A. Nitzan, Tel Aviv University The Abdus Salam African College on Science at the Nanoscale Cape Town, November 2007 1. Relaxation, reactions and electron

A. Nitzan, Tel Aviv University

The Abdus Salam African College on Science at the

Nanoscale Cape Town, November 2007

1. Relaxation, reactions and electron 1. Relaxation, reactions and electron transfer in condensed molecular systemstransfer in condensed molecular systems

2. Fundamentals of molecular conduction2. Fundamentals of molecular conduction

3. 3. Inelastic effects in electron transfer and Inelastic effects in electron transfer and molecular conductionmolecular conduction

Molecular conductionMolecular conduction

m o lecule

Molecular Rectifiers

Arieh Aviram and Mark A. RatnerIBM Thomas J. Watson Research Center, Yorktown Heights, New

York 10598, USADepartment of Chemistry, New York New York University, New

York 10003, USA

Received 10 June 1974Abstract

The construction of a very simple electronic device, a rectifier, based on the use of a single organic molecule is discussed. The molecular rectifier consists of a donor pi system and an acceptor pi system, separated by a sigma-bonded (methylene) tunnelling bridge. The response of such a molecule to an applied field is calculated, and rectifier properties indeed appear.

Xe on Ni(110)

Moore’s “Law”

Cornell group

m ole c ule

•Fabrication

•Stability

•Characterization

•Funcionality

•Control

FABRICATION

Nanopore

Reed et al. APL 71 (97)

Molecule lying on a surface

Molecule between two electrodes:Break junction:

Dorogi et al. PRB 52 (95) @ Purdue

Au(111)

Pt/Ir Tip

SAM1 nm

~1-2 nm

Self-assembled monolayers

Adsorbed molecule addressed by STM tip

C60 on gold

Joachim et al. PRL 74 (95)

STMtip

Au

Nanotube on Au

Lieber et al. Nature 391 (98)

Van Ruitenbeek, Wittenberg Lectures 2004

•Fabrication

•Stability

•Characterization

•Funcionality

•Control

H O M O

L U M O

E F

W o rkfu n c tio n

System open to electrons and energy

THE MOLECULE

Nonequilibrium

Electron-vibration coupling

Heat generation

Relaxation

Strong electric field

AN, Oxford University Press, 2006

Cape Town, November 2006

(1a) Relaxation and reactions in condensed molecular systems•Timescales•Relaxation•Solvation•Activated rate processes•Low, high and intermediate friction regimes•Transition state theory•Diffusion controlled reactions

(1b) 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

(2) 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

Chapter 13-15Chapter 16Chapter 17

(3) Inelastic effects in molecular conduction•Dependence on nuclear configurations•Electron-vibration coupling•Timescales•Coherent and incoherent transport•Heating•Current induced nuclear changes•Heat conduction•Inelastic tunneling spectroscopy

1A1A

Relaxation and reactions Relaxation and reactions in molecular systemsin molecular systems

The importance of The importance of timescalestimescales

Molecular processes in Molecular processes in condensed phases and condensed phases and

interfacesinterfaces•Diffusion

•Relaxation

•Solvation

•Nuclear rerrangement

•Charge transfer (electron and xxxxxxxxxxxxxxxxproton)

•Solvent: an active spectator – energy, friction, solvation

Molecular timescales

Electronic 10-16-10-15s

Vibraional period 10-14s

Vibrational xxxxrelaxation 1-10-12s

Diffusion D~10-5cm2/s 10nm 10-7 - 10-8 s

Chemical reactions xxxxxxxxx1012-10-12s

Rotational period 10-12s

Collision times 10-12s

Molecular vibrational Molecular vibrational relaxationrelaxation

D

/~ DcVRk e

Relaxation in the X2Σ+ (ground electronic state) and A2Π (excite electronic state) vibrational manifolds of the CN radical in Ne host matrix at T=4K, following excitation into the third vibrational level of the Π state. (From V.E. Bondybey and A. Nitzan, Phys. Rev. Lett. 38, 889 (1977))

22 ˆ ˆ( ) (0)ifi tf i T

k dte F tV F

µ

Golden RuleFourier transform of bath correlation function

Molecular vibrational Molecular vibrational relaxationrelaxation

The relaxation of different vibrational levels of the ground electronic state of 16O2 in a solid Ar matrix. Analysis of

these results indicates that the relaxation of the < 9 levels is dominated by radiative decay and possible transfer to impurities. The relaxation of the upper levels probably takes place by the multiphonon mechanism. (From A. Salloum, H. Dubust, Chem. Phys.189, 179 (1994)).

Frequency dependent Frequency dependent frictionfriction

consˆ ˆ~ ( ) (0) tantifi tf i T

t

k dte F t F

ˆ ˆ~ ( ) (0)ifi t

f i Tk dte F t F

1

DWIDE BAND APPROXIMATION

MARKOVIAN LIMIT

Dielectric solvationDielectric solvation

q = + e q = + eq = 0

a b c

C153 / Formamide (295 K)

Wavelength / nm

450 500 550 600

Rel

ativ

e E

mis

sion

Int

ensi

ty

ON O

CF3

Emission spectra of Coumarin 153 in formamide at different times. The times shown here are (in order of increasing peak-wavelength) 0, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, and 50 ps (Horng et al, J.Phys.Chem. 99, 17311 (1995))

2 11 1 2eV (for a charge)

2 s

q

a

Born solvation energy

Continuum dielectric theory of Continuum dielectric theory of solvationsolvation

D 4

(r, ) ( ) (r, )t

D t dt t t E t

D E

( ) ( ) 4 ( )

( ) ( ) ( )

1

4

D E P

P E

D(r, ) r ' (r r ', )E(r ', )εt

t d dt t t t

1 2

( )1

s ee

Di

How does solvent respond to a sudden change in the molecular charge distribution?

Electric displacement

Electric field

Dielectric function

Dielectric susceptibility

polarizationDebye dielectric relaxation model

Electronic response

Total (static) response

Debye relaxation time

Poisson equation

Continuum dielectric theory of Continuum dielectric theory of solvationsolvation

0 0( )

0

tE t

E t

1; 0s

D D

dDD E t

dt

1( ) ( 4) ;e s

D

dE

dDD D E

t

/ /( ) (1 )D Dt ts eD t e e E

0 0( )

0

tD t

D t

1; 0s

e D s

dE E D t

dt

/1 1 1( ) Lt

s e s

E t D De

eL D

s

WATER:

D=10 ps L=125 fs

““real” solvationreal” solvationThe experimental solvation function for water using sodium salt of coumarin-343 as a probe. The line marked ‘expt’ is the experimental solvation function S(t) obtained from the shift in the fluorescence spectrum. The other lines are obtained from simulations [the line marked ‘Δq’ –simulation in water. The line marked S0 –in a neutral atomic solute with Lennard Jones parameters of the oxygen atom]. (From R. Jimenez et al, Nature 369, 471 (1994)).

“Newton”

dielectric

Electron solvationElectron solvationThe first observation of hydration dynamics of electron. Absorption profiles of the electron during its hydration are shown at 0, 0.08, 0.2, 0.4, 0.7, 1 and 2 ps. The absorption changes its character in a way that suggests that two species are involved, the one that absorbs in the infrared is generated immediately and converted in time to the fully solvated electron. (From: A. Migus, Y. Gauduel, J.L. Martin and A. Antonetti, Phys. Rev Letters 58, 1559 (1987)

Quantum solvation

(1) Increase in the kinetic energy (localization) – seems NOT to affect dynamics

(2) Non-adiabatic solvation (several electronic states involved)

C153 / Formamide (295 K)

Wavelength / nm

450 500 550 600

Rel

ativ

e E

mis

sion

Int

ensi

ty

ON O

CF3

Electron tunneling Electron tunneling through waterthrough water

E F

W o rkfu n ct io n( in wa te r)

W A T E R

12

3

Polaronic state (solvated electron)

Transient resonance through “structural defects”


Time (ms)

STM current in pure waterSTM current in pure waterS.Boussaad et. al. JCP (2003)S.Boussaad et. al. JCP (2003)

Chemical reactions in Chemical reactions in condensed phasescondensed phases

Bimolecular

Unimolecular

diffusion

4k DR

Diffusion controlled

rates

Bk TD

mR

2

1

k1 2 k2 1

k2

excitation

reaction

21 2

12 2

k Mkk

k M k

k

M

Thermal interactions

Unimolecular reactions (Lindemann)

Activated rate processesActivated rate processes

E B

r e ac t i o nc o o r di nate

KRAMERS THEORY:

Low friction limit

High friction limit

Transition State theory

0 /

2B B

TSTE k Tk e

0 /

2B BB B

TSTE k Tk e k

/0

B BE k TB

B

k J ek T

(action)

4k DR


rates

Bk TD

m

0

B

Effect of solvent frictionEffect of solvent friction

A compilation of gas and liquid phase data showing the turnover of the photoisomerization rate of trans stilbene as a function of the “friction” expressed as the inverse self diffusion coefficient of the solvent (From G.R. Fleming and P.G. Wolynes, Physics Today, 1990). The solid line is a theoretical fit based on J. Schroeder and J. Troe, Ann. Rev. Phys. Chem. 38, 163 (1987)).

TST

The physics of transition The physics of transition state ratesstate rates

0

2BEe

0

( ,TST B f BP xk d P x

v v v v)

212

212

0 1

2

m

m

d e

md e

v

v

vv

v

20exp

( )2exp ( )

B

B

B EB E

E mP x e

dx V x

Assume:

(1) Equilibrium in the well

(2) Every trajectory on the barrier that goes out makes it

E B

0

B


The (classical) transition The (classical) transition state rate is an upper state rate is an upper

boundbound

E B


•Assumed equilibrium in the well – in reality population will be depleted near the barrier

•Assumed transmission coefficient unity above barrier top – in reality it may be less

R *

a b

diabatic

R *

1

1

2

Adiabatic

*

0

( , )k dR R P R R

Quantum considerations

1 in the classical case( )b aP R











The importance of timescales


E F

W o r kfunc t i o n( i n w ate r )

W A T E R

1

2

3

Polaronic state (solvated electron)

Transient resonance through “structural defects”

Scoles

Activated rate processesActivated rate processes

E B


KRAMERS THEORY:

Low friction limit

High friction limit

Transition State theory

0 /

2B B

TSTE k Tk e

0 /

2B BB B

TSTE k Tk e k

/0

B BE k TB

B

k J ek T

(action)

4k DR


rates

Bk TD

m

0

B

The physics of transition The physics of transition state ratesstate rates

0

2BEe

0

( ,TST B f BP xk d P x

v v v v)

Assume:

(1) Equilibrium in the well

(2) Every trajectory on the barrier that goes out makes it

E B

0

B


R *

a b

diabatic

R *

1

1

2

Adiabatic

*

0

( , )k dR R P R R

Quantum considerations

1 in the classical case( )b aP R

PART 1B

Electron transfer


Chapter 16


Theory of Electron TransferTheory of Electron Transfer

Rate – Transition state theoryRate – Transition state theory

0

( ,TST B abP xk d P

v v v v)

Transition rate

BoltzmannBoltzmannActivation Activation energyenergy

Transition Transition probabilityprobability

Electron transfer in polar Electron transfer in polar mediamedia

•Electrons are much faster than nuclei

Electronic transitions take place in fixed nuclear configurations

Electronic energy needs to be conserved during the change in electronic charge density

a

q = 0

b

q = + e

c

q = + e

Electronic transition

Nuclear relaxation

q = 1q = 0 q = 0q = 1

Electron transfer

Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations

Nuclear motion

Nuclear motion

q= 0q = 1q = 1q = 0

Electron transferElectron transfer

E a

E b

E

e ne r g y

ab

X a X tr X b

Solvent polarization coordinate

EA

Transition state theoryTransition state theory of of electron transferelectron transfer

Adiabatic and non-adiabatic ET processesE

R

E a(R )

E b(R )

E 1(R )

E 2(R )

R *

tt= 0

V ab

Landau-Zener problem

*

0

( , ) ( )b ak dRR P R R P R

2,

*

2 | |( ) 1 exp a b

b a

R R

VP R

R F

*

2,| |

2Aa b E

NAR R

VKk e

F

(For harmonic diabatic surfaces (1/2)KR2)

Electron transfer – Electron transfer – Marcus theoryMarcus theory

(0) (0) (1) (1)B BA Aq q q q (0) (0) (1) (1)

B BA Aq q q q

D 4

E D 4 P

eP P Pn

1

4e

eP E

4s e

nP E

They have the following characteristics:(1) Pn fluctuates because of thermal motion of solvent nuclei.(2) Pe , as a fast variable, satisfies the equilibrium relationship (3) D = constant (depends on only)Note that the relations E = D-4P; P=Pn + Pe are always satisfied per definition, however D sE. (the latter equality holds only at equilibrium).

We are interested in changes in solvent configuration that take place at constant solute charge distribution

D Es

q = 1q = 0 q = 0q = 1

q= 0q = 1q = 1q = 0

Electron transfer – Electron transfer – Marcus theoryMarcus theory

0 (0) (0)BAq q

(0) (0) (1) (1)B BA Aq q q q (0) (0) (1) (1)

B BA Aq q q q

Free energy associated with a nonequilibrium fluctuation of Pn

“reaction coordinate” that characterizes the nuclear polarization

q = 1q = 0 q = 0q = 1

q= 0q = 1q = 1q = 0

1 (1) (1)A Bq q

The Marcus parabolasThe Marcus parabolas

0 1 0( ) Use as a reaction coordinate. It defines the state of the medium that will be in equilibrium with the charge distribution . Marcus calculated the free energy (as function of ) of the solvent when it reaches this state in the systems =0 and =1.

20 0( )W E 2

1 1( ) 1W E

21 1 1 1 1

2 2e s A B AB

qR R R

Electron transfer: Electron transfer: Activation energyActivation energy

2[( ) ]

4b a

A

E EE

21 1 1 1 1

2 2e s A B AB

qR R R

E aE A

E b

E

e ne r g y

ab

a= 0 trb= 1

2( )a aW E

2( ) 1b bW E

Reorganization energy

Activation energy

Electron transfer: Effect of Electron transfer: Effect of Driving (=energy gap)Driving (=energy gap)

Experimental confirmation of the inverted regime

Marcus papers 1955-6

Marcus Nobel Prize: 1992

Miller et al, JACS(1984)

Electron transfer – the Electron transfer – the couplingcoupling

• From Quantum Chemical Calculations

•The Mulliken-Hush formula max 12DA

DA

VeR

• Bridge mediated electron transfer

2 4~

ab

B

E

k Tet abk V e

Bridge assisted electron Bridge assisted electron transfertransfer

D A

B 1 B 2 B 3

D A

12

3V D 1

V 1 2 V 2 3

V 3 A

1

1 1

1

, 1 , 11

ˆ

1 1

1 1

N

D j Aj

D D AN NA

N

j j j jj

H E D D E j j E A A

V D V D V A N V N A

V j j V j j

, 1 /,j B j j B D AE E V E E

EB

VDB

D A

BVAD E

D A

Veff DB ABeff

V VV

E

VDB

D A

B1

VAD

D A

E

Veff

1eff DB N ABV V G V

B2 BN…V12

12 23 1,1

... N NN N

V V VG

E

1

1

1exp (1 / 2) '

NB

N NB

VG N

E V

' 2 ln / BE V D A

12

3V D 1

V 1 2 V 2 3

V 3 A

Green’s Function

1ˆG E E H

Marcus expresions for non-Marcus expresions for non-adiabatic ET ratesadiabatic ET rates

2

2 (1

2)1 ( )

|

)2

| ( )

(

2

BD

DA

D

D A AD

N ANA D

V

V

E

GV E

k

E

F

F

2 / 4

( )4

BE k T

B

eE

k T

F

Bridge Green’s Function

Donor-to-Bridge/ Acceptor-to-bridge

Franck-Condon-weighted DOS


Bridge mediated ET rateBridge mediated ET rate

~ ( , )exp( ' )ET AD DAk E T RF

’ (Å-1)=

0.2-0.6 for highly conjugated chains

0.9-1.2 for saturated hydrocarbons

~ 2 for vacuum

Bridge mediated ET rateBridge mediated ET rate(J. M. Warman et al, Adv. Chem. Phys. Vol 106, 1999).

Incoherent hoppingIncoherent hopping

........

0 = D

1 2 N

N + 1 = A

k 2 1

k 1 0 = k 0 1 e x p (-E 1 0 ) k N ,N + 1 = k N + 1 ,N e x p (-E 1 0 )

0 1,0 0 0,1 1

1 0,1 2,1 1 1,0 0 1,2 2

1, 1, , 1 1 , 1 1

1 , 1 1 1,

( )

( )N N N N N N N N N N N N

N N N N N N N

P k P k P

P k k P k P k P

P k k P k P k P

P k P k P

constant STEADY STATE SOLUTION

ET rate from steady state ET rate from steady state hoppinghopping

........

0 = D

1 2 N

N + 1 = A

k


k k0,1 2,1 1 1,0 0 1,2 2

1, 1, ,

1 1,

1 1

0 ( )

0 ( )N N N N N N N

N N N N

N

k k P k P k P

k k P k P

P k P

0D Ak P

/

1,0

1

1

B BE k T

D A N

N A D

kek k

k kN

k k

Dependence on Dependence on temperaturetemperature

The integrated elastic (dotted line) and activated (dashed line) components of the transmission, and the total transmission probability (full line) displayed as function of inverse temperature. Parameters are as in Fig. 3 .

The photosythetic reaction The photosythetic reaction centercenter

Michel - Beyerle et al

Dependence on bridge Dependence on bridge lengthlength

Ne

11 1up diffk k N

DNA (Giese et al 2001)DNA (Giese et al 2001)

ELECTROCHEMISTRYELECTROCHEMISTRY

A

C

R

DA

, ,( ) 1 ( )b a b ak E E f E

Rate of electron transfer to metal in vacuum

Rate of electron transfer to metal in electrolyte solution

,( ) 1 ( ) b ak dE E f E F E E

2

, , ,2

( ) ( )b a D M M b aE V E

Transition rate to a continuum (Golden Rule)

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

2 / 4

( )4

BE k T

B

eE

k T

F

M

EF


Chapter 16













q = 1q = 0 q = 0q = 1

Electron transfer

Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations

Nuclear motion

Nuclear motion

q= 0q = 1q = 1q = 0

Electron transfer rateElectron transfer rate

• Bridge mediated electron transfer

2 4~

ab

B

E

k Tet abk V e

Marcus expresions for non-Marcus expresions for non-adiabatic ET ratesadiabatic ET rates

2

2 (1

2)1 ( )

|

)2

| ( )

(

2

BD

DA

D

D A AD

N ANA D

V

V

E

GV E

k

E

F

F

2 / 4

( )4

BE k T

B

eE

k T

F

Bridge Green’s Function

Donor-to-Bridge/ Acceptor-to-bridge

Franck-Condon-weighted DOS


1 exp ' / 2NG N

Incoherent hoppingIncoherent hopping

........

0 = D

1 2 N

N + 1 = A

k 2 1


/

1,0

1

1

B BE k T

D A N

N A D

kek k

k kN

k k

2

Molecular conduction

(2) 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•Electron transfer by hopping•Inelastic tunneling spectroscopy

Chapter 17AN, Oxford University

Press, 2006


Steady state evaluation Steady state evaluation of ratesof rates

Rate of water flow depends linearly on water height in the cylinder

Two ways to get the rate of water flowing out:

(1) Measure h(t) and get the rate coefficient from k=(1/h)dh/dt

(1) Keep h constant and measure the steady state outwards water flux J. Get the rate from k=J/h

= Steady state rate

h

Steady state quantum Steady state quantum mechanicsmechanics

{ }l l0

V0l

Starting from state 0 at t=0:

P0 = exp(-t)

= 2|V0l|2L (Golden Rule)

Steady state derivation:

0 0 0 sl ll

dC iE C i V C

dt

0( ) ( ) 0 ( )ll

t C t C t l 0( / )

0 0i E tC c e

0 0 ; alll l l ld

C iE C iV C ldt

l

0( / )0 0

i E tC c e

0 0 ; alll l l ld

C iE C iV C ldt

(1 / 2) lC

0( / ) 0

0

( ) ;/ 2

i E t lsl l l

l

V cC t c e c

E E i

2 2 20 0 2 2

0

2 200 0 0

//

/ 2

2

l ll l l

l ll

J C C VE E

C V E E

0

2 20 0 0 02

0

2 2/

l

l l l LE El

Jk V E E V

C

pumping damping

Resonance scatteringResonance scattering

{ }l1

V1r

r l

V1l

0

0H H V

0 0 10

0 0 1 1 l rl r

H E E E l l E r r

0,1 1,0 ,1 1, ,1 1,0 1 1 0 1 1 1 1l l r r

l r

V V V V l V l V r V r


0( / )0 0

i E tC c E

( / 2) rC

( / 2) lC

1 1 1 1,0 0 1, 1,

,1 1

,1 1

l l r rl r

r r r r

l l l l

C iE C iV C i V C i V C

C iE C iV C

C iE C iV C

0( ) exp ( / )j jC t c i E t j = 0, 1, {l}, {r}

For each r and l

0 0 0 0,1 1C iE C iV C

0 0( ) ( / ) exp ( / )j jC t i E c i E t


0

0 1 1 1,0 0 1, 1,

0 ,1 1

0 ,1 1

( / )0 0

( /

0

)

/

0

0

2

( 2)

l l r rl r

r r r

l

i E t

r

l ll

i E E c iV c i V c i

C c e

c

c

V c

i E E c iV c

i E E c iV c

For each r and l

,1 1

0 / 2r

rr

V cc

E E i

1, 1 0 1( )r r RrV c iB E c

1 1

21 1

21

1

( ) (1 / 2) ( )

( ) 2 | | ( )

| | ( )( )

r

R R

R r R r E E

r R rR r

r

E i E

E V E

V EE PP dE

E E

1 1

wide ban

( ) (

d approximatio

)

n

1/ 2R RB E i

21

1| |

( ) rR

rr

VB E

E E i

SELF ENERGY

0 1 1 1,0 0 1, 1,0 l l r rl ri E E c iV c i V c i V c

0 ,1 10 ( / 2)r r r ri E E c iV c c

1,0 01

0 1 1 0( / 2) ( )

V cc

E E i E

1 0 1 0 1 0( ) ( ) ( )L RE E E

2 2 2,1 1,0 02 22 2 2 2

0 0 1 1 0

| | | | | || | | |

( ) ( / 2) ( ) / 2

rr r

r

V V cc C

E E E E E

22 1,00 21 0

0 02 20 1 1 0

| | ( )/ | |

( ) / 2

RR r

r

V EJ c c

E E E

21 02 r rV E E

{ }l1

V1r

r l

V1l

0

Resonant tunnelingResonant tunneling

21,0 21

0 02 20 1 1

| || |

/ 2

RR

VJ c

E E

|1 >

|0 >

x

V (x )

RL

. . . .

. . . .

. . . .

. . . .

. . . . . . . .

(a )

(b)

( c)

L

R

{ }l1

V1r

r l

V1l

0V10

Resonant TunnelingResonant Tunneling|1 >

|0>

x

V (x )

RL

. . . .

. . . .

. . . .

. . . .

. . . . . . . .

(a )

(b)

( c)

L

R

21,0 21

0 02 20 1 1

| || |

/ 2

RR

VJ c

E E

2 00 0 0 0incident flux ( ) | | ( )R

pJ E c E

mL T TTransmission Coefficient

102 ( )L E

1 0 1 0

0 2 20 1 1 0

1 1 1

( ) ( )( )

( ) / 2

L R

R L

E EE

E E E

T

Resonant Transmission – 3dResonant Transmission – 3d

|1 >

|0 >

x

V (x )

RL

. . . .

. . . .

. . . .

. . . .

. . . . . . . .

(a )

(b)

( c)

R

L 1 0 1 0

0 2 20 1 1 0

1 1 1

( ) ( )( )

( ) / 2

L R

R L

E EE

E E E

T

1d

0

21 0 1 002 2

0 1 1 0

( ) ( ) ( )1| |

2 ( ) / 2

L R L R

E E

dJ E E Ec

dE E E E

3d: Total flux from L to R at energy E0:

If the continua are associated with a metal electrode at thermal equilibrium than 12

0 0 0( ) exp ( ) / 1Bc f E E k T

(Fermi-Dirac distribution)

CONDUCTIONCONDUCTION

0

20 0

( ) 1( ) | |

2L R

E E

dJ EE c

dE

T

( ))2

( ) (L Rf Ee

I d fE EE

T

1 1

2 21 1

( ) ( )( )

( ) / 2

L RE EE

E E E

T

( ) ( ) ( )f E e f E e E 2

( )2

eI E

T

2 spin states

2

( )e

I E

T

Zero bias conduction ( 0)g

L R

–e|

f(E0) (Fermi function)

Landauer formulaLandauer formula2

( 0) ( ) ; Fermi energye

g E

T

( ) ( ) ( )L R

eI dE f E f E E

T ( )

dIg

d

1 1

2 21 1

( ) ( )( )

( ) / 2

L RE EE

E E E

T

(maximum=1)

2

112.9

eg K

Maximum conductance per channel

For a single “channel”:

( ))2

( ) (L Rf Ee

I d fE EE

T

fL(E) – fR(E) T(E)

e

fL(E) – fR(E)

T(E)e

I

Weber et al, Chem. Phys. 2002

g

Molecular level structure Molecular level structure between electrodesbetween electrodes

en erg y

LUMO

HOMO

GATINGGATING

IONIC GATING

“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”.

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

General caseGeneral case

( )† ( )B

ˆ ˆˆ ˆ(E)=Tr ( ) ( ) ( ) ( )B BE G E E G E T

( ) ( )

( ), , ', '

( ) (1 / 2)

2

R R

Rn R R n Rn n

B E i

H H

( ) ( ) ( )L R

eI dE f E f E E

T

1( ) ( ) ( )ˆ( ) IB B BG E E H

( )

, ' , ', 'B

n n n nn nH H B

Unit matrix in the bridge space

Bridge Hamiltonian

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

Wide band approximation

1 1 1

21

2 21

1 1

( ) ( ) ( ) ( )( )

( ) / (2 1 / 2)

L R L RE E E EE

E E E E E i

T

The N-level bridge (n.n. The N-level bridge (n.n. interactions)interactions)

0

{ r }{ l}

RL

1 . . . . N + 1

2

( )e

g E

T

( ) ( )20, 1 0 1( ) | ( ) | ( ) ( )L R

N NE G E E E T

( ) ( ) ( )L R

eI dE f E f E E

T

01 12 , 10, 1

1 10

1 1 1 1ˆ ( ) ...B N NN

N N

G E V V VE E E E E EE E

0 0

1

1

2 LE E i

1 1,

1

1

2N N RE E i

G1N(E)

ET vs ConductionET vs Conduction

2

01 ,(11

2

2)

2| |

2

(

(( )

)

)

AD

NBN D

A DA

AN

D

DV V

E

G E

k

E

V

F

F

01 , 1

( ) ( )0

( ) ( )

1

0 1

( ) ( )0

2

2

( )

1

1

2

1

0,

2

2

| ( ( ) ( )

( )1 1

)

)

2

(

2

( )

|

N N

L RD

L RN

L RNN

A

B

N

N

eg

e V V

E E i E

E E

E

G

GE

EE

E

i

........

0 = D

1 2 N

N + 1 = A

E

2 / 4

( )4

BE k T

B

eE

k T

F

A relation between g and A relation between g and kk

2

2 ( ) ( )

8D AL R

D A

eg k

F

conduction Electron transfer rate

MarcusDecay into electrodes

Electron charge

A relation between g and A relation between g and kk

2

2 ( ) ( )

8D AL R

D A

eg k

F

1

4 exp / 4B Bk T k T

F

eV( ) ( ) 0.5L RD A eV

2 13 1

17 1 1

~ / 10 ( )

10 ( )

D A

D A

g e k s

k s

Comparing conduction Comparing conduction to ratesto rates

(M. Newton, 2003)(M. Newton, 2003)

2-level bridge (local 2-level bridge (local representation)representation)

( ) ( ) 22

1,21 2

2( ) ( ) 2

1 2 1,21 2

( ) ( ) | |( )

(1 / 2) ( ) (1 / 2) ( ) | |

L R

L R

E E Veg E

E E i E E E i E V

1

{ r }{ l}

RL

2

V 1 2

•Dependence on:

•Molecule-electrode coupling L

, R

•Molecular energetics E1, E2

•Intramolecular coupling V1,2

-1

0

1

2

3

4

5

6

-1 -0.5 0 0.5 1

I /

arb

. u

nit

s

0.0 - 0.5

0.5

I

V (V)

Ratner and Troisi, 2004

Temperature and chain Temperature and chain length dependencelength dependence

Giese et al, 2002

Michel-Beyerle et al

Xue and Ratner 2003

M. Poot et al (Van der Zant), Nanolet 2006

Barrier dynamics effects on Barrier dynamics effects on electron transmission electron transmission

through molecular wiresthrough molecular wires

•HEAT CONDUCTION -- RECTIFICATION

•INELASTIC TUNNELING SPECTROSCOPY

•STRONG e-ph COUPLING: (a) resonance inelastic tunneling spectroscopy (b) multistability and hysteresis

•NOISE

•RELEVANT TIMESCALES

•INELASTIC CONTRIBUTIONS TO CURRENT

•DEPHASING AND ACTIVATION

•HEATING

h

INELSTIC ELECTRON TUNNELING SPECTROSCOPY

V

V

V

h0h0

incident scattered

Light Scattering

o utin-0in

o utin-0in

o utin-0in

Localization of Inelastic Tunneling and the Determination of Atomic-Scale Structure with Chemical Specificity

B.C.Stipe, M.A.Rezaei and W. Ho, PRL, 82, 1724 (1999)

STM image (a) and single-molecule vibrational spectra (b) of three acetylene isotopes on Cu(100) at 8 K. The vibrational spectra on Ni(100)are shown in (c). The imaged area in (a), 56Å x 56Å, was scanned at 50 mV sample bias and 1nA tunneling current

Electronic Resonance and Symmetry in Single-Molecule Inelastic Electron Tunneling

J.R.Hahn,H.J.Lee,and W.Ho, PRL 85, 1914 (2000)

Single molecule vibrational spectra obtained by STM-IETS for 16O2 (curve a),18O2 (curve b), and the clean Ag(110)surface (curve c).The O2 spectra were taken over a position 1.6 Å from the molecular center along the [001] axis.

The feature at 82.0 (76.6)meV for 16O2 (18O2) is assigned to the O-O stretch vibration, in close agreement with the values of 80 meV for 16O2 obtained by EELS.The symmetric O2 -Ag stretch (30 meV for 16O2) was not observed.The vibrational feature at 38.3 (35.8)meV for 16O2 (18O2)is attributed to the antisymmetric O2 -Agstretch vibration.

Inelastic Electron Tunneling Spectroscopy of

Alkanedithiol Self-Assembled Monolayers W. Wang, T. Lee, I. Kretzschmar and M. A. Reed (Yale, 2004)

Inelastic electron tunneling spectra of C8 dithiol SAM obtained from lock-insecond harmonic measurements with an AC modulation of 8.7 mV (RMS value) at a frequency of 503 Hz (T =4.2 K).Peaks labeled *are most probably background due to the encasing Si3N4

Nano letters, 2004

h

INELSTIC ELECTRON TUNNELING SPECTROSCOPY

V

V

V

h

INELASTIC ELECTRON TUNNELING SPECTROSCOPY

V

Conductance of Small Molecular JunctionsN.B.Zhitenev, H.Meng and Z.Bao

PRL 88, 226801 (2002)

Conductance of the T3 sample as a function of source-drain bias at T =4.2 K. The steps in conductance are spaced by 22 mV. Left inset: conductance vs source-drain bias curves taken at different temperatures for the T3 sample (the room temperature curve is not shown because of large switching noise). Right inset: differential conductance vs source-drain bias measured for two different T3 samples at T = 4.2 K.

38mV

22

125

35,45,24


SUMMARY

(1a) 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


(2) 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•Electronic conduction by hopping•Inelastic tunneling spectroscopy

Chapter 13-15Chapter 16Chapter 17

THANK YOUA. Nitzan, Tel Aviv University

INTRODUCTION TO ELECTRON TRANSFER AND MOLRCULAR

CONDUCTION

Download:

http://atto.tau.ac.il/~nitzan/Cape Town 07.ppt

Documents

A. Nitzan, Tel Aviv University The Abdus Salam African College on Science at the Nanoscale Cape Town, November 2007 1. Relaxation, reactions and electron