Friedel Oscillations: wherein we learn that the electron has a size

For some reason, everyone who learns quantum mechanics is taught about the double slit experiment.  Certainly, it is an excellent example to use with students who are just being introduced to quantum mechanics.  The experimental setup is simple, and it has surprising results that are plainly contradictory to classical physics.

I know of another experimental phenomenon, though, that is similarly surprising and simple in its setup.  This is the phenomenon of “Friedel Oscillations”: the formation of a strange rippling pattern of electrons around a stationary charge.  Strangely, Friedel Oscillations are never mentioned until upper-level graduate courses in quantum mechanics.  What’s worse,  they are given very little conceptual discussion.  While the double slit experiment is used to elucidate the strange nature of quantum objects, Friedel Oscillations are generally derived using the strictest and most opaque formalisms of quantum mechanics.  Finally, when the instructor/textbook reaches the surprising conclusion, there is a brief comment to the effect of “quantum mechanics…  sure is wacky sometimes, isn’t it?”  But the example is never really used to teach us anything about what quantum mechanics means.

Luckily for me, I have an advisor who is a prodigy at distilling the main idea from a complicated derivation.  And when I told him that I didn’t understand Friedel Oscillations, he gave me this explanation, which I think is worth sharing.

To describe what a Friedel Oscillation is, I need to first discuss a more basic idea: screening of electric charge.  Imagine for a moment, a portion of space filled with a large number of positive and negative charges that move around freely.  This is a good description of lots of different things, like salt water (where positive sodium and negative chlorine ions float freely through the water) or a chunk of metal (where negative electrons wander freely around a periodic array of negative nuclei).  I’ll call this material the “charge sea”; it is made of an equal number of mobile positives and negatives.  What happens if you bring a big, heavy, external charge and put it in the middle of the charge sea?  I’ll call this the “impurity” in the charge sea.

The answer is that the impurity pushes around the members of the charge sea.  A positive impurity repels other positives and pulls negatives toward it.  As a result, a “cloud” of opposite charge forms around the impurity.  Something like this:

A charge impurity and its screening atmosphere

The cloud is called the “screening atmosphere”.  If the impurity is positive, then it draws negative charges in to it.  The first negative charges are drawn in very strongly, and form a dense coating around the surface.  Later negative charges are drawn in less strongly because the impurity now has a smaller effective charge (because of the negatives sitting on its surface).  As a result, the screening atmosphere is densest at the surface of the impurity and becomes sparser as you move away — more distant charges are attracted less strongly to the impurity.  Taken as a whole, the screening atmosphere completely compensates for the impurity charge: the impurity gets completely “screened”.  In most situations, the screening atmosphere is simple and well-behaved.  Its density decays exponentially with the distance from the impurity.

Put an impurity in a cold piece of metal, though, and something funny happens.  The positive impurity draws negative electrons to itself, as you would expect, but they don’t just form a nice decaying pattern.  Rather, the electrons form a funny rippling structure circling the impurity.  At the surface of the impurity is a region with high a concentration of negative charge, as you would expect, but it is followed by a region with positive charge, then another negative region, then a positive, and so on in an alternating sequence.  The funny rippling pattern is called a “Friedel Oscillation”.  Here’s a picture of one on a two-dimensional surface:

Friedel Oscillations

The images you’re seeing are visualizations of the electron density surrounding a positive impurity, which sits in the middle of each picture.  Dark areas represent high concentrations of electrons whereas light areas indicate regions where the electron density is low.  In regions where the electron density is very low, the nuclei of the atoms in the middle are left “exposed”, so that the net charge is positive.  Funny to have rings of positive charge surrounding a positive impurity!  Different pictures correspond to different electron energies.  Notice that on the left, where electrons have a small amount of energy, the wavelength of the ripples is fairly long.  As the electron energy increases, the ripples around the impurity have a smaller wavelength.

So what’s going on here?  The answer is that there ‘s a problem with thinking that the electron is just a little negative dot that gets drawn into the positive impurity.  The electron has a size to it which we call its “wavelength”.  The wavelength of the electron is a property of its energy: more energetic electrons have shorter wavelengths.  You can think of the wavelength as the size of the wave that the electron “surfs on” as it moves through space, or you can think that the electron is itself some kind of wave with a particular size.  Either way, it doesn’t make sense to say that an electron sits at an exact point in space.  An electron occupies a region of space, and the size of this region is called its wavelength.  In a cold metal, all the mobile electrons have nearly the same energy, and therefore nearly the same size.

But how does that explain the ripples?  To answer that question, let’s imagine a simplified version of this problem.  Suppose that we have a positively charged surface with a “charge sea” on one side of it.  The positive surface draws a negative screening atmosphere to it, something like this:

A positive surface is screened by negative points

In the land of quantum mechanics, though, we have to remember that each of those negative points has a size to it.  Each negative charge is not sitting at exactly one spot, but is “smeared out” over a region of space.  This is something like screening by negatively charged rods:

A positive surface is screened by a negatively charged rods

Instead of concentrated points of negative charge, the “screeners” are now regions of smeared-out negative charge.  Let’s say that each of them has a length , and we’ll call the distance from the surface x.

Maybe from here you can see why the funny ripples form.  Negative rods are initially pulled strongly toward the surface.  This determines the charge density not just at the surface, but for the next distance after it.  This creates a charge density that is correct for the surface, but is too strong for further distances.  The charge density is “supposed” to decay significantly between and , but it can’t.  Deciding on the density of rods near the surface influences the density for the next distance .  It’s a bit like letting the richest men in America decide the tax code: it may be right for the guys up front, but it’s too damn much for the people that come later!

The fact that every “decision” about how densely to place your charges determines the density for the next distance sets up a cycle of overcompensation and correction.  As a result, you get a rippling density of charge.

For those who think this argument is a bit wishy-washy (and it is), I invite you to solve the “screening by charged rods” problem for yourself.  You can use the so-called Poisson-Boltzmann equation, which dictates how charges distribute themselves around other charges at a given temperature.  Decomposing the equation by Fourier transform shows the existence of an oscillating mode.  Here it is, solved numerically:

A solution to the "screening by charged rods" problem. Negative density means there are more positives than negatives.

You can see the regions of “overcompensation” (high density) followed by regions of “correction” (low density) repeated on and on forever, with gradually decreasing amplitude.  The screening atmosphere that makes up a Friedel Oscillation generally occupies much more space than it would for screening by point charges.  These oscillations go on and on (decaying only like ), rather than dying out exponentially.

The reason that you only see Friedel Oscillations at low temperature is because higher temperatures result in a wide range of electron wavelengths.  If every electron has a different wavelength, then there is no cycle of “collective overcorrection” because every charged rod is a different length.  So Friedel Oscillations, like every other quantum phenomenon, only appear at small temperatures.

In the end, I guess this example is more complicated than the double-slit experiment.  But it also has a straightforward message: the electron has a size, and that size is called the wavelength.  When electrons act in concert, you shouldn’t be surprised to see evidence of their size.

UPDATE:

The argument of this post may be a little misleading.  In the case of “screening by charged rods”, the oscillations in charge density happen with wavelength , where is the length of the rod.  In observed Friedel oscillations, however, the wavelength of density oscillations is .  Specifically, the charge density n is roughly

where is the electron wavelength at the Fermi level, r is the distance from the charge being screened, and D is the dimensionality of the problem (D = 1 for one dimension, = 2 for 2-D, = 3 for 3-D).

The reason for the discrepancy comes from thinking about one wavelength of the electron as a single “lump” of density.  In reality, since the density of the electron n is proportional to the wave amplitude squared (), each wavelength corresponds to two density “lumps”, like this:

A schematic representation of the one wavelength of the electron wavefunction (left) and the corresponding density of charge (right)

So each “screening rod” consists of two density lumps, and as a result the charge density oscillates with every half-wavelength of the screening electrons.