Lecture 5Lecture 5

A. Nitzan, Tel Aviv University

SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED

SYSTEMS

Boulder, Aug 2007

Coming March 2006

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

Coming March 2006

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

Coming March 2006

(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

General caseGeneral case

( )† ( )B

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

( ) ( ) ( )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

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

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

Reasons for switchingReasons for switching Conformational changesConformational changes

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

Tsai et. al. PRL 1992: RTS in Me-SiO2-Si junctions

Transient chargingTransient charging

time

Polaron formationPolaron formation

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

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

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

Potential distributionPotential distribution

NEGF - HF calculationNEGF - HF calculation

PART E

Inelastic effects in molecular conductions

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

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

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.

{ }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 Tunneling supporting supporting

structures in structures in waterwater

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.

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

Electron transmission through water: Resonance Lifetimes

Configurations

Resonance

(eV)energy Decay time (fsec)

0ps )4.5029- ,0.0541( 6

0ps )4.6987- ,0.0545( 6

50ps )4.4243- ,0.0424( 7.6

50ps )4.8217 ,0.0463( 7

Traversal time for tunneling?

A B

1

2

3

4

Traversal Time

1

2

0 ||lim c

c

C 1 (-)= 1C 2 (-)= 0

C 1 (+)= C 1

C 2 (+)= C 2

"Tunnelling Times"

02( )B

m DU E

NE

D

E 0 U B........

0

1 2 N

E {r}

Estimates

~ 0.2fs02( )B

m DU E

~ 2fsNE

D=10A (N=2-3)

UB-E = E~1eV

m=me

For:

Notes:

Both time estimates are considerably shorter than vibrational period

Potential problem: Near resonance these times become longer

Tunneling time and transmission probability

Vacuum barrier

Instantaneous normal modes Instantaneous normal modes for waterfor water

The density ρ of instantaneous normal modes for bulk water systems at 60K (full line) and 300K (dotted line) shown together with the result for a water layer comprised of three monolayers of water molecules confined between two static Pt(100) surfaces, averaged over 20 configurations sampled from an equilibrium (T=300K)(dashed line). The densities of modes shown are normalized to 1. The usual convention of displaying unstable modes on the negative frequency axis is applied here.

Solvation correlation functions for electron in water

Linearized INM and MD solvation response functions for upward (a) and downward (b) transitions. The solid lines are the MD results obtained from the fluctuations of the energy gap, the red lines are results of INM calculation using stable normal modes,and the blue lines stand for a calculation with all modes included.

(Chao- Yie Yang, Kim F. Wong, Munir S. Skaf, and Peter J. Rossky; J. Chem. Phys. 2001)

Fig. 5The ratio between the inelastic (integrated over all transmitted energies) and elastic components of the transmission probability calculated for different instantaneous structures of a water layer consisting of 3 monolayers of water molecules confined between two Pt(100) surfaces.

Vacuum barrier

0 0 0†

0

( ', ) ( ) ( ) ( ')

( ) ( ' ) (( ) ( 1) ( '') ( )')

r aL R L R

r a a rL R

T E E G E G E E E

MG E G E M N E E N E EG E G E

•HEAT CONDUCTION -- RECTIFICATION

•INELASTIC TUNNELING SPECTROSCOPY

•MULTISTABILITY AND HYSTERESIS

•LIGHT NOISE

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

molecular wiresmolecular wires•Relevant timescales

•Inelastic contributions to the tunneling current

•Dephasing and activation

•Heating of current carrying molecular wires

h

INELSTIC ELECTRON TUNNELING SPECTROSCOPY

V

V

V

What is typically observed

Negative signals possible too

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

Recall: van Ruitenbeek et al (Pt/H2)- dips

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, in press

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

Nanomechanical oscillations in a single C60 transistor

H. Park, J. Park, A.K.L. Lim, E.H. Anderson, A. P. Alivisatos and P. L. McEuen [NATURE, 407, 57 (2000)]

Vg(Volt)

Vsd(mV)

Two-dimensional differential conductance (I/V)plots as a function of the bias voltage (V) and the gate voltage (Vg ). The dark triangular regions correspond to the conductance gap, and the bright lines represent peaks in the differential conductance.

A contour map of d2I/dV2 plotted against the source-drain, and gate, potentials, obtained from a simple junction model with on-site e-e interaction U. This characteristic “diamond’ structure shows the thresholds for conduction with varying VSD and VG, and satellite peaks associated with vibrational transitions (inelastic contributions to conduction). Lower panels: An experimental realization (42) with an oligophenylenevinylene molecule an gold electrodes with an Al2O3 gate.

Experimental (black) and computed (red) IETS spectra for the molecule in the inset. Blue lines indicate the computed frequency and IETS intensity of the individual modes. (A. Troisi et. al, submitted to PNAS)

Parameters

electrons

Molecular vibrations

Thermal environment

M

U

L R

0

V

M – from reorganization energy (~M2/0)

U – from vibrational relaxation rates

Constant in the wide band approximation

†, ' '( , ') ( ') ( )n n n nG t t i a t a t

†, ' '( , ') ( ) ( ')n n n nG t t i a t a t

†, ' '( , ') ( ') ( ), ( ')r

n n n nG t t i t t a t a t

({ }=anticommutator)

†, ' '( , ') ( ' ) ( ), ( ')a

n n n nG t t i t t a t a t

( ) ( ) ( ) ( )2

q dEI Tr E G E E G E

1( ) Im ( )r

j jjE G E

1( ) Im ( )

2j jjn E G E

r aG G G r aG G G

0 0( ) ( ) ( ) ( ) ( )r r r r rG G G G

0 0( ) ( ) ( ) ( ) ( )a a a a aG G G G

0,' '

1( )r

jj jjj

Gi

0,' '

1( )a

jj jjj

Gi

0,, ' , '( ) 2 ( ) ( )j j j j j jG if

0,, ' , '( ) 2 1 ( ) ( )j j j j j jG i f

2

2r rph

r r

dE iM D G E

D G E D G E

2

2phd

E iM D G E

NEGF

electrons

vibrationsM

A1A2M

A3M2

2 2

2 2 2 213 31

21A AA MA M AA AM

elastic inelastic elastic

Changing position of molecular resonance:

Changing tip-molecule distance

IETS (intrinsic?) linewidth

electrons

Molecular vibrations

Thermal environment

M

U

L R

0

V

M – from reorganization energy (~M2/0)

U – from vibrational relaxation rates

IETS linewidth1=1eV

L=0.5eV

R=0.05eV

0=0.13eV

M2/0=0.7eV

•HEATING AND HEAT CONDUCTION

•INELASTIC TUNNELING SPECTROSCOPY

•MULTISTABILITY AND HYSTERESIS

•LIGHT NOISE

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

molecular wiresmolecular wires•Relevant timescales

•Inelastic contributions to the tunneling current

•Dephasing and activation

Elastic transmission vs. maximum heat generation:

HeatingHeating

Thermal conduction by Thermal conduction by moleculesmolecules

T L T R

“a single phonon mode can at most contribute a quantum of 2kB

2T/3h to the

thermal conductance.”Prediction: Rego & Kirczenow, PRL (1998)

Observation for electrons - O. Chiatti et al PRL (1998)

Observation for photons – Meschke et al, Nature 2006)

NATURE|VOL 404 | 27 APRIL 2000

R. Y. Wang, R. A. Segalman, A. Majumdar

Ultrafast Flash ThermalConductance of Molecular ChainsZ. Wang, J. A. Carter, A. Lagutchev, Y. K. Koh, N-H. Seong, D. G. Cahill, D. D. Dlott

SCIENCE VOL 317 10 AUGUST 2007

SCIENCE, VOL 315, 16 MARCH 2007

The quantum heat fluxThe quantum heat flux

( ) ( ) ( )h L RI n n d T

( ) ( ), ' , ' †

', ' '

( ) ( )1( ) ( ) ( )

8

R Lk k k k

k kk k k k

T A A

Bose Einstein populations for left

and right baths.

Transmission coefficient at frequency

With Dvira Segal and Peter Hanggi

J. Chem. Phys. 119, 6840-6855 (2003)

2 20 0 , ' 0 '

0, 0

'

( )k k k

k

k kk

A i A

V A

Heat current vs. chain length from classical simulations. Full line: harmonic chain; dashed line: anharmonic chain using the alkane force field parameters; dash-dotted line: anharmonic chain with unphysically large (x 30) anharmonicity

Anharmonicity effects

Heat conduction in alkanesHeat conduction in alkanes

D.Schwarzer, P.Kutne, C.Schroeder and J.Troe,

J. Chem. Phys., 2004Segal, Hanggi, AN, J. Chem. Phys (2003)

Thermal conduction vs. Thermal conduction vs. alkane chain lengthalkane chain length

Dashed line: T=0.1K; Blue dotted line: T=1K; Full line: T=10K; Red- dotted line: T=100K; Line with circles: T=1000K. c=400

cm-1 ,VL=VR=50 cm-2.

Rectification of heat Rectification of heat transporttransport

0 0.2 0.4 0.6 0.8 1

0

0.02

0.04

0.06

0.08

J /

J 0

0 40 8050

100

150

T i [K

]

0 40 80150

200

250

T i [K

]

site

1

1

1 2( )1 2

1 12 2( ) ( )

(2 ) 1

1 1

eqi i

N

N Nx x x

ii i

x a b x

H m p D e

D e D e

1 11

1 ; 2, 3, ... - 1

1( )

1( )

ii

L L

N R N RN

Hx i N

m x

Hx x F t

m x

Hx x F t

m x

(1 )L

The asymmetry in the thermal conduction plotted as a function of χ. parameters used: D=3.8/c2 eV, α=1.88c Å-1, xeq=1.538 Å and m=m_carbon (c=1 is from standard carbon-carbon force field in alkanes). Here we artificially increase the system anharmonicity by taking c=6. Full, dashed, dotted and dashed-dotted lines correspond to N=10, N=20, N=40 and N=80, respectively, with =50 ps-1, Th = 300K and Tc = 0K. The inset presents the temperature profile for the N=80, χ=0.5 case with TL=Tc,; TR=Th (full), TL=Th ; TR=Tc (dashed).

22

( ) ( )2

K j jj Km

0

harmonic oscillator:

L RL R

R L

J n n

0 1 2 1 2

two level system:

L R L R

L L R R

n nJ

n n

1/( ) 1B Kk T

Kn e

Result:Result:

HEATING AND HEAT HEATING AND HEAT CONDUCTIONCONDUCTION

•HEATING•HEAT CONDUCTION

•HEAT RECTIFICATION

•HEAT ENGINES

Molecular heat Molecular heat pumppump

A heat pump is a device that transfers heat from a low temperature reservoir to a high temperature reservoir by applying an external work that modulates the system parameters

WORK

HEATD. Segal & AN (PRE 02/06)

PRL 99, 027203 (2007)

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

molecular wiresmolecular wires

•HEAT CONDUCTION -- RECTIFICATION

•INELASTIC TUNNELING SPECTROSCOPY

•MULTISTABILITY AND HYSTERESIS

•LIGHT

•Relevant timescales

•Inelastic contributions to the tunneling current

•Dephasing and activation

•Heating of current carrying molecular wires

Negative differential Negative differential resistanceresistance

J. Chen, M. A. Reed, A. M. Rawlett, and J. M. Tour, Science 286: 1550-1552 (1999)

Negative differential Negative differential resistanceresistance

(Color) Representative current–voltage characteristics (a) for molecule 1 (red/ blue curves) and molecule 2 (black curve). Molecule 1 (red/ blue curves) exhibits both the negative differential resistance peak and a wide range of background ohmic currents. The distribution of resistances is shown by the histogram inset (b). In contrast, molecule 2 (black curve) shows no NDR-like features and resistances in the ohmic region are much more tightly clustered [51.6±18 G , N = 15, see histogram inset and resistances in the ohmic region are much more tightly clustered [51.6±18]Gohm

A.M.Rawlett et al. Appl. Phys. Lett. 81 , 3043 (2002)

HysteresisHysteresis

Typical I–V curves of molecular devices. (a), (b), and (c) correspond to molecules a, b, and c shown in Fig. 2, respectively

C.Li et al. Appl. Phys. Lett. 82 , 645 (2003)

a, Diagram of STM I/V experiment. The tip is positioned over the gold nanoparticle to measure the properties of an individual BPDN molecule inserted into the C11 alkane matrix. b, I/V measurement of an isolated BPDN molecule from the Type II STM experiment.

Blum et al, Nature Materials, 2005

Neutral

M Charged

M(-)EF

Self consisten equation for Self consisten equation for electronic populationelectronic population

2 20

2

1

0

/

0

2 ; ,

2

1

/

(

2

) K B

L R

K k kk K

E

L L R R

k TK

nn

V E K L

f E f E

R

E

f E e

d

E

†ˆ ˆr rc c †ˆ ˆllc c

•Obvious feedback mechanism on the mean field level

•Is mean field good enough?

•Timescale considerations critical

NDR

Summary:Summary: Barrier dynamics Barrier dynamics effects on electron transmission effects on electron transmission

through molecular wiresthrough molecular wires

•HEAT CONDUCTION -- RECTIFICATION

•INELASTIC TUNNELING SPECTROSCOPY

•MULTISTABILITY AND HYSTERESIS

•LIGHT

•Relevant timescales

•Inelastic contributions to the tunneling current

•Dephasing and activation

•Heating of current carrying molecular wires

g=7 D e=31+/-1.5 D

g=5.5 D e=15.5+/-1.5 D

g=7 D e=30+/-1.5 D

CHARGE TRANSFER TRANSITIONS

S. N. Smirnov & C. L. Braun, REV. SCI. INST. 69, 2875 (1998)

Current induced light emission Current induced light emission and light induced current in and light induced current in

molecular tunneling junctionsmolecular tunneling junctions

{ |l> } { |r > }

|1 >

|2 >

L R

M. Galperin &AN , cond- mat/ 0503114, 4 Mar 2005-03-23

†0

{ , }

( ) †

†

1,2

†12

† †1 1

,, 1,2

†

0 2 2

;

( ) †', '

, '

( ) ( )0

0

ˆ ˆ ˆ

ˆ ˆ . .

ˆ ˆ ˆ . .

ˆ .

ˆ ˆ

ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ .ˆ ˆˆ

k kkk L R

MKM k m k

K L R m k K

NKN kk k k

K L R k k

m m mm

m

K

P PR

H c c

V V c h c

V

a

V c c h

c c

c

c c

c c

c

V V h c V h c

a

a a c c

light

Light induced current

E21=2eV

M,1=0.2eV

M,2=0.3eV, 0.02eV

N=0.1eV

Incident light =108 W/cm2

Current induced light

E21=2eV

M,1=M,2=0.1eV

N=0.1eV

Observations: Flaxer et al, Science 262 , 2012 (1993),

Qiu et al, Science 299 , 542 (2003).

Yield

Intensity

Emission yield from 9-10 dichloroanthracene on a quartz lens coated with ITO (Indium Tin Oxide), a transparent conductor.

Flaxer et all, Science, 262, 2012 (1993)

Summary:Summary: Barrier dynamics Barrier dynamics effects on electron transmission effects on electron transmission

through molecular wiresthrough molecular wires

•HEAT CONDUCTION -- RECTIFICATION

•INELASTIC TUNNELING SPECTROSCOPY

•MULTISTABILITY AND HYSTERESIS

•LIGHT

•Relevant timescales

•Inelastic contributions to the tunneling current

•Dephasing and activation

•Heating of current carrying molecular wires

THANK YOUA. Nitzan, Tel Aviv University

SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED

SYSTEMS

Boulder, Aug 2007