Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates
- Ehsan Rahimi^{1}Email author and
- Shahram Mohammad Nejad^{1}
https://doi.org/10.1186/1556-276X-7-274
© Rahimi and Mohammad Nejad; licensee Springer. 2012
Received: 29 March 2012
Accepted: 25 April 2012
Published: 30 May 2012
Abstract
Molecular quantum-dot cellular automata (mQCA) has received considerable attention in nanoscience. Unlike the current-based molecular switches, where the digital data is represented by the on/off states of the switches, in mQCA devices, binary information is encoded in charge configuration within molecular redox centers. The mQCA paradigm allows high device density and ultra-low power consumption. Digital mQCA gates are the building blocks of circuits in this paradigm. Design and analysis of these gates require quantum chemical calculations, which are demanding in computer time and memory. Therefore, developing simple models to probe mQCA gates is of paramount importance. We derive a semi-classical model to study the steady-state output polarization of mQCA multidriver gates, directly from the two-state approximation in electron transfer theory. The accuracy and validity of this model are analyzed using full quantum chemistry calculations. A complete set of logic gates, including inverters and minority voters, are implemented to provide an appropriate test bench in the two-dot mQCA regime. We also briefly discuss how the QCADesigner tool could find its application in simulation of mQCA devices.
Keywords
Electron transfer reactions Molecular electronics Molecular gates Molecular quantum-dots Quantum cellular automataBackground
Methods
Two-dot molecular QCA test bench
The MinV gate is an alternative to the MV gate in the four-dot QCA, where the output is inverted. Compared to MV and INV gates in the four-dot architecture, which require 16 and 28 quantum-dots correspondingly, the MinV and INV gates require only 8 and 4 quantum-dots in the two-dot mQCA regime. Consequently, these gates provide a small two-dot mQCA test bench, which make high level quantum chemical calculations feasible. The MinV gate can perform NAND and NOR logical operations, as shown in Figure 3d, and provides a functionally complete logic set to implement any logic function in the two-dot mQCA framework. Additionally, it is possible to implement multi-input (or multidriver) MinV gates, which in turn decrease the total number of gates required to implement a logic circuit. It is important to note that since the MinV gate is not a planar gate, circuits implemented in the two-dot mQCA regime are not planar circuits. We highlight that the practical QCA circuits require clocked-control cells and clocking schemes [21, 27–29], which are not addressed in this paper.
Two-state model for molecular QCA gates
where E′_{ i,j } and E_{ i,j } denote the electrostatic energy of cells i and j having opposite and same polarizations correspondingly.
where P^{—}_{ j } is the sum of the polarizations of the neighboring four-dot QCA cells. Equation 7 is currently used in the nonlinear and two-state simulation engine of QCADesigner to solve the metallic-based QCA circuits. It is important to note that mQCA utilizes non-abrupt clocking to reduce the probability of Kink, the property that is not currently present in the QCADesigner as it is based on metallic QCA. In mQCA, the tunneling barriers can be controlled by external electric field [27]. It is demanding to enhance the tool to be able to simulate mQCA circuits. As a primary step towards this end, we present how a similar equation to (7) can be derived directly from the two-state approximation in electron transfer theory [34, 35] for two-dot mQCA. We then discuss how these approximations affect the results compared to those obtained from full quantum chemistry calculations.
The additivity relation in Equation 20 originates from the additivity of electrostatic potential energy in Equation 13 for diabatic states.
Results and discussion
Two-state model parameters for the used molecules
Molecule (cation) | ^{1}H_{ab}(eV) | ^{2}H_{ab}(eV) | l(nm) | E_{k}(eV) | |μ| | ^{ 3 }Δζ(Å) |
---|---|---|---|---|---|---|
1,6-heptadiene | 0.310 | 0.368 | 0.56 | −0.7531 | 1.023 | 0.06969 |
1,8-nonadiene | 0.14 | 0.12 | 0.83 | −0.5081 | 2.117 | 0.07002 |
1,4-diallyl butane | 0.00707 | 0.00693 | 0.7 | −0.6025 | 43.04 | 0.00905 |
INV gates
INV gates
1,6-heptadiene | 1,8-nonadiene | 1,4-diallyl butane | ||||
---|---|---|---|---|---|---|
P _{ d } | P _{ o } | P _{ o } ^{ * } | P _{ o } | P _{ o } ^{ * } | P _{ o } | P _{ o } ^{ * } |
0.0 | −0.068 | 0 | −0.058 | 0 | −2.3 E-05 | 0 |
0.1 | −0.126 | −0.101 | −0.217 | −0.207 | −0.987 | −0.974 |
0.2 | −0.191 | −0.200 | −0.398 | −0.389 | −0.987 | −0.993 |
0.3 | −0.260 | −0.293 | −0.534 | −0.536 | −0.989 | −0.997 |
0.4 | −0.323 | −0.378 | −0.630 | −0.646 | −0.990 | −0.998 |
0.5 | −0.378 | −0.455 | −0.695 | −0.726 | −0.992 | −0.998 |
0.6 | −0.423 | −0.523 | −0.740 | −0.785 | −0.994 | −0.999 |
0.7 | −0.460 | −0.582 | −0.771 | −0.828 | −0.996 | −0.999 |
0.8 | −0.490 | −0.633 | −0.793 | −0.861 | −0.997 | −0.999 |
0.9 | −0.513 | −0.677 | −0.808 | −0.885 | −0.998 | −0.999 |
1.0 | −0.531 | −0.715 | −0.818 | −0.904 | −0.999 | −0.999 |
RMSE^{ * } | 0.104 | 0.050 | 0.006 |
Equation 30 shows that the saturation polarization of the output increases with the increase of μ. This is also evident from the results in Table 2.
Two-driver devices
Two-driver devices
P _{d1} | P _{d2} | 1,6-heptadiene | 1,8-nonadiene | 1,4-diallyl butane | |||
---|---|---|---|---|---|---|---|
P_{o}(P_{d1},P_{d2}) | P_{o}(P_{d1} + P_{d2}) | P_{o}(P_{d1},P_{d2}) | P_{o}(P_{d1} + P_{d2}) | P_{o}(P_{d1},P_{d2}) | P_{o}(P_{d1} + P_{d2}) | ||
0 | 0 | −0.068 | −0.068 | −0.058 | −0.058 | −0.001 | −0.001 |
0.2 | 0.2 | −0.316 | −0.323 | −0.625 | −0.630 | −0.998 | −0.992 |
0.4 | 0.2 | −0.431 | −0.423 | −0.753 | −0.740 | −0.994 | −0.987 |
0.6 | −0.2 | −0.326 | −0.323 | −0.651 | −0.630 | −0.998 | −0.992 |
0.8 | 0.2 | −0.560 | −0.531 | −0.840 | −0.768 | −0.985 | −0.979 |
1 | −0.4 | −0.383 | −0.423 | −0.718 | −0.740 | −0.995 | −0.988 |
0.4 | −0.2 | −0.204 | −0.191 | −0.454 | −0.398 | −0.999 | −0.993 |
1 | −0.2 | −0.469 | −0.490 | −0.788 | −0.793 | −0.990 | −0.984 |
0.6 | −0.8 | 0.090 | 0.191 | 0.075 | 0.398 | 0.997 | 0.991 |
RMSE | 0.038 | 0.112 | 0.005 | ||||
RMSE^{ * } | 0.137 | 0.108 | 0.014 |
Three-input MinV gates
Three-driver MinV gates
P _{ d1 } | P _{ d2 } | P _{ d3 } | 1,6-heptadiene | 1,8-nonadiene | 1,4-diallyl butane | |||
---|---|---|---|---|---|---|---|---|
P _{ o } | P _{ o } ^{ * } | P _{ o } | P _{ o } ^{ * } | P _{ o } | P _{ o } ^{ * } | |||
0 | 0 | 0 | −0.005 | 0 | −0.015 | 0 | −0.001 | 0 |
0.2 | 0.2 | 1 | −0.605 | −0.819 | −0.826 | −0.947 | −0.993 | −0.999 |
0.4 | 0.2 | 0.6 | −0.596 | −0.775 | −0.825 | −0.930 | −0.996 | −0.999 |
0.6 | −0.2 | 1 | −0.610 | −0.819 | −0.826 | −0.947 | −0.993 | −0.999 |
0.8 | 0.2 | −1 | −0.064 | 0 | −0.289 | 0 | −0.992 | 0 |
1 | −0.4 | 1 | −0.617 | −0.853 | −0.828 | −0.959 | −0.989 | −0.999 |
0.4 | −0.2 | −0.2 | −0.061 | 0 | −0.143 | 0 | −0.954 | 0 |
1 | −0.2 | 1 | −0.627 | −0.878 | −0.832 | −0.967 | −0.984 | −0.999 |
0.6 | −0.8 | −0.2 | 0.197 | 0.378 | 0.377 | 0.646 | 0.988 | 0.998 |
−0.4 | −0.8 | −0.8 | 0.639 | 0.898 | 0.837 | 0.973 | 0.993 | 0.999 |
0.6 | 0.8 | 1 | −0.617 | −0.950 | −0.836 | −0.981 | −0.969 | −0.999 |
1 | 1 | 1 | −0.634 | −0.926 | −0.830 | −0.987 | −0.957 | −0.999 |
RMSE^{*} | 0.213 | 0.162 | 0.397 | |||||
RMSE^{**} | 0.244 | 0.153 | 0.019 |
Conclusions
Molecular QCA gates are the building blocks of more complex modules. Probing molecular devices requires quantum chemical calculations, which are challenging as the molecular system grows in size. A semi-classical model was derived directly from the two-state approximation in the ET theory, serving as a device for studying mQCA gates. This model is very similar to the two-state model which is currently the core of the QCADesigner simulation engine for solving circuits based on metallic QCA. The range of applications and limitations of this model for mQCA gates was investigated carefully. The parametric TSM can be used to study more complex mQCA gates composed of practical candidate mixed-valence molecules, where exploiting the SA/CASSCF method is of high computational cost. A complete set of logic gates were implemented within the two-dot mQCA framework. These gates include INV and MinV gates, which provide a small molecular test bench, making further analysis by quantum chemistry methods, particularly SA/CASSCF, practical. The INV gate was studied as a nucleus of all other gates. It was also presented that output polarizations of all other gates can be derived from extrapolating the results obtained from inverters based on the additivity relation. We compared the results obtained from the TSM to those obtained from SA/CASSCF calculations for INV and MinV gates. The degree of agreement between the TSM and quantum chemical calculations is highly dependent on the μ parameter and the symmetry of the head groups. Additionally, application of the additivity relation for CASSCF method can in turn reduce the computational cost. It is important to note that we did not address questions of surface attachment, input/output, clocked control, layout, and patterning, which are the requirements of a practical QCA system. Moreover, we did not consider the relaxation of nuclear degrees of freedom associated with electron transfer. It is presented that for mQCA, the electron localization and Coulombic interactions play the key roles, and nuclear positions can be considered frozen (nuclear relaxation even assists charge localization) [4]. Although we limited our focus on the two-dot mQCA, it merits highlighting that the model can also be used for four-dot cells, since they can be considered as double two-dot cells. Our focus was on the mQCA gates as building blocks of circuits. The two-state model may be applied to simulate mQCA circuits as well, as it is currently used iteratively for simulation of metallic QCA circuits in the QCADesigner. However, to determine the additive error resulting from exploiting the two-state model for solving mQCA circuits, further quantum chemical calculations on the mQCA clocked circuits composed of several molecules are required, which are extremely challenging at the time, and have not been addressed in this paper.
Declarations
Acknowledgment
ER was affiliated with Norwegian University of Science and Technology. Calculations presented in this work have been carried out on Stallo. ER also thanks Professor Sven Larsson for many enlightening discussions on electron transfer theory at Chalmers University of Technology.
Authors’ Affiliations
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