What are heavy fermions?

I'm surprised that I haven't written about heavy fermions as a separate post before, so here we go. (It's a break from thinking about science and politics, anyway.)

I've written before about "effective mass" for electronic excitations in solids (wiki page here).  From classical physics, we are used to the idea that inertial mass \(m\) is the ratio between an external force \(\mathbf{F}\) and the acceleration \(\mathbf{a}\) of some object, \(\mathbf{F} = m\mathbf{a}\), which is also the rate of change of momentum, \(d\mathbf{p}/dt\).  Kinetic energy (for a nonrelativistic particle) is \(p^{2}/2m\).  Electrons in crystalline solids "feel" the lattice, so in general their kinetic energy \(\epsilon\) can be a more complicated function of their (crystal) momentum, and we can try do define an effective mass as \(1/m* \equiv d^{2}\epsilon/dp^{2}\).  So, if the kinetic energy is very weakly dependent on \(p\), this corresponds to having a very large effective mass.  TL/DR:  the periodic lattice can strongly alter how an electronic excitation accelerates in the presence of a force from, e.g., an electric field, compared to a free particle.  This isn't too surprising.  

Interestingly, in most semiconductors and metals, \(m*\) for electrons in the conduction band (or holes in the valence band) is not thaaaaat different than the free electron mass \(m_{0}\).  The lightest effective mass I know (leaving aside graphene and other Dirac systems when \(\epsilon\) is approximately linear in \(p\)) is electrons in InSb, about \(0.014 m_{0}\).  Holes tend to be a bit heavier.  Also, \(m*\) in molecular organic semiconductors like pentacene tends to be a bit larger, since hopping from molecule to molecule is comparatively weak.  There are ways to measure effective mass, including cyclotron resonance, electronic transport including Shubnikov-de Haas oscillations, magnetic susceptibility and de Haas/van Alphen oscillations, and specific heat measurements.  The electronic specific heat contribution for a metal is linear in the temperature at low \(T\), and the constant of proportionality includes the density of electronic states at the Fermi energy, which can be written in terms of \(m*\).  I've left out a lot of the complications of real anisotropic materials with complicated band structures, but generally the different measurements give consistent results. 

Therefore, it was a big surprise in 1975 when investigators found a material, CeAl3, in which the heat capacity implied an effective mass tens to hundreds of times larger than \(m_{0}\).  They knew right away that this had something to do with the very localized \(4f\) electrons of the Ce atoms.  Because those electrons are very localized, their energy is almost independent of \(p\), implying a large effective mass.  (Some heavy fermion materials also superconduct at temperatures surprisingly high given their effective masses.)

Heavy fermions, adapted from here.  (a) At high temperatures, the 

conduction  electrons are not well coupled to the unpaired local 4f 

moments.  (b) At low enough temperatures, Kondo scattering

hybridizes the f electrons with the conduction  electrons, boosting 

the carrier density.  (c) The hybridized energy-momentum relation 

is much flatter near the Fermi energy leading to a large effective mass.  

So what's the physics?  I wrote about the Kondo effect here, where "ordinary" conduction electrons scatter in a nontrivial way from local magnetic moments (such as partially filled \(4f\) states), and well below a characteristic temperature \(T_{\mathrm{K}}\), the conduction electrons hybridize with the impurities, screening out the unpaired spin.  In the heavy fermion compounds, instead of impurities, there is a whole crystal lattice of local magnetic moments. At sufficiently low temperatures, thanks to that Kondo scattering process, those otherwise localized electrons hybridize with the conduction electrons, boosting the effective density of charge carriers (see figure) and greatly increasing the effective mass.  See this figure, adapted from excellent lecture notes by Piers Coleman.  

So, two key ingredients for heavy fermions are itinerant conduction electrons and a periodic array of comparatively localized, unpaired electrons that have magnetic moments. It turns out that this combination can also be achieved in moiré lattice materials.  There are no \(f\) electrons here, but the moiré lattice can localize spins.  Apologies for not linking to all the relevant papers, but a couple of key theory results are here, here, and here, and a key experimental result is here.  The tunability of the 2D material-based systems is an excellent feature for digging down into the detailed physics.

Update:  Now some added insight from Prof. Andrew Millis:

Hi Doug:

An addendum to your very nice post on heavy fermions, to draw attention to what I think were important experimental results: Frank Steglich’s 1979  Phys. Rev. Lett. 43, 1892–1896 reporting superconductivity in CeCu2Si2 and Louis Taillefer and Gil Lonzarich’s 1988 determination of the quasiparticle mass and fermi surface in UPt3.

Prior to Steglich’s paper we knew that some rare earth/actinide intermetallics (e.g. CeAl3) had a very enhanced specific heat coefficient at low temperatures and that the entropy implied by  this specific heat was  derived from the magnetic moments of the rare earth ions. But  while it was plausible, there was no direct evidence that this enhanced specific heat was associated with heavy-mass fermions, so the physical relevance of the Kondo lattice concept remained uncertain.

Steglich observed that in CeCu2Si2 the specific heat jump at the superconducting transition (which in BCS theory is basically the same size as the electronic specific heat at Tc) was about as big as the normal state specific heat coefficient, thus showing that the spin entropy had been transmuted into something that could go superconducting. Then (I think in subsequent experiments) Steglich observed that the rest of the superconducting thermodynamics in Cecu2Si2 was also consistent with pairing of heavy mass entities. This, I believe, is what convinced everyone that the spin entropy from the rare earth moments had been converted into heavy mass electrons—in other words, that the lattice Kondo effect was real. 

A few years after this, Louis Taillefer and Gil Lonzarich’s quantum oscillation study of UPt3 (Phys. Rev. Lett. 60, 1570 ) showed indeed that the U-f electrons (which appear as local moments at higher temperatures) were included in the Fermi surface at low T and had heavy masses, providing direct experimental confirmation of the Kondo lattice concept.

Cheers

Andy Millis