# Impact of Molecular Electrostatics on Field-Coupled Nanocomputing and Quantum-Dot Cellular Automata Circuits

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## Abstract

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## 1. Introduction

## 2. Background

#### 2.1. The General QCA Paradigm and the Molecular Field-Coupled Nanocomputing

#### 2.2. Modelling of Molecular FCN Electrostatics

#### 2.3. Molecular Candidates and Species

## 3. Methodology

#### 3.1. Reference Molecular Model

#### 3.2. Energy Modeling

- For ${W}_{00}$, polarizations are ${P}_{DrC}=-1$ and ${P}_{CUT}=-1$
- For ${W}_{01}$, polarizations are ${P}_{DrC}=-1$ and ${P}_{CUT}=+1$
- For ${W}_{10}$, polarizations are ${P}_{DrC}=+1$ and ${P}_{CUT}=-1$
- For ${W}_{11}$, polarizations are ${P}_{DrC}=+1$ and ${P}_{CUT}=+1$

#### 3.3. Simulation Tools

## 4. Results: Basic Interactions

#### 4.1. Adjacent Cell-to-Cell Interaction

#### 4.2. Single Diagonal Cell-to-Cell Interaction

#### 4.3. Double Diagonal Cell-to-Cell Interaction

## 5. Results: Devices Analysis

#### 5.1. Device Crosstalk

#### 5.2. Majority Voter

#### 5.3. Fanout

#### 5.4. Single-Branch Inverter

#### 5.5. Double-Branch Inverter

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AC | Aggregated Charge |

CMOS | Complementary Metal-Oxide Semiconductor |

CUT | Cell Under Test |

DrC | Driver Cell |

FCN | Field-Coupled Nanocomputing |

GUI | Graphical User Interface |

MEP | Minimum Energy Path |

MoSQuiTo | Molecular Simulator Quantum-dot cellular automata Torino |

SCERPA | Self-Consistent Electrostatic Potential Algorithm |

QCA | Quantum-dot Cellular Automata |

VACT | Vin-Aggregated Charge Transcharacteristics |

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**Figure 1.**Information encoding, propagation, and elaboration. (

**a**) Information encoding in the general QCA paradigm: two electrons are positioned on the antipodal quantum dots of a QCA cell; (

**b**) information encoding in molecular FCN: the charge distributions of two adjacent molecules, and the consequent antisimmetric dipole moments ${\mu}_{1}$ and ${\mu}_{2}$, are positioned to mimic the configuration of a QCA cell; (

**c**) Molecular FCN wire: the information is propagated in a wire composed by six molecular cells divided into three clock zones; (

**d**) Molecular FCN majority voter.

**Figure 2.**Layouts implementing possible operations in molecular FCN. (

**a**) The fanout: the device replicates the information on three different output branches to enable information split; (

**b**) the single-branch inverter: the inversion is performed thanks to a single diagonal interaction occurring between the clock zone 1 and the clock zone 2; (

**c**) the double-branch inverter: the inversion is performed thanks to two diagonal interactions obtained between the clock zone 2 and the clock zone 3.

**Figure 3.**Derivation of the Aggregated Charge model through ab initio calculation. (

**a**) The bis-ferrocene molecule composed by two ferrocenes and the carbazole; (

**b**) the atomic charges obtained by fitting the electrostatic potential generate by the bis-ferrocene molecule that may be evaluate through ab initio calculation. The circles represent the group of atomic charges used to evaluate the aggregated charges; (

**c**) aggregated charge distribution obtained by summing the atomic charges in the specific groups.

**Figure 4.**Geometry of the reference molecule used in this work with the corresponding VACT for each molecular species. (

**a**) VACT of the neutral molecule; (

**b**) VACT of the oxidized molecule; (

**c**) VACT of the zwitterionic molecule.

**Figure 5.**Molecular FCN cell modeling. (

**a**) Molecular FCN cell composed by two molecules M1 and M2, with a highlight on the logical dot charges; (

**b**) two cells considered as Driver Cell (DrC) and Cell Under Test (CUT).

**Figure 6.**Molecular FCN switching and crosstalk fields for fundamental cell-to-cell interactions. (

**a**) Horizontal adjacent interaction; (

**b**) vertical adjacent interaction; (

**c**) single diagonal interaction; (

**d**) double diagonal interaction.

**Figure 7.**Energy analysis for the vertical adjacent interaction for the different molecular species. (

**a**) The four limit configurations analyzed to evaluate the energy of the system, showing the normalized dipole moments of the CUT molecules; (

**b**) energy of the two-cell system based on neutral molecules as a function of $\overline{{\mu}_{1}}$ and $\overline{{\mu}_{2}}$. Each line represents a 0.05 eV energy increment; (

**c**) energy of the two-cell system based on oxidized molecules as a function of $\overline{{\mu}_{1}}$ and $\overline{{\mu}_{2}}$; each line represents a 0.05 eV energy increment; (

**d**) energy of the two-cell system based on zwitterionic molecules as a function of $\overline{{\mu}_{1}}$ and $\overline{{\mu}_{2}}$. Each line represents a 0.05 eV energy increment.

**Figure 8.**Energy analysis for the single diagonal interaction for the different molecular species. (

**a**) Energy of the system based on neutral molecules as a function of the input–output polarization. Each line represents a 0.005 eV energy increment; (

**b**) energy of the system based on oxidized molecules as a function of the input–output polarization. Each line represents a 0.02 eV energy increment; (

**c**) Kink Error (${\Delta}_{zw}$) associated with the system based on zwitterionic molecules as a function of the counterion position; the Kink Error is always in the range $[{\Delta}_{n},{\Delta}_{ox}]$.

**Figure 9.**Energy analysis for the double diagonal interaction for neutral, oxidized and zwitterionic molecular species: energy of the layout of Figure 6d as a function of the DrC and CUT polarizations. Each line represents a 0.005 eV energy increment.

**Figure 10.**Crosstalk field analysis. (

**a**) System used to analyze the crosstalk. The crosstalk field is evaluated on a line orthogonal to a fully-polarized molecular wire, the position on the evaluation line represent the eventual center-center distance of a fictitious molecular wire; (

**b**) comparison among the generated crosstalk field as a function of the distance D implemented with different molecular species.

**Figure 11.**SCERPA simulation of a majority voter, the figures show the electrostatic potential evaluated 0.2 nm above the active dot plane: the spots indicate the aggregated charges. (

**a**) Majority voter made with oxidized molecules for four significant input configurations; (

**b**) majority voter made with neutral molecules; (

**c**) majority voter made with zwitterionic molecules with counterion position $z=-5$ Å.

**Figure 12.**Simulation of the majority voter performed with QCADesigner. (

**a**) Schematic of the majority voter drawn in QCADesigner. The colours are slightly changed from the original for the sake of clarity; (

**b**) QCADesigner simulation results showing the cell polarization versus the simulation step.

**Figure 13.**SCERPA simulation of a fanout circuit, the figures show the electrostatic potential evaluated 0.2 nm above the active dot plane: the spots indicate the aggregated charge. (

**a**) Fanout circuit made with oxidized molecules for both the input configurations; (

**b**) fanout circuit made with neutral molecules for both the input configurations; (

**c**) fanout circuit made with zwitterionic molecules for both the input configurations and with counterion position $z=-5$ Å; (

**d**) fanout circuit made with zwitterionic molecules for both the input configurations and with counterion position $z=5$ Å (same plane of the logic dots).

**Figure 14.**Simulation of the fanout circuit performed with QCADesigner. (

**a**) Schematic of the fanout drawn in QCADesigner. The colors are slightly changed from the original for the sake of clarity; (

**b**) QCADesigner simulation results showing the cell polarization versus the simulation step. For the sake of simplicity, only a single output is reported. The three outputs are equal.

**Figure 15.**SCERPA simulation of a single-branch inverter, the figures show the electrostatic potential evaluated 0.2 nm above the logic dot plane: the spots indicate the aggregated charge. The circle encases the most involved aggregated charges in the diagonal interaction. (

**a**) Single-branch inverter made with neutral molecules with input logic ‘1’; (

**b**) single-branch inverter made with neutral molecules with input logic ‘0’; (

**c**) single-branch inverter made with oxidized molecules with input logic ‘1’; (

**d**) single-branch inverter made with oxidized molecules with input logic ‘0’; (

**e**) single-branch inverter made with zwitterionic molecules with counterion position $z=5$ Å (the same plane of the logic dots); (

**f**) single-branch inverter made with zwitterionic molecules with counterion position $z=0$ Å; (

**g**) single-branch inverter made with zwitterionic molecules with counterion position $z=-5$ Å.

**Figure 16.**Simulation of the single-branch inverter performed with QCADesigner. (

**a**) Schematic of the single-branch inverter drawn in QCADesigner. The colors are slightly changed from the original for the sake of clarity; (

**b**) QCADesigner simulation results showing the cell polarization versus the simulation step.

**Figure 17.**SCERPA simulation of a double-branch inverter, the figures show the electrostatic potential evaluated 0.2 nm above the logic dot plane: the spots indicate the aggregated charge. (

**a**) Double diagonal interaction involved in the double-branch inverter made with oxidized molecules with input logic ‘0’ (

**left**) and with input logic ‘1’ (

**right**); (

**b**) double-branch inverter made with neutral molecules with input logic ‘1’; (

**c**) double-branch inverter made with oxidized molecules with input logic ‘1’; (

**d**) double-branch inverter made with zwitterionic molecules with input logic ‘1’ and counterion position $z=-5$ Å.

**Figure 18.**Simulation of the double-branch inverter performed with QCADesigner. (

**a**) Schematic of the double-branch inverter drawn in QCADesigner. The colors are slightly changed from the original for the sake of clarity; (

**b**) QCADesigner simulation results showing the cell polarization versus the simulation step.

Interaction | Molecular Species | ${\mathit{W}}_{00}$ (eV) | ${\mathit{W}}_{01}$ (eV) | ${\mathit{W}}_{10}$ (eV) | ${\mathit{W}}_{11}$ (eV) | ${\mathit{E}}_{\mathit{k}0}$ (eV) | ${\mathit{E}}_{\mathit{k}1}$ (eV) | $\Delta $ * |
---|---|---|---|---|---|---|---|---|

Parallel Horizontal | Neutral | −0.147 | 0.147 | 0.147 | −0.147 | −0.294 | −0.294 | 0.000 |

Oxidized | 2.914 | 3.208 | 3.208 | 2.914 | −0.294 | −0.294 | 0.000 | |

Zwitter | 0.272 | 0.567 | 0.567 | 0.272 | −0.294 | −0.294 | 0.000 | |

Single diagonal | Neutral | 0.033 | −0.033 | −0.033 | 0.033 | 0.066 | 0.066 | 0.000 |

Oxidized | 2.376 | 2.087 | 2.087 | 1.929 | 0.289 | −0.158 | 0.447 | |

Zwitter | 0.148 | 0.044 | 0.044 | 0.071 | 0.104 | 0.027 | 0.077 | |

Double diagonal | Neutral | 0.061 | −0.070 | −0.070 | 0.061 | 0.131 | 0.131 | 0.000 |

Oxidized | 5.762 | 5.631 | 5.631 | 5.762 | 0.131 | 0.131 | 0.000 | |

Zwitter | 0.240 | 0.109 | 0.109 | 0.240 | 0.131 | 0.131 | 0.000 |

**Table 2.**Energy analysis of the adjacent vertical interaction when the DrC encodes a fixed logic ‘0’. The subscript numbers refer to the normalized dipole moment of the two molecules composing the CUT, as Figure 7a schematically shows. The values in parentheses are the energy without considering the electrostatic contribution of charges within the same molecule.

Molecular Species | ${\mathit{W}}_{-1,1}$ (eV) | ${\mathit{W}}_{-1,-1}$ (eV) | ${\mathit{W}}_{1,1}$ (eV) | ${\mathit{W}}_{1,-1}$ (eV) |
---|---|---|---|---|

Neutral | −1.71 (−0.27) | −1.44 (0.00) | −1.44 (0.00) | −2.01 (−0.57) |

Oxidized | 5.24 (5.24) | 6.28 (6.28) | 4.76 (4.76) | 4.95 (4.95) |

Zwitterionic | −7.54 (0.61) | −7.09 (1.05) | −7.44 (0.71) | −7.83 (0.31) |

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**MDPI and ACS Style**

Ardesi, Y.; Beretta, G.; Vacca, M.; Piccinini, G.; Graziano, M. Impact of Molecular Electrostatics on Field-Coupled Nanocomputing and Quantum-Dot Cellular Automata Circuits. *Electronics* **2022**, *11*, 276.
https://doi.org/10.3390/electronics11020276

**AMA Style**

Ardesi Y, Beretta G, Vacca M, Piccinini G, Graziano M. Impact of Molecular Electrostatics on Field-Coupled Nanocomputing and Quantum-Dot Cellular Automata Circuits. *Electronics*. 2022; 11(2):276.
https://doi.org/10.3390/electronics11020276

**Chicago/Turabian Style**

Ardesi, Yuri, Giuliana Beretta, Marco Vacca, Gianluca Piccinini, and Mariagrazia Graziano. 2022. "Impact of Molecular Electrostatics on Field-Coupled Nanocomputing and Quantum-Dot Cellular Automata Circuits" *Electronics* 11, no. 2: 276.
https://doi.org/10.3390/electronics11020276