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### Lecture 4

A. Nitzan, Tel Aviv University

ELECTRON TRANSFER AND TRANSMISSION IN MOLECULES AND MOLECULAR JUNCTIONS

AEC, Grenoble, Sept 2005

- (1) Relaxation and reactions in condensed molecular systems
- Kinetic models
- Transition state theory
- Kramers theory and its extensions
- Low, high and intermediate friction regimes
- Diffusion controlled reactions

Coming March 2006

Chapter 13-15

- (2) Electron transfer processes
- Simple models
- Marcus theory
- The reorganization energy
- Adiabatic and non-adiabatic limits
- Solvent controlled reactions
- Bridge assisted electron transfer
- Coherent and incoherent transfer
- Electrode processes

Coming March 2006

Chapter 16

- (3) Molecular conduction
- Simple models for molecular conductions
- Factors affecting electron transfer at interfaces
- The Landauer formula
- Molecular conduction by the Landauer formula
- Relationship to electron-transfer rates.
- Structure-function effects in molecular conduction
- How does the potential drop on a molecule and why this is important
- Probing molecules in STM junctions
- Electron transfer by hopping

Coming March 2006

Chapter 17

Donor gives an electron and goes from state “a” (reduced) to state “b” (oxidized). Eb,a=Eb-Ea is the energy of the electron given to the metal

ELECTRODE PROCESSES

Transition rate to a continuum (Golden Rule)

D

A

EF

Rate of electron transfer to metal in vacuum

M

Rate of electron transfer to metal in electrolyte solution

Reorganization energy here – from donor only (~0.5 of “regular” value)

General case

Unit matrix in the bridge space

Bridge Hamiltonian

B(R) + B(L) -- Self energy

Wide band approximation

Cui et al (Lindsay), Science 294, 571 (2001)

“The resistance of a single octanedithiol molecule was 900 50 megaohms, based on measurements on more than 1000 single molecules. In contrast, nonbonded contacts to octanethiol monolayers were at least four orders of magnitude more resistive, less reproducible, and had a different voltage dependence, demonstrating that the measurement of intrinsic molecular properties requires chemically bonded contacts”.

A relation between g and k

Electron charge

conduction

Electron transfer rate

Decay into electrodes

Marcus

A relation between g and k

l0.5eV

Issues in molecular conductions

- (3) Molecular conduction
- Structure-function effects in molecular conduction
- The role of contacts
- How does the potential drop on a molecule and why this is important
- Probing molecules in STM junctions
- Electron transfer by hopping
- Charging
- Switching

2-level bridge (local representation)

- Dependence on:
- Molecule-electrode coupling GL , GR
- Molecular energetics E1, E2
- Intramolecular coupling V1,2

Reasons for switching

- Conformational changes

- Transient charging

- Polaron formation

time

Tsai et. al. PRL 1992: RTS in Me-SiO2-Si junctions

STM under waterS.Boussaad et. al. JCP (2003)

I. Inoue et al, Journal of Physiology 541.3, pp. 769-778(2002)

[Ca+2]=1x10-6M

Single (K+) channel currents from Schwann cells isolated enzymatically from the giant axons of the squids Loligo forbesi, Loligo vulgaris and Loligo bleekeri. The channel conductance was 43.6 pS when both internal and external solutions contained 150 mM K+. Activity was weakly dependent on membrane voltage but sensitive to the internal Ca2+ concentration.

Michel-Beyerle et al

Xue and Ratner 2003

Selzer et al 2004

Temperature and chain length dependenceConjugated vs. Saturated Molecules: Importance of Contact Bonding

Au//

S/Au

Au/S

S/Au

Kushmerick et al., PRL (2002)

Au/S(CH2)8SAu

2- vs. 1-side Au-S bonded conjugated system gives at most 1 order of magnitude current increase compared to 3 orders for C10 alkanes!

Au//CH3(CH2)7S/Au

Xue, Ratner (2003)

Potential profile

Galperin et al JCP 2003

Where does the potential bias falls, and how?- Image effect
- Electron-electron interaction (on the Hartree level)

Vacuum

Galperin et al 2003

Experiment

Theoretical Model

Experimental (Sek&Majda)

aCurrent at the negative bias refers to the measurement with the Hg side of the junction biased negative relative to the Au side.

TIMESCALE CONSIDERATIONS

Does the tunneling electron interact with other degrees of freedom and what are the possible consequences of this interaction?

The case of electron tunneling in water

Overbarrier electron transmission through water (D2O on Pt(1,1,1)

Effective Barrier

The effective one-dimensional barrier obtained by fitting the low energy tunneling probability to the analytical results for tunneling through a rectangular barrier. Solid, dotted, and dashed lines correspond to the polarizable, nonpolarizable, and bare barrier potentials, respectively.

The numerical problem

- Get a potential
- Electrostatics
- Generate Water configurations
- Tunneling calculations
- Integrate to get current

Potentials for electron transmission through water

Water-Water.....................RWKM, SPC/E

Electron-Water..............Barnett et al +correction for many body polarizability

Water-Wall........Henziker et al (W-Pt), Hautman et al (W-Au)

Electron-Wall..............Square Barrier

Earlier studies – Tunneling through static water configurations

STM model

Fig. 1. A model system used to compute electron transmission between two electrodes, L and R separated by a narrow spatial gap (M) containing a molecular species. The surface S1 of L is shaped to mimic a tip. The lines A'B', C'D' and AB and CD are projections of boundary surfaces normal to the transmission direction (see text for details). The numerical solution is carried on a grid (Shown).

Potential distribution

A cut of the external potential distribution between the tip and the flat substrate for a voltage drop of 0.5V between these electrodes

The image potential along different lines normal to the flat electrode: (1) x=0 (a line going through the tip axis); (2) x=11.96au (distance from the tip axis); (3) x=23.92au.

MOLECULAR DYNAMICS TO GENERATE WATER CONFIGURATIONS

Figure - Ohmine et al

CALCULATION OF TRANSMISSION FACTORS

Absorbing boundary conditions Green's function method: Replace by i(r), smoothly rising towards edges of M system, provided LM and MR boundaries are set far enough

For nearest neighbor coupling:

The self energy - 2Tunneling current in water

Current against bias voltage in a biased tip-planar electrode junction under water. Upper and lower lines are results for single water configurations characterized by tip-substrate separation of 5.85Å (2 water monolayers) and 12.15Å(4 water monolayers), respectively. The intermediate group of lines are results for 5 different water configurations at tip-substrate separation 9Å (3 water monolayers).

Transmission through several water configurations (equilibrium, 300K)

A compilation of numerical results for the transmission probability as a function of incident electron energy, obtained for 20 water configurations sampled from an equilibrium trajectory (300K) of water between two planar parallel Pt(100) planes separated by 10Å. The vacuum is 5eV and the resonance structure seen in the range of 1eV below it varies strongly between any two configurations. Image potential effects are disregarded in this calculation.

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