferredtoasthe wave-functionanddenotedbyt ,isalinearcombinationofthesetwostates,i.e.t c0t 0 c1t 1.Thecoefcients, c0tand c1t,arefunctionsoftime.The expectedvalueofanyobservable, At ,canbeexpressedintermsofthewavefunctionas A t At t orequivalentlyasTr At t t ,whereTrdenotesthetrace operation,Tr 00 11.Thetermt t isknownasthedensity operator, t.Expectedvalueofanyobservableofaquantumsystemcanbecomputedif tisknown. Theentriesofthedensitymatrix, ijt,isdenedby citcjtor tctct wheredenotestheconjugatetransposeoperation.NotethatthedensitymatrixisHermitian,i.e. tt .Eachdiagonalterm, iit cit 2,representsthe probability of ndingthesysteminstatei.Itcanbeeasilyshownthat 00t11t1.InQCA devicemodelingliterature,oneusestheconceptof polarization s ,tocharacterizethestate ofacellandissimply 11t 00t,thedifferenceofthetwoprobabilities.Itranges from-1to1. Thedensityoperatorisafunctionoftime, t,anditsdynamicsiscapturedbythe LoiuvilleequationorthevonNeumannequation,whichcanderivedfromthebasicSchrodinger 56

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equationsthatcapturetheevolutionofthewavefunctionovertime, t. i h tt H t tH (4.1) where H isa2by2matrixrepresentingtheHamiltonianofthecell.Forarrangementsof QCAcells,itiscommontoassumeonlyCoulombicinteractionsbetweencellsandusethe Hartree-FockapproximationtoarriveatthematrixrepresentationoftheHamiltoniangiven by[36] H 1 2iEksifi1 2iEksifi 1 2S1 2S (4.2) wherethesumsareoverthecells. Ekistheenergycostoftwoneighboringcellswith oppositepolarizations;thisisalsoreferredtoasthe"kinkenergy". fiisthegeometric factorcapturingelectrostaticfalloffwithdistancebetweencells. siisthepolarizationof the i -thneighboringcell.Thetunnelingenergybetweenthetwostatesofacell,whichis controlledbytheclockingmechanism,isdenotedby .Fornotationalsimplication,we willuse S todenotethetotalkinkenergyduetothepolarizedneighbors. ToarriveatamorecompactmathematicalrepresentationweusetheBlochformulation oftheSchrodingerequationthatexpressestheevolutionofquantumsystemsinoperator spaces.ThedensityoperatorcanbeexpressedasalinearcombinationoftheSU(2),the Pauli'sspinoperators is t3i1ii(4.3) Here iarethePauli'sspinmatricesgivenby: 1 01 10 2 0110 3 10 01 (4.4) 57

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Thecombinationcoefcientsformthecoherencevector( )canbeexpressedby iTr i (4.5) ThetwostateHamiltoniancanbeprojectedontothebasisofgeneratorstoformareal three-dimensionalenergyvector ,whosecomponentsare Tr H h1 h 2 0S (4.6) TheBlochequationgoverningtheevolutionofthecoherencevectorcanbederived fromtheLiouvilleequationtobe d dt (4.7) Thisformulationdoesnotaccountfortheeffectofdissipativecouplingwithheatbath. Onereasonableapproximationistoaddaninhomogeneouslineartermtothisequationto accountfordamping. d dt (4.8) Wechoosetheparameters and sothattheyrepresentinelasticdissipativeheatbath coupling(openworld),withthesystemmovingtowardsthegroundstate[36,132].1 ssand 1 00 01 0 001 (4.9) wheressisthesteady-statecoherencevectorand istheenergyrelaxationtime.If itrepresentstheabsenceofanydissipation.Lowerthevalueof ,fastertheheatdissipation awayfromthecell.Thesteady-statecoherencevectorcanbederivedfromthesteady-state 58

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densitymatrixatthermalequilibrium. sseHkT TreHkT(4.10) where k istheBoltzmanconstantand T isthetemperature.Thecorrespondingsteadystate coherencevectorisgivenbyssTrss tanh (4.11) where kT,isthethermalratio,with 4 2S2,theenergyterm(alsoknownas theRabifrequency). TheexpectedvalueoftheHamiltonianateachtimeinstantisgivenby E H h 2 (4.12) Theequationfortheinstantaneouspowerisgivenby Ptotald dt E h 2d dt h 2 d dt(4.13) Thersttermcapturesthepowerinandoutoftheclockandcelltocellpowerow.The secondtermrepresentsthedissipatedpower.Itisthisquantitythatweareinterestedin. Pdisst h 2t d dtt (4.14) 59

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0 20 40 60 80 100 1 0.5 0 0.5 1 Time (hbar/Ek)PolarizationTemp = 5.00K, tau = 3, Tc = 75 (in units of hbar/Ek) Driver ss 0 20 40 60 80 100 0 0.01 0.02 0.03 0.04 Time (hbar/Ek)Power Dissipated (eV/s) Clock ( ) Pdiss 0 20 40 60 80 10 0 1 0.5 0 0.5 1 Time (hbar/Ek) Polarization Temp = 5.00K, tau = 3, Tc = 75 (in units of hbar/Ek) Driver ss 0 20 40 60 80 10 0 0 0.01 0.02 0.03 0.04 Time (h bar /Ek)Power Dissipated (eV/s) Clock ( ) Pdiss (a)(b) Figure4.1.Polarizationchange(topplot)andpowerloss(bottomplot)inasinglecellwhen itspolarizationchangesfrom(a)-1to1(or0to1logic)and(b)-1to-1(remainsatstate 0)duringaquasi-adiabaticclockingscheme. 4.3UpperBoundforPowerDissipation CouplingtheexpressionforpowerdissipationwiththedampedBlochequationwesee that Pdisst h 2 t t sst (4.15) Iftheinstantaneouscoherencevectortracksthesteadystatecoherencevectorforthattime instant,i.e.t sstthenthepowerdissipatedisverylow.Fig.4.1.(a)(bottomplot) showstheinstantaneouspowerdissipationwhenthedriverpolarizationswitchesfrom-1to 1.ThetopplotofFig.4.1.(a)showsthechangeinthedriverpolarizationandtheassociated changeinthesteadystatepolarizationss 3t.Notethatinthiscase,thecellpolarization3ttracksthesteadystatevaluequitewell.ThebottomplotofFig.4.1.(a)showsthe clockenergychangeandthepowerdissipated.Wenotethatthereisonlyveryslightloss ofpowerduringtheswitchingofthedriverpolarization.Fig.4.1.(b),whichisforthecase ofnochangeindriverpolarization,alsotellsthesamestoryoflowpowerloss. 60

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0 20 40 60 80 10 0 1 0.5 0 0.5 1 Time (hbar/Ek)PolarizationTemp = 5.00K, tau = 3, Tc = 75 (in units of hbar/Ek) Driver ss 0 20 40 60 80 10 0 0 0.01 0.02 0.03 0.04 Time (hbar/Ek)Power Dissipated (eV/s) Clock ( ) Pdiss 0 20 40 60 80 10 0 1 0.5 0 0.5 1 Time (hbar/Ek)PolarizationTemp = 5.00K, tau = 3, Tc = 75 (in units of hbar/Ek) Driver ss 0 20 40 60 80 10 0 0 0.01 0.02 0.03 0.04 Time (hbar/Ek)Power Dissipated (eV/s) Clock ( ) Pdiss (a)(b) Figure4.2.Polarizationchange(topplot)andpowerloss(bottomplot)inasinglecellwhen itspolarizationchangesfrom(a)-1to1(or0to1logic)and(b)-1to-1(nochangeinstate) duringnon-adiabaticclockingscheme Highdissipationsituationariseswhentlagsthechangingsst.FromEq.4.11 wecanseethatsschangeswhenevertheunderlyingHamiltonianchanges,whichhappens when(i)clockgoesfromlowtohigh( LH)soasto"depolarize"acell,(ii)inputor cellsinpreviousclockzoneswitchesstates( S S),and(iii)clockchangesfromhigh tolow( HL),latchingandholdingthecellstatetothenewstate.Fig.4.2.(a)shows theswitchingbehaviorandpowerdissipationforabruptchangeindriverpolarizationand clocking.Aswecanseefromthegraph,thesteadystatepolarization3tofthecellis notabletofollowthecorrespondingsteadystatepolarization.Thereissomelagandripple associatedwiththechange.Thisleadstopowerloss,whichisshowninthebottomplot ofFig.4.2.(a)Notethatthereispowerlossforallthethreeevents.Fig.4.2.(b)shows thesameswitchingbehaviorandpowerdissipationinthecellduringanonadiabaticevent evenwhenthedriverpolarizationremainsthesame(-1to-1switching).Aswecansee fromthegraphs,thetotalpowerdissipatedbythecelloccursnotonlywhenitspolarization changes,butasignicantamountofpowerlossalsooccurswhentheclockenergybarriers areraisedandlowered.Thefasterthechangesinvolvedintheseevents,themorethepower 61

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dissipation.Toarriveattheupperboundofthepowerloss,weconsiderthelimitingcase ofinstantaneouschange;wemodeltheseeventsusingasstepfunctions. Wederivetheenergydissipatedforeachofthesethreeeventsbyintegratingaround them.Withoutlossofgenerality,lettheeventunderconsiderationbecenteredat t0. Weintegrateover DD,suchthat D ,i.e.theintegrationtimeperiodismuch largerthantherelaxationtimeconstant.This,ofcourse,placeslimitontheclockspeed. Thisconstraintisnaturalalsoforcorrectoperation;clockperiodshouldbelargerthanthe relaxationtimeconstantsotherwiseerrorswillarise.Energydissipatedoveratimeperiod DDcanbearrivedatbyintegrating Pdisst. Ediss h 2DDd dt dt h 2 DD DDd dt dt h 2 DDd dt dt(4.16) whereweusethenotationandtodenote DandD,andsimilarlyfor Thisdissipatedpowertothebathwillbemaximumwhentherateofchangeof isthe maximum,i.e.non-adiabatic.Wemodelthismathematicallyusingthedeltafunction. d dt t(4.17) whereandarethevaluesoftheHamiltonian"before"andthe"after"thetransition. Usingthismodelandtheintegralpropertyofthedeltafunction,fttdtf0,we have Ediss h 2 0 (4.18) 62

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Table4.1.BlochHamiltonianbeforeandafterachangeinclockortheneighboringpolarization ClockUpDriverPolarizationClockDown LHS SHL =1 h 2 L0S 1 h 2 H0S 1 h 2 H0S 1 h 2 H0S 1 h 2 H0S 1 h 2 L0S Asmentionedbefore,weassumethat D ,i.e.thesystemisinequilibriumwith theheatbathat t D and tD .Insuchcase,wehave 0 ssand ss. Usingtheseobservations,wecanshowthat Ediss h 2 ss ss (4.19) PEdiss Tc h 2 Tc tanh h kT tanh h kT (4.20) wherewehavecharacterizedthepowerdissipationastheenergyperclockcycle; Tcis theclockperiod.Asisevident,thepowerupperboundcanbederivedoncewehavethe beforeandafterHamiltonianforthethreepowerdissipatingevents.Thesevaluesofthe HamiltonianareasshowninTable4.2.The"leakage"powerdissipated(energyperclock cycle)istheenergydissipatedduringtherstandthethirdeventassociatedwithclock change.And,the"switching"power(energyperclockcycle)istheenergylossduetothe secondevent. 4.4EnergyDissipatedperClockCycleinaQCACircuit SincethephysicsgoverningthepowerdissipationateachcellinaQCAcircuitissimilar,wecancomputethetotalpower(energyperclockcycle)byaggregatingthepowercomputedforeachcell.Theeffectofcellsoneachotheriscapturedthroughtheelectrostatic 63

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kinkenergybetweenthem.Letusconsideracircuitwith N cells,denotedby X1 XN, withtherst r ofthemrepresentingtheinputcells.Letthepolarizationofthe i -thcellbe denotedby xi 11.Foreachswitchingoftheinputcells,wecomputethepowerby keepingtrackofthebeforeandafterpolarizationofthecells.Let xikbethepolarizationof the i -thcellforthe k -thpossibleinputcombination.Wecancomputethispolarizationusing anyofthesimulationmethodsthatareavailableforQCAcircuits[42].Inourexperiments weusetheBayesianNetworkmodelingtechniquein[25]toprobabilisticallydeterminethe polarizationsinanefcientmanner.Sinceinthiswork,weareinterestedinahardupper boundonthetotalpowerdissipatedinacircuit,henceweactuallyroundoffthecomputed polarizationofeachcelltothenearestpurevalue,i.e.-1or1value. Tocomputethepowerdissipatedateachcell,weneedtocomputetheeffectivekink energyofrestofthecells, Siand Siastheinputswitchesfrom k -thcombinationtothe m -thcombination.Thisiseasilycomputedas Si jNeXiEkfjxjkand Si jNeXiEkfjxjm(4.21) wherethesumcanberestrictedtoalocalneighborhoodofthecellsincethedistance relatedterm, fj,fallsoffas5-thpowerofthedistancefromthecell.Usingthesevalues, andknowledgeofthelowandthehighclockenergies, Land H,wecancomputethe leakagePleak ikmandtheswitchingPswitch ikmpower(energyperclockcycle)boundsateach cell(Eq.4.20).Giventheseestimateswecancomputedifferentdesignrelatedparameters asoutlinedbelow.Notethatthequantitieswecomputeareactuallyboundsoftherespective quantities;wedonotemphasizetheboundaspecttoreducenotationalclutter. 64

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TotalDissipatedPower:fortransitionfromthe k -thinputstatetothe m -thinputstate isgivenby Ptot kmNir1Pleak ikmPswitch ikm(4.22)AveragePower(overallinputtransitions):isgivenby Pavg1 2rkmPtot km(4.23)MaximumPower(overallinputtransitions):isgivenby PmaxmaxkmPtot km(4.24)HotSpots:Powerisnotuniformlydissipatedateachcell.Itisimportantfroma thermalerroranalysispointofviewtoidentifythecellsinadesignwherethepower dissipationishigh.Oncewecomputetheaveragepowerdissipationateachcellover allinputtransitions,wecanidentifythehot-spotasthecellswith k maximumpower dissipation. argk-max1 2rkmPikmir1N(4.25) 4.5Results Werstpresentempiricalvalidationofthepowerboundsbycomputingexactpower ofoneQCAcellunderdifferentclockingconditionsandshowthattheboundholds.We followthisbyshowingexamplesofhowthisboundestimatecanbeusedforQCAdesign automation.ThesizeofQCAcellsusedinthisstudyis20nmx20nmwithagridspacingof 20nm.WecomputepowerdissipationboundsforsomebasicQCAlogicelementssuchas themajoritygate,inverter,ANDgate,ORgate,crossbarandclockedmajoritygate.Since 65

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104 103 102 101 10 0 0 1 2 3 4 5 6 7 8 9 Clock SmoothingTotal Switching Power Dissipated (eV/s) /Ek=0.25 /Ek=0.5 /Ek=0.75 104 103 102 101 10 0 0 1 2 3 4 5 6 7 Clock SmoothingTotal Leakage Power Dissipated (eV/s) /Ek=0.25 /Ek=0.5 /Ek=0.75 (a)(b) Figure4.3.Variationof(a)switchingpowerand(b)leakagepowerdissipatedinasinglecell withdifferentamountofclocksmoothingfordifferentclockenergy levels.Adiabaticity oftheswitchingprocessiscontrolledbysmoothnessoftheclocktransition.Thehorizontal lineplotstheupperboundsforeachcaseascomputedusingthederivedexpressions. powerisdependentontheinputs,weshowthemaximum,minimumandaveragepower dissipatedineachofthesecircuitsoverallpossibleinputtransitions.Further,wemake useofthepowermodeltoestimatepowerdissipatedintwodifferentdesignsofsinglebit addersandthethermallayoutforbothdesigns.Finally,wedemonstratethemodelforsome largecircuitsthe4x1multiplexerandasinglebitALUdesign[131].TheALUdesign consistsofseveninputsandtwooutputs.ThesinglebitALUcanbeusedtoperformlogical operationssuchaAND,ORandinversion.Itcanalsoperformmathematicaloperations suchasadditionandsubtractionbetweentwosinglebitnumbers. 4.5.1EnergyDissipationperClockCycleinaSingleQCACell Thepowerdissipatedateachcellisafunctionoftherateofchangeoftheclockandthe clockenergy.Weestimatedtheactualpowerdissipatedusingquantummodelforvarious valuesoftheseparameterandcomparedthemwiththepowerbounds.Fig.4.3.(a)and(b) showsthevariationofswitchingandleakagepowerdissipationwithvaryingamountclock 66

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0 0.5 1 1.5 2 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 H/EkEnergy (eV)Temp = 5.00K, Tc = (in units of hbar/Ek) 0 to 0 0 to 1 1 to 0 1 to 1 Figure4.4.Dependenceofenergydissipated(upperbound)inacellwithclockenergyfor differentclocktransitions.(a)00(b)01(c)10and(d)11.Notethattheplotsfor cases(a)and(d)overlapcompletelyandsodoestheplotsforcases(b)and(c). smoothingandfordifferentvaluesofclockenergy.Thepowerbounds,whicharefunctions oftheclockenergy( ),areshownashorizontallines.Adiabaticityofthesystemisdirectly proportionaltotheamountofclocksmoothing.Higherclocksmoothingimpliesmore adiabaticity.Weseethatboundsdoindeedholdandarereachedwhentheclocksmoothing iszero,i.e.abruptclockchanges,representingthefullynon-adiabaticcase. Fig.4.4.showshowthedissipatedenergyboundisdifferentfordifferentstatetransitions(a)00(b)01(c)10and(d)11,astheclockenergysuppliedtothecellis increasedfrom005 Ekto2 Ek.Notethatenergyisdissipatedevenifthestateofacelldoes notchange,i.e.forcases(a)and(d).Thisisbecausethehighclockstateonly partially depolarizesacellandthereischangeinthispartialpolarizationwithinputchange.Asthe highclockenergyisincreased,thecellgetsdepolarizedtoagreaterextentandthecontributiontooveralldissipationduetoswitchingstatesisless.However,asweseeinFig.4.4., thetotaldissipatedenergyalsoincreases;thisisduetothecontributionofdissipativeevent associatedwithclocktransitions,i.e."leakagepower."So,eventhoughhighclockenergy isdesirabletodepolarizethecellandensurewhentheclockenergysuppliedtothecellis 67

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increasedfrom005 Ekto2 Ekforcorrectoperation,ithastobelimitedfrompowerconsiderations.Hencethereisatrade-offbetweenpoweranderrorwhenchoosingtheclock energy. 4.5.2EnergyDissipationperClockCycleinBasicQCACircuits WeconsiderarrangementsofQCAcellsimplementingcrucialQCAcircuitelements. InTable4.2.foreachcircuit,wevisualizetheenergydissipatedateachcell,averagedover differentinputtransitions.Weusegrayscaleshadingtovisualizethedissipationateach celldarkerthecell,morethedissipation.Wewillrefertothiskindofvisualizationasthe thermallayout .Notethatthedissipationscaleforeachcircuitisdifferent.Wecanclearly seethatnotallthecellsofthecircuitsdissipatesameamountofenergy. Inadditiontotheenergydissipation,averagedoverallinputcombinations,wealso showthemaximumdissipationoverallinputconditionsandtheminimumdissipationover allinputconditions.Theminimumenergydissipationcaseiswhentheinputcellsdo notswitch.Thesethreequantitiesconveysomeideaabouttheoverallvariabilityofthe dissipationwithinput.Wehavetabulatedtheseresultsforthreevaluesof Ek. Thenumberofcellsinthetablerefertothenumberofcellsthatparticipateinenergy dissipation.Wedonotincludeinputcellsincalculatingthetotalenergydissipation.We cansee,thatincaseofaclockedmajoritygateshowninTable4.2.(a)eventhoughthetotal energydissipatedismuchhigherthanthatofaninvertershowninTable4.2.(b),weakspots intheinverterdesigndissipatehigheramountofenergythaninamajoritylogic.Thisis evidentfromthescaleassociatedwiththecolorcode.Hencetheinverterdesignismore susceptibletothermalbreakdown. WecanalsoseethateventhoughtheenergydissipatedforthecircuitslistedinTable4.2. greatlydependsonthenumberofcellsforeachdesign,stilltheaverage(overallinput 68

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Table4.2.ThermallayoutvisualizingtheenergydissipatedateachcellaveragedoverdifferentinputtransitionsforsomebasicQCAlogicelements.Darkerthecolor,morethe dissipation. (a)ClockedMajority (b)Inverter (c)Crossbar No.ofCells 16 9 10 Thermal Layoutat )Tj/T1_2 1 Tf9.9626 0 0 9.9626 123.1201 561.66 Tm(EK=0.5 (Energy Dissipation scaleisin termsof 10)Tj/T1_0 1 Tf7.3723 0 0 7.3723 129.84 505.5 Tm(3eV) 0 0.5 1 1.5 2 2.5 0. 5 1 1. 5 2 2. 5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 )Tj/T1_2 1 Tf9.9626 0 0 9.9626 123.1201 455.58 Tm(EK 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 Avg Ediss(meV) 32.91 39.10 47.48 17.41 21.26 26.16 17.58 26.01 35.70 Max Ediss(meV) 71.99 73.84 77.76 31.76 33.54 36.58 28.52 33.67 41.36 Min Ediss(meV) 4.27 13.86 25.49 3.06 8.97 15.75 6.97 18.37 29.98 (d)SimpleMajority (e)ANDGate (f)ORGate No.ofCells 3 4 4 Thermal Layoutat )Tj/T1_2 1 Tf9.9626 0 0 9.9626 123.1201 304.5 Tm(EK=0.5 (Energy Dissipation scaleisin termsof 10)Tj/T1_0 1 Tf7.3723 0 0 7.3723 129.84 248.34 Tm(3eV) 0.5 1 1.5 2 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 1. 6 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 1. 6 )Tj/T1_2 1 Tf9.9626 0 0 9.9626 123.1201 230.46 Tm(EK 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 Avg Ediss(meV) 5.99 7.19 8.78 6.20 8.09 10.46 6.20 8.09 10.46 Max Ediss(meV) 14.71 15.03 15.70 18.72 18.61 19.21 18.72 18.61 19.21 Min Ediss(meV) 0.75 2.46 4.57 0.99 3.30 6.16 0.99 3.30 6.16 69

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Average Energy Dissipation per cell at Clock_High = 0.5 Ek0 0.0005 0.001 0.0015 0.002 0.0025ALU MUX 4x 1 Add e r 1 Add e r 2 Clock e d M a j Cr o ss ba r I n ve r t e r Major i ty OR ANDEnergy Dissipated (eV / cell) Average Energy Dissipation / cell Figure4.5.EnergydissipationboundspercellfordifferentQCAlogicelements,averaged overdifferentinputcombinations.Thenumberofcellsforeachcircuitreferstothenumber ofcellsthatdissipateenergyduringaswitchingevent.Thegraphshownhereisfor EK=0.5.Notethatthecolormappingscaleforeachcircuitisdifferent. combinations)energydissipation percell forclockedmajoritygate,inverter,crossbarand simplemajoritygatedoesnotvarygreatlyascanbeseenfromthegraphshowninFig4.5. 4.5.3EnergyDissipationperClockCycleinQCAAdderCircuits Table4.3.showsthecomparativestudyofenergydissipatedintwodifferentQCA addersdesigns.Aswecanseefromthetable,Adder-1hasmuchhigherenergydissipation, asithas359energydissipatingQCAcellspresentinitslayoutascomparedtoAdder-2 designthathasonly165suchcells.Wecanseefromthetablethatthermalenergylayout foreachdesignshowsthatthehighestaverageenergydissipationforanyparticularcellin bothdesignsisalmostthesame,eventhoughtheAdder-2designhascomparativelylarger numberofsuch'highenergydissipation'dissipatingcellspresentinitslayout.Wecanalso seefromthegraphinFig.4.5.thateventhoughthetotalenergydissipationforbothdesigns mayvarygreatly,stilltheaverage(overallinput)powerdissipation percell isalmostthe 70

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Table4.3.Thermallayoutvisualizingtheenergydissipationateachcell,averagedoverall possibleinputcombinationsfortwoQCAadderdesigns. (a)Adder1 (b)Adder2 No.ofCells 359 165 Thermal Layoutat )Tj/T1_2 1 Tf9.9626 0 0 9.9626 123.1201 452.7 Tm(EK=0.5 (Energy Dissipation scaleisin termsof 10)Tj/T1_0 1 Tf7.3723 0 0 7.3723 129.84 396.54 Tm(3eV) 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 )Tj/T1_2 1 Tf9.9626 0 0 9.9626 123.1201 346.62 Tm(EK 0.5 1.0 1.5 0.5 1.0 1.5 Avg Ediss(meV) 857.74 1110.45 1421.66 379.56 499.44 645.18 Max Ediss(meV) 1524.42 1655.32 1868.33 671.80 736.98 840.29 Min Ediss(meV) 203.82 576.53 984.33 97.42 271.06 458.34 71

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sameforbothdesigns.Forbothdesigns,themaximumenergydissipationoccurredwhen theinputcombinationsswitchedfrom000111. Thisresultseemtobeinterestingbecauseithasbeenalreadyshownin[27]thatAdder-2 designismorepronetoerrorthanAdder-1design.Whereas,wecanseethatwhenitcomes topower,Adder-2ismoreenergyefcienteventhoughithasmorehot-spotspresentinits layout. 4.5.4EnergyDissipationperClockCycleinLargeQCACircuits Inordertodemonstratethatthisworkisapplicabletoevenlargerdesigns,wealso presenttheresultsfora4x1multiplexerandasinglebitALUdesigns.TheALUdesign consistsofover800QCAcells.Fig.4.6.showsthethermallayoutforaveragepower dissipatedateachcellina4x1multiplexerdesignandFig.4.7.showsthethermallayout forasinglebitadderdesign.Wecanclearlyseethethermalhot-spotsinbothdesigns. Thesehotspotsdissipatelargepower,averagedoverallinputcombinations,andinorderto makethedesignslesssusceptibletothermalbreakdowns,designerscantargettheseweak spotsinthedesignforfurtherreinforcements.Thefabricationscientistscanalsousethese resultstoselectdifferenttypesofdevices. InordertoevaluatethemultiplexerandALUdesignweranasimulationtomodelall possibleinputvectorcombinationsanddeterminetheaveragepowerdissipationoverall possibleinputvectorsettransitions.Incaseof4x1multiplexertherewere6inputsand hencewehaveavectorsetcomprisingof64inputvectors.IncaseofALUsincethereare seveninputs,hencewehave128possibleinputcombinations.Table4.4.showstheaverage (overallinputcombinations),maximum(overallinputcombinations),andminimum(over allinputcombinations)powerdissipationboundsforthesedesignsat 05 EK, 10 EKand 15 EK.SincetheALUdesignhasmuchlargernumberofcellsascomparedtothe multiplexerdesign,itobviouslydissipatesmoreenergycomparedtothemultiplexer.Itcan 72

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure4.6.ThermalLayoutforaverageenergydissipatedineachcellofa4x1MUXcircuit. Thedarkspotsaretheonesthatdissipatelargeramountofenergyonanaverage.Thelayout wasobtainedbysimulatingoverallpossibleinputswitchingcombinationsfrom000000111111for )Tj/T1_3 1 Tf11.9552 0 0 11.9552 174.6051 459.9 Tm(EK=0.5.Theenergydissipationscaleforeachcellisintermsof10)Tj/T1_1 1 Tf8.9664 0 0 8.9664 508.68 464.22 Tm(3eV. 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 Figure4.7.ThermalLayoutforaverageenergydissipatedineachcellofasinglebitALU circuit.Thedarkspotsaretheonesthatdissipatelargeramountofenergyonanaverage. Thelayoutwasobtainedbysimulatingoverallpossibleinputswitchingcombinationsfrom 00000001111111for )Tj/T1_3 1 Tf11.9552 0 0 11.9552 240.3058 133.26 Tm(EK=0.5.Theenergydissipationscaleforeachcellisinterms of10)Tj/T1_1 1 Tf8.9664 0 0 8.9664 139.92 123.18 Tm(3eV. 73

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Table4.4.Statisticsoftheenergydissipationpercellfora4x1MUXandasinglebitALU overallpossibleinputcombinationsandfordifferentpossibleclockenergies.Weshowthe average,maximum,andminimumenergypercelloverallinputcombinations. (a)SingleBitALU (b)4x1MUX No.ofCells 801 270 EK 0.5 1.0 1.5 0.5 1.0 1.5 Avg Ediss(meV) 1781.97 2370.33 3083.37 668.67 850.92 1080.00 Max Ediss(meV) 3192.91 3525.86 4030.15 1174.97 1274.37 1432.77 Min Ediss(meV) 456.62 1279.19 2185.83 136.69 404.03 707.49 beseenthatthepercellenergydissipationstillremainsmoreorlessthesameforboth designs(inFig.4.5.).Wealsoseefromthethermallayoutofthetwodesignsthatsome ofthecellsinmultiplexerdissipatemuchhigherenergyonanaveragethananycellinthe ALUdesign. ApartfromcalculatingthethermallayoutfortheaverageenergydissipationinanALU design,wealsostudiedthethermalenergylayoutincaseofmaximumandminimumenergy,overallinputtransitions.InFig4.8.(a)and(b)weshowthethermallayoutofALU circuitforthemaximumandminimumenergydissipationcases,respectively.Thedark spotsaretheonesthatdissipatelargeramountofenergy.Thelayoutwasobtainedby simulatingworstcaseandbestcaseinputswitchingvectorsat EK=0.5.Theenergy dissipationscaleinFig4.8.(b)ismuchsmallerthanthatinFig4.8.(a)sinceenergyis dissipatedonlyduetoleakagecomponentandhenceismuchlessthanFig4.8.(a)where switchingenergyplaysadominantroleintotalenergydissipationofacell.Itcanbeclearly seenfromthelayoutsthattheenergydissipatedinalmostallcellsofFig.4.8.(a)ismore thanthatofcelldissipatinghighestenergyinFig4.8.(b)OnanaverageeachcellinFig 4.8.(a)dissipatesamagnitudehigherenergythanthatincaseofFig4.8.(b)Thereasonbehindthisisthatincaseofminimumpowerdissipation,noneoftheinputcellsswitchstate. Andthetotalenergydissipatedateachcellinthiscaseisonlytheleakageenergy(which 74

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0 2 4 6 8 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) (b) Figure4.8.ThermalLayoutforenergydissipatedineachcellofanALUcircuitfor(a) Maximumenergydissipatinginputcombinationand(b)forleastenergydissipatinginput combination.Energydissipationscaleisinmultiplesof10)Tj/T1_1 1 Tf8.9664 0 0 8.9664 396.12 420.54 Tm(3. 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 H/EkEnergy (eV)Temp = 5.00K, Tc = (in units of hbar/Ek) Avg. Switching Energy Avg. Leakage Energy Total Avg Energy Dissipation 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 H/EkEnergy (eV)Temp = 5.00K, Tc = (in units of hbar/Ek) Max Avg Energy Dissipation Min Avg Energy Dissipation (a) (b) Figure4.9.GraphsshowingenergydissipatedinaQCAALUcircuit(a)Showsthevariationofleakageandswitchingcomponentsofenergydissipatedforvariousvaluesof )Tj/T1_3 1 Tf11.9552 0 0 11.9552 524.9573 154.86 Tm(EK(b)Showsthevariationinmaximumandminimumenergydissipatedforvariousvaluesof )Tj/T1_3 1 Tf11.9552 0 0 11.9552 118.9199 125.94 Tm(EK75

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isquitelowcomparedtotheswitchingcomponentofenergy).However,thisconclusionis notvalidforhigherclockenergies. InFig.4.9.(a)weplotthevariationofthedissipation(averagedoverallinputtransitions)withclockenergy.Weseethatswitchingcomponentofenergyreduceswhenwe increasetheclockenergy,buttheleakagecomponentincreasesmuchmoresignicantly, resultinginoverallincreaseinpowerdissipation.At 09 EK,theleakagecomponentof energyandswitchingcomponentcontributeequallytothetotalenergydissipationofthe circuit.Beyondthisvalueof ,theleakagecomponentofenergydissipationcontributes morethantheswitchingcomponenttowardstotalenergydissipation.Thisresultwillbeof greatusetodesignersorevencircuitfabricatorstochoosethemostoptimumclockenergy tobesuppliedtoaQCAcircuit.Fig4.9.(b)showsthevariationofmaximumandminimum energydissipationinaQCAALUdesignwithrespecttotheclockenergy.Wecanseethat whileitisdesirabletohavehigherclockenergyinordertoreduceerrorsinQCAoperation,itcanbeseenclearlyfromtheresultsthatiftheclockenergyisraisedsignicantly, theenergydissipationishigh. 76

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CHAPTER5 HIERARCHICALDESIGNINQCAUSINGPROBABILISTIC MACROMODELING 5.1Introduction Timeisripetolookbeyondjustdevicelevelresearchinemergingdevicessuchas QCAandexplorecircuitlevelissuessoastoscopeoutthetypesofcircuitsthatcanbe built[133,86,134,87,110,111].However,QCAmodelingtoolsavailableforsuchdesignshavebeenatthelayoutlevel.Thereareseveralapproximatesimulatorsavailable atthelayoutlevel,suchasthebistablesimulationengineandthenonlinearapproximationmethods[135,136,42].Thesemethodsareiterativeanddonotproducesteadystate polarizationestimates.Inotherwords,theyestimatejuststateassignmentsandnotthe probabilitiesofbeinginthesestates.Thecoherencevectorbasedmethod[126,42]does explicitlyestimatethepolarizations,butitisappropriatewhenoneneedsfulltemporal dynamicssimulation(Blochequation),andhenceisextremelyslow;forafulladderdesignwithabout150cellsittakesabout500secondsfor8inputvectors.Perhaps,theonly approachthatcanestimatepolarizationforQCAcells,withoutfullquantum-mechanical simulationisthethermodynamicmodelproposedin[121],butitisbasedonsemi-classical Isingapproximation.In[24,137,25],itwasshownthatlayout-levelQCAcellprobabilities canbemodeledusingBayesianprobabilisticnetworks. ToadvancedesignwithQCA,itisnecessarytolookbeyondthelayoutlevel.Hierarchicaldesignatmultiplelevelsofabstraction,suchasarchitectural,circuit,layout,and devicelevels,hasbeenasuccessfulparadigmforthedesignofcomplexCMOScircuits.It 77

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isonlynaturaltoseektobuildasimilardesignstructureforemergingtechnology.Henderson etal. [26]proposedanhierarchicalCMOS-liketop-downapproachforQCAblocks thatareanalyzedwithrespecttotheoutputlogicstates;thisissomewhatsimilartofunctionallogicvericationperformedinCMOS.Wealsoadvocatebuildinganhierarchical designmethodologyforQCAcircuits.However,suchanhierarchyshouldbebuiltbased onnotjustthefunctionalityofthecircuit,butitshouldalsoallowtheabstractionofimportantnano-deviceparameters.ItisnotsufcientjusttoabstractaQCAcircuitinterms of0-1booleanlogicbasedmajoritygatesandotherlogiccomponents,wehavetoalso representtheprobabilisticnatureoftheoperations.Thus,foreachlogicvariable X ,we havetoassigntheprobabilitiesassociatedwiththelogicvalues,i.e. PX1or PX0. IntheparlanceofQCA,thespecicdesignvariableisthe"polarization"ofcell,whichis PX1 PX0.Theseprobabilities(orpolarizations),whicharegovernedbyquantummechanics,aredependentontemperature,whichisanimportantdesignvariablefor QCAsthatneedstoberepresentedatupperdesignlevels.Anotherneedforprobabilistic representationsariseduetothenatureoftheQCAoperations.QCAcircuitsaredesigned sothattheintendedlogicismappedtothelowest-energy(groundstate)ofthecellarrangement.So,itisimportantthatthecircuitbekeptneargroundstateduringoperations, usingmechanismssuchasfour-phasedadiabaticclocking.LogicalerrorsinQCAcircuits canariseduetothefailuretothesettletothegroundstate.Itisimportanttocompute thedifferencebetweentheprobabilityoflowest-energystatecongurationthatresultsin correct outputandthelowest-energystatecongurationthatresultsin erroneous output. Itwouldindeedbeusefultobeabletocomputetheseerroneouscongurationsathigher levelsofdesign.Buildingadevice-levelcharacterizationsensitivemacromodelwillfacilitateansweringthefollowingkindsofquestionsathigherdesignlevelsofabstractionitself. Whatisexpectedpolarizationoftheoutputs?Howdoesitchangewithtemperature?How sensitiveisthedesignwithrespecttooperationalerrors? 78

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Inthischapter[27,28],weformulateaprobabilisticframeworkforhigherlevelofabstractionofQCAcircuitsthatwouldenableonetocharacterizedesignswithrespectto thermalprolesanderrors,thetwomostimportantdesignissuesinnano-circuitdesign. StandardQCAcircuitelementssuchasmajoritylogic,lines,wire-taps,cross-overs,inverters,andcornersarerepresentedusingconditionalprobabilitydistributionsdenedoverthe outputstates given theinputstates.TheprobabilisticmacromodelsallowustomodelQCA circuitsatanabstractlevelabovethecurrentpracticeoflayoutlevel;wetermthishigher levelasthe"circuit"level.Thefullcircuitlevelmodelisconstructedbychainingtogether theindividuallogicelementmacromodels.Thiscircuitrepresentedusingthegraphical probabilisticmodelsknownasBayesiannetworks,wherethenodesofthegraphsarethe individualmacromodelsandthelinksrepresenttheconnectionbetweenthem.Thenodes arequantiedbythemacromodelconditionalprobabilities.Thecompletenetworkrepresentsajointprobabilitydistributionoverthewholecircuit.Sinceconditionaldistribution overtheinputsandoutputsareobtainedbasedonquantummechanicalprobabilisticcharacterization,thecircuitlevelmodelisalsofaithfultotheunderlyingquantum-mechanical phenomena. Computationsusingthemacromodeltranslatestodifferentkindsofprobabilisticinferenceproblems.Forinstance,computationofgroundstatepolarizationisdoneusingthe average likelihoodpropagationonthebuiltBayesiannetworkmacromodel.Similarly,the most-likelycongurationoftheinternalnodescorrespondingtorst-excited,alsocalled near-groundstateorthemostlikelyerrorstateattheoutputs,canbeisolatedatthemacromodelcircuitlevelitselfusing maximum likelihoodpropagationonthesameBayesian networkmacromodel.Wedemonstrateandvalidateourmodelusingcommonlystudied QCAcircuitsandelements,whosebehaviorsareprettywellunderstoodbyothers.First, weshowthatthegroundstatepolarizationprobabilitiesoftheoutputnodesaswellasthe intermediatenodesinthemacromodeloftheQCAlogiccircuitcloselymatchwiththose 79

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obtainedfromafulllayoutlevelimplementation[24]atdifferenttemperatures.Weshow examplesofcharacterizationofthermalbehaviorofaQCAlogiccircuitthatcanbecarried out.Second,wedemonstratethatboththegroundandthenextexcited(error)statecongurationofthemacromodelexactlymatchthecorrespondingcongurationsofthedetailed layoutcells.Themismatchbetweenthegroundandthenextexcitederrorstatecongurationcanbeusedtoidentifyweakspotsincircuitdesign.Usingthemacromodel,thiscan nowbedoneatanhigherlevelofabstraction.Isolationoferror-pronecomponentswould beusefulinapplyingredundancyselectivelytothenecessarycomponentsratherthantothe wholecircuit.Third,weusethecircuitlevelimplementationtovetbetweenalternatedesignchoices.Weshowexamplesofthisdesignspaceexplorationprocesswiththeexample oftwoadders.Wendthatoneadderdesign,Adder-1,inspiteofitslargerarea,isbetterin termsofpolarizationwhichisanextremelyimportantmeasurefortheQCAcircuits.Also, weseethatforAdder-1,numberoferror-pronecomponentsislessthanasecondadder design,Adder-2,andhencetheneededredundancymeasureswouldbelessforAdder-1. Theorganizationofthischapterisasfollows.InSection5.2,webeginbyexplaining thehierarchicalmodelingschemeusedinthiswork.Thenweproceedinsubsection5.2.1 tosummarizethequantum-mechanicalnatureoftheprobabilitiesassociatedwiththeQCA cells.InSection5.2(5.2.2),weshowhowanarrangementofQCAcellscanbemodeledbyajointprobabilityfunction,representedasaBayesiannetwork.Furtherdownin Section5.2(5.2.3)wepresentthetheorybehindthemacromodels.Wedemonstratehow usingthesemacromodelswecan(i)modelfullcircuitsSection5.2(5.2.4),(ii)exploredesignspaceexplorationinQCAcircuitlayouts(Section5.4(5.4.3)),and(iii)conducterror studies(Section5.3).Wecommentonthecomputationaladvantageofthecircuitlevel representationoverthelayoutleveloneinSection5.4andweconcludewithSection7. 80

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5.2ModelingTheory Inthissection,weexplainthehierarchicalmodelingscheme.Wefocusontwolevels: thelayoutlevelandthecircuitlevel,wheregroupsofQCAcells,correspondingtoabasic logicelement,arerepresentedasonemacroblock.Forboththeselevels,wewillusethe graphicalprobabilisticmodelcalledBayesianNetworkstorepresenttheunderlyingjoint probabilityoftheentiresetofnodes.Notethatprobabilisticrepresentationisessentialto capturetheinherentlyuncertainnatureofthecomputingwithQCAs. BayesianNetworks[122]areefcientrepresentationsofthejointprobabilitydistributionoverasetofrandomvariablesusingaDirectedAcyclicGraph(DAG).Eachrandom variableofinterestisrepresentedasanodeandlinksbetweenthenodesdenotedirectdependencies(cause-effectinteractions)betweentherandomvariables.Forourproblem,the randomvariablesarethestatesoftheQCAcellsatthelayoutlevelortheI/Ostatesofthe macromodels.Thelinksareguidedbytheinteractionneighborhoodofthecellsandthe logicalowofinformationfrominputstotheoutputs.ForQCAcircuitsthesecause-effect directionswouldbedeterminedbydirectionofpropagationofquantum-mechanicalinformationpropagationwithchangeininput.Clocksdeterminethecausalorderbetweencells. Withineachclockzone,orderingisdeterminedbythedirectionofpropagationofthewave function[36].SincetheCoulombicinteractionbetweencellsfallofffasterthanthefth powerofthedistancebetweenthem,weneedtoconsiderlinksbetweencellsthatarewithin asmallneighborhoodofeachother,typically2celldistance. InFig5.1.(a),weshowtheQCAlayoutofaNANDgate.Fig5.1.(b)showsthelayout levelBayesianrepresentation.Notethatwehave18randomvariablesrepresentingthe stateof18QCAcells.Fig5.1.(c)showsthecircuitlevelabstractionofaNANDgate.The BayesianrepresentationofcircuitlevelabstractionasshowninFig.5.1.(d)hasfewercells. Notethateachnodeatthecircuitlevelisthecollectionofcellsfromthelayoutlevel. 81

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X3 = 0 X1 X2 X18 Y3Y4 Y5 (a)(b) AND INV LINE A B Out Y1 Y2 Y3 Y4Y5 (c)(d) Figure5.1.ANANDlogicgate(a)QCAlayout(b)BayesianmodelofQCAlayout(c) Macromodelblockdiagram(d)Bayesiannetworkofmacromodelblockdiagram. Inthiswork,wewilluse X torepresenttherandomvariabledenotingthestatesof aQCAcellatthelayoutlevel(Fig.5.1.(b)).Theinputcellstateswillbedenotedby X1 Xr,thenon-inputQCAcellswillbe Xr1 XNand Xswilldenoteoneoftheoutput cellwhere r1sN .Similarlyforthecircuitlevel,wewilluse Y torepresentthe randomvariabledenotingthelinestates.The Y1 Yraresetofinputcells, Yr1 YMarethenon-inputQCAcellsand Ysdenotesoneoftheoutputcellwhere r1sM ThenodesoftheBayesiannetworkarequantiedbytheconditionalprobabilities.At thelayoutlevel,weneedtospecifytheconditionalprobabilityofthestateofacellgiven thestatesofparentneighbors,i.e. PxpaXwhere PaXarethedirectcausesofthe randomvariable X ortheparentsofthenode X inthedirectedgraphrepresentation.We uselowercasetoindicatevalueofarandomvariable.i.e. Pxdenotestheprobabilityof theevent Xx or PXx ).Weestimatethisusingthequantummechanicalmodeling ofQCAcells.Atthecircuitlevel,weneedtospecifytheconditionalprobabilityofthe outputstatesofamacromodelgiventhestatesoftheinputs, PyPaY.Theseconditional probabilitiesareestimatedfromtheconditionalprobabilitiesforinthelayoutlevelmodel oftheQCAcellscomprisingthemacromodel,atdifferenttemperatures. 82

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Ingeneral,aBayesiannetworkencodesthejointprobabilityfunctionasasetoffactored conditionalprobabilities,ofminimalrepresentationalcomplexity.Proofofminimalitycan befoundinstandardBayesiannetworktextssuchas[122]. Px1 xnmk1Pxkpaxk(5.1) Intheconditionalprobabilityterm PxpaX, paXrepresentsthevaluestakenon bytheparentset, PaX. InferenceorcomputationwithBayesiannetworksexploitsthesparselyconnectedgraph structure.Themostcommonschemesinvolvepassingmessagesamongthenodes.Aswe shallsee,forwewillneedtoconductbothaveragecaseandmaximumlikelihoodinferences.Forboththe average and maximum likelihoodpropagation,weadoptthecluster basedexactinferencescheme.Wereferthereaderto[122,138,137]fordetailsonthe inferencescheme.However,itsufcestonotethatthepropagationschemesarebasedon messagepassingandaresimilar,differingonlyinthekindsofmessagesthatarepassed. TheoriginalBayesiannetwork,whichisaDAGstructure,isrsttransformedintoajunctiontreeofcliquesandthenmarginalprobabilitiesarecomputedbylocalmessagepassing betweentheneighboringcliques.Thesemethodsresultinexactinferenceofprobabilities. Intherestofthissection,weprovidedetailsoftheprocess.Westartwithdiscussionof themacromodelconstructionprocessbytheBayesiannetworkmodelatthelayoutlevel, whichwasproposedin[137].Then,wepresenttheconstructionofthemacromodelsand circuitlevelBayesianrepresentation. 83

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5.2.1QuantumMechanicalProbabilities WesketchhowthestateprobabilitiesofaQCAcellaredependentonthestateprobabilitiesofitslayoutneighbors,distancetotheneighbors,andtemperature.Eachcellhas 2electronsthatcanoccupy4possibledots.Amongallthepossibleoccupancycongurations,therearetwolowestenergycongurationscorrespondingtothediagonaloccupancy ofthecells.Theserepresentthetwologicalstates,0or1.So,followingTougawand Lent[36]andothersubsequentworksonQCA,weusethetwo-stateapproximatemodel ofasingleQCAcell.Wedenotethetwopossible,orthogonal,eigenstatesofacellby1and0.Thestateattime t ,whichisreferredtoasthewave-functionanddenotedbyt ,isalinearcombinationofthesetwostates,i.e.t c1t 1 c2t 0.Note thatthecoefcientsarefunctionoftime.Theexpectedvalueofanyobservable, At ,can beexpressedintermsofthewavefunctionas A t At t orequivalentlyas Tr At t t ,whereTrdenotesthetraceoperation,Tr 11 00. Thetermt t isknownasthedensityoperator, t.Expectedvalueofanyobservableofaquantumsystemcanbecomputedif tisknown. A2by2matrixrepresentationofthedensityoperator,inwhichentriesdenotedby ijtcanbearrivedatbyconsideringtheprojectionsonthetwoeigenstatesofthecell,i.e. ijt i t j.Thiscanbesimpliedfurther. ijt i t j it t j it jt citcjt(5.2) ThedensityoperatorisafunctionoftimeandusingLoiuvilleequationswecancapturethe temporalevaluationof tinEq.5.3. 84

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i h ttH t tH (5.3) whereHisa2by2matrixrepresentingtheHamiltonianofthecellandusingHartree approximation.ExpressionofHamiltonianisshowninEq.5.4[36]. H 1 2iEkPifi1 2iEkPifi 1 2Ek P1 2Ek P (5.4) wherethesumsareoverthecellsinthelocalneighborhood. Ekisthe"kinkenergy"or theenergycostoftwoneighboringcellshavingoppositepolarizations. fiisthegeometric factorcapturingelectrostaticfalloffwithdistancebetweencells. Piisthepolarizationof the i -thcell.And, isthetunnelingenergybetweentwocellstates,whichiscontrolledby theclockingmechanism.Thenotationcanbefurthersimpliedbyusing P todenotethe weightedsumoftheneighborhoodpolarizations iPifi.UsingthisHamiltonianthesteady statepolarizationisgivenby Pss ss 3ss 11ss 00Ek P E2 k P24 2tanh E2 k P242 kT(5.5) Eq.5.5canbewrittenas PssE tanh(5.6) where E05 iEkPifi,thetotalkinkenergy, E2 k P242,theRabifrequency, and kTisthethermalratio.Weusetheaboveequationtoarriveattheprobabilitiesof observing(uponmakingameasurement)thesystemineachofthetwostates.Specically, PX1ss 11051Pssand PX0ss 00051Pss,wherewemadeuseof thefactthat ss 00ss 111. 85

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X2 X4 X3 X5 X1 MAJ Y1 Y2 Y3 Y4 0.4 0.6 0.8 1 123456789Temperature (K)Pout Input 000 Input 001 Input 010 Input 011 (a)(b)(c)(d) Figure5.2.Majoritylogic(a)QCAcelllayout(b)Bayesiannetworkmodel(c)Macromodel (d)Probabilityofthe correct outputvaluefo ra5cellmajoritygateatdifferenttemperatures andfordifferentinputs. 5.2.2LayoutLevelModelofCellArrangements Toenableustoformmacromodelsofvariouscellarrangements,weneedtorepresent thejointstateprobabilitiesofacollectionofcellsatthelayoutlevel.Inthissection,we summarizehowthisjointprobabilitycanbeefcientlyrepresentedusingBayesiannetworks,asshownin[137,24].WewillusethemajoritylogicarrangementofQCAcellsin Fig.5.2.(a)toillustratetheprocess. Eachcellisrepresentedbyarandomvariable,takingontwopossiblevalues,shownin theBayesiannetworkinFig.5.2.(b).Eachnodeinthenetworkhasaconditionalprobability table(CPT),capturingtheprobabilitiesofthatnode,giventhestatesoftheparent(cause) nodes.Forexample,thecenternode X 4,willbeassociatedwiththeconditionalprobability Px 4x 1x 2x 3.TheproductoftheseCPTsdeterminethejointprobabilitydistribution overallthevariablesinthenetwork.Thus,thejointprobability Px 1x 2x 3x 4x 5Px 4x 1x 2x 3Px 5x 4x 3x 2x 3.Thepolarizationoftheoutputcell X 5isafunction oftheremainingfourcellsinthelayout.Thecenternode X 4isactuallytheonewhich getspolarizedbasedonthemajorityofinputs.Theoutputcelldepictedherereceives thepolarizationofthecentralcell X 4andalsothethreeinputs, X 1, X 2,and X 3.The 86

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interactionbetweentheoutputcellandthecentralcellwillbemuchmorethantheinputs. Thisisbecausethekinkenergy(whichdeterminestheamountofinteractionbetweentwo neighboringcells),decaysasthefthpowerofdistance. Foragivensetofpossibleparentnodeassignments,theconditionalprobabilityvalues arecomputedusingtheHartree-Fockapproximation,appliedlocally.Theparentstatesare constrainedtobeasspeciedintherequiredconditionalprobability.Wexthechildren states(orpolarization)soastomaximize E2 k P242,whichwouldminimizethe groundstateenergyoverallpossiblegroundstatesofthecell.Thus,thechosenchildren statesare ch XargmaxchXargmaxchXi PaX ChXEk P (5.7) Thesteadystatedensitymatrixdiagonalentries(Eq.5.6withthesechildrenstateassignmentsareusedtodecideupontheconditionalprobabilitiesintheBayesiannetwork(BN). PX0paXss 00paX ch XPX1paXss 11paX ch X(5.8) Notethatoncetheconditionalprobabilitiesbetweenthenodesanditsparentsareobtained theBayesianNetworkisquantiedcompletely.Someoftheimportantparametersused inthismodelthateffectthepolarizationofacellapartfromtemperatureare: relative permitivity=12.9,radiusofeffect=4,celldimension=20nm,celltocellpitch=10nm, CLOCK HIGH61102eVandCLOCK LOW191015eV 5.2.3Macromodel ThebasiccircuitelementsofaQCAcircuitconsistsoftypicallogicelements,such asMajority,NAND,AND,OR,andNOT,andQCAspecicelementssuchaswiresand crossbars.Themacromodelsofdifferentcircuitelementsaretheconditionalprobabilityof 87

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Table5.1.Macromodeldesignblocks Macromodel QCALayout BayesianModel BlockDiagram ThermalProperties (a) Clocked Majority B A B CCM Out 04 05 06 07 08 09 1 123456789Temperature (K)Pout nput 000 nput 001 nput 010 nput 011 (b)Inverter INV Out In 04 06 08 1 123456789Temperature (K)Pout Input = 0/1 (c)Corner COIn Out 04 05 06 07 08 09 1 123456789Temperature (K)Pout Input = 0/1 (d)Line LineIn Out 04 05 06 07 08 09 1 123456789Temperature (K)Pout Input = 0/1 (e)Inverter Chain ICIn Out 04 05 06 07 08 09 1 123456789Temperature (K)Pout nput = 0/1 88

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Table5.2.Macromodeldesignblocks Macromodel QCALayout BayesianModel BlockDiagram ThermalProperties (a)Even Tap o Input ET Out 04 05 06 07 08 09 1 123456789Temperature (K)Pout Input = 0/1 (b)Odd Tap OT Input Out 04 05 06 07 08 09 1 123456789Temperature (K)Pout Input = 0/1 (c)Crossbar O2 CBIn1Out1 In2 Out2 04 05 06 07 08 09 1 123456789Temperature (K)Pout Horizontal Line Vertical Line (c)And Gate AND A B Out 04 05 06 07 08 09 1 11 123456789Temperature(K)Pout Input 00 Input 01 Input 10 Input 11 (c)OR Gate OR A B Out 04 05 06 07 08 09 1 123456789Temperature(K)Pout Input 00 Input 01 Input 10 Input 11 89

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outputcellsgiventhevaluesoftheinputcells.Wecomputethisbymarginalizingoverthe internalcells.Theunderlyingpremiseofthemacromodelingisthatifthejointprobability distributionfunction Px1 xnoverallthe n cellsinthelayoutisavailable,usingthe processoutlinedintheprevioussubsection5.2.2,thenwecanalwaysobtaintheexact distributionoversubsetofcellsbymarginalizingtheprobabilitiesoverrestofthevariables. Forinstance,thejointprobabilityoverjustthreecells, xixjand xk,canbeobtainedby Pxixjxkxmm ijkPx1 xn(5.9) Hence,atthecircuitlevel,wedonotrepresentallthe m internalcells.Notethatatcircuitlevel,weonlyrepresent Pxixjxkandrepresentthemwithdifferentvariable Y whichessentiallycapturestheinput-outputdependencebutisfaithfultothelayoutlevel quantuminteractionsincethemacromodelisbuiltbymarginalizingthelayoutlevelcells. Thismarginalizingisachievedbyconducting average likelihoodinference[122,138]on theBayesiannetworkrepresentationoverallthecellsinthemacromodelunit.Notethat Eq.5.9willyielddifferentresultsatdifferenttemperaturesandwestoretheconditional probabilitiesatvarioustemperaturepoints. Fig.5.2.(d)showsthethermalmodelsforthemajoritygateinFig.5.2.(a).Themacromodelprobabilitydistributionisdenedovertheoutputandthe3inputnodes.Atatemperatureof1K,ifinputsare0,0and0thentheprobabilityofoutputnodeisatstate0is "0.999963".Asthetemperatureisincreased,thisprobabilitydecreases.Wealsonotice thatthethermalbehaviorisdependentontheinputvalues.Notethat,forcorrectoperation, theprobabilityof correct outputshouldbegreaterthan0.5. Intherestofthissection,wepresentresultsforotherbasicbuildingblocks:clocked majoritygate(Table.5.1.(a)),inverter(Table.5.1.(b)),line(Table.5.1.(c)),corner(Table.5.1.(d)),inverterchain(Table.5.1.(e)),eventap(Table.5.2.(a)),oddtap(Table.5.2.(b)), 90

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crossbar(Table.5.2.(c)),ANDgate(Table.5.2.(d))andORgate(Table.5.2.(e)).Foreach macro-cell,weshowtheQCAlayout,layoutlevelBayesianmodel,circuitlevelinputoutputrelationandmagnitudeofpolarizationdropwithtemperature.Alltheconditional probabilitiesarestoredatvariouspointoftemperatures. Wemakethreeimportantobservations.First,aclockedmajoritygate,whichisnecessarytosynchronizealltheinputsignalsreachingthemajoritygate,hasweakerpolarization athighertemperaturecomparedtothesimplemajorityshowninFig.5.2.(d)asnumberof cellsarehigherintheclockedmajoritygate.Henceifinputstoamajoritygatearearrive atthesametime,thensimplemajorityyieldsbetterpolarizationsathighertemperatures. Second,invertershavelargerdropofpolarizationovertheodd-tapstructureathighertemperatures.Third,thecrossbarstructure,whichallowstwosignaltocrosseachotherina coplanarway,hasadifferentdropforthetwosignals. 5.2.4CircuitLevelModeling Table5.3.listsallthesymbolsusedformacromodeldesignblocksthatwehaveused inourdesigns.Amacromodellibrarystorestheinput-outputcharacteristics(outputnode probabilitiesforeachinputvectorset)ofeachmacromodelblockbasedontemperature. Thatmeansforeachtemperature,wehavealibraryofmacromodelblockslistedinthe Table5.3..Onceweknowthelogiccomponentsrequiredtobuildacircuit,wesimply extractthemacromodellogicblocksandtherequiredconnectivityblocks(e.g.Line,Corner,InverterChain,etc.)fromthelibraryatagiventemperatureandusethemtobuildthe logiccircuit.WeformaBayesianmacromodelusingtheinput-outputprobabilitiesofeach block.Theoutputfromonemacromodelblockisfedtotheinput(s)ofnextmacromodel block. Weillustratetheprocessusingthefulladdercircuit,Adder-1,showninFig.5.3.(a). Itconsistsofvemajoritygateswithnoinverters.Fig.5.3.(b)showsthecorresponding 91

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Sum Carry Out (a) (b) ABC i n ET IC IC CB ETCout CMLINE ETCB CB OTCB CB ET ET CB CB OT CB IC ET CB IC ET CB CMCO LNE IC ET CB ET CB OT IC IC IC CM CM Sum MAJ LNE CO A0 B0 C0 A1 A2 A3 A4 B1 B2 B3 B4 B5 B6 B7 B8 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20 G21 G22 G23 G24 G25 G26 G27 G28 G29 G30 G31 G32 G33 G34 (c) (d) Figure5.3.Afulladdercircuit(Adder-1)(a)QCAcelllayout(b)LayoutlevelBayesian networkrepresentation.(c)Circuitlevelrepresentation.(d)CircuitlevelBayesiannetwork macromodel.Note:Nodeelementsaregeneric. 92

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Table5.3.AbbreviationsusedforMacromodelBlocksfordesigningQCAarchitecturesof FullAddersandMultiplier Symbol Macromodel Maj SimpleMajorityGate CM ClockedMajority Gate Inv Inverter Line LineSegment CO Corner IC InverterChain OT OddTap ET EvenTap CB Crossover AND AndGate OR OrGate ZL z-line layoutlevelBayesiannetwork.WemodelthecircuitlevelQCAmacromodelshownin Fig.5.3.(c)whichisthecircuitlevelabstractionofFig.5.3.(a).TheBayesianmacromodel isshowninFig.5.3.(d).Eachsignal(node)caneitherbeaprimaryinput,oranoutputcell ofamacroblocklikeline,inverteretc.Thelinksaredirectedfromtheinputtotheoutputof eachmacroblockandarequantiedbythedevicemacromodels.Thus,wearriveatdirected acyclicgrapheasilyfromthecircuitmodelinFig.5.3.(c). 5.3ErrorComputation ApartofthecomputationofthepolarizationofeachQCAcellormacromodelline, whichwecanarriveatbyusing average casepropagation,anotheranalysisofinterestwhen comparingdesignsisthecomparisonoftheleastenergystatecongurationthatresults incorrectoutputversusthosethatresultinerroneousoutputs.Whatistheprobability oftheminimumenergycongurationthatresultsin error attheoutput, xs,foragiven inputassignment, x1 xr?Thiscanbearrivedatbyconditionalmaximumlikelihood propagation.Inessence,wecomputeargmaxx1x2 xrPxr1 xNx1 xrxsandthe 93

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minimumenergycongurationofallthecellsthatgeneratestheerroneousoutput xsisxe 1xe 2 xe r1 xe N.Thiscongurationcorrespondstothemostlikelyerrorstateatthe output xs.Wheneverwehave xg i xe i,the ithcellisconsideredsensitivetoerroratoutput xs(alsotermedasweakspots). Theabovecomputationalproblemofmaximizationofaproductofprobabilityfunctionscanbefactoredasproductofthemaximizationovereachprobabilityfunctions,these maximizationscanalsobecomputedbylocalmessagepassing[122].Theexactmaximumlikelihoodinferenceschemeisbasedonlocalmessagepassingonatreestructure, whosenodesaresubsets(cliques)ofrandomvariablesintheoriginalDAG[138].Thistree ofcliquesisobtainedfromtheinitialDAGstructureviaaseriesoftransformationsthat preservetherepresenteddependencies.Thedetailsoftheinferenceschemecanbefound in[137].Atthistransformedpoint,wehaveatreeofcliqueswhereeachcliqueisasub-set ofrandomvariables.Twoadjacentcliquesthatshareafewcommonvariableplayakey roleininference.Thejointprobabilityofallthevariablescanbeproventobetheproductofindividualcliqueprobabilities.Sincetheproblemofmaximizationofaproductof probabilityfunctionscanbefactoredasproductofthemaximizationovereachprobability functions,thismaximizationcanalsobecomputedbylocalmessagepassing[138].The overallmessagepassingschemeinvolvestheneighboringcliquesusingthemaximumoperatorwherethecliqueprobabilitiesareupdatedtillthemarginalprobabilityoftheshared variablesarethesame. Thiskindofmaximumlikelihoodanalysiscanbeconductedbothatthelayoutand thecircuitlevels.Letussaythatthecircuitlevelmacroblockshave Y1 Yrasinputs and Yr1 YMasinternalcircuitlevellines(nodes).Letussaythatthegroundstate macroblockcellpolarizationsaredenotedbyyg 1yg 2 yg r1 yg M.Withrespecttothe theerroneousoutput ys,lettheminimumenergycongurationisye 1ye 2 ye r1 ye M. 94

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Asinthecaseoflayout,wheneverwehave yg j ye j,the j -thcellisconsideredsensitiveto erroratoutput ys. Inthenextsection,wewillpresentsresultsthatshowthattheerrormodesofthecircuit andlayoutlevelsmatch.Thatis,whenever Yjissensitivetotherst-excitederrorstatefor output Ys,thecorrespondinglayoutlevelmodel,showsthesetofXithatconstitutedthe macroblock Yjisalsosensitive.Thisisanextremelyimportantndingthatindicatesthat weakspotinthedesigncanbeidentiedatthecircuitlevelitselfwithoutobtainingthecell layout.Alsothisisanimportantdesignmetricsandcanbeusedtovetonedesignoverand abovethethermalproleoftheoutputpolarization. 5.4Results Wepresentresultsusingthefulladderdesign,whichhasbeenwidelystudiedbyothers. Wealsouseamultiplierdesign,whichisasomewhatlargerdesign.First,wewillshowthat thegroundstatepolarizationprobabilitiesoftheoutputnodesaswellastheintermediate nodesinthemacromodeloftheQCAlogiccircuitcloselymatchwiththoseobtainedfrom afulllayoutlevelimplementation[24]atvarioustemperatures.Second,wedemonstrate thatboththegroundandthenextexcited(error)statecongurationofthemacromodelexactlymatchthecorrespondingcongurationsofthedetailedlayoutcellsfortwofulladders designs.Third,weusethecircuitlevelimplementationtovetbetweenalternatedesign choices.Weshowexamplesofthisdesignspaceexplorationprocesswiththeexampleof twoadders. 5.4.1Polarization Fig.5.4.plotsthepolarizationestimatesatthelayoutandthecircuitlevelsforvarious temperature,andfordifferentinputsforAdder-1architectureshowninFig.5.3.a(layout 95

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0 02 04 06 08 1 1234512345 Sum Cout emperature (K)Output Probability Macro-model Layout 0 02 04 06 08 11234512345 Sum Cout emperature (K)Output Probabiity Macro-model Layout 0 02 04 06 08 1 1234512345 Sum Cout emperature (K)Output Probability Macro-model Layout 0 02 04 06 08 1 1234512345 Sum Cout emperature (K)Output Probability Macro-model Layout (a) (b) (c) (d) Figure5.4.ProbabilityofcorrectoutputforsumandcarryofAdder-1basedonthelayoutlevelBayesiannetmodelandthecircuitlevelmacromodel,atdifferenttemperatures,for differentinputs(a)(0,0,0)(b)(0,0,1)(c)(0,1,0)(d)(0,1,1). A B Carry In S C ET CO C IC C ET IC ET IC Cout MAJ CO MAJ INV LINE LINE ETCB NV CB CB Sum MAJ ETCB CO ABC i n A0 B0 C0 A1 A2 B1 B2 B3 C1 C2 C3 C4 C5 C6 G1 G3 G2 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G18 G19 G20(a) (b) (c) Figure5.5.AQCAFullAddercircuit(Adder-2)(a)QCAFulladdercelllayout(b)Macromodelrepresentation(c)MacromodelBayesiannetwork.Note:Nodeelementsaregeneric. 0 02 04 06 08 1 1234512345 Sum Cout emperature (K)Output Probability Macro-model Layout 0 02 04 06 08 1 1234512345 Sum Cout emperature (K)Output Probabilty Macro-model Layout 0 02 04 06 08 1 1234512345 Sum Cout emperature (K)Output Probability Macro-model Layout 0 02 04 06 08 1 1234512345 Sum Cout emperature (K)Output Probability Macro-model Layout (a) (b) (c) (d) Figure5.6.ProbabilityofcorrectoutputforsumandcarryofAdder-2basedonthelayoutlevelBayesiannetmodelandthecircuitlevelmacromodel,atdifferenttemperatures,for differentinputs(a)(0,0,0)(b)(0,0,1)(c)(0,1,0)(d)(0,1,1). 96

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level)andFig.5.3.c(circuitlevel).Fig.5.5.(a)showssecondadderarchitecture(Adder-2), consistingofthreemajoritygatesandtwoinverters[139].Fig.5.6.plotsthepolarization estimatesatthelayoutandthecircuitlevelsforvarioustemperature,andfordifferentinputs.Weseethatthedifferenceinprobabilityofcorrectoutputnodebetweencircuitand layoutlevelmodeldesignislowforboththeadders.Wealsoseethatinbothlayoutand circuitleveldesigns,theprobabilityoftheoutputnodeisdependentontheinputvectorset. Similartrendsisalsoseenforthe2x2multipliercircuitshowninFig.5.7.(a).The multipliercircuitissomewhatlargerthanthefulladdercircuitandconsistsoftwoAND gatesandtwohalfadders.WemadeuseofahalfaddersimilartoAdder-2fulladder design,forthesimplereasonthatitoccupieslessarea.Thepolarizationoftheoutput nodesinthemultiplierlayoutisalmostsimilartothatobtainedattheoutputsofmultiplier circuitdesignedusingthemacromodelblocks.InFig.5.9.and5.10.,weshowthevariation ofoutputnodesC0,C1,C2andC3ofthemultiplierwithrespecttotemperatureforboth layoutandmacromodeldesign. 5.4.2ErrorModes Wecomputethenear-groundstatecongurationsthatresultsinerrorintheoutputcarry bit CoutoftheQCAfulladders(Adder-1andAdder-2)usingboththelayoutandcircuit levelmodels.TheseareshowninFig.5.11.and5.12.andFig5.13.and5.14.Weshow fourcases,forinputvectors(0,0,0),(1,0,0),(0,1,0)and(1,1,1).Theotherfourinputvector setswillhavesimilarresultsduetosymmetryindesign.Weuseredmarkertopointtothe componentsthatareweak(higherrorprobabilities)inboththelayoutandcircuitlevel.We caneasilyseethatthenodeswithhigherrorprobabilitiesinQCAlayoutaretheonesthat areclusteredtoformanerroneousnodeinthemacromodelcircuitdesign.Inotherwords, ifanode(amacromodelblock)inmacromodelcircuitlayoutishighlyerrorpronefora giveninputset,thensomeoralltheQCAcellsformingthatmacromodelblockarehighly 97

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pronetoerror.Thisindicatesthatweakspotinthedesigncanbeidentiedearlyinthe designprocess,atthecircuitlevelitself. 5.4.3DesignSpaceExploration Weshowthatevenatthemacromodelcircuitlevel,wehavetheabilitytoexplorethe designspacewithrespecttodifferentcriteria.Inaddition,toobviouscriteriasuchasgate count,wecanusepolarizationasadesignmetric.Theprobabilisticmacromodelallowsus veryfastestimatesofpolarizationthatcorrelateverywellwithlayoutlevelestimates.As anexampleweusethetwoaddersinFig.5.3.(a)andFig.5.5.(a).Thetwoaddersshown herehavebeendesignedusingdifferentmacromodelblocks,occupyingdifferentdesign areas. TheoutputsofAdder-1circuitisgivenby SumABCin A BCin AB CinA B Cinmm ABCin mAB Cin mA BCinCoutmABCin(5.10) where mABCinisthemajoritygatecontainingA,Band Cinasinputs.Similarly,for Adder-2circuittheoutputsaregivenby[139] Summ CoutCinmAB CinCoutmABCin(5.11) WeseethatAdder-1circuitusesvemajoritygatesandthreeinvertersforimplementationwhileAdder-2circuitusesthreemajoritygatesandtwoinverters.Hencethedesign circuitdesignofAdder-2iscertainlysuperiortoAdder-1intermsofarea.However,asit canbeseenfromthethermalstudy,inverterhasoneoftheworstpolarizationdropwith respecttotemperatureandinvertersinseriespathwillreducetheoverallpolarizationbya 98

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A0 B0 C C0 C3 C2 C1 IC Line CO ET CB INV AND MAJMacromodels A1A0B0B1 Cin1 Cin2 C0 C1 C2 C3 Figure5.7.AQCA2x2Multipliercircuit(a)QCAmultipliercelllayout(b)Macromodel representation 99

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A1 B1 C1 A0 B0 C0 G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17 G19 G22 G23 G24 G25 G26 G27 G28 G29 G30 G31 G32 G33 G35 G36 G37 G39 G40 G41 G43 G44 G45 G46 G47 G48 G49 G50 G51 G52 G53 G54 G55 G56 G57 G58 G59 G60 G61 G62 G63 G64 G65 G66 G67 G68 G69 G70 G71 G72 G73 G74 G75 G76 G77 G78 G79 G80 G81 G82 G83 G85 G86 G87 G88 G89 G90 G91 G93 G94 G95 G96 G97 G98 G99 G100 G101 G102 G103 G104 G105 G107 G108 G109 G110 G111 G112 G113 G114 G115 G116 G117 G118 G119 G121 G122 G123 G124 G125 G126 G127 G128 G129 G130Figure5.8.MacromodelBayesiannetworkofaQCA2x2Multipliercircuit.Note:Node elementsaregeneric. 0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.533.5 C3C2C1C0 Temperature (K)Output Probability Layout Macromodel 0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.533.5 C3C2C1C0 Temperature(K)Output Probability Layout Macromodel (a)(b) 0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.53 C3C2C1C0 Temperature(K)Output Probability Layout Macromodel 0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.533.5 C3C2C1C0 Temperature(K)Output Probability Layout Macromodel (c)(d) Figure5.9.Probabilityofcorrectoutputatthefouroutputnodesof2x2Multipliercircuit basedonthelayout-levelBayesiannetmodelandthecircuitlevelmacromodel,atdifferent temperatures,fordifferentinputs(a)(0,0),(0,1)(b)(0,0),(1,1)(c)(0,1),(0,1)(d)(0,1),(1,1) 100

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0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.533.5 C3C2C1C0 Temperature(K)Output Probability Layout Macromodel 0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.533.5 C3C2C1C0 Temperature(K)Output Probability Layout Macromodel (a)(b) 0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.533.5 C3C2C1C0 Temperature(K)Output Probability Layout Macromodel 0.0 0.2 0.4 0.6 0.8 1.0 11.522.533.511.522.533.511.522.533.511.522.533.5 C3C2C1C0 Temperature(K)Output Probability Layout Macromodel (c)(d) Figure5.10.Probabilityofcorrectoutputatthefouroutputnodesof2x2Multipliercircuit basedonthelayout-levelBayesiannetmodelandthecircuitlevelmacromodel,atdifferent temperatures,fordifferentinputs(a)(1,0),(0,1)(b)(1,0),(1,1)(c)(1,1),(0,1)(d)(1,1),(1,1). 101

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S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X X X X X X X s0s s1s s2s s3s s4s s5s s6s s7s s8s s9s s10s s11s s12s s13s s14s s15s s16s s17s s18s s19s s20s s21s s22s s23s s24s s25s s26s s27s s28s s29s s30s s31s s32s s33s s34s s35s s36s s37s s38s s39s s40s s41s s42s s43s s44s s45s s46s s47s s48s s49s s50s s51s X52X s53s s54s s55s s56s X57X s58s (a)(b) S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X X X X X X X X X X s0s s1s s2s s3s s4s s5s s6s s7s s8s s9s s10s s11s s12s s13s s14s s15s s16s s17s s18s s19s s20s s21s s22s s23s s24s s25s s26s s27s s28s s29s s30s s31s s32s s33s s34s s35s s36s s37s s38s s39s s40s s41s s42s s43s s44s s45s s46s s47s s48s s49s s50s s51s X52X s53s s54s s55s s56s X57X s58s (c)(d) Figure5.11.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-1Circuit anditsMacromodeldesign.Itcanbeseenthattheerroneousnodesinthelayoutare effectivelymappedinthemacromodeldesign.Inputvectorsetfor(a)and(b)is(0,0,0)and thatfor(c)and(d)is(1,0,0).Note:Nodeelementsaregeneric. 102

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S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X X X X X X X s0s s1s s2s s3s s4s s5s s6s s7s s8s s9s s10s s11s s12s s13s s14s s15s s16s s17s s18s s19s s20s s21s s22s s23s s24s s25s s26s s27s s28s s29s s30s s31s s32s s33s s34s s35s s36s s37s s38s s39s s40s s41s s42s s43s s44s s45s s46s s47s s48s s49s s50s s51s X52X s53s s54s s55s s56s X57X s58s (a)(b) S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X X X X X X X s0s s1s s2s s3s s4s s5s s6s s7s s8s s9s s10s s11s s12s s13s s14s s15s s16s s17s s18s s19s s20s s21s s22s s23s s24s s25s s26s s27s s28s s29s s30s s31s s32s s33s s34s s35s s36s s37s s38s s39s s40s s41s s42s s43s s44s s45s s46s s47s s48s s49s s50s s51s X52X s53s s54s s55s s56s X57X s58s (c)(d) Figure5.12.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-1Circuit anditsMacromodeldesign.Itcanbeseenthattheerroneousnodesinthelayoutare effectivelymappedinthemacromodeldesign.Inputvectorsetfor(a)and(b)is(0,1,0)and thatfor(c)and(d)is(1,1,0).Note:Nodeelementsaregeneric. 103

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S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X S S S S X X X X X X S S S S X X X S X X X S S S S S S S s0s s1s s2s s3s ss s5s s6s s7s s8s s9s s10s s11s s12s s13s s1s s15s s16s s17s s18s s19s s20s s21s s22s X23X s2s s25s s26s s27s s28s X29X s30s X31X X32X s33s (a) (b) S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X S S S S S S S S S S S S X S X S X S S S S X X X X X X S S S S X X X S X X X S S X X X X X s0s s1s s2s s3s ss s5s s6s s7s s8s s9s s10s s11s s12s s13s s1s s15s s16s s17s s18s s19s X20X s21s s22s X23X s2s s25s s26s s27s s28s X29X X30X X31X X32X X33X (c) (d) Figure5.13.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-2Circuit anditsMacromodeldesign.Itcanbeseenthattheerroneousnodesinthelayoutare effectivelymappedinthemacromodeldesign.Inputvectorsetfor(a)and(b)is(0,0,0)and thatfor(c)and(d)is(1,0,0).Note:Nodeelementsaregeneric. 104

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S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X S S S S X X X X X X S S S S X X X S X X X S S X X X X X s0s s1s s2s s3s ss s5s s6s s7s s8s s9s s10s s11s s12s s13s s1s s15s s16s s17s s18s s19s X20X s21s s22s X23X s2s s25s s26s s27s s28s X29X X30X X31X X32X X33X (a) (b) S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S X S S S S X X X X X X S S S S X X X S X X X S S X X X X X s0s s1s s2s s3s ss s5s s6s s7s s8s s9s s10s s11s s12s s13s s1s s15s s16s s17s s18s s19s s20s s21s s22s X23X s2s s25s s26s s27s s28s X29X X30X X31X X32X X33X (c) (d) Figure5.14.Error-pronenodesforrst-excitedstateatcarryoutputQCAAdder-2Circuit anditsMacromodeldesign.Itcanbeseenthattheerroneousnodesinthelayoutare effectivelymappedinthemacromodeldesign.Inputvectorsetfor(a)and(b)is(0,1,0)and thatfor(c)and(d)is(1,1,0).Note:Nodeelementsaregeneric. 105

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greatextent.Henceforlargercircuits,adesigncriteriamightlookatAdder-1inadifferent light. Notethatinthecontextoferrormodes,presentedearlier,wesawthatAdder-1again shows less numberoferror-pronenodesthanAdder-2(Fig.5.11.showserror-pronenodes forrst-excitedstateatcarryoutput)formostlikelyerrorsintheoutputs.Notethat,ideally thisconclusionrequiresthedetailedlayout,however,maximum-likelihoodpropagationof thecircuitlevelBayesianNetworkyieldsthesameerrormodesasthedetailedlayout.This measureindicatesthatcostofadditionerrorcorrectionrequiredforAdder-2wouldbemore thanthatofAdder-1. Lastbutnottheleast,weobservethatanoddtapshowninSection5.2.isagoodtarget foroneinverterasthepolarizationlossislessthananinverterandaneventapworksbetter thananevennumberofinverterchains.Themultiplierdesignthatweshow,utilizesthese factstoarriveatbetterdesignwithrespecttooutputpolarizationandthis,inturn,improves themultiplier'sthermalcharacteristics. 5.4.4ComputationalAdvantage Toquantifythecomputationaladvantageofacircuitlevelmacromodelwithalayout levelmodel,weconsiderthecomplexityoftheinferencebasedontheBayesiannetmodelsforeachofthem.Aswementionedearlier,inthecluster-basedinferencescheme,the BayesianNetworkisconvertedintoajunctiontreeofcliquesandtheprobabilisticinference isperformedonthejunctiontreebylocalcomputationbetweentheneighboringcliquesof thejunctiontreebylocalmessagepassing[122,24].SpacecomplexityofBayesianinferenceis On2Cmax where n isthenumberofvariables,Cmaxisthenumberofvariables inthelargestclique.Timecomplexityis Op2Cmax ,wherepisthenumberofcliquesin thejunctiontree.WetabulatethecomplexitytermsforthetwoadderdesignsinTable5.4., alongwiththecorrespondingvaluesfor n p andCmax.Wecanseethatmacromodelis 106

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Table5.4.Layoutandmacromodeltime( Tc)andspace( Ts)complexities.Pleaseseetext foranexplanation Cmax, n ,and p Adder1 Adder2 Multiplier Parameters Layout model Macromodel Layout model Macromodel Layout model Macromodel Cmax 15 8 10 5 15 5 p 215 57 96 30 436 119 n 278 64 125 34 539 130 Tcp2Cmax 7045120 14592 98304 960 14286848 3808 Tsn2Cmax 9109504 16384 128000 1088 17661952 4160 orderofmagnitudefasterespeciallyduetothereductioninCmaxwhichwouldbeimportantinsynthesizinglargernetworksofQCAcells.AnotherobservationisthatAdder2is lessexpensiveintermsofcomputationeventhoughpolarizationdropsaremoreduetothe presenceofinverters. AswecanseefromtheTable5.5.,thesimulationtimerequiredtoevaluateacircuitis ordersofmagnitudelowerthanthatinQCADesignertool.Moreover,weseethatthesimulationtimingforbayesianmacromodelsoftheaddercircuitaremuchlowerthanbayesian fulllayoutmodel.ThegraphsdepictedinFig.5.4.,Fig.5.6.,Fig.5.9.andFig.5.10.present thecruxofthiswork.Thedroopingcharacteristicofoutputnodepolarizationwithrisein temperatureisauniversallyknownfact.Whatwehaveshowninthiswork(asdepicted inthesegraphs)isthatthepolarizationoftheoutputnodeinourmacromodeldesignis showingthesamedroopingcharacteristicsandisalmostthesameasthatofthefulllayout. Wecanseethatmacromodelisorderofmagnitudefasterspeciallyduetothereduction inCmaxwhichwouldbeimportantinsynthesizinglargernetworksofQCAcells.AnotherobservationisthatAdder2islessexpensiveintermsofcomputationeventhough polarizationdropsaremoreduetothepresenceofinverters. 107

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Table5.5.Comparisonbetweensimulationtiming(inseconds)ofaFullAdderandMultipliercircuitsinQCADesigner(QD)andGenieBayesianNetwork(BN)ToolforFullLayout andMacromodelLayout SimulationTime Adder-1 Adder-2 2x2Multiplier 278cells 125cells 539cells QDCoherenceVector 566 253 966 QDBistableApprox. 5 3 15 QDNonlinearApprox. 3.5 2 8 BNFullLayoutmodel 0.240 0.030 0.801 BNMacromodelLayout 0.010 0.000 0.08 108

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CHAPTER6 EFFECTOFKINKENERGYINQCADESIGN Inchapter3weshowedhowtocalculatethegroundstatepolarizationprobabilitiesand buildagraphicalprobabilisticmodelbasedonthat.Weusedthesegraphicalprobabilistic modelstodeteminethermalerrorattheoutputatdifferenttemperatures.In[140],an efcientmethod,basedongraphicalprobabilisticmodelswaspresented,tocomputethe N-lowestenergymodesofaclockedQCAcircuit.InQCA,anerroneousstatemayresult duetothefailureoftheclockingschemetoswitchportionsofthecircuittoitsnewground statewithchangeininput.Thiserrorstateofasinglecellinturncausestheerrorinthe neighboringcellsresultinginanerroneousoutput.Duetothequantummechanicalnature ofoperationofaQCAdevice,temperatureplaysanimportantroleindeterminingthe groundstatepolarizationofeachcell.PowerdissipationinaQCAcircuitprimarilyresults duethetheapplicationofanon-adiabaticclockingscheme.Wehavealsoseeninchapter4, howclockenergyaffectstheoverallpowerdissipationinaQCAcircuit. InthischapterweperformstudiestodeterminetheerrorandpowertradeoffinaQCA circuitdesignbystudyingtheeffectofkinkenergyontheoutputerrorandpowerdissipationinaQCAcircuit.WeusethreedifferentsizesofQCAcellsandgridspacingtostudy thepolarizationandpowerdissipationforbasicQCAcircuitsusingthesecells. WerstsimulateanumberofbasicQCAcircuitssuchasmajoritygateandinverterto studythepolarizationerrorattheoutputforeachinputvectorset.Wealsodeterminethe powerdissipationinthesecircuitsfordifferentkinkenergies.Allotherparameterssuch astemperatureandclockenergyarekeptconstant.Weshowhowthisstudycanbeused 109

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P = +1 P = -1 P = +1 P = +1 minus Energy ofEnergy of Kink Energy (EK) = Figure6.1.KinkenergybetweentwoneighboringQCAcells bycomparingtwosinglebitadderdesigns.Thestudywillbeofgreatusetodesigners andfabricationscientiststochoosethemostoptimumsizeandspacingofQCAcellsto fabricateQCAlogicdesigns. 6.1KinkEnergy TwoelectronsinaasimplefourdotQCAcelloccupydiagonallyoppositedotsinthe cellduetomutualrepulsionoflikecharges.AQCAcellcanbeinanyoneofthetwo possiblestatesdependingonthepolarizationofchargesinthecell.Thetwopolarized statesarerepresenteda sP=+1andP=-1. Electrostaticinteractionbetweenchargesin twoQCACellsisgivenas: Em1 4 or 4i1 4j1qi mqj k ri mrj k(6.1) Thisinteractionisdeterminesthekinkenergybetweentwocells. EkinkEopppolarizationEsamepolarization(6.2) Kinkenergy(Fig6.1.)istheenergycostoftwoneighboringQCAcellshavingopposite polarization.KinkenergybetweentwocellsdependsonthedimensionoftheQCAcellas wellasthespacingbetweenadjacentcells.Itdoesnotdependonthetemperature. 110

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Table6.1.DifferenttypesofQCAcellsandgridspacingusedinthisstudy QCAcell Size GridSpacing AssociatedKinkEnergy Cell-1 10nm 5nm Ek 14 Ek Cell-2 20nm 10nm Ek 22 Ek Cell-3 40nm 20nm Ek 3Ek 6.2Results Inthissectionwepresenttheresultsobtainedfromthestudyofvariationofkinkenergy onerrorandpowerdissipatedinthecircuits.Weobtainedtheseresultsbysimulatingeach ofthecircuitsataconstanttemperatureof2K.Thethreedifferenttypesofcellsizesused inthisstudyareelaboratedinTable6.1. Here Ek 1isthemaximumkinkenergyforthecelllayoutwithsmallestcelldimensions (andgridspacing).Similarly, Ek 3isthemaximumkinkenergyfortheQCAlayoutwith largestcelldimensions(andgridspacing).Aswecanseefromthetable, Ek 12 Ek 24 Ek 3. 6.2.1NodePolarizationError Wequantifytheerrorinacircuitasameasureofitsoutputnodepolarization.In chapter3,usingtemperatureasavariableandkeepingthekinkenergyconstantwehave shownhowtheoutputnodepolarizationdropssteadilywithriseintemperatureleadingto moreerroneousoutputs.Thiseffectbecomesmoreandmoresignicantwiththeincrease inthenumberofcellsinadesign.Hencetwodifferentdesignsrepresentingsimilarlogic buthavingunequalnumberofcellswillhavedifferentpolarizationsattheoutputnodes. Similarly,inthisstudy,byvaryingthekinkenergyofthecircuitandkeepingthetemperatureconstantweseethatthegain(drop)inoutputnodepolarizationofacircuitis directlyproportionaltotheincrease(decrease)inmaximumkinkenergy( Ek)ofthecircuit. 111

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Table6.2.OutputnodepolarizationofasimplemajoritygatefordifferentKinkEnergies MaximumKinkEnergy( Ek) Input Ek 3=0.75meV Ek 2=1.5meV Ek 1=3.0meV 000 0.9278 0.9999 1.0000 001 0.9880 0.9999 1.0000 010 0.9880 0.9999 1.0000 011 0.9075 0.9999 1.0000 100 0.9075 0.9999 1.0000 101 0.9880 0.9999 1.0000 110 0.9880 0.9999 1.0000 111 0.9278 0.9999 1.0000 Table6.3.OutputnodepolarizationofaQCAInverterfordifferentKinkEnergies MaximumKinkEnergy( Ek) Input Ek 3=0.75meV Ek 2=1.5meV Ek 1=3.0meV 0 0.9750 0.9998 1.0000 1 0.9843 0.9998 1.0000 Hereincreasein EkreferstodecreaseinQCAcellsizeandgridspacing.Similareffectwas seenfordifferentvaluesoftemperature. Asanexample,refertotheoutputnodepolarizationofasimplemajoritygateshown inTable6.2.Aswehaveshownearlier,werstformaBayesiannetworkoftheQCA circuitanduseagraphicalsimulatortoobtainthepolarizationprobabilityforeachQCA cell(representedasanode)inthedesign.Wecanseethatthepolarizationprobabilityatthe outputoftheBayesiannetworkriseswiththeincreaseinkinkenergy.Hence,wecaninfer thatdesignswithlowervalueofmaximumkinkenergyaremorepronetoerrorandthis errorismoresignicantwhenthenumberofcellsinadesignincreaseorthetemperature israised.Table6.3.showstheoutputnodepolarizationprobabilityofaQCAinverter.We wouldliiketomakeaclaricationontheterm error usedinthisstudy.Inchapter5,we usedtheterm error tosignifytherstexcitedstateofaQCAcell.Here error referstothe dropinpolarizationprobabilityattheoutputnodeofaQCAdesign. 112

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Table6.4.PowerdissipationinQCAmajoritygatefordifferentKinkEnergies MaximumKinkEnergy( Ek) Ek 3=3.0meV Ek 2=1.5meV Ek 1=0.75meV Max Ediss(inmeV) 0.0294 0.0147 0.0051 Avg Ediss(inmeV) 0.0120 0.0060 0.0018 Min Ediss(inmeV) 0.0015 0.0008 0.0002 Avg Eleak(inmeV) 0.0018 0.0009 0.0003 Avg Esw(inmeV) 0.0102 0.0051 0.0015 Table6.5.PowerdissipationinQCAInverterfordifferentKinkEnergies MaximumKinkEnergy( Ek) Ek 3=3.0meV Ek 2=1.5meV Ek 1=0.75meV Max Ediss(inmeV) 0.0785 0.0392 0.0196 Avg Ediss(inmeV) 0.0425 0.0213 0.0106 Min Ediss(inmeV) 0.0066 0.0033 0.0016 Avg Eleak(inmeV) 0.0066 0.0033 0.0016 Avg Esw(inmeV) 0.0359 0.0180 0.0090 6.2.2SwitchingPower Weperformedanexhaustivestudyontheeffectofvaryingkinkenergyonthepower dissipatedduringaswitchingeventinaQCAcircuit.Whilewehavepresentedtheresult ofpowerdissipatedinaQCAcircuitwithvaryingclockenergyinchapter4,inthischapter weintendtoanalyzetheeffectofthesizeofaQCAcellandthekinkenergyassociatedwith itonthepowerdissipatedinthecircuit.AscanbeseenfrominTable6.4.,increasingthe valueofkinkenergyinacircuitleadstoanincreaseintheoverallaveragepowerdissipated inthecircuit.Table6.5.showstheenergydissipationinaQCAinverterfordifferentvalues ofkinkenergy. SomeveryinterestingobservationswereobtainedfromthisstudyofeffectofkinkenergyontheoverallpowerdissipationandprobabilityoferrorinQCAcircuitdesign.We haveseenthatwhileitisdesirabletodesigncircuitswithlowererrorprobabilities(by increasingthekinkenergybetweencells),itinadvertentlyincreasesthepowerdissipated inthecircuit.Thiseffectismorepronouncedinlargercircuitssuchassinglebitadders. 113

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Table6.6.OutputnodepolarizationatSUMoutputnodeofAdder-1andAdder-2QCA designs Ek 3=1.09meV Ek 2=2.18meV Ek 1=4.36meV Input Adder-1 Adder-2 Adder-1 Adder-2 Adder-1 Adder-2 000 0.9110 0.8095 0.9998 0.9964 1.0000 1.0000 001 0.7311 0.8058 0.9935 0.9965 1.0000 1.0000 010 0.7440 0.6833 0.9944 0.9667 1.0000 0.9991 011 0.7090 0.6312 0.9931 0.9569 1.0000 0.9989 100 0.7090 0.6312 0.9931 0.9569 1.0000 0.9989 101 0.7440 0.6833 0.9944 0.9667 1.0000 0.9991 110 0.7311 0.8058 0.9935 0.9965 1.0000 1.0000 111 0.9110 0.8095 0.9998 0.9964 1.0000 1.0000 Table6.7.Non-AdiabaticEnergydissipationinAdder-1andAdder-2QCAdesigns Ek 1=4.36meV Ek 2=2.18meV Ek 3=1.09meV Adder-1 Adder-2 Adder-1 Adder-2 Adder-1 Adder-2 Max Ediss(inmeV) 3.0939 1.3556 1.5404 0.6778 0.8127 0.3389 Avg Ediss(inmeV) 1.7398 0.7650 0.8665 0.3825 0.4556 0.1912 Min Ediss(inmeV) 0.4083 0.1949 0.2038 0.0974 0.1041 0.0487 Avg Eleak(inmeV) 0.4089 0.1956 0.2041 0.0978 0.1043 0.0489 Avg Esw(inmeV) 1.3309 0.5693 0.6624 0.2847 0.3513 0.1423 Table6.6.comparestheresultsofoutputpolarizationatSUMnodeoftwoaddersfordifferentkinkenergies.AswecanseethateventhoughAdder-2hasamoreefcientdesign anduseslessnumberofcells,thepolarizationatitsoutputisworsethanthatofAdder-1 fordifferentinputvectorsets.Similarly,Table6.7.comparestheenergydissipationinthe twoadderdesigns.PowerdissipationinAdder-2isgreaterthanthatofAdder-1sinceit hassignicantlymorenumberofcells.However,wedoseethattheenergydissipationin aQCAcircuitisalmostlinearlyproportationaltothemaximumkinkenergyofthecircuit. Aswecanseefromtheresultstheoutputnodepolarizationerror improves whilepower dissipation deteriorates whenthekinkenergyisincreased.Hencedesignersneedtochoose thesizeofQCAcellsbasedoncircuitrequirementstooptimizepoweranderror.Thisis differentfromthermalstudiesperformedonQCAcircuitswhichresultedinincreasein outputerrorandpowerdissipationathighertemperatures.Fromtheresultsobtainedfor 114

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polarizationandpowerdissipationinsmallandbigQCAcircuits,wehaveclearlyseenthat kinkenergyisanimportantfactortodesignmostoptimumcircuitsatagiventemperature andclockenergy.Hencedesignersneedtomakecarefuluseofkinkenergyasparameter fordesigningQCAcircuitstooptimizeerrorandpower. 115

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CHAPTER7 CONCLUSIONANDFUTUREWORKS Inthisdissertation,weproposedanefcientBayesianNetworkbasedprobabilistic schemeforQCAcircuitdesignthatcanestimatecellpolarizations,groundstateprobability, andlowest-energyerrorstateprobability,withouttheneedforcomputationallyexpensive quantum-mechanicalcomputations.Bayesianmodelingcapturestheinherentcausalnature ofQCAdevicesandoffersafastapproximationbasedmethodtoestimateerror,powerand reliabilityinQCAdesign. Someofthelimitationsandscopeofthisworkarelistedbelow:Inhierarchicalmacromodelingitisassumedthatthedesignerhassomeideathe layoutleveldesignofthesamecircuitInthepowermodelweavenottakenintoaccountthepowerdissipationintheclock circuititself.Inerror-powertradeoffstudybyvariationofmaximumkinkenergywehaveassumed thatthismodelwillaccuratelycapturealltheeffectsevenatasmallerscale.Finally,thescopeofthisworkislimitedtoa4-dotelectronicQCAimplementation. Themodelwillbedifferentforothertypesofimplementationssuchasmolecularand magneticQCAs. 116

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Forthepurposeofthisworkweveriedthegroundtruthusingthecoherencevector basedmethodinQCADesignersimulator.Asummaryofimportantcontributionsofthis dissertationissummarizedbelowTothebestofourknowledge,ournon-adiabaticpowerdissipationmodelingscheme istherstworkinaQCAdesignwhichprovidesarealisticestimateofworstcase powerdissipatedduringaswitchingevent.ThismodelcanbeusedtoquicklycomputetheworstcasepowerdissipatedateachindividualcellinaQCAlayoutforany inputvectortransition.Thisenablesustolocatecellsinalayout,earlyoninthedesignprocess,thatarecriticalintermsofpowerdissipationandalsoidentifytheinput vectortransitionsthatresultinlargepowerdissipations.Wehavealsodemonstrated theeffectofclockenergyonoverallpowerdissipatedinaQCAdesign.Tothebestofourknowledge,themacromodeldesignschemeistherstworkto modelQCAdesignsatahierarchicalcircuitlevel.Ourresultsdemonstratedthat boththepolarizationandtheerrormodeestimatesatthecircuitlevelmatchthose atthelayoutlevel.Thedevelopedmodelsinthisworkcanbeusedtoselectively identifyweakcomponentsinadesignearlyinthedesignprocess.Itwouldthen bepossibletoreinforcethoseweakspotsinthedesignusingreliabilityenhancing strategies.StudytheeffectofKinkenergyoncircuitdesign.Weperformederror-powertradeoff studiestobyvaryingthekinkenergyofaQCAcircuit.Wefoundthattheoutputnode polarizationerroraswellasthepowerdissipationdecreasewhenthekinkenergyis increased.Thisisdifferentfromthermalstudieswhichresultedinincreaseinoutput errorandpowerdissipationathighertemperatures. 117

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OneinterestingfuturestudyusingthismodelcouldbetoseetheeffectofhighlyerrorpronecellsonthethermalhotspotsinaQCAdesign.Thatistoseeifthehighlyerror-prone cellsarethesamecellsthatcausethemaximumenergydissipation. AnotherpossiblefuturedirectionofthisworkinvolvestheextensionoftheBNmodel tohandlesequentiallogic.ThisispossibleusinganextensioncalledthedynamicBayesian networks,whichhavebeenusedtomodelswitchinginCMOSsequentiallogic[141]. Thereisalsoavastscopetoconductprobabilisticmodelingtoestimateerror,power andotherdesignrelatedissuesonotheremergingnanotechdevicessuchasmagneticand molecularQCA,spintronics,nano-CMOSandphotonicdevices. 118

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ABOUTTHEAUTHOR SaketcompletedhisB.TechinElectricalandElectronicsEngineeringfromRegional EngineeringCollege,Trichy,India(nowknownasNationalInstituteofTechnology,Trichy) in2003.HejoinedinadirectPhDprograminElectricalEngineeringatUniversityofSouth Florida,Tampa.Hehasco-authoredintwojournals(IEEETransactionsinComputersand IEEETransactionsonEducation).Hisworkinnon-adiabaticpowerdissipationmodelfor Quantum-dotCellularAutomataisunderrevise-resubmitinIEEETransactionsinNanotechnology.Hehasco-authoredvepeerreviewedconferencepublicationsandoneother publication.Twooftheconferencepublicationswerepublishedineducationalconferences.HehasalsoreviewedpapersforanumberofIEEEconferencessuchasISCASand VLSI-Design.HewasnominatedfortheProvost'sAwardforOutstandingTeachingbya GraduateTeachingAssistantin2007.