## 3 Proposed QCA Circuits
### Physical Proof of the Proposed 2-to-1 Multiplexer  
Equation (8) is a formula for calculating the potential energy due to the interaction between electrons. \(U\) is the potential energy, \(k\) is the Coulomb constant, \(q_{1}\) and \(q_{2}\) are the magnitudes of the charge of the electron, and \(r\) is the distance between the two electrons. At this time, since \(k\), \(q_{1}\), and \(q_{2}\) are constants, the numerator can be treated as a constant in Equation (8). The corresponding constant value is called \(A\) and is expressed as in Equation (9). In addition, the potential energy of the cell's electronic configuration is expressed as the sum of the potential energies of two electrons in the cell (\(U_{T}\)) and is shown in Equation (10) [36]. \[U=\frac{kq_{1}q_{2}}{r} \tag{8}\]  
\[A=kq_{1}q_{2}=9\times 10^{9}\times\left(1.6^{2}\right)\times 10^{-38}=23.04 \times 10^{-29} \tag{9}\]  
\[U_{T}=\sum_{i=1}^{2}U_{i} \tag{10}\]  
Before making a physical proof, some assumptions are necessary. First, it is assumed that all cells are square, with a side length of 18 nm and a distance between cells of 2 nm. Second, the quantum dots of the QCA cell are located at the vertices of the cell, and electrons are also located at the center of the QCA cell. Third, the maximum distance for the interaction between electrons is 80 nm. In other words, if the distance between electrons exceeds 80 nm, they cannot affect each other. For the multiplexer proposed in this paper to output the value of A, SEL = 1 is required. In order to physically prove this, it is assumed that A, SEL = 1, and B = \(-\)1, and physical proof is performed. Figure 17 is physical proof of the proposed multiplexer. There are five cells whose polarization is unknown. Each cell is numbered and the polarization corresponding to each cell is obtained. Since the proposed multiplexer outputs the polarization opposite to that of cell 5, the output value of the multiplexer can be determined by finding the polarization of cell 5. Physical proof was performed using the proposed MXL and showed that left-right reversal does not affect the overall result. Figure 16: Dissipated power maps of RAM cells (0.5 Ek): (**a**) [20]; (**b**) [21]; (**c**) MXLR; (**d**) MXCR. The results of the comparison showed that since the potential energy when the polarization of cell 1 = +1 is smaller than the potential energy when the polarization is \(-1\), the polarization of cell 1 is determined to be +1. The polarization of cell 2 is +1 because two cells with the same polarization are adjacent to each other. For the polarization of cell 3, the potential energy generated by the electrons inside the fixed cell +1, \(-1\), input cells S, A, B, and cell 1 and cell 2 is obtained, which is the same as Equations (15) and (16). \[P=+1:U_{T}=U_{X}+U_{Y}=1.02\times 10^{-19}+1.035\times 10^{-19}=2.055\times 10^{-19} \tag{15}\]  
\[P=-1:U_{T}=U_{X}+U_{Y}=1.865\times 10^{-19}+2.082\times 10^{-20}=3.947\times 10^{-19} \tag{16}\]  
The results of Equations (15) and (16) show that when the polarization of cell 3 is +1, the potential energy is lower, thus the polarization of cell 3 is determined to be +1. The polarization of cell 4 can also be obtained using the potential energy with the 7 cells mentioned above, and the results are obtained using Equations (17) and (18), and the polarization of cell 4 is +1, based on the results. Now, the polarization of cell 5 is obtained through the potential energy generated between the above-mentioned 7 cells, cell 3, and cell 4.