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

NSF, National Science Board, and the politics of staying quiet

As I mentioned previously, the National Science Board was summarily fired on April 25.  The NSB nominally advises the National Science Foundation.  There have been a number of pieces written about this:

• Going back in time to 2022, this essay is interesting to read, about the history of the NSF and the NSB, and the compromises put in place with the administrative structure.  Short version: Initially there was a real tension between the Director (reporting to the President) and the NSB.  Over time, the NSB was made subordinate to the director (1968).  Senatorial confirmation of board members was waived by the Senate in 2011.  
• Many professional organizations issued statements expressing grave concern about this wholesale dismissal of the board.  This AIP news article has a summary.  The CEO of the APS wrote this, the ACS leadership wrote this, the AAS wrote this, etc.
• The presidents of the National Academy of Sciences, National Academy of Engineering, and National Academy of Medicine issued this joint statement.  That has to set some kind of record for blandness, as it somehow does not even mention that the NSB was fired.  I fully understand that the Academies have a number of federal contracts, as one of their key responsibilities is leveraging their membership to do authoritative studies, with federal agencies usually being the customers.  I have no inside knowledge, but it sure looks like they are trying to walk a line of not raising the administration's ire.  (Surely this raises the question:  If it's never acceptable to say anything that might upset the administration, then how can the objectivity of their reports relating to policy ever be trusted?)
• In contrast to the leadership, a lot of Academy membership has signed an open letter to Congress demanding the reinstatement of the board.
• Scientific American has very good reporting on this, including a no-holding-back statement by my colleague Neal Lane.
• Update:  Here is Dan Garisto's reporting in Science about letters sent by House Democrats and by Senate Democrats demanding action on this.  That article includes a statement by the fired head of the NSB, basically saying they were dismissed for defending the NSF budget from OMB.  I'm glad these letters were sent, but without the R majority signing on, I'm not holding my breath.

Meanwhile, the pace of NSF awards continues to be glacial, even compared to last year.  See this plot from Grant Witness. 

We are 7 months into the fiscal year, and obligated dollars are less than half at this time last year, and more like 27% of those at this time in "normal" year.  It's hard to look at that and not wonder whether someone is aiming for a pocket rescission, regardless of what Congress appropriated.  NSF looks like an outlier here, by the way.  As badly hit as NIH has been, their award curves look much closer to last year.

Other related things worth reading:  

• This is a sobering and interesting article about what the author describes as the anti-science movement.
• When is staying quiet effectively giving tacit support to administration policies?  There is an article in The Nation (not readable without a free signup) talking about what they term "Vichy Science".
• Update:  Essay by Holden Thorp, EIC of Science, including a link to a half-hour interview with Tim Snyder, author of "On Tyranny", about how resistance can work.  

Back to science in my next post.

Energy storage in the internal states of molecules - old and new

A science story first, then a US research ecosystem story later.

When we think about using molecules to store energy, it's usually in the context of food or fuel, so that chemical reactions take place - bonds are broken and remade, and in an exothermic reaction, the products end up with more kinetic energy (center of mass motion, molecular vibrations and rotations) than the initial reactants.  However, there are other ways that molecules can store energy.  I read about a cool example of this last week, but first I want to give tell you an old and very quantum mechanical story that I learned about in grad school when I did very low temperature physics.

Diatomic hydrogen, H2, is the simplest molecule there is, just two electrons and two protons.  Roughly speaking, the \(1s\) orbitals of the H atoms hybridize to form \(\sigma\) bonding and \(\sigma*\) antibonding molecular orbitals.  The lowest electronic state is the two electrons in a spin singlet, \((1/\sqrt{2})(|\uparrow \downarrow\rangle - |\downarrow \uparrow\rangle)\) in the \(\sigma\) molecular orbital.  Remember, the electrons are fermions, so the electronic wavefunction has to be antisymmetric (pick up a minus sign) under exchange of the electrons. The spin singlet is antisymmetric under exchange, the \(\sigma\) orbital is spatially symmetric under exchange, so the full electronic wavefunction (product of the spin and spatial components) is appropriately antisymmetric.  

That's not all there is to it, though, as explained thoroughly here.  The protons (while being made up of quarks and gluons, etc.) are (composite) fermions, so we have to think about the quantum wavefunction that describes them, too.  There are two possibilities.  In the "para" configuration, the proton spins are in a singlet (antisymmetric), meaning that the spatial wavefunction of the protons must be symmetric under exchange.  The spatial state of the bound protons can have some orbital angular momentum \(\mathbf{L}\), and the simplest, lowest energy situation is with quantum numbers \(\ell =0\) and therefore \(m_{\ell} = 0\).  In contrast, in the "ortho" configuration, the proton spins form a triplet state (symmetric under exchange), meaning that the spatial wavefunction must be antisymmetric, \(\ell = 1\).  Approximating the H2 molecule as a rigid barbell-like rotor with some moment of inertia \(I\), then ortho molecule has a rotational energy \(\hbar^2/2I\) larger than the para case.  That works out to about 15 meV of energy per molecule.  So, para-hydrogen is the true ground state.  It turns out that the ortho/para spin isomer energy difference makes liquefying hydrogen a challenge, since the latent heat of vaporization for H2 is only 9.4 meV.  That is, every time an ortho-hydrogen molecule converts to para-hydrogen through some collisional process, it releases enough energy to kick a hydrogen molecule out of the liquid.  I learned about this in my thesis work playing around at ~ 1 mK temperatures - any H2 adsorbed or otherwise stuck in the apparatus could result in detectable long-term heating effect as it slowly converted from ortho to para.  Bottom line:  Energy can be stored in the internal states of molecules.

From Fig. 1 of this paper.

This seems very esoteric, but the idea of storing energy in some internal state of a molecule for later release shows up elsewhere.  Last week, in this article in Science, for example, the authors report a molecule inspired by aspects of DNA that can be put via UV exposure into a distorted form ("Dewar isomer") where it is metastable at room temperature (half-life of 481 days).  It can be induced to pop back into the undistorted isomer by heat, acid exposure, or via a catalyst, and when it does, each molecule releases the stored energy (2.36 eV per molecule!) into vibrations and rotations that heat its surroundings.  The stored energy density in this stuff is about 4% of the releasable energy density of gasoline, which is not too shabby.  The authors propose a system where exposure to sunlight can store energy in the molecules, and this can later be released on demand via catalyst.  They demonstrated that the heat release from enough dissolved molecules can readily boil water.  Very neat stuff.