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TU04-C
A. Nitzan ([email protected]; http://atto.tau.ac.il/~nitzan/nitzan.html)
Introduction to electron transport in
molecular systems
PART A: Introduction: electron transfer
in molecular systems
PART B: Molecular conduction: Main
results and phenomenology
PART C: Inelastic effects in molecular
conduction
http://atto.tau.ac.il/~nitzan/tu04-X.ppt
© A. Nitzan, TAU
(X=A,B,C)
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) 

dI
g ( ) 
d
1 L ( E )1 R ( E )
E  E1

2
  1 ( E ) / 2 
(maximum=1)
2
1  1 R  1 L
Maximum conductance per channel
g
© A. Nitzan, TAU
e

2
  12.9 K  
1
Molecular level structure between
electrodes
energy
LUMO
HOMO
© A. Nitzan, TAU
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
© A. Nitzan, TAU

1
2
Unit matrix in
the bridge space
Bridge Hamiltonian
B(R) + B(L)
--
Self energy
BARRIER DYNAMICS EFFECTS ON ELECTRON
TRANSMISSION THROUGH MOLECULAR WIRES
AND LAYERS
•?Using frozen configurations in transmission
calculations?
•Relevant timescales
•Inelastic contributions to the tunneling current
•Dephasing and activation - transition from
coherent transmission to activated hopping
•Heating of current carrying molecular
wires
•Inelastic tunneling spectroscopy
© A. Nitzan, TAU
Electron transfer/transmission
Bridge
D
2
2
2
2
A k

V
V
G
(
E
)
F
D A
D1 NA
1N
D
N
........
1
E
2
........
1
N
g( E ) 
E
0=D
N+1 = A
© A. Nitzan, TAU
e2

| GDA ( E ) |2 (DL ) ( E )(AR ) ( E )
BARRIER DYNAMICS EFFECTS ON ELECTRON
TRANSMISSION THROUGH MOLECULAR WIRES
AND LAYERS
•?Using frozen configurations in transmission
calculations?
•Relevant timescales
•Inelastic contributions to the tunneling current
•Dephasing and activation - transition from
coherent transmission to activated hopping
•Heating of current carrying molecular
wires
•Inelastic tunneling spectroscopy
© A. Nitzan, TAU
ENERGY PARAMETERS:
{r}
{l}
ΔE (energy gap) ; VB (intersite coupling,
bandwidth) ; Γ (coupling of edge sites to leads);
U (charging energy); λ (reorganization energy interaction with environment); kBT (thermal
energy), eΦ (voltage drop)
EFL

VB
TIME SCALES:
Traversal time
energy relaxation time
dephasing time
environmental correlation time
hopping time
lifetime of electron on edge site
© A. Nitzan, TAU

EFR
Traversal time for tunneling?
1
2
B
A
© A. Nitzan, TAU
Traversal Time

C1(-)=1
C2(-)=0

c2 
  lim 



|

|
c
 0 
1 
© A. Nitzan, TAU
C1(+)=C1
C2(+)=C2
"Tunnelling Times"
D
E0

UB
2
........
1
m
D
2(U B  E0 )
N
E
{r}
0
© A. Nitzan, TAU
 
N
E
Estimates

m
D
2(U B  E0 )
~ 0.2fs
N
 
E
For:
~ 2fs
D=10A (N=2-3)
UB-E = E~1eV
Notes:
m=me
For resonance
transmission
same expression
applies except
that E becomes
the bandwidth
Both time estimates are considerably shorter than
vibrational period
Potential problem: Near resonance these times become
© A. Nitzan, TAU
longer
Transmission through several water
configurations (equilibrium, 300K)
Galperin, AN &
Benjamin, J. Phys.
Chem., 106, 1079096 (2002)
Vacuum
barrier
1. 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
© A. Nitzan,
TAU
configurations. Image potential effects
are disregarded
in this calculation.
BARRIER DYNAMICS EFFECTS ON ELECTRON
TRANSMISSION THROUGH MOLECULAR WIRES
AND LAYERS
•?Using frozen configurations in transmission
calculations?
•Relevant timescales
•Inelastic contributions to the tunneling current
•Dephasing and activation - transition from
coherent transmission to activated hopping
•Heating of current carrying molecular
wires
•Inelastic tunneling spectroscopy
© A. Nitzan, TAU
Water: Tunneling time and
transmission probability
© A. Nitzan, TAU
Galperin, AN, Peskin J. Chem. Phys. 114,
9205-08 (2001)
Vacuum
barrier
TL R ( E ', E )   LG r ( E ) RG a ( E ) ( E  E ')
Fig. 5
 MG r ( E ) LG a ( E ) M †G a ( E ') RG r ( E ')
  N 0 ( E  E '  0 )  ( N 0  1) ( E  E '  0 )
The ratio between
the inelastic
(integrated over all
transmitted
energies) and elastic
components of
the transmission probability calculated
for different instantaneous structures of
a water layer consisting of 3 monolayers
of water molecules confined between
© A. Nitzan, TAU
two Pt(100) surfaces.
Vacuum
barrier
Galperin & AN, J. Chem. Phys.
115, 2681 (2001)
BARRIER DYNAMICS EFFECTS ON ELECTRON
TRANSMISSION THROUGH MOLECULAR WIRES
AND LAYERS
•?Using frozen configurations in transmission
calculations?
•Relevant timescales
•Inelastic contributions to the tunneling current
•Dephasing and activation - transition from
coherent transmission to activated hopping
•Heating of current carrying molecular
wires
•Inelastic tunneling spectroscopy
© A. Nitzan, TAU
Coupling to thermal environment
E1
activated
coherent
E0
Molecule
Lead
•Energy resolved transmission:
T ( E0 , E ) 
1 L 1 R
( E1  E0 )  ( 1 / 2)
2
2
 ( E0  E ) 
  / 2  e   ( E  E ) 
2
( E1  E )2   1 / 2  
© A. Nitzan, TAU
1
0
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. © A. Nitzan, TAU
The photosythetic reaction center
© A. Nitzan, TAU
Selzer et al, JACS
(2003)
© A. Nitzan, TAU
Dependence on bridge length
e
 N
 k
1
up
© A. Nitzan, TAU
k
1
diff
N 
1
DNA (Giese et al 2001)
© A. Nitzan, TAU
Tunneling through n-hexane
L.G. Caron et. al. (L.
Sanche) Phys. Rev. B
33, 3027 (1986)
© A. Nitzan, TAU
BARRIER DYNAMICS EFFECTS ON ELECTRON
TRANSMISSION THROUGH MOLECULAR WIRES
AND LAYERS
•?Using frozen configurations in transmission
calculations?
•Relevant timescales
•Inelastic contributions to the tunneling current
•Dephasing and activation - transition from
coherent transmission to activated hopping
•Heating of current carrying molecular
wires
•Inelastic tunneling spectroscopy
© A. Nitzan, TAU
Elastic transmission vs. maximum heat
generation:

© A. Nitzan, TAU
Heating
© A. Nitzan, TAU
The quantum heat flux
I h   T ( )  nL ( )  nR ( ) d
Transmission
coefficient at
frequency 
Bose Einstein
populations for left
and right baths.
© A. Nitzan, TAU
J. Chem. Phys. 119, 68406855 (2003)
Anharmonicity effects
Heat current vs. chain
length from classical
simulations. Full line:
harmonic chain; dashed
line: anharmonic chain
using the alkane force field
parameters; dash-dotted
line: anharmonic chain
with unphysically large (x
30) anharmonicity
© A. Nitzan, TAU
Heat conduction in alkanes of
different chain length
The thermal conductance vs.
c=400 cm-1 , VL=VR=200
the chain length for Alkanes,
cm-2. Black: T=50K;
c=400 cm-1 , VL=VR=50 cm-2.
Red: T=300K;
Black: T=50K; Red: T=300K;
Blue: T=1000K.
© A. Nitzan, TAU
Blue: T=1000K
Density of normal modes in
alkanes
Low frequency group: 0-600cm-1;
Intermediate frequency group: 700-1500cm-1
© A. Nitzan, TAU
High frequency group:
2900-3200cm-1
Heat conduction by the lower
frequency groups
© A. Nitzan, TAU
Thermal conduction vs. alkane
chain length
© A. Nitzan, TAU
Dashed line:
T=0.1K; Blue dotted
line: T=1K; Full line:
T=10K; Red- dotted
line: T=100K; Line
with circles:
T=1000K. c=400
cm-1 ,VL=VR=50 cm-2.
D.Schwarzer, P.Kutne, C.Schroeder and J.Troe, J. Chem. Phys., 2004
Intramolecular vibrational energy redistribution in bridged azuleneanthracene compounds:Ballistic energy transport through molecular
© A. Nitzan, TAU
chains
BARRIER DYNAMICS EFFECTS ON ELECTRON
TRANSMISSION THROUGH MOLECULAR WIRES
AND LAYERS
•?Using frozen configurations in transmission
calculations?
•Relevant timescales
•Inelastic contributions to the tunneling current
•Dephasing and activation - transition from
coherent transmission to activated hopping
•Heating of current carrying molecular
wires
•Inelastic tunneling spectroscopy
© A. Nitzan, TAU
Light Scattering
in
incident
h 0
in-0
out
scattered
in
in-0
in
in-0
out
h 0
© A. Nitzan, TAU
out
INELSTIC ELECTRON TUNNELING SPECTROSCOPY
h
V
h
h
© A. Nitzan, TAU
h
Localization of Inelastic Tunneling and the Determination
of Atomic-Scale Structure with Chemical Specificity
B.C.Stipe, M.A.Rezaei and W. Ho, PRL, 82, 1724 (1999)
STM image (a) and single-molecule vibrational spectra (b) of three
acetylene isotopes on Cu(100) at 8 K. The vibrational spectra on
Ni(100)are shown in (c). The imaged area in (a), 56Å x 56Å, was
scanned at 50 mV sample bias and 1nA tunneling current
Recall: van Ruitenbeek et al (Pt/H2)- dips
© A. Nitzan, TAU
Electronic Resonance and Symmetry in SingleMolecule Inelastic Electron Tunneling
J.R.Hahn,H.J.Lee,and W.Ho, PRL 85, 1914 (2000)
Single molecule vibrational spectra
obtained by STM-IETS for 16O2 (curve
a),18O2 (curve b), and the clean
Ag(110)surface (curve c).The O2 spectra
were taken over a position 1.6 Å from the
molecular center along the [001] axis.
The feature at 82.0 (76.6)meV for 16O2 (18O2)
is assigned to the O-O stretch vibration, in
close agreement with the values of 80 meV
for 16O2 obtained by EELS.
The symmetric O2 -Ag stretch (30 meV for
16O2) was not observed.The vibrational
feature at 38.3 (35.8)meV for 16O2 (18O2)is
attributed to the antisymmetric O2 -Ag
stretch vibration.
© A. Nitzan, TAU
Inelastic Electron Tunneling Spectroscopy of
Alkanedithiol Self-Assembled Monolayers
W. Wang, T. Lee, I. Kretzschmar and M. A. Reed (Yale, 2004)
Inelastic electron tunneling spectra of C8 dithiol SAM obtained from lock-in
second harmonic measurements with an AC modulation of 8.7 mV (RMS value) at
a frequency of 503 Hz (T =4.2 K).Peaks labeled *are most probably background
due to the encasing Si3N4
© A. Nitzan, TAU
Nano letters, in press
Nanomechanical oscillations in a single C60
transistor
H. Park, J. Park, A.K.L. Lim, E.H. Anderson, A. P. Alivisatos and P. L.
McEuen [NATURE, 407, 57 (2000)]
Vsd(mV)
Two-dimensional
differential conductance
(I/V)plots as a function of
the bias voltage (V) and the
gate voltage (Vg ). The dark
triangular regions
correspond to the
conductance gap, and the
bright lines represent
peaks in the differential
conductance.
Vg(Volt)© A. Nitzan, TAU
Conductance of Small Molecular Junctions
N.B.Zhitenev, H.Meng and Z.Bao
PRL 88, 226801 (2002)
38mV
22
125
35,45,24
Conductance of the T3 sample as a function of source-drain bias at T
=4.2 K. The steps in conductance are spaced by 22 mV.
Left inset: conductance vs source-drain bias curves taken at different
temperatures for the T3 sample (the room temperature curve is not
shown because of large switching noise).
Right inset: differential conductance vs source-drain bias measured
TAU
for two different T3 samples at©TA.=Nitzan,
4.2 K.
MODEL
Hˆ 0   0cˆ †cˆ 
Hˆ 1 

kL, R


kL, R
 kL, R dˆk†dˆk  0aˆ †aˆ    m bˆm† bˆm
m

ˆ Bˆ  MA
ˆ cˆ †cˆ
Vki dˆk†cˆ  h.c .   U m A
m
m
ˆ  aˆ †  aˆ ;
A

dˆ l dˆ l
B̂  bˆ †  bˆ


cˆ †cˆ
ˆa†aˆ

bˆm† bˆm

© A. Nitzan, TAU
dˆr dˆr

Constant in the
wide band
approximation
Parameters
L
1
R
electrons
M
Molecular
vibrations
0
U
Thermal
environment
© A. Nitzan, TAU
V
M – from reorganization
energy (~M2/0)
U – from vibrational
relaxation rates
NEGF


Gn,n 'r (t , t ')   i(t  t ') an (t ), an '† (t ')
Gn,n 'a (t , t ') 
i(t ' t )

G r ( )  G0r ( )  G r ( )r ( )G0r ( )
Ga ( )  G0a ( )  G a ( )a ( )G0a ( )
G   G r  G a

an (t ), an '† (t ')
G   G r  G a
({ }=anticommutator)

an '† ( t ')an ( t )

an ( t )an '† ( t ')
Gn,n ' ( t , t ')   i
Gn,n ' ( t , t ')   i
r
G 0,
jj ' ( )
 j (E)  
1
Im G rjj ( E )
 rph  E   iM 2 
d  
r
D

G


E 
2 
 D     G   E     D r    G r  E    
ph  E   iM 2 
  jj '
1
   j  i
d 
D   G   E   
2
a
G 0,
jj ' ( )
  jj '
1
   j  i
G 0,j , j ' ( )   j , j ' 2 if ( j ) (   j )


1
G 0,
( )    j , j ' 2 i 1  f ( j )  (   j )

j
,
j
'
nj (E) 
Im G jj ( E )
2
q dE  

I 
Tr  ( E )G  ( E )©A.Nitzan,
( ETAU
)G  ( E ) 
2
electrons
M
vibrations
A1
A2M
A3M2

A1  A2 M  A3 M
2

2

 A12  M 2 A2 2  A1 A3
elastic
© A. Nitzan, TAU
inelastic
elastic

© A. Nitzan, TAU
© A. Nitzan, TAU
Changing position of molecular
resonance:
© A. Nitzan, TAU
Changing
tip-molecule
distance
© A. Nitzan, TAU
IETS (intrinsic?) linewidth
L
1
R
electrons
M
Molecular
vibrations
0
U
Thermal
environment
© A. Nitzan, TAU
V
M – from reorganization
energy (~M2/0)
U – from vibrational
relaxation rates
IETS linewidth
1=1eV
L=0.5eV
R=0.05eV
0=0.13eV
M2/0=0.7eV
© A. Nitzan, TAU
© A. Nitzan, TAU
BARRIER DYNAMICS EFFECTS ON ELECTRON
TRANSMISSION THROUGH MOLECULAR WIRES
AND LAYERS
•?Using frozen configurations in transmission
calculations?
•Relevant timescales
•Inelastic contributions to the tunneling current
•Dephasing and activation - transition from
coherent transmission to activated hopping
•Heating of current carrying molecular
wires
•Inelastic tunneling spectroscopy
© A. Nitzan, TAU
A. Nitzan ([email protected])
http://atto.tau.ac.il/~nitzan/nitzan.html
Introduction to electron transport in
molecular systems
PART A: Introduction: electron transfer in
molecular systems
PART B: Molecular conduction: Main
results and phenomenology
PART C: Inelastic effects in molecular
conduction
THANK YOU !
© A. Nitzan, TAU
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