What are "quantum oscillations"?

For the first time in a couple of decades, I was visiting the Aspen Center for Physics, which is always a fun, intellectually stimulating experience.  (Side note: I sure hope that the rapidly escalating costs of everything in the Aspen area don't make this venue untenable in the future, and that there are growing generous financial structures that can allow this to be accessible for those of limited funding.)  One of the topics of discussion this week was "quantum oscillations" in insulators, and I thought it might be fun to try to explain, on some accessible level, just how weird those observations are.  

Historically, quantum oscillations are observed in metals and (doped) semiconductors, and they have been a great tool for understanding electronic structure in conductive materials, a topic sometimes called "fermiology".   First, I need to talk about Fermi surfaces.

Annoyingly, it's easiest to describe the electronic states in a crystal in terms of "reciprocal space" or \(\mathbf{k}\)-space, where the wave-like electronic states are labeled by some wavevector \(\mathbf{k}\), and have some (crystal) momentum given by \(\hbar \mathbf{k}\).  ( Near the bottom of an energy band, the energy of such a state is typically something like \(E_{0} + (\hbar^2 k^2)/2m^{*}\), where \(m^{*}\) is an effective mass.)

At low temperatures, the electrons settle into their lowest energy states (toward low values of \(\mathbf{k}\)), but they stack up in energy because of the Pauli principle, so that there is some blob (possibly more than one) of filled states in \(\mathbf{k}\)-space, with a boundary called the Fermi surface, surrounded by empty states.  Because the relationship between energy and momentum, \(E(\mathbf{k})\), depends on the atoms in the material and the crystal structure, the Fermi surface can be complicated and have funny shapes, like the one shown in the link.  "Fermiology" is the term for trying to figure out, experimentally, what Fermi surfaces look like.  This matters because knowing which electronic states are the highest occupied affects many properties that you might care about.  The electrons in states right "at" the Fermi surface are the ones that have energetically nearby empty states and thus are the ones that respond to perturbations like electric fields, temperature gradients, etc.

Now turn on a magnetic field.  Classically, free electrons in a magnetic field \(B\) with some velocity perpendicular to the field will tend to move in circles (in the plane perpendicular to the field) called cyclotron orbits, and that orbital motion has a characteristic cyclotron frequency, \(\omega_{c} = eB/m\).  In the quantum problem, free electrons in a magnetic field have allowed energies given by \((n+1/2)\hbar \omega_{c}\).  Since there are zillions of conduction electrons in a typical chunk of conductor, that means that each of these Landau levels holds many electrons.  

An electron with wavevector
\(\mathbf{k}\) in a magnetic 
field \(\mathbf{B}\) will trace
out an orbit (yellow) in
\(\mathbf{k}\)-space.

For electrons in a conducting crystal, the idea of cyclotron motion still works, though the energy of an electronic state involves both the magnetic field and the zero-field band structure.  For an electron with wavevector \(\mathbf{k}\), one can define a velocity \(\mathbf{v}= (1/\hbar) \nabla_{\mathbf{k}}E(\mathbf{k})\) and use that in the Lorentz force law to figure out how \(\mathbf{k}\) varies in time.  It turns out that an electron at the Fermi surface will trace out an orbit in both real space and \(\mathbf{k}\)-space.  (Of course, for this physics to matter, the system has to be sufficiently free of disorder and at sufficiently low temperatures that the electrons are unlikely to scatter as they trace out orbits.)

Now imagine sweeping the magnetic field.  As \(B\) is ramped up, discrete cyclotron energy levels will sweep past the energy of the highest occupied electronic states, the Fermi surface.  That coincidence, when there are a lot of electronic states at the Fermi energy coinciding with a cyclotron level, leads to a change in the number of electronic states available to undergo transitions, like scattering to modify the electrical resistance, or shifting to different spin states because of an external magnetic field.  The result is, quantities like the resistance and the magnetization start to oscillate, periodic in \(1/B\).    (It's a bit more  complicated than that for messy looking Fermi surfaces - oscillations in measured quantities happen when "extremal orbits" like the ones shown in the second figure are just bracketed by contours of cyclotron energy levels.  The period in \(1/B\) is inversely proportional to the area in \(\mathbf{k}\)-space enclosed by the orbit.).  

Fermi surface of Cu.  If a magnetic field
is directed as shown, there are two orbits
(purple) that will contribute oscillations
in resistivity and magnetization.

Bottom line:  in clean conductors at low temperatures and large magnetic fields, it is possible to see oscillations in certain measured quantities that are periodic in \(1/B\), and that period allows us to infer the cross-sectional area of the Fermi surface in \(\mathbf{k}\)-space.  Oscillations of the resistivity are called Shubnikov-De Haas oscillations, and oscillations of magnetization are called De Haas-van Alphen oscillations. 

These quantum oscillations, measured as a function of field at many different field orientations, have allowed us to learn a lot about the Fermi surfaces in many conducting systems.   

Imagine the surprise when De Haas-van Alphen oscillations were found in a material whose bulk is expected to be electrically insulating!  More on this soon.