Introduction to electron transport in molecular systems

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Lecture 3
A. Nitzan, Tel Aviv University SELECTED
TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS
Boulder, Aug 2007
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Boulder Aug 2007

• (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

Chapter 13-15
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Boulder Aug 2007

• (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

Chapter 16
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Theory of Electron Transfer
Transition rate
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Dielectric solvation
Born solvation energy
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Electron transfer Activation energy
Reorganization energy
Activation energy
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Electron transfer Effect of Driving (energy gap)
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Experimental confirmation of the inverted regime
Marcus papers 1955-6
Miller et al, JACS(1984)
Marcus Nobel Prize 1992
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Marcus expresions for non-adiabatic ET rates
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Bridge mediated ET rate
Charge recombination lifetimes in the compounds
shown in the inset in dioxane solvent. (J. M.
Warman et al, Adv. Chem. Phys. Vol 106, 1999).
The process starts with a photoinduced electron
transfer a charge separation process. The
lifetimes shown are for the back electron
transfer (charge recombination) process.
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ET rate from steady state hopping
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The photosythetic reaction center
Michel - Beyerle et al
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DNA (Giese et al 2001)
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ELECTROCHEMISTRY
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Donor gives an electron and goes from state a
(reduced) to state b (oxidized). Eb,aEb- Ea
is the energy of the electron given to the metal
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
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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
• Measure h(t) and get the rate coefficient from
  k(1/h)dh/dt
• Keep h constant and measure the steady state
  outwards water flux J. Get the rate from kJ/h
• Steady state rate

h
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PART C
Molecular conduction
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Steady state quantum mechanics
Starting from state 0 at t0 P0 exp(-G0t)
G0 2pV0l2rL (Golden Rule)
V0l
Steady state derivation
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pumping
damping
V0l
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Resonance scattering
V1r
V1l
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Resonance scattering
For each r and l
j 0, 1, l, r
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Resonance scattering
For each r and l
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SELF ENERGY
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Resonant tunneling
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Resonant Tunneling
Transmission Coefficient
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How is current generated?
Occupation probabilities (Fermi functions)
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Resonant Transmission 3d
1d
3d Total flux from L to R at energy E0
If the continua are associated with a metal
electrode at thermal equilibrium than
(Fermi-Dirac distribution)
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CONDUCTION
R
L
m
m ef
f(E0) (Fermi function)
2 spin states
Zero bias conduction
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Landauer formula
For a single channel
(maximum1)
Maximum conductance per channel
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Current from classical kinetics
I/e
0 at steady state
Find P1 and insert into I
Quantum mechanical resalt
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Molecular level structure between electrodes
LUMO
HOMO
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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.
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General case
Unit matrix in the bridge space
Bridge Hamiltonian
B(R) B(L) is Self energy
Wide band approximation
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The N-level bridge (n.n. interactions)
G1N(E)
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ET vs Conduction
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A relation between g and k
Electron charge
conduction
Electron transfer rate
Decay into electrodes
Marcus
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A relation between g and k
l?0.5eV
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Conductance (g (O-1)) vs Kinetics ( k0 (s-1) )
for alkane spacers Marshal Newton
Alkane Bridge X(CH2)n-2 low bias limit low bias limit low bias limit
Alkane Bridge X(CH2)n-2 I / V in nano-pore junctions Reed et al (monothiolates) STM / break junctions Tao et al (dithiolates) Scaled k0 5 x 10-19 a k0/DOS Nitzan M(D?BA?)M model ( D? and A? chemisorbed to M)
n8 5.0 E-11 1.9 E-8 4.1 a E-8
n10 5.7 E-12 1.6 E-9 6.8 a E-9
n12 6.5 E-13 1.3 E-10 4.6 a E-10

• Conclusions
• conductance data of Tao et al (g) and
  rate constant
• data (k0) correspond to within 1-2
  orders of magnitude
• results from the 2 sets of conductance
  measurements
• differ by gt 2 orders of magnitude

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TOMORROW
FACTORS AFFECTING
MOLECULAR CONDUCTION