Skip to main content

Realization of Complex Logic Operations at the Nanoscale

  • Conference paper
  • First Online:
Book cover Architecture and Design of Molecule Logic Gates and Atom Circuits

Part of the book series: Advances in Atom and Single Molecule Machines ((AASMM))

Abstract

The principles underlying the implementation of complex logic operations at the molecular scale are outlined. Different types of logic machines can be implemented. The simplest ones are combinational circuits, in which logic gates are connected in order to compute a logic function. We discuss several physical realizations of combinational circuits operating on Boolean or multivalued variables, as well as cascade thereof, implemented in a solid state or in a biochemical environment. The next level of complexity in logic machines is that of finite-state machines, which, in addition to a combinational unit, possess a memory unit so that the outputs depend not only on the inputs but also on the state of the memory. They therefore offer the possibility to implement parallel logic operations. Physical realizations of electrically and optically addressed finite-state machines are discussed. Special emphasis is given to electrical addressing which is currently able to implement logic on a single atom and even to concatenate.

Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Steinitz, D., Remacle, F., Levine, R.D.: Spectroscopy, control and molecular processing. Chem. Phys. Chem. 3, 43–51 (2002)

    Article  Google Scholar 

  2. Remacle, F., Levine, R.D.: Inter- and intra- molecular level logic devices. In: Waser, R. (ed.) Information Technology IV, vol. 4. Nanotechnology, pp. 213–248. Wiley, Weinheim (2008)

    Google Scholar 

  3. De Silva, A.P., Gunaratne, H.Q.N., McCoy, C.P.: A molecular photoionic AND gate based on flourescent signaling. Nature 364(6432), 42–44 (1993)

    Article  ADS  Google Scholar 

  4. de Silva, A.P., Gunaratne, H.Q.N., McCoy, C.P.: Molecular photoionic and logic gates with bright fluorescence and “off-on” digital action. J. Am. Chem. Soc. 119(33), 7891–7892 (1997)

    Article  Google Scholar 

  5. de Silva, A.P., Uchiyama, S.: Molecular logic and computing. Nature Nano. 2(7), 399–410 (2007)

    Article  ADS  Google Scholar 

  6. Szacilowski, K.: Digital information processing in molecular systems. Chem. Rev. 108(9), 3481–3548 (2008)

    Article  Google Scholar 

  7. Andreasson, J., Pischel, U.: Smart molecules at work-mimicking advanced logic operations. Chem. Soc. Rev. 39(1), 174–188 (2010)

    Article  Google Scholar 

  8. de Ruiter, G., van der Boom, M.E.: Surface-confined assemblies and polymers for molecular logic. Accounts Chem. Res. 44(8), 563–573 (2011)

    Article  Google Scholar 

  9. Joachim, C., Renaud, N., Hliwa, M.: The different designs of molecule logic gates. Adv. Mater. 24(2), 312–317 (2012)

    Article  Google Scholar 

  10. Gust, D., Andreasson, J., Pischel, U., Moore, T.A., Moore, A.L.: Data and signal processing using photochromic molecules. Chem. Comm. 48(14), 1947–1957 (2012)

    Article  Google Scholar 

  11. Benenson, Y., Adar, R., Paz-Elizur, T., Livneh, Z., Shapiro, E.: DNA molecule provides a computing machine with both data and fuel. Proc. Natl. Acad. Sci. USA 100(5), 2191–2196 (2003)

    Article  ADS  Google Scholar 

  12. Benenson, Y., Paz-Elizur, T., Adar, R., Keinan, E., Livneh, Z., Shapiro, E.: Programmable and autonomous computing machine made of biomolecules. Nature 414(6862), 430–434 (2001)

    Article  ADS  Google Scholar 

  13. Shapiro, E., Benenson, Y.: Bringing DNA computers to life. Sci. Am. 294(5), 44–51 (2006)

    Article  ADS  Google Scholar 

  14. Xie, Z., Liu, S.J., Bleris, L., Benenson, Y.: Logic integration of mRNA signals by an RNAi-based molecular computer. Nucleic Acids Res. 38(8), 2692–2701 (2010)

    Article  Google Scholar 

  15. Lederman, H., Macdonald, J., Stefanovic, D., Stojanovic, M.N.: Deoxyribozyme-based three-input logic gates and construction of a molecular full adder. Biochem. 45(4), 1194–1199 (2006)

    Article  Google Scholar 

  16. Pei, R., Matamoros, E., Liu, M., Stefanovic, D., Stojanovic, M.N.: Training a molecular automaton to play a game. Nature Nanotechnology 5(11), 773–777 (2010)

    Article  ADS  Google Scholar 

  17. Stojanovic, M.N., Stefanovic, D.: A deoxyribozyme-based molecular automaton. Nature Biotech. 21(9), 1069–1074 (2003)

    Article  Google Scholar 

  18. Qian, L., Winfree, E.: Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034), 1196–1201 (2011)

    Article  ADS  Google Scholar 

  19. Qian, L., Winfree, E., Bruck, J.: Neural network computation with DNA strand displacement cascades. Nature 475(7356), 368–372 (2011)

    Article  Google Scholar 

  20. Seelig, G., Soloveichik, D., Zhang, D.Y., Winfree, E.: Enzyme-free nucleic acid logic circuits. Science 314(5805), 1585–1588 (2006)

    Article  ADS  Google Scholar 

  21. Elbaz, J., Lioubashevski, O., Wang, F., Remacle, F., Levine, R.D., Willner, I.: DNA computing circuits using libraries of DNAzyme subunits (vol 5, pg 417, 2010). Nature Nanotechnology 6(3), 190–190 (2011)

    Article  ADS  Google Scholar 

  22. Gupta, T., van der Boom, M.E.: Redox-active monolayers as a versatile platform for integrating Boolean logic gates. Angew. Chem. Int. Ed. 47(29), 5322–5326 (2008)

    Article  Google Scholar 

  23. Duchemin, I., Joachim, C.: A quantum digital half adder inside a single molecule. Chem. Phys. Lett. 406(1–3), 167–172 (2005)

    Article  ADS  Google Scholar 

  24. Joachim, C., Martrou, D., Rezeq, M., Troadec, C., Jie, D., Chandrasekhar, N., Gauthier, S.: Multiple atomic scale solid surface interconnects for atom circuits and molecule logic gates. J. Phys. Cond. Mat. 22(8) (2010)

    Google Scholar 

  25. Renaud, N., Ito, M., Shangguan, W., Saeys, M., Hliwa, M., Joachim, C.: A NOR-AND quantum running gate molecule. Chem. Phys. Lett. 472(1–3), 74–79 (2009)

    Article  ADS  Google Scholar 

  26. Stadler, R., Ami, S., Joachim, C., Forshaw, M.: Integrating logic functions inside a single molecule. Nanotechnology 15(4), S115-S121 (2004)

    Article  ADS  Google Scholar 

  27. Prins, F., Shaikh, A.J., van Esch, J.H., Eelkema, R., van der Zant, H.S.J.: Platinum-nanogaps for single-molecule electronics: room-temperature stability. Phys. Chem. Chem. Phys. 13(32), 14297–14301 (2011)

    Article  Google Scholar 

  28. Manheller, M., Trellenkamp, S., Waser, R., Karthaeuser, S.: Reliable fabrication of 3 nm gaps between nanoelectrodes by electron-beam lithography. Nanotechnology 23(12) (2012)

    Google Scholar 

  29. Lansbergen, G.P., Rahman, R., Wellard, C.J., Woo, I., Caro, J., Collaert, N., Biesemans, S., Klimeck, G., Hollenberg, L.C.L., Rogge, S.: Gate-induced quantum confinement transition of a single dopant atom in a silicon FinFET. Nature Phys. 4, 656–661 (2008)

    Article  Google Scholar 

  30. Sellier, H., Lansbergen, G.P., Caro, J., Rogge, S., Collaert, N., Ferain, I., Jurczak, M., Biesemans, S.: Transport spectroscopy of a single dopant in a gated silicon nanowire. Phys. Rev. Lett. 97, 206805 (2006)

    Article  ADS  Google Scholar 

  31. Fuechsle, M., Miwa, J.A., Mahapatra, S., Ryu, H., Lee, S., Warschkow, O., Hollenberg, L.C.L., Klimeck, G., Simmons, M.Y.: A single-atom transistor. Nat Nano 7(4), 242–246 (2012)

    Article  Google Scholar 

  32. Klein, M., Lansbergen, G.P., Mol, J.A., Rogge, S., Levine, R.D., Remacle, F.: Reconfigurable logic devices on a single dopant atom-operation up to a full adder by using electrical spectroscopy. Chemphyschem 10(1), 162–173 (2009)

    Article  Google Scholar 

  33. Mol, J.A., Verduijn, J., Levine, R.D., Remacle, F., Rogge, S.: Integrated logic circuits using single-atom transistors. Proc. Natl. Acad. Sci. USA 108(34), 13969–13972 (2011)

    Article  ADS  Google Scholar 

  34. Klein, M., Mol, J.A., Verduijn, J., Lansbergen, G.P., Rogge, S., Levine, R.D., Remacle, F.: Ternary logic implemented on a single dopant atom FET in Si. Appl. Phys. Lett. 96, 043107 (2010)

    Article  ADS  Google Scholar 

  35. Mol, J.A., van der Heijden, J., Verduijn, J., Klein, M., Remacle, F., Rogge, S.: Balanced ternary addition using a gated silicon nanowire. Appl. Phys. Lett. 99(26) (2011)

    Google Scholar 

  36. Kompa, K.L., Levine, R.D.: A Molecular Logic Gate. Proc. Natl. Acad. Sci. USA 98, 410–414 (2001)

    Article  ADS  Google Scholar 

  37. Kuznetz, O., Salman, H., Eichen, Y., Remacle, F., Levine, R.D., Speiser, S.: All optical Full-adder Based on Intramolecular Electronic Energy transfer in the Rhodamine-Azulene Bichromophoric system. J. Phys. Chem. C 112, 15880–15885 (2008)

    Article  Google Scholar 

  38. Remacle, F., Speiser, S., Levine, R.D.: Intermolecular and intramolecular logic gates. J. Phys. Chem. B 105, 5589–5591 (2001)

    Article  Google Scholar 

  39. Kling, M.F., Vrakking, M.J.J.: Attosecond electron dynamics. Ann. Rev. Phys. Chem. 59(1), 463–492 (2008)

    Article  ADS  Google Scholar 

  40. Krausz, F., Ivanov, M.: Attosecond physics. Rev. Modern Phys. 81(1), 163–234 (2009)

    Article  ADS  Google Scholar 

  41. Woerner, H.J., Bertrand, J.B., Fabre, B., Higuet, J., Ruf, H., Dubrouil, A., Patchkovskii, S., Spanner, M., Mairesse, Y., Blanchet, V., Mevel, E., Constant, E., Corkum, P.B., Villeneuve, D.M.: Conical intersection dynamics in NO(2) probed by homodyne high-harmonic spectroscopy. Science 334(6053), 208–212 (2011)

    Article  ADS  Google Scholar 

  42. Remacle, F., Levine, R.D.: An electronic time scale for chemistry. Proc. Natl. Acad. Sci. USA 103(May 2), 6793–6798 (2006)

    Article  ADS  Google Scholar 

  43. Beil, F., Buschbeck, M., Heinze, G., Halfmann, T.: Light storage in a doped solid enhanced by feedback-controlled pulse shaping. Phys. Rev. A 81(5) (2010)

    Google Scholar 

  44. Heinze, G., Rudolf, A., Beil, F., Halfmann, T.: Storage of images in atomic coherences in a rare-earth-ion-doped solid. Phys. Rev. A 81(1) (2010)

    Google Scholar 

  45. Beil, F., Halfmann, T., Remacle, F., Levine, R.D.: Logic operations in a doped solid driven by stimulated Raman adiabatic passage. Phys. Rev. A 83(3) (2011)

    Google Scholar 

  46. Hurst, S.L.: Multiple-valued logic - its status and its future. IEEE Trans. Comp. C-33, 1160–1179 (1984)

    Article  Google Scholar 

  47. Kohavi, Z.: Switching and Finite Automata Theory. Tata McGraw-Hill, New Delhi (1999)

    Google Scholar 

  48. Gill, A.: Linear Sequential Circuits. McGraw Hill, New York (1966)

    MATH  Google Scholar 

  49. Remacle, F., Levine, R.D.: Towards parallel computing: representation of a linear finite state digital logic machine by a molecular relaxation process. Eur. Phys. J. D 42, 49–59 (2007)

    Article  ADS  Google Scholar 

  50. Torres, E.A., Kompa, K.L., Remacle, F., Levine, R.D.: Ultrafast vibrational spectroscopy and relaxation in polyatomic molecules: Potential for molecular parallel computing. Chem. Phys. 347(1–3), 531–545 (2008)

    Article  ADS  Google Scholar 

  51. Kaye, P., Laflamme, R., Mosca, M.: An Introduction to Quantum Computing. Oxford University Press, New York (2007)

    MATH  Google Scholar 

  52. Klein, M., Levine, R.D., Remacle, F.: Principles of design of a set-reset finite state logic nanomachine. J. Appl. Phys. 104, 044509–044511 (2008)

    Article  ADS  Google Scholar 

  53. Margulies, D., Melman, G., Shanzer, A.: A molecular full-adder and full-subtractor, an additional step toward a moleculator. J. Am. Chem. Soc. 128(14), 4865–4871 (2006)

    Article  Google Scholar 

  54. Remacle, F., Weinkauf, R., Levine, R.D.: Molecule-based photonically-switched half and full adder. J. Phys. Chem. A 110(1), 177–184 (2006)

    Article  Google Scholar 

  55. Mano, M.M., Kime, C.R.: Logic and Computer Design Fundamentals. Prentice Hall, Upper Saddle River, NJ (2000)

    Google Scholar 

  56. ELin, J.F., Hwang, Y.T., Sheu, M.H.A., Ho, C.C.: A novel high-speed and energy efficient 10-transistor full adder design. IEEE Trans. Circ. Syst. 54(5), 1050–1059 (2007)

    Google Scholar 

  57. Shlyahovsky, B., Li, Y., Lioubashevski, O., Elbaz, J., Willner, I.: Logic gates and antisense DNA devices operating on a translator nucleic acid scaffold. ACS Nano 3(7), 1831–1843 (2009)

    Article  Google Scholar 

  58. Soloveichik, D., Seeliga, G., Winfree, E.: DNA as a universal substrate for chemical kinetics. Proc. Natl. Acad. Sci. USA, 107, 5393–5398 (2010)

    Article  ADS  Google Scholar 

  59. Wang, Z.-G., Elbaz, J., Remacle, F., Levine, R.D., Willner, I.: All-DNA finite-state automata with finite memory. Proc. Natl. Acad. Sci. USA 107(51), 21996–22001 (2010)

    Article  ADS  Google Scholar 

  60. Elbaz, J., Wang, F., Remacle, F., Willner, I.: pH-Programmable DNA logic arrays powered by modular DNAzyme libraries. Nano Lett.(published on web), DOI: 10.1021/nl300051g (2012)

    Google Scholar 

  61. Breaker, R.R., Joyce, G.F.: A DNA enzyme with Mg2 + dependent RNA phosphoesterase activity. Chem. Biol. 2(10), 655–660 (1995)

    Article  Google Scholar 

  62. Bajec, I.L., Zimic, N., Mraz, M.: The ternary quantum-dot cell and ternary logic. Nanotechnology 17(8), 1937–1942 (2006)

    Article  ADS  Google Scholar 

  63. Chattopadhyay, T.: All-optical symmetric ternary logic gate. Optic Laser Tech. 42(6), 1014–1021 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  64. Jin, X., Li, C., Liu, J., Jiang, X., Zeng, X.: Ternary logic dynamic CMOS comparators. In, vol. 317–319. pp. 1177–1182 (2011)

    Google Scholar 

  65. Lin, S., Kim, Y.B., Lombardi, F.: CNTFET-based design of ternary logic gates and arithmetic circuits. IEEE Trans. Nanotechnology 10(2), 217–225 (2011)

    Article  ADS  Google Scholar 

  66. Uemura, T., Yamamoto, M.: Three-valued magnetic tunnel junction for nonvolatile ternary content addressable memory application. J. Appl. Phys. 104(12) (2008)

    Google Scholar 

  67. Medalsy, I., Klein, M., Heyman, A., Shoseyov, O., Remacle, F., Levine, R.D., Porath, D.: Logic implementations using a single nanoparticle-protein hybrid. Nature Nanotechnology 5(6), 451–457 (2010)

    Article  ADS  Google Scholar 

  68. Klein, M., Rogge, S., Remacle, F., Levine, R.D.: Transcending binary logic by gating three coupled quantum dots. Nano. Lett. 7, 2795–2799 (2007)

    Article  ADS  Google Scholar 

  69. Halpern, I., Yoeli, M.: Ternary arithmetic unit. Proc. IEE 115(10), 1385–1388 (1968)

    Google Scholar 

  70. Yoeli, M., Rosenfeld, G.: Logic design of ternary switching circuits. IEEE Trans. Elect. Comp. EC-14, 19–29 (1965)

    Article  Google Scholar 

  71. Remacle, F., Levine, R.D.: Configuration interaction between covalent and ionic states in the quantal and semiclassical limits with application to coherent and hopping charge migration. J. Phys. Chem. A 104(11), 2341–2350 (2000)

    Article  Google Scholar 

  72. Remacle, F., Levine, R.D., Schlag, E.W., Weinkauf, R.: Electronic control of site selective reactivity: a model combining charge migration and dissociation. J. Phys. Chem. A 103, 10149–10158 (1999)

    Article  Google Scholar 

  73. Pierre, M., Wacquez, R., Roche, B., Jehl, X., Sanquer, M., Vinet, M., Prati, E., Belli, M., Fanciulli, M.: Compact silicon double and triple dots realized with only two gates. Appl. Phys. Lett. 95(24), 242107–242103 (2009)

    Article  ADS  Google Scholar 

  74. Remacle, F., Levine, R.D.: Towards molecular logic machines. J. Chem. Phys. 114(23), 10239–10246 (2001)

    Article  ADS  Google Scholar 

  75. Chattopadhyay, T., Reis, C., Andre, P., Teixeira, A.: Theoretical analysis of all-optical clocked D flip-flop using a single SOA assisted symmetric MZI. Optic Comm. 285(9), 2266–2275 (2012)

    Article  ADS  Google Scholar 

  76. Remon, P., Balter, M., Li, S.M., Andreasson, J., Pischel, U.: An all-photonic molecule-based d flip-flop. J. Am. Chem. Soc. 133(51), 20742–20745 (2011)

    Article  Google Scholar 

  77. Sun, J., Wallin, D., He, Y., Maximov, I., Xu, H.Q.: A sequential logic device realized by integration of in-plane gate transistors in InGaAs/InP. Appl. Phys. Lett. 92(1), (2008)

    Google Scholar 

  78. Remacle, F., Heath, J.R., Levine, R.D.: Electrical addressing of confined quantum systems for quasiclassical computation and finite state logic machines. Proc. Natl. Acad. Sci. USA 102, 5653–5658 (2005)

    Article  ADS  Google Scholar 

  79. Elbaz, J., Moshe, M., Willner, I.: Coherent activation of DNA tweezers: A “SET-RESET” logic system. Angew. Chem. Int. Ed. 48(21), 3834–3837 (2009)

    Article  Google Scholar 

  80. Elbaz, J., Wang, Z.-G., Orbach, R., Willner, I.: pH-stimulated concurrent mechanical activation of two DNA “Tweezers”. A “SET-RESET” logic gate system. Nano Lett. 9(12), 4510–4514 (2009)

    Google Scholar 

  81. Baron, R., Onopriyenko, A., Katz, E., Lioubashevski, O., Willner, I., Sheng, W., Tian, H.: An electrochemical/photochemical information processing system using a monolayer-functionalized electrode. Chem. Comm. (20), 2147–2149 (2006)

    Article  Google Scholar 

  82. Periyasamy, G., Collin, J.-P., Sauvage, J.P., Levine, R.D., Remacle, F.: Electrochemically driven sequential machine: an implementation on copper rotaxanes. Chem. Eur. J. 15(6), 1310–1313 (2009)

    Article  Google Scholar 

  83. De Ruiter, G., Tartakovsky, E., Oded, N., Van Der Boom, M.E.: Sequential logic operations with surface-confined polypyridyl complexes displaying molecular random access memory features. Angew. Chem.– Int. Ed. 49(1), 169–172 (2010)

    Google Scholar 

  84. Remacle, F., Levine, R.D.: All optical digital logic: full addition or subtraction on a three-state system. Phys. Rev. A 73, 033820–033827 (2006)

    Article  ADS  Google Scholar 

  85. Yan, Y., Mol, J.A., Verduijn, J., Rogge, S., Levine, R.D., Remacle, F.: Electrically addressing a molecule-like donor pair in silicon: an atomic scale cyclable full adder logic. J. Phys. Chem. C 114(48), 20380–20386 (2010)

    Article  Google Scholar 

  86. MacMillen, D.B., Landman, U.: Variational solutions of simple quantum systems subject to variable boundary conditions. II. Shallow donor impurities near semiconductor interfaces: Si, Ge. Phys. Rev. B 29(8), 4524 (1984)

    Google Scholar 

  87. Calderon, M.J., Koiller, B., Hu, X., Das Sarma, S.: Quantum control of donor electrons at the Si-SiO[sub 2] interface. Phys. Rev. Lett. 96(9), 096802–096804 (2006)

    Google Scholar 

  88. Cullum, J.K., Willoughby, R.A.: Lanczos Algorithm for Large Symmetric Eigenvalues Computations. Birkhauser, Boston (1985)

    Google Scholar 

  89. Andresen, S.E.S., Brenner, R., Wellard, C.J., Yang, C., Hopf, T., Escott, C.C., Clark, R.G., Dzurak, A.S., Jamieson, D.N., Hollenberg, L.C.L.: Charge state control and relaxation in an atomically doped silicon device. NanoLett. 7(7), 2000–2003 (2007)

    Article  ADS  Google Scholar 

  90. Barrett, S.D., Milburn, G.J.: Measuring the decoherence rate in a semiconductor charge qubit. Phys. Rev. B 68(15), 155307 (2003)

    Article  ADS  Google Scholar 

  91. Bergmann, K., Theuer, H., Shore, B.W.: Coherent population transfer among quantum states of atoms and molecules. Rev. Mod. Phys. 70(3), 1003–1025 (1998)

    Article  ADS  Google Scholar 

  92. Halfmann, T., Bergmann, K.: Coherent population transfer and dark resonance in SO2. J. Chem. Phys. 104(18), 7068–7072 (1996)

    Article  ADS  Google Scholar 

  93. Vitanov, N.V., Fleischhauer, M., Shore, B.W., Bergmann, K.: Coherent manipulation of atoms and molecules by sequential pulses. Adv. At. Mol. Opt. Phys. 46, 55–190 (2001)

    Article  ADS  Google Scholar 

  94. Vitanov, N.V., Halfmann, T., Shore, B.W., Bergmann, K.: Laser-induced population transfer by adiabatic passage technique. Ann. Rev. Phys. Chem. 52, 763–809 (2001)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was partially funded by the FP7 EC NANOICT project MOLOC and by the FP6 FET open project MOLDYNLOGIC. We thank our colleagues in these projects, in particular Sven Rogge, Itamar Willner, Thomas Halfmann, and Karl Kompa, and the students and post-doc that contributed to the work reported here, in particular Johann Elbaz, Michael Klein, Dr. Gabriel Lansberger, Jan Mol, Dr. Elva Torres, Arjan Verduijn, and Dr. Yonghong Yan. F. R. is a Director of Research with Fonds National de la Recherche Scientifique, Belgium.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to F. Remacle .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Levine, R.D., Remacle, F. (2013). Realization of Complex Logic Operations at the Nanoscale. In: Lorente, N., Joachim, C. (eds) Architecture and Design of Molecule Logic Gates and Atom Circuits. Advances in Atom and Single Molecule Machines. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33137-4_16

Download citation

Publish with us

Policies and ethics