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Mind the pseudogap

Physicists are still searching for a convincing theory of high-temperature superconductivity. But at least the nature of the puzzle is becoming clearer. Mark Buchanan weighs the odds of a breakthrough in understanding.

We can blame Georg Bednorz and Alex Müller. Fifteen years ago, physicists thought they understood superconductivity — the tendency of some materials to conduct current with zero resistance at temperatures of around 10 to 20 kelvin. But in 1986, working at the IBM Research Laboratory in Zurich, Bednorz and Müller unveiled a new class of ceramic materials that pulled off the trick at much higher temperatures1. More than 100,000 research papers later, and with the record for high-temperature superconductivity now standing at some 150 K, theorists are still baffled.

“We now probably know more experimentally about this class of materials than any other,” says Peter Littlewood, a condensed-matter physicist at the University of Cambridge. “But we still have a fundamental conundrum.”

In the past few years, however, as physicists have found ways to follow what electrons are doing inside high-temperature superconductors with an accuracy once thought impossible, there is growing optimism that this conundrum will be resolved. “There will be a glorious theory of physics at the end of the journey,” predicts Jan Zaanen, a theoretical physicist at Leiden University in the Netherlands. And the key may be an understanding of how these ceramics behave when heated just beyond the temperatures at which they superconduct — in particular a curious phenomenon known as the 'pseudogap'.

The X-factor

Ordinary superconductors are simple metals. But the high-temperature variety discovered by Bednorz and Müller are more complicated. Known informally as cuprates, they are made up of parallel two-dimensional sheets containing copper and oxygen atoms. Sandwiched between these sheets are other kinds of atoms — in the material La2−xSrxCuO 4, for example, these atoms are lanthanum and strontium. Within the two-dimensional sheets, each copper atom has the potential to relinquish one loosely bound electron which can then move from one copper atom to another and carry electricity. What actually happens depends on the value of x .

With no strontium (x=0), the material is a poor conductor and is known as a 'Mott insulator'. Because electrons repel one another quite strongly, the energy needed to force a second electron onto a copper atom that has not already relinquished its loosely bound electron is prohibitive. So with one loosely bound electron on virtually every copper atom, the resulting electronic traffic jam makes the material a poor conductor. This 'undoped' material is also antiferromagnetic — an ordered state with no bulk magnetization. This occurs because each electron carries a spin and acts as a tiny magnet. The forces between these magnets tend to make neighbours point in opposite directions, so the spins point alternately up and down across the material.

Things get more interesting when strontium atoms replace some of the lanthanum atoms. Strontium has a greater affinity for electrons and so attracts the loosely bound electrons from the copper atoms. With some copper sites now vacant, electrons can hop between the copper atoms and so carry electricity. When the material is doped with strontium, and x increases from zero, La2−xSrxCuO4 first turns from an antiferromagnetic insulator into a reasonable conductor. At larger values of x, the material gains the ability to be superconducting when cooled. The transition temperature, Tc, at which superconductivity arises depends on the value of x, reaching a peak at around x=0.15 (see figure, below).

Figure 1
figure 1

DAVID PARKER/SPL

Bridge that gap: the temperatures at which a cuprate superconducts are affected by doping.

All the cuprates show this same basic behaviour, and a good theory of high- Tc superconductivity would explain why. No such theory yet exists, but when it does, it is sure to be very different from the theory of conventional superconductors. This was created in the late 1950s by John Bardeen of the University of Illinois, Leon Cooper of Brown University in Rhode Island and J. Robert Schrieffer of the University of Pennsylvania. Known as the BCS theory, it stands firmly on the shoulders of an earlier theory of 'quantum liquids', devised by Lev Landau of the Institute of Physical Problems in Moscow.

Storm clouds

Inside an ordinary metal, a cloud of electrons moves about within a crystal lattice of atoms and ions. These electrons interact strongly with one another, like the particles in a liquid, and the movement of any one will stir up a local storm in the motions of others. This made theoretical analysis seem nearly impossible. But Landau showed that, within the setting of a quantum liquid, the electron plus the storm would act as a 'quasi-particle' in its own right. So the electronic properties of metals can be understood fairly simply in terms of a collection of quasi-particles.

Bardeen, Cooper and Schrieffer used this picture to unravel the mystery of superconductivity. They realized that in some metals the motion of one quasi-particle might set the lattice of atoms and ions vibrating in such a way as to exert an attractive force on other quasi-particles, even binding the quasi-particles together into pairs. This has far-reaching consequences. On their own, the quasi-particles obey the Pauli exclusion principle — no two can occupy the same quantum state. But when paired together they no longer face this prohibition. And, at low enough temperatures, the pairs all condense into the same quantum state. This condensation is the key to superconductivity. Because the condensate of pairs is a single quantum entity, individual pairs are no longer scattered by localized impurities in the material. They can therefore glide along without losing energy.

Got it taped: superconducting ribbons can carry as much current as copper cables with 100 times the cross-sectional area. Like all superconductors, cuprates can be levitated over a magnet (opposite). Credit: AMERICAN SUPERCONDUCTOR

Unfortunately, this nice picture falls to pieces when faced with the cuprates. Here, too, the charge carriers — which might be electrons, or the 'holes' left behind when electrons are displaced — are generally thought to move around in pairs. But there is a big difference. In the BCS theory, when the lattice vibrations force the pairs together, they form without any internal angular momentum — the charge carriers within each pair do not rotate about one another. In the cuprates, there is angular momentum. In the theorists' jargon, this is 'd-wave', as opposed to 's-wave' superconductivity. In addition, lattice vibrations should not be strong enough to keep the pairs together and give superconductivity much above 30 K — way too low to explain what happens in the cuprates.

Shooting gallery

To probe the properties of cuprates, condensed-matter physicists turned to a technique called angle-resolved photo-emission (ARPES). In this method, researchers shoot high-energy photons at a material, and then measure the energy and momentum of any electrons that get knocked out. These values are nearly identical to that of the stimulating photon, but tiny differences reveal how much energy and momentum the material absorbed. These differences also characterize the material's natural quasi-particles. Given the precise measurements needed, says Andy Millis, a condensed-matter theorist at Rutgers University in New Jersey, “it is amazing that it works”. But work it does — and in the past five years or so, the technique has revealed strange behaviour in the cuprates.

When superconducting, any ordinary superconductor has an 'energy gap'. If you fire photons at the material in an effort to excite quasi-particles, nothing much happens until the input of energy rises above a minimum value known as Eg — this being the energy needed to break apart one of the charge-carrier pairs. Not surprisingly, this energy gap goes away when the temperature rises above Tc and the pairs again split into single particles.

The cuprates also have an energy gap, but things work out rather differently. Below Tc, ARPES experiments reveal that the superconducting gap depends on the direction in which the photon is fired into the material. There is a large gap in some directions and a small gap in others2,3,4. This just reflects the 'd-wave' nature of the pairs and fits in well with Landau's theoretical picture. But things go terribly awry when you heat the materials up above Tc — the energy gap does not go away5,6.

This anomaly is the pseudogap, and it makes a shambles of the conventional BCS theory. Other features of the ARPES results for cuprates are even more disquieting. Below Tc, if you put in enough energy to overcome the energy gap, you can excite quasi-particles. These show up as a sharp peak in the data, as the material absorbs energy in the conceptually simple way that Landau said it should5,6. But above Tc, the sharp peak becomes broad and confused, as if the quasi-particle concept does not apply7. “Well-defined quasi-particles have not been observed in the normal state of any of the cuprates,” says Zaanen.

This picture has emerged as data from dozens of different high-temperature superconductors have come to tell the same story. In the past, says John Tranquada, an experimentalist at the Brookhaven National Laboratory on Long Island, New York, “it has often been easier to attribute disturbing properties to 'bad samples' than to change one's perspective”. But physicists are now facing up to the cuprates' undeniable peculiarities. And with the traditional BCS theory in ruins, they are looking elsewhere for guidance.

Coherent thoughts

Unfazed: Steve Kivelson sees quantum wave coherence as key to high-temperature superconductivity.

The pseudogap may provide one key. In 1995, Vic Emery of the Brookhaven National Laboratory and Steve Kivelson of the University of California at Los Angeles pointed out that superconductivity requires more than just paired charge carriers — it also requires 'phase coherence' between those pairs8. Loosely speaking, each pair has a quantum wave associated with it, and for the pairs to condense into the superconducting state all the waves have to be in phase with one another. As the pseudogap exists almost up to room temperature, it could be that some feature of cuprate structure makes it possible for pairs to form at high temperatures, well above Tc. If so, the onset of superconductivity would signify not the formation of pairs, but the setting in of phase coherence below Tc.

In this view, cuprate superconductivity may break down above T c because the pairs have so much thermal energy that they can no longer maintain phase coherence. Emery and Kivelson suggest that, just above Tc, superconductivity should therefore become fragmented or fluctuating — it should be possible to find evidence of superconductivity in the material, but only over very short distances or timescales.

Pioneers: Bednorz (top) and Müller received a Nobel prize for their discovery of high-temperature superconductors, materials that have caused many theoretical physicists sleepless nights. Credit: EPA/EPA

In 1999, this idea won support from experiments by a team led by Joe Orenstein of the University of California at Berkeley9. Orenstein and his colleagues studied how another type of cuprate — Bi2Sr2CaCu2O8+δ — responded to an electrical field that was alternating very rapidly. This allowed them to test Emery and Kivelson's idea by probing the material's properties over the appropriate distances and timescales. They found exactly the sort of fluctuations that Emery and Kivelson had predicted. Above Tc, the higher levels of thermal energy appeared to churn up small 'vortices' — regions in which the material became non-superconducting. And the number of these vortices grew with temperature, ultimately taking over and destroying superconductivity completely.

In August last year, Shin-ichi Uchida of the University of Tokyo and his colleagues found further evidence for such vortices10. They placed a thin cuprate film into a magnetic field, and made one edge warmer than the other. Magnetic fields generally will not pass through a superconductor, but they can penetrate through non-superconducting vortices. By looking for evidence of the magnetic field in the material, Uchida and his colleagues showed that vortices formed above Tc. This again suggests that the charge carriers in cuprates do become paired at high temperatures, and then turn superconducting when things get colder.

So what might the pairing mechanism be? Most theorists are now focusing on the physics that takes place in a Mott insulator in which just a few electrons have been removed from the copper atoms, creating a small number of charge 'holes'. “Magnetic interactions play an important role in every theory,” says Millis. In undoped cuprates, the magnetic interactions between the electrons force an antiferromagnetic arrangement of their spins. These interactions must change when electrons start to move around, with the pairing of charge carriers being one consequence. The puzzle is to find out why this occurs.

Spins and stripes

Some researchers believe that fluctuations in electron spins might play a role similar to the lattice vibrations in the conventional BCS theory — the spin of one moving electron might distort nearby spins in such a way as to exert an attractive force on other electrons, binding them together into pairs11,12. But others suspect that something more exotic may be going on. One popular idea centres on theoretical calculations that suggest that in a lightly doped Mott insula tor — in which a few electrons have been removed from the copper atoms to break the electronic traffic jam — the charges and spins will undergo a natural process of segregation.

Charge holes can move from atom to atom, but have trouble doing so in a largely antiferromagnetic material. “Any movement tends to create strings of misaligned spins,” says Emery. This tends to force the holes into 'stripes' — linear regions with lots of holes, like rivers of charge, running parallel to other regions with few holes and electron spins locked into the antiferromagnetic pattern.

Based on this picture, Emery and Kivelson theorize that superconductivity would emerge in several stages13. Being forced together into the stripes, the holes can become bound together into pairs even at high temperatures — thus generating the pseudogap. At a somewhat lower temperature, the pairs within each stripe would then condense into single quantum states, giving rise to superconductivity — but only in the stripes. Finally, at Tc and below, the weakening of thermal noise would permit the pairs to hop between stripes, giving phase coherence and full superconductivity.

In neutron-scattering experiments, researchers have now seen stripes of this sort in several different types of cuprates, doped to a variety of levels at a range of different temperatures14. Yet for the Emery–Kivelson theory to work, when the material is superconducting, the stripes have to vibrate or meander within the two-dimensional copper-oxide planes. Theoretical calculations suggest that stripes should work against superconductivity if they are more or less stationary15. And although some experiments offer hints that fluctuating stripes may exist in the cuprates16, the significance of this for superconductivity remains unclear. “Most people seem to accept that stripes can occur in certain compounds,” says Tranquada. “But there is still considerable scepticism as to whether fluctuating stripes are a universal feature of the cuprates.” Littlewood agrees: “The real question is whether the stripes are central to the problem or instead represent a red herring.”

Fractional progress

One of the most exciting recent alternatives to the stripe picture comes from theorists Matthew Fisher and Todadri Senthil of the University of California at Santa Barbara. Fisher and Senthil suggest that, when you give energy and momentum to an electron, it splinters into constituents which behave as free particles — some, called 'spinons', carrying spin and others, called 'chargons', carrying charge. This 'fractionalization' of the electron would explain why cuprates in their non-superconducting state appear to lack well-defined quasi-particles. This view also predicts that charge and spin should come back together with the onset of superconductivity, and so sharp quasi-particles should then exist again — as they do, at least in the direction where the pseudogap is largest. “No other theory seems able to account for this,” says Millis.

The basic idea of electron fractionalization has been around for years17. But some physicists believe that Fisher and Senthil have found a more natural theoretical framework in which it might occur18,19. “Many of us see this as a demystification of the notion of spin–charge separation,” says Zaanen.

One of the theory's most intriguing aspects is that it predicts superconductivity might even take place without any pairing between charge carriers. Superconductivity requires the charge carriers to condense into a single quantum state — and the Pauli exclusion principle means that pairing is usually the only way to achieve this. But the separation of charge and spin releases the chargons from this restriction, allowing them to condense into a single quantum state without pairing ever taking place.

Fisher and Senthil say that their ideas can be put to the test. “They have proposed a spectacular experiment which can prove or disprove it,” says Zaanen. In addition to the chargons and spinons, the theory proposes that electron fractionalization would cause the formation of a third type of excitation called 'visons' — vortex-like anomalies that can associate with magnetic fields. If visons exist, it should be possible to trap them by heating and cooling a cylindrical cuprate sample with a hole drilled down the centre, and turning a magnetic field on and off, all in a precise sequence. The visons' presence would be betrayed by a small portion of magnetic field becoming trapped in the central hole.

Although both stripes and electron fractionalization have appeal, no one yet knows whether the right ideas are yet on the table. What is clear is that the high-temperature superconductors live well outside the regime of traditional theories. And it could be that there will never be a simple way to understand them. “We might ultimately find a theory that everyone believes,” says Littlewood, “even though no one will be able to use it to calculate anything.”

But after all this effort, most theorists would prefer to think that someone, someday, will come up with a new perspective that will suddenly make everything fall into place. “I wait with bated breath for some young kid to come up with a completely new idea,” says Littlewood.

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  1. Mark Buchanan is a science writer in Normandy, France.

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Buchanan, M. Mind the pseudogap. Nature 409, 8–11 (2001). https://doi.org/10.1038/35051238

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