If the molecule-electrode coupling
is strong (i.e. comparable to the spacing of
molecular orbitals) the combined system of electrodes and molecule can
lower its free energy by hybridizing states from the electrodes and the
molecule. From the point of view of the molecular states, this means
that the formerly sharp states become
, i.e. they acquire a finite life time, as electrons in
the new hybrid states will spend some time in the electrode and not on
the molecule. The broadening will be differential, as some states, i.e.
that are initially localized in the middle of the molecule, will
hybridize only weakly to the electrodes, while other states, like
some localized at the edges or extended states, will hybridize
strongly, see the figure above.
In addition to the broadening, there will be a (differential) shift of
the molecular orbitals due to the fact in contact a spatially uniform electrochemical potential
will be established. This results in charge transfer
(typically from the
elctrode to the molecule) and a corresponding electrostatic
potential at the interface region, similar to a Schottky barrier
at semiconductor-metal interfaces.
The electronic transport in this situation can be understood by
scattering theory, in the sense that the current is carried by scattering states
that extend from
deep in the left electrode over the molecule deep into the right
electrode. The molecule in this approach is considered as a (dynamic)
scatterer that mixes the Bloch eigenstates of the left and right
electrodes. This is called the "Landauer Approach". To compute the
electronic current one needs to know the energy and bias
dependent transmission function
which is integrated in the bias "window"
spanned by the Fermi distribution functions of the electrodes.
Simplistically, the current can be understood as a product of a
transmission rate and an effective continuous density of states on the
molecule as depicted in the figure above. As the electrochemical
potential will often lie in the region of low density of states that
was formerly the HOMO-LUMO gap, the conductance G(V) = dI(V)/dV will be
low at small bias. As the bias is increased, at some bias a resonance
due to the broadened
molecular orbital will be captured by the bias window. The current will
increase rapidly and the conductance will show a peak. If one ignores
the influence of the bias on the electronic states, the conductance
peak will show at a bias approximately twice the energy difference of
the electrochemical potential and the closest resonance (typically, but
not always. the resonance related to the former HOMO).
Naturally, the problem is how to compute the transmission function in a
quantitative way. The transmission function is expressably in
terms of a non-equilibrium Green functions. Nowadays still the
best method to compute these
Green functions is based on Density Functional Theory (DFT). However,
DFT was never meant to be used for transport through nanoscopic
systems. It is by now well known that its application to transport has
intrinsic shortcomings, because the DFT Green functions are by no means
assured to capture all the physics. In general, DFT
calculations show a low bias conductance that is orders of magnitude
too high in comparison to experiment.
At larger bias, the agreement with experiment is qualitatively better,
but still insufficient for quantitative predictions.
Weak molecule-electrode coupling
Weak molecule-electrode coupling effectively means high tunnel barriers
electrode and (parts of ) the molecule. In this case one can
ignore the broadening of the molecular states to first approximation,