1. 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. 2 Collaboration l Derek Walter, PhD. Student (UCLA) l Prof. Eran Rabani, Tel Aviv University l Oded Hod, PhD. student (Tel Aviv U) l Acknowledgments: n Israel Science Foundation n Fritz Haber center for reaction dynamics

3. 3 Overview l Molecular electronics is interesting l Formalism l O(N 3 ) algorithm: non-interacting electrons l Possible O(N) algorithm l Electron correlation: O(N 2 ) algorithm

4. 4 Introduction Why are coherent molecular wires interesting?

5. 5 Conductance of C60 (a) I QD V R 1,C 1 R 2,C 2 Voltage [V] dI/dV [a.u] 1.0 0.5 0.0 -1.0 0.01.0 T=4.2 K STM tip Tunnel Junction 1 Tunnel Junction 2 (b) D. Porath and O. Millo, J. Appl. Phys. 81, 2241 (1997).

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

7. 7 Conductance of C 6 H 4 S 2 Reed et al, Science 278,252 (1997) Chen et al, Science 286,1550 (1998)

8. 8 Coherent electronics Size: ~ 10 13 logic gates/cm 2 (10 8 ) Response times: 10 -15 sec (10 -9 ) l Quantum effects: n Interference n Uncertainty n Entanglement n Inclonability

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

10. 10 A simple wire W: Huckel parameters  S   D  M: chain of 20 “gold” atoms,  G   G  l MW coupling = b l Expect: current should grow with b Units: eV 30 Carbons long MLML MRMR V b b

11. 11 Sometimes more is less Inversion

12. 12 Current from transmittance Landauer current formula 30 Carbons long MLML MRMR V

13. 13 Just because of the coupling…

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

15. 15 Current-Voltage Destructive Constructive

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

17. 17 XOR gate based on interference V1V1 V2V2 Current I V1V1 V2V2 I 000 110 011 101

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

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

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

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

22. 22 Calculating conductance Non-interacting particle formalism 4 step O(N 3 ) algorithm

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

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

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

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

27. 27 Efficient O(N 3 ) Implementation N(E) is spikyIntegrate energy analytically

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

29. 29 Including electron correlation Time Dependent Density Functional Theory

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

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

32. 32 The setup for C 3 a) Dynamic density b) Frozen density c) Total Density d) Kohn-Sham potential

33. 33 Conductance of C 3

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

35. 35 Summary l Molecular electronics l Theory of conductance l Linear scaling calculation of conductance l Importance of electrson-electron correlations l TDDFT is expensive and at least O(N 2 )