Conductance is Transmission

My latest story in Physical Review Focus describes measurements of electrical conduction between two "buckyballs," or C₆₀
molecules. This sort of characterization is a prerequisite for the sort of understanding and control that would be needed for future "molecular electronics."

The electrical conductance (the inverse of the resistance) in such tiny systems is limited to values of the order of 2e²/h, where e is the electron charge and h is Planck's constant, which sets the scale for quantum phenomena. This combination goes by the name of "conductance quantum," or G₀.

Unlike other quanta like photons, however, the conductance is often not generally required to come in discrete packets. Under special experimental circumstances, however, such as in "quantum point contacts," the conductance can take on reasonably stable values that are simple multiples of G₀.

Still, the idea that conductance has special value was quite jarring when it became popular in the 1980s. Most materials have a well-defined conductivity determined by number of electrons and how frequently they scatter from imperfections of atomic motion. The conductance, which is just the total current divided by the voltage, is then calculated from the conductivity by multiplying by the cross-sectional area of a piece of material, and dividing by its length.

In very small devices, however, electrons move as a wave from one end to the other. The conductance is then determined by the likelihood that they propagate to the far end. The visionary IBM researcher Rolf Landauer laid the groundwork for this view in a 1957 article in the IBM Journal of Research and Development.

Only a quarter-century later in the 1980s, however, did experiments start to catch up. Researchers had been doing experiments at very low temperatures in clean semiconductor systems, where the electrons propagate cleanly as waves over many microns. Lithographic patterning can easily create structures that are smaller than this distance, and comparable to the wavelength of the electrons themselves (typically a few hundred angstroms, or a few hundredths of a micron).

In the semiconductor samples, electrons are free to flow only in a thin sheet near the surface. Researchers can apply a voltage a metal film on top of a semiconductor so that the electrons have to avoid the region under the metal. If there is a small gap in a line of metal, electrons can squeeze through this quantum point contact between them. This is the situation where the quantum effects become important.

The usual derivation goes like this (feel free to skip over this long paragraph): on the two sides, electrons fill up the available states equally, so filled states on one side face filled states on the other and have no way to move across. Applying a voltage V raises the energy of electrons on one side, so the top ones now face empty states on the other. The number of such exposed states is the energy change, eV, times the number of states in each energy interval. Here's the magic: the number of states, for the special case where they are one dimensional waves, is determined by how their energy E varies with wave vector k: dE/dk. Their group velocity--the rate at which they impinge on the contact--is (1/h) times dk/dE. Each carries a charge of e, and there are two electrons in each state because there are two spin states. Presto: the total current is eV(dE/dk)(1/h)(dk/dE)2e = 2e²/h x V.

To me it's rather unsatisfying to go through these shenanigans to get a simple answer like 2e²/h. It doesn't seem right that we have to introduce all these extra quantities just to have them cancel out. Is there an easier way to get to this answer?

In any case, it is now clearly established that each quantum state has an overall conductance of G₀=2e²/h, multiplied by the transmission coefficient, which is the probability of a particular wavelike electron making it to the other side. This result applies to any quantum transmission, whether it's in an engineered semiconductor or a single C₆₀ molecule.