© A. Nitzan, TAU How Molecules Conduct Introduction to electron transport in molecular systems I. Benjamin, A. Burin, B. Davis, S. Datta, D. Evans, M. Galperin, A. Ghosh, H. Grabert, P. Hänggi, G. Ingold, J. Jortner, S. Kohler, R. Kosloff, J. Lehmann, M. Majda, A. Mosyak, V. Mujica, R. Naaman, U. Peskin, M. Ratner, D. Segal, T. Seideman, H. Tal-Ezer Thanks Reviews: Annu. Rev. Phys. Chem. 52, 681– 750 (2001) [ Science, 300, (2003); MRS Bulletin, 29, (2004); Bulletin of the Israel Chemical Society, Issue 14, p (Dec 2003) (Hebrew)

© A. Nitzan, TAU Part A INTRODUCTION A. Nitzan

© A. Nitzan, TAU Molecules as conductors

© A. Nitzan, TAU Aviram Ratner Abstract The construction of a very simple electronic device, a rectifier, based on the use of a single organic molecule is discussed. The molecular rectifier consists of a donor pi system and an acceptor pi system, separated by a sigma- bonded (methylene) tunnelling bridge. The response of such a molecule to an applied field is calculated, and rectifier properties indeed appear. Molecular Rectifiers Arieh Aviram and Mark A. Ratner IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA Department of Chemistry, New York New York University, New York 10003, USA Received 10 June 1974

© A. Nitzan, TAU Xe on Ni(110)

© A. Nitzan, TAU Feynman Chad Mirkin (DPN)

© A. Nitzan, TAU Moore’s “Law”

© A. Nitzan, TAU Moore’s 2nd law

© A. Nitzan, TAU Molecules get wired

© A. Nitzan, TAU Cornell group

© A. Nitzan, TAU

April 2002

© A. Nitzan, TAU The Hedonistic Imperative outlines how genetic engineering and nanotechnology will abolish suffering in all sentient life. The abolitionist project is hugely ambitious but technically feasible. It is also instrumentally rational and ethically mandatory. The metabolic pathways of pain and malaise evolved because they served the fitness of our genes in the ancestral environment. They will be replaced by a different sort of neural architecture. States of sublime well-being are destined to become the genetically pre-programmed norm of mental health. The world's last unpleasant experience will be a precisely dateable event.

© A. Nitzan, TAU

Feynman For a successful Technology, reality must take precedence over public relations, for nature cannot be fooled

© A. Nitzan, TAU First Transport Measurements through Single Molecules Single-wall carbon nanotube on Pt Dekker et al. Nature 386(97) Nanopore Reed et al. APL 71 (97) Break junction: dithiols between gold Molecule lying on a surface Molecule between two electrodes Dorogi et al. PRB 52 Purdue Au(111) Pt/Ir Tip SAM 1 nm ~1-2 nm Self-assembled monolayers Adsorbed molecule addressed by STM tip C 60 on gold Joachim et al. PRL 74 (95) STM tip Au Reed et al. Science 278 Yale Nanotube on Au Lieber et al. Nature 391 (98)

© A. Nitzan, TAU Park et. al. Nature 417, (2002) Datta et al

© A. Nitzan, TAU Weber et al, Chem. Phys. 2002

© A. Nitzan, TAU Electron transfer in DNA

© A. Nitzan, TAU Electronic processes in molecular systems Electron transfer Electron transfer Electron transmission Electron transmission Conduction Conduction Parameters that affect molecular conduction Parameters that affect molecular conduction Eleastic and inelastic transmission Eleastic and inelastic transmission Coherent and incoherent conduction Coherent and incoherent conduction Heating and heat conduction Heating and heat conduction

© A. Nitzan, TAU Theory of Electron Transfer The physics of activated rate process The physics of activated rate process The physics of electron transfer in a nuclear environment The physics of electron transfer in a nuclear environment A simple model – parabolic potential surfaces A simple model – parabolic potential surfaces The Marcus parabolas The Marcus parabolas The reorganization energy The reorganization energy Adiabatic and non adiabatic rates Adiabatic and non adiabatic rates Solvent control Solvent control Bridge length dependence (“super-exchange”) Bridge length dependence (“super-exchange”) Transition to hopping Transition to hopping Experimental implications Experimental implications

© A. Nitzan, TAU Activated rate processes KRAMERS THEORY: Low friction limit High friction limit Transition State theory (action) Diffusion controlled rates

© A. Nitzan, TAU The physics of transition state rates Assume: (1) Equilibrium in the well (2) Every trajectory on the barrier that goes out makes it

© A. Nitzan, TAU Electron transfer in polar media Electron are much faster than nuclei  Electronic transitions take place in fixed nuclear configurations  Electronic energy needs to be conserved during the change in electronic charge density Electronic transition Nuclear relaxation

© A. Nitzan, TAU Electron transfer Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations Nuclear motion

© A. Nitzan, TAU Electron transfer

© A. Nitzan, TAU Transition state theory of electron transfer Adiabatic and non-adiabatic ET processes Landau-Zener problem Alternatively – solvent control

© A. Nitzan, TAU Electron transfer – Marcus theory They have the following characteristics: (1) P n fluctuates because of thermal motion of solvent nuclei. (2) P e, as a fast variable, satisfies the equilibrium relationship (3) D = constant (depends on  only) Note that the relations E = D-4  P; P=P n + P e are always satisfied per definition, however D   s E. (the latter equality holds only at equilibrium). We are interested in changes in solvent configuration that take place at constant solute charge distribution 

© A. Nitzan, TAU The Marcus parabolas Use  as a reaction coordinate. It defines the state of the medium that will be in equilibrium with the charge distribution  . Marcus calculated the free energy (as function of  ) of the solvent when it reaches this state in the systems  =0 and  =1.

© A. Nitzan, TAU Electron transfer: Activation energy

© A. Nitzan, TAU Electron transfer: Effect of Driving (=energy gap)

© A. Nitzan, TAU Experimental confirmation of the inverted regime Miller et al, JACS(1984) Marcus Nobel Prize: 1992

© A. Nitzan, TAU Electron transfer – the coupling From Quantum Chemical Calculations The Mulliken-Hush formula Bridge mediated electron transfer

© A. Nitzan, TAU Bridge assisted electron transfer

© A. Nitzan, TAU BEWARE: equations are coming!

© A. Nitzan, TAU Effective donor-acceptor coupling

© A. Nitzan, TAU Marcus expresions for non-adiabatic ET rates Bridge Green’s Function Donor-to-Bridge/ Acceptor-to-bridge Franck-Condon- weighted DOS Reorganization energy

© A. Nitzan, TAU Bridge mediated ET rate  ’ ( Å -1 ) = for highly conjugated chains for saturated hydrocarbons ~ 2 for vacuum

© A. Nitzan, TAU Incoherent hopping constant STEADY STATE SOLUTION

© A. Nitzan, TAU ET rate from steady state hopping

© A. Nitzan, TAU Dependence on temperature The integrated elastic (dotted line) and activated (dashed line) components of the transmission, and the total transmission probability (full line) displayed as function of inverse temperature. Parameters are as in Fig. 3.

© A. Nitzan, TAU The photosythetic reaction center

© A. Nitzan, TAU Dependence on bridge length

© A. Nitzan, TAU DNA (Giese et al 2001)

© A. Nitzan, TAU Steady state evaluation of rates Rate of water flow depends linearly on water height in the cylinder Two ways to get the rate of water flowing out: (1)Measure h(t) and get the rate coefficient from k=(1/h)dh/dt (2)Keep h constant and measure the steady state outwards water flux J. Get the rate from k=J/h -- Steady state rate h

© A. Nitzan, TAU Steady state quantum mechanics V0lV0l Starting from state 0 at t=0: P 0 = exp(-   t) G = 2  |V sl | 2  L (Golden Rule) Steady state derivation:

© A. Nitzan, TAU pumping damping

© A. Nitzan, TAU Resonance scattering V 1r V 1l

© A. Nitzan, TAU Resonance scattering j = 0, 1, {l}, {r} For each r and l

© A. Nitzan, TAU SELF ENERGY

Resonant tunneling V 1r V 1l

© A. Nitzan, TAU Resonant Tunneling Transmission Coefficient

© A. Nitzan, TAU Resonant Transmission – 3d 1d 3d: Total flux from L to R at energy E 0 : If the continua are associated with a metal electrode at thermal equilibrium than (Fermi-Dirac distribution)

© A. Nitzan, TAU CONDUCTION 2 spin states Zero bias conduction L R  –  e|   f(E 0 ) (Fermi function)

© A. Nitzan, TAU Landauer formula (maximum=1) Maximum conductance per channel For a single “channel”: