Saturday, January 24, 2009

What is a polaron?

This is another attempt to explain a condensed matter physics concept in comparatively nontechnical language. Comments are, as always, appreciated.

One common example of a quasiparticle is the polaron. When a charge carrier (an electron or hole) is placed into a solid, the surrounding ions can interact with it (e.g., positive ions will be slightly attracted to a negatively charged carrier). The ions can adjust their positions slightly, balancing their interactions with the charge carrier and the forces that hold the ions in their regular places. This adjustment of positions leads to a polarization locally centered on the charge carrier. The combo of the carrier + the surrounding polarization is a polaron. There are "large" and "small" polarons, defined by whether or not the polarization cloud is much larger than the atomic spacing in the material. Polarons are a useful way of thinking about carriers in ionic crystals, materials with "soft" vibrational modes (such as the manganites), and organic semiconductors (very squishy, deformable systems held together by van der Waals rather than covalent bonding).

Not content to let the general relativity fans have all the fun, I can describe this with a ball-rolling-on-a-rubber-sheet analogy. The ball is the charge carrier; the deformation of the rubber sheet is the polarization "cloud". Consider tilting the rubber sheet - this is analogous to applying an electric field to the material. The ball will roll in response to the tilt, but it will be slowed down compared to how it would roll on a hard tilted surface, since it has to put energy into deforming the sheet. In real materials, this shows up as a correction to the effective mass of the charge carrier. All other things being equal, polarons are heavy compared to bare quasiparticles.

We can carry this analogy further. Suppose we have two balls on the rubber sheet. In this classical picture, if the balls are so close together that their sheet deformations touch, the balls will be attracted together and end up in one deformation, held apart by their mutual hard-core repulsion. This is a crude analogy for bipolaron formation, which does happen in real materials. (Though, in real bipolarons the (purely quantum mechanical) spins of the individual polarons are important to stabilizing the bipolaron. The spins form a singlet....) Furthermore, suppose the rubber sheet takes some time to respond to the balls, and takes some time to restore itself to its undeformed state once a ball passes by. You can picture a ball rolling in some direction, leaving behind itself a little groove in the sheet that "fills in" at some rate. This would lower the energy of some other ball if that other ball were traveling in, say, the exact opposite direction of the first ball. This is a very crude way of thinking about the attractive pairing interaction between electrons in low temperature superconductors.

Finally, suppose the rubber sheet is really stretchy. A ball dropped on the sheet will pull the sheet down so far that it'll look like a little punching bag. Now if you try to tilt the sheet, the sheet will have stretched so tightly that the ball won't want to roll at all. Instead, the little punching bag will hang there at an angle relative to the sheet. Something analogous to this can happen in real materials, too - polarons can self-trap. That is, the charge carrier deforms the local environment so much that it basically digs itself such a deep potential well that it can't move anymore. Chemists have their own name for this, by the way. A molecule that deforms to self-trap an extra electron is a radical anion, and a molecule that deforms to self-trap a hole is a radical cation.

11 comments:

CarlBrannen said...

I'm having great difficulty imagining how these deformed regions are going to find each other attractive. I guess I'm thinking of the deformation as a sort of cloud of opposite charges which renormalizes the bare charge.

What bothers me here is that I have trouble seeing how I'm basically going to arrange for electromagnetism to become attractive. I'm looking forward to your further explanation.

Doug Natelson said...

Carl, yes, you really should think of it that way. All the little dipoles in a material rearrange themselves, changing the local charge density. So, a negatively charged electron (say) is surrounded by a slight excess of positive charge. In the rubber sheet analogy, two balls that produce dips will be attracted to each other, since one will "feel" the bending of the sheet due to the other.

Things along these same lines happen for colloids in solutions, by the way. You can have individual particles that are negatively charged in solution, and find that two such particles will experience an effective attractive interaction. The origin of this attraction is that each particle interacts not just with the other particle, but with the screening cloud of ions in solution that surround that other particle. There was a big hullabaloo about this a few years ago. See here, for example. I agree that it's counterintuitive, and it's not at all clear that the net interaction has to end up being attractive.

Massimo (formerly known as Okham) said...

My PhD advisor used to talk about "particles attracting as they fall in the same depression of the mattress", or sometimes that "criminals attract each other, as they all end up in a single jail"... physics humor, I guess... whatever.

Mobile holes in quantum antiferromagnets also attract at short distances, as both alter the local antiferromagnetic order with their motion, and it is energetically convenient for the system to privilege configurations in which holes are near each other, as the damage to the background caused by each of them can be altogether reduced in this way (they share a common cloud of disorder).

CDrone said...

Thanks a bunch!! you're gorgeous.

Honestly your explanation is so simplistic and genuine.

Anonymous said...

thanks a lot. it really helped to understand.

CrazyStitcher said...

Thanks so much! This was really helpful in explaining what a polaron is in layman's terms.

muhammad said...

Excellent ....thank you very much...

Anonymous said...

Very nice analogy

sunny said...

Very lively explaination

Anonymous said...

The analogy helped a lot. Thank you so much!

Anonymous said...

I was struggling to understand what a polaron is and the relevant effect it has on the surroundings, and the explanation really gives me very clear ideas for these terms. Thank you so much!