In this work we present an error-power tradeoff study in a Quantum-dot Cellular Automata (QCA) circuit design. Device parameter variation to optimize performance is a very crucial step in the development of a technology. In this work we vary the maximum kink energy of a QCA circuit to perform an error-power tradeoff study in QCA design. We make use of graphical probabilistic models to estimate polarization errors and non-adiabatic energy dissipated in a clocked QCA circuit and demonstrate the tradeoff studies on the basic QCA circuits such as majority gate and inverter. We also show how this study can be used by comparing two single bit adder designs. The study will be of great use to designers and fabrication scientists to choose the most optimum size and spacing of QCA cells to fabricate QCA logic designs

Bhanja, S.

Sarkar, S.

Srivastava, S.

English

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Error-Power Tradeoffs in QCA Design

University of Lincoln Institutional Repository

1Error-Power Tradeoffs in QCA DesignSaket Srivastava∗, Sudeep Sarkar† and Sanjukta Bhanja∗∗ Electrical Engineering,† Computer Science and EngineeringUniversity of South Florida, Tampa, Florida.Emails: ssrivast@mail.usf.edu, sarkar@cse.usf.edu, bhanja@eng.usf.eduAbstract— In this work we present an error-power tradeoffstudy in a Quantum-dot Cellular Automata (QCA) circuit design.Device parameter variation to optimize performance is a verycrucial step in the development of a technology. In this work wevary the maximum kink energy of a QCA circuit to performan error-power tradeoff study in QCA design. We make useof graphical probabilistic models to estimate polarization errorsand non-adiabatic energy dissipated in a clocked QCA circuitand demonstrate the tradeoff studies on the basic QCA circuitssuch as majority gate and inverter. We also show how this studycan be used by comparing two single bit adder designs. Thestudy will be of great use to designers and fabrication scientiststo choose the most optimum size and spacing of QCA cells tofabricate QCA logic designs.I. INTRODUCTIONQuantum-dot Cellular Automata (QCA) is one of the mostpromising technologies being currently researched. It uses aninteresting charge coupled computing paradigm to performcomputation [1]. A lot of progress has been made recently tofabricate QCA using magnetic dots [2], [3] and molecules [4],[5] that will enable it to perform room temperature operations.While there have been experimental studies related to defectand fault tolerance in QCA [6], [7], [8], not much work hasbeen done to study the effects of variation in device parameterson error and power in QCA design. Similar studies have beenthe hallmark of CMOS research over the years that contributedsignificantly in the development of CMOS technology. It isnatural to perform such a study with respect to parametervariations in QCA. We perform a study of error and powerdissipation in a clocked QCA design by varying one of themost crucial parameter in QCA design; the kink energy. Kinkenergy is the energy cost of keeping two adjacent cells inopposite polarization and varies with the size of a QCA celland the grid spacing. We analyze the effects of kink energywith a design perspective to help designers and fabricationscientists to choose the most optimum size of QCA cell andspacing between adjacent QCA cells.In [9] it has been shown how to calculate the ground statepolarization probabilities and build a graphical probabilisticmodel based on that. We used these graphical probabilisticmodels to detemine thermal error at the output at differenttemperatures. In [10], an efficient method, based on graphicalprobabilistic models was presented, to compute the N-lowestenergy modes of a clocked QCA circuit. In QCA, an erroneousstate may result due to the failure of the clocking scheme toswitch portions of the circuit to its new ground state withchange in input. This error state of a single cell in turn causesthe error in the neighboring cells resulting in an erroneousTABLE IDIFFERENT TYPES OF QCA CELLS AND GRID SPACING USED IN THISSTUDYQCA cell Size Grid Spacing Associated Kink EnergyCell-1 10nm 5nm Ek1 = 4EkCell-2 20nm 10nm Ek2 = 2EkCell-3 40nm 20nm Ek3 = Ekoutput. Due to the quantum mechanical nature of operationof a QCA device, temperature plays an important role indetermining the ground state polarization of each cell. Powerdissipation in a QCA circuit primarily results due the theapplication of a non-adiabatic clocking scheme. We have alsoseen in [11], how clock energy affects the overall powerdissipation in a QCA circuit.In this work we perform studies to determine the error andpower tradeoff in a QCA circuit design by studying the effectof kink energy on the output error and power dissipation in aQCA circuit. We use three different sizes of QCA cells andgrid spacing to study the polarization and power dissipationfor basic QCA circuits using these cells. The three differenttypes of cell sizes used in this study are elaborated in Table I.Here Ek1 is the maximum kink energy for the cell layout withsmallest cell dimensions (and grid spacing). Similarly, Ek3 isthe maximum kink energy for the QCA layout with largestcell dimensions (and grid spacing). It can be seen from TableI, Ek1 = 2Ek2 = 4Ek3.We first simulate a number of basic QCA circuits such asmajority gate and inverter to study the polarization error atthe output for each input vector set. We also determine thepower dissipation in these circuits for different kink energies.All other parameters such as temperature and clock energyare kept constant. We show how this study can be used bycomparing two single bit adder designs. The study will beof great use to designers and fabrication scientists to choosethe most optimum size and spacing of QCA cells to fabricateQCA logic designs.II. KINK ENERGYTwo electrons in a simple four dot QCA cell occupydiagonally opposite dots in the cell due to mutual repulsionof like charges. A QCA cell can be in any one of the twopossible states depending on the polarization of charges in thecell. The two polarized states are represented as P = +1 and2P = +1 P = - 1 P = +1 P = +1minusEnergy of Energy ofKink Energy  (EK) =Fig. 1. Kink energy between two neighboring QCA cellsP = -1. Electrostatic interaction between charges in two QCACells is given as:Em =14πεoεr4∑i=14∑j=1qimq jk|rim − r jk| (1)This interaction determines the kink energy between twocells.Ekink = Eopp.polarization −Esamepolarization (2)Kink energy (Fig 1) is the energy cost of two neighboringQCA cells having opposite polarization. Kink energy betweentwo cells depends on the dimension of the QCA cell as wellas the spacing between adjacent cells. It does not depend onthe temperature.III. ESTIMATION OF ERRORA QCA system can be erroneous if it fails to settle downto ground state. In order to estimate error in a QCA circuit,we use the probabilistic error model presented in [10]. Thismodel arrives at the probability of error in a QCA systemusing ground state polarization probabilities and using it toestimate error at the output of a QCA circuit represented bya graphical probabilistic network. A brief description of thismodel is provided below.The steady state polarization of a QCA cell can be derivedfrom the Hamiltonian of the cell using Hartree approximation.Expression of Hamiltonian is shown in Eq. 3 [12].H =[ − 12 ∑i EkPi fi −γ−γ 12 ∑i EkPi fi]=[ − 12 Ek ¯P −γ−γ 12 Ek ¯P](3)where the sums are over the cells in the local neighborhood.Ek is the “kink energy” . fi is the geometric factor capturingelectrostatic fall off with distance between cells. Pi is thepolarization of the i-th cell. And, γ is the tunneling energybetween two cell states, which is controlled by the clockingmechanism. The notation can be further simplified by using ¯Pto denote the weighted sum of the neighborhood polarizations∑i Pi fi. Using this Hamiltonian the steady state polarization isgiven byPss =−λss3 = ρss11 −ρss00 =Ek ¯P√E2k ¯P2+4γ2tanh(√E2k ¯P2/4+ γ2kT )(4)Eq. 4 can be written asPss =EΩ tanh(Δ) (5)where E = 0.5∑i EkPi fi, total kink energy and Rabi frequencyΩ =√E2k ¯P2/4+ γ2 and Δ = ΩkT is the thermal ratio. Wewill use the above equation to arrive at the probabilities ofobserving (upon making a measurement) the system in eachof the two states. Specifically, ρss11 = 0.5(1+Pss) and ρss00 =0.5(1−Pss), where we made use of the fact that ρss00+ρss11 = 1.Where ρss11 (ρss00) is the probability of observing the system(here a QCA cell) in state 1 (state 0).The value of steady state polarization proabability is used toestimate error in a QCA circuit using graphical probabilisticmodel [10].IV. ESTIMATION OF POWER DISSIPATIONWe make use of the non-adiabatic power dissipation modelpresented in [11], to estimate switching power losses is a QCAcircuit. This model was derived from the the quasi-adiabaticmodel presented by Timler et.al. [13]. According to thismodel, the equation for the instantaneous power is given by:Ptotal =ddt E =h¯2(ddtΓ)·λ+ h¯2Γ ·(ddtλ)(6)Whereλ the coherence vector and Γ is the three-dimensionalenergy vector. The first term captures the power in and outof the clock and cell to cell power flow. We are concernedwith Pdiss which represents the instantaneous dissipated power.Power dissipated during switching can thus be calculated byintegrating Pdiss over time.Pdiss(t) =h¯2Γ(t).(ddtλ(t))(7)The energy vector Γ of a QCA cell varies with kink energythereby affecting the overall power dissipation . Please see [13]for more details about these terms.V. RESULTSIn this section we present the results obtained from thestudy of variation of kink energy on error and power dissipatedin the circuits. We obtained these results by simulating eachof the circuits at a constant temperature of 2K. We usedQCADesigner [14] tool to design QCA circuits and thenanalyzed it using our graphical probabilistic modeling scheme.A. Output Node Polarization ErrorWe quantify the error in a circuit as a measure of its outputnode polarization. In prior work [9], [15], using temperatureas a variable and keeping the kink energy constant, it has beenshown how the output node polarization drops steadily withrise in temperature leading to more erroneous outputs. Thiseffect becomes more and more significant with the increase inthe number of cells in a design. Hence two different designsrepresenting similar logic but having unequal number of cellswill have different polarizations at the output nodes.Similarly, in this study, by varying the kink energy ofthe circuit and keeping the temperature constant we see thatthe gain (drop) in output node polarization of a circuit isdirectly proportional to the increase (decrease) in maximum3(a) (b)Fig. 2. (a) QCA Majority Gate (b) InverterTABLE IIOUTPUT NODE POLARIZATION OF A SIMPLE MAJORITY GATE FORDIFFERENT KINK ENERGIESMaximum Kink Energy (Ek)Input Ek3 = 0.75meVEk2 = 1.5 meV Ek1 = 3.0 meV0 0 0 0.9278 0.9999 1.00000 0 1 0.9880 0.9999 1.00000 1 0 0.9880 0.9999 1.00000 1 1 0.9075 0.9999 1.00001 0 0 0.9075 0.9999 1.00001 0 1 0.9880 0.9999 1.00001 1 0 0.9880 0.9999 1.00001 1 1 0.9278 0.9999 1.0000kink energy (Ek)of the circuit. Here increase in Ek refers todecrease in QCA cell size and grid spacing. Similar effect wasseen for different values of temperature.As an example, refer to the output node polarization of asimple majority gate (Fig. 2) shown in Table II. As we haveshown earlier, we first form a Bayesian network of the QCAcircuit and use a graphical simulator to obtain the polarizationprobability for each QCA cell (represented as a node) in thedesign. We can see that the polarization probability at theoutput of the Bayesian network rises with the increase in kinkenergy. Hence, we can infer that designs with lower value ofmaximum kink energy are more prone to error and this error ismore significant when the number of cells in a design increasesor the temperature is raised. Table III shows the output nodepolarization probability of a QCA inverter.TABLE IIIOUTPUT NODE POLARIZATION OF A QCA INVERTER FOR DIFFERENTKINK ENERGIESMaximum Kink Energy (Ek)Input Ek3 = 0.75meVEk2 = 1.5 meV Ek1 = 3.0 meV0 0.9750 0.9998 1.00001 0.9843 0.9998 1.0000TABLE IVPOWER DISSIPATION IN QCA MAJORITY GATE FOR DIFFERENT KINKENERGIESMaximum Kink Energy (Ek)Ek3 = 0.75meVEk2 = 1.5 meV Ek1 = 3.0 meVMax Ediss (meV) 0.0051 0.0147 0.0294Avg Ediss (meV) 0.0018 0.0060 0.0120Min Ediss (meV) 0.0002 0.0008 0.0015Avg Eleak (meV) 0.0003 0.0009 0.0018Avg Esw (meV) 0.0015 0.0051 0.0102TABLE VPOWER DISSIPATION IN QCA INVERTER FOR DIFFERENT KINK ENERGIESMaximum Kink Energy (Ek)Ek3 = 0.75meVEk2 = 1.5 meV Ek1 = 3.0 meVMax Ediss (meV) 0.0196 0.0392 0.0785Avg Ediss (meV) 0.0106 0.0213 0.0425Min Ediss (meV) 0.0016 0.0033 0.0066Avg Eleak (meV) 0.0016 0.0033 0.0066Avg Esw (meV) 0.0090 0.0180 0.0359SumCarry OutA B Carry InSC(a) (b)Fig. 3. Single bit QCA Adder layout(a) Adder-1 (b) Adder-2B. Switching PowerWe performed an exhaustive study on the effect of varyingkink energy on the power dissipated during a switching eventin a QCA circuit. In prior works, it has been shown how powerdissipated in a QCA circuit varies with clock energy [11]. Inthis work we intend to analyze the effect of the size of aQCA cell and the kink energy associated with it on the powerdissipated in the circuit.The results of variation on power dissipation of a simplemajority gate are shwon in Table IV. It can be seen thatincreasing the value of kink energy in a circuit leads to anincrease in the overall average power dissipated in the circuit.Table V shows the energy dissipation in a QCA inverter fordifferent values of kink energy.Some very interesting observations were obtained from thisstudy of effect of kink energy on the overall power dissipationand probability of error in QCA circuit design. We have seenthat while it is desirable to design circuits with lower errorprobabilities (by increasing the kink energy between cells),it inadvertently increases the power dissipated in the circuit.This effect is more pronounced in larger circuits such assingle bit adders (shown in Fig. 3). Table VI compares theresults of output polarization at SUM node of two addersfor different kink energies. As we can see that even thoughAdder-2 has a more efficient design and uses less numberof cells, the polarization at its output is worse than that ofAdder-1 for different input vector sets. Similarly, Table VIIcompares the energy dissipation in the two adder designs.Power dissipation in Adder-2 is greater than that of Adder-1 since it has significantly more number of cells. However,we do see that the energy dissipation in a QCA circuit is4TABLE VIOUTPUT NODE POLARIZATION AT SUM OUTPUT NODE OF ADDER-1 AND ADDER-2 QCA DESIGNSEk3 = 1.09meV Ek2 = 2.18meV Ek1 = 4.36meVInput Adder-1 Adder-2 Adder-1 Adder-2 Adder-1 Adder-20 0 0 0.9110 0.8095 0.9998 0.9964 1.0000 1.00000 0 1 0.7311 0.8058 0.9935 0.9965 1.0000 1.00000 1 0 0.7440 0.6833 0.9944 0.9667 1.0000 0.99910 1 1 0.7090 0.6312 0.9931 0.9569 1.0000 0.99891 0 0 0.7090 0.6312 0.9931 0.9569 1.0000 0.99891 0 1 0.7440 0.6833 0.9944 0.9667 1.0000 0.99911 1 0 0.7311 0.8058 0.9935 0.9965 1.0000 1.00001 1 1 0.9110 0.8095 0.9998 0.9964 1.0000 1.0000TABLE VIINON-ADIABATIC ENERGY DISSIPATION IN ADDER-1 AND ADDER-2 QCA DESIGNSEk1 = 4.36meV Ek2 = 2.18meV Ek3 = 1.09meVAdder-1 Adder-2 Adder-1 Adder-2 Adder-1 Adder-2Max Ediss (in meV) 3.0939 1.3556 1.5404 0.6778 0.8127 0.3389Avg Ediss (in meV) 1.7398 0.7650 0.8665 0.3825 0.4556 0.1912Min Ediss (in meV) 0.4083 0.1949 0.2038 0.0974 0.1041 0.0487Avg Eleak (in meV) 0.4089 0.1956 0.2041 0.0978 0.1043 0.0489Avg Esw (in meV) 1.3309 0.5693 0.6624 0.2847 0.3513 0.1423almost linearly proportational to the maximum kink energy ofthe circuit.It can be seen from the results that the output node polariza-tion error improves while power dissipation deteriorates whenthe kink energy is increased. Hence designers need to choosethe size of QCA cells based on circuit requirements to optimizepower and error. This is different from thermal studies per-formed on QCA circuits which resulted in increase in outputerror and power dissipation at higher temperatures. From theresults obtained for polarization and power dissipation in smalland big QCA circuits, we have clearly seen that kink energyis an important factor to design most optimum circuits at agiven temperature and clock energy. Hence designers need tomake careful use of kink energy as parameter for designingQCA circuits to optimize error and power.VI. CONCLUSIONIn this work we performed error-power tradeoff studies forQCA circuits. We can see from the results that output nodepolarization error improves while power dissipation deterio-rates when the kink energy is increased. Hence designersneed to choose the size of QCA cells based on circuit re-quirements to optimize power and error. This is different fromthermal studies performed on QCA circuits which resultedin increase in output error and power dissipation at highertemperatures. From the results obtained for polarization andpower dissipation in small and big QCA circuits, we haveclearly seen that kink energy is an important factor to designmost optimum circuits at a given temperature and clock energy.Hence designers need to make careful use of kink energy asa parameter for designing QCA circuits to optimize error andpower.REFERENCES[1] C. Lent and P. Tougaw, “A Device Architecture for Computing withQuantum dots,” in Proceeding of the IEEE, vol. 85-4, pp. 541–557,April 1997.[2] R. P. Cowburn and M. E.Welland, “Room temperature magnetic quantumcellular automata,” Science, vol. 287, Feb 2000.[3] M. Parish and M. Forshaw, “Magnetic Cellular Automata (MCA)systems,” IEEE Proceedings of Circuits, Designs and Systems, vol. 151,pp. 480–485, 2004.[4] C. Lent and B. 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Error-power tradeoffs in QCA design

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