Lecture 4

Lecture 4Lecture 4

A. Nitzan, Tel Aviv University

SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS

Boulder, Aug 2007

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

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

Boulder Aug 2007

(3) 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 17

DA

, , ,( ) 1 ( )b a 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

ELECTRODE PROCESSES

Reorganization energy here – from donor only (~0.5 of “regular” value)

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”:

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

Molecular level structure Molecular level structure between electrodesbetween electrodes

en erg y

LUMO

HOMO

“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)

ET vs ConductionET vs Conduction

2

01 ,(11

2

2)

2| |

2

(

(( )

)

)

AD

NB

D A

D A DA

N N 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

G

E

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

ET rate from steady state ET rate from steady state hoppinghopping

........

0 = D

1 2 N

N + 1 = A

k

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 )

k k

/

1,0

1

1

B BE k T

D A N

N A D

kek k

k kN

k k

Incoherent hoppingIncoherent hopping

D A1 N

2/ BE k T

D AB

eg e k

k T

LARGE N:

1 12 0 1/

1 1( ) ( )0 1

1

1

BD N AE k T

D AL RB

N

k k k Neg e k

k T k N

Or at T=300K.

/1 18 1( ) 5 10 ( )BE k TD Ag e k s

Current from classical Current from classical kineticskinetics

11 1 1 1

1 1 1 1

1 1

1 1

L L L L

R R R R

dPf P f P

dt

f P f P

1 1

1 1

L RL R

L RI e f f

( ) ( ) ( )L R

eI dE f E f E E

T

= 0 at steady state

1 1

1 1 12 21 1

( ) ( )( )

( ) / 2

L RL R

E EE E E E

E E E

T

Quantum mechanical resalt:

L M R

PART D

Issues in molecular conductions

A. Nitzan

Boulder Aug 2007

Molecular conduction•Structure-function effects in molecular conduction•The role of contacts•How does the potential drop on a molecule and why this is important•Probing molecules in STM junctions•Electron transfer by hopping•Charging•Switching

(Original picture from Datta et al)

THE IDEAL EXPERIMENT

T1T2

Inject and detect Light

Temperature measurement and control

Control type and number of molecule

Molecule(s)

REPRODUCIBLE!

A

Current measurement vs. V and Vg

G

S D DS

Source-Drain potential (V) Gate potential VG

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

““Switching”Switching”

SwitchingSwitching Conformational changesConformational changes

Tsai et. al. Appl.Phys.Lett 1992: Random telegraph signals in Me-SiO2-Si junctions

Transient chargingTransient charging

STM under waterSTM under waterS.Boussaad et. al. JCP S.Boussaad et. al. JCP (2003)(2003)

time

Polaron formationPolaron formation

Time (s)

Tip

hei

gh

t

Moore et al (P.S. Weiss) Conduction switching in Oligo(phenylene ethynylene) molecules (nitro functionalized)

Dynamics of current voltage switching response of single bipyridyl-dinitro oligophenylene ethynylene dithiol (BPDN-DT) molecules between gold contacts. In A and B the voltage is changed relatively slowly and bistability give rise to telegraphic switching noise. When voltage changes more rapidly (C) bistability is manifested by hysteretic behavior

Lortscher et al (Riel), Small, 2, 973 (2006)

Single (K+) channel currents from Schwann cells isolated enzymatically from the giant axons of the squids Loligo forbesi, Loligo vulgaris and Loligo bleekeri. The channel conductance was 43.6 pS when both internal and external solutions contained 150 mM K+. Activity was weakly dependent on membrane voltage but sensitive to the internal Ca2+ concentration.

[Ca+2]=1x10-6M

I. Inoue et al, Journal of Physiology I. Inoue et al, Journal of Physiology 541.3, 541.3, pp.pp. 769- 769-778778(2002) (2002)

Chem. Commun., 2006, 3597 - 3599, DOI: 10.1039/b609119a

Uni- and bi-directional light-induced switching of diarylethenes on gold nanoparticles

Tibor Kudernac, Sense Jan van der Molen, Bart J. van Wees and Ben L. Feringa

“In conclusion, photochromic behavior of diarylethenesdirectly linked to gold nanoparticles via an aromatic spacer hasbeen investigated. Depending on the spacer, uni- (3) or bidirectionality(1,2) has been observed.”

Switching with light

Current–voltage data (open circles) for (a) openmolecules 1o and (b) closed molecules 1c

Nanotechnology 16 (2005) 695–702Switching of a photochromic molecule on gold electrodes: single-moleculemeasurementsJ. He, F. Chen, P. Liddell, J. Andr´easson, S D Straight, D. Gust, T. A. Moore,A. L. Moore, J. Li, O. F Sankey and S. M. Lindsay

Temperature and chain Temperature and chain length dependencelength dependence

Giese et al, 2002

Michel-Beyerle et al

Selzer et al 2004

Xue and Ratner 2003

V. J. Langlais et al, PRL 83, 2809 (1999)

Electron transfer in DNAElectron transfer in DNA

DNA-news-1DNA-news-1

DNA-news-4DNA-news-4

DNS-news-3DNS-news-3

DNA-news-2DNA-news-2

Conjugated vs. Saturated Molecules: Importance of Contact Bonding

Kushmerick et al., PRL

(2002)

2 -vs. 1-side Au-S bonded conjugated system gives at most 1 order of magnitude current increase compared to 3 orders

for C10 alkanes !

SS S/AuAu/S

10-4

10-3

10-2

10-1

100

101

102

0.0 0.2 0.4 0.6 0.8 1.0

Current (nA)

Tip bias (V)

Curr

ent

(nA

)

SS S/AuAu//

Au//CH3(CH2)7S/Au

Au/S(CH2)8SAu

Positive bias

negative bias

Lindsay & Ratner 2007

Where does the potential Where does the potential bias falls, and how?bias falls, and how?

•Image effect

•Electron-electron interaction (on the Hartree level)Vacuu

mExcess electron density

Potential profile

Xue, Ratner (2003)

Galperin et al 2003

L

Galperin et al JCP 2003

Why is it important?Why is it important?D. Segal, AN, JCP 2002 Heat Release on junction

Tian et al JCP 1998

ExperimentExperiment Theoretical Model

Experimental i/V behaviorExperimental i/V behavior

Experimental (Sek&Majda)Experimental (Sek&Majda)junction Ratio of current:

i(-1.0 V)/i(+1.0 V)a Hg-SC12/C12S-Au 0.98 0.13

Hg-SC12/C10S-Au 1.03 0.07

Hg-SC16/C12S-Au 1.22 0.16

Hg-SC12/C9S-Au 1.44 0.20

Hg-SC16/C10S-Au 1.34 0.19

Hg-SC16/C9S-Au 2.03 0.27

aCurrent at the negative bias refers to the measurement with the Hg side of the junction biased negative relative to the Au side.

Potential distributionPotential distribution

NEGF - HF calculationNEGF - HF calculation

HS - CHHS - CH22CHCH22CHCH22CHCH22CHCH22CHCH33 . . . CH . . . CH33CHCH22 - SH- SH

MO Segment Orbital

A

B

A

B

L

With Galperin, Ingold and Grabert

J. Chem. Phys., 117, 10837-41 (2002)

Single molecule vs. molecular layer

(D) A conductance histogram obtained from 1000 measurements shows peaks near 1 , 2 , and 3 0.01 G0 that are ascribed to one, two, and three molecules, respectively. (F) In the absence of molecules, no such steps or peaks are observed within the same conductance range.

Xu and Tao, Science, 301, 1221 (2003)

Cui et al (Science 2001):

The sulfur atoms (red dots) ofoctanethiols bind to a sheet of gold atoms (yellow dots), and theoctyl chains (black dots) form a monolayer. The second sulfuratom of a 1,8-octanedithiol molecule inserted into themonolayer binds to a gold nanoparticle, which in turn is contacted by the gold tip of the conducting AFM.

J. G. Kushmerick et al., Nano Lett. 3, 897 (2003). A. S. Blum, J. G. Kushmerick, et al., The J. Phys. Chem. B 108, 18124 (2004).

A. Salomon, D. Cahen, S. M. Lindsay, et al., Advanced Materials 15, 1881 (2003).

1-nitro-2,5-di(phenylethynyl-4’-mercapto)benzene

Y. Selzer et al., Nano Letters 5, 61 (2005).

Red – single molecule; black – molecular layer. Dashed black is molecular layer per molecule

Red – single molecule; black – molecular layer per molecule

3,4,9,10-perylenetetracarboxylicacid-Dianhydride (PTCDA) on silver(111)

Analysis yields: effective mass meff =0.43mee

Temirov et al, Nature Vol 444 (2006)

[1] Yaliraki S N and Ratner M A, Molecule-interface coupling effects on electronic transport in molecular wires J. Chem. Phys. 109 5036-43 (1998)

[2] Magoga M and Joachim C, Conductance of molecular wires connected or bonded in parallel Phys. Rev. B-Condens Matter 59 16011-21(1999)

[3] Lang N D and Avouris P, Electrical conductance of parallel atomic wires Phys. Rev. B-Condens Matter 62 7325 (2000)

[4] Kim Y-H, Tahir-Kheli J, Schultz P A, Goddard W A and Iii 2006 First-principles approach to the charge-transport characteristics of monolayer molecular-electronics devices: Application to hexanedithiolate devices Phys. Rev. B (Condensed Matter and Materials Physics) 73 235419

* Weak effect * strong effect

Probes of different sizes see different numbers of molecule

Molecular layers and islands

Observations of single molecule behavior does not necessarily imply single molecules

Slopes: 1.15, 1.48, and 1.55 are found for the “small”, “medium”, and “large” clusters, respectively. The larger than unity slope indicates a dipole enhancement, with the effect increasing with increasing cluster size.”

D. Deutsch, A. Natan, Y. Shapira and L. Kronik, JACS, 2/2007

Natan, A.; Zidon, Y.; Shapira, Y.; Kronik, L. Phys. ReV. B 2006, 73, 193310.

The smaller than unity slope indicates a dipole reduction, with the effect increasing with increasing coverage

TIMESCALE TIMESCALE CONSIDERATIONSCONSIDERATIONS

Does the tunneling electron interact with other degrees of freedom and what are the possible consequences of this

interaction?

The case of electron tunneling in water

Overbarrier electron Overbarrier electron transmission through water transmission through water

(D(D22O on Pt(1,1,1)O on Pt(1,1,1)

A look from above on a water film

Effective BarrierEffective Barrier

The effective one-dimensional barrier obtained by fitting the low energy tunneling probability to the analytical results for tunneling through a rectangular barrier. Solid, dotted, and dashed lines correspond to the polarizable,

nonpolarizable, and bare barrier potentials, respectively.

The numerical problem

z

W A T E RP tP t

1 2

S 1 S 2

d

(1) Get a potential

(2) Electrostatics

(3) Generate Water configurations

(4) Tunneling calculations

(5) Integrate to get current

Potentials for electron Potentials for electron transmission through transmission through

waterwaterWater-Water.....................RWKM, SPC/E

Electron-Water..............Barnett et al +correction for many body polarizability

Water-Wall........ Henziker et al (W-Pt), Hautman et al (W-Au)

Electron-Wall..............Square Barrier

Earlier studies – Tunneling through static water configurations

STM modelSTM model

Z

L

C '

D '

A

B

C

D

A '

B '

S 1 S 2

RM

Fig. 1. A model system used to compute electron transmission between two electrodes, L and R separated by a narrow spatial gap (M) containing a molecular species. The surface S1 of L is shaped to mimic a tip. The lines A'B', C'D'

and AB and CD are projections of boundary surfaces normal to the transmission direction (see text for details). The numerical solution is carried on a grid (Shown).

Potential distribution

A cut of the external potential distribution between the tip and the flat substrate for a voltage drop of 0.5V between these electrodes

The image potential along different lines normal to the flat electrode: (1) x=0 (a line going through the tip axis); (2) x=11.96au (distance from the tip axis); (3) x=23.92au.

MOLECULAR DYNAMICS TO GENERATE WATER CONFIGURATIONS

Figure - Ohmine et al

CALCULATION OF TRANSMISSION FACTORS

SY ST E M (M )

RL

0

0

L LM

ML M MR

RM R

H HH H H H

H H

( ) ( )1( )

M

M L RM M

G EE H

( ) ( )†

,( ) ( ) ( ) ( )L R

M M M M Mlrl r

E G E E G ETr

T

†i

CALCULATION OF TRANSMISSION FACTORS

1( )L R

MM i i

G EE H

†

,( ) ( ) (r) ( ) (r)M M L RMlr

l rE G E G ETr

T

L R

SY ST E M (M )

Absorbing boundary conditions Green's function method: Replace by i(r), smoothly rising towards edges of M system, provided LM and MR

boundaries are set far enough

Tunneling current in water

Current against bias voltage in a biased tip-planar electrode junction under water. Upper and lower lines are results for single water configurations characterized by tip-substrate separation of 5.85Å (2 water monolayers) and 12.15Å (4 water monolayers), respectively. The intermediate group of lines are results for 5 different water configurations at tip-substrate separation 9Å (3 water monolayers).

0

( ) ( ) ( )L R

eI dE f E f E e E

T

{ }l

1

V1r

r l

V1l

|1 >

|0 >

x

V (x )

RL . . . .

Resonant tunneling?Resonant tunneling?

Resonance transmission through water

3 4 5 6 70.0

0.2

0.4

0.6

0.8

P

E (eV)

Tunneling supporting

structures in water

Transmission through several water Transmission through several water configurations (equilibrium, 300K)configurations (equilibrium, 300K)

A compilation of numerical results for the transmission probability as a function of incident electron energy, obtained for 20 water configurations sampled from an equilibrium trajectory (300K) of water between two planar parallel Pt(100) planes separated by 10Å. The vacuum is 5eV and the resonance structure seen in the range of 1eV below it varies strongly between any two configurations. Image potential effects are disregarded in this calculation.