Transcript Introduction to electron transport in molecular systems
A. Nitzan, Tel Aviv University ELECTRON TRANSFER AND TRANSMISSION IN MOLECULES AND MOLECULAR JUNCTIONS AEC, Grenoble, Sept 2005
Lecture 4
Grenoble Sept 2005 Coming March 2006 Chapter 13-15 (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
Grenoble Sept 2005 Coming March 2006 Chapter 16 (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
Coming March 2006 Chapter 17 Grenoble Sept 2005 (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
Donor gives an electron and goes from state “a” (reduced) ELECTRODE PROCESSES b,a b a is the energy of the electron given to the metal Transition rate to a continuum (Golden Rule)
(
E
)
2
V
2
M
(
E
) D Rate of electron transfer to metal in vacuum )
1
)
E F A M Rate of electron transfer to metal in electrolyte solution
k
dE
Reorganization energy here – from donor only (~0.5 of “regular” value)
E
F
e
E
2 / 4
4
I
g
(
e
0)
Landauer formula
e
2 T (
E
) ;
Fermi energy
L
f R
T
g
dI d
For a single “channel”: T
E
1
E
1
L
2
1
R
1
2 (maximum=1) Maximum conductance per channel
g
e
2
12.9
K
1
General case
I
e
L
f R
T T T (E)=Tr
E
B
1
E
1
L
G
2
( )
1
R
1
E
I
H
H
2
2
B E
1
L
1
R
E
1
1 ( 1 / 2)
i
2 Unit matrix in the bridge space Bridge Hamiltonian
B
(1 / 2)
i
2
H H
R
B (R) + B (L) - Self energy Wide band approximation
Molecular level structure between electrodes
energy LUMO HOMO
Cui et al (Lindsay), Science 294, 571 (2001) “The resistance of a single 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”
.
ET vs Conduction
0 = D 1 2
E
........
N N+1 = A
g
e
2
|
G
0,
N
1 (
E
) | 2
0
( )
N
1 2
e
2
E
E D
1 2
i
0
1
E
E A
1 2
i
N
1
G
1
N
(
E
) 2
0
k D
A
2
|
V DA
| 2
F
(
E AD
)
2
N
,
N
1 2
G
( 1
B N
) (
E D
) 2
F
(
E A D
)
F
( )
N
1
e
E
2 / 4
4
A relation between g and k
Electron charge
g
8
e
2 2
(
D L
)
(
A R
)
F
k D
A
conduction Decay into electrodes Marcus Electron transfer rate
A relation between g and k
g
8
e
2 2
(
D L
)
(
A R
)
F
k D
A
F
0.5
eV 4
k T B
1 exp
/ 4
k T B
(
D L
)
(
A R
)
0.5
eV g
~
e
2 /
10
13
k D
A
(
s
1 )
10
17
k D
A
(
s
1 )
1
ET rate from steady state hopping
1 k k 10 =k 01 exp(-
E 10 ) 0 = D 2 k
........
k N k N,N+1 =k N+1,N exp(-
E 10 )
k D
A
k N
1,0
ke
E B
/
k k N
A
k k
1
D
N+1 = A
N
Incoherent hopping
D 1 N A
g
e
2
k T B e
B k
0
k
0
k
1
( 1
L D
)
1
1
k N
N
(
R
)
A
1
1
N N
1 1
k D
A
LARGE N:
g
e
2
k T B e
g
( Or at T=300K.
1 )
18
e
E
/
B k D
A k D
A
(
s
1 )
PART D
Issues in molecular conductions
Grenoble Sept 2005 (3) Molecular conduction
•
Structure-function effects in molecular conduction
•
The role of contacts
•
How does the potential drop on a molecule and why this is important
•
Probing molecules in STM junctions
•
Electron transfer by hopping
•
Charging
•
Switching
2-level bridge (local representation)
{l} {r}
R
L 1 V 12 2
e
2
E
E
1
(1 / 2)
i
1
1
E
2
E
2
V
1,2 | 2
(1 / 2)
i
2
•Dependence on: •Molecule-electrode coupling
L ,
R
•Molecular energetics E
1 , E 2
•Intramolecular coupling V
1,2
|
V
1,2 | 2 2
Ratner and Troisi, 2004 I
5 4 6 3 2 1 0 -1 -1 0
0.0
1
V (V)
“Switching”
Reasons for switching
Conformational changes
Transient charging
Polaron formation time STM under water S.Boussaad et. al. JCP (2003) Tsai et. al. PRL 1992: RTS in Me-SiO 2 -Si junctions
I. Inoue et al, Journal of Physiology 541.3, pp.
769-778 (2002) [Ca +2 ]=1x10 -6 M Single (K+) channel currents from Schwann cells isolated enzymatically from the giant axons of the squids Loligo forbesi, Loligo vulgaris and Loligo bleekeri. The channel conductance was 43.6 pS when both internal and external solutions contained 150 mM K+. Activity was weakly dependent on membrane voltage but sensitive to the internal Ca2+ concentration.
Temperature and chain length dependence
Michel Beyerle et al Giese et al, 2002 Selzer et al 2004 Xue and Ratner 2003
V. J. Langlais et al, PRL 83, 2809 (1999)
Electron transfer in DNA
DNA-news-1
DNA-news-4
DNS-news-3
DNA-news-2
“Prediction is very difficult, Especially of the future
”
attributed to Niels Bohr
Conjugated
vs.
Bonding Saturated Molecules: Importance of Contact
S
S/Au
S
S/Au
Kushmerick
et al
.,
PRL
(2002) 2- vs. 1-side C 10 alkanes! Au-S bonded conjugated system gives at most 1 order of magnitude current increase compared to 3 orders for
Au/S (CH 2 ) 8 SAu Au//CH 3 (CH 2 ) 7 S/Au
Where does the potential bias falls, and how?
•Image effect •Electron-electron interaction (on the Hartree level)
Vacuum
Excess electron density L Xue, Ratner (2003) Potential profile Galperin et al JCP 2003 Galperin et al 2003
Why is it important?
D. Segal, AN, JCP 2002 Heat Release on junction Tian et al JCP 1998
Experiment
Theoretical Model
Experimental i/V behavior
Experimental (Sek&Majda)
junction Hg-SC 12 /C 12 S-Au Hg-SC 12 /C 10 S-Au Hg-SC 16 /C 12 S-Au Hg-SC 12 /C 9 S-Au Hg-SC 16 /C 10 S-Au Ratio of current: i(-1.0 V)/i(+1.0 V) a 0.98
0.13 1.03
0.07 1.22
0.16 1.44
0.20 1.34
0.19 2.03
0.27 Hg-SC 16 /C 9 S-Au a Current at the negative bias refers to the measurement with the Hg side of the junction biased negative relative to the Au side.
Potential distribution
NEGF - HF calculation
HS - CH 2 CH 2 CH 2 CH 2 CH 2 CH 3 . . . CH 3 CH 2 - SH MO Segment Orbital
B A A B
TIMESCALE CONSIDERATIONS
Does the tunneling electron interact with other degrees of freedom and what are the possible consequences of this interaction?
The case of electron tunneling in water
Overbarrier electron transmission through water (D
2
O on Pt(1,1,1)
A look from above on a water film
Effective 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.
The numerical problem
1
2 Pt WATER d Pt z S 1 (1) Get a potential (2) Electrostatics (3) Generate Water configurations S 2 (4) Tunneling calculations (5) Integrate to get current
Potentials for electron transmission through water
Water-Water.....................
RWKM, SPC/E Electron-Water..............
Barnett et al +correction for many body polarizability Water-Wall........
Henziker et al (W-Pt), Hautman et al (W-Au) Electron-Wall..............
Square Barrier Earlier studies – Tunneling through static water configurations
A' L
STM model
A C S 1 S 2 M R C' B' B D D' Z Fig.
1.
A model system used to compute electron transmission between two electrodes, L and R separated by a narrow spatial gap (M) containing a molecular species. The surface S 1 of L is shaped to mimic a tip. The lines A'B', C'D' and AB and CD are projections of boundary surfaces normal to the transmission direction (see text for details). The numerical solution is carried on a grid (Shown).
Potential distribution
A cut of the external potential distribution between the tip and the flat substrate for a voltage drop of 0.5V between these electrodes The image potential along different lines normal to the flat electrode: (1) x=0 (a line going through the tip axis); (2) x=11.96au (distance from the tip axis); (3) x=23.92au.
MOLECULAR DYNAMICS TO GENERATE WATER CONFIGURATIONS Figure - Ohmine et al
CALCULATION OF TRANSMISSION FACTORS SYSTEM (M) L R
H
H
0
L H ML H LM H M H RM H
0
H MR R
G M
(
E
)
E
H M
1
M
(
R
)
M
T
lr
(
E
)
Tr M
G M
(
E
)
(
L
)
M
( †
M
(
E
)
(
R
)
M
i
†
CALCULATION OF TRANSMISSION FACTORS SYSTEM (M) L R Absorbing boundary conditions Green's function method: Replace
i
(r), smoothly rising towards edges of M system, provided LM and by MR boundaries are set far enough
G M
(
E
)
E
H M
1
i
L
i
R
T
lr
(
E
)
Tr M
G M
(
E
)
L
(r)
G
†
M
(
E
)
R
(r)
The self energy
- 2 H
R
H
s R
H
RS
H
SR
H
R
E
)
E
H
R
i
1
S S
E
H
s R
1
H
SR
E
H
R
i
1 H
RS
V S For nearest neighbor coupling:
E
)
H
SR E
H
s R
E
)
1 H
RS
I
e
Tunneling current in water
dE f L
(
E
)
f R
(
E e
)
0
T (
E
) Current against bias voltage in a biased tip-planar electrode junction under water. Upper and lower lines are results for single water configurations characterized by tip-substrate separation of 5.85Å (2 water monolayers) and 12.15Å (4 water monolayers), respectively. The intermediate group of lines are results for 5 different water configurations at tip-substrate separation 9Å (3 water monolayers).
0.8
0.6
0.4
0.2
0.0
3
Resonance transmission through water
4 5 E (eV) 6 7
Tunneling supporting structures in water
Transmission through several water 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.