Electronic transport
through
single molecules
Matthias H. Hettler
Forschungszentrum Karlsruhe
Institut für Nanotechnologie (INT)
Collaborators:
M. R. Wegewijs, H. Schoeller: RWTH Aachen
W. Wenzel, A. Thielmann, P. Stampfuss: INT
J. Heurich, J.-C. Cuevas, G. Schön:Uni Karlsruhe
Outline
Motivation
Experimental setups
Physical Picture: strong vs. weak coupling
Strong coupling (brief)
Weak coupling: methodology
Tunneling transport through benzene
Conclusions
Molecular electronics
Attractive features
Large energy scale (meV
Difficulties
eV)
Chemically designable
geometric/electronic structure
“Moving parts” /
Conformational changes
Biological assembly possible?
“Size does matter”
Contacting
a single molecule
Stability
Interface properties largely
unknown (contacts)
Gating ?
Experimental Setups
Controllable break junction
STM over molecular layer
Top view
Bumm et al., Science 271 (1996)
Courtesy of H. Weber, INT
Nanopore Experiments
Chen, Reed, Rawlett and Tour, Science 286 (1999)
Negative differential conductance
(NDC)
Peak current per molecule of order 10-13
A!
Planar electrodes
e.g. J. Park, A.N. Pasupathy et al., Nature 417 (2002)
''Transition metal holder''
2,2':6',2''-terpyridine
symmetry restricts
coupling
buffer L
Buffer R
d-like
Co states
Strong Coulomb
correlations
Co
Physical Picture:
Separate components
MO: Molecular Orbital
Physical picture:
Strong molecule-electrode coupling (1)
: Molecule-electrode coupling
Physical picture:
Strong molecule-electrode coupling (2)
Molecular Orbitals severely broadened
on order of coupling energy
Electrons stay short time on molecule
Only static e-e interactions important
single particle description suffices
L
R
L
T(E,V): Transmission function, eV
f E
f E
dE T E , V
2e h
F(E): Fermi function
Landauer Approach: I V
If
charging energy
Transport via
coherent scattering states
R
Strong coupling methodology
Density Functional Theory (DFT) treatment of “extended molecule”
Couple the resulting Kohn-Sham eigenstates to bulk electrodes
“Diagonalize”, compute electron density on molecule, and iterate
Many groups, many variations in many aspects of theory
“extended molecule”
Figures from Heurich,Cuevas, Wenzel and Schön, PRL 88 (2002)
Physical picture:
Weak molecule-electrode coupling
Molecular Orbitals stay sharp
Electrons stay long time on molecule
Dynamic e-e interactions important
< temperature,
If
many-body description necessary
<< charging energy
Transport via sequential tunneling
Benzene-(1,4)-dithiolate/Au system
M.A.Reed et al., Science 278 (1997): First single molecule experiment?
C
S
C
C
C
C
S
Small current?
C
Bias Voltage (V)
Theory
M. DiVentra, S.T. Pantelides, N.D. Lang, PRL 84, 979 (2000)
P.S. Damle, A. W. Gosh, S. Datta, PRB 64, 201403 (2001)
E. G. Emberly, G. Kirczenow, PRB 64, 235412 (2001)
2 benzenes, each one connected to different electrode!
How to weakly coupled molecules?
H
H
C
Au
Buffer
Mol.
C
C
H
Tunnel contact
C
C
Buffer
Mol.
C
H
“Effective”
Tunnel contact
Molecular Device
Sequential tunneling on/off molecule,
transitions between many-electron states
Example: J. Park, A.N. Pasupathy et al., Nature 417 (2002)
Au
and
C
C
H
Sp2
C
C
z
C
H
P
H
H
electron system
Benzene
H
C
H
High symmetry, good energetic separation of
Small enough to test theoretical methods
and
(C atoms) electron system
(C,H atoms) and
Prototype aromatic molecule
systems
'
ck
'
cl
'
ijkl
†
cj
†
U ijkl c i
cj
ij
†
ij c i
H mol.
Interacting Model
Electronic structure
Effective interacting Hamiltonian for
Create orthogonalized Wannier-states
i
localized at C-atom i = 1,...,6 on the ring
electrons
Low energy spectrum 0-5 eV well-reproduced
for neutral states molecule (exp. & theory)
less accurate for the charged states (anion) (not much known)
Wavefunction amplitudes
Dipole moments
Tunneling
Emission and Absorption of Photons
radiative relaxation of many-body states
'
cl
eff.
ext
j
e.g. partial drop
of bias potential
Full unscreened Coulomb interaction
polarization)
for electrons (no
kl
ij
e2
x
x'
jk
U ijkl
e2
ij
kl
x x'
eff. il
V
Freeze “core” electrons
effective potential
for remaining electrons
V
V
nucl.
V
H kin
ck
'
'
i
ij
†
cj
†
U ijkl c i
ijkl
ij
cj
ij
ci
†
H
Effective Hamiltonian
External Bias Potential
rx
L
V
R
r
L
j
r
V
R
x
L
V
L
V
R
L
2
V bias
V
ext
i
V
r V
V
ext
d3r
cj
ext
ij
†
ci
V
ij
ext
ij
V
e
electrons to external linear ramp potential
H bias
Adjustment of
0.4 nm
“Worst case”: All the bias drops between the tunnel barriers
c †l a k
h.c.
e ,ak
e lk
2
leads
H mol.
Electron Tunneling
:
Density of states and operators
in electrode α=L,R
ti
c
3
3
3
Es
3
E s'
s'
s ci
ti
E s'
s'
2
i
s
4e
d
s'
2
2
i
Es
f
s
1
s'
s ci
Es'
N Es
Es
s
f
s'
Transition Rates
E s'
s d s'
j
:single-particle states
N
E
1
1
E
e
E
i
1
N E
N
r
r er
i
d3r
d ij
s,s'
: many-particle states, determined by effective Hamiltonian
higher energies => relaxation mostly (photons)
lower energies => relaxation + excitation (phonons)
2
s'
s
0
s'
dipole transitions
affect occupations,
thereby the current
Rates
s'
tunneling
in/out (+/-), lead α =L, R
directly enters into current
s'
P s'
s
Current
e
s'
LR p
s
d
s
s'
s
p
Rates determine
non-equilibrium
occupations P
I
Ps
Ps'
t
s'
s
s'
Ps
Stationary master equation
s
Ps
d
4
eV
k T 2.5 10
10
p
ss '
2
eV @ RT
Current vs. Bias Voltage
para-position
para-position + relaxation
z
y
x
I [ e/h]
meta position + relaxation
eV
Symmetric bias voltage
Coulomb
blockade
strong
NDC
VL [V]
NDC: simplest case
Hettler,Schoeller and Wenzel, EPL 57 (2002)
in
N+1
out
populated fast – decays slowly
Blocking state:
N
out
I~e
I
NDC occurs when a transition
to a blocking state becomes
energetically allowed
I~e
slow process dominates current
occupation probability
1
“occupied most of the time”
(compare: population inversion)
in
out
V
dI/dV
NDC
V
NDC with
electrons in benzene?
Molecular orbitals
Many-electron states
Single-particle states
Molecular orbitals
Top View: symmetry adapted
linear combinations of pz AOs
HF-groundstate
z
3D View:
pz Atomic
Orbitals
y
x
I
para-positions
no overlap
with contacts
Meta-positions
always overlap
with contacts
I
Many-electron states
(dominant configurations)
3.4eV
2.1eV
1.7eV
VL
....
High
voltage
1.8eV
3.4eV
Blocking
state
+
2.1eV
1.7eV
Low
voltage
Neutral Molecule
Negative Ion
Robustness
Tunneling in and out of
orbitals?
channel “in series” : slowest process dominates
Slow rotations about
bonds to the buffer groups?
orbitals not involved in bonding rotation possible
Redistribution of
electrons to due to bias potential?
smaller but sizable NDC voltage window
Response to the external bias
“Worst case”
bias effect
I [ e/h]
para-position + relaxation
para-position + relaxation
+ bias effect
z
y
x
extra weak
NDC effects
reduced
charging
smaller
NDC
window
VL [V]
“Message”
In weakly coupled molecules
e-e interactions
Relaxation
symmetry (side groups)
can lead to the occurrence of blocking states.
Possible spoilers : (
Molecule not orthogonal to electrode
Electric field not // to transport axis
Mixing of y-symmetric and y-antisymmetric states
Will have at least quantitative effect
Low-lying anionic Rydberg states
Extra electron occupies a diffuse orbital ?
Spontaneous relaxation to lower Rydberg many-particle state ?
Manipulate with symmetric side groups
Vibrations
Could provide processes to help escape blocking state
Adiabatic change of nuclear lattice in blocked state
Slower than electron motion
Conclusions
Molecular Electronics – Approach
Weakly coupled benzene - Effective
Strong vs. weak coupling picture
electron model
Current-Voltage Characteristics
Blocking state and spatial symmetry
Spoilers/Loopholes
Thanks to the organizers and a great audience!
LONG
Hartree-Fock + Frozen core
augmented double-zeta atomic natural orbitals (ANO)
truncated beyond 2p shell
e-e interaction => effective potential for electrons
“freeze”/integrate out states
Excitation spectrum up to 5 eV well described by interacting
electron model
State
Selecting MRD-CI Method
(Multi-Reference Double-excitation Configuration-Interaction)
HF orbitals
1 determinant (single reference)
select an “active space”
orbitals which can be excited to/from
create all possible single + double excitations
multiple references
create all possible single + double excited states from the references
configurations
perturbation theory
select most important configurations (threshold)
diagonalize 1-electron density matrix
natural orbitals
most important configurations states with optimized orbitals
basis for CI matrix
variational many-particle
wavefunction
EXTRA
Comparison of many-particle “ states”
Molecule
2nd Triplet 1st Singlet opposite order
Electron Affinity to small
Anion
Effective
1st Excitation to big
model
MRD-CI /cc-pVDZ
MRD-CI /aug-cc-pVDZ
(without diffuse functions)
(with diffuse functions)
4.0
3.9
1.4
4.0
3.8
1.7
2.7
1.7
4.1
2.4
0.9 1.5
EXTRA
I-V scenarios
4.8
1.8
5.0
3.9
1.7
2.1
2.4
4.1
4.1
0.9
2.4
1.5
1.8
3.9
2.1
3.8
1.7
zy 1 photon transition
x dipole perpendicular to transport x-axis
1 photon transition
dipole parallel to transport x-axis
EXTRA
EOE
OEE
States
Molecule Many Particle States
EOE
EOE
OEE
anti-symmetric w.r.t. x-z plane refl.
symmetric w.r.t. x-z plane refl.
Anion Many Particle States
destroys the NDC
EOE
OEO
x OEE
x
EEO
y
z
destroys NDC
OOO
z
y
OEE
x
EEE y y OOE
x
x
OEE
EOE
x
y
x
EEE
y
EEE
Strongly coupled benzene
H
C
Au
S
H
C
C
C
C
S
Au
C
H
H
Effective “extended” molecule
Orbitals extended over device + electrodes
Interface effects are dominant
Transmission through coherent channels (Landauer Approach)
Method & Assumptions
Born-Oppenheimer
Approximation
Equilibrium
structure
Hartree Fock
Molecular
Orbitals
Quantum transport
Paralell plate
V drops
over molecule
External bias potential
V-dependent Hamiltonian
with -electron reaction
Electronic structure
Exact diagonalization
C-atoms at Discrete many-body states
1,4 (para) or
Neglect
Tunneling Hamiltonian
1,3
(meta)
polariz.
Radiation field
screening Spontaneous
repeat for
decay (dipole)
Transition rates each V
“Freeze” states eff. potential
Constant DOS
Master equation
Localized Wannier states
Tunn. strength/
Broadening
Nonequilibrium
40meV
Effective interacting
occupations
T Room
electron Hamiltonian
temperature
Full Hamiltonian
in MO basis
Non-linear Current-Voltage
Recovery of transport
3.4eV
I
2.1eV
1.7eV
Tunneling
V
sym.
asym.
1.8 eV
sym.
3.4 eV
2.1 eV
sym.
asym.
1.7 eV
sym.
Neutral benzene
Negative Ion