Introduction to electron transport in molecular systems

Introduction to electron transport in molecular systems

A. Nitzan, Tel Aviv University IAS HU Tutorial: Electron transfer Jerusalem, July 2012 1. Relaxation, reactions and timescales 2. Electron transfer in condensed molecular systems 3. Fundamentals of molecular conduction IAS Workshop 2012 (1) 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 The importance of timescales Molecular processes in condensed phases and Molecular timescales

interfaces Diffusion Electronic 10-16-10-15s Relaxation Vibraional period 10-14s Solvation Vibrational xxxxrelaxation 1-10-12s Nuclear rerrangement Charge transfer (electron and xxxxxxxxxxxxxxxxproton) Solvent: an active spectator energy, friction, solvation Diffusion D~10-5cm2/s 10nm 10-7 - 10-8 s Chemical reactions xxxxxxxxx1012-10-12s Rotational period 10-12s Collision times 10-12s Frequency dependent

friction i iif t t (0) ~ e 1 / D iif if F t( t)F k kf i ~~~dte ddte te F (Ft )(Ft )(0) constant F (0) ff T ii T T

t MARKOVIAN LIMIT WIDE BAND APPROXIMATION 1 D Molecular vibrational relaxation Relaxation in the X2+ (ground electronic state) and A2 (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 state. (From V.E. Bondybey and A. Nitzan, Phys. Rev. Lett. 38, 889 (1977)) Dielectric solvation C153 / Formamide (295 K) q=0 Relative Emission Intensity CF3 q=+e

Born solvation energy Emission spectra of a b q=+e N O O c Coumarin 153 in q2 1 formamide at different 1 1 2eV (for a charge) 2a s times. The timesshown here are (in order of increasing peakwavelength) 0, 0.05, 0.1, 0.2, 0.5, 1, 2, 5, and 50 ps (Horng et al, J.Phys.Chem.

450 500 99, 17311 (1995)) 550 Wavelength / nm 600 real solvation Newton dielectric The 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 q q 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)).

Electron solvation C153 / Formamide (295 K) Relative Emission Intensity CF3 N 450 500 550 Wavelength / nm O O 600 The 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) Activated rate processes B E 0 KRAMERS THEORY: Low friction limit High friction limit Transition State theory B r e ac tio n c o o r d in ate

k 4 DR k 0 J B e E B / k BT kBT (action) 0 B EB / kBT k e kTST B 2 0 EB / kBT kTST e 2 Effect of solvent friction TST 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)). The physics of transition

state rates Assume: E (1) Equilibrium in the well 0 B B r e actio n c o o rd in ate (2) Every trajectory on the barrier that goes out makes it kTST d v v P ( x B , v ) v f P x B 0 d vv e 0

d v e 12 mv 2 12 mv 2 0 EB e 2 2 m 1 0 EB P ( x ) e B EB 2 2 m

dx exp V ( x ) exp E B The (classical) transition state rate is an upper bound E B r e ac tio n c o o r d in ate 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

Quantum considerations 2 a 1 b 1 R * R A d ia b a t ic d ia b a t ic * k dR R P ( R , R )Pb a ( R ) 0 *

1 in the classical case IAS Tutorial 2012 (1) 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 IAS Tutorial 2012 (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 Theory of Electron Transfer

Rate Transition state theory kTST d v v P ( x B , v ) Pab v 0 Boltzmann Transition Activation probability energy Electron transfer in polar media 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 q =0

a q =+e q =+e b Electronic transition c Nuclear relaxation Electron transfer q=0 q=1 q=1 q= 0 Nuclear motion Nuclear motion

q=0 q=1 q=1 q=0 Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations Electron transfer a b energy EA E a E E b X a X tr X b

Solvent polarization coordinate Transition state theory of electron transfer Adiabatic and non-adiabatic ET processes E E 2(R ) Landau-Zener problem E a(R ) ab P ( R* , R ) P ( R ) k dRR b a E 1(R ) 2 | Va ,b |2 Pb a ( R ) 1 exp F

R R R* V 0 E b(R ) R * t= 0 R t k NA | Va ,b |2 K 2 F e EA

R R* (For harmonic diabatic surfaces (1/2)KR2) Electron transfer Marcus theory (0) (0) (1) (0) (1) (1) (0) q A qB q A qB q=0 D 4 E D 4 P P Pe Pn e 1 Pe E 4 s e Pn E 4

q=0 D s E q=1 q A qB q A qB(1) q=1 q= 0 We are interested in changes in solvent configuration that take place at constant solute charge distribution They have the following characteristics: (1) Pn fluctuates because of thermal motion of solvent nuclei. q=1 q=1 q=0 (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 Electron transfer Marcus

theory (0) (0) (1) (0) (1) (1) (0) q A qB q=0 q=0 q=1 q=1 q A qB q A qB(1) q A qB q=1 q=1 Free energy associated with a nonequilibrium fluctuation of Pn

q= 0 q=0 0 (0) (0) q A qB 1 (1) (1) q A q B reaction coordinate that characterizes the nuclear polarization The 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. W0 ( ) E0 2

2 W1 ( ) E1 1 1 1 1 1 1 e s 2 RA 2 RB RAB 2 q Electron transfer: Activation energy Wa ( ) Ea 2 Wb ( ) Eb 1 a 2 b

energy E A E a E E b [( Eb Ea ) ]2 EA 4 1 1 1 1 1 e s 2 RA 2 RB RAB a= 0 2 q tr

b= 1 Reorganization energy Activation energy Electron transfer: Effect of Driving (=energy gap) Experimental confirmation of the inverted regime Marcus papers 1955-6 Miller et al, JACS(1984) Marcus Nobel Prize: 1992 Electron transfer the coupling 2 ket ~ Vab e Eab 4 k BT

From Quantum Chemical Calculations The Mulliken-Hush formula VDA max 12 eRDA Bridge mediated electron transfer Bridge assisted electron transfer B V D V B 1 12 2 1 B 2 V

3 23 V D 1 EB A 3 3A A D E j E B , V j , j 1 E B E D / A N H E D D D E j j j EA A A j 1

VD1 D 1 V1 D 1 D V AN A N VNA N N 1 V j , j 1 j 1 j j 1 V j , j 1 j 1 j A B VDB VAD D A Veff D A

E Veff VDBVAB E B1 E VDB DD Veff B2 BN VAD V12 Veff VDBG1 NVAB A A

N V 1 ...V Greens Function V 12 23 N 1, N V 1 1 B G G1 N1 N exp G(1 E ' N H E/ 2) N E N E VB V 1 ' 2 ln E / VB V D

D 1 12 2 V 23 3 V 3A A Marcus expresions for nonadiabatic ET rates 2 | VDA |2 F ( E AD ) 2 2 2 (B) VD1VNA G1 N ( E D ) F ( E AD ) k D A

Donor-to-Bridge/ Acceptor-to-bridge F (E) Bridge Greens Function e 2 E / 4 k BT 4 k B T Reorganization energy Franck-Condonweighted DOS Bridge mediated ET rate k ET ~ F ( E AD , T )exp( ' RDA ) (-1)= 0.2-0.6 for highly conjugated chains 0.9-1.2 for saturated hydrocarbons ~2 for vacuum

Bridge mediated ET rate (J. M. Warman et al, Adv. Chem. Phys. Vol 106, 1999). Incoherent hopping k 10 =k 01 e x p (- E 10 21 2 N ........ k 1

) k N ,N + 1 0 = D =k N + 1 ,N e x p (- E 10 ) N +1 = A P0 k1,0 P0 k0,1 P1 constant P1 ( k0,1 k2,1 ) P1 k1,0 P0 k1,2 P2 STEADY STATE SOLUTION

P N ( k N 1, N k N 1, N ) PN k N , N 1 PN 1 k N , N 1 PN 1 P N 1 k N , N 1 PN 1 k N 1, N PN ET rate from steady state hopping k 10 =k 01 e x p (- E 10 2 k ........ k 1 ) k

N k N ,N + 1 =k 0 = D N + 1 ,N e x p (- E N +1 = A k D A k N 1,0 ke k kN A E B / k BT k k1 D 1 N

10 ) Dependence on temperature 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 center Michel - Beyerle et al Dependence on bridge length e N 1 1 kup kdiff N 1 DNA (Giese et al 2001)

IAS Tutorial 2012 (2) Electron transfer processes AN, Oxford University Press, 2006 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 IAS Tutorial 2012 (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 Molecular conduction m o le c u le Molecular Rectifiers Arieh Aviram and Mark A. Ratner IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, USA Department of Chemistry, New York New York University, New York 10003, USA Received 10 June 1974 Abstract 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 sigmabonded (methylene) tunnelling bridge. The response of such a molecule to an applied field is calculated, and rectifier properties indeed appear. Xe on Ni(110) m o le c u le

Fabrication Characterization Stability Funcionality Control Fabrication Stability Characterization Funcionality Control THE MOLECULE Strong electric field System open to electrons and energy LUM O Nonequilibrium Relaxation Electron-vibration coupling Heat generation HO M O Work function

EF Landauer formula 2 e g ( 0) T ( E ) ; I e Fermi energy f R ( E )T ( E ) dE f L ( E ) dI g ( ) d For a single channel: T (E)

1 L ( E )1 R ( E ) E E 1 2 1 ( E ) / 2 Maximum conductance per channel 2 2 (maximum=1) e 1 g 12.9 K I e

dE f L ( E ) 2 f R ( E )T ( E ) e e T(E) fRT(E) (E) fL(E)fL(E) fR(E) I g Weber et al, Chem. Phys. 2002 The N-level bridge (n.n. interactions) {r} { l} 1 0 ....

N +1 L R I e dE f L ( E ) f R ( E )T ( E ) e2 g T (E ) 1 L ( E )1 R ( E ) T (E) ( R) 2 (2L )

2 T ( E ) E| G 0, E 1N 1( E 1)( E| ) / 2 0 ( E ) N 1 ( E ) G B ( E ) 0, N 1 1 1 1 1 V01 V12 ... VN , N 1 E EN E E N 1

E E0 E E1 1 1 1 E E i 0 2 0L 1 E E i

N 1 2 N 1, R G1N(E) 1 2 N ........ Electron Transfer vs Conduction E 0 = D N +1 = A e2 g | G0, N 1 ( E ) |2 (0L ) ( E )(NR)1 ( E ) 2 V01VN , N 1

e2 1 ( L) 1 ( R) E E i E E i N 1 D 0 A 2 2 2 (B) G1 N ( E )

2 k D A | VDA |2 F ( E AD ) 2 2 2 (B) V01VN , N 1 G1 N ( E D ) F ( E AD ) (0L ) ( E )(NR)1 ( E ) F (E) e 2 E / 4 k BT 4 k B T A relation between g and k Electron charge 2 8e

g 2 ( L) ( R ) k D A D A F conduction Decay into electrodes Marcus Electron transfer rate A relation between g and k 8e 2 g 2 ( L) ( R ) k D A D A F F 4 k BT exp / 4k BT eV

1 2 g ~ e / ( L) D 10 13 ( R) A 0.5eV 1 k D A ( s ) 17 1 1

10 k D A ( s ) Conductance (g (-1)) vs Kinetics ( k0 (s-1) ) for alkane spacers [Marshal Newton] low bias limit Alkane Bridge X(CH2)n-2 I / V in nanopore junctions STM / break Scaled k0: junctions 5 x 10-19 k0/DOS* Reed et al Tao et al (monothiolates) (dithiolates) Nitzan M(DBA)M model ( D and A chemisorbed to M) n=8

5.0 E-11 1.9 E-8 4.1 E-8 n=10 5.7 E-12 1.6 E-9 6.8 E-9 n=12 6.5 E-13 1.3 E-10 4.6 E-10 Conclusions: conductance data of Tao et al (g) and rate constant data (k0) correspond to within ~ 1-2 orders of magnitude results from the 2 sets of conductance measurements differ by > 2 orders of magnitude Temperature and chain length dependence

Miche lBeyerl e et al M. Poot et al (Van der Zant), Nanolet 2006 Giese et al, 2002 Xue and Ratne r 2003 Barrier dynamics effects on electron transmission through molecular wires RELEVANT TIMESCALES INELASTIC CONTRIBUTIONS TO CURRENT DEPHASING AND ACTIVATION HEATING

HEAT CONDUCTION -- RECTIFICATION INELASTIC TUNNELING SPECTROSCOPY STRONG e-ph COUPLING: (a) resonance inelastic tunneling spectroscopy (b) multistability and hysteresis NOISE SUMMARY THANK YOU (1) and reactions in (3) Molecular (2) Relaxation Electron conduction transfer Simple models for molecular condensed molecular systems processes conductions Kinetic models Simple models electron transfer at

Factors affecting Transition state theory Marcus theory interfaces theory andenergy its extensions A. Nitzan, Tel Aviv University Kramers TheLandauer reorganization The formula Low, highconduction and Molecular by the Landauer Adiabatic and intermediate non-adiabatic friction regimes formula limits INTRODUCTION TO ELECTRON TRANSFER AND Diffusion to electron-transfer

controlled reactionsrates. Relationship Solvent controlled reactions MOLRCULAR CONDUCTION Structure-function effects in molecular conduction Bridge assisted electron transfer Download: AN, Oxford University Electronic Press, 2006 Coherentconduction and incoherent http://atto.tau.ac.il/%7Enitzan/Molecular%20Electronics-HU-2012.ppt by hopping transfer Chapter Chapter 16 Inelastic tunneling spectroscopy Chapter13-15 17 Electrode processes

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