Electronic Tunneling through Dissipative Molecular Bridges

1
Electronic Tunneling through Dissipative
Molecular Bridges

• Uri Peskin
• Department of Chemistry,
• Technion - Israel Institute of Technology

Musa Abu-Hilu (Technion) Alon Malka
(Technion) Chen Ambor (Technion) Maytal Caspari
(Technion) Roi Volkovich (Technion) Darya Brisker
(Technion) Vika Koberinski (Technion) Prof.
Shammai Speiser (Technion)
Thanking
2
Outline

• Motivation
• Controlled electron transport in molecular
  devices and in biological systems.
• Background
• ET in Donor-Acceptor complexes The Golden Rule,
  the Condon approximaton and the spin-boson
  Hamiltonian.
• ET in Donor-Bridge-Acceptor complexes
  McConnells formula for the tunneling matrix
  elements.
• The problem
• Electronic-nuclear coupling at the molecular
  bridge and the breakdown of the Condon
  approximation.
• The model system
• Generalized spin-boson Hamiltonians for
  dissipative through-bridge tunneling.
• Results
• The weak coupling limit Langevin-Schroedinger
  formulation, simulations and interpretation of ET
  through a dissipative bridge
• Beyond the weak coupling limit An analytic
  formula for the tunneling matrix element in the
  deep tunneling regime.
• Conclusions
• Promotion of tunneling through molecular barriers
  by electronic-nuclear coupling.
• The effect of molecular rigidity.

3
Motivation Electron Transport Through
Molecules
Molecular Electronics
Tans, Devoret, Thess, Smally, Geerligs, Dekker,
Nature (1997)
Reichert, Ochs, Beckmann, Weber, Mayor,
Lohneysen, Phys. Rev. Lett. (2002).
Resonant tunneling through molecular junctions
4
Long-range Electron Transport In Nature
Tunneling pathway between cytochrome b5 and
methaemoglobin
The Photosynthetic Reaction Center
Electron transfer is controlled by molecular
bridges
Deep (off-resonant) tunneling through molecular
barriers
5
Controlled tunneling through molecules?
Resonant tunneling
Deep (off resonant) tunneling
Why Off-Resonant (deep) Tunneling ?

• Minor changes to the molecular electronic density

• High sensitivity (exponential) to the molecular
  parameters

• A potential for a rational design based on
  chemical knowledge

6
Electron Transfer in Donor-Acceptor Pairs
Donor
Acceptor
Electronic tunneling matrix element
Nuclear factor Frank-Condon weighted density of
states
The case of through bridge tunneling
The role of electronic nuclear coupling?
7
Theory Electron Transfer in Donor-Acceptor Pairs
The electronic Hamiltonian
Diabatic electronic basis functions
The Hamiltonian matrix
8
Theory Electron Transfer in Donor-Acceptor Pairs
The Harmonic approximation
A Spin Boson Hamiltonian
9
Theory Electron Transfer in Donor-Acceptor Pairs
Donor
Acceptor
The golden rule expression for the rate
The Condon approximation
A nuclear factor
An electronic tunneling matrix element
10
Long Range Electronic Tunneling
Donor
Acceptor
The direct tunneling matrix element vanishes
McConnell (1961) Introducing a set of bridge
electronic states
The donor and acceptor sites are connected via an
effective tunneling matrix element through the
bridge
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McConnells FormulaA tight binding model
The deep tunneling regime First order
perturbation theory
A simple expression for the effective tunneling
matrix element
12
Superexchange dynamics througha symmetric
uniform bridge
Tunneling oscillations at a frequency
H. M. McConnell, J. Chem. Phys. 35, 508 (1961)
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Deep tunneling through a molecular bridge

• The role of bridge nuclear modes?
• Validity of the Condon approximation?

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Electronic nuclear coupling at the bridge
Breakdown of the Condon approximation!

Molecules 1-5
Rigid bridges enable highly efficient electron
energy transfer
Charge transfer is gated by bridge vibrations
Davis, Ratner and Wasielewski (J.A.C.S. 2001).
Lokan, Paddon-Row, Smith, La Rosa, Ghiggino and
Speiser (J.A.C.S. 2001).
15
Structural (promoting) bridge modes
Electronically active (accepting) bridge modes
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A generalized spin-boson model

• Harmonic nuclear modes
• Linear e-nuclear coupling in the bridge modes
• The e-nuclear coupling is restricted to the
  bridge sites

The nuclear potential energy surface changes at
the bridge electronic sites
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A Dissipative Superexchange Model A symmetric
uniform bridge
The nuclear frequencies 10-500 (1/cm) are larger
than the tunneling frequency!!
M. A-Hilu and U. Peskin, Chem. Phys. 296, 231
(2004).
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Coupled Electronic-Nuclear Dynamics
A mean-field approximation
The coupled SCF equations
Mean-fields
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The Langevin-Schroedinger equation
Electronic bridge population
A non-linear, non Markovian dissipation term
Fluctuations
Initial nuclear position and momentum
At zero temperature, R(t) vanishes
U. Peskin and M. Steinberg, J. Chem. Phys. 109,
704 (1998).
20
Numerical Simulations Weak e-n coupling
The tunneling frequency increases!
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Simulations Strong e-n Coupling
The tunneling is suppressed !
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Interpretation a time-dependent Hamiltonian
The Instantaneous electronic energy
A time-dependent McConnell formula
Weak coupling Energy dissipation into nuclear
vibrations lowers the barrier for electronic
tunneling

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Interpretation a time-dependent Hamiltonian
The Instantaneous electronic energy
Resonant Tunneling
Weak coupling Energy dissipation into nuclear
vibrations lowers the barrier for electronic
tunneling
Strong coupling Irreversible electronic
energy dissipation

24
Numerically exact simulations for a single bridge
mode

• Tunneling acceleration at weak coupling

• Tunneling suppression at strong coupling

25
A dissipative-acceptor model
The acceptor population
Introducing a bridge mode
Dissipation leads to a unidirectional ET
The tunneling rate Increases with e-n coupling at
the bridge!
A. Malka and U. Peskin, Isr. J. Chem. (2004).
26
Interpretation Nuclear potential energy surfaces
A dimensionless measure for the effective
electronic-nuclear coupling
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Entangled electronic-nuclear dynamics beyond the
weak coupling limit
The symmetric uniform bridge model
Deep tunneling weak electronic inter-site
coupling
A small parameter
M. A.-Hilu and U. Peskin, submitted for
publication (2004).
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A Rigorous Formulation
The Donor/Acceptor Hamiltonian
The Bridge Hamiltonian
The coupling Hamiltonian (purely electronic!)
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Introducing vibrational eigenstates
Diagonalizing the tight-binding operator
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In the absence of electronic coupling the ground
state is degenerate
Regarding the electronic coupling as a (second
order) perturbation
The energy splitting temperature reads
Frank-Condon overlap factors
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The energy splitting
Expanding the denominators in powers ofand
keeping the leading non vanishing terms gives
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Interpretation
McConnells expression
Effective barrier for tunneling
Effective electronic coupling
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Summation over vibronic tunneling pathways

• Lower barrier for tunneling

• Multiple Dissipative pathways

The effective tunneling barrier decreases
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An example (N8)
1/cm
The tunneling frequency increases by orders of
magnitude with increasing electronic nuclear
coupling
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The slow electron adiabatic limit
Considering only the ground nuclear vibrational
state
A condition for increasing the tunneling
frequency by increasing electronic-nuclear
coupling
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An example (N8)
The slow electron approximation
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Flexible vs. Rigid molecular bridges
Molecular rigidity small deviations from
equilibrium configuration
?Increasing rigidity ?
A consistency constraint
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Langevin-Schroedinger simulations
The tunneling frequency increases with bridge
rigidity
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A rigorous treatmentThe slow electron limit
Rigidity larger Frank Condon factor!
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Summary and Conclusions

• The effect of electronic-nuclear coupling in
  electronically active molecular bridges was
  studied using generalized McConnell models
  including bridge vibrations.

• Mean-field Langevin-Schroedinger simulations of
  the coupled electronic-nuclear dynamics suggest
  that weak electronicnuclear coupling promotes
  off-resonant (deep) through bridge tunneling

• A rigorous calculation of electronic tunneling
  frequencies beyond the weak electronic-nuclear
  coupling limit, predicts acceleration by orders
  of magnitudes for some molecular parameters

• An analytical approach was introduced and a
  formula was derived for calculations of tunneling
  matrix elements in a dissipative McConnell model.
  A comparison with approximate methods for
  studying open quantum systems is suggested.
• The way for rationally designed, controlled
  electron transport in molecular devices is
  still long

41
Long-range Electron Transport In Nature
Electron transfer is controlled by molecular
barriers
Deep (off-resonant) tunneling through molecular
bridges
42
Long-range Electron Transport In Nature
This enzyme is used by the bacterium to allow it
to inhabit areas of low oxygen concentration when
it leads to infections in humans. It contains a
calcium ion which appears to be crucial in the
control of electron transfer. (Fig 9)
Fig. 9. The calculated route of electron transfer
between the two haem groups of cytochrome c
peroxidase is shown ( in green) together with the
close proximity of the bound calcium ion (grey
sphere).