Analytical method for cell displacement defect quantum-dot cellular automata primitive

Considering the problems existing in the current complementary metal oxide semiconductor (CMOS) technology, such as short channel effects, ultra-thin gate oxides, high power density, quantum effects, complex lithography processes, and interconnection in the complex design, investigation of alternative technologies is urgently needed [1]. As a result, several new devices for computation are being explored to extend the historical integrated circuit (IC) scaling and to sustain the performance gain beyond CMOS scaling [2,3]. The International Technology Roadmap for Semiconductors (ITRS) 2015 described “Beyond CMOS” and mentioned that the quantum-based technology and computation paradigm are suitable candidates for the “Beyond CMOS” device technology [3].

Quantum-dot cellular automata (QCA) is an array of cells, in which each cell carries the binary information and the Coulombic interaction between cells is responsible for the information flow [4]. The main logic primitive in QCA is the majority voter (MV) and inverter [5]. Owing to its extremely small geometry, at the molecular scale, fabrication defects are oblivious in QCA compared with the existing CMOS technology. The fabrication of QCA includes two phases: Synthesis and deposition [6]. An individual cell is fabricated in the synthesis phase and deposited on the substrate to form a device in the deposition phase. In the deposition phase, defects including cell displacement, cell rotation, cell misalignment, missing cell, and additional cell are the most prominent. These defects could alter the functionality of QCA devices and hence circuits. Therefore, it is necessary to analyze these defects to ensure the devices work as desired.

The studies of QCA defects have been reviewed in Ref. [7] and the state-of-the-art was reported recently [8]. In Ref. [9] and Ref. [10], the displacement defect analysis was carried out using extensive simulation. In Ref. [11], the effects of cell movement were examined. In Ref. [12], the random cell displacement defect in QCA building blocks was analyzed. However, the mathematically analytical method for analyzing the displacement defect has not been available so far. In this paper, the kink energy based mathematical method for the displacement cell defect is proposed, which has been used to mathematically analyze the multiple missing cells (MMCs) defect in our earlier published work [13]. First, single-cell and multiple-cell displacement defects are analyzed by extensive simulation to act as a reference to validate the developed mathematically analytical method. Then, the mathematical model is applied to analyze the cell displacement defect and the result is compared with that of the simulation. The results reveal that this mathematically analytical method also works well for the cell displacement defect.

The rest of the paper is organized as follows: Section 2 presents the fundamentals of QCA, its defects, and some terminologies. In Section 3, simulation-based displacement defects in MV are analyzed. Section 4 presents the kink energy based mathematical model, and compares the result with the simulation result. The conclusion and further scope are given in Section 5.

The QCA cell schematic with four quantum dots is shown in Fig. 1 (a). Each cell consists of two highly free electrons which decide the cell polarization according to their positions. The cell polarization P is given by (1):

Figure 1.  QCA cell: (a) schematic, (b) polarization P = –1, and (c) polarization P = +1 [4].

Here ρ_i is the expectation value, where i is the quantum dot number as marked in Fig. 1 (a). Based on the positions of free electrons, the polarization of cells shown in Figs. 1 (b) and (c) is P = –1 and P = +1, respectively [4,14–17].

The basic QCA primitive is MV. MV with a five-cell structure is shown in Fig. 2 (a). MV with corresponding cell Cartesian coordinates is shown in Fig. 2 (b), which is useful to describe the displacement defect analysis in the later section. The Boolean function of MV is F = AB + BC + AC [5].

Figure 2.  QCA main primitive MV: (a) cell structure and (b) Cartesian coordinates.

Being at the molecular level, the existence of defects cannot be ignored in QCA. The possible defects induced in the synthesis and deposition phases in the QCA main primitive, MV, are summarized in Fig. 3. Fig. 3 (a) shows the missing electron/dot defect which results from the occurrence of the missing electron/dot or the extra electron/dot. It is very rare due to the ease of small inorganic molecules purification [18]. Also, it is easy to detect as it causes a fatal error. Fig. 3 (b) illustrates the cell misalignment defect, which is formed when the cell of MV is displaced in the horizontal direction from its original position. While when the cell of MV is vertically displaced from its actual position, it causes the displacement defect as shown in Fig. 3 (c). Both misalignment and displacement defects have been simulated and analyzed in Ref. [9] and Ref. [10]. A cell may get shifted as an angular position as shown in Fig. 3 (d), resulting in a cell rotation defect [19]. Additionally, the extra or missing cell is also possibly occurred due to improper removal of resist during the lithography process. These missing and additional cell defects are shown in Figs. 3 (e) and (f), respectively, and have been analyzed in Ref. [20].

Figure 3.  Cell defects in MV: (a) missing dot/electron, (b) cell misalignment, (c) cell displacement, (d) cell rotation, (e) missing cell, and (f) additional cell [13].

If the cells i and j are considered, electrostatic energy between these cells can be calculated by (2):

where ε₀ is the permittivity of free space, ε_r is the relative permittivity of a material, $ q_n^i $ is the charge in the dot n of the cell i, $ q_m^j $ is the charge in the dot m of the cell j, $ r_n^i $ is the position of the dot n in the cell i, and $ r_m^j $ is the position of the dot m in the cell j. Therefore, $ \left| {r_n^i} \right. - r_m^j\left. {} \right| $ is the distance between the dot n in the cell i and the dot m in the cell j. Since all the parameters except the distance between two dots, where the charges are localized, are constant, it can be concluded from (2) that the energy value is inversely proportional to the distance.

Kink energy ($E_{{\rm{kink}}} $) describes the Coulombic interaction between two cells. As shown in Fig. 4, it is determined by taking the difference of the electrostatic energy values of two cells, which are calculated when these two cells are having the opposite polarization or the same polarization, respectively.

Figure 4.  Interpretation of kink energy.

where $E_{{\text{opposite}}}^{i{\rm{,}}j}{\text{ }}$ is the energy between the cell i and the cell j with the opposite polarization and $E_{{\text{same}}}^{i{\rm{,}}j}{\text{ }}$ is that with the same polarization. Kink energy depends upon the dimensions of cells of a device. If the calculated kink energy value is positive (+ve), then the cell j adopts the same polarization as the cell i, whereas in the negative case (–ve), the cell j adopts the opposite polarization. The output cell polarization of any QCA device can be calculated and decided by kink energy. Hence, it is also useful to analyze the effect of the displacement defect in QCA primitives.

The parameter “radius of effects” is used to decide that whether cell/cells will interact with the reference cell or not [4]. All other cells which locate within the radius of effects of the reference cell (the middle cell) will interact with it, while the ones residing outside of the radius of effects will not, as shown in Fig. 5.

Figure 5.  Interpretation of the parameter, radius of effects [13].

In this section, displacement defects are analyzed by using very well-known QCADesigner [21], open-source QCA layout software, and the simulation tool developed by the Advanced Technology Information Processing Systems (ATIPS) Laboratory, University of Calgary. Although simulation-based displacement defects have been analyzed in Ref. [9] and Ref. [10], here they are presented again and compared with the mathematical analysis, as described in the following sections.

The QCADesigner’s bistable approximation engine is used and the simulation parameters are kept as depicted in Fig. 6. The other simulation parameters like the number of samples, convergence tolerance, and relative permittivity are kept as default. The simulation result of defect-free MV is shown in Fig. 7 (a). It is observed that, for all the eight combinations of inputs A, B, and C, the output is the majority of the inputs. To analyze the displacement defect in MV, each cell shown in Fig. 2 (b) is displaced by different distances from their original positions, including a single cell as well as multiple cells. Correspondingly, extensive simulation is carried out.

Figure 6.  Cell parameters and values for simulation.

Figure 7.  Simulation results of (a) defect-free MV and (b) MV with the cell displacement defect.

The effects of single-cell displacement are summarized below:

1) Cell (2, 3), cell (1, 2), or cell (2, 1): If any one of these cells is displaced by d < 20 nm (d is the displacement distance), then the desired output is obtained, whereas in the case of d > 20 nm, the output of MV is the same as the input B.

The effects of multiple-cell displacement are as follows:

1) Cells (2, 3) and (2, 1): Desired functioning is observed for d < 5 nm, but in the case of d > 5 nm, the output is the same as the input B.

2) Cells (2, 3) and (1, 2): Desired functioning is observed for d < 5 nm, but in the case of d > 5 nm, the output is the same as the input C.

3) Cells (1, 2) and (2, 1): Desired functioning is observed for d < 5 nm, but in the case of d > 5 nm, the output is the same as the input A.

4) Cells (2, 3), (1, 2), and (2, 1): Desired functioning is observed for d < 20 nm, but in the case of d > 20 nm, the output polarization is very low, i.e., the polarization is neither +1 nor –1.

The simulation result for the cell (2, 3) displacement by 30 nm is shown in Fig. 7 (b). The displacement of the cell (2, 3) causes the output of MV, which is the same as the input B.

It is observed that, in the single-cell displacement defect, MV functions properly if the distance d < 20 nm. Even though the cell is displaced by the distance d < 20 nm, it can interact with other cells due to the radius of effects. The effects of the cell displacement defect in MV for various displacement distances are summarized in Table 1.

In this section, a kink energy based mathematically analytical method is applied to mathematically analyze displacement defects in MV, which is also useful for other defects like cell misalignment and missing cell defects. Such a method can calculate the output polarization of any existing and new QCA devices with and without the defect. It is also useful to calculate the output polarization of defect-free MV and the MMCs defects in MV as described in Ref. [13].

In this subsection, two energy values of E_opposite and E_same are calculated by Case 1 and Case 2, respectively, to determine the final E_kink value and thus to analyze the effect of the single-cell displacement in MV. Here, the input cell A, i.e. (2, 3), of MV shown in Fig. 2 (b) is considered, which is displaced by a distance of d nm.

Case 1 is shown in Fig. 8 (a), where inputs A = 0, B = 0, and C = 1 are considered. According to the functionality of MV, the device cell polarization must be P = –1 or logic 0. Therefore, the undesired device cell polarization, i.e. P = +1 or logic 1, is considered to calculate E_opposite in Case 1. The dot labeling for each input cell A, B, and C and the device cell (middle cell) are shown in Fig. 8 (b). Four quantum dots of the input cell A are labeled as a, aa, a', and aa' clockwise. Similar nomenclature is applied for the other two cells B and C and the clockwise numbers of 1, 2, 3, and 4 are allotted to the device cell. The polarization of the cell A is P = –1. For the polarization of P = –1, the electrons are residing in the dot a and the dot a'. Similarly, the electrons of the device cell for P = +1 are localized in the dot 1 and the dot 3. Thus, a1 denotes the distance between the dot a of the input cell and the dot 1 of the device cell. In this way, the distances for E_opposite in Fig. 8 (a) can be calculated.

Figure 8.  MV with the cell A displaced by d nm for (a) E_opposite, (b) cell dimensions, and (c) E_same.

Equation (2) shows that, to obtain the energy value, the distance between the input cell and the device cell in which charges are located needs to be calculated firstly. Here the distance between the dot a of the input cell A and the dot 1 of the device cell, a1, is considered because charges are located in these dots. According to the cell parameters mentioned in Fig. 6 and Fig. 8 (b), the distances along the horizontal (x) and vertical (y) directions are 9 nm and 60 nm, respectively, for a1. So, based on the Pythagoras theorem, a1 is calculated as 60.67 nm, and the corresponding energy value between the cell A and the device cell is 0.01648k J, where k is a constant. In this way, the other distances from the cell B to the device cell and the cell C to the device cell are calculated for the given input combination A = 0, B = 0, and C = 1, and the obtained results are shown in Table 2. Further, all of these distances are summed up, thus achieving the final energy (${E}_{\text{opposite}} $) value:

Case 2 is depicted in Fig. 8 (c) for the $E_{{\rm{same}}} $ calculation. As shown in Fig. 8 (c), the electrons locate in the dots 2 and 4 of the device cell when its polarization P = –1 or logic 0. The cell parameters show that the distances for a2 (the distance between the dot a of the input cell and the dot 2 of the device cell) are 9 nm and 69 nm in the x and y directions, respectively. So, a2 is calculated as 69.584 nm and its corresponding energy value, based on (2), is calculated as 0.01437k J. Similarly, the other values of the distance and energy can be calculated and the results are shown in Table 3. Similar to E_opposite, E_same is given by (5) with all the distances mentioned in Table 3 considered.

Therefore, $E_{{\rm{kink}}} $ for the given example when A = 0, B = 0, and C = 1 can be achieved and it is the difference of $E_{{\rm{opposite}}} $ and $E_{{\rm{same}}} $, i.e., $E_{{\rm{kink}}} = E_{{\rm{opposite}}} - E_{{\rm{same}}} $ = 0.00005k J. In a similar way, the values of $E_{{\rm{opposite}}} $, $E_{{\rm{same}}} $, and $E_{{\rm{kink}}} $ for all eight combinations of inputs can be determined, as summarized in Table 4. On the basis of $E_{{\rm{kink}}} $, the output polarization (or logic value) of a particular input combination can be decided, which is shown in the last column of Table 4. It is observed that the output of MV in the case of the input cell (2, 3) with the displacement defect (d = 40 nm) is the same as the input B.

The results are also shown in Fig. 9 for better visual observation. As discussed in subsection 2.3, if E_kink has a +ve value, the output cell polarization is P = –1, i.e., logic 0, contrarily for –ve, the output cell polarization is P = +1, i.e., logic 1. As shown in Fig. 9, for the corresponding input combination depicted on the x axis, if the E_kink value is above the baseline then the polarization is considered as +ve and vice versa. From the values of polarization, it is observed that the output is always the same as the input B. Obviously, the result obtained by the mathematically analytical method matches well with that obtained by simulation shown in Fig. 7 (b).

Figure 9.  Energy values of E_opposite, E_same, and E_kink for the displacement defect in the case of eight input combinations.

Similarly, the values of E_opposite, E_same, and E_kink can be calculated for various displacement distances of d < 5 nm, d > 5 nm, d < 20 nm, d > 20 nm, and so on, as well as for single-cell and multiple-cell displacement defects. Similar results as mentioned in Table 1 are obtained by the developed mathematically analytical method.

In this paper, the kink energy based mathematically analytical method was adopted to calculate the polarization of defect-free MV in the cases of all input combinations. Further, this method was applied to investigate the effects of the cell displacement defect in MV. It is proven that the results obtained by simulation and the kink energy based mathematically analytical method match well for cell displacement defects in MV, which demonstrates the effectiveness and feasibility of this kink energy based mathematically analytical method in the analysis of cell displacement defects. The developed method can be extended to analyze the effects of cell rotation with various angles, misalignment of multiple cells, multiple additional cell defects, etc. Also, the fault model can be developed by the defect analysis for further testing of QCA circuits.

The authors declare no conflicts of interest.