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).

Numerical Simulations: Weak e-n coupling

The tunneling frequency increases!

The tunneling is suppressed !

Simulations: Strong e-n Coupling

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:

An example (N=8) 100000 EEB 1000BT

8000 T2000

The slow electron approximation

Spectral densities

Molecular rigidity = small deviations from equilibriumconfiguration

2

max

)(Q

QB

Flexible vs. Rigid molecular bridges

Increasing rigidity

A consistency constraint:

Langevin-Schroedinger simulations:

The tunneling frequency increases with bridge rigidity

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