Molecular Electronics Theory: What is it about?

The central quest of molecular electronics theory is to understand electronic transport through single molecules and molecular layers. There are two main difficulties that make this problem particularly hard. 

1) Transport takes place through a  "hybrid" system, consisting of (mostly) metallic electrodes and a  "semiconducting"  molecule  (i.e.  a  system  with  a gapped excitation spectrum). Bringing these systems into contact, interface physics like charge transfer and the resulting barriers  (similar  to Schottky barriers) as well as the possibility of Coulomb blockade complicate matters,  since transport is strongly influenced by these effects.  Therefore, knowledge of the separated system parts (molecule and electrodes) in general is  not by itself sufficient to describe transport through the combined system.

2) One needs to know more of the molecular electronic structure than necessary for the description of  thermodynamic or equilibrium properties.  To describe transport, one needs to know the true single-particle excitation spectrum of the molecule in contact to the electron reservoirs of the electrodes that are not in equilibrium, i.e.  that differ in their chemical potential by the applied bias V.
  1. Posing the problem
  2. Strong molecule-electrode coupling
  3. Physical picture
  4. Weak molecule-electrode coupling
Posing the problem
To elucidate these issues, consider a situation as depicted on the left figure below. The sketch shows two gold (Au) electrodes on a substrate, separated by a gap of size about 2 nanometers. Above the gap a molecule is approaching the electrodes (protection groups prevent unwanted chemical reactions to take place). The electronic structure of the three separated parts (left electrode, right electrode and the molecule) is depicted in the picture on the right side. The electrodes can be considered as "Fermi seas" of electrons with a continuous density of states filled to the corresponding chemical potential (Fermi energy).

Two metal electrodes with an approaching molecule. Electronic structure of the separated system

The molecule in isolation is a finite quantum system. As such, the molecule has a spectrum of discrete quantum states (Molecular Orbitals, MO)  that are either occupied or  unoccupied. For simple organic molecules, the MOs are occupied by two electrons (spin up and down) up  to the Highest Occupied MO (HOMO). All MOs above the HOMO are unoccupied, starting with the lowest unoccupied MO (LUMO). Between the HOMO and the LUMO the electronic spectrum has a gap (HOMO-LUMO gap) that can be associated to the light absorption spectrum of the molecule in isolation, since the lowest energy process for absorption if a photon would be an excitation where one kicks an electron out of the HOMO and deposits it in the LUMO. However, as we will see below, the HOMO-LUMO gap has not much to do with gaps appearing in electronic transport.

Animation of molecule approach

In the animation above, the molecule "falls  down" into the gap between the electrodes. The thiol groups ( -S- ) prefer to  a covalent bond to the gold atoms of the electrodes, so the protection groups are removed and will diffuse away.
Now the molecule is bound covalently to both electrodes, meaning that the fomerly separated electronic systems
are coupled.  Depending on the strength of coupling (and on the molecule)  two "extreme" scenarios of transport can be considered.

Strong molecule-electrode coupling Gamma

Strong molecule-electrode coupling smears and shifts the molecular orbitals

If the molecule-electrode coupling Gamma 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 broadened, 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  T(E,V) which  is integrated in  the bias "window" spanned by the Fermi distribution functions of the electrodes.

Landauer approach to transport

Physical picture
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 Gamma

Weak molecule-electrode coupling effectively means high tunnel barriers between the electrode and (parts of )  the molecule. In this case one can ignore the broadening of the molecular states to first approximation, see below.

Weak coupling means tunnel barriers

At first glance, not much has changed from the physical picture of transport described above for strong coupling. One still would expect low (exponentially small) conductance at small bias and a rapid rise of the current and a peaked conductance as soon as the first resonance (that is now very close to the molecular state of the separated components) is captured by the bias window. However, this picture almost certainly fails for the molecules of interest in the weakly coupled situation. The underlying reason  for this failure is the phenomenon of  "Coulomb blockade", also known from transport through mesoscopic quantum dots.