Fault tolerance properties in quantum-dot cellular automata devices

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2006 J. Phys. D: Appl. Phys. 39 1489

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 39 (2006) 1489–1494 doi:10.1088/0022-3727/39/8/006

Fault tolerance properties in quantum-dotcellular automata devicesM Khatun1, T Barclay1, I Sturzu1 and P D Tougaw2

1 Department of Physics and Astronomy, Center for Computational Nanoscience, Ball StateUniversity, Muncie, IN 47306, USA2 Department of Electrical and Computer Engineering, Valparaiso University, Valparaiso, IN46383, USA

E-mail: mkhatun@bsu.edu

Received 29 September 2005, in final form 23 December 2005Published 30 March 2006Online at stacks.iop.org/JPhysD/39/1489

AbstractWe present a study of the joint influence of temperature and fabricationdefects on the operation of quantum-dot cellular automata (QCA) devices.Canonical ensemble, a Hubbard-type Hamiltonian and the inter-cellularHartree approximation were used, and a statistical model has beenintroduced to simulate defects in the QCA devices. Parameters such assuccess rate and breakdown displacement factor (BDF) were defined andcalculated numerically. Results show the thermal dependence of BDFvalues of the QCA devices. The BDF values decrease with temperature. Thejoint influence of randomly missing dots and temperature was also studied.

1. Introduction

In recent years a general effort was made to create a newparadigm for computation on the nanoscale. As the computingindustry moves forward, faster as well as smaller machinesare developed, each more capable than the last. However,these machines are reaching an apparent limit to the numberof transistors that can be placed on a microprocessor. Thelimit is given by the encroaching genuine quantum effectswhich come in play on the nanoscale. Also, the increasingdensity of devices is exacerbating the present-day problem ofheat generation. Hence, the main problems come from genuinequantum mechanical effects and power dissipation limitations.A new technological paradigm must be elaborated for the trendof higher speeds and smaller circuits to continue.

Quantum-dot cellular automata (QCA) are a class of nano-electronic circuits in which the computation circuit functionsare realized through cooperative, quantum mechanical andelectrostatic interactions between electrons confined in arraysof quantum dots. The propagation of the logical valuesthroughout the circuit is due to the electrostatic interactionbetween the free electrons from the QCA cells. Therefore,QCA architecture is an alternative technique for transferringand processing information without the need for chargetransport [1, 2]. Within the QCA concept, the binaryrepresentation of information replaces the traditional currentswitch by a cell having a bistable charge configuration.A typical QCA cell is composed of at least four dots, positioned

at the corners of a square and has two free electrons which cantunnel between dots but not outside the cell. The dots can berealized in several ways: quantum dots formed in depletionregions a multi-layer semiconductor material, small metallicislands connected by tunnel junctions or redox centres inspecific molecules. In the present paper, we will report resultsfor the semiconductor implementation. Electrons are confinedin dots by potential modulation between the dots. They areallowed to tunnel between the dots in the same cell, but notbetween different cells. The degenerated ground state of acell is given by the electron configurations of the cell wherethe two free electrons are localized at opposite corners. Theelectrostatic quadrupole field of the charge configuration ofone cell alters the charge configuration of the other cells. Allcomputation tasks can be achieved using specific QCA devices[3, 4]. Theoretical investigations on various topics, such ascell–cell response functions, logic circuits [5], clocking [6,7],memory [8], power gain and dissipation [9,10] and many othershave been done. Experimental results for QCA cells composedof small metallic islands, connected by tunnel junctions andoperating at sub-Kelvin temperatures have been reported[11–13]. Also, some candidates for QCA molecules have beensynthesized and shown to exhibit switching behaviour [14].

In order to develop a viable, understandable and usableQCA architecture, the robustness of the device has to bestudied. It is necessary to study how to control data flow in adevice, then determine how temperature and manufacturingdefects will influence the proper output of the device.

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M Khatun et al

Theoretically a ‘quantum wire’ of perfectly aligned QCA cellsat zero Kelvin temperature has been examined in the past.QCA processors will not operate at zero temperature andinherently the manufacturing process may introduce varioustypes of defects into the system. Fabrication and designingdefects could occur both at the device and individual celllevels. Defects associated within a small array of cells isreferred to as device defects. Defects in the placement of a cellcan be caused by horizontal and/or vertical translation or byrotation. An imperfect cell can have asymmetric dot locationsand/or sizes, or missing dots. Several researchers haveinvestigated fault tolerance properties and robustness of QCAlogic gates during the past few years [15–19]. Recently, someof our investigations have focused on general fault toleranceproperties of QCA wires [20, 21]. In QCA systems, both thethermal as well as manufacturing faults are very important.With QCA being on such a small scale, the manufacturingprocess has to be highly precise. Any defect could lead topossible failure within the QCA structure. Therefore, thesepossible manufacturing defects and thermal effects need tobe studied in some detail before QCA can be implemented.In this paper, manufacturing fault tolerance properties andthe thermal effect on the operation of QCA wires will bediscussed. Specifically, randomly dot and cell displacementsas well as randomly missing dots are considered examples ofmanufacturing faults.

In a QCA cell, an extended Hubbard model is usedto describe the system of the two free electrons. For anarray of N cells, one can use either the full Hamiltonian[22] or an approximation method, such as inter-cellularHartree approximation (ICHA) [4]. ICHA is a ground stateself-consistent method: for every iteration step, quantumcalculations are done at the level of a given cell, updating thevalues for the dot charges using the values corresponding tothe ground state (the eigenstate of the cell Hamiltonian whichhas the smallest value), while the charge configuration fromthe other cells is used self-consistently. One can write theICHA-adapted Hamiltonian for a cell, α, as

Hα =∑i,σ

E0nαi,σ +

∑i

EQnαi,↑nα

i,↓

+ VQ

[1

2

∑β(β �=α)

∑i, jσ, µ

ρβ

j −ρ0

| �R(α)i − �R(β)

j |(nα

i,σ−ρ0)

+∑

i,j (i �=j)

(n(α)i,↑ − ρ0)(n

(α)j,↓ − ρ0)

| �R(α)i − �R(α)

j |

+∑i,j,σ

ρ(dr)j − ρ0

| �R(α)i − �R(0)

j |(n

(α)i,σ − ρ0)

+∑i, j, σ(i > j)

ti,j

(a

(α)+i,σ a

(α)j,σ + a

(α)+j,σ a

(α)i,σ

)]}

α, β = 1, . . . , N i, j = 1, . . . , 4. (1)

Here, the usual second-quantization notations are used.α and β are used to represent the cells, while i and j are usedfor the dots. E0(=130.6 meV) is the on-site energy of each dotand EQ(=846 meV) the energy needed to put two electrons ofopposite spin in the same dot. ρ

β

j is the self-consistent value

for the quantum probability of occupation of site j from thecell β, VQ is the electrostatic constant (VQ = e2/(4πε0)), ρ0

is the background positive charge per dot (ρ0 = 2/s) andρ

(dr)j is the quantum probability of occupation of site j in

the driver cell. The last term accounts for electron tunnellingbetween dots within the same cell. The coefficients representthe hopping energies between dots i and j . For an ideal 4 dotcell, the values for the tunnelling energies are, respectively,−0.290 meV and −0.008 meV between adjacent and diagonalpairs of dots. Using the canonical occupation number basis forthe Hilbert space, the matrices of 2nd quantization operatorsare calculated and the quantum averages are numericallycomputed. Quantum averages for the number operatorsN

(α)j = n

(α)j,↑ + n

(α)j,↓ are the probabilities of localization for

specific dots:ρ

(α)j = Tr(N (α)

j ρα). (2)

Here we have introduced the temperature influence viathe canonical ensemble: ρα = (1/Tr(e−Hα/(kBT )))e−Hα/(kBT ),where kB is the Boltzmann’s constant, T is the temperatureand α is the current cell. The updated charge configurationis given by the canonical ensemble averages rather than theground state quantum averages (which, however, is the limitof the former for T0). An a posteriori justification for thisprocedure is given in appendix A. We have shown that thecombination of ICHA and canonical ensemble works well forsmaller temperatures and high values of the driver polarization.

The simulation of manufacturing defects will be based ona Monte Carlo technique. In order to study fault toleranceproperties due to displaced dots and missing dots, defectparameters are introduced. Operation of the device has beenstudied as a joint function of temperature and the defectparameters.

In section 2, we present a description of the numericalmethod. Section 3 presents the results and discussions.Summary and conclusions are given in section 4.

2. Numerical methodology

We study a planar QCA array of N cells, each cell havings(=4) quantum dots and two extra mobile electrons of oppositespins. In this study, we have chosen the radius of a dotas 5 nm, the distance between two adjacent dots as 20 nm,and the separation between two cells as 42 nm. A schematicrepresentation of a QCA wire is shown in figure 1.

Using an occupation number basis for the Hilbert spaceof one cell and after some mathematical manipulation, thegeneral expression for the ‘diagonal’ matrix elements of theHamiltonian, Hα , is written as

〈�iαjα|Hα|�iαjα

〉 = 〈�iαjα|H0|�iαjα

〉 = 2E0 + EQδ(iα, jα)

+ VQ

∑β(β �= α)

j

(ρ

β

j −ρ0

)(1

| �R(α)iα

− �R(β)

j |+

1

| �R(α)jα

− �R(β)

j |

)

+ VQ

1 − δ(iα, jα)

| �R(α)iα

− �R(α)jα

|+ VQ

∑j

(ρ

(dr)j − ρ0

)(1

| �R(α)iα

− �R(0)j |

+1

| �R(α)jα

− �R(0)j |

)− ρ0VQ

[2

∑i,j

ρ(dr)j − ρ0

| �R(α)i − �R(0)

j |

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Fault properties in quantum-dot cellular automata devices

Figure 1. A schematic representation of a QCA wire.

+ 2∑

β(β �= α)j

(ρβ

j − ρ0)

(1

| �R(α)iα

− �R(β)

j |+

1

| �R(α)jα

− �R(β)

j |

)

+∑

jj �= iα

1

| �R(α)iα

− �R(α)j |

+∑

jj �= jα

1

| �R(α)jα

− �R(α)j |

]. (3)

The first term represents the total on-site energy of the electronsof the given cells. The second term shows the total energywhen both electrons are in the same dot, and the third termshows electron interactions between localized electrons fromthe given cells with the charge configuration from the otherdriven cells. The fourth term represents the interaction withinthe given cell. The interactions between the driver and theelectrons from the given cell are shown in the fifth term.Interactions between the positive background charges and thedriver cell are shown in the sixth term. The last two termsrepresent the interactions between the positive backgroundcharge and localized electrons within the driven cells. It shouldbe mentioned here that all the (non-significant) constant termsin the Hamiltonian have been dropped.

The off-diagonal elements, which are associated with thetunnelling effect, are given by⟨

�iαjα

∣∣ H1

∣∣�mαnα

⟩ = tiα,mαδjα,nα

+ tjα,nαδiα,mα

. (4)

Here the tunnelling parameters are the usual Hubbard hoppingenergies. For the ICHA method, a global convergenceparameter for the spectra of the cells is set,

εi =√∑

α

∑n

(Eiα,n − Ei−1

α,n )2, (5)

where Eiα,n is the nth energy eigenvalue for the cell α after the

ith iteration.QCA arrays can be used for logical devices if a binary

logical value is associated with a physical magnitude of everycell. Cell polarization, given by

P (α) = ρ(α)1 + ρ

(α)3 − ρ

(α)2 − ρ

(α)4

ρ(α)1 + · · · + ρ

(α)s

, (6)

is a good candidate because it gives a good estimation of thequadrupolar electric moment of the cell. The logical value canbe obtained from the relation, L(α) = (1 + P (α))/2.

Any fabrication-type defect study of a QCA device implieschanges in the electrostatic energy, the tunnelling energyand/or the on-site energy of the electron. In this paper, we arediscussing fault tolerance properties at the cell level where thedots are displaced or are missing from their ideal locations, buttheir sizes are considered to be perfect. Therefore, here, onlythe electrostatic and tunnelling energies are changed becauseof defects, while the first two terms in equation (3) do not

Figure 2. Tunnelling energy versus inter-dot distance.

change and are not considered in the simulation. For thequantitative estimation of the tunnelling energies, we useda simple double-well model for a pair of quantum dots andcomputed the tunnelling energy numerically for every givenvalue of the distance between dots. The results are shown infigure 2. One can note that for distances greater than 30 nmthe tunnelling energy is practically zero.

The numerical simulation for the fabrication defect study(dot and cell displacements) is carried out using normaldistribution where we have utilized the property of the errorfunction. The random locations of the dot or the cell aregenerated with the help of the error function and the parametercalled ‘displacement factor.’ The mathematical expressionfor individual random values of any dot coordinate withinthe array is given by xid + σ · erf−1(ξ) · |xid|. Here xid isthe corresponding ideal value, ξ is a random number chosenuniformly in the interval (-1,1) and erf is the usual errorfunction, while the dimensionless parameter σ is proportionalto the standard deviation. For the dot displacement case, σ

is a defect parameter is referred to here as the ‘displacementfactor.’ The case of the cell displacement within the array isconsidered almost identically, with the sole difference beingthat only transversal displacements are allowed. For the caseof randomly missing dot study, the defect parameter is definedas the ‘number of missing dots.’ A given number of dots are‘taken out’ from cells in a uniform random manner, using asingle stochastical selection step, so that they can be from anycell, except from the driver or the output cells. Care is taken,however, that no two dots will be taken from the same cell. Theeffect of temperature for the given values of these parametershave been studied.

In order to define a measure for operationality, for a giveninput Pin, the actual output Pout is compared with the expectedoutput Pout ideal. If, for a given trial of the random spreading

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M Khatun et al

(a) (b)

Successful operation

Unsuccessful operation

Figure 3. (a) Success rate versus the displacement factor of the dots within cells, for different temperatures. (b) Phase diagram of thesuccessful operation in the plane of temperature and displacement factor (dots within cells).

of quantum dots, the difference, δ = |Pout ideal − Pout|, is lessthan a given threshold parameter then the trial is considered tobe successful. Here, the threshold parameter is taken as 0.15.However, it has been tested that the results do not dependqualitatively on the value of the threshold parameter. Thisvalue corresponds to the output polarization Pout = 0.85 whenthe expected ideal value is Pout ideal = 1. The success rate, RS,as a measure of the operationality, is defined as the fraction ofsuccessful trials.

3. Results and discussions

We report the results of numerical simulations for a QCA wire.Simulation has been performed for a wire with nine cells.The success rate has been calculated for different values ofthe temperature and either for the displacement factor or fordifferent values of the ‘number of missing dots’ parameter.In order to avoid enormous computation time, ICHA hasbeen used jointly with the canonical ensemble. The globalconvergence in the full spectrum of all cells has been soughtand the tolerance parameter has been set to 10−3.

Figure 3(a) shows results for the numerical simulationof positional defects within cells. The dots from the cellsof the QCA wire (except for the input and output cells) aredisplaced randomly around the ideal positions using a normaldistribution. The success rate is plotted as a function of theparameter σ for different values of temperature.

For small temperatures, the QCA operation is successfulif the displacement factor is smaller than a breakdown value,which depends on temperature, as is shown in figure 4. Figure3(b) is a phase diagram in the plane (σ, T ) separating thestates with successful operationality. It is clearly seen that theoperation of the device is almost independent of temperaturefor temperature values smaller than ∼3.4 K because thebreakdown displacement fact is almost constant (∼0.1).

Figure 5(a) shows results for the numerical simulationof positional defects of cells within the wire. The cellsof the QCA wire (except for the input and output cells)are displaced vertically around the ideal positions using anormal distribution. The success rate is plotted versus thedisplacement factor for different values of temperature. Thegeneral trend of the curves is very similar to the previous case.For small temperatures, the QCA operation is successful if thedisplacement factor is smaller than a breakdown value.

Figure 4. Breakdown displacement factor (BDF) of the dots withincells versus temperature.

The breakdown displacement factor (BDF) dependson temperature according to the similar pattern for dotdisplacement, as shown in figure 6. Note that the temperatures-dependence of operation starts at values greater than 3.4 K. Inthis case the BDF is σ ≈0.6. The displacement factor here ismuch greater than the dot displacement case. This is becausethe functionality of a device should be more sensitive to the celllevel defect than to the array level. Figure 5(b) shows a phasediagram in the plane (σ, T ) separating the states of successfuloperation .

Either for dot or cell displacement, for highertemperatures, the thermal effect becomes considerable and fortemperatures higher than 6.5 K the device does not operateeven for the ideal case (σ = 0).

Figure 7 shows results for the numerical simulation of thedefects of cells when some dots are missing in the array. Agiven number of dots are ‘taken out’ from cells in a randommanner. The success rate is plotted as a function of temperaturefor different values of the ‘number of missing dots’ parameter.Results are compared with those corresponding to the idealcase. Even if, as figure 7 shows, the success rate is never equalto unity, which means that the device is not working properly,the presented results are important because they exhibit theinfluence of the fabrication quality. It is expected that theresponse on missing dot faults can be drastically improvedwhen using multi-line arrays. However, due to methodological

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Fault properties in quantum-dot cellular automata devices

(a)(b)

Successful operation

Unsuccessful operation

Figure 5. (a) Success rate versus the displacement factor of the cells in the wire, for different temperatures. (b) Phase diagram of thesuccessful operation in the plane of temperature and displacement factor (cells in array).

Figure 6. BDF of the cells within the array versus Temperature.

constraints given by the limitation to ICHA approximation,we were not able to study these cases here. The successrate is almost constant for temperature less than a breakdownvalue. The breakdown temperature increases slightly with thenumber of missing dots. For values less than the correspondingbreakdown temperatures, the value of the success ratedecreases with the number of missing dots. For temperaturesgreater than a given breakdown value the success rate fallsabruptly. One can notice that these breakdown values are veryclose to the critical value Tc = 3.36 K, which is the temperaturecorresponding to the tunnelling energy between dots.

4. Summary and conclusions

Using an extended Hubbard-type Hamiltonian, the canonicaldistribution, a Monte-Carlo generator of positional defectswithin or for QCA cells and the ICHA method a success ratefor QCA wires was numerically calculated as a function oftemperature and a defect parameter. The defect parameteris either a stochastic displacement factor or a ‘number ofmissing dots’ value. The values of the BDFs or temperatureswere numerically calculated. The domains of successfuloperation states have been found. More investigations needto be done in order to incorporate all types of manufacturingand operational defects. This method provides a general

Figure 7. Success rate versus temperature for different values of thenumber of missing dots parameter.

foundation for studying the fault tolerance properties of a QCAdevice. Therefore, the method can be implemented for morecomplex QCA devices.

Acknowledgment

This work was supported by the Indiana 21st Century Researchand Technology Fund, Grant No 04-492.

Appendix A. Validity of inter-cellular Hartreeapproximation method in canonical ensemblecalculation

A comparison between ICHA and the full basis quantumcalculation, when using the canonical ensemble, is shown here.By the self-consistent procedure the individuality of cells ispartially lost. Therefore, the average of polarization for allcells in the array is more appropriate for the description ofthe system. We calculate the polarization averages by bothmethods and compared them using the relative error:

e = |〈Pα〉 − 〈P ICHAα 〉|

|〈Pα〉| .

Here 〈Pα〉 is the average of the polarization for allcells obtained using full basis method and 〈P ICHA

α 〉 is the

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M Khatun et al

Figure 8. Relative error when using ICHA compared with full basis for a 2-cell QCA array: (a) for a sharp input polarization Pd = 1 versustemperature and (b) versus input polarization and Temperature (only values smaller than 0.1 are shown).

corresponding value from the self-consistent method. Thedependence of the relative error on temperature and inputpolarization was investigated. We have investigated a QCAwire made of two, three and four cells. In figure 8 the caseof a two-cell array is shown. In figure 8(a) the dependence ofthe relative error on temperature for a sharp input polarizationPd = 1 is presented. One may note that the relative erroris less than 5% for T < 4 K and less than 25% for highertemperatures. It is also noted that at T = 2.89 K the relativeerror becomes zero and gradually starts to rise. At T = 6.55 Ka small discontinuity is observed. Further investigationsare needed for clarifying these two issues. In figure 8(b)dependence of relative error on both input polarization andtemperature is shown (only the values of e smaller than 0.1are exhibited). The results for three- and four-cell systems arealmost identical (not shown here).

From this error calculation, one can see that the relativeerror is almost negligible at very high input polarization valuesand low temperatures. Therefore, we conclude that one canuse the ICHA method jointly with the canonical ensemble, toobtain thermal averages for the cells’ polarization for sharpinput polarization and low temperatures.

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