What are magnons?

Another in my continuing series trying to explain some condensed matter concepts in comparatively jargon-free language. So far I've talked about electron-like quasiparticles, phonons, and plasmons. Now we consider magnons, also known as "spin waves". A magnon is another collective excitation, like a phonon or a plasmon, that may be described by a wavelength (or equivalently a wavevector) and an accompanying frequency. In phonons, we were interested in the pattern of atomic displacements away from their equilibrium positions, and we thought about this in a balls-and-springs picture of solids. Magnons, as the name suggests, are intimately related to magnetism. In many materials there are magnetic moments associated with (some or all of) the atoms in the material, and you can think of these moments as little arrows. In a material with "ferromagnetic interactions", the system can lower its energy by having the moments tend to align with each other. In a true ferromagnetic state all of the moments spontaneously align - all of the arrows point in the same direction. Flipping one arrow 180 degrees around would cost quite a bit of energy, since that arrow would then be antialigned with its neighbors. On the other hand, it costs much less energy to move one arrow just a little bit out of alignment with its neighbors. A magnon is a collective excitation where the relative alignment between neighboring magnetic moments is spatially described by some wavelength (That is, start at some arrow. Translating over by one magnon wavelength takes you back to an arrow tilted the same way as the initial arrow.).

Now, when you tilt a magnetic moment in a magnetic field, that moment will feel a torque that will cause it to precess. This is completely analogous to a tilted gyroscope precessing when it feels a gravitational torque. So, each little moment participating in the magnon is precessing around, giving a time-dependence to the local moment orientation.

This has been a very classical description. Quantum mechanics enters in a couple of ways when talking about real materials. First, there are quantum mechanical restrictions on what we can say about different components of an electron's magnetic moment at any one time. Second, like phonons, one can think of these magnons a bit like harmonic oscillators - a given magnon mode with angular frequency \omega can only exchange energy in chunks of size \hbar \omega.

What is a phonon?

In hindsight, I suppose that I should have addressed phonons earlier. A phonon is a quantized sound wave - a collective vibrational mode of a solid (or liquid). In a crystalline solid, the idea is that the atoms in the solid are displaced, at any given instant, from their equilibrium positions. For a single phonon, the instantaneous displacements are periodic in space (that is, there is some wavelength, where atoms separated by an integer number of wavelengths are displaced the same amount). The displaced atoms feel restoring forces due to their interactions with neighbors, and will tend to oscillate in time around their equilibrium positions. When the wavelength is much longer than the interparticle separation, the frequency of those oscillations times the wavelength gives the speed of sound for the material - phonons propagate along at the speed of sound. In general, the speed of sound can depend on the direction of propagation as well as the direction of the direction of the displacement. If the displacement is along the direction of propagation, the sound is longitudinal; if the displacement is normal to the direction of propagation, the sound is transverse.

The quantum nature of phonons comes in when one discusses their energy content. In a classical mechanical oscillator, you can dump in as much energy as you want; the energy content is proportional to the square of the amplitude of the oscillation, and that can be varied continuously. In a quantum mechanical oscillator of frequency f, the energy content of that oscillator can only take on discrete values, (n + 1/2)hf, where n is a nonnegative integer. This is a subtle yet hugely important distinction. Mathematically it explains a major contribution to the heat capacity of crystalline solids at low temperatures (and it's very strongly related to the form of blackbody radiation when one is worrying about photons rather than phonons).

Because they have a wavelength and therefore a wavevector (and an effective momentum) as well as an energy, one can think about processes that involve the emission, propagation, and scattering of phonons - they have particle-like attributes in that sense.

(For a layperson discussion, I'm avoiding subtle distinctions like acoustic vs. optical phonons. If you really care, in acoustic phonons all the atoms within a unit cell move together, while for optical phonons different atoms within a single unit cell move by different amounts.)

What is a plasmon?

Continuing my series of posts trying to describe condensed matter topics in relatively non-technical language....

As I've mentioned before, in condensed matter physics, we tend to give particle-like names (that is, ones that end in "-on") to excitations of systems that have well-defined particle-like attributes, like momentum, energy, and angular momentum (such as spin). Plasmons are another example of this, and lately they've become extremely fashionable because it's increasingly clear that they can be technologically useful.

A plasmon is a collective excitation of the electronic "fluid" in a piece of conducting material, like ripples on the surface of a pond are a collective mode of the water molecules of the liquid. The simile here isn't too far off, because like water, the electronic fluid in a metal is pretty close to incompressible. If you push down on the surface of a pond somewhere with a float, the density of the water doesn't change; instead the water elsewhere is displaced, because the water molecules have finite volume and push each other out of the way. The electronic fluid acts similarly, not because of any finite size or even the Coulomb repulsion of the electrons, but mostly because of the Pauli exclusion principle, which tends to keep the electrons out of each others' way.

These electronic ripples can have a well-defined wavelength (which quantum mechanics tells us is related to their momentum). What makes them have a frequency? That is, what makes the plasmon waves wave? When the electrons are displaced, the positive charge left behind exerts an attractive force on the electrons, trying to pull them back to their original positions. This interaction is what makes the plasmons oscillate once they're excited, and these Coulomb interactions are also why plasmons cost energy to excite. These Coulomb interactions with the positive background charge also force plasmons to obey certain boundary conditions at the edges of the host metal. As a result, nanoparticles can have discrete allowed plasmonic modes strongly influenced by particle shape, while larger structures (e.g., thin metal films) can have propagating plasmon modes over a broad range of wavelengths. Typical plasmon frequencies are comparable to the frequencies of visible light (i.e., ~ 10¹⁵ Hz). Plasmons decay (into incoherent electron-hole pair excitations), eventually dissipating their energy as the sloshing electrons scatter instead of oscillating smoothly, and as oscillating electric dipoles (and other multipoles) radiate.

Plasmons have gotten so much attention lately for several reasons. They may offer a way of shuttling information around on computer chips that naturally interfaces with optics. Plasmons are also associated with large local electric fields at metal surfaces, which can be very useful for certain kinds of spectroscopies and things like optical trapping. Finally, in properly designed materials, plasmon properties can be manipulated so that the overall optical response of a conducting system can be tuned, leading to lots of hope and hype about "perfect lenses" and "invisibility cloaks".

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.

What are quasiparticles?

The word quasiparticle is a term of art that condensed matter physics types throw around quite a bit. What does is it really mean? I'll describe one analogy that may be useful, and then give a more rigorous definition. Suppose you had a bin filled up to some height with rubber balls of uniform size. The lowest energy ("ground") state of this would be the one with the balls pretty much forming a close-packed structure, all stacked up. If you took one ball from somewhere and set it on top of the others, that would cost a little bit of energy, because the ball has some mass acted on by gravity and it takes work to lift it up. This one ball popped up above the rest is not exactly a quasiparticle. Notice that it's not really the same as an isolated ball. It's a bit deformed from interactions with the balls underneath it, since it has weight and the balls are all a little squishy. Similarly, if you took a step back and looked really carefully, you'd see that the balls right under that one have all had to rearrange themselves a little. The whole package (popped-up ball + deformations + rearrangement of the positions of the neighboring balls) is analogous to a quasiparticle, since you can't really have some parts without the others. In condensed matter physics, a fancier scientific definition would be: "a low energy excitation of a system, possessing a set of quantum numbers and/or well-defined expectation values of certain operators (position, charge, momentum, angular momentum, energy) often associated with isolated particles."

More postings soon, but looming deadlines may mean a slow-down.

What does it mean for a material to be a "metal"?

Continuing on from my earlier posts about insulators, it's worth thinking about what we mean by a "metallic" state. Colloquially, people have an image of what they think is a metal: a material that is shiny, electrically conducting, and probably relatively ductile and malleable. Let's not discuss the elastic properties at the moment, since their origin is rather subtle. The electrical conduction is what really stands in contrast to insulators, and the shiny surface is a consequence of the electrical conduction at high frequencies (optical, ~ 10¹⁵ Hz). (By the way, for those interested in why some metals have color to them, this site has a pretty nice explanation. The short answer: interband transitions alter the absorption at short wavelengths.)

It's important to understand that, from the condensed matter physicist's perspective, there's a big difference between a substance that is merely electrically conductive and one that is a "real" metal. In a real metal, the electrical resistivity decreases as temperature is decreased. There are conduction mechanisms (e.g., ionic conduction in glasses; hopping conduction in doped organic semiconductors) that become much less effective at lower temperatures - those systems are not metals, just moderately conducting at room temperature. Similarly, lightly doped semiconductors aren't metals either; as T approaches 0 they have no mobile charge carriers. It would be nice to be able to find a ground state property that lets us decide whether something is a metal or an insulator rather than worrying about temperature dependences. Fortunately, there is. As discussed here (a nice pdf that I found while learning more about what Peter had written in the comments to the previous post), when placed between capacitor plates at T = 0, a metal develops only a surface charge, while an insulator develops a bulk dielectric polarization (dipole moment per unit volume) throughout itself.

There are different types of metals. Conventional metals are Landau Fermi liquids. The low energy electronic excitations of Fermi liquids are "quasiparticles" that act very much like non-interacting electrons - they have spin-1/2, charge -e, and have a lifetime much longer than h/k_BT. In bulk Fermi liquids, electronic excitations can have arbitrarily low energies. The spectrum of these excitations is said to be gapless. The hallmark of Fermi liquids is that they have properties that look much like those we find in undergrad statistical mechanics treatments of noninteracting Fermi gases. For example, their heat capacities vary at low temperatures as T, and their resistivities vary at low temperatures as T².

There are other metallic states known variously as bad metals or strange metals. The classic example of a bad metal is the normal state of optimally doped high temperature superconductors. These systems have a metallic ground state, but near T = 0, their resistivities vary linearly in T rather than quadratically. This may not seem like a big deal, but it has major implications. It implies that the low energy electronic excitations of these materials are not well described as quasiparticles; they must somehow involve collective excitations of many correlated electrons, and may not have easily intuitive quantum numbers. That is, they're non-Fermi liquids. Trying to understand these systems and their excitations is a major outstanding challenge in condensed matter physics today. It's hard because it involves understanding excitations of a system of many strongly interacting quantum particles, and also because our intuition has been shaped by our classical ideas about simple quasiparticles. By the way, this idea of excitations that are complicated and lack particle-like quantum numbers has come into vogue in high energy physics in the form of "unparticles".

More about insulators

I've been thinking more about explaining what we mean by "insulators", in light of some of the insightful comments. As I'd said, we can think about three major classes of insulators: band insulators (a large gap due to single-particle effects (more below) exists in the ladder of electronic states above the highest occupied state); Anderson insulators (the highest occupied electronic states are localized in space, rather than extending over large distances; localization happens because of disorder and quantum interference); and Mott insulators (hitherto neglected electron-electron interactions make the energetic cost of moving electrons prohibitively high).

The idea of an energy gap (a big interval in the ladder of states, with the states below the gap filled and the states above the gap empty) turns out to be a unifying concept that can tie all three of these categories together. In the band insulator case, the states are pretty much single-particle states (that is, the energy of each state is dominated by the kinetic energies of single electrons and their interactions with the ions that supply the electrons). In the Anderson insulator case, the gap is really the difference in energy between the highest occupied state and the nearest extended state (called the mobility edge). In the Mott case, the states in question are many-body states that have a major contribution due to electron-electron interactions. The electron-electron interaction cost associated with moving electrons around is again an energy gap (a Mott gap), in the ladder of many-body (rather than single-particle) states.

I could also turn this around and talk in terms of the local vs. extended character of the highest occupied states (as Peter points out). In the ideal (infinite periodic solid) band insulator case, all (single-particle) electronic states are extended, and it's the particular lattice arrangement and electronic population that determines whether the highest occupied state is far from the nearest unoccupied state. In the Anderson case, quantum interference + disorder leads to the highest occupied states looking like standing waves - localized in space. In the Mott case, it's tricky to try to think about many-body states in terms of projections onto single-particle states, but you can do so, and you again find that the highest relevant states are localized (due, it turns out, to interactions). Like Peter, I also have been meaning to spend more time thinking hard about insulators.

Coming soon: a discussion of "metals".

What does it mean for a material to be an "insulator"?

I've been thinking for a while about trying to explain some physics concepts on, well, a slightly more popular level. This is a first pass at this, focusing on electrical insulators. Feedback is invited. I know that this won't be perfect for nonscientists at a first cut.

Very often we care about the electrical properties of materials. Conceptually, we imagine hooking the positive terminal of a battery up to one end of a material, hooking the negative terminal up to the other end, and checking to see if any current is flowing. We broadly lump solids into two groups, those that conduct electricity and those that don't. Materials in the latter category are known as insulators, and it turns out that there are at least three different kinds.

• Band insulators. One useful way of thinking about electrons in solids is to think about the electrons as filling up single-particle states (typically with two electrons per state). This is like what you learn in high school chemistry, where you're taught that there are certain orbitals within atoms that get filled up, two electrons per orbital. Helium has two electrons in the 1s orbital, for example. In solids, there are many, many states, each one with an associated energy cost for being occupied by an electron, and the states are grouped into bands separated by intervals of energy (band gaps) with no states. (Picture a ladder with groups of closely spaced rungs, and each rung has two little divots where marbles (the electrons) can sit.) Now, in clean materials, you can think of some states as corresponding to electrons moving to the left. Some states correspond to electrons moving to the right. In order to get a net flow of electrons when a battery is used to apply a voltage difference across a slab of material, there have to be transitions that, for example, take electrons out of left-moving states and put them into right-moving states, so that more electrons are going one way than the other. For this to happen, there have to be empty states available for the electrons to occupy, and the net energy cost of shifting the electrons around has to be low enough that it's supplied by the battery or by thermal energy. In a band insulator, all of the states in a particular band (usually called the valence band) are filled, and the energetically closest empty states are too far away energetically to be reached. (In the ladder analogy, the next empty rung is waaay far up the ladder.) This is the situation in materials like diamond, quartz, and sapphire.
  
• Anderson insulators. These are materials where disorder is responsible for insulating behavior. In the ladder analogy above, each rung of the ladder corresponded to what we would call an "extended" state. To get a picture of what this means, consider looking at a smooth, grooved surface, like a freshly plowed field, and filling it partially with water. Each furrow would be an extended state, since on a level field water would extend along the furrow from one end of the field to the other. Now, a disordered system in this analogy would look more like a field pockmarked with hills and holes. Water (representing the electrons) would pool in the low spots rather than forming a continuous line from one end of the field to the other. These local low spots are defects, and the puddles of water correspond to localized states. In the real quantum situation things are a bit more complicated. Because of the wavelike nature of electrons, even weak disorder (shallow dips rather than deep holes in the field) can lead to reflections and interference effects that can cause states to be localized on a big enough "field". Systems like this are insulating (at least at low temperatures) because it takes energy to hop electrons from one puddle to another puddle. For small applied voltages, nothing happens (though clearly if one imagines tilting the whole field enough, all the water will run down hill - this would correspond to applying a large electric field.). Examples of this kind of insulating behavior include doped polymer semiconductors.
  
• Mott insulators. Notice that nowhere in the discussion of band or Anderson insulators did I say anything at all about the fact that electrons repel each other. Electron-electron interactions were essentially irrelevant to those two ways of having an insulator. To understand Mott insulators, think about trying to pack ping-pong balls closely in a 2d array. The balls form a triangular lattice. Now the repulsion of the electrons is represented by the fact that you can't force two ping-pong balls to occupy the same site in the 2d lattice. Even though you "should" be able to put two balls (electrons) per site, the repulsion of the electrons prevents you from doing so without comparatively great energetic cost (associated with smashing a ping-pong ball). The result is, for exactly 1 ball (electron) per site ("half-filled band") in this situation dominated by ball-ball interactions ("on-site repulsion"), no balls are able to move in response to an applied push (electric field). To get motion (conduction) in this case, one approach is to remove some of the balls (electrons) to create vacancies in the lattice. This can be done via chemical doping. Examples of Mott insulators are some transition metal oxides like V₂O₃ and the parent compounds of the high temperature superconductors.

Often when speaking of "metals" vs. "insulators", we are interested in the ground state of the material, the state that would describe the material in equilibrium as T approaches 0. Materials with an electrical resistivity that tends toward infinity as T approaches 0 are insulators in this sense.