Charge Transport 101:
A Few Major Mechanisms
Prof. Geoffrey Hutchison
Department of Chemistry
University of Pittsburgh
[email protected]
July 24, 2009
http://hutchison.chem.pitt.edu
Outline
Superexchange
Basic Hopping
Metallic Transport (e.g., Drude Model)
Coulomb Blockade
Kondo Effect
Negative Differential Resistance
Electron-Vibrational Effects
Landauer Coherent Conduction
At the nanoscale, conductance is scattering...
Tii= transition probability
in the ith transverse channel
T=1
T=0
Superexchange
Looks Like Tunneling
(Almost) Everything is Particle-In-A-Box...
Finite probability of
moving through barrier
Greater probability for E
close to V0.
Wavefunction falls off
e-!x
β=
!
2m(V0 − E)
!2
What are “normal” ß values?
Exponential Dependence
What are “normal” ß values?
In Vacuum
~3–5 Å-1
Insulating
(alkanes, protein, etc.)
~0.8–1.2 Å-1
Conjugated
(alkenes, alkynes...)
~0.2–0.5 Å-1
Gray, Winkler, PNAS 2005 vol. 102 p. 3534-3539
Outline
Superexchange
Basic Hopping
Metallic Transport (e.g., Drude Model)
Coulomb Blockade
Kondo Effect
Negative Differential Resistance
Electron-Vibrational Effects
Hopping: Activated Transport
“Pay the Boltzmann Price”
1/R dependence
−Ea /kb T
Ae
D
B1
B2
B3
A
Superexchange to Hopping
PNAS 2005 (102) p. 3540-3545
Marcus Theory
kCT = Ae
0
2
−(∆G +λ) /4λkb T
Free Energy
What is the maximum rate?
"G0 = 0
"G0 = -#
Reaction Coordinate
Marcus Inverted Region
Experimental Proof: Inverted Region Exists
Miller, et. al. JACS 1984 106 p.3047.
QM Treatment of Marcus Theory
From Rossky lecture:
J. Phys. Chem. 1996, 100 p13148.
13154 J. Phys. Chem., Vol. 100, No. 31, 1996
Th
un
fa
ba
Th
po
tio
a
ex
th
tra
Th
V!p
re
th
)
20, 2009
.1021/jp9605663
X (
pr
er
os
SV
pr
tio
qu
of
Reorganization Energies
λ = λouter + λinner
Environment
Solvent
e.g., Rossky lecture
Dielectric
Continuum
Molecule
e.g., $(vibrational modes)
1
1
1 1
1
λs ≈ (∆e) [
+
− ][
− ]
2rd
2ra
R "∞
"0
2
Marcus-Hush Charge-Transfer Theory
Consider h+ or e- transfer:
Change in electronic structure
! (neutral charge)
Energy
Change in geometry
! reorganization of bonds,
! & environment
Cation
Neutral
A +B →A+B
+
+
A +B →A+B
−
−
Nuclear Configuration
−(∆G0 +λ)2 /4λkb T
kCT = Ae
"%
Reorganization Energies: Geometric Relaxation
M +M →M +M
+
S
S
S
S
+
Energy
n
Cation
#+
S
Neutral
S
S
S
n
S
S
S
S
n
Nuclear Configuration
Intermolecular Interaction:
Orbital Splitting Energies
For self-exchange:
2ß
k=
!
π
λkb T
" 12
2
Hab −λ/4kb T
!
e
Note: electronic coupling has
exponential fall-off
Alternate View of Hab
Outline
Superexchange
Basic Hopping
Metallic Transport (e.g., Drude Model)
Coulomb Blockade
Kondo Effect
Negative Differential Resistance
Electron-Vibrational Effects
Metallic Transport:
“Electron Sea” Model of Metals
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
+ + + + + +
Evolution of Band Structure
S
S
n/2
6
4
Energy (eV)
2
0
-2
-4
-6
-8
-10
1
2
3
4
N
Hutchison, et. al. Phys. Rev. B 2003 68 no.035204.
5
6
!
"
Interpretation of Band Diagrams
trans-cisoid-polyacetylene
S
Diagram shows energy as
a function of Brillouin zone
Broad bands: delocalized
Thin bands: highly
localized (e.g. S atom)
Direct band gap
Curvature of bands:
effective mass of h+ or e-
Hutchison, et. al. Phys. Rev. B 2003 68 no.035204.
!
"
!
S
nn / 2
"
3D Band Structure: Possibly Anisotropic
Drude Model
Limit on econductance is
scattering
Treat as “electron gas”
of non-interacting
electrons
Collision between eand positive ions,
creates “drag” !
γ = m/τ
F = ma
d
! − γ!!v "
m !!v " = q E
dt
steady state:
qτ !
!
!!v " =
E = µE
m
Effective Masses
F = ma
1 d2 ε
from quantum mechanics:
a = 2 · 2 qE
! dk
F = qE
simplifying...
from Drude model...
1
=
m∗
!
d2 ε
dk2
!2
"
qτ !
!
!!v " =
E = µE
m
Comparisons of Effective Masses
Material
Hole
Electron
Si
0.56
1.08
Ge
0.37
0.55
P(Thio)
0.14
0.15
P(Pyr)
0.21
0.24
(unit is fraction of electron mass)
Hutchison, et. al. Phys. Rev. B 2003 68 no.035204.
Outline
Superexchange
Basic Hopping
Metallic Transport (e.g., Drude Model)
Coulomb Blockade
Kondo Effect
Negative Differential Resistance
Electron-Vibrational Effects
Coulomb Blockade
We saw this already...
Source
V
(Pt)
N
N
N
N
N
N
Ru
N
N
N
N
N
Ru
N
N
N
N
N
N
N
Gate Oxide (Al2O3)
Gate (Al)
Vg
Source
Molecule
Capacitance
Drain
Drain
(Pt)
Coulomb Blockade in
+4
[Ru2(tppz)3]
4K Temp. Single-Molecule
)
V
Coulomb
Blockade
Transistor
m
-
e
(
Source
Drain
s
a
i
B
7.5
Source
Vsd 0.0
Drain
-7.5
0.00
Charge passes when source, drain
and molecule bridge electronic
states are aligned
e-
0.02
Gate (V)
0.04
Charge does not pass when
molecule
bridge
electronic
states
0
1250
2500
areConductance
above or below
(nS)
Coulomb Blockade I/V Curves
)
V
m
(
Current (A)
s
a
i
B
7.5
0.0
-7.5
0.00
0.02
0.04
Gate (V)
Potential (Vsd)
0
1250
Conductance (nS)
2500
n-equidistant way along the V g axis, implying that effects
han simple Coulomb charging dominate the properties of
between two states with char
2
electron charge (21.6 £ 10
Be Careful... Multiple Coulomb Blockade?
Nature (2003)
425a,p.Measurements
698-701
Experimental
results.
of the differential conductance
representative I s-d–V g trace. b, Exam
single OPV5 molecule obtained at dif
-d) as function of V s-d and V g. All red lines, and all blue lines, have identical
How
know
this shows
is one
molecule?
discussed in the text. The
full soliddo
line you
at the top
of the figure
a
Curves are shifted vertically for clarity