A. Nitzan, Tel Aviv University
ELECTRON TRANSFER AND
TRANSMISSION IN MOLECULES
AND MOLECULAR JUNCTIONS
AEC, Grenoble, Sept 2005
Lecture 3
Grenoble Sept 2005
(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
Coming March 2006
Chapter 13-15
Grenoble Sept 2005
(2) Electron transfer
processes
•Simple models
•Marcus theory
•The reorganization energy
•Adiabatic and non-adiabatic
limits
•Solvent controlled reactions
•Bridge assisted electron transfer
Coming March 2006 •Coherent and incoherent
transfer
Chapter 16
•Electrode processes
Theory of Electron Transfer
q=0
q=1
q=1
a
q=0
b
energy

EA
Ea
E
q=0
q=1
q=1
q=0
Eb
Xa
Xtr
Xb
Transition rate

kTST   d v v P ( x B , v ) Pab  v 
0
S
S
q=0
q=1
q=0
q=1
q=1
q=1
q=0
S
q=0
S
Dielectric solvation
C153 / Formamide (295 K)
Relative Emission Intensity
CF3
450
N
O
O
q=0
q=+e
q=+e
a
b
c
Born solvation energy
500
550
Wavelength / nm
600
 q2  
1
  1    1  2eV
 2a    s 
(for a molecular charge)
Electron transfer: Activation energy
W0 ( )  E0   2
W1 ( )  E1   1   
a
2
b
energy

EA
Ea
E
Eb
[( Eb  Ea )   ]2
EA 
4
 1
1  1
1
1
    


  e  s   2 RA 2 RB RAB
a tr
 2
 q

b
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
Marcus expresions for non-adiabatic
ET rates
k D A 

2
2
| VDA |2 F ( E AD )
VD1VNA
2
2
(B)
G1 N ( E D )
F ( E AD )
Bridge Green’s
Function
Donor-to-Bridge/
Acceptor-to-bridge
    E  / 4  k BT
2
F (E) 
e
4 k B T
Reorganization energy
Franck-Condonweighted DOS
Bridge mediated ET rate
Charge recombination
lifetimes in the
compounds shown in
the inset in dioxane
solvent. (J. M. Warman
et al, Adv. Chem. Phys.
Vol 106, 1999). The
process starts with a
photoinduced electron
transfer – a charge
separation process. The
lifetimes shown are for
the back electron
transfer (charge
recombination) process.
ET rate from steady state hopping
k
2
k
k
........
1
k10=k01exp(-E10)
N
kN,N+1=kN+1,Nexp(-E10)
0=D
k D A  k N 1,0 
N+1 = A
ke
k
kN  A
  E B / k BT 

k
k1 D
1 N
The photosythetic reaction center
Michel - Beyerle et al
DNA (Giese et al 2001)
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
(1) Keep h constant and measure the
steady state outwards water flux
J. Get the rate from k=J/h
= Steady state rate
ELECTROCHEMISTRY
R

C

A
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
Transition rate to a continuum
(Golden Rule)
2
2
( Eb,a ) 
VD, M  M ( Eb,a )
D
A
EF
Rate of electron transfer to metal
in vacuum

k ( E )   Eb,a
1  f ( Eb,a )
Rate of electron transfer to metal
in electrolyte solution

k   dE ( E ) 1  f ( E )  F Eb,a  E
M

    E  / 4  k BT
2
F (E) 
e
4 k B T
PART C
Molecular conduction
Steady state quantum mechanics
l 
Starting from state 0 at t=0:
P0 = exp(-0t)
0
{l }
V0l
0 = 2|V0l|2 lL (Golden Rule)
Steady state derivation:
 ( t )  C0 ( t ) 0   C l ( t ) l
l
d  ( i / ) E0t
C0  c0 eC0   iE0C0  i Vsl Cl
dt
l
d
Cl   iEl Cl  iVl 0C0
dt
; all l
C0  c0 e
 ( i / ) E0t
pumping
damping
d
Cl   iEl Cl  iVl 0C0 (1/ 2) Cl
dt
Cl ( t )  cl e
 ( i / ) E0 t
  Cl
J   /
2
 C0
Vls c0
cl 
E0  El  i / 2
;
2
l
 Vl 0

 C0
k
J
C0
2

/
2
 E0  El    / 2 
2
l
 0
2
2
2
2
 Vl 0   E0  El  
l
2
 Vl 0   E0  El 
l
2
; all l
2

2
Vl 0  L

El  E0
 0 /
Resonance scattering
r
l 
1
{l }
V1l
V1r
0
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
   Vr ,1 r
r
r r
r
1  V1,r 1 r

Resonance scattering
 ( i / ) E0t
CC00  c0iEE0C0  iV0,1C1
C1   iE1C1  iV1,0 C0  i  l V1,l C l  i  r V1,r C r
C r   iEr C r  iVr ,1C1  ( / 2)Cr For each r
C l   iEl C l  iVl ,1C1  ( / 2)Cl
C j ( t )  c j exp  ( i / ) E0 t 
and l
j = 0, 1, {l}, {r}
C j ( t )  ( i / ) E0 c j exp  ( i / ) E0 t 
Resonance scattering
C 0  c0 E
 ( i / ) E0 t
0  i  E0  E1  c1  iV1,0 c0  i  l V1,l cl  i  r V1,r cr
0  i  E0  Er  cr  iVr ,1 c1  ( / 2)cr
0  i  E0  El  cl  iVl ,1 c1  ( / 2)cl
cr 
Vr ,1 c1
E0  Er  i / 2
cl 
For each r
and l
Vl ,1 c1
E0  El  i / 2
0   i  E1  E0  c1  iV1,0 c0  i  l V1,l cl  i  r V1,r cr
cr 
Vr ,1 c1
E0  Er  i / 2
 r V1,r cr  iB1R ( E0 )c1
2
|
V
|
1r
| V1r B
|
B1 R ( E )  
E )  (1 / 2)i 1 R ( E )
1 R (E) 1
R (
r E  Er  i
r E  E1  i
2
B1 R ( E )  (1 / 2)i 1 R


wide band approximation
1 R ( E )  2 | V1r |2  R ( Er )

Er  E
| V1r |2  R ( Er )
 1 R ( E )  PP  dEr
E  Er

SELF
ENERGY
c1 
V1,0 c0
E0  E1  ( i / 2)1 ( E0 )
1 ( E0 )  1 L ( E0 )  1 R ( E0 )
0  i  Er  E0  cr  iVr ,1c1  ( / 2)cr
| cr |2 | C r |2 
J 0 R   /
| Vr ,1 |2
| V1,0 |2 | c0 |2
( E0  Er )2  ( / 2)2
 E0  E1    1 ( E0 ) / 2 
  cr
r
2
2
| V1,0 |2
 0



l   E
E
0
1

2
 1 R ( E0 )
  1 ( E0 ) / 2 
1
{ l }
2 Vr 1   E0  Er 
2
V1l
0
V1r
2
r
2
| c0 |2
Resonant tunneling
r
l 
J 0 R 
1
{l }
V1l
V1r
| V1,0 |2
 E0  E1    1 / 2 
2
1 R
2
0
V(x)
(a)
L
|1>
R
....
L
....
|0>
....
(b)
R
....
....
x
(c)
....
| c0 |2
Resonant Tunneling
V(x)
(a)
L
|1>
R
....
| V1,0 |2
L
....
|0>
....
(b)
R
....
....
....
(c)
J 0 R 
 E0  E1  2   1 / 2  2
1 R
| c0 |2
x
Transmission Coefficient
J 0 R
p0
  incident flux   T ( E0 ) | c0 |
T ( E0 )
mL
2
T ( E0 ) 
 2
1 L ( E0 )1 R ( E0 )
 E0  E1    1 ( E0 ) / 2 
2
1  1 R  1 L
2
 L ( E0 ) 
1
Resonant Transmission – 3d
V(x)
(a)
L
|1>
R
T ( E0 ) 
....
|0>
....
(b)
L
....
....
1 L ( E0 )1 R ( E0 )
....
R
(c)
....

E0  E1

2
  1 ( E0 ) / 2 
1  1 R  1 L
x
2
1d
3d: Total flux from L to R at energy E0:
1
 dJ L R ( E ) 



dE

 E  E0  2
1 L ( E0 )1 R ( E0 )
  E0  E1  2   1 ( E0 ) / 2  2
| c0 |2
If the continua are associated with a metal electrode at
thermal equilibrium than c0 2  f ( E0 )  exp  ( E0   ) / k BT   1 1

(Fermi-Dirac distribution)

CONDUCTION
L
R
 –|e|

f(E0) (Fermi
function)
1
 dJ L R ( E ) 



dE

 E  E0  2

I
2
T ( E0 ) | c0 |
e
2
T (E) 
1 L ( E )1 R ( E )
 E  E1    1 ( E ) / 22
2

 dE  f L ( E )  f R ( E ) T ( E )

f ( E  e  )  f ( E )  e  ( E   )
e2
I
T (E  ) 
2 spin states
2
I
e2

T (E  ) 
Zero bias conduction
g (  0)
Landauer formula
g(  0) 
I
e

e

2
T (E  ) ;
  Fermi energy

 dE  f L ( E )  f R ( E ) T ( E )

For a single “channel”:
T (E) 
1 L ( E )1 R ( E )
 E  E1    1 ( E ) / 2
2
Maximum conductance per channel
g
2
e

2
dI
g( ) 
d
(maximum=1)
  12.9 K  
1
Molecular level structure between
electrodes
energy
LUMO
HOMO
Cui et al (Lindsay),
Science 294, 571 (2001)
“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”.
General case
I
e


 dE  f L ( E )  f R ( E ) T ( E )

ˆ( E()E
ˆ1(L (EE ))G1ˆR((BE))( E )

1)RG
(ˆE()B )† ( E )
TT(E)=Tr

1 L
B 
(E) 


 E  E1    1 ( E ) / 2
2
2

E  E1  (1/ 2)i 
G ( B ) ( E )  EI ( B )  Hˆ ( B )
( B)
H n ,n '
 H n,n '  Bn,n '
B ( R ) ( E )  (1 / 2)i  ( R )
 (nR,n)'  2 H n, R H R ,n '  R
Wide band approximation

1
2
Unit matrix in
the bridge space
Bridge Hamiltonian
B(R) + B(L)
--
Self energy
The N-level bridge (n.n.
interactions)
{r}
{l}
1
0
....
L
N+1
R
e
I


 dE  f L ( E )  f R ( E ) T ( E )

g
T ( E ) | G0, N 1 ( E ) |
2

Gˆ B ( E )

0, N 1


T (E  )
( L)
( R)
 0 ( E ) N 1 ( E )
1
1
1
1
V01
V12 ...
V N , N 1
 E  EN 
 E  E N 1 
 E  E0   E  E1 
1
1


E

E

i


0
2 0 L 

e2
1
G1N(E)
1


E

E

i


N 1
2 N 1, R 

ET vs Conduction
2
........
1
N
E
0=D
g
e2

N+1 = A
| G0, N 1 ( E ) |2  (0L ) ( E ) (NR)1 ( E )
2

e2

V01VN , N 1
1 ( L)  
1 ( R) 

E

E

i

E

E

i  N 1 

D
A
0 
2
2



k D A 

2
2
2
(B)
G1 N ( E )
 (0L ) ( E ) (NR)1 ( E )
| VDA |2 F ( E AD )
    E  / 4  k BT
2
V01VN , N 1
2
2
(B)
G1 N ( E D )
F ( E AD )
F (E) 
e
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 kBT
0.5eV

g ~ e2 / 

1
exp   / 4kBT 
 (DL )   (AR )  0.5eV

1013 k D  A ( s 1 )
 1017 k D  A ( s 1 )   1

Comparing conduction to rates
(M. Newton, 2003)