Signal Energy in Quantum-Dot Cellular Automata Bit Packets

Delivered by Ingenta toCraig Lent

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Copyright copy 2011 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol 8 972ndash982 2011

Signal Energy in Quantum-Dot CellularAutomata Bit Packets

Enrique Pacis Blair1 Mo Liu2 and Craig S Lent2lowast1United States Naval Academy

2University of Notre Dame

Quantum-dot cellular automata is a novel paradigm for computing at the nanoscale Cells are thebasic computing element in quantum-dot cellular automata and function as structured charge con-tainers rather than as current switches Computing with quantum-dot cellular automata is enabledby quantum-mechanical tunneling and Coulomb interactions The use of molecules as cells torealize quantum-dot cellular automata may make possible nanometer-scale devices and ultra-highdevice densities without excessive heat dissipation Molecular quantum-dot cellular automata can beclocked using an external electric field A time-varying clock can be used to drive data flow throughlayouts of cells Together the clock and the device layout define a computational architecture wheredata flows through the circuitry in the form of bit packets Here we analyze the energetics of QCAbit packets We find a heuristic model based on cell-cell interactions works well Bit packet energiesin general scale with the packet length It may however be possible to design a cell geometry sothat the energy is packet-length independent Fan-out and fan-in can be understood as investingenergy from the clock in the signal and then returning the energy back to the clock

Keywords Molecular Quantum-Dot Cellular Automata QCA Bit Packet Cost Fan-OutMolecular Electronics Recoverable Energy

1 INTRODUCTION

11 Quantum-Dot Cellular Automata

The quantum-dot cellular automata (QCA) paradigm is anovel approach to computing at the nanoscale12 which isenabled by quantum-mechanical tunneling and Coulombinteractions but does not rely on the flow of current3 Bitsare encoded in the charge configuration of cells composedof quantum dots QCA devices can be clocked using anexternal electric field4 A time-varying clock can be usedto drive data flow through arrays of cells Together theclock and the device layout define the circuit and compu-tational architecture in the QCA paradigm5

Linear arrays of QCA cells form binary wires throughwhich bit packets move not ballistically but under clockedcontrol Computation occurs when bits collide in cellsarranged to form logic gates Here our focus is on the ener-getics of QCA bit packets themselves We calculate theenergy required to form bit packets using two methods(a) a heuristic approach based on the pair-wise interac-tion of cells and characterized by several basic interactionenergies and (b) a direct Coulomb calculation that sums

lowastAuthor to whom correspondence should be addressed

the interactions between charges in the system The agree-ment between these two approaches justifies the use of theheuristic model In general bit packet energy scales with thelength of the packet but we find that cells can be designedsuch that the signal energy is constant with respect to thepacket length The signal energy cost is paid just at theldquosurfacesrdquo of the bit packet We examine the energeticsof fan-out and fan-in We find that again the notion of apacket energy is useful the energetic cost of replicatinga bit through fan-out is just the cost of creating new bitpackets That energy is returned to the clock in a fan-inIt is important to note that QCA is not quantum comput-

ing Though quantum mechanics provides the all-importantldquogreaserdquo of tunneling that makes switching possible infor-mation is encoded in classical degrees of freedommdashthequadrupole moment of the cell charge configurationmdashnotin the coefficients of a quantum mechanical superpositionstate This makes computation more robust though limitedto conventional bits rather than q-bits QCA is aimed atgeneral-purpose computing

12 QCA Implementations

Functioning QCA cells and assemblies of cells form-ing more complex circuits have been synthesized by the

972 J Comput Theor Nanosci 2011 Vol 8 No 6 1546-195520118972011 doi101166jctn20111777


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Blair et al Signal Energy in Quantum-Dot Cellular Automata Bit Packets

Notre Dame group6ndash24 In these experiments aluminumislands act as the dots coupled by aluminum oxide tun-nel junctions The islands and junctions are formed bye-beam lithography and two angled metal evaporationswith an intervening oxidation step Overlaps between thetwo metal layers create the tunnel junctions Because theshadow evaporation technique produces capacitances thatare still relatively large these experiments are performedat cryogenic temperatures (lt1 K)Using these metal-dot QCA cells and number of impor-

tant circuit elements have been demonstrated function-ing individual cells6 logic gates14 clocked QCA cells1623

inverters22 memories18 and shift registers20 In additionpower gain greater than 3 has been measured25 Powergain is important so that small energy losses due toinevitable inelastic processes can be restored In QCA theenergy lost to dissipative processes is restored from theclock Fan-out of QCA signals has also been demonstratedexperimentally26

In addition to the metal-dot QCA work at NotreDame other groups have succeeded in building QCAcells in semiconductors Mitic et al reported fabricationof QCA cells in silicon using dots formed by few-atomimplantation of phosphorus donors27 Perez-Martinez et alcreated QCA cells using gate-defined dots in a GaAs two-dimensional electron gas28 As with the metal dot sys-tems lithographically defined dots whether in silicon orGaAs are relatively large so that the Coulomb energiesinvolved are small and operating QCA cells can only func-tion at cryogenic temperatures QCA can also be imple-mented in magnetic systems comprised of small permalloyislands2930

The Wolkow group at the University of Alberta recentlyfabricated a QCA cell using single-atom dangling bondson a H-passivated Si surface as the quantum dots31 Indi-vidual hydrogen atoms were removed from the surface toexpose the Si dangling bonds (DBs) using STM lithog-raphy Coulombic interactions between nearby DB dotswas apparent in the brightness of the STM images Byelectrostatically perturbing a symmetric four-dot cell thepolarization of the QCA cell was apparent These roomtemperature results confirm the viability of molecular-scaleQCA Work is underway to construct room-temperatureQCA cells from molecules designed for the purpose32ndash37

2 QCA CELLS AND DEVICES

21 Dots Cells and Logic Gates

The elementary computing device in QCA circuitry is theQCA cell38 a charge-containing structure with multiplesites quantum dots which localize mobile charges Deviceswitching is enabled by quantum-mechanical tunnelingA cell with two mobile charges and four dots arranged in asquare has two preferred states with electrons in antipodal

ldquo1rdquoldquo0rdquo

Fig 1 The two states of a four-dot QCA cell with two mobile chargesand four dots A quantum dot is represented by a white disc and the reddisc within a white disc represents a mobile charge occupying a dot

dots (see Fig 1) These states are degenerate in energy andcan be used to represent binary ldquo0rdquo and ldquo1rdquo respectivelyThe basis of QCA circuitry is intercellular interaction

via Coulomb repulsion The interaction between cells liftsthe degeneracy between the 0 and 1 states and determinesthe state of the cell Broadside coupling causes neighbor-ing cells to align (Fig 2) This is the basis for the binarywire (Fig 3)39 Signal inversion can be achieved via diag-onal coupling (Fig 3) The majority gate is a three-inputgate having a single output which is determined by thebit dominating the three inputs (Fig 3) One of the threeinputs can be used as a control input to make the majoritygate function as a programmable AND-OR gate This isseen in the truth table for the majority gate (Table I) TheQCA paradigm is heirarchical and QCA-based adders andeven a Simple-12 ALU processor have been designed338

22 Clocked QCA Cells

Clocking QCA cells provides power gain for the restora-tion of weakened signals minimizes power dissipation2540

and facilitates large-scale circuit operation41 QCA cellscan be clocked if two additional dots called ldquonull dotsrdquoare added and the voltage on the null dots can be con-trolled The null dots add an additional state called theldquonull staterdquo which does not convey information (theinformation-bearing states are now called ldquoactive statesrdquo)as shown in Figure 4 The voltage of the null dots can beused to drive the cell to an active state or force the cell intothe null state A cell in the null state does not lift the degen-eracy of the active states in a neighboring cell Clockeddevices such as shift registers have been demonstrated42

The state of a six-dot cell can be characterized by twoquantities polarization P and activation A The quanti-ties P and A originate in a three-state quantum-mechanical

Fig 2 Broadside coupling causes cells to align

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Signal Energy in Quantum-Dot Cellular Automata Bit Packets Blair et al

Fig 3 Basic QCA devices binary wire (top) inverter (middle) major-ity gate (bottom) The majority gate has three inputs (A B and C) Theinputs ldquovoterdquo at the device cell (in the red dashed box) and the output(D) is copied from the device cell Here two input zeros dominate asingle one and the output is zero

description of a six-dot QCA cell as discussed in Appen-dices 7 and 71 P and A respectively are functions ofthe expectation values of 7 and 8 two of the eight gen-erators of SU(3) sometimes called the Gell-Mann matri-ces (see Appendix 71) used in calculating the quantumdynamics of a QCA cell coupled to a dissipative heatbath43 P and A are defined by

P = minus7 (1)

A = 2minusradic384

(2)

Table I The truth table for a MAJORITY gate is partitioned to high-light the ability of MAJORITY gates to function as programmable two-input AND-OR gates Let A be the control input If A= 0 then D= BCif A= 1 then D = B+C

A B C D =MABC

0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1

Fig 4 Three states of a six-dot cell

The polarization and activity can also can be expressedmore directly as functions of the charge on the dots of aQCA cell With dots numbered as in Figure 5 P and Aare

P = q1+q4minus q3+q6Q (3)

A= 1minus q2+q5Q (4)

where qi is the charge on the ith dot and Q=sumqi These

expressions are valid regardless of the sign of the mobilecharge A ranges from 0 to 1 P ranges from minus1 to 1 andmust satisfy P le A A fully active cell has A= 1 A cellin the null state has PA = 00 A fully active com-pletely polarized cell has PA = 11 (correspondingto the ldquo1rdquo bit) or PA = minus11 (corresponding to theldquo0rdquo bit)

23 Clocking Molecular QCA Cells

Molecular QCA allow room-temperature operation Fixedcharge in a molecular QCA cell (ligand charges) can bal-ance the mobile charge to give the molecule a net zerocharge Numerous possibilities exist for the configurationand placement of neutralizing charge and add to the designspace for QCA cells364445

Clocked molecular QCA cells can be designed withcoplanar active dots and null dots which lie outside of thisplane Such molecules can be clocked without individualelectrical connections to each cell but rather using the ver-tical component of an electric field4 as shown in Figure 6Such an electric field can be created using an array ofwires buried in the substrate below the device plane onwhich the QCA devices lie A multi-phase clock signalcan be applied to the conductors yielding regions of thedevice plane where cells are active and other regions arenull as depicted in Figure 7 Moreover the time-varyingnature of the clock will cause ldquoactive domainsrdquo to flowthrough the QCA circuitry in a controlled manner Asactive domains move throughout the circuitry they carry

Fig 5 A six-dot cell with dots numbered for reference

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Fig 6 A six-dot cell can be clocked using a conductor buried belowthe substrate Here a cellrsquos six dots are represented by blue spheresmobile chargesmdashin this case two electronsmdashare represented by red cir-cles When the conductor is negatively charged (left conductor coloredred) it creates an electric field that drives the mobile charge upward andthe cell takes an active state When positively charged (right conductorshown in green) the conductorrsquos electric field draws the electrons downto the null dots and the cell takes the null state

Fig 7 Charged clocking wires activate some molecular QCA cells onthe substrate while driving other cells to the null state thus formingactive and null domains

Fig 8 A moving active domain pushes a bit packet (in this case a ldquo1rdquo)rightward through a binary wire The color of the background representsthe vertical component of the electrical field The white region is themoving active domain and the black region is the null domain Threesnapshots in time are shown arranged early to late from top to bottom

signals in the form of bit packets as in Figure 8 Nulldomains are regions where cells are not active and thefrontier between the null domains and active domains is atransition region in which computation occurs5

3 THE ENERGETICS OF QCA BIT PACKETS

31 Interactions Between Neighboring Cells

An expression for the cost of a bit packet can be devel-oped by accounting for changes in the basic intercellularinteractions present The states of interest in an isolatedcell are illustrated in Figure 9 The active states PA=plusmn11 are degenerate and the energy Ec0 of the null state[PA= 00] is driven by the clock biasFor a cell with a single neighbor in the null state

(depicted in Fig 10) the active states remain degener-ate although they both may be raised by Ehn1 the energyof broadside interaction between a polarized cell and anull cell The null state energy will be raised by Ehnnthe energy of electrostatic broadside interaction betweentwo null cells In Figure 10 Ehn1 and Ehnn arbitrarily aredepicted as having negative and positive values respec-tively but this need not be the caseWhen a cell has broadside coupling with a single neigh-

bor in a fully polarized state as in Figure 11 the degen-eracy in the cellrsquos active-state energy levels is lifted Thecellrsquos preferred (ground) state matches its neighborrsquos stateThe preferred statersquos energy shifts by an amount Ehthe horizontal broadside coupling energy between alignedactive cells The non-preferred active state shifts by anamount Eh+Ek where Ek is identified as the kink energyEk is the cost of having two adjacent cells in opposingstates relative to the energy of the same cells in an alignedconfiguration The cellrsquos null state is raised by Ehn1 (in thiscase Ehn1 lt 0 for consistency with Fig 10)

Fig 9 Energy levels for an isolated QCA cell In this figure and sub-sequent figures the null dots are drawn smaller than the active dots toindicate that they are on a plane below that of the active dots



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Fig 10 Energy levels for a QCA cell having a single neighbor in thenull state The neighboring cellrsquos null state does not lift the degeneracyof the active states but it raises the null state of the active cell (identifiedwith a red dashed box) by the horizontal null-null energy Ehnn (here Ehnn

arbitrarily is shown as positive and Ehn1 is shown as negative)

32 The Energy of a Bit Packet of Length N

321 The Reference State

The energy cost of a bit packet will be referenced to theenergy Eref of a wire with all cells in the null state Thusit is helpful to understand the interactions present in sucha wire Consider only interactions between adjacent cells(nearest-neighbor interactions) in a L-cell wire Each cellin the null state with no additional clock contributes minusEc0

to the system cost and each of the Lminus1 nearest-neighbornull-null interactions raise the system energy by Ehnn sothat the reference energy of the wire is

Eref =minusLEc0+ Lminus1Ehnn (5)

Fig 11 Energy levels for a QCA cell having a single polarized neigh-bor The polarized neighbor lifts the degeneracy of the cellrsquos active statesand the null-state energy is shifted up by the energy Ehn1 of interac-tion between a null cell and a polarized cell (in this case Ehn1 lt 0 forconsistency with Fig 10)

322 Heuristic Model

We first construct a model of the energy of a bit packed bysumming the various relevant interaction The advantageof this heuristic approach is itrsquos generality It is appropriatefor many types of QCA implementations including metal-dot molecular and even magnetic because the model isderived from cell-cell interaction energies Since our focushere is on clocked molecular QCA we validate the modelby direct summation of the Coulomb energiesConsider activating a bit packet N cells long anywhere

along the wire of length L cells except at the edges Toactivate the first cell in the array costs Ec0 however twohorizontal null-null interactions each are replaced by aninteraction between a null cell and a polarized cell ofenergy Ehn1 The total cost E of activating the first cell inthe bit packet is

E = Ec0minus2Ehnn+2Ehn1 (6)

Each of the N minus1 subsequent cells activated in the N -cellpacket costs an additional Ec0 saves another Ehnn andintroduces Eh the interaction energy between two alignedfully active cells One of the previously-introduced inter-actions having an energy of Ehn1 is removed but is reestab-lished at the new end of the bit packet The total cost Eof the N -cell bit packet (N ge 1) is

E = NEc0minus N +1Ehnn+ N minus1Eh+2Ehn1 (7)

Ehnn Enh1 Eh and various other interactions with theircorresponding energy value are illustrated and catalogedin Figure 12

323 Direct Coulomb Model

Figure 13 shows that packet costs predicted using Eq (7)match values calculated by considering all the pairwiseCoulombic energies in the system Calculated packet costswere determined by summing the energy input from theclock and the electrostatic potential developed by nearest-neighbor and next-nearest-neighbor intercellular interac-tions as follows

Ecalculated =msumj=1

qjnullVj clock +msumj=1

sjsumi=1

qj iVj ineighbors (8)

where m is the number of cells in the system qjnull is thecharge on the null dots of the jth cell Vj clock is the elec-trostatic potential at the null dots of the jth cell due to theclock Vj clock is the sum of two components a bias Vc0 andan additional variable voltage value Vc used to activate ordeactivate cells For calculating intercellular interactionsthe jth cell is represented by sj point charges in space ofwhich qj i is the ith qj i is either a mobile charge occu-pying one of the six quantum dots of the jth cell or aneutralizing charge at a fixed location within the jth cell



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Fig 12 Basic nearest-neighbor broadside and next-nearest-neighbordiagonal QCA interactions and associated energy identifiers are catalogedand illustrated For each energy given degenerate states are not shownThe subscripts h v and d indicate horizontal broadside coupling verti-cal broadside coupling and diagonal coupling respectively The numberof ldquonrdquos occurring in a subscript indicate the number of cells in the inter-action that are in the null state Numbers and stars appearing in thesubscript differentiate between similar interactions

Vj ineighbors is the electrostatic potential at the location ofeach charge qj i of the jth cell due the charges compris-ing cells which are neighbors to the jth cell To avoiddouble-counting neighbor interactions all cells are givena numeric index and only those nearest or next-nearestneighbors having an index k such that k gt j contribute toVj ineighbors Vj ineighbors is calculated

Vj ineighbors =1

40

msumk=j+1

sksump=1

qkp

rip(9)

1 2 3 4 50

2

4

6

8

10

Packet length N (Cells)

Pack

et c

ost (

EE

k)

Packet cost versus packet length

Predicted costCalculated cost

Fig 13 The cost of a bit packet of length N predicted using (7) agreeswith values calculated from the direct Coulomb model Energies are givenin multiples of the kink energy Ek

where qkp is the pth point chargemdashfixed or mobilemdashofthe kth cell a total of sk charges comprise the kth cellrip is the distance between qj i and qkp the entire arrayof cells in consideration has a total of m cells and 0 isthe permittivity of free spaceFigure 14 illustrates the cell geometry The origin is

taken to be a point on the substrate at the center of thecell The active dots form an a-by-a square centered abovethe origin and they lie in a plane that is zact above thenull dots The null dots are assumed to be on the liney = 0 at a height of z = 0 and each lies a distance xnullfrom the origin For the predicted and calculated packetcosts reported in Figure 13 cells having a xnull zact =07035035 nm were used and one positive (neutral-izing) elementary charge is located at each null dot Cellswere spaced such that adjacent cells with broadside cou-pling have a center-to-center spacing of 2a The clock biasVc0 was set such that the energy Ec0 of an isolated cellrsquosnull state is Ec0 =minusEk

33 Design of Constant-Energy Bit Packets

Equation (7) may be written

E = NEc0minusEhnn+Ehminus Ehnn+Ehminus2Ehn1 (10)

The design space for QCA cells can be used to achieveparticular behaviors We can choose Ec0 and cell geom-etry so that Eh = minusEc0 +Ehnn This removes the length-dependence from the cost E of a bit packet reducingthe cost to a constant E = minusEhnn + Eh minus 2Ehn1 Thisconstant minusEhnn +Eh minus 2Ehn1 may be understood as thesurface area cost of having the interface at either endof the bit packet each costing minusEhnn +Eh minus 2Ehn12A specific cell design that exhibits this length indepen-dence has a xnull zact= 07017500875 nm (reason-able molecular dimensions) Its length-independent packetcost is seen in Figure 15 For this design one positiveelementary half-charge is fixed at each active dot crudely

Xnull

z

aa

X

y

x

Zact

Fig 14 The geometry of a QCA cell The origin indicated by a greenldquoxrdquo is taken to be in the plane of the lower dots at the cell center



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1 2 3 4 5minus05

0

05

1

15

2


Pack

et C

ost (

EE

k)Packet cost versus packet length


Fig 15 QCA cells can be designed to have an energy cost independentof the length of the bit-packet The energy cost is shown as predicted bythe heuristic model and calculated by the direct Coulomb model

approximating a possible neutral molecule A center-to-center spacing of 14 nm was used for adjacent cells hav-ing horizontal coupling The clock bias Vc0 was set suchthat Ec0 =minus0673Ek

4 THE ENERGETICS OF FAN-OUT ANDFAN-IN

QCA fan-out is the branching of a single bit packet intotwo identical packets as it is pushed along a wire whichbranches into two wires This is accomplished using a fan-out circuit depicted in Figure 16 The additional cost offan-out is the cost of the second packet which is equal tothe cost of the first packet (Fig 17)A bit packet can be fanned out into two packets and

then fanned in back to one using a circuit as shown inFigure 18 Work equal to the cost of replicating the orig-inal bit packet is done on the clock when fan-in occurs(Fig 19)

Fig 16 A fan-out circuit receives a right-ward propagating bit packetand fans it out into two identical packets

Energy input from clock

Epacket

Eclock 0

ColumnA

Fig 17 The energy input from the clock to the cells identified withldquoColumn Ardquo (Fig 16) is summed over one clock period The additionalenergy required to fan a bit packet out into two identical bit packets isthe cost of the original packet

Fig 18 A fanout circuit receives a right-ward propagating bit packetand fans it out into two identical packets and then the two packets arecombined into one

The cost of fanning a packet out into multiple copiesis the cost of one packet times the number of copies pro-duced A three-level fanout circuit (depicted in Fig 20)produces seven bit-packet copies and clock input over the


Epacket

Eclock 0

1

ndash1

ColumnA

ColumnB

Fig 19 The energy input from the clock over one clock period requiredto fan a bit packet out into two is the cost of the original packet but anequal amount of work is done on the clock when the two bit packets arefanned in to one (fan-in) in column B (Fig 18)



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Fig 20 A three-level fan-out circuit receives a right-ward propagatingbit packet and fans it out into eight identical packets

fanout circuit is seven times the cost of the original packet(Fig 21)These results are verified in a dynamic simulation of

the fanout circuits46 in which energy input into the cellsis monitored and reported over one full clock period (thedynamic simulation reported in46 did not include interac-tions with null cells)

5 ENERGETICS OF BIT PACKETS IN WIDEWIRES

QCA circuits can be designed for robustness by using mul-tiple QCA cells to widen binary wires and QCA devices46


Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

Fig 21 The energy input from the clock over one period required tofan a bit packet out into eight bit packets is the sum of work done oncolumns A B and C (Fig 20) The total work done by the clock is seventimes the cost of the original packet

This concept takes advantage of vertical broadside cou-pling in addition to horizontal broadside couplingConsider a bit packet that is N cells long and spans the

width of a binary wire R-cells wide as in Figure 22 Inaddition to the change in energy levels due to horizontalbroadside coupling vertical broadside interactions and avariety of next-nearest-neighbor diagonal interactions mustbe considered The energy Eref of a wire of length L andwidth R with all cells in the null state is

Eref =minusLREc0+ Lminus1REhnn+LRminus1Evnn

+2Rminus1Lminus1Ednn (11)

In activating the first column of R cells spanning thewidth of the wire the following interactions are removed2R horizontal null-null interactions with energy Ehnn4Rminus 1 diagonal null-null interactions having energyEdnn and Rminus1 vertical broadside null-null interactions ofenergy Evnn This comes at the cost of REc0 with Rminus 1vertical interactions having energy Ev and 2R interactionshaving energy Ehn1 and 2Rminus 1 diagonal interactionsbetween a null cell and a polarized cell having energiesEdn1 and Edn2 Thus the first column of cells costs

E = REc0+2REhn1+ Rminus1Ev+2Rminus1Edn1+Edn2

minus2REhnnminus Rminus1Evnnminus4Rminus1Ednn (12)

Each additional column of cells costs another REc0 +Rminus1Ev+ Rminus1Edn1+Edn2 and introduces horizontalinteractions costing REh it also yields a savings of REhnn+Rminus 1Evnn + 2Rminus 1Ednn + Rminus 1Edn1 +Edn2 Thereare N minus 1 additional columns so that the total cost of theR-by-N bit packet in the interior of an R-by-L wire is

E = NREc0+ N minus1REh+NRminus1Ev

+ N minus1Rminus1Ed1+Ed2 middot middot middot+2REhn1

+2Rminus1Edn1+Edn2 middot middot middotminus N +1REhnn

minusNRminus1Evnnminus2N +1Rminus1Ednn (13)

Fig 22 A bit packet packet of data N cells long and R cells wideCells are color-coded to identify different groups of cells Corner cells areshown in yellow Skin cells comprise the outer border of the packet Thehorizontally-oriented skin is identified in red and the vertically orientedskin is in green Bulk cells are color-coded blue



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This cost may also be understood in terms of activatingcells of different groups and written in the form

E = XcornerEcorner +XskinhEskinh+Xskin vEskinv+XbulkEbulk

(14)

where Xcorner Xskinh Xskinv and Xbulk respectively arethe number of cells at the corner of the packet the numberof cells forming the horizontal edges of the packet (notcounting the corner cells) the number of cells formingthe vertical edges of the packet (not counting the cornercells) and the number of cells internal to the packet andEcorner Eskinh Eskinv and Ebulk respectively are the per-cell energy costs associated with each group The per-cellactivation cost for each group are as follows

Ecorner = Ec0+Ehn1+12Eh+Ev+Edn1+Edn2

+ 14Ed1+Ed2middotmiddotmiddotminus

32Ehnn+Ednnminus

12Evnn (15)

Ebulk=Ec0+Eh+Ev+Ed1+Ed2minusEhnnminusEvnnminus2Ednn

(16)

Eskinh=Ec0+Eh+12Ev+Ed1+Ed2minusEhnnminus

12EvnnminusEdnn

(17)

Eskinv = Ec0+Ev+12Eh+Ed1+Ed2+Ehn1+Edn1+Edn2 middotmiddotmiddot

minus32EhnnminusEvnnminus3Ednn (18)

The number of cells in each group is given by

Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2

R = 3Predicted costCalculated cost

Fig 23 The cost of a multi-row packet as predicted by the heuristicmodel using Equations (14)-(22) and calculated from the direct Coulombmodel for various values of packet width R

Xskinh = 2N minus2 (20)

Xskinv = 2Rminus2 (21)

Xbulk = N minus2Rminus2 (22)

Equations (14)ndash(22) are valid when both R gt 2 andN gt 2 Figure 23 shows that Eq (13) [or equivalentlyEqs (14)ndash(22) when valid] accurately predict the cost ofa multi-row packet For these predictions and calculationscells had a xnull zact= 07017500875 nm and onepositive elementary half-charge is located at each activedot A center-to-center spacing of 14 nm was used foradjacent cells having horizontal or vertical broadside cou-pling The clock bias Vc0 was set such that Ec0 =minusEk

6 ENERGY DISSIPATION

The energy cost we are considering here is the energy thatmust be supplied by the clock to create a QCA bit packetThis energy can be considered ldquoinvestedrdquo in the QCA sig-nal but it need not be lost to the environment As longas the clocking period is long enough for the individualcells to be always near their instantaneous ground state thecondition known as adiabatic clocking the energy investedin a bit packet is in principle recoverable ie it can bereturned to the clocking circuit rather than being dissi-pated as heat For molecular systems where the intrinsicswitching time is the time it takes for an electron to movefrom one molecular orbital to another adiabaticity can bemaintained at very high clock rates indeedIn a real physical system of course some energy is

always lost to irreversible processes (molecular vibronssubstrate phonons or plasmons for example) In addition ifa logically irreversible calculation is performed there mustbe dissipation of at least kB ln2 in accordance with Lan-dauerrsquos Principle These considerations and how to mini-mize dissipation in practical systems have been discussedelsewhere47 though it should be noted that LandauerrsquosPrinciple itself remains somewhat controversial48ndash53

7 CONCLUSIONS

Computation with QCA cells instead of transistors requiresdeveloping new concepts and abstractions Conceivingcomputation as essentially the motion and interaction of bitpackets is a helpful abstraction one level higher than indi-vidual QCA cell switching The energetics of such packetsis important because after all it is the disastrously highenergy dissipation (per unit area) of transistors that pre-vents them from being useful in large arrays when sim-ply scaled to molecular dimensions We have analyzedthe energy investment that must be made to create a bitpacket In most instances the energy required scales withthe length of the packet though we found that one coulddesign length-independent cells (whether such a design



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is realizable using synthetic chemistry is another matter)A heuristic model built from just cell-cell interaction ener-gies works very well Fan-out and fan-in can be simplyunderstood as borrowing or returning bit packed energyto the clocking circuit We have focused here on the sig-nal energy and not the energy dissipated due to residualnon-adiabaticity or bit erasure

APPENDIX A THE THREE-STATEAPPROXIMATION AND COHERENCEVECTOR FORMALISM

The Three-State Approximation

A three-state model uses three basis vectors minus1+1 +1 each corresponding to one of the activestates [PA= ∓11] or the null state [PA= 00]In an array of cells the Hamiltonian H for the jth cell

is given

Hj =

⎛⎜⎜⎝Eminus1 0 minus13

0 E+1 minus13

minus13 minus13 Enull

⎞⎟⎟⎠ (A1)

where each of the diagonal energies include the electro-static assembly energy of the cell intercellular interac-tions and the applied clock These matrix elements canbe obtained from microscopic models of for example themolecular orbitals of a QCA molecule35

71 Coherence Vector Formalism

The elements i of the coherence vector for a cell arethe projections of the cellrsquos density matrix onto the gen-erators of SU(3)54

i = Tri (A2)

i represents the the ith generator of SU(3) as given below

1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠

This is useful because the time evolution of the systemscan be expressed in terms of the 8 real degrees of freedomcontained in the components of 46

References

1 C Lent P Tougaw and W Porod Appl Phys Lett 62 714 (1993)2 C Lent P Tougaw W Porod and G Bernstein Nanotechnology

4 49 (1993)3 C Lent and P Tougaw Proc of the IEEE 85 541 (1997)4 K Hennessy and C Lent J Vac Sci Technol 19 1752 (2001)5 E Blair and C Lent An architecture for molecular computing using

quantum-dot cellular automata IEEE C Nanotechnol IEEE (2003)Vol 1 pp 402ndash405

6 A O Orlov I Amlani G H Bernstein C S Lent and G L SniderScience 277 928 (1997)

7 I Amlani A O Orlov G L Snider C S Lent and G H BernsteinAppl Phys Lett 72 2179 (1998)

8 G L Snider A O Orlov I Amlani G H Bernstein C S LentJ L Merz and W Porod Semiconductor Science and Technology13 A130 (1998)

9 G L Snider A O Orlov I Amlani G H Bernstein C S LentJ L Merz and W Porod Japanese Journal of Applied Physics Part1-Regular Papers Short Notes amp Review Papers 38 7227 (1999)

10 G L Snider A O Orlov I Amlani X Zuo G H Bernstein C SLent J L Merz and W Porod J Appl Phys 85 4283 (1999)

11 G L Snider A O Orlov I Amlani G H Bernstein C S LentJ L Merz and W Porod Microelectronic Engineering 47 261(1999)

12 G L Snider A O Orlov I Amlani X Zuo G H BernsteinC S Lent J L Merz and W Porod Journal of Vacuum Science ampTechnology A-vacuum Surfaces and Films 17 1394 (1999)

13 I Amlani A O Orlov G L Snider C S Lent W Porod andG H Bernstein Superlattices and Microstructures 25 273 (1999)

14 I Amlani A O Orlov G Toth G H Bernstein C S Lent andG L Snider Science 284 289 (1999)

15 A O Orlov I Amlani G Toth C S Lent G H Bernstein andG L Snider Appl Phys Lett 74 2875 (1999)

16 A O Orlov I Amlani R K Kummamuru R RamasubramaniamG Toth C S Lent G H Bernstein and G L Snider Appl PhysLett 77 295 (2000)

17 I Amlani A O Orlov R K Kummamuru G H Bernstein C SLent and G L Snider Appl Phys Lett 77 738 (2000)

18 A O Orlov R K Kummamuru R Ramasubramaniam G TothC S Lent G H Bernstein and G L Snider Appl Phys Lett78 1625 (2001)

19 M Niemier and P Kogge Exploring and exploiting wire-levelpipelining in emerging technologies 28Th Annual InternationalSymposium on Computer Architecture Proceedings ACM SigarchComputer Architecture News Gothenburg Sweden June (2001)pp 166ndash177

20 A O Orlov R Kummamuru R Ramasubramaniam C S LentG H Bernstein and G L Snider J Nanosci Nanotechnol 2 351(2002)


22 R K Kummamuru A O Orlov R Ramasubramaniam C S LentG H Bernstein and G L Snider International Electron Devices2002 Meeting Technical Digest (2002) p 95

23 A O Orlov R Kummamuru R Ramasubramaniam C S LentG H Bernstein and G L Snider Surface Science 532 1193 (2003)

24 R K Kummamuru A O Orlov R Ramasubramaniam C S LentG H Bernstein and G L Snider IEEE Transactions on ElectronDevices 50 1906 (2003)



IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

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25 R K Kummamuru J Timler G Toth C S LentR Ramasubramaniam A O Orlov G H Bernstein and G LSnider Appl Phys Lett 81 1332 (2002)

26 K K Yadavalli A O Orlov J P Timler C S Lent and G LSnider Nanotechnology 18 375401 (2007)

27 M Mitic M C Cassidy K D Petersson R P Starrett E GaujaR Brenner R G Clark A S Dzurak C Yang and D N JamiesonAppl Phys Lett 89 013503 (2006)

28 S J Chorley C G Smith F Perez-Martinez J Prance P AtkinsonD A Ritchie and G A C Jones Microelectronics Journal 39 674(2008) 6th International Conference on Low Dimensional Structuresand Devices (LDSD 2007) San Andres Colombia (2007)

29 R P Cowburn and M E Welland Science 287 1466 (2000)30 A Imre G Csaba L Ji A Orlov G H Bernstein and W Porod

Science 311 205 (2006)31 M B Haider J L Pitters G A DiLabio L Livadaru J Y Mutus

and R A Wolkow Phys Rev Lett 102 046805 (2009)32 C S Lent Science 288 1597 (2000)33 H Qi S Sharma Z Li G L Snider A O Orlov C S Lent and

T P Fehlner J Am Chem Soc 125 15250 (2003)34 H Qi Z H Li G L Snider C S Lent and T P Fehlner Abstracts

of Papers of the American Chemical Society 226 101 (2003)35 Y H Lu M Liu and C Lent J Appl Phys 102 034311

(2007)36 C Lent and B Isaksen IEEE T Electron Dev 50 1890 (2003)37 Y H Lu and C S Lent Nanotechnology 19 155703 (2008)38 P Tougaw and C Lent J Appl Phys 75 1818 (1994)39 C Lent and P Tougaw J Appl Phys 74 6227 (1993)40 J Timler and C Lent J Appl Phys 91 823 (2002)

41 M Niemier and P M Kogge Int Conf Elec Circ Sys 1211(1999)


43 J Timler and C Lent J Appl Phys 94 1050 (2003)44 M Lieberman S Chellamma B Varughese Y Wang C Lent G H

Bernstein G Snider and F C Peiris Ann N Y Acad Sci 960 225(2002)

45 C Lent B Isaksen and M Lieberman J Am Chem Soc 125 1056(2003)

46 M Liu Robustness and power dissipation in quantum-dot cellularautomata PhD Thesis U of Notre Dame (2006)

47 C S Lent M Liu and Y H Lu Nanotechnology 17 4240 (2006)48 H Leff and A Rex Maxwellrsquos Demon 2 Entropy Classical

and Quantum Information Computing Taylor and Francis London(2002)

49 V V Zhirnov R K Cavin J A Hutchby and G I BourianoffProceedings of the IEEE 91 1934 (2003)

50 V V Zhirnov and R K Cavin Nanotechnology 18 298001 (2007)51 R K Cavin V V Zhirnov J A Hutchby and G I Bourianoff

Fluctuations and Noise in Photonics and Quantum Optics III5846 IX (2005)

52 J Earman and J D Norton Studies In History and Philosophy ofModern Physics 30B 1 (1999)

53 J D Norton Studies In History and Philosophy of Modern Physics36B 375 (2005)

54 G Mahler and V A Weberruss Quantum Networks Dynamics ofOpen Nanostructures Springer New York (1995)

Received 3 April 2010 Accepted 8 May 2010



IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

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Notre Dame group6ndash24 In these experiments aluminumislands act as the dots coupled by aluminum oxide tun-nel junctions The islands and junctions are formed bye-beam lithography and two angled metal evaporationswith an intervening oxidation step Overlaps between thetwo metal layers create the tunnel junctions Because theshadow evaporation technique produces capacitances thatare still relatively large these experiments are performedat cryogenic temperatures (lt1 K)Using these metal-dot QCA cells and number of impor-

tant circuit elements have been demonstrated function-ing individual cells6 logic gates14 clocked QCA cells1623

inverters22 memories18 and shift registers20 In additionpower gain greater than 3 has been measured25 Powergain is important so that small energy losses due toinevitable inelastic processes can be restored In QCA theenergy lost to dissipative processes is restored from theclock Fan-out of QCA signals has also been demonstratedexperimentally26

In addition to the metal-dot QCA work at NotreDame other groups have succeeded in building QCAcells in semiconductors Mitic et al reported fabricationof QCA cells in silicon using dots formed by few-atomimplantation of phosphorus donors27 Perez-Martinez et alcreated QCA cells using gate-defined dots in a GaAs two-dimensional electron gas28 As with the metal dot sys-tems lithographically defined dots whether in silicon orGaAs are relatively large so that the Coulomb energiesinvolved are small and operating QCA cells can only func-tion at cryogenic temperatures QCA can also be imple-mented in magnetic systems comprised of small permalloyislands2930

The Wolkow group at the University of Alberta recentlyfabricated a QCA cell using single-atom dangling bondson a H-passivated Si surface as the quantum dots31 Indi-vidual hydrogen atoms were removed from the surface toexpose the Si dangling bonds (DBs) using STM lithog-raphy Coulombic interactions between nearby DB dotswas apparent in the brightness of the STM images Byelectrostatically perturbing a symmetric four-dot cell thepolarization of the QCA cell was apparent These roomtemperature results confirm the viability of molecular-scaleQCA Work is underway to construct room-temperatureQCA cells from molecules designed for the purpose32ndash37

2 QCA CELLS AND DEVICES

21 Dots Cells and Logic Gates

The elementary computing device in QCA circuitry is theQCA cell38 a charge-containing structure with multiplesites quantum dots which localize mobile charges Deviceswitching is enabled by quantum-mechanical tunnelingA cell with two mobile charges and four dots arranged in asquare has two preferred states with electrons in antipodal

ldquo1rdquoldquo0rdquo

Fig 1 The two states of a four-dot QCA cell with two mobile chargesand four dots A quantum dot is represented by a white disc and the reddisc within a white disc represents a mobile charge occupying a dot

dots (see Fig 1) These states are degenerate in energy andcan be used to represent binary ldquo0rdquo and ldquo1rdquo respectivelyThe basis of QCA circuitry is intercellular interaction

via Coulomb repulsion The interaction between cells liftsthe degeneracy between the 0 and 1 states and determinesthe state of the cell Broadside coupling causes neighbor-ing cells to align (Fig 2) This is the basis for the binarywire (Fig 3)39 Signal inversion can be achieved via diag-onal coupling (Fig 3) The majority gate is a three-inputgate having a single output which is determined by thebit dominating the three inputs (Fig 3) One of the threeinputs can be used as a control input to make the majoritygate function as a programmable AND-OR gate This isseen in the truth table for the majority gate (Table I) TheQCA paradigm is heirarchical and QCA-based adders andeven a Simple-12 ALU processor have been designed338

22 Clocked QCA Cells

Clocking QCA cells provides power gain for the restora-tion of weakened signals minimizes power dissipation2540

and facilitates large-scale circuit operation41 QCA cellscan be clocked if two additional dots called ldquonull dotsrdquoare added and the voltage on the null dots can be con-trolled The null dots add an additional state called theldquonull staterdquo which does not convey information (theinformation-bearing states are now called ldquoactive statesrdquo)as shown in Figure 4 The voltage of the null dots can beused to drive the cell to an active state or force the cell intothe null state A cell in the null state does not lift the degen-eracy of the active states in a neighboring cell Clockeddevices such as shift registers have been demonstrated42

The state of a six-dot cell can be characterized by twoquantities polarization P and activation A The quanti-ties P and A originate in a three-state quantum-mechanical

Fig 2 Broadside coupling causes cells to align



IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

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P = minus7 (1)

A = 2minusradic384

(2)


A B C D =MABC

0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1













IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE














IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE










322 Heuristic Model











sjsumi=1





IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




Vj ineighbors =1

40

msumk=j+1

sksump=1

qkp

rip(9)

1 2 3 4 50

2

4

6

8

10


Pack

et c

ost (

EE

k)










Xnull

z

aa

X

y

x

Zact




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE


1 2 3 4 5minus05

0

05

1

15

2


Pack

et C

ost (

EE










Epacket

Eclock 0

ColumnA





Epacket

Eclock 0

1

ndash1

ColumnA

ColumnB




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE








Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

















IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE






is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE





























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




P = minus7 (1)

A = 2minusradic384

(2)


A B C D =MABC

0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1













IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE














IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE










322 Heuristic Model











sjsumi=1





IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




Vj ineighbors =1

40

msumk=j+1

sksump=1

qkp

rip(9)

1 2 3 4 50

2

4

6

8

10


Pack

et c

ost (

EE

k)










Xnull

z

aa

X

y

x

Zact




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE


1 2 3 4 5minus05

0

05

1

15

2


Pack

et C

ost (

EE










Epacket

Eclock 0

ColumnA





Epacket

Eclock 0

1

ndash1

ColumnA

ColumnB




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE








Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

















IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE






is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE





























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE














IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE










322 Heuristic Model











sjsumi=1





IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




Vj ineighbors =1

40

msumk=j+1

sksump=1

qkp

rip(9)

1 2 3 4 50

2

4

6

8

10


Pack

et c

ost (

EE

k)










Xnull

z

aa

X

y

x

Zact




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE


1 2 3 4 5minus05

0

05

1

15

2


Pack

et C

ost (

EE










Epacket

Eclock 0

ColumnA





Epacket

Eclock 0

1

ndash1

ColumnA

ColumnB




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE








Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

















IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE






is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE





























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE










322 Heuristic Model











sjsumi=1





IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




Vj ineighbors =1

40

msumk=j+1

sksump=1

qkp

rip(9)

1 2 3 4 50

2

4

6

8

10


Pack

et c

ost (

EE

k)










Xnull

z

aa

X

y

x

Zact




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE


1 2 3 4 5minus05

0

05

1

15

2


Pack

et C

ost (

EE










Epacket

Eclock 0

ColumnA





Epacket

Eclock 0

1

ndash1

ColumnA

ColumnB




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE








Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

















IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE






is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE





























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




Vj ineighbors =1

40

msumk=j+1

sksump=1

qkp

rip(9)

1 2 3 4 50

2

4

6

8

10


Pack

et c

ost (

EE

k)










Xnull

z

aa

X

y

x

Zact




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE


1 2 3 4 5minus05

0

05

1

15

2


Pack

et C

ost (

EE










Epacket

Eclock 0

ColumnA





Epacket

Eclock 0

1

ndash1

ColumnA

ColumnB




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE








Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

















IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE




(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE






is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

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IP 98228125146Wed 20 Apr 2011 173657

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1 2 3 4 5minus05

0

05

1

15

2


Pack

et C

ost (

EE










Epacket

Eclock 0

ColumnA





Epacket

Eclock 0

1

ndash1

ColumnA

ColumnB




IP 98228125146Wed 20 Apr 2011 173657

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Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

















IP 98228125146Wed 20 Apr 2011 173657

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(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

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is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

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IP 98228125146Wed 20 Apr 2011 173657

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Epacket

Eclock

4

2

1

ColumnA

ColumnB

ColumnC

















IP 98228125146Wed 20 Apr 2011 173657

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(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

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is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

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IP 98228125146Wed 20 Apr 2011 173657

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(14)




32Ehnn+Ednnminus

12Evnn (15)


(16)


12EvnnminusEdnn

(17)




Xcorner = 4 (19)

1 2 3 4 50

2

4

6

8

10

12

14


Pack

et c

ost (

EE

k)


R = 1

R = 2










7 CONCLUSIONS




IP 98228125146Wed 20 Apr 2011 173657

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ARTIC

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is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE





























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

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is given

Hj =


0 E+1 minus13


⎞⎟⎟⎠ (A1)




i = Tri (A2)


1 =

⎛⎜⎜⎝0 1 0

1 0 0

0 0 0

⎞⎟⎟⎠ 2 =

⎛⎜⎜⎝0 0 1

0 0 0

1 0 0

⎞⎟⎟⎠

3 =

⎛⎜⎜⎝0 0 0

0 0 1

0 1 0

⎞⎟⎟⎠ 4 =

⎛⎜⎜⎝

0 i 0

minusi 0 0

0 0 0

⎞⎟⎟⎠

5 =

⎛⎜⎜⎝

0 0 i

0 0 0

minusi 0 0

⎞⎟⎟⎠ 6 =

⎛⎜⎜⎝0 0 0

0 0 i

0 minusi 0

⎞⎟⎟⎠

7 =

⎛⎜⎜⎝minus1 0 0

0 1 0

0 0 0

⎞⎟⎟⎠ 8 =

1radic3

⎛⎜⎜⎝minus1 0 0

0 minus1 0

0 0 2

⎞⎟⎟⎠


References

























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

ARTIC

LE





























IP 98228125146Wed 20 Apr 2011 173657

RESEARCH

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Documents

Signal Energy in Quantum-Dot Cellular Automata Bit Packets