Gordon Conference

I am going to be at the Gordon Research Conference on correlated electrons for the next few days. Should be fun, but blogging about such meetings is generally frowned upon (don't want to discourage people from frank discussions and showing brand new, untried stuff).  There are rules about confidentiality for these meetings.  I'll write more later in the week on other topics.

FeSe on SrTiO3: report of 100 K superconductivity

I'd heard rumors about this for a while.  I presume that the posting of this on the arxiv means that some form of this paper is in submission out there to a suitably glossy, high impact journal that requires reference citations in its abstracts.  Background:  Bulk FeSe superconducts below around 8 K at ambient pressure (see here).  Under pressure, that transition can be squeezed up beyond 35 K (see here).  The mechanism for superconductivity in this material is up for debate, as far as I know (please feel free to add a reference or two in the comments). 

These investigators have a very fancy ultrahigh vacuum system, in which they are able to grow single layer FeSe on top of SrTiO₃ (with the substrate doped with niobium in this case).  This material is not stable in air, and apparently doesn't do terribly well even when coated with some protective layer.  However, these folks have a multi-probe scanning tunneling microscope system in their chamber, along with a cold stage, so that they can perform electrical measurements in situ without ever exposing their single layer to air.  They find that the electrical resistance measured in their four-point-probe configuration drops to zero below around 100 K (as high as 109 K, depending on the sample).  One subtle point that clearly worried them:  SrTiO₃ is know to have a structural phase transition (the onset of ferroelasticity - see here) at around 105 K, so they wanted to be sure that what they saw wasn't somehow an artifact of that substrate effect.  (Makes me wonder what happens to superconductivity in the FeSe depending on the ferroelastic domain orientation underneath it.)  For the lay audience:  liquid nitrogen boils at ambient pressure at 77 K.  This would be the first iron-based superconductor to cross that threshold, a domain previously limited to the copper oxides.   Remember, if the bulk transition is at 8 K and the single layer case exceeds 100 K, it doesn't seem crazy to hope for some related system with an additional factor of three or four that takes us beyond room temperature.

Important caveats:  Right now, they have resistance measurements and tunneling spectroscopy measurements.  Because of the need for in situ measurement they don't have Meissner data.  It's also important to realize that the restrictions here (not air stable; only happens in single layer material when ultraclean) are not small.  At the same time, this is potentially very exciting, and hopefully it holds up well and can be the foundation for more exciting materials.

750th post - blog demographics

This is the 750th post since this blog's inception.  Fun facts gleaned from google analytics:

1) Unsurprisingly, the US leads in blog hits over that time, with 270,648.  In second place, the UK with 26,698.

2) According to google's tracking, over the last nine years there have been hits from every country in North, Central, and South America, as well as Europe.  In Asia, the only countries with zero hits are Turkmenistan and New Guinea.  In Africa, I'm missing about a dozen, basically the sub-Saharan region plus Somalia. 

3) In the US, the state with the least hits is South Dakota (84 visits over nine years), narrowly edging out Wyoming (88) and Alaska (91).   The states with the most hits are Texas, California, New York, and Massachusetts.

4) Most common browser, by a wide margin, is Firefox, followed by Chrome.  I like the idea that someone has read the blog on a PlayStation 3, and someone else on a PlayStation Portable.  Disappointed (and showing my age by that fact) that no one used lynx or emacs.  

5) Most-viewed post of all time was the meme contest.  Most-viewed physics posts were these on plasmons and polarons. 

Thank you all for reading!

"Seeing" chemical bonds with sub-molecular resolution

Chemists (and physicists) often draw molecular bonds as little lines connecting atoms, but actually imaging the bonds themselves is very hard.  With the advent of the scanning tunneling microscope, it's become almost commonplace to be able to image the position of atoms.  STM images the ability of electrons to enter or leave a conducting surface, and since an atom on the surface carry electrons within itself, the presence of an atom on the surface strongly modulates the STM signal.  This doesn't show anything direct about bonding between atoms, however.

Wilson Ho's group at UC Irvine has published another gem.  The paper is here (unfortunately behind the Science paywall), and the news release is here.  The new STM-based imaging technique, "itProbe", is based in inelastic tunneling, which I've described before.  (One advantage in being an ancient blogger - I can now refer back to my old stuff, with google helping me remember what I wrote.)  The Ho group deliberately attaches a CO molecule to their STM tip.  The CO molecule has a couple of very sharp vibrational (and "hindered translational") modes at low energies that can be seen electrically through inelastic electron tunneling spectroscopy (IETS) - basically sharp features in (the second derivative of) the tunneling current-voltage curve.  In the itProbe technique, the experimenters map out spatially what happens to those modes.  The idea is, as the CO molecule interacts with the sample close by, the precise energies of those vibrational modes shift - the environment of the CO molecule tweaks the effective spring constant for the CO's motion.  Imaging in this way, they find that maps of the inelastic signal seem to show the bonds between the atoms in an underlying molecule, rather than the atom positions themselves.  I admit I don't understand the precise mechanism here, but the images are eye-popping.  A similar idea, involving atomic force microscopy with CO attached to an AFM tip, was demonstrated before (here and here, for example).  In those experiments, the investigators looked at how interactions between the CO on the tip and the sample affected the mechanical properties of the tip as a whole.   

This is an example of a tour de force experiment that can be accomplished by long, sustained effort - the Ho group has been refining their IETS measurements for nearly two decades, and it's really paid off.  Hopefully these kinds of efforts will not become even less common as research funding seems to be focused increasingly on short time horizons and rapid changes in fashion.

Bad physics as a marker for tracking text recycling

A colleague of mine was depressed to find, in a reasonably high impact journal, a statement that magnetic nanoparticles obey Coulomb's law, and thus can be manipulated by external magnetic fields.  As far as physics goes, this is just wrong.  Coulomb's law is the mathematical relationship that says that the force between two charges is proportional to the product of their charges and inversely proportional to the distance between them.  This has nothing to do with magnetic nanoparticles. 

I was curious - where did this weird, incorrect statement come from?  I turned to google to find out.  The earliest result I can find is from this paper by Pankhurst, Connolly, Jones, and Dobson.  The paper seems quite good, and the (strange to me) Coulomb's Law language appears to be some shorthand for a physically sound description of the interactions of magnetic materials with magnetic fields.  The Pankhurst paper includes the following sentence:  "Second, the nanoparticles are magnetic, which means that they obey Coulomb’s law, and can be manipulated by an external magnetic field gradient." This is part of a paragraph that lists three virtues of magnetic nanoparticles for biological applications.  


For fun, try copy/pasting that sentence into google.  Look at how many times that sentence (indeed, that whole introductory paragraph with very minimal changes) shows up nearly verbatim in other publications.  At the risk of saying something actionable, this is plagiarism.   This tends to happen in obscure proceedings, edited book chapters, etc., rather than high impact literature.  The proliferation of shady publication houses and vanity press journals only aggravates this situation.  Very depressing.

My views on teaching "nano"

Blatant self-promotion time:  I was grateful for the invitation to write an editorial about teaching "nano" for Nature Nanotechnology.  The full text is available for free at the above link, and comments and feedback are invited below.  (As a blog reader, you get the added bonus of reading the analogy I made that was cut due to space constraints.  When I advise becoming an expert in a traditional discipline first before tackling an interdisciplinary field, I had written:  "To make a food analogy, it would be very difficult to become an expert at Korean/Mexican fusion cuisine if you did not first know Korean and/or Mexican cooking at a high level.") 

What is chemical potential?

I've been meaning to do a post on this for a long time.  Five years ago (!) I wrote a post about the meaning of temperature, where I tried to go from the intuitive understanding given by common experience (temperature has something to do with energy content, and that energy flows from hot things to cold things) to a deeper view (that flow of energy comes from the tendency of the universe to take on macroscopic configurations that correspond to the most common ways of arranging microscopic degrees of freedom - the 2nd law of thermodynamics, basically).  I wasn't very satisfied with how the post turned out, but c'est la vie.

Chemical potential is a similar idea, but with added complications - while touch gives us an intuition for relative temperatures, we have no corresponding sense for chemical potential; and the rigorous definition of chemical potential is more complicated.  (For another take on this, see this article, available in full text via google from a variety of sources.)

Let's reason by analogy with temperature.  Energy tends to flow from a system at high temperature to a system at low temperature; when systems with identical temperatures are brought into contact so that they may exchange energy (e.g., by thermal conduction), there is no net flow of energy.  Now suppose systems are able to exchange particles as well as energy.  If two systems are at the same temperature, then particles will tend to flow from the system of higher chemical potential (one of the several parameters denoted by the symbol \(\mu\)) to that of lower chemical potential.  If two systems have identical chemical potentials for a particular kind of particle, there will be no net flow of particles.  In general, particles tend to flow from regions of high \(\mu/T\) to regions of low \(\mu/T\).  The classic example of this is the case of a closed bottle of perfume in a room full of (non-perfumed) air.  The perfume molecules have a high \(\mu\) in the bottle relative to the rest of the room.  When the bottle is opened, perfume molecules will tend to diffuse out of the bottle, simultaneously lowering their \(\mu)\) in the bottle and increasing their \(\mu\) in the room.  This will continue until the chemical potentials equalize.  From the point of view of entropy, there are clearly very many more arrangements of molecules with them roughly spread throughout the room+bottle than the number of arrangements with the molecules happening to occupy just the bottle.  Hence, the universe tends toward the macroscopic configuration corresponding to the most microscopic configurations.  Bottom line:  equilibrium between two systems that can exchange particles requires equal temperatures and equal chemical potentials. 

Where this also gets tricky is that thermodynamics tells us that \(\mu\) also corresponds to the energy per particle required to add (or remove) one particle from the system at constant temperature and pressure (!).  This identity is not at all obvious from the above description, but it's nevertheless true.  This latter way of thinking about chemical potential means that when particles can couple to some "real" potential (gravitational, electrical), it is possible to tune their total \(\mu\).  The connection to the entropic picture is the idea that particles will tend to "fall downhill" (there are usually fewer configurations of the combined system that have some particles "stacked up" in a region of high potential energy with others in a region of low potential energy, than the situation when the energy gets spread around among all the particles). 

Prize season again - updated w/ Kavli winners

Once again the Breakthrough Prize and New Horizons Prize in fundamental physics are seeking nominations.  See here.  I have very mixed feelings about these prizes, given how the high energy theory components seem increasingly disconnected from experiment (and consider that a feature rather than a bug).

On a related note, the Kavli Prizes are being awarded this Thursday.  Past nanowinners are Millie Dresselhaus (2012), Don Eigler (love his current affiliation) and Nadrian Seeman (2010), and Louis Brus and Sumio Iijima (2008).   Not exactly a bunch of underachievers.  Place your bets.  Whitesides?  Alivisatos and Bawendi?

Update:  Thomas Ebbeson (extraordinary transmission through deep sub-wavelength apertures, thanks to plasmons), Stefan Hell (stimulated emission depletion microscopy, for deep subwavelength fluorescence microscopy resolution), and John Pendry (perfect lenses and cloaking).  Congratulations all around - all richly deserved.  I do think that the Kavli folks are in a sweet spot for nano prizes, as there is a good-sized pool of outstanding people that has built up, few of whom have been honored already by the Nobel.  This is a bit like the early days of the Nobel prize, though hopefully with much less political infighting (see this book if you really want to be disillusioned about the Nobel process in the early years).

Workshop on structural and electronic instabilities in oxide nanostructures

I've spent the last two days at a fun "Physics at the Falls" workshop at the University of Buffalo.  It's been cool learning about the impressive variety of physics at work in these systems.  A few takeaways:

• With enough stainless steel and high tech equipment you can grow (and in situ characterize with everything from electron diffraction to photoemission, angle-resolved and otherwise) just about anything these days!
• There's a lot of pretty work getting done growing epitaxial complex oxides down to the single unit cell level, and a lot of accompanying extremely high resolution transmission electron microscopy.
• Untangling thermal effects from optical effects in nonequilibrium experiments can be tricky.  Interesting to see that lower energy photons can be more efficient at kicking systems from one phase to another than photons much more energetic than any energy gap.
• There does seem to be some convergence on understanding LAO/STO oxide heterojunctions.
• We still don't understand superconductivity in strontium titanate, even though it's been known for decades.
• Orbitals really matter, when you are dealing with relatively localized electrons.
• Niagara Falls is very impressive!

Slow blogging + interesting links

The end of our academic year + travel + some major writing has cramped my blogging of late.  Things should pick back up to a more regular pace in a couple of weeks.  In the meantime, here are some links that caught my eye lately:

• On Sir Harold Kroto's website, here are some interesting lectures by Richard Feynman.  It's absolutely worth browsing around the rest of the site, too - lots of cool videos.
• This preprint by Sean Hartnoll looks very interesting.  There are materials out there that act like metals (in the sense of having lots of low energy excitations available, and an electrical resistivity that falls with decreasing temperature), but the electrons interact so strongly and in such a complex way that it no longer makes sense to think about "quasiparticles" that act basically like ordinary electrons.  The challenge is, if the quasiparticle picture (which works spectacularly well for materials like gold, copper, aluminum, doped semiconductors) fails, what's the right way to treat these systems?  This paper tries to look at what features would have to be there in such a system.
• This video is cute.  The material used in this LED has a bandgap that apparently increases a fair bit upon cooling.  As a result, the light emitted from the diode shifts toward the blue when the device is dipped in liquid nitrogen, and comes back toward the red when it's warmed.
• We still really don't understand triboelectricity, the "static electricity" you see when you rub a balloon against your hair or rub a glass rod with rabbit fur.  News story here.  It's amazing to me that we still don't know how this kind of charge transfer works, given that it was discovered thousands of years ago.  (As Pauli said, "God made the bulk; surfaces were the work of the devil.")

What are the Kramers-Kronig relations, physically?

Let me pose a puzzle.  Suppose you are in a completely dark room.  You know that at some point in the future, someone will turn on a light in that room for a few minutes, and then turn it off later.  Being a mathematically sophisticated person, it occurs to you that you could think about the time dependence of the electric field in the room.  It's zero for a while, oscillating (b/c that's what happens when there is light there) for a few minutes, and then zero again.  Being clever, you think about Fourier transforming that time dependence, and thinking about all the frequencies in there - the fact that the room right now is dark is actually because of the amazing cancellation of a whole bunch of frequency components!  Therefore, you should be able to put on glasses that are frequency-filtering, block out some of those components, and suddenly be able to see in a dark room!  Except that totally doesn't work, even in a completely classical world without photons.  Why not? 

Think about a material placed in a time-varying (say, harmonically varying, because that's what physicists like) electric field.  The material responds in some way - electrons rearrange themselves within the material in response to that electric field; if the field is slow enough, atoms or groups of atoms can even shift their positions.  The result is a polarization density (electric dipole moment per unit volume) \(\mathbf{P} \equiv \chi_{e}\mathbf{E}\).  Here \( \chi_{e}\) is the electric susceptibility (generally a tensor, meaning that \(\mathbf{P}\) and \(\mathbf{E}\) don't have to point in the same direction).  The dielectric function of a material is defined \(\epsilon \equiv \epsilon_{0}(1 + \chi_{e})\).  In general, the response of the material depends on the frequency \(\omega\) of the electric field, and it can be out of phase with the external electric field.  This is described in mathematical shorthand by considering \(\epsilon(\omega)\) to be complex, having real and imaginary components.

The Kramers-Kronig relations are fairly intimidating looking integral expressions that describe relationships that have to be obeyed between the real and imaginary components of \(\epsilon(\omega)\).  These relationships come from the fact that \(\mathbf{P}\) now can only depend on \(\mathbf{E}\) in the past, up until now.  This restriction of causality, plus the properties of Fourier transforms, are what leads to the K-K integrals.  The wikipedia page about this actually has a very nice description here.   So, while the math is not something that most people would think of as obvious, the basic idea (electromagnetic fields influence materials in a causal way, and that places constraints on how materials can respond as a function of frequency) is not too surprising.

National Nano Infrastructure Network - feedback requested

I wrote before about the saga of the NNIN and how painful the outcome was this year - no awards made, after thousands of person-hours invested on the writing and reviewing of proposals.  Well, NSF is requesting input, in part because they want guidance on structuring the new solicitation to come out this autumn.  So, please give your input if you're in the US and think this kind of support for shared infrastructure is valuable.  By the way, if it seems like NSF had already gone through an exercise like this before the last solicitation (the one where they didn't fund anyone), you're right - they even had a two-day workshop and produced a report.

Recurring themes in (condensed matter/nano) physics: Fermi's Golden Rule

Very often in condensed matter (or atomic) physics we are interested in trying to calculate the rate of some quantum process - this could be the absorption of photons by an isolated atom or a solid, for example.  In (advanced) undergraduate quantum mechanics, we can apply time-dependent perturbation theory to do such a calculation.  Typically you assume that the system starts in some initial state \( |i\rangle \), is subjected to some perturbation \(V\) that turns on at time \(t = 0\), and ends up in final state \( |f\rangle \).  If \(V\) has a harmonic time dependence with some (angular) frequency \(\omega\), then you can do a nice bit of math that calculates the rate at which this process happens.  You discover that at long times the only allowed transitions are the ones where the energies of the initial and final states differ by \(\hbar \omega\), and that the rate of that process is \( (2\pi/\hbar) |\langle i |V| f\rangle|^{2} \rho \), where \(\rho\) is the number of states per unit energy per unit volume that satisfy the energy constraint.

This result, associated with Enrico Fermi, shows up over and over, with some common motifs in condensed matter and nanoscale physics, at least in spirit (that is, sometimes people apply heuristically even though the perturbation may not be harmonic, for example).  First, the \( |\langle i |V| f\rangle|^{2} \) term is what gives us selection rules.  If you think about optical transitions in atoms, this is why you get electric dipole transitions from the 2\(p\) state of hydrogen to the 1\(s\) state, rather than from the 2\(s\) state; in the latter, this quantity is zero.  In crystalline solids, if the initial and final states are Bloch waves, it's the periodicity of the lattice that makes this quantity zero unless (crystal) momentum is conserved.  This is the root of the idea that processes ordinarily forbidden in macroscopic crystals can sometimes take place in nanocrystals or at surfaces.

Similarly, meso- and nanoscale systems can greatly constrain \( \rho \).  One reason that you can get very long mean free paths for charge carriers in semiconductor nanowires, carbon nanotubes, graphene, at the edges of quantum Hall systems, etc., is that the density of states available into which carriers can scatter is very restricted.  Similarly, enhancing \(\rho\) can pay dividends - this is the source of the Purcell effect, where radiative transition rates can be greatly enhanced by increasing the photon density of states, and is part of the reason for enhanced rates of optical processes near plasmonic nanostructures.

updated - Physics education and real world perspective

I was absolutely horrified to read about this story, about how more than 200 girls were kidnapped in Nigeria by a radical Islamic group because they had the temerity to show up for a physics test (emblematic of getting a Western education).  I fervently hope that there is enough public outrage about this to get some positive action there, though it's hard to be optimistic.

Stories like this should remind many of us how petty our concerns (departmental rankings; referee comments; grant reviews; tenure decisions; grad school admissions decisions) really are, even if they seem stressful and important.  It is terrible that in the 21st century there are still large segments of the world where modern learning is expressly forbidden, on pain of death.

Update:  Uggh.  It would appear that the abducted girls have been forcibly married off.  There are so many things wrong here that it's hard to know where to begin.

Informal survey: How important are departmental rankings?

Coincident with the annual US graduate school admission season, I've had a few conversations in recent days where the topic of departmental rankings has come up.  I've written about this general topic before (here and here, for example).  I want to perform a non-serious (in the sense that it's a self-selected survey population that may well be atypical) survey here of those applying to grad school, currently in grad school, or recently (say within the last 4 years) completing grad school:  How important were official rankings (e.g., US News; NRC) of graduate programs in your grad school application process (where you decided to apply) and in your eventual decision?  

Recurring themes in (condensed matter/nano) physics: spatial periodicity

A defining characteristic of crystalline solids is that their constituent atoms are arranged in a spatially periodic way.  In fancy lingo, the atomic configuration breaks continuous translational and rotational invariance (that is, it picks out certain positions and orientations in space from an infinite variety of possible choices), but preserves discrete translational invariance (and other possible symmetries).  

The introduction of a characteristic spatial length scale, or equivalently a spatial frequency, is a big deal, because when other spatial length scales in the physical system coincide with that one, there can be big consequences.  For example, when the wavelength of x-rays or electrons or neutrons is some integer harmonic of the (projected) lattice spacing, then waves scattered from subsequent (or every second or every third, etc.) plane of atoms will interfere constructively - this is called the Bragg condition, is what gives diffraction patterns that have proven so useful in characterizing material structures.  Another way to think about this:  The spatial periodicity of the lattice is what forces the momentum of scattered x-rays (or electrons or neutrons) to change only by specified amounts.

It gets better.  When the wavelength of electrons bound in a crystalline solid corresponds to some integer multiple of the lattice spacing, this implies that the electrons strongly "feel" any interaction with the lattice atoms - in the nearly-free-electron picture, this matching of spatial frequencies is what opens up band gaps at particular wavevectors (and hence energies).  Similar physics happens with lattice vibrations.  Similar physics happens when we consider electromagnetic waves in spatially periodic dielectrics.  Similar physics happens when looking at electrons in a "superlattice" made by layering different semiconductors or a periodic modulation of surface relief.

One other important point.  The idea of a true spatial periodicity really only applies to infinitely large periodic systems.  If discrete translational invariance is broken (by a defect, or an interface), then some of the rules "enforced" by the periodicity can be evaded.  For example, momentum changes forbidden for elastic scattering in a perfect infinite crystal can take place at some rate at interfaces or in defective crystals.  Similarly, the optical selection rules that must be rigidly applied in perfect crystals can be bent a bit in nanocrystals, where lattice periodicity is not infinite.

Commensurate spatial periodicities between wave-like entities and lattices are responsible for electronic and optical bandgaps, phonon dispersion relations, x-ray/electron/neutron crystallography, (crystal) momentum conservation and its violation in defective and nanoscale structures, and optical selection rules and their violations in crystalline solids.  Rather far reaching consequences!

End of an era.

As long as we're talking about the (alleged) end of science, look at this picture (courtesy of Don Monroe).  This is demolition work being done in Murray Hill, NJ, as Alcatel-Lucent takes down a big hunk of Building 1 of Bell Labs. 

This building and others at the site were the setting for some of the most important industrial research of the 20th century.  (Before people ask, the particular lab where the transistor was first made is not being torn down here.)  I've written before about the near-demise of long-term basic research in the industrial setting in the US.  While Bell Labs still exists, this, like the demise of the Holmdel site, are painful marks of the end of an era.

John Horgan: Same old, same old.

John Horgan writes about science for National Geographic.  You may remember him from his book, The End of Science.   His thesis, 17 years ago, was that science is basically done - there just aren't going to be too many more profound discoveries, particularly in physics, because we've figured it all out and the rest is just details.  Well, I'll give him this for consistency:  He's still flogging this dead horse 17 years later, as seen in his recent column.  I disagree with his point of view.  Even if you limit yourself to physics, there are plenty of discoveries left to be made for a long time to come - things only look bleak if (a) you're only a reductionist; and (b) you limit your interest in physics to a narrow range of topics.  In other words, possibly looking for supersymmetric partners at the LHC might not be a great bet, but that doesn't mean that all of science is over.

A video interview for an online nano course

Two of my colleagues (Dan Mittleman and Vicki Colvin) put together a Coursera class this past year, "Nanotechnology:  The Basics", and as part of that they interviewed several Rice faculty about different bits and pieces.   They spoke to me about nanoelectronics, but the conversation ended up ranging into a discussion of hype in science and the importance of communicating to a general audience.  The video is now up online here, and the hype/science presentation discussion starts at around 18:38. 

Recurring themes in (condensed matter/nano) physics: hybridization

Suppose I have two identical systems, such as two copies of a mass attached to a spring (anchored to an immovable wall).  Each system by itself has some characteristic response, like a frequency of motion, and those responses are identical because the independent systems are identical.  Now consider coupling the two systems together, such as linking the two masses by another (weak) spring, and ask what the total coupled system response looks like.  With classical oscillators like our example, we would say that we find the "new normal modes" of the coupled system - instead of writing separate equations to describe Newton's laws for each mass separately, we can do some kind of change of variables and consider new coordinates that combine the motions of the two masses.  When we do this, we end up again with two characteristic frequencies (basically two effectively independent oscillators), but now the frequencies differ a bit, one being higher and one being lower than the original independent oscillator frequency.  You can generalize this to \(N\) oscillators and find \(N\) new normal modes with a band of frequencies, with the bandwidth determined by the strength of the couplings.

This coupling+splitting the modes crops up again and again.  You can use this to make an electronic band pass filter by capacitively coupling together a bunch of \(LC\) circuits.  You can make a mechanical band pass filter by elastically coupling together a bunch of mechanical oscillators.  You can make a miniband of electronic states by coupling together a bunch of quantum wells that each individually would have identical bound states.  You can make a band of electronic states by coupling together the single-particle atomic states of a periodic array of atoms.  The stronger the coupling, the wider the resulting bandwidth.  This basic idea (taking noninteracting systems, coupling them, and re-solving for new effectively noninteracting modes) is so powerful that when it doesn't work sometimes, it can be jarring.