Defect-tolerance analysis of fundamental quantum-dot cellular automata devices

: Quantum-dot cellular automata (QCA) is a burgeoning technology at the nano-scale range, with the potential for lower power con-sumption, smaller size and faster speed than conventional complementary metal – oxide semiconductor-based technology. Because of its ultra-density integration and its inherent physical properties, fault-tolerance is an important property to consider in the research and manufacture of QCA. In this paper, one type of defect, in which displacement and misalignment occur coinstantaneously, is investigated in detail on rudi-mentary QCA devices (majority voter (MV), inverter, wire) with QCADesigner. Another MV with rotated cells is also proposed, and it is more robust than the original one. Simulation results present the defect-tolerance of these devices, that is, the maximum precise region the defective cell can be moved moreover, with correct logical function. These conclusions have a meaningful guiding signi ﬁ cance for QCA physical implementation and fault-tolerance research.


Introduction
Current transistor-based technique is approaching the physical limits [1], and this system has found several drawbacks such as high leakage current, high lithography cost and limitation of frequency in gigahertz range [2]. New technology must be taken into account for enhancing the overall performances of the system. Quantum-dot cellular automata (QCA), which is a novel technology and promising alternative to complementary metaloxide semiconductor, was first proposed by Lent et al. [3]i n 1993 and experimentally verified [4] in 1997.
QCA implements logical operations and propagates information by unalloyed Coulombic interactions of electrons within QCA cells between the cells rather than transport of charge such as in a conventional transistor [5,6]. Conventional binary information is represented by the electron configurations confined in QCA cell. The basic QCA devices are three-input majority gate and inverter [6]. In addition, wire crossings, including coplanar wire crossing and multilayer crossing, play an important role in logical design of digital system [7,8].
Dysart et al. [9][10][11][12][13] examined the possible defects that may occur in the fabrication of metal-dots and molecular implementations in QCA systems. Fault models and methods for improving defect tolerance have also been developed; as have complete joint probability models based on Bayesian network, as presented in [14][15][16][17]. This method is for abstracting a probabilistic operation of the behaviour of circuit components in QCA circuit. Yang et al. [18] specifically investigated rotation defect of the QCA fundamental devices (straight wire, bend wire, fanout, inverter and majority voter (MV)). Tahoori et al. [19][20][21][22][23] systematically researched all possible misalignment on the direction of the one-dimensional. However, the movement of cells is in one direction in each trial. It cannot determine the precise defect-tolerance. In this paper, based on three kinds of fundamental devices, we examined the two-dimension fault-tolerance by moving the cells in horizontal and in vertical directions at the same time.
The rest of this paper is as follows: in Section 2, the background of QCA technology is presented; Section 3 outlines the QCA theoretical foundation of bistable approximation engine; in Section 4 simulation method and defect device models are presented; Section 5 shows the simulation results of these defective devices; finally, this paper is concluded in Section 6.

Background
In this section, QCA background will be described.

QCA cell
A QCA cell is a square figure with four quantum dots confining two mobile electrons, which are able to tunnel between adjacent dots but not between adjacent cells [6]. These four dots which are located at the corners of one cell and the two free electrons tend to occupy diagonally opposed dots for obtaining the maximum distance because of Coulombic repulsion. The electrons are polarised in either +1.00 polarisations (binary 1) or −1.00 polarisations (binary 0) [6,24], as shown in Fig. 1a.
The polarisation is defined as (1) where ρ i denotes the electronic charge at dot i.

Logic device
In QCA, different geometrical arrangements of cells correspond to different logic gates. The fundamental logic gates, that are three-input MV and inverter, are described in previous works. This type of QCA structure is a highly irregular array of cell [3]. The device configurations are shown in Figs. 1b and c. MV is described as MV (A, B, C) = AB + AC + BC. The output is the more one of inputs. AND gate and OR gate can be implemented with three-input majority gates by fixing the polarisation of one input as logic '1' or logic '0'.
Inverter is described as a conversion of the input.

Clock
Clock, to synchronise and control the direction of the information flow, plays an important role in QCA circuit. As shown in Fig. 2, it has four clock zones, each shifted in phases by 90°. And each clock zone has four states: relax, switch, hold and release. Multi-phased clocking mechanism is used in the QCA circuit not only for information flow but provides energy as power dissipation of circuits.

Theoretical calculation
This section outlines the QCA theoretical foundation of bistable approximation engine used in this paper for simulating the displacement and misalignment defects.

Radius of effect
The radius of effect is the distance of two cell centres, and refers to the Coulombic interaction of these two. The interaction exists within the radius of effect while disappears beyond it as shown in Fig. 3.

Kink energy
The interaction between any two cells is the only Coulombic interaction that is described by the kink energy E kink of the inter-cellular Hartree-Fork approximation. E kink is the electrostatic energy difference of two adjacent fully polarised cells with opposite polarisation and the same polarisation [25]. The electrostatic energy between cells i and j is given by where ε 0 is the permittivity of vacuum, ε r is the relative dielectric constant of the material system, q i k is the charge in dot k of cell i, r i k − r j l expresses the distance between the kth dot in the cell i and the lth dot in the cell j. Note that the distance between the centre of cell i and the centre of cell j must be less than the radius of effect. Then the kink energy between cell i and cell j is The kink energy only related to the size of the QCA cells and the spacing between the considered cells but it does not depend on the temperature [26,27].

Inter-cellular Hartree-Fork approximation
The inter-cellular Hartree-Fork approximation assumes that cells interact only through expectation values ignoring inter-cellular entanglement effects altogether, and solve the time-independent Schrödinger equation for each individual cell separately. In this assumption, an N-cell system is divided into a set of N single-cell subsystems. Then, the Hamiltonian for a single cell, i, is simplŷ where E k i, j is the kink energy between cell i and cell j, that is, neighbours within the radius of effect of cell i. P j is the polarisation of the jth neighbouring cell. The polarisation of any cell i can be evaluated using [28] Equation (5) is the mathematical expression of the non-linear cell-cell response function as shown in Fig. 4. A very small polarisation in cell i induces electrostatically cell j to be polarised completely. This non-linear response is the basis of representation of binary information in QCA cells.

Physical modelling simulation setup
In this section, simulation method and defect device models are presented. To acquire the maximum region that a cell can be moved, and with correct logic function, considerable simulations are then conducted.

Defect modelling
For QCA, defects are possible unexpected occurrences in both the synthesis phase (which may cause extra or missing electrons or/and dots in one cell) and the deposition phase (which will result in cell displacement) in the present stage of QCA manufacturing, but mainly on the deposition phase because of its inherent property [20,[28][29][30][31][32]. The number of manufacturing defects has been predicted to be as high as 50% of the devices [9]. Therefore only the following cell defects in the deposition phase are considered in this paper: † Cell displacement: The direction of the defective cell is misplaced but no misalignment. † Cell misalignment: The defective cell is not properly aligned with other cells.
In this paper, we assume that only the following defect is considered and simulated for QCA device with faultless manufacture and operation: cell displacement and cell misalignment both occur concurrently in only one cell defect in order to figure out the maximum region this cell can be moved around in. In each simulation, each cell is then moved in the horizontal and vertical directions by steps of 0.1 nm. The coordinates are negative for cells moved left  or down, and positive for cells moved right or up based on the original cells centre with free defect. We define a threshold polarisation 0.5 (or −0.5). If output polarisation >0.5, then the output is a correct logic 1, and if output polarisation <−0.5, the output is a correct logic 0. If not, the trial is considered to be unsuccessful and the distance the defective cell moved is greater than the effective. The different provisions of different models are described in each kind of device. MV defect: A different MV with rotated cell is proposed as shown in Fig. 5a. Fig. 5b is the simulation result of the MV with 18 × 18 nm 2 cell size, 2 nm cell distance and 5 nm dot size. Table 1 shows the polarisations of different physical properties of original MV and rotated cells MV. As we can see from these data, the polarisations of the latter are slightly larger than the former. Fig. 6 shows the possible defective models of MVs described earlier. The input A1 (A2) can be moved left both up to A1′ (A2′). We do not consider cell C1 (C2) because of it is A1 (A2) symmetrical cell. The input B1 (B2) can be moved left both up (or both down) to B1′ (B2′). Similarly, cell F1 (F2) can be moved to F1′ (F2′). And the device cell is a fault-free cell.
Inverter defect: Fig. 7 shows the different defective models of inverter. We just consider cells A, 1, 2, 4, 6 and F because of the inverter is a symmetrical geometrical arrangement of cells. And cell 1 just can be moved to right.   Wire defect: For wire defects, just double binary wires and double inverter chains have been investigated. These two faultless wires are shown in Fig. 8. The cell-centre to cell-centre distance of two parallel wires is 40 nm. The cells investigated have been labelled A1, 11,12,13,14,15 and FA1 in double binary wires and A2, 21, 22, 23, 24, 25 and FA2 in double inverter chains correspondingly. Fig. 9 shows the defects in wires. For five middle cells, the defects are simulated just by moving one cell in the upper wire towards the lower wire to investigate the effect of defects in the wire to the information transmission. It must be ensured that the two output polarisations of parallel defective wires are both correct or the simulation will be classified as a failure.

Simulation parameter
The values for the physical parameters shown later were the defaults of QCADesigner (version 2.0.3). Cell size for displacement and misalignment defect simulations is 18 × 18 nm 2 ; the cell-to-cell distance is 2 nm and dot size is 5 nm. † Radius of effect = 65 nm. This value is used to determine the local neighbours for a cell. The cell-centre to cell-centre distance exceeding in this value will not work. † γ = 3.8 × 10 −23 . This is ∼0.16 E k , E k is the ideal kink energy between two non-defective cells, and for the cell used here E k is ∼0.0015 eV.

Physical modelling simulation result
The simulation results of defect models of fundamental QCA devices modelling proposed in Section 4 have been presented in this section. Clearly, the area of F2 can be moved is greater than the area of F1 can be moved. And these regions are all symmetric because of the symmetry of QCA device cells. Combined with the polarisations of two MVs, we may conclude that the fault-tolerance of RCMV is stronger than the MV's in a manner.

Inverter defect simulation result
For cell 1, it just can be moved right 1.9 nm. And cells 4 and 6 can be moved down 14.2 and 17.5 nm, respectively but these two cells can be moved up to infinity because they are just in one branch of inverter. The cases of input cell A and output cell F of inverter are more complicated. In Fig. 11a, the two areas surrounded by red lines are the failure areas because cell A and cell 2 (or cell 3) can constitute an inverter in these areas. The border of cell F with correct output has been divided into three regions. The cause of this case and cell A's is very much alike. The area of cell 1 can be moved is very limited, and the maximum distance in horizontal or vertical direction do not exceed 3 nm. In addition, the symmetrical cells A and F of QCA device have symmetrical fault-tolerance regions but cell 1.

Wire defect simulation result
Cell 11 can be moved down 20 nm maximal both right 6.9 nm in the meanwhile and cells 12, 13, 14, 15 can be only moved down 19.7, 18.8, 18.3 and 17.6 nm in double binary wires, respectively. And cells 21, 22, 23, 24, 25 can be moved down 5.3, 4.9, 4.9, 5.6 and 6.5 nm, respectively. As shown in Fig. 12, the region of A1 can be moved is very larger than A2 can be moved but the differences of output cells are smaller. From these data, we can draw the conclusion that the double binary wires more stable than the double inverter chains in that way. Moreover, the regions of input cells are symmetrical but the output cells regions are not. This shows that the lower wire of two parallel wires in double binary wires or in double inverter chains has some effect to the upper wire, so that the regions of outputs are not symmetrical.

Conclusion
Quantum-dot cellular automata is a promising novel computing pattern in the nano size range. However, as a rising technology, various fabrication defects are expected to be 50% of the device. Therefore, in this paper, the behaviours of cell displacement and misalignment occurring simultaneously in basic QCA devices are simulated in order to determine the maximum region the defective cell can be moved while ensuring that the logic function of the defect device remains correct. The results show that different QCA devices (MV, inverter and wire) have different fault-tolerance regions. We find that the fault-tolerance of RCMV is stronger than the MV's based on the polarisations and maximum regions of these two devices. Simulation results show that the situations of inverter are more complicated because of an inverter can be made up with two cells. Therefore the graphics are very irregular. And in wires, it has some effect between two parallel wires to outputs.

Acknowledgment
This work was supported by the National Natural Science Foundation of China (grant no. 61271122).