© a. nitzan, tau introduction to electron transport in molecular systems i. benjamin, a. burin, b....
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© A. Nitzan, TAU© A. Nitzan, TAU
Introduction to electron Introduction to electron transport in molecular systemstransport in molecular systems
I. Benjamin, A. Burin, B. Davis, S. Datta, D. Evans, M. Galperin, A. Ghosh, H. Grabert, P. Hänggi, G. Ingold, J. Jortner, S. Kohler, R. Kosloff, J. Lehmann, M. Majda, A. Mosyak, V. Mujica, R. Naaman, U. Peskin, M. Ratner, D. Segal, T. Seideman, H. Tal-Ezer
Thanks
Reviews: Annu. Rev. Phys. Chem. 52, 681– 750 (2001) [http://atto.tau.ac.il/~nitzan/nitzanabs.html/#213]
Science, 300, 1384-1389 (2003); MRS Bulletin, 29, 391-395 (2004); Bulletin of the
Israel Chemical Society, Issue 14, p. 3-13 (Dec 2003) (Hebrew)
© A. Nitzan, TAU
PART BPART B: Molecular : Molecular conduction: Main results and conduction: Main results and phenomenologyphenomenology
A. Nitzan ([email protected];
http://atto.tau.ac.il/~nitzan/nitzan.html)
PART APART A: Introduction: electron : Introduction: electron transfer in molecular systemstransfer in molecular systems
PART CPART C: Inelastic effects in : Inelastic effects in molecular conductionmolecular conduction
Introduction to electron transport in molecular systems
http://atto.tau.ac.il/~nitzan/tu04-X.ppt (X=A,B,C)
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Molecules as conductorsMolecules as conductors
m o lecule
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Aviram Ratner
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 sigma-bonded (methylene) tunnelling bridge. The response of such a molecule to an applied field is calculated, and rectifier properties indeed appear.
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 1974
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Xe on Ni(110)
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Feynman
Chad Mirkin (DPN)
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Moore’s “Law”
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Moore’s 2nd law
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Molecules get wired
© A. Nitzan, TAUCornell group
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Science, Vol 302, Issue 5645, 556-559 , 24 October 2003MOLECULAR ELECTRONICS:
Next-Generation Technology Hits an Early Midlife CrisisRobert F. Service
Researchers had hoped that a new revolution in ultraminiaturized electronic gadgetry lay almost within reach. But now some are saying the future must wait
Two years ago, the nascent field of molecular electronics was riding high. A handful of research groups had wired molecules to serve as diodes, transistors, and other devices at the heart of computer chips. Some had even linked them together to form rudimentary circuits, earning accolades as Science's 2001 Breakthrough of the Year (Science, 21 December 2001, p. 2442). The future was so bright, proponents predicted that molecular electronics-based computer chips vastly superior to current versions would hit store shelves in 2005
Now, critics say the field is undergoing a much-needed reality check. This summer, two of its most prominent research groups revealed that some of their devices don't work as previously
thought and may not even qualify as molecular
electronics. And skeptics are questioning whether
labs will muster commercial products within the next
decade, if at all
© A. Nitzan, TAU
IEEE TRANSACTIONS ON ELECTRON DEVICES VOL.43OCTOBER 1996 1637
Need for Critical AssessmentRolf Landauer,Life Fellow,IEEE
AbstractAdventurous technological proposals are subject to inadequate critical assessment. It is the proponents who organize meetings and special issues. Optical logic, mesoscopic switching devices and quantum parallelism are used to illustrate this problem.
This editorial,disguised as a scientific paper, is obviously a plan for more honesty. We do not, in the long run, build effective public support for science and technology by promising more than we can deliver.
© A. Nitzan, TAU
Feynman
• For a successful Technology, reality must take precedence over public relations, for nature cannot be fooled
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First Transport Measurements through Single Molecules
Single-wall carbon nanotube on Pt
Dekker et al. Nature 386(97)
Nanopore
Reed et al. APL 71 (97)
Break junction: dithiols between gold
Molecule lying on a surface Molecule between
two electrodes
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
Reed et al. Science 278 (97) @ Yale
Nanotube on Au
Lieber et al. Nature 391 (98)
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Park et. al. Nature 417,722-725 (2002)
Datta et al
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Weber et al, Chem. Phys. 2002
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Electron transfer in DNA
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loge of GCGC(AT)mGCGC conductance vs length (total number of base pairs). The solid line is a linear fit that reflects the exponential dependence of the conductance on length. The decay constant, , is determined from the slope of the linear fit. (b) Conductance of (GC)n vs 1/length (in total base pairs).
Xu et al (Tao), NanoLet (2004)
=0.43Å-1
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Electron transmission Electron transmission processes in molecular processes in molecular
systemssystems Electron transferElectron transfer Electron transmissionElectron transmission ConductionConduction Parameters that affect molecular Parameters that affect molecular
conductionconduction Eleastic and inelastic transmissionEleastic and inelastic transmission Coherent and incoherent conductionCoherent and incoherent conduction Heating and heat conductionHeating and heat conduction
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Activated rate processesActivated rate processes
E B
0
B
- frictio nr 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
/
2B Bo B B
TSTE k Tk e k
/0
B BE k TB
B
k J ek T
(action)
4k DR
Diffusion controlled
rates
Bk TD
m
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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
r e ac t i o nc o o r di nate
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Theory of Electron TransferTheory of Electron Transfer
Activation energyActivation energy Transition probabilityTransition probability Rate – Transition state theory or Rate – Transition state theory or
solvent controlledsolvent controlled
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Electron transfer in polar Electron transfer in polar mediamedia
•Electron 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
c
q = + e
b
q = + e
a
q = 0
Electronic transition
Nuclear relaxation
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q = 1q = 0 q = 1q = 0
Electron transfer
Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations
Nuclear motion
Nuclear motion
q= 0q = 1q = 1q = 0
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Electron transferElectron transfer
E aE A
E b
E
e ne r g y
ab
X a X tr X b
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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
Alternatively – solvent control
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Electron transfer – Marcus Electron transfer – Marcus theorytheory
(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 1
4sP E E
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
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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
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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 tr b
20 0( )W E
21 1( ) 1W E
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Electron transfer: Effect of Electron transfer: Effect of Driving (=energy gap)Driving (=energy gap)
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Experimental confirmation of the inverted regime
Miller et al, JACS(1984)
Marcus Nobel Prize: 1992
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Electron transfer – the Electron transfer – the couplingcoupling
• From Quantum Chemical Calculations
•The Mulliken-Hush formula
max 12| |DADA
HeR
• Bridge mediated electron transfer
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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
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1
1 1 12 1
21 2 23 2
32
0 0 0
0 0
00
0 0 0
0 0 0
D D D
D
NA
NA A A
N N
E E V c
E E V c
V
V E E V c
V
E E V
V E c
c
E
1 12 1
21 2 23 2
32
1 1,
,
1
1
0 0
00 0
0
0 0
N N N
NN N
D
NA AN
DV cE E V cV E E V c
V
E E V
cV E E V c
1 1 0
0
D D D
A A AN N
E E c V c
E E c V c
ˆ I c uB B BH E
( )ˆc uBB G
1( )ˆ ˆIBB BG E H
D D A A j jjc c c
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Effective donor-acceptor Effective donor-acceptor couplingcoupling
0
0
D DA
A AD
D A
A D
E c c
V c
E V
E E c
( ) ( )1 111
ˆ ˆ;B BD D D D A A AN NN NAE E V G V E E V G V
( ) ( ) *1 11 1
ˆ ˆ;B BDA D NA AD AN D DAN NV V G V V V G V V
12 23 1,1
/ 1 / 2 / 1 /
1 1 1 1ˆ ...B N NN
D A D A D A N D A N
G V V VE E E E E E E E
(1/ 2) '1
0/
N
NbD NA BDA
B D A B
V V VV V e
V E E
/
2' ln B
D A B
V
b E E
0 0 0 0 0 0 ...G G G VG G VG VG
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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
Reorganization energy
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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
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Incoherent hoppingIncoherent hopping
........
0 = D
1 2 N
N + 1 = A
k 1 2
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
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ET rate from steady state ET rate from steady state hoppinghopping
0,1 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
/
1,0
1
1
B BE k T
D A N
N A D
kek k
k kN
k k
0D Ak P
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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.
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The photosythetic reaction The photosythetic reaction centercenter
Michel - Beyerle et al
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Dependence on bridge Dependence on bridge lengthlength
Ne
11 1up diffk k N
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DNA (Giese et al 2001)DNA (Giese et al 2001)
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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
(2)Keep h constant and measure the steady state outwards water flux J. Get the rate from k=J/h -- Steady state rate
h
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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
© A. Nitzan, TAU
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
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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
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Resonance scatteringResonance scattering0
1 1 1 1,0 0 1, 1,
,1 1
,
( /
1
)0
1
0
( / 2)
( / 2)
l l r rl r
r r r r
l
i E t
r
l ll l
C iE C iV C i V C i V C
C iE C iV C
C iE C iV C
C c E
C
C
0( ) exp ( / )j jC t c i E t j = 0, 1, {l}, {r}
,1 1
0 / 2r
rr
V cc
E E i
,1 1
0 / 2l
ll
V cc
E E i
For each r and l
© A. Nitzan, TAU
,1 1
0 / 2r
rr
V cc
E E i
1, 1 0 1( )r r RrV c iB E c
1 1
21
2
1
21
1
11
1
( ) (1 / 2) ( )
( ) 2 | | ( )
| | (
| |( )
)( )
r
R R
R r R r
rR
E E
r R rR
r
rr
VB E
E EE i E
E V E
V EE
i
PP dEE E
1 0 1 1,0 0 1, 1,0 l l r rl ri E E c iV c i V c i V c
1 1
wide ban
( ) (
d approximatio
)
n
1/ 2R RB E i
21
1| |
( ) rR
rr
VB E
E E i
SELF
ENERGY
© A. Nitzan, TAU
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
© A. Nitzan, TAU
Resonant tunnelingResonant tunneling
{ }l1
V1r
r l
V1l
0
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
© A. Nitzan, TAU
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
© A. Nitzan, TAU
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)
© A. Nitzan, TAU
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)
© A. Nitzan, TAU
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”: