What is weak localization?

A few days ago I wrote about localization, where waves in a medium can become trapped due to interference by scattering off disorder.  This is an extremely general phenomenon that applies to light, sound, and electronic waves in solids.  

Now I want to write about a phenomenon that is specific to electrons (or at least wavepackets that carry electronic charge, if we want to be very general).  Rather than the completely general arguments about conductivity scaling, now we are going to consider particular sets of trajectories in the weak scattering limit.  

We can define "weak" scattering here in terms of the ratio of the mean free path \(\ell\), the typical distance a wavepacket of electrons travels between being redirected by elastic scattering off disorder (vacancies, impurities, surfaces, grain boundaries), and the Fermi wavelength of the electrons, \(\lambda_{\mathrm{F}}\).  If \(\ell/\lambda_{\mathrm{F}} \gg 1\), then the scattering is weak.  (If you have some measurement that allows you to calculate that ratio for a given system and you find instead that you get \(\ell/\lambda_{\mathrm{F}} \ll 1\), then the disorder is so strong that the model of propagating electronic waves really fails and you have to worry about conduction by something like thermally assisted hopping between localized states.)

Electron wavepackets scattering around
a loop trajectory clockwise (red) or 
counterclockwise (blue).  Gray circles
are scattering sites.  Magnetic field \(B\)
is shown pointing out of the page.

In weak localization, as initially explored here, we consider electronic wavepackets bopping through a disordered environment, as shown.  There are many possible trajectories for the electrons, and bouncing off disorder (symbolized here as gray circles) leads to a shift in the phase of the waves as well as a direction change, but it's all deterministic and reversible.  An electron can bounce around a particular loop trajectory from defect to defect in two ways, clockwise or counterclockwise.  The reversibility means that whatever phase the wavepacket racks up going clockwise, it would accumulate the same phase if it went counterclockwise.  This means that there is constructive interference from the loop trajectories for the electron to end up back where it started - that tends to localize the electrons.  Each particular loop trajectory has its own amount of accumulated phase, but all of them have this "constructive interference for back-scattering" issue.

How can we tell this is really going on?  We can turn on a magnetic field \(\mathbf{B} = \nabla \times \mathbf{A}\) that threads flux through the loops.  As I described here, the propagating electrons then pick up an additional phase \(\delta \varphi = (q/\hbar)\int \mathbf{A}\cdot d\mathbf{r}\) as they go along a trajectory.  This means that the clockwise and counterclockwise versions of the loop trajectories are now offset in phase by an amount proportional to the magnetic flux through the loop and in general no longer interfere constructively for back-scattering.   

How large of loops do we need to consider?  Because of inelastic interactions with other electrons, lattice vibrations, etc., the phase of the electronic waves gets scrambled on a characteristic coherence timescale \(\tau_{\phi}\), and a corresponding coherence length scale \(L_{\phi} = \sqrt{D \tau_{\phi}}\), where \(D\) is the diffusion constant for the electrons.  (See here.)

The result of all this is a positive magnetoconductance (equivalently a negative magnetoresistance), since applying the magnetic field suppresses the back-scattering.  The magnetic field scale over which the zero-field conductance dip gets suppressed is on the order of \(B_{c} \sim (h/e)/L_{\phi}^{2}\), though the detailed functional form of \(\delta \sigma (B)\) depends on the relative size of \(L_{\phi}\) and the sample dimensions.  (See here for a key reference if you want details.) Weak localization is one of the main techniques used to infer coherence properties of metals and semiconductors.  A classic review by Gerd Bergmann is here.  Note that this is also closely related to the physics of universal conductance fluctuations.

(One additional point for experts.  I hadn't mentioned spin or spin-orbit coupling.  It turns out that in the strong spin-orbit coupling limit (\(\tau_{\mathrm{so}} \ll \tau_{\phi}\)), the accumulated phases for the time-reversed loop trajectories are no longer of the same sign, but instead are of opposite signs.  The result is destructive interference for back-scattering, and therefore a negative magnetoconductance and "weak antilocalization" (WAL), where the analytic expressions for WAL differ from the WL forms by a factor of -1/2.)