A. Nitzan: Lecture 1

How Molecules Conduct
Introduction to electron transport in molecular
systems
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)
Thanks
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,
© A. Nitzan, TAU
U. Peskin, M. Ratner, D. Segal, T. Seideman, H. Tal-Ezer
.
A. Nitzan
Part A
INTRODUCTION
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Molecules as conductors
molecule
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Molecular Rectifiers
Aviram Ratner
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.© A. Nitzan, TAU
Xe on Ni(110)
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Feynman
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Chad
Mirkin
(DPN)
Moore’s “Law”
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Moore’s 2nd law
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Molecules get wired
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Cornell group
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http://www.hemdweb.co/
April 2002
© A. Nitzan, TAU
The Hedonistic Imperative outlines how
genetic engineering and nanotechnology
will abolish suffering in all sentient life.
The abolitionist project is hugely ambitious but
technically feasible. It is also instrumentally
rational and ethically mandatory. The metabolic
pathways of pain and malaise evolved because
they served the fitness of our genes in the
ancestral environment. They will be replaced by a
different sort of neural architecture. States of
sublime well-being are destined to become the
genetically pre-programmed norm of mental
health. The world's last unpleasant experience will
© A.event.
Nitzan, TAU
be a precisely dateable
© 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
Adsorbed molecule
addressed by STM tip
Molecule between
two electrodes
Self-assembled monolayers
Break junction:
dithiols between gold
Molecule lying
on a surface
Single-wall carbon
nanotube on Pt
Pt/Ir Tip
~1-2 nm
1 nm
SAM
Au(111)
Dorogi et al. PRB 52 (95) @
Purdue
Reed et al. Science 278 (97) @ Yale
Nanopore
C60 on gold
STM
tip
Joachim et al. PRL 74 (95)
Dekker et al. Nature 386(97)
Nanotube on Au
Au
Nitzan,
Reed et©
al.A.
APL
71 (97)TAU
Lieber et al. Nature 391 (98)
Park et. al. Nature 417,722-725 (2002)
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Datta et al
Weber et al, Chem. Phys. 2002
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Electron transfer in DNA
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Electronic processes in molecular
systems
„
„
„
„
„
„
„
Electron transfer
Electron transmission
Conduction
Parameters that affect molecular conduction
Eleastic and inelastic transmission
Coherent and incoherent conduction
Heating and heat conduction
© A. Nitzan, TAU
Theory of Electron Transfer
„
„
„
„
„
„
„
„
„
„
The physics of activated rate process
The physics of electron transfer in a nuclear
environment
A simple model – parabolic potential surfaces
The Marcus parabolas
The reorganization energy
Adiabatic and non adiabatic rates
Solvent control
Bridge length dependence (“super-exchange”)
Transition to hopping
Experimental implications
© A. Nitzan, TAU
Activated rate processes
ωB
EB
ω0
γ - friction
KRAMERS THEORY:
Low friction limit
High friction limit
Transition State
theory
k=
γ
reaction
coordinate
ω 0 J B e − E B / k BT
Diffusion
controlled
rates
k = 4π DR
kBT
D=
mγ
kBT
(action)
ωoω B − EB / kBT
ωB
k=
e
= kTST
2πγ
γ
ω 0 − E B / k BT
kTST =
e
© A. Nitzan, TAU
2π
The physics of transition state rates
Assume:
ωB
(1) Equilibrium in the well
EB
ω0
reaction
coordinate
(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
∞
∫
dv e
− 12 β mv 2
=
− 12 β mv 2
1
2πβ m
P ( xB ) =
ω0 − β EB
=
e
2π
exp ( − β E B )
EB
∫−∞ dx exp ( − β V ( x ))
=
β mω02 − β EB
e
2π
−∞
© A. Nitzan, TAU
Electron transfer in polar media
•Electron are much faster than nuclei
•Î Electronic transitions take place in fixed nuclear
configurations
•Î Electronic energy needs to be conserved during the
change in electronic charge density
q=0
a
Electronic
transition
q=+e
b
q=+e
c
Nuclear
relaxation
© A. Nitzan, TAU
Electron transfer
q=0
q=1
q=1
q=0
Nuclear
motion
Nuclear
motion
q=0
q=1
q=0
q=1
Electron transition takes place in unstable nuclear
configurations obtained via thermal fluctuations
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Electron transfer
a
b
energy
λ
EA
Ea
∆E
Eb
Xa
Xtr
Xb
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Transition state theory of electron
Alternatively –
transfer
solvent control
Adiabatic and non-adiabatic ET processes
E
E2(R)
Landau-Zener
problem
Ea(R)
∞
& & P ( R* , R& ) P ( R& )
k = ∫ dRR
b←a
Vab
0
 2π | Va ,b |2 
Pb←a ( R& ) = 1 − exp  −
& ∆F 
h
R

 R = R*
Eb(R)
E1(R)
*
R
t=0
R
t
k NA =
πβ K
2
| Va ,b |2
h ∆F
R = R*
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e− β EA
Electron transfer – Marcus theory
(0) (0)
q A qB
→
(1) (1)
q A qB
∇ ⋅ D = 4πρ
E = D − 4π P
εs − 1
P = χE =
E
4π
εe − 1
Pe =
E
4π
ε s − εe
Pn =
E
4π
(0)
qA
+
(0)
qB
=
(1)
qA
+
(1)
qB
We are interested in changes in solvent
configuration that take place at
constant solute charge distribution ρ
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 ρ only)
Note that the relations E = D-4πP;
P=Pn + Pe are always satisfied per
definition, however D ≠ εsE. (the latter
equality holds only at equilibrium).
© A. Nitzan, TAU
The Marcus parabolas
ρθ = ρ0 + θ ( ρ1 − ρ0 )
Use θ as a reaction coordinate. It defines the state of the
medium that will be in equilibrium with the charge
distribution ρθ. Marcus calculated the free energy (as
function of θ) of the solvent when it reaches this state in the
systems θ =0 and θ=1.
W0 (θ ) = E0 + λθ
2
W1 (θ ) = E1 + λ ( 1 − θ )
 1
1  1
1
1
+
−
λ =  − 
 ε e ε s   2 RA 2 RB RAB
2
 2
 ∆q

© A. Nitzan, TAU
Electron transfer: Activation energy
W0 (θ ) = E0 + λθ 2
W1 (θ ) = E1 + λ ( 1 − θ )
a
2
b
energy
λ
EA
Ea
∆E
[( Eb − Ea ) + λ ]2
Eb
EA =
4λ
 1
1  1
1
1
λ =  − 
+
−
 ε e ε s   2 RA 2 RB RAB
θa θtr
 2
 ∆q

θb
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Electron transfer: Effect of 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 coupling
• From Quantum Chemical Calculations
•The Mulliken-Hush formula
| H DA |=
hωmax µ12
eRDA
• Bridge mediated electron transfer
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Bridge assisted electron transfer
B2
B1
V12
D
2
1
B3
V23
3
A
V3A
VD1
A
D
N
Hˆ = E D D D + ∑ E j j
j =1
E j − E B , V j , j +1
j + EA A A
+ VD1 D 1 + V1 D 1 D + VAN A N + VNA N
+
N −1
∑ (V j , j + 1
j =1
j
EB − ED / A
j + 1 + V j , j +1 j + 1 j
A
)
© A. Nitzan, TAU
 ED − E

 V1 D

0

0


M

0

VD1
E1 − E
V21
0
M
0
0
V12
0
0
E2 − E V23
V32
O
L
 E1 − E

 V21
 0

 M
 0

O
0
O
EN − E
VNA
V12
0
E2 − E V23
O
V32
L
  cD 
 
  c1 
  c2 
M
  = 0
0  M 
VNA   c N 
 
E A − E   c A 
0
0
L
L
0
L
E N −1 − E
V N , N −1
( E D − E ) c D + VD1 c1 = 0
( E A − E ) c A + VAN c N = 0
BEWARE:
equations are
coming!
  c1

M
  c2
0  M

VN −1, N   M
E N − E   c N
0
(

 V1D c D 



0



 = − M 



0




V c 

 NA A 
)
Hˆ B − EI B c B = u
(B)
ˆ
c B = −G u
(
Gˆ ( B ) = EI B − Hˆ B
)
−1
© A. Nitzan, TAU
Effective donor-acceptor coupling
( E% D − E ) cD + V%DA c A = 0
( E% A − E ) c A + V%AD cD = 0
(B)
(B)
E% D = E D + VD1Gˆ 11
VD1 ; E% A = E A + V AN Gˆ NN
VNA
( )
Gˆ B
V%DA
*
V%DA = VD1Gˆ 1( BN)VNA ; V%AD = VAN Gˆ N( B1)V1 D = V%DA
G = G0 + G0VG0 + G0VG0VG0 + ...
1N
=
1
( E D / A − E1 )
VD1VNA
=
VB
V12
1
( ED / A


VB


−
E
E
B)
 ( D/ A
1
1
V23 ...
VN −1, N
− E2 )
( E D / A − E N −1 )
( ED / A − EN )
N
= V0 e − (1 / 2) Nbβ '
VB
2
β ' = ln
b ED / A − EB
© A. Nitzan, TAU
Marcus expresions for non-adiabatic
ET rates
2π
| VDA |2 F ( E AD )
h
2
2π
2
(B)
=
VD1VNA G1 N ( E D ) F ( E AD )
h
k D→ A =
Donor-to-Bridge/
Acceptor-to-bridge
F (E) =
Bridge Green’s
Function
e
− ( λ + E ) / 4 λ k BT
2
4πλ k B T
Reorganization energy
Franck-Condonweighted DOS
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Bridge mediated ET rate
k ET ~ F ( E AD , T )exp( − β ' RDA )
β’ (Å-1)=
0.2-0.6 for highly conjugated chains
0.9-1.2
for saturated hydrocarbons
~2
for vacuum
© A. Nitzan, TAU
Incoherent hopping
2
N
........
k12
1
k10=k01exp(-βE10)
kN,N+1=kN+1,Nexp(-βE10)
0=D
N+1 = A
P&0 = − k1,0 P0 + k0,1 P1
constant
P&1 = − ( k0,1 + k2,1 ) P1 + k1,0 P0 + k1,2 P2
STEADY STATE
SOLUTION
M
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
© A. Nitzan, TAU
ET rate from steady state hopping
0 = − ( k0,1 + k2,1 ) P1 + k1,0 P0 + k1,2 P2
M
0 = − ( k N −1, N + k N +1, N ) PN + k N , N −1 PN −1
P&N +1 = k N +1, N PN
k D → A ≡ k N +1,0 =
ke
k
kN → A
− ( E B / k BT )
+
k
k1→ D
−1+ N
© A. Nitzan, TAU
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.
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The photosythetic reaction center
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Dependence on bridge length
e−β N
−1
−1
 kup
+ kdiff
N 
−1
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DNA (Giese et al 2001)
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Steady state evaluation of rates
Rate of water flow depends linearly
on water height in the cylinder
h
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
© A. Nitzan, TAU
Steady state quantum mechanics
{l}
Starting from state 0 at t=0:
P0 = exp(-Γ0t)
0
{l }
V0l
G = 2π|Vsl|2ρ{L } (Golden Rule)
l
Steady state derivation:
ψ ( t ) = C0 ( t ) 0 + ∑ C l ( t ) l
l
d − ( i / h ) E0 t
h
C 0 = c0 eC0 = − iE0 C0 − i ∑ Vsl C l
dt
l
d
h C l = − iEl C l − iVl 0 C 0
dt
; all l
© A. Nitzan, TAU
C 0 = c0 e
pumping
− ( i / h ) E0 t
damping
d
h C l = − iEl C l − iVl 0 C 0 − (1 / 2)η C l
dt
C l ( t ) = cl e
− ( i / h ) E0 t
J = (η / h ) ∑ C l
2
= C0
Vls c0
cl =
E0 − El + iη / 2
;
2
l
∑ Vl 0

→ C0
k=
J
C0
2
2π
=
h
η/h
2
( E0 − El ) + (η / 2 )
2
l
η →0
∑ Vl 0
l
2
; all l
2
2π
h
∑ Vl 0
l
2
2
δ ( E0 − E l )
(
2π
2
δ ( E0 − E l ) =
Vl 0 ρ L
h
)
E l = E0
= Γ0 / h
© A. Nitzan, TAU
Resonance scattering
{r}
{l}
1
V1l
V1r
0
{l }
H = H0 + V
H 0 = E0 0 0 + E1 1 1 + ∑ El l l +
(
l ≠0
V = V0,1 0 1 + V1,0 1 0 + ∑ Vl ,1 l 1 + V1,l 1 l
l
∑ Er
r r
r
) + ∑ (Vr ,1 r
1 + V1,r 1 r
r
© A. Nitzan, TAU
)
Resonance scattering
C 0 = c0 E − ( i / h ) E0t
hC& 1 = − iE1C1 − iV1,0 C 0 − i ∑ l V1,l C l − i ∑ r V1,r C r
hC& r = − iEr C r − iVr ,1C1 − (η / 2)C r
hC& l = − iEl C l − iVl ,1C1 − (η / 2)C l
C j ( t ) = c j exp ( −( i / h ) E0 t )
cr =
Vr ,1 c1
E0 − Er + iη / 2
For each r
and l
j = 0, 1, {l}, {r}
cl =
Vl ,1 c1
E0 − El + iη / 2
© A. Nitzan, TAU
0 = − i ( E1 − E0 ) c1 − iV1,0 c0 − i ∑ l V1,l cl − i ∑ r V1,r cr
cr =
Vr ,1 c1
E0 − Er + iη / 2
∑ r V1,r cr = − iB1R ( E0 )c1
2
|
V
|
1r
| V1r B
|
≡
(
E
)
E ) − (1 / 2)i Γ1 R ( E )
B1 R ( E ) ≡ ∑
1 R = Λ 1 R (∑
r E − E r + iη
r E − E1 + iη
2
B1 R ( E ) = − (1 / 2)i Γ1 R
(
)
wide band approximation
Γ1 R ( E ) = 2π | V1r |2 ρ R ( Er )
∞
Er = E
| V1r |2 ρ R ( Er )
Λ 1 R ( E ) = PP ∫ dEr
E − Er
−∞
SELF
ENERGY
© A. Nitzan, TAU
c1 =
V1,0 c0
E0 − E% 1 + ( i / 2)Γ1 ( E0 )
Γ1 ( E0 ) = Γ1 L ( E0 ) + Γ1 R ( E0 )
0 = − i ( Er − E0 ) cr − iVr ,1 c1 − (η / 2)cr
| cr |2 =| C r |2 =
| Vr ,1 |2
| V1,0 |2 | c0 |2
(
( E0 − Er )2 + (η / 2)2
J 0→ R = (η / h ) ∑ cr
2
r
) + ( Γ1 ( E0 ) / 2 )
2
| V1,0 |2
η →0

→
E0 − E% 1
(
E0 − E% 1
)
2
+ ( Γ1 ( E0 ) / 2 )
2
2
Γ1 R ( E0 )
| c0 |2
h
Vr 1 δ ( E0 − Er )
2
© A. Nitzan, TAU
Resonant tunneling
{r }
{l }
J 0→ R =
1
{l }
V1r
V1l
| V1,0 |2
( E0 − E1 ) + ( Γ1 / 2 )
2
2
Γ1 R
| c0 |2
h
0
V(x)
L
(a)
|1>
R
....
R
....
|0>
....
(b)
L
....
....
x
(c)
....
© A. Nitzan, TAU
Resonant Tunneling
V(x)
L
(a)
|1>
R
....
| V1,0 |2
R
....
|0>
....
(b)
L
....
....
(c)
....
J 0→ R =
( E0 − E1 ) 2 + ( Γ1 / 2 ) 2
Γ1 R
| c0 |2
h
x
Transmission Coefficient
J 0→ R
p0
= ( incident flux ) × T ( E0 ) =| c0 |
T ( E0 )
mL
2
T ( E0 ) =
(
( 2π hρ L ( E0 ) )
Γ1 L ( E0 )Γ1 R ( E0 )
E0 − E% 1
) + ( Γ1 ( E0 ) / 2 )
2
Γ1 = Γ1 R + Γ1 L
2
© A. Nitzan, TAU
−1
Resonant Transmission – 3d
V(x)
L
(a)
|1>
R
T ( E0 ) =
....
|0>
....
(b)
L
....
....
Γ1 L ( E0 )Γ1 R ( E0 )
....
R
(c)
....
x
(
E0 − E% 1
)
2
Γ1 = Γ1 R + Γ1 L
+ ( Γ1 ( E0 ) / 2 )
2
1d
3d: Total flux from L to R at energy E0:
Γ1 L ( E0 )Γ1 R ( E0 )
1
 dJ L→ R ( E ) 
2
|
|
c
=
0


2
2
dE
2
π
h
) E0 − E% 1 + ( Γ1 ( E0 ) / 2 )

 E = E0 (
(
)
If the continua are associated with a metal electrode at
thermal equilibrium than c0 2 = f ( E0 ) = exp ( ( E0 − µ ) / k B T ) + 1 −1

(Fermi-Dirac distribution)

© A. Nitzan, TAU
CONDUCTION
L
R
µ –|e|φ
µ
f(E0) (Fermi
function)
1
 dJ L→ R ( E ) 
=
T ( E0 ) | c0 |2


dE

 E = E0 ( 2π h )
∞
I=
e
T (E) =
(
Γ1 L ( E )Γ1 R ( E )
E − E% 1
) + ( Γ1 ( E ) / 2 ) 2
2
∫ dE ( f L ( E ) − f R ( E ) )T ( E )
2π h −∞
f ( E + e φ ) − f ( E ) = e φδ ( E − µ )
e2
I=
T (E = µ) ×φ
2 spin states
2π h
e2
I=
T (E = µ) ×φ
πh
Zero bias conduction
g (φ = 0)
© A. Nitzan, TAU
Landauer formula
g (φ = 0) =
I=
e
∞
e
2
πh
T (E = µ) ;
µ − Fermi energy
∫ dE ( f L ( E ) − f R ( E ) )T ( E )
π h −∞
For a single “channel”:
T (E) =
(
Γ1 L ( E )Γ1 R ( E )
E − E% 1
) + ( Γ1 ( E ) / 2 )
2
Maximum conductance per channel
g=
e
2
2
πh
dI
g (φ ) =
dφ
(maximum=1)
= ( 12.9 K Ω )
© A. Nitzan, TAU
−1