YOU probably take it for granted, and why shouldn’t you? Electrical conductivity, the ability of electrons to move virtually unimpeded through the dense thicket of atoms that comprise a chunk of metal, is surely one of the most familiar and useful things in your life. Discovered by Michael Faraday, who presciently declared that governments would one day tax it, it has now become the basis of almost all modern technology. Life without electricity flowing through metals is almost unimaginable now.
Well, don’t worry – electrical conductivity isn’t going away. But a growing number of physicists are realising that their understanding of conductivity is on shaky ground, to say the least. “The conventional theory of metals is in crisis,” says Philip Phillips, a professor of physics at the University of Illinois at Urbana-Champaign. Given how long we have thought we understood what makes metals metallic – it was an early triumph of quantum mechanics – this might seem silly. But Phillips isn’t joking. “Things are really strange,” he says. “We’ve never had a time like this before.”
This crisis centres on a striking mismatch between theoretical predictions about the properties of conductors and the results of experiments designed to confirm those theories. So far the only way to resolve the mismatch is to invoke the existence of an entirely new state of matter, which would mean the existence of a new kind of metal. And though plenty of people have been looking, no one has yet found either the new state of matter or the new kind of metal.
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Before 1928, the fact that a hunk of copper or iron conducted electricity at all was a deep mystery. The atoms in a metal are arranged in a stunningly regular crystalline lattice, and neighbouring atoms sit close enough together to be able to swap their outer electrons, which thus become liberated to form an “electron sea”. But it was not clear how these electrons could travel any distance without colliding with the remaining densely packed atoms.
In 1928, the Swiss physicist Felix Bloch found the answer. According to quantum mechanics, electrons can behave like waves. He showed that if the electrons’ wavelength matched the spacing of the atoms, they could surf through the lattice without any trouble.
Bloch considered perfect crystals, but in the real world crystals are pocked with defects. When the crystal forms, the atoms often fail to stack into a perfect lattice, rather like a display of oranges built by a careless grocer. Or the crystal might contain impurities, like lemons among the oranges. Nevertheless, physicists were confident that the electrons could adjust their wavelengths, and this would explain why defects and impurities don’t destroy the conductivity of metals.
In 1979, the subject of defects and conductivity came under scrutiny again, this time from Philip Anderson, perhaps the greatest condensed matter physicist of his generation – and hot off winning his 1977 Nobel prize – along with Elihu Abrahams, T. V. Ramakrishnan and Don Licciardello. The “gang of four”, as they were called, looked at two-dimensional crystals, in which the electrons are essentially confined to move in just two dimensions, and considered what happens as they are cooled to absolute zero (from now on assume that we are talking about materials at absolute zero unless stated otherwise). Anderson calculated that even an extremely small number of defects would destroy a metal’s conductivity.
That is because in two dimensions an electron wanders about the lattice like a drunk around a lamp post. The effect, now known as Anderson localisation, shows that eventually the drunk will stumble back to the lamp post. So an electron scattering off a defect will eventually find its way back to the defect: it becomes localised, or stuck in the lattice (see Diagram). Unless the metal is a perfect crystal, at absolute zero it should be an insulator in two dimensions. In other words, there are no two-dimensional metals.
Anderson’s result provided a way to reduce things down to a place where we could test our fundamental theory of metals. It was based on mathematics so clear and elegant that it seemed it had to be right. However, when experimenters finally had the technology to check it they were in for a surprise. Using the techniques developed to manufacture semiconductors, physicists learned to experiment on very thin films. In 1994 Sergey Kravchenko and his collaborators at Northeastern University in Boston, Massachusetts, found that contrary to the theory, a thin film of silicon appeared to become a metal when cooled almost to absolute zero. Of course, it would be physically impossible for Kravchenko to cool his sample to precisely absolute zero, but he got to within thousandths of a degree, and extrapolating the results clearly indicated that the film would behave as a metal. The experiment has been refined and repeated over the years, and the effect has consistently appeared. Phillips says the results mean that theorists “have some explaining to do”.
As is their wont, theorists have not been slow to concoct theories, but so far little consensus has emerged. Anderson localisation was based on what physicists call perturbation theory. The idea is to start from a theoretical perfect crystal, which you can analyse exactly, and then slowly add disorder to the crystal, either by disrupting the positions of the atoms or by adding impurities. For the Kravchenko experiments, it is also necessary to include the interactions between the electrons.
Critics say that this works fine for a small amount of disorder and electron interactions, but it becomes unwieldy and breaks down as the amount of disorder and electron interaction increases. They suggest that what is needed is an analysis that starts from a highly interacting state. But the perturbative theorists deny there is anything wrong with their techniques, and say that the experimenters’ extrapolations are wrong: once they cool their films close enough to absolute zero they will see that they become insulators. Of course, you can’t disprove this line of argument, since you can never reach absolute zero. “Welcome to hell,” says physicist Gergely Zimanyi at the University of California, Davis.
The crisis in our understanding of conductivity would be bad enough if it were confined to metals. But there is also a problem with superconductors. The theory says that at absolute zero, two-dimensional superconducting films should either remain superconducting or become insulators. But experiments done by Allen Goldman and his colleagues at the University of Minnesota in Minneapolis once again seem to contradict the theory: they indicate that these materials turn into metals.
In a superconductor, electrical charge is carried by electrons loosely bound into what are known as Cooper pairs. Electrons possess a quantum mechanical property called spin, analogous to the spin of a top. An electron possesses a spin of 1/2 (the units are Planck’s constant divided by 2π), and particles like this with “half-integer” spin are called fermions. In contrast, particles with integer spin such as photons, which have spin 1, are called bosons.
Quantum mechanics forbids any two fermions from sharing the same quantum state (which is why the electrons in atoms don’t all simply collapse into a single orbit), but when two electrons form a Cooper pair, adding their spins means they effectively become a boson. And just as the boson nature of photons gives lasers their power, the boson nature of Cooper pairs is the secret behind the power of superconductivity. Where does the power come from? From the fact that bosons like to be in the same state.
The Cooper pairs of electrons all “condense” into the same quantum state, like a gas becoming a liquid. This change creates order from chaos, allowing the pairs to move through the crystal like a well-rehearsed marching band. “They have the same phase,” Phillips says. “When they march out of step, the result is an insulator.”
In fact, the famous Heisenberg uncertainty principle says that those are the only two possibilities for this sea of bosons – insulating or superconducting. In its most familiar form, the uncertainty principle says that you cannot simultaneously measure the exact position and momentum of a particle; the more accurately you know one quantity, the more uncertain the other becomes. But in its most general form, the information trade-off of the uncertainty principle extends to any pair of what physicists call conjugate variables, such as energy and time.
In the case of a sea of particles, the phase and the particle number form a pair of conjugate variables. If the phase of the particles is known perfectly – that is, if the particles are all marching in lockstep as bosons can – then the number of particles is infinitely uncertain. “What that means is if you take snapshots of one region of a superconductor, you’ll see that the particle number in the region you are looking at changes wildly,” Phillips says.
A fluctuating particle number indicates a flow of electrons, just as a constantly changing number of leaves on the surface of a stream would indicate it was flowing. If the number of leaves you count in a section of stream remained fixed, you would conclude that the leaves were languishing on a stationary surface. A wildly fluctuating particle number is what gives a superconductor its zero resistance to the flow of electrons.
But if the particles fall out of lockstep and begin milling around like a marching band on a break, the phase becomes unknown, which means the particle number is known exactly. “The particle number is not fluctuating, which means they are not moving,” Phillips says. So there can be only two states: superconducting or insulating.
So much for the theory. Experiments seem to indicate that – in direct contradiction to the uncertainty principle – superconductivity does indeed disappear at absolute zero and is replaced not by an insulator, but by a metallic state consisting of bosons.
Phillips and others have been struggling to reconcile the existence of these “Bose metals” with the uncertainty principle. As with the silicon film, which was predicted to become an insulator near absolute zero but in fact behaved like a metal, one approach is to simply say that the experiments cannot yet have got close enough to absolute zero. But others argue that the Cooper pairs that produce the superconductivity are embedded in a metal-like sea of electrons. At temperatures clear of absolute zero, they say, the superconducting Cooper pairs mask the material’s metallic character by effectively short-circuiting it. As the material cools and the superconductivity vanishes, as the uncertainty principle requires, the underlying metal is revealed.
Though this describes what is happening pretty well, it does not deal with the difficult question of how a material can be transformed from a metal to a superconductor, a question rather like asking how a material could be both a liquid and a solid.
Phillips’s own explanation is that the bosons condense into a glass-like state. Unlike crystals, glasses are flowing, dynamic structures. The atoms in a glass have no overall order and no unique ground state, and slowly move by nudging each other aside. To explain how this leads to metallic behaviour, Phillips returns to the marching band analogy. Imagine that the band is marching up a very steep hill, and as the musicians tire at different rates, they fall out of step at different rates. “Although the whole band is out of step, there will be local regions of order where groups of musicians still march in step,” Phillips says.
This is a state in which there is local order – pockets of musicians marching in step – but global disorder. It is a state that the conventional models of superconductivity have not considered. The phase (and therefore particle number) of this state is neither entirely unknown or completely certain, so the material is somewhere between a superconductor and an insulator: a metal.
A surprising confirmation of this idea comes from the work of Zimanyi. He was interested in what happens when superconductors are penetrated by a magnetic field. A superconductor usually expels magnetic fields, but if the fields are strong enough, small bundles of magnetic “flux” can punch right through the superconductor. The flux bundles are surrounded by small vortices of normal current. Usually, these vortices have two options: they can either move freely, thereby destroying superconductivity, or become gridlocked together in a triangular lattice known as an Abrikosov vortex lattice. But Zimanyi showed that there is a third possibility: that the vortices slowly slide past one another in what he calls a vortex molasses.
At first glance, Zimanyi’s system seems very different from Phillips’s: one deals with vortices in a superconductor and the other with a sea of bosons. But in 1988, David Nelson, a theorist at Harvard University, showed a surprising connection between vortices and bosons. By using a formulation of quantum mechanics proposed by the late Caltech physicist Richard Feynman, Nelson demonstrated a conceptual link between these systems. He showed there is a simple way to translate results learned from studying vortex lines into statements about bosons. You don’t need any additional mathematics, you just have to transpose the words.
So theorems that Nelson and others derived for vortex lines in superconductors become equally valid statements about bosons, simply by substituting the word “boson” for every occurrence of the term “vortex line” and making a few other simple changes. Theorems about two seemingly very different phenomena – the motion of vortex lines in superconductors and the possible existence of a Bose metal – prove to be intimately related.
Zimanyi’s results on vortex molasses relate directly to Phillips’s work on Bose metals. The vortices fixed in an Abrikosov lattice do not dissipate and hence preserve superconductivity in Phillips’s theory. The freely moving vortices disrupt superconductivity and hence produce an insulating state, and the vortex molasses are Bose metals. Though this connection remains suggestive, experiments carried out by Chris Lobb at the University of Maryland and N. C. Yeh at Caltech also help support the idea.
There’s still a long way to go before the crisis is properly resolved, but Phillips is optimistic; it could prove unexpectedly fruitful, he says. After all, even the smallest crisis can prefigure revolutionary changes. A tiny discrepancy between the observed orbit of the planet Mercury and that computed by Newton was eventually settled by Einstein’s theory of gravity. All we need now is the Einstein of electricity.