1. Institut für Theoretische Festkörperphysik (Universität Karlsruhe) and Universidad Autonoma de Madrid (Spain)

2. Molecular Electronics: Experimental Techniques 1. Breakjunctions 2. Nanopores 3. Electrochemical methods 4. Electromigration

3. Molecular Electronic: Functional Structures 1. Diode: Au-SAM-Ti-Au (Nanopore) 4-thioacetatebiphenyl, M. Reed, APL (1997) 2. Swicht: Nanopore (60 K) M. Reed et al., Science (1999) 3. Reconfigurable Swicht: Catanane J.R. Heath et al., Science (2000) 4. Single-electron transistor: Park et al., Nature (2003).

4. Molecular Electronics: Goal for the Theory.  Understanding of the transport mechanisms at the molecular scale.  Quantitative description of the transport properties.

5. Outline of this tutorial 1)Description of the elastic current: Landauer approach. i. Resonant tunneling ii. Temperature dependence of the conductance iii. Symmetry of the IVs: Rectification iv. Calculation of the transmission: Green´s functions v. Two level system: hydrogen molecule vi. Length dependence of the conductance vii. Ab initio calculations 2)Inelastic current: role of the vibration modes in the conduction i. Experimental motivation ii. Simple theoretical model: different transport regimes 3)Other transport mechanisms: correlation effects, hopping, transport, etc. (very brief) 4) Challenges and open problems

6. 1. Landauer approach to electron transport in molecular contacts Relation between electronic structure and electronic transport

7. real system S a(E)t(E)a(E) r(E)a(E) Landauer formula scattering problem T(E)+R(E) = |t(E)| 2 + |r(E)| 2 =1 S electron reservoirs scattering region EFEF E F +eV spin degeneracy Landauer approach to electron transport

8. 1.1 Resonant tunneling (single level) Energy scheme of a molecular contact Single-level model Current formula

9. Resonant tunneling (single level) Off-resonant transport: molecules as tunneling juncions 

10. Single molecules as tunnel junctions Cui et al. (Lindsay), Science 294, 571 (2001)

11. 1.2 Tunneling: temperature dependence Voltage dependence: Again a tunnel junction! Current independent of the temperature Wang, Lee and Reed, PRB 68, 035416 (2003)

12. Tunneling: temperature dependence Conclusion  Off-resonant transport  T independent  On-resonant transport  T dependent (as long as T ~  )

13. 1.3 Symmetry of the IVs: Rectification “Molecular rectifiers” Arieh Aviram and Mark A. Ratner (Chem. Phys. Lett., 1974) “The construction of a very simple electronic device, a rectifier, based on the used of a single organic molecule is discussed. The molecular rectifier consists of a donor pi system and a acceptor pi system, separated by a sigma-bonded (methylene) tunneling bridge. The response of such a molecule to an applied field is calculated, and rectifier properties indeed appear.” R. Metzger et al., JACS 1997 (… 23 years later)

14. Molecular conductivity takes shape J. Reichert et al., PRL 88, 176804 (2002) (INT Karlsruhe) asymmetric molecule symmetric molecule

15. Symmetry of the IVs: Rectification Molecular conductivity takes shape J. Reichert et al., PRL 88, 176804 (2002) Single-level model: asymmetric coupling

16. Beware, Green‘s functions are coming!

17. 1.4 Calculation of the transmission: Green‘s functions AtomsOrbitals Green´s function Equation of motion? Three subsystems: left (L), right (R) and center (C)

18. 1.4 Calculation of the transmission: Green‘s functions

19. 1.5 Two-level conduction: Conductance of a hydrogen molecule Smit et al., Nature (2002); Leiden University  The hydrogen molecule forms a stable bridge between Pt electrodes.  The conductance is G ~ G 0 and it is largely dominated by a single conduction channel.

20. Tunneling: two sites Bonding and antibonding states Transmission:

21. Conductance of a hydrogen molecule

22.  (i) Charge transfer between H 2 and the Pt leads and (ii) strong hybridization between the molecule and the electrodes.  The Transport is dominated by the binding orbital. DFT conductance calculation JCC, J. Heurich, F. Pauly, W. Wenzel, G. Schön, Nanotechnology (2003)

23. 1.6 Length dependence of conductance The conductance decays exponentially with the length of the molecule Wang, Lee and Reed, PRB 68, 035416 (2003)

24. N-level bridge: n.n. interaction G 1N (E)

25. 1.7 Ab initio calculation (DFT) J. Heurich, JCC, W. Wenzel, G. Schön, Phys. Rev. Lett. 88, 256803 (2002) Three subsystems: central cluster and leads  Central cluster: Density functional calculation (DFT)  Leads: The reservoirs are modeled as two perfect semi-infinite crystals using a tight- binding parametrization (Papaconstantopoulos 1986).  Coupling: Hopping between the lead orbitals and the molecular orbitals of the central cluster. Kohn-Sam energies Molecular orbitals

26. Theory: linear regime. Total transmission and total density of states of the two molecules At the Fermi energy: EFEF

27. From molecular orbitals to conduction channels J. Heurich, JCC, W. Wenzel, G. Schön, Phys. Rev. Lett. 88, 256803 (2002) Charge density of four molecular orbitals: (a) HOMO; (c) LUMO The contribution to the transport of an individual molecular orbital depends on its character: extended or localized, weak coupling or strong coupling, etc.

28. I-V characteristics: symmetric molecule

30. 2. Inelastic current Role of the molecular vibrations in the electrical conduction

31. 2.1 Experimental motivation: Inelastic tunneling spectroscopy Ho and coworkers: ``Single molecule spectroscopy“ J. Chem. Phys. (2002)

32. 2.1 Experimental motivation: hydrogen molecule Smit et al., Nature (2002): measurement of the conductance of a hydrogen molecules between Pt leads

33. 2.1 Experimental motivation: gold atomic chains N. Agrait et al., PRL (2002): onset of energy dissipation in ballistic atomic wires (See talk by Nicolas Agrait this afternoon)

34. 2.1 Experimental motivation: nanotubes Satellite peaks: signature of phonon-assisted tunneling

35. 2.2 Inelastic current: toy model    electron-phonon coupling constant Second order perturbation theory:

36. Simple model: resonant case

37. Simple model: resonant tunneling

38. Weak coupling regime: multiphonon peaks See for instance S. Braig and K. Flensberg, PRB 68, 205324 (2003) Rate equations:

39. Weak coupling regime: multiphonon processes Theory: S. Braig and K. Flensberg PRB 68, 205324 (2003)

40. Microscopic models Janne Viljas and JCC (TB model, unpublished) See talk by Thomas Frederiksen: T. Frederiksen et al. PRL 93, 256601 (2004)

41. 3. Other mechanisms

42. 3.1 Correlation effects

43. 3.2 Incoherent hopping constant STEADY STATE SOLUTION

44. ET rate from steady state hopping

45. 3.3 Conformation changes

46. 3.4 Molecules as optoelectronic devices

47. 4. Challenges and open problems  Quantitative agreement with experimental results.  Development of methods that describe the electronic transport through strongly correlated systems: interpolation between the weak coupling regime and strong coupling regime.  More extensive work on gating of molecular junctions.  Characterizing transport junctions behavior in the presence of radiation.  Effects of changing chemistry and doping on the bridge – can mechanisms be altered by chemical change, as in conducting polymers, and can we predict and control such behavior?  Elucidating the change in behavior from a single molecule conductance through junctions comprising a few molecules to molecular film conductors.  Understanding noise  Understanding heating, heat conduction and current induced chemical changes

48. Universität Karlsruhe Univ. Autónoma de Madrid Prof. Gerd Schön Prof. Alvaro Martín Priv. Doz. Dr. Wolfgang Wenzel (INT) Prof. Alfredo Levy Dr. Jan E. Heurich Prof. Nicolas Agrait Fabian Pauly Prof. Gabino Rubio Michael Häfner Dr. Carlos Untiedt Dr. Janne Viljas Universität Konstanz Quantronics Group (Saclay) Leiden University Prof. Elke Scheer C. Urbina Prof. J. Van Ruitenbeek Prof. Peter Nielaba D. Esteve Dr. Bas Ludoph Markus Dreher M.H. Devoret (now in Yale)