Quasi-classical modeling of molecular quantum-dot cellular automata multidriver gates
- Ehsan Rahimi^{1}Email author and
- Shahram Mohammad Nejad^{1}
DOI: 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
References
- Lent CS: Bypassing the transistor paradigm. Science 2000, 288: 1597–1599. 10.1126/science.288.5471.1597View Article
- Niemier M, Kogge P, Murphy R, Rodrigues A, Dysart T, Frost S: Data flow in molecular QCA: logic can “sprint”, but the memory wall can still be a "hurdle". University of Notre Dame: Technical report; 2005.
- Aviram A: Molecules for memory, logic, and amplification. J Am Chem Soc 1988, 110: 5687–5692. 10.1021/ja00225a017View Article
- Lent CS, Isaksen B, Lieberman M: Molecular quantum-dot cellular automata. J Am Chem Soc 2003, 125: 1056–1063. 10.1021/ja026856gView Article
- Orlov AO, Amlani I, Kummamuru RK, Ramasubramaniam R, Toth G, Lent CS, Bernstein GH, Snider GL: Experimental demonstration of clocked single-electron switching in quantum-dot cellular automata. Appl Phys Lett 2000, 77: 295–297. 10.1063/1.126955View Article
- Smith C: Realization of quantum-dot cellular automata using semiconductor quantum dots. Superlatt Microst 2003, 34: 195–203. 10.1016/j.spmi.2004.03.009View Article
- Gardelis S, Smith C, Cooper J, Ritchie D, Linfield E, Jin Y: Evidence for transfer of polarization in a quantum-dot cellular automata cell consisting of semiconductor quantum dots. Phys Rev B 2003, 67: 033302.View Article
- Lent CS, Isaksen B: Clocked molecular quantum-dot cellular automata. IEEE T Electron Dev 2003, 50: 1890–1896. 10.1109/TED.2003.815857View Article
- Qi H, Sharma S, Li Z, Snider GL, Orlov AO, Lent CS, Fehlner TP: Molecular quantum cellular automata cells. electric field driven switching of a silicon surface bound array of vertically oriented two-dot molecular quantum cellular automata. J Am Chem Soc 2003, 125: 15250–15259. 10.1021/ja0371909View Article
- Jiao J, Long GJ, Grandjean F, Beatty AM, Fehlner TP: Building blocks for the molecular expression of quantum cellular automata. Isolation and characterization of a covalently bonded square array of two ferrocenium and two ferrocene complexes. J Am Chem Soc 2003, 125: 7522–7523. 10.1021/ja035077cView Article
- Jiao J, Long GJ, Rebbouh L, Grandjean F, Beatty AM, Fehlner TP: Properties of a mixed-valence (feII)2(feIII)2 square cell for utilization in the quantum cellular automata paradigm for molecular electronics. J Am Chem Soc 2005, 127: 17819–17831. 10.1021/ja0550935View Article
- Qi H, Gupta A, Noll BC, Snider GL, Lu Y, Lent C, Fehlner TP: Dependence of field switched ordered arrays of dinuclear mixed-valence complexes on the distance between the redox centers and the size of the counterions. J Am Chem Soc 2005, 127: 15218–15227. 10.1021/ja054508jView Article
- Li Z, Fehlner TP: Molecular QCA cells. 2. Characterization of an unsymmetrical dinuclear mixed-valence complex bound to a Au surface by an organic linker. Inorg Chem 2003, 42: 5715–5721. 10.1021/ic026255qView Article
- Lu Y, Lent CS: Theoretical study of molecular quantum-dot cellular automata. J Comput Elec 2005, 4: 115–118. 10.1007/s10825-005-7120-yView Article
- Braun-Sand SB, Wiest O: Theoretical studies of mixed-valence transition metal complexes for molecular computing. J Phys Chem A 2002, 107: 285–291.View Article
- Walus K, Dysart TJ, Jullien GA, Budiman RA: QCADesigner: a rapid design and simulation tool for quantum-dot cellular automata. IEEE T Nanotechnol 2004, 3: 26–31. 10.1109/TNANO.2003.820815View Article
- Vetteth A, Walus K, Dimitrov SV, Jullien GA: Quantum-dot cellular automata carry-look-ahead adder and barrel shifter. Richardson, TX: IEEE Conf on Emerging Telecom Tech; 2002.
- Frost SE, Rodrigues AF, Janiszewski AW, Rausch RT, Kogge PM: Memory in Motion: a Study of Storage Structures in QCA. 1st Workshop on Non-Silicon Computation 2002.
- Niemier MT, Kogge PM: Logic in wire: using quantum dots to implement a microprocessor. ICECS '99 6th IEEE Int Conf Circuits and Systems 1999, 3: 1211–1215.View Article
- Walus K, Vetteth A, Jullien GA, Dimitrov VS: RAM design using quantum-dot cellular automata. Nanotechnology Conf and Trade Show 2003, 2: 160–163.
- Lent CS, Tougaw PD: A device architecture for computing with quantum dots. Proc of the IEEE 1997, 85: 541–557. 10.1109/5.573740View Article
- Rahimi E, Nejad SM: Secure clocked QCA logic for implementation of quantum cryptographic processors. Int Conf on Applied Electronics 2009, 217–220.
- Rahimi E, Nejad SM: Quantum-Dot Cellular ROM: A nano-scale level approach to digital data storage. 6th IEEE Int Conf on Communication Systems, Networks and Digital Signal Processing 2008, 618–621.
- Rahimi E, Nejad SM: A novel architecture for quantum-dot cellular ROM. 7th IEEE Int Conf on Communication Systems, Networks and Digital Signal Processing 2010, 347–350.
- Tougaw PD, Lent CS: Logical devices implemented using quantum cellular automata. J Appl Phys 1994, 75: 1818–1825. 10.1063/1.356375View Article
- Mohammad Nejad S, Rahimi E: QCA: The prospective technology for digital telecommunication systems. In Nanotechnology for Telecommunications. CRC Press; 2010:275–307.
- Hennessy K, Lent CS: Clocking of molecular quantum-dot cellular automata. J Vac Sci Technol B 2001, 19: 1752–1755.View Article
- Orlov AO, Kummamuru R, Ramasubramaniam R, Lent CS, Bernstein GH, Snider GL: Clocked quantum-dot cellular automata shift register. Surf Sci 2003, 532–535: 1193–1198.View Article
- Karim F, Walus K, Ivanov A: Analysis of field-driven clocking for molecular quantum-dot cellular automata based circuits. J Comput Elec 2010, 9: 16–30. 10.1007/s10825-009-0300-4View Article
- Lent CS, Tougaw PD, Porod W, Bernstein GH: Quantum cellular automata. Nanotechnology 1993, 4: 49. 10.1088/0957-4484/4/1/004View Article
- Timler J, Lent CS: Power gain and dissipation in quantum-dot cellular automata. J Appl Phys 2002, 91: 823–831. 10.1063/1.1421217View Article
- Tougaw PD, Lent CS: Dynamic behavior of quantum cellular automata. J Appl Phys 1996, 80: 4722. 10.1063/1.363455View Article
- Géza T: Correlation and coherence in quantum-dot cellular automata. PhD Thesis. University of Notre Dame; 2000.
- Newton MD: Quantum chemical probes of electron-transfer kinetics: the nature of donor-acceptor interactions. Chem Rev 1991, 91: 767–792. 10.1021/cr00005a007View Article
- Newton MD: Control of electron transfer kinetics: models for medium reorganization and donor–acceptor coupling. Adv Chem Phys 2007, 106: 303–375.
- Zener C: Non-adiabatic crossing of energy levels. Proc R Soc A 1932, 137: 696–702. 10.1098/rspa.1932.0165View Article
- Zener C: Discussion of excited diatomic molecules by external perturbations. Proc R Soc A 1933, 140: 660–668. 10.1098/rspa.1933.0095View Article
- Larsson S, Braga M: Transfer of mobile electrons in organic molecules. Chem Phys 1993, 176: 367–375. 10.1016/0301-0104(93)80247-7View Article
- Li X-Y, Tang X-S, Xiao S-Q, He F-C: Application of Koopmans’ theorem in evaluating electron transfer matrix element of long-range electron transfer. J Mol Struct (THEOCHEM) 2000, 530: 49–58. 10.1016/S0166-1280(00)00327-4View Article
- Li X-Y, Tang X-S, He F-C: Electron transfer in poly (p-phenylene) oligomers: effect of external electric field and application of Koopmans theorem. Chem Phys 1999, 248: 137–146. 10.1016/S0301-0104(99)00239-6View Article
- Rodriguez-Monge L, Larsson S: Conductivity in polyacetylene. IV. Ab initio calculations for a two-site model for electron transfer between allyl anion and allyl. Int J Quantum Chem 1997, 61: 847–857.
- Hush NS: Intervalence-transfer absorption. Part 2. Theoretical considerations and spectroscopic data. Prog Inorg Chem Vol 2007, 8: 391–444.View Article
- Hush NS: Homogeneous and heterogeneous optical and thermal electron transfer. Electrochim Acta 1968, 13: 1005–1023. 10.1016/0013-4686(68)80032-5View Article
- Bailey SE, Zink JI, Nelsen SF: Contributions of symmetric and asymmetric normal coordinates to the intervalence electronic absorption and resonance raman spectra of a strongly coupled p-phenylenediamine radical cation. J Am Chem Soc 2003, 125: 5939–5947. 10.1021/ja021343vView Article
- Coropceanu V, Gruhn NE, Barlow S, Lambert C, Durivage JC, Bill TG, Nöll G, Marder SR, Brédas J-L: Electronic couplings in organic mixed-valence compounds: the contribution of photoelectron spectroscopy. J Am Chem Soc 2004, 126: 2727–2731. 10.1021/ja039263uView Article
- Hush NS, Wong AT, Bacskay GB, Reimers JR: Electron and energy transfer through bridged systems. 6. Molecular switches: the critical field in electric field activated bistable molecules. J Am Chem Soc 1990, 112: 4192–4197. 10.1021/ja00167a014View Article
- Cabrero J, Calzado CJ, Maynau D, Caballol R, Malrieu JP: Metal−ligand delocalization in magnetic orbitals of binuclear complexes. J Phys Chem A 2002, 106: 8146–8155. 10.1021/jp0204410View Article
- Lu Y, Quardokus R, Lent CS, Justaud F, Lapinte C, Kandel SA: Charge localization in isolated mixed-valence complexes: an STM and theoretical study. J Am Chem Soc 2010, 132: 13519–13524. 10.1021/ja105958pView Article
- Lu Y, Lent CS: A metric for characterizing the bistability of molecular quantum-dot cellular automata. Nanotechnology 2008, 19: 155703. 10.1088/0957-4484/19/15/155703View Article
- Malmqvist P-Å, Roos BO: The CASSCF state interaction method. Chem Phys Lett 1989, 155: 189–194. 10.1016/0009-2614(89)85347-3View Article
- Yamamoto N, Vreven T, Robb MA, Frisch MJ, Bernhard Schlegel H: A direct derivative MC-SCF procedure. Chem Phys Lett 1996, 250: 373–378. 10.1016/0009-2614(96)00027-9View Article
- Koopmans T: Über die zuordnung von wellenfunktionen und eigenwerten zu den einzelnen elektronen eines atoms. Physica 1934, 1: 104–113. 10.1016/S0031-8914(34)90011-2View Article
- Ruedenberg K, Cheung LM, Elbert ST: MCSCF optimization through combined use of natural orbitals and the brillouin–levy–berthier theorem. Int J Quantum Chem 1979, 16: 1069–1101. 10.1002/qua.560160511View Article
- Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, et al.: Gaussian 09. Revision B.01. 2009.
- Newton MD, Cave RJ: Molecular Control of Electron and Hole Transfer Processes: Theory and Applications. Mol Electron 1997., 73:
- Farazdel A, Dupuis M, Clementi E, Aviram A: Electric-field induced intramolecular electron transfer in spiro.pi.-electron systems and their suitability as molecular electronic devices. A theoretical study. J Am Chem Soc 1990, 112: 4206–4214. 10.1021/ja00167a016View Article
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