What are universal conductance fluctuations?

Another realization I had at the Gordon Conference:  there are plenty of younger people in condensed matter physics who have never heard about some mesoscopic physics topics.   Presumably those topics are now in that awkward purgatory of being so established that they're "boring" from the research standpoint, but they are beyond what is taught in standard solid state physics classes (i.e., they're not in Ashcroft and Mermin or Kittel).  Here is my attempt to talk at a reasonably popular level about one of these, so-called "Universal Conductance Fluctuations" (UCF).

In physics parlance, sometimes it can be very useful to think about electrons in solids as semiclassical, a kind of middle ground between picturing them as little classical specks whizzing around and visualizing them as fuzzy, entirely wavelike quantum states.  In the semiclassical picture, you can think of the electrons as following particular trajectories, and still keep in mind their wavelike aspect by saying that the particles rack up phase as they propagate along.  In a typical metal like gold or copper, the effective wavelength of the electrons is the Fermi wavelength, \( \lambda_{\mathrm{F}} \sim 0.1~\)nm.  That means that an electron propagating 0.1 nm changes its quantum phase by about \(2 \pi\).  In a relatively "clean" metal, electrons propagate along over long distances, many Fermi wavelengths, before scattering.  At low temperatures, that scattering is mostly from disorder (grain boundaries, vacancies, impurities).

The point of keeping track of the quantum phase \(\phi\) is that this is how we find probabilities for quantum processes.  In quantum mechanics, if there are two paths to do something, with (complex) amplitudes \(A_{1}\) and \(A_{2}\), the probability of that something is \(|A_{1} + A_{2}|^{2}\), which is different than just adding the probabilities of each path, \(|A_{1}|^{2}\) and \(|A_{2}|^{2}\).  For an electron propagating, for each trajectory we can figure out an amplitude that includes the phase.  We add up all the (complex) amplitudes for all the possible trajectories, and then take the (magnitude) square of the sum.  The cross terms are what give quantum interference effects, such as the wavy diffraction pattern in the famous two-slit experiment.  This is how Feynman describes interference in his great little book, QED. 

Electronic conduction in a disordered metal then becomes a quantum interference experiment.  An electron can bounce off various impurities or defects in different sequences, with each trajectory having some phase.  The exact phases are set by the details of the disorder, so while they differ from sample to sample, they are the same within a given sample as long as the disorder doesn't change.  The conduction of the electrons is then something like a speckle pattern.  The typical scale of that speckle is a change in the conductance \(G\) of something like \(\delta G \sim e^{2}/h\).  Note that inelastic processes can change the electronic wavelength (by altering the electron energy and hence the magnitude of its momentum) and also randomize the phase - these "dephasing" effects mean that on length scales large compared to some coherence length \(L_{\phi}\), it doesn't make sense to worry about quantum interference.

Now, anything that alters the relative phases of the different trajectories will lead to fluctuations in the conductance on that scale (within a coherent region).  A magnetic field can do this, because the amount of phase racked up by propagating electrons depends not just on their wavelength (basically their momentum), but also on the vector potential, a funny quantity discussed further here.  So, ramping a magnetic field through a (weakly disordered) metal (at low temperatures) can generate sample-specific, random-looking but reproducible, fluctuations in the conductance on the order of \(e^{2}/h\).  These are the UCF. 

By looking at the UCF (their variation with magnetic field, temperature, gate voltage in a semiconductor, etc.), one can infer \(L_{\phi}\), for example.  These kinds of experiments were all the rage in ordinary metals and semiconductors in the late 1980s and early 1990s.  They enjoyed a resurgence in the late '90s during a controversy about coherence and the fate of quasiparticles as \(T \rightarrow 0\), and are still used as a tool to examine coherence in new systems as they come along (graphene, atomically thin semiconductors, 2d electron gases in oxide heterostructures, etc.).