# Effectiveness of Molecules for Quantum Cellular Automata as Computing Devices

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## Abstract

**:**

## 1. Introduction

## 2. Background: Molecular QCA

## 3. Methodology: A Quantitative Approach for MQCA Characterization

**Ab initio simulation**: accurate ab initio simulations based on quantum chemistry, are performed to analyze the electronic structure and the physical properties of the molecule. External stimuli are inserted in the simulation to study the molecule in different conditions: electric fields and point charges emulate the synchronizing clock, switching fields and driver.**Post processing of ab initio simulation**: from ab initio simulations we derive the key features, as figures of merit, describing the electrostatic behavior of the molecule which are used to describe the interaction with other molecules and to provide the quantitative information necessary for the analysis of molecular circuits. The figures of merit encase all the physical-chemical properties of the molecules in quantities which can be used by engineers to understand, simulate, design and fabricate MQCA devices.**System-level analysis**: the figures of merit defined in the second step of MosQuiTo enable the study of molecular circuits at the system level. The information propagation in MQCA circuits can be evaluated considering parameters which are closer to the electronic engineering techniques: the engineer does not have to deal with the physics of the device and can analyze it from a higher level of abstraction. An algorithm for the evaluation of molecular propagation in large MQCA circuits, which exploits the figures of merit, has been developed and published in [29].

**switching capability**of the molecules when subjected to an electric field generated by external electrodes or other molecules; (ii) the effectiveness of the switched molecule in potentially influencing the nearby molecule, and

**the effectiveness in transferring the information**. We thus define new macro-quantities, derived by the post-processing of ab initio results: simple quantities provide a complete characterization of molecules when embedded in real systems. These figures of merit are presented and discussed in the following sub-sections.

#### 3.1. Aggregated Charge (AC)

**Definition AC:**We compute the atomic charges of the three molecules using the ESP approximation [32] and we define the so-called Aggregated Charge (AC) of the dots simply by summing the charges of the atoms constituting the dot, as shown in Figure 2a–c for the three considered molecules.

#### 3.2. Electric-Field Generated at the Receiver (EFGR)

**Definition EFGR:**As sketched in Figure 3b, we suppose to have a fictitious nearby molecule, named Receiver, at the ideal distance from the MUT (distance that leads to a square-shaped QCA cell). We compute the electric field, generated by the charge distribution of the MUT, that impacts the virtual Receiver molecule. The computed electric field is the electric field generated at the Receiver, herein EFGR.

#### 3.3. Vin–Vout Transcharacteristics (VVT) and Vin–AC Transcharacteristics (VACT)

**Definition Vout:**The voltage at the MUT named input voltage or Vin, is evaluated by integrating the electric field that influences the molecule along the segment connecting the active dots of the MUT. Vin is a practical ‘measure’ of the field that causes the MUT to change its charge distribution. The field which is considered in the evaluation might be generated by nearby molecules, as shown in Figure 3a, or by a switching field inserted in the ab initio simulation, as shown in Figure 3b.

**Clarification:**The evaluation of the potential could be performed also using the dipole moment, that can be obtained from the Aggregated Charge distribution using Equation (1), instead of integrating the electric field generated by the Aggregated Charge distribution. The potential generated by a dipole moment $\overrightarrow{\mu}$ can be evaluated in a generic point $\overrightarrow{r}$ as

**Definition VACT:**We define the Vin–AC Transcharacteristics (VACT) as the relation between the input voltage and the Aggregated Charge. The VACT is an $\mathbb{R}\to {\mathbb{R}}^{N}$ function, where N is the number of Aggregated Charges describing the molecule. The VACT is independent of the Receiver position (i.e., the VACT can be used to evaluate the field generated by the molecule, when inserted in an electric field, in any point of the space), therefore, it generalizes the study of the intermolecular interaction.

**Definition Vout:**The voltage at the Receiver, named output voltage or Vout, is evaluated by integrating the electric field generated by the MUT (EFGR), computed using the Aggregated Charges, on the segment that connects the Receiver active dots (Dot 1 and Dot 2). Vout, again, is a practical measure of the field generated by the MUT. The output voltage is in evidence in Figure 3a,b.

**Definition VVT:**The relation between the two voltages, Vin and Vout, is the Vin–Vout transcharacteristics (VTT) and it provides a description of the molecule operation. Clearly, the VTT can be obtained from the VACT.

**Clarification:**A similar relation has been proposed for a two-dot molecule in [7] following a probabilistic approach, using Two-State Approximation (TSA). The polarization of the MUT is evaluated analytically approximating the molecule as a two-state quantum system. The two states that represent the logic states are named ${\psi}_{A}$ and ${\psi}_{B}$ and the wave function of the molecular system is written as $\psi ={c}_{a}{\psi}_{A}+{c}_{b}{\psi}_{B}$. The molecule polarization is defined as ${P}_{2}={c}_{a}^{2}-{c}_{b}^{2}$.

#### 3.4. Vout Maps (VOM) and MQCA Cell Working Zone (CWZ)

**Definition VOM:**The VOM is a map of the input voltage Vin that the Receiver experiences when placed in a surrounding position with respect to the MUT. An example of VOM is shown in Figure 3c.

**Definition CWZ:**We define the Cell Working Zone (CWZ) as the intersection between the two regions where the Receiver can encode the two logic values when induced by the proper EF. The CWZ is the real working zone of the Receiver

## 4. Results: Neutral Molecules

## 5. Results: Oxidized Molecules

#### 5.1. Aggregated Charge–AC/VACT

#### 5.2. Generated Field

#### 5.3. Generated Field at the Receiver–EFGR

#### 5.4. Molecule Transcharacteristics–VVT

#### 5.5. Output Voltage Maps–VOM

#### 5.6. MQCA Cell Working Zone–CWZ

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC | Aggregated Charge |

CMOS | Complementary Metal-Oxide Semiconductor |

CV | Cyclic-Voltammetry |

CWZ | Cell Working Zone |

DFT | Density Functional Theory |

EF | Electric Field |

EFGR | Electric Field Generated at the Receiver molecule |

HF | Hartree-Fock |

MQCA | Molecular Quantum-dot Cellular Automata |

MUT | Molecule Under Test |

QCA | Quantum-dot Cellular Automata |

TSA | Two-State Approximation |

VACT | Vin–Aggregated Charge Transcharacteristics |

VOT | VOuT Map |

VVT | Vin–Vout Transcharacteristics |

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**Figure 1.**Six dots molecular QCA cell. (

**a**) The free charges are confined in the main dots along a diagonal, encoding thus the logic states 1 and 0; (

**b**) When the free charges are forced in the central dot the cell is in the so-called NULL state.

**Figure 2.**Structure of the QCA candidate molecules. (

**a**) Diallyl-butane: the two allyl groups represent the dots. Here a third fictitious dot, including the butane bridge, is sketched just for charge balancing reason; (

**b**) Decatriene: the three ethylene groups are the dots; in particular, dot1 and dot2 are the main dots responsible for the binary encoding; (

**c**) Bis-ferrocene: the ferrocenes represent the two main dots, while the carbazole bridge acts as the third dot.

**Figure 3.**Basic schematic for the derivation of figures of merit. (

**a**) Vin–Vout scheme, evaluated in presence of a switching field; (

**b**) Vin–Vout scheme, evaluated in presence of nearby molecules; (

**c**) Basic schematic for the derivation of VOMs; (

**d**) Example of VOM.

**Figure 4.**Bis-ferrocene molecule analysis. (

**a**) Comparison of the potential generated by a bis-ferrocene molecule computed using: molecular dipole moment, atomic charges and the proposed Aggregated Charge. The potential is evaluated on a segment parallel to the active segment connecting the two ferrocenes at a distance of 1 nm (typical Receiver position). In the potential calculation that uses the dipole, an additive term equal to $e/\left(4\pi {\epsilon}_{0}r\right)$ is inserted to consider the oxidation of the molecule; (

**b**) Comparison in the Vin–Vout Transcharacteristics evaluated using the Two-State Approximation (TSA) theory and computed with ab initio.

**Figure 5.**Dot charges as a function of the applied electric field for the three molecules (VACT). (

**a**) Neutral diallyl-butane; (

**b**) Neutral decatriene; (

**c**) Neutral bis-ferrocene; (

**d**) Oxidized diallyl-butane; (

**e**) Oxidized decatriene; (

**f**) Oxidized bis-ferrocene.

**Figure 6.**Electric field generated by the charge distribution of the oxidized bis-ferrocene at the equilibrium and computed along the three spatial components. (

**a**) EF along x, computed with the atomic charges; (

**b**) EF along y, computed with the atomic charges; (

**c**) EF along z, computed with the atomic charges; (

**d**) EF along x, computed with the Aggregated Charges; (

**e**) EF along y, computed with the Aggregated Charges; (

**f**) EF along z, computed with the Aggregated Charges.

**Figure 7.**Electric field generated by the charge distribution of molecules at the equilibrium and computed along the working dot axis. (

**a**) EF generated by the oxidized diallyl-butane computed with the atomic charges, considering the atomic charges; (

**b**) EF generated by the oxidized decatriene computed with the atomic charges, considering the atomic charges; (

**c**) EF generated by the oxidized diallyl-butane computed with the atomic charges, considering the Aggregated Charges; (

**d**) EF generated by the oxidized decatriene computed with the atomic charges, considering the Aggregated Charges.

**Figure 8.**Electric field generated by the charge distribution of the three oxidized molecules at the equilibrium. The curves are computed at the position of the virtual Receiver and along the working dot axis.

**Figure 9.**Electric field generated by the charge distribution of the three oxidized molecules under the effect of the minimum switching field that localizes all the positive charge in one of the two working dots [33]. The curves are computed at the position of an ideal Receiver and along the working dot axis. Vertical bars underline the range of interest between the two dots. Null, positive and negative electric fields are applied as biasing condition at the MUT. (

**a**) Diallyl-butane; (

**b**) Decatriene; (

**c**) Bis-ferrocene.

**Figure 10.**Equivalent voltage at the Receiver (Vout) as a function of the voltage applied to the molecule (Vin, given by the switching field), for different molecule-receiver distance d. (

**a**) Diallyl-butane; (

**b**) Decatriene; (

**c**) Bis-ferrocene.

**Figure 11.**Vout maps (VOM) for the bis-ferrocene molecule. (

**a**) VOM for input encoding a Logic ‘1’ $Vin=+0.5$ V; (

**b**) VOM at the equilibrium $Vin=0$ V; (

**c**) VOM for input encoding a Logic ‘0’ $Vin=-0.5$ V.

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**MDPI and ACS Style**

Ardesi, Y.; Pulimeno, A.; Graziano, M.; Riente, F.; Piccinini, G.
Effectiveness of Molecules for Quantum Cellular Automata as Computing Devices. *J. Low Power Electron. Appl.* **2018**, *8*, 24.
https://doi.org/10.3390/jlpea8030024

**AMA Style**

Ardesi Y, Pulimeno A, Graziano M, Riente F, Piccinini G.
Effectiveness of Molecules for Quantum Cellular Automata as Computing Devices. *Journal of Low Power Electronics and Applications*. 2018; 8(3):24.
https://doi.org/10.3390/jlpea8030024

**Chicago/Turabian Style**

Ardesi, Yuri, Azzurra Pulimeno, Mariagrazia Graziano, Fabrizio Riente, and Gianluca Piccinini.
2018. "Effectiveness of Molecules for Quantum Cellular Automata as Computing Devices" *Journal of Low Power Electronics and Applications* 8, no. 3: 24.
https://doi.org/10.3390/jlpea8030024