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Molecular electronics: a new challenge for O(N) methods

Molecular electronics: a new challenge for O(N) methods. Roi Baer and Daniel Neuhauser (UCLA) Institute of Chemistry and Lise Meitner Center for Quantum Chemistry The Hebrew University of Jerusalem, Israel. IPAM, April 2, 2002. Collaboration. Derek Walter , PhD. Student (UCLA)

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Molecular electronics: a new challenge for O(N) methods

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  1. Molecular electronics: a new challenge for O(N) methods Roi Baer and Daniel Neuhauser (UCLA) Institute of Chemistry and Lise Meitner Center for Quantum Chemistry The Hebrew University of Jerusalem, Israel IPAM, April 2, 2002

  2. Collaboration • Derek Walter, PhD. Student (UCLA) • Prof. Eran Rabani, Tel Aviv University • Oded Hod, PhD. student (Tel Aviv U) • Acknowledgments: • Israel Science Foundation • Fritz Haber center for reaction dynamics

  3. Overview • Molecular electronics is interesting • Formalism • O(N3) algorithm: non-interacting electrons • Possible O(N) algorithm • Electron correlation: O(N2) algorithm

  4. Introduction Why are coherent molecular wires interesting?

  5. Tunnel Junction 1 STM tip dI/dV [a.u] I 1.0 0.5 0.0 T=4.2 K R1,C1 Tunnel Junction 2 -1.0 0.0 1.0 Voltage [V] QD V R2,C2 (b) (a) Conductance of C60 D. Porath and O. Millo, J. Appl. Phys. 81, 2241 (1997).

  6. Conductance of a nanotube S. Frank and W. A. de Heer et al, Science 280, 1744 (1998).

  7. Chen et al, Science 286,1550 (1998) Conductance of C6H4S2 Reed et al, Science 278,252 (1997)

  8. Coherent electronics • Size: ~ 1013 logic gates/cm2 (108) • Responsetimes: 10-15 sec(10-9) • Quantum effects: • Interference • Uncertainty • Entanglement • Inclonability

  9. Interference effects • de-Broglie: electrons are waves • Interference • Nonlocal particle nature • Electrons are not photons! • Fermions: cannot scatter into “any energetically open state” • Correlated: inelastic collisions, Coulomb blockade… • Tunneling: reducing/killing interference effects, sensitive

  10. V b ML MR b 30 Carbons long A simple wire • W: Huckel parameters a = -6.6, bS = -2.55,bD = -2.85 • M: chain of 20 “gold” atoms, aG =-6.6bG =-2.7 • MW coupling= b • Expect: current should grow with b Units: eV

  11. Sometimes more is less Inversion

  12. V ML MR 30 Carbons long Current from transmittance Landauer current formula

  13. Just because of the coupling…

  14. A switch based on interference • Simplest model of interference effects

  15. Current-Voltage Constructive Destructive

  16. l=2a Band top: Totally non bonding Band middle Half filling lF=4a l=L Band bottom Totally bonding a=CC Fermi Wave length

  17. V2 Current I V1 XOR gate based on interference

  18. Sensitivity DFT electronic structure. Molecule connected to gold wire, acting as a lead Current (nA) Bias (Volt)

  19. hR=1 hR=0 Wire L R IR Quantum conductance formalism R. Baer and D. Neuhauser, submitted (2002).

  20. Weak Bias: Linear Response Conductivity is a current-current correlation formula R. Kubo, J Phys. Soc. Japan 12, 570 (1957).

  21. Non-interacting Electrons NlR(E) =cumulative transmission probability (from l to R) R. Landauer, IBM J. Res. Dev. 1, 223 (1957).

  22. Calculating conductance Non-interacting particle formalism 4 step O(N3) algorithm

  23. Left slab Right slab + + + + + + + - - - - - - - s s Step #1: Structure under bias • Use SCF model like DFT/HF etc. • Optimize structure and e-density

  24. + + + + + + + - - - - - - - s s Step #2: Add Absorbing boundaries Left slab Right slab D. Neuhauser and M. Baer, J. Chem. Phys 90, 4351 (1989)

  25. Step #3: Trace Formula T. Seideman and W. H. Miller, J. Chem. Phys. 96, 4412 (1992).

  26. Step #4: Current formula (Landauer formula)

  27. Efficient O(N3) Implementation N(E) is spiky Integrate energy analytically

  28. O(N) Algorithm • N(E) is averaged over E→A sparse part of G needed • The trace can be computed by a Chebyshev series • All energies computed in single sweep: integration is trivial R. Baer, Y. Zeiri, and R. Kosloff, Phys. Rev. B 54 (8), R5287 (1996).

  29. Including electron correlation Time Dependent Density Functional Theory

  30. - + - + - + - + Uniform, weak, time dependent electric field: Linear response

  31. Large jellium sandwich Dynamic system (w+contacts) Embed small in large Imaginary potential Imaginary potential Frozen Jellium (leads) Building the model Small jellium sandwich

  32. The setup for C3 • Dynamic density • Frozen density • Total Density • Kohn-Sham potential

  33. Conductance of C3

  34. Are correlations important? • Conductance is smaller by a factor 10. • Possible reason: the same reason that causes DFT to underestimate HOMO-LUMO gaps

  35. Summary • Molecular electronics • Theory of conductance • Linear scaling calculation of conductance • Importance of electrson-electron correlations • TDDFT is expensive and at least O(N2)

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