Saturday, July 06, 2024

What is a Wigner crystal?

Last week I was at the every-2-years Gordon Research Conference on Correlated Electron Systems at lovely Mt. Holyoke.  It was very fun, but one key aspect of the culture of the GRCs is that attendees are not supposed to post about them on social media, thus encouraging presenters to show results that have not yet been published.  So, no round up from me, except to say that I think I learned a lot.

The topic of Wigner crystals came up, and I realized that (at least according to google) I have not really written about these, and now seems to be a good time.

First, let's talk about crystals in general.  If you bring together an ensemble of objects (let's assume they're identical for now) and throw in either some long-range attraction or an overall confining constraint, plus a repulsive interaction that is effective at short range, you tend to get formation of a crystal, if an object's kinetic energy is sufficiently small compared to the interactions.  A couple of my favorite examples of this are crystals from drought balls and bubble rafts.  As the kinetic energy (usually parametrized by a temperature when we're talking about atoms and molecules as the objects) is reduced, the system crystallizes, spontaneously breaking continuous translational and rotational symmetry, leading to configurations with discrete translational and rotational symmetry.  Using charged colloidal particles as buiding blocks, the attractive interaction is electrostatic, because the particles have different charges, and they have the usual "hard core repulsion".  The result can be all kinds of cool colloidal crystal structures.

In 1934, Eugene Wigner considered whether electrons themselves could form a crystal, if the electron-electron repulsion is sufficiently large compared to their kinetic energy.  For a cold quantum mechanical electron gas, where the kinetic energy is related to the Fermi energy of the electrons, the essential dimensionless parameter here is \(r_{s}\), the Wigner-Seitz radius.  Serious calculations have shown that you should get a Wigner crystal for electrons in 2D if \(r_{s} > \sim 31\).  (You can also have a "classical" Wigner crystal, when the electron kinetic energy is set by the temperature rather than quantum degeneracy; an example of this situation is electrons floating on the surface of liquid helium.)

Observing Wigner crystals in experiments is very challenging, historically.  When working in ultraclean 2D electron gases in GaAs/AlGaAs structures, signatures include looking for "pinning" of the insulating 2D electronic crystal on residual disorder, leading to nonlinear conduction at the onset of "sliding"; features in microwave absorption corresponding to melting of the crystal; changes in capacitance/screening, etc.  Large magnetic fields can be helpful in bringing about Wigner crystallization (tending to confine electronic wavefunctions, and quenching kinetic energy by having Landau Levels).  

In recent years, 2D materials and advances in scanning tunneling microscopy (STM) have led to a lot of progress in imaging Wigner crystals.  One representative paper is this, in which the moiré potential in a bilayer system helps by flattening the bands and therefore reducing the kinetic energy.  Another example is this paper from April, looking at Wigner crystals at high magnetic field in Bernal-stacked bilayer graphene.   One aspect of these experiments that I find amazing is that the STM doesn't melt the crystals, since it's either injecting or removing charge throughout the imaging process.  The crystals are somehow stable enough that any removed electron gets rapidly replaced without screwing up the spatial order.  Very cool.

Two additional notes:


Anonymous said...

I've always been puzzled by the moire based wigner crystal claims. How is a sparse distribution of electrons *each localized in a moire potential well* a wigner crystal....?
Electrons sitting in lattice potential wells is not a wigner crystal?

Douglas Natelson said...

Anon, I believe the idea is, the actual e-e repulsion dominates (so in that sense it’s a real Wigner crystal), but the moiré potential pins the crystal commensurately. An example where this is discussed:

Stefan Bringuier said...

Love this post Prof. Natelson, super interesting to connect the Wigner-Seitz radius (r_s/a_0) we learn about in SS physics to some phenomena. One thing I'm not following is whether the Wigner crystal phase only occurs in the strongly e-e correlated regime. My understanding is that in this case, given the electron K.E. << e-e repulsion, the electron density is low and therefore r_s is large, leading to crystallization. In contrast, in the weakly correlated regime, electron density is high, resulting in small r_s, and the electrons behave more like a Fermi gas. I guess I ask because I assume there are regimes in the mean-field approximation where strong repulsion occurs, yet it is not the same origin and Wigner crystal will not form?

Anonymous said...

Regarding e-e repulsion dominating: that should mean that increasing the density one should be able to get a Wigner crystal that is not commensurate with the moire lattice. I don't think that has been shown.
I still am not convinced the energetics are proper Wigner phase.