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IN THE early 20th century, Heike Kamerlingh Onnes, a physicist at the University of Leiden, watched the electrical resistance of purified mercury drop abruptly at a temperature close to absolute zero. Onnes had discovered the extraordinary property of superconductivity. Since then, researchers have found as many as 20 elements and thousands of metallic compounds that can conduct electricity without experiencing any resistance.

Superconductivity has fascinated many physicists because it shows that, under certain conditions, some electrons within a metal can behave in unexpected ways. Normally, electrons behave as if they were free particles, seeming not to interact with one another. In superconducting materials, however, the electrons ‘condense’ into a new state in which they behave cooperatively. As physicists broadened their search to new, more exotic materials, they realised that superconductivity came in yet more extraordinary forms. Researchers are now enthusiastically searching for new and increasingly more exotic forms of this strange phenomenon.

The history of the remarkable collection of metallic compounds known as ‘heavy-electron’ systems began during such a search. These materials are making physicists re-examine their theories of how electrons behave in solids – a problem that they thought they had solved more than 30 years ago.

In the late 1970s, while investigating which combinations of elements could become superconducting, Frank Steglich at the University of Darmstadt in West Germany noticed that the somewhat unlikely alloy of the elements cerium, copper and silicon (CeCu₂Si₂ was superconducting. What was remarkable about this discovery was that the superconductivity seemed to arise from an exotic group of hypothetical negative charge carriers with masses closer to those of neutrons or protons than to electrons. This aroused the interest of the physics community; heavy-electron systems were soon firmly on the map.

Since Steglich’s initial work, researchers have discovered several more heavy-electron superconductors. These are all alloys containing an actinide or rare-earth element. In 1983, Hans Ott and his colleagues working at the Swiss Federal Institute of Technology in Zurich with Jim Smith and Zak Fish working at the Los Alamos National Laboratory in New Mexico detected superconductivity in the compound of uranium and beryllium (UBe₁₃). Meanwhile, Greg Stewart and colleagues at Los Alamos made a similar discovery in a uranium-platinum alloy, UPt₃. These compounds have more than lived up to their initial promise. They show a new type of superconductivity. Perhaps more exciting still is the conceptual challenge that these materials are presenting to physicists. What are these exotic charge carriers that seem to behave like electrons but can be several 100 times more massive? These are early days in the search for a theory to explain such bizarre systems. But an approach called the Fermi liquid theory, first proposed in the late 1950s by the Soviet physicist Lev Landau, provides a useful starting conceptual framework for discussing the problem. To appreciate the subtlety of Landau’s ideas, we need to look at where our traditional picture of a metal breaks down and why.

In the early 1900s, just a few years after J. J. Thomson discovered the electron, Paul Drude at the Institute of Physics at Giessen in Germany developed a theory to describe how metals conduct heat and electricity. Drude thought of a metal as consisting of a crystalline lattice of atoms. These atoms release one or more of their negative electrons, which then wander through the lattice of fixed and massive positive ions, conducting both heat and electric charge.

A model for metals

Remarkably, Drude found that he could model many of the electronic properties of typical metals if he assumed that the conducting electrons behaved like a classical ideal gas. Drude’s electron-gas model ignores collisions between electrons and the massive ions of the lattice. It seems rather strange, however, that negatively charged electrons which, in metals, are at least 1000 times as concentrated as a typical gas, should act as independent particles.

Nevertheless, to many scientists at the time, the success of Drude’s model was persuasive if somewhat unaccountable. The model did have some disturbing failures, however. According to the theory, heating the metal or applying a magnetic field should affect the electrons such that they provide a large contribution to the specific heat and magnetic susceptibility. In fact, this is not the case. It was also clear from experiments that electrons could travel much further in the metal than the simple picture of collisions predicted.

With the development of quantum mechanics in the early 1920s, however, it finally seemed possible to interpret the properties of theses materials and to account for the apparent successes of the earlier Drude model.

The crucial concepts contained in the quantum picture were developed by studying the atomic spectra of atoms with many electrons. It was clear that the electrons, bound by the atomic nucleus, occupied discrete energy states. What is more, no more than one electron could exist in any one of the allowed states. The latter feature, which became known as the Pauli exclusion principle, is characteristic of a general class of particles known as Fermi particles, or fermions, of which the electron is an example. This principle is a vital key to our understanding of electrons in metals. By the late 1920s, Arnold Sommerfeld, at the University of Munich, had applied the same ideas to predict the distribution of energies, or momenta, of Drude’s gas of electrons confined within the metal. The results were startling.

At absolute zero, when there is no thermal energy available to the electrons, the momentum states can be filled up with electrons, starting with those of lowest energy. In an ordinary gas, the distribution of energies among the molecules is given by a statistical result, the so-called Maxwell-Boltzmann distribution. Quantum mechanics gives a new distribution known as the Fermi-Dirac distribution (see Figure 1). Physicists have developed a useful theoretical construction from this description. If we imagine the momentum states as making up a grid in three dimensions, then at absolute zero, we can define a hypothetical surface that separates filled and unfilled states. This is called the Fermi surface and is shown in Figure 2.FIG-mg17174202.GIF

Sommerfeld’s work revealed that at the electron densities found in typical metals, the final electron velocities reached at these momenta were enormous. In sodium, for example, the value is around a million metres a second. The maximum momentum is called the Fermi momentum and the corresponding kinetic energy is the Fermi energy. The Fermi energy of these electrons is equivalent to the thermal energy of a body heated to a temperature known as the Fermi temperature which for simple metals is around 40,000 K.

Armed with the Pauli principle, Sommerfeld could resolve many of the problems of Drude’s model. Most notably, he could account for the size and temperature dependence of the electronic contribution to the heat capacity and the magnetic susceptibility. It seemed that the electronic properties of typical metals were determined almost entirely by the behaviour of a tiny fraction of the electron population having energies close to the Fermi energy. The reason is that only these electrons have empty energy states into which they can move and so are free to modify their behaviour. The low heat capacity that had caused such alarm was no longer a problem. What is more, Sommerfeld predicted that the heat capacity would increase linearly with temperature. This, together with a magnetic susceptibility that is independent of temperature well below the Fermi temperature and an electrical conductivity that increases with temperature, have become the normally accepted signatures of metallic behaviour.

While going beyond the Drude model in incorporating the effects of the Pauli principle, Sommerfeld’s theory still ignored interactions among the electrons and between the electrons and the lattice of positive metal ions. Shortly afterwards, Felix Bloch improved this picture by including the effects of the metallic lattice and of the average distribution of charge of the conducting electrons. Bloch considered the lattice as a regular array of positive charges – a static periodic electric potential. He then looked at the effect that this would have on individual conducting electrons, which he described in terms of waves propagating through the structure.

By this time, physicists understood well how electromagnetic waves travel in regular structures such as crystals. As early as 1913, William Bragg, then Cavendish Professor of Physics at the University of Leeds, had used X-rays to look at the structure of crystals. He found that these waves were not scattered by a regular structure unless the wavelength of the incident X-rays was related in a special way to the spacing of the atomic planes of the crystal. It came as less of a surprise to physicists, therefore, when Bloch predicted that the only effect of the metal lattice on conducting electrons would be to change slightly the energy states available to them. Bloch’s electrons were behaving in qualitatively the same way as Sommerfeld’s electrons, but their energies had been modified by the lattice; they seemed to be moving as if they possessed an effective mass that differed from that of the bare electron.

Interpreting the data for heat capacity and magnetic susceptibility within this framework, which became known as the single-electron model, generates values for the effective masses of electrons in these materials. In all metals, it appears that the effective mass, known as the band mass, differs from that of the isolated electron that Thomson first measured in his experiments. These band masses calculated from the heat capacity according to the single-electron model are considerably larger, in some cases an order of magnitude larger than the bare mass. The enormous masses inferred from the heat capacity of heavy-electron systems (or heavy-fermion systems as they are also called), however, cannot be explained by the effects of the static underlying metal lattice. Instead, they are the signature of a very different picture in which the interactions among the electrons play a subtle and crucial role.

The evidence for this state of affairs has, in fact, been building up over several decades. In some transition metal compounds, such as those made of zirconium and zinc (ZrZn₂) and titanium and beryllium (TiBe₂), the effective mass is about five times greater than the calculated band mass. In rare-earth and actinide heavy-electron metals, the dis crepancy can be 10 times as large again. Considering that the starting point for this enhancement is the band mass, we end up with a tentative picture of fermion (or electron-like) ‘particles’ as much as several hundred times as massive as the bare electron. The difference between these heavy particles and bare electrons is so striking that it raises the ques tion of whether they are merely a theoretical construction for interpreting, for example, the heat capacity and magnetic susceptibility at low temperatures. It seemed that the only way to resolve the question was to detect directly the signa ture of fermion particles in a solid – a Fermi surface.

The strange world of quasiparticles

Measuring the Fermi surface is a difficult experiment. It demands very pure materials and extremely sensitive detectors. Nonetheless, in 1986, Gilbert Lonzarich and Louis Taillefer, working at the Cavendish Laboratory in Cambridge, succeeded in growing crystals of uranium-platinum (UPt3) of high purity. Below a certain temperature, the alloy was superconducting. Above this temperature, they observed a Fermi surface as shown in Figure 2. This was the first heavy-electron superconducting material in which a Fermi surface had been measured and the results caused tremendous excitement. At around the same time, Mike Springford and Peter Reinders at the University of Sussex made the first measurements on a heavy-electron alloy, cerium-copper (CeCu6), prepared by a Canadian team at the National Research Council Laboratory in Ottawa. These results confirmed that the strange metallic behaviour of these materials arises from fermion ‘particles’ normally called quasiparticles.

The true nature of these quasiparticles remains an enigma. It appears that in these materials, the positive metallic lattice slows down the motion of the electrons to such an extent that they can scatter strongly from one another. This novel scattering produces a ‘dynamic’ contribution to the electrical potential that the electron feels. It is the principle origin of the enormous mass of the quasiparticles. This factor underlies the breakdown of Bloch’s single-electron picture of a metal. Remarkably, however, we can still think of the electrons as single particles. The ‘particles’ are fermions with the same charge as electrons, but otherwise have dramatically different properties. The Bloch framework for describing electrons in metals is replaced by what is known as the Fermi liquid model.

The Russian theorist, Lev Landau, first developed the Fermi liquid model in the 1950s to explain some of the bizarre properties of liquid helium. He suggested that you could think of a gas of interacting fermions as a bunch of negative quasiparticles and positive ‘holes’. These, he believed, were created as the temperature is raised, rather like the quanta of light energy, or photons, that are produced when a black body is heated. The crucial property of these particles is that within a narrow range of low temperatures, they do not scatter from one another. In this one respect, they are like the ordinary electrons of the Bloch model apart form having a different mass. However, although these quasiparticles behave as discrete entities, their total energy is not simply the sum of their individual energies. From the standpoint of classical physics, this is bizarre behaviour indeed, for it implies that the quasiparticles interact without scattering off each other.

It turns out that these predictions of the Fermi liquid theory are a direct consequence of Pauli’s exclusion principle. What is more, this state is highly precarious. As the tendency for the quasiparticles to scatter from each other grows stronger, they can lose their individuality; the normal Fermi liquid description becomes invalid. The most dramatic example of this breakdown is when the heavy-electron metal suddenly becomes superconducting.

In heavy-electron superconductors, it seems that, at low temperatures around 0.5 to 1 K, quasiparticles start to attract each other. Initially, a few bind together in pairs to form ‘extended molecules’ – analogous in some respects to the pairs of electrons (Cooper pairs) responsible for superconductivity in conventional superconductors. As in traditional superconductivity, as more pairs form, their binding energy increases. This produces a rapid avalanche into the cooperative superconducting state.

Superconductivity in heavy-fermion materials is unusual not only because massive quasiparticles, rather than electrons, form the Cooper pair states, but also because the internal electronic structure of these molecules are of a novel kind.

Of the heavy-fermion superconductors discovered to date, uranium-platinum (UPt₃) is the best studied. Physicists have built-up a clear picture of how heavy-fermion metals become superconducting. In 1989, an American-French collaboration involving Norman Phillips at the University of California at Berkeley, Louis Taillefer and Jacques Flouguet at the National Centre for Scientific Research in Grenoble and Jim Smith at Los Alamos National Laboratory, revealed unique features that show two distinct superconducting transitions (see Figure 3). The subtle intricacies of this novel phase of matter are still being unravelled.FIG-mg17174203.GIF

Steglich investigated new alloys because he believed that a whole range of new and exotic condensed states may exist. So far, physicists have studied only a tiny fraction of the possible metallic alloys. With 70 elements in the periodic table, heavy-fermion systems have just opened the window on to a whole new world of physics.

Sarah Law is on the editorial staff of Physics World.