Electronic Tunneling through Dissipative
Molecular BridgesUri 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:
OutlineMotivation: • 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: McConnell’s 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.
Motivation: Electron Transport Through Molecules
Molecular Electronics
Resonant tunneling through molecular junctions
Tans, Devoret, Thess, Smally, Geerligs, Dekker, Nature (1997)
Reichert, Ochs, Beckmann, Weber, Mayor, Lohneysen, Phys. Rev. Lett. (2002).
SS
Long-range Electron Transport In Nature
The Photosynthetic Reaction Center
Deep (off-resonant) tunneling through molecular barriers
Electron transfer is controlled by molecular bridges
Tunneling pathway between cytochrome b5
and methaemoglobin
Controlled tunneling through molecules?
• Minor changes to the molecular electronic density
• High sensitivity (exponential) to the molecular parameters
• A potential for a rational design based on chemical knowledge
Resonant tunneling
Deep (off resonant) tunneling
Why Off-Resonant (deep) Tunneling ?
)](|)()(|)([||2
'
2',
*,
',
2, lll ADlAlD
llDADET EEqqdqEfHk
Electron Transfer in Donor-Acceptor Pairs
Donor Acceptor
Electronic tunneling matrix element
Nuclear factor:
Frank-Condon weighted
density of states
The role of electronic nuclear coupling?
The case of through bridge tunneling :
Theory: Electron Transfer in Donor-Acceptor Pairs
DAAD HqHqHqH ˆ)()()( )()(
The electronic Hamiltonian:
Diabatic electronic basis functions:
DDDDH )(
AAAAH )(
The Hamiltonian matrix:
qADA
DAqD
TqEqH
qHTqEH
)()(
)()(
Theory: Electron Transfer in Donor-Acceptor Pairs
A Spin Boson Hamiltonian:
The Harmonic approximation:
2)()( )(2
1)()( DD
DD qqqEqE
qADA
DAqD
TqEqH
qHTqEH
)()(
)()(
2)()( )(2
1)()( AA
AA qqqEqE
zqxDAzE qIqpHH
)(2
2 22
10
01
2
1z
01
10
2
1x
nenucelec HHHH
][||2 2
, FCHk ADET
Theory: Electron Transfer in Donor-Acceptor Pairs
The Condon approximation
Donor Acceptor
The golden rule expression for the rate
)(|)()()(|)(2
'
2'
*
',lll ADlDAl
llDET EEqqHqdqEfk
dqqqHqqHqdq llDAlDAl )()()()()( '*
'*
qADA
DAqD
TqEqH
qHTqEH
)()(
)()(
An electronic tunneling matrix element
A nuclear factor
01
,,)(,
N
n nD
AnnDeffAD E
TTH
McConnell (1961): Introducing a set of bridge electronic states;
0, ADH
The direct tunneling matrix element vanishes
e
Donor Acceptor
Long Range Electronic Tunneling
The donor and acceptor sites are connected via an effective tunneling matrix element through the bridge
11 0
)(
N
N
k k
keffDA t
EE
tH1
0
EE
T
k
k
McConnell’s Formula:A tight binding model
The deep tunneling regime: First order perturbation theory
A simple expression for the effective tunneling matrix element
11
1
11
1
3
322
211
10
0
0
NN
NNN
NNN
N
Et
tEt
tEt
tt
tEt
tEt
tE
H
NB
NB
DA EE
TTH
)(
)(
0
12
0
011 Ttt N
Tunneling oscillations at a frequency :
Superexchange dynamics througha symmetric uniform bridge
NB
NB
EE
T
h
T
)(
)(2
0
120
NkTt Bk ,..,3,2
H. M. McConnell, J. Chem. Phys. 35, 508 (1961)
Deep tunneling through a molecular bridge
• The role of bridge nuclear modes?
• Validity of the Condon approximation?
][||2 2
, FCHk effADET
Davis, Ratner and Wasielewski (J.A.C.S. 2001).
Molecules 1-5
Charge transfer is gated by bridge vibrations
Electronic nuclear coupling at the bridge:
Rigid bridges enable highly efficient electron
energy transfer Lokan, Paddon-Row, Smith, La Rosa,
Ghiggino and Speiser (J.A.C.S. 2001).
Breakdown of the Condon approximation!
Structural (promoting) bridge modes:
Electronically active (accepting) bridge modes:
)(0
)()(
)()(
0)(
3
322
211
10
Qt
QtEQt
QtEQt
QtE
H
3
322
211
10
0
)(
)(
0)(
t
tQEt
tQEt
tQE
H
3
322
211
10
00
0
0
00
t
tEt
tEt
tE
H elec
A generalized “spin-boson” model:
N
jjjj
N
jjj
jelec QPQHH
11
22 ˆˆ)ˆˆ(2
ˆˆ
000
000
000
0000
||ˆ2,
1,
,1 j
j
njQ
QN
nQj nn
The nuclear potential energy surface changes at the bridge electronic sites
• Harmonic nuclear modes • Linear e-nuclear coupling in the bridge modes • The e-nuclear coupling is restricted to the bridge sites
nenucelec HHHH
A Dissipative Superexchange Model:
A symmetric uniform bridge
?eVTTeVEN BB 1,,2.3,8 0
Introducing nuclear modes with an Ohmic ( ) spectral density ce /
The nuclear frequencies: 10-500 (1/cm) are larger than the tunneling frequency!!
BQ
N
nQj
jjnn ˆ||ˆ
1
and a uniform electronic-nuclear coupling :
M. A-Hilu and U. Peskin, Chem. Phys. 296, 231 (2004).
Coupled Electronic-Nuclear Dynamics
N
jjQjB
N
jjj
jelec QPQHH
j11
22 ˆˆ)ˆˆ(2
ˆˆ
A mean-field approximation:
N
jjQjBelecelec ttQHt
ti
j1
)(|])(ˆˆ[)(|
),(])(ˆ)ˆˆ(2
[),( 22 tQtQQPtQt
i jjBQjjjjj
jj j
j
jjelec tQtt ),()(|)(|
The coupled SCF equations:
))'(sin()'(')sin()0()cos()0()(0
tttdttPtQtQ jB
t
Qjjjjjj j
Mean-fields:
N
BridgenelecB tnt 2|)(||)(
The Langevin-Schroedinger equation
)]sin()0()cos()0([)(1
tPtQtR jjjj
N
jjQ j
)(|)]}()([ˆˆ{)(| ttRtFHtt
i elecBelecelec )(tR
A non-linear, non Markovian dissipation term
FluctuationsAt zero temperature, R(t) vanishes
)(tF
t
BjQj
N
j
dtttttFj
0
22
1
')'())'(sin()(
Initial nuclear position and momentum
Electronic bridge population
U. Peskin and M. Steinberg, J. Chem. Phys. 109, 704 (1998).
Interpretation: a time-dependent Hamiltonian
)(ˆˆ)(ˆ tFHtH Beleceff
)()()()(ˆ )()( tUttUtH ll
leff
The Instantaneous electronic energy:
Weak coupling: Energy dissipation into nuclear vibrations lowers the barrier for
electronic tunneling NB
NB
EtEh
TTt
])([
)(2)(
0
120
A time-dependentMcConnell formula
Interpretation: a time-dependent Hamiltonian
)(ˆˆ)(ˆ tFHtH Beleceff
)()()()(ˆ )()( tUttUtH ll
leff
The Instantaneous electronic energy:
Weak coupling: Energy dissipation into nuclear vibrations lowers the barrier for
electronic tunneling
Strong coupling:“Irreversible” electronic
energy dissipation
Resonant Tunneling
Numerically exact simulations for a single bridge mode
•Tunneling suppression at strong coupling
•Tunneling acceleration at weak coupling
A dissipative-acceptor model:
The acceptor population:
Dissipation leads to a unidirectional ET
The tunneling rate Increases with e-n coupling at the bridge!
Introducing a bridge mode
A. Malka and U. Peskin, Isr. J. Chem. (2004).
A dimensionless measure for the effective electronic-nuclear coupling:
Interpretation: Nuclear potential energy surfaces
0
2 2/
EEB
Q
Deep tunneling = weak electronic inter-site coupling
Entangled electronic-nuclear dynamics beyond the weak coupling limit
A small parameter: 1||
||
0
0 EE
T
B
)(
)(
)(0
0)(
)(
)(
00
0
0
00
QET
TQET
TQET
TT
TQET
TQET
TQE
H
BB
BBB
B
B
BBB
BB
The symmetric uniform bridge model:
M. A.-Hilu and U. Peskin, submitted for publication (2004).
nenucelec HHHH ˆˆˆˆ
.).|||(|ˆ10/ ccT ANDBAD H
.).||(||ˆˆ1
1, ccTH n
BridgennBn
N
nnQBB
H
A Rigorous Formulation
BADBAD //ˆˆˆˆ HHΗH
The Donor/Acceptor Hamiltonian
The Bridge Hamiltonian
The coupling Hamiltonian (purely electronic!)
|)||(|ˆˆ,0/ AADDQAD H H
Introducing vibrational eigenstates:
N
jjjl
llllQ lEH
10,0,0,0,0,0 )
2
1(;||ˆ
N
jjjBBlBlB
llBlBQB lEEEH
10,,,,, )
2
1()(;||ˆ
l
llADn
nnlAD ||||ˆ,0,0
,,0/ H
l
N
mnlBlBmnmnlBmnBB T
1,,,,,1, ||||)(ˆ H
llBlBnn
N
nlnB ||||ˆ
,,1
, H
Diagonalizing the tight-binding operator:
)1
cos(2,,
N
nTBlBln
mm
n N
mn
N |)
1sin(
1
2|
1
l
N
nBlB
lSA
Nn
T
NNn
Nn
NTEE1
0,0,
2,0
20
)1
cos(2
)1
sin()1
sin()1
2(
2
',,0', | lBlll
Regarding the electronic coupling as a (second order) perturbation
In the absence of electronic coupling the ground state is degenerate:
1||
||
0,
0 ET
lB
The energy splitting temperature reads:
0,01
, |)|(|2
| ADAS BAD HΗH0
ˆˆˆ/
BAD /0ˆˆˆ HΗH
.).|||(|ˆ10/ ccT ANDBAD H
Frank-Condon overlap factors
l
N
nBlB
lSA
Nn
T
NNn
Nn
NTEE1
0,0,
2,0
20
)1
cos(2
)1
sin()1
sin()1
2(
2
The energy splitting:
Expanding the denominators in powers of
and keeping the leading non vanishing terms gives
0,0,
)1
cos(2
lB
B Nn
T1
||
||
0,
0 ET
lB
1
0,0,0,0,
2,0
20 )(
2
N
lB
B
l lB
lSA TT
hh
EE
Interpretation:
1
0,0,0,0,
2,0
20
)(2
N
lB
B
l lB
l TT
h
N
jjjBlBlEE
100,0,
)1)((
N
j
l
j
Bj
j
EE
lj
N
j j
Bj
EE
le
1
0
)(
2,0
])(
[!
11
0
1
00
202
N
B
B
B EE
T
EE
T
h
Effective electronic coupling
Effective barrier for tunneling
McConnell’s expression:
)1)(( 0 EEB
Summation over vibronic tunneling pathways:
•Lower barrier for tunneling
•Multiple “Dissipative” pathways
)( 0EEB
The effective tunneling barrier decreases
An example (N=8)
The tunneling frequency increases by orders of magnitude
with increasing electronic nuclear coupling
100000 EEB 1000BT8000 T2000
1/cm
0
2 2/
EEB
Q
The “slow electron” adiabatic limit
1
0,0,0,0,
2,0
20
)(2
N
lB
B
l lB
l TT
h
NB
NB
EE
ad EE
TeT
h
N
j j
Bj
)]1)([(
)(2
0
1)(
20
1
0
0])(
)1([ 0
j
Bad
j
ad EEN
Considering only the ground nuclear vibrational state:
N
EEBj
)1)(( 0
A condition for increasing the tunneling frequency by increasing electronic-nuclear coupling:
Spectral densities
Molecular rigidity = small deviations from equilibriumconfiguration
2
max
)(Q
QB
Flexible vs. Rigid molecular bridges
Increasing rigidity
A consistency constraint:
A rigorous treatment:The “slow electron” limit
NB
NB
EE
E
TeT
h
B
)]1([
)(2 1)(2
0
0
Rigidity = larger Frank Condon factor!
Summary and Conclusions
• 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…
• 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 electronic–nuclear coupling promotes off-resonant (deep) through bridge tunneling