New experiments fuel debate over the nature of high-T _c superconductors

The parents of the copper oxide high-temperature superconductors are antiferromagnetic insulators. It’s thought that they would conduct electricity were it not for the very strong interactions among their electrons. Just dope those insulators to a level of 5% or more by chemically adding or removing electrons, and you get the cuprate compounds that superconduct at critical temperatures T _c up to 140 K. When T _c is plotted as a function of doping, it encloses a dome-shaped superconducting region, as seen in figure 1(a). Abutting the dome on the low-doping side is a mysterious “pseudogap” region, in which materials exhibit a gap in the density of states near the Fermi energy even though they are not superconducting. Theorists seeking the mechanism underlying high-T _c superconductors must account for the strongly interacting parents and the pseudogap neighbors.

The plot in figure 1(a) has two temperature scales: T _c, the temperature at which electrical resistance drops to zero, and the strikingly high T*, above which the pseudogap disappears. A heated debate has raged for years concerning the significance of the two temperature scales. ¹ Do they imply the existence of two different phenomena, each characterized by a distinct energy gap? Or is the pseudogap an extension of the superconducting gap?

The two-gap scenario is consistent with a class of theories holding that competing orders vie for dominance in the copper oxide compounds, one ordered state being a superconductor and the other something like a spin or charge density wave. In such theories, the pseudogap is associated with the competing order and may be essentially different from the superconducting gap. Another class of theories favors a one-gap scenario, in which the two temperature scales are part of the same phenomenon, with electron pairs forming at the higher temperature T* but not condensing into coherent, superconducting pairs until the temperature drops to T _c. Two temperature scales are seen, for example, in antiferromagnetic materials such as manganese oxide: The magnetic moments develop at a high temperature and order at a much lower one. By contrast, conventional superconductors are characterized by only one temperature, T _c, at which pairs both form and condense.

Continuous progress has been made in the sophistication, resolution, and ingenuity of many experimental techniques aimed at probing the nature of the pseudogap region of the phase diagram, but to date no consensus has been reached. Different techniques are applied to different cuprate superconductors and are sensitive to different properties of the samples. Results are also affected by the quality of the sample and the inherent disorder in the materials, especially for underdoped samples. Some researchers report phenomena that hint at a second superconducting gap, while other measurements find no such evidence. Nevertheless, the increasingly precise results are discovering rich new details about the key regions in and around the superconducting dome.

Among the experiments that have added important insights are three recent results from scanning tunneling microscopy (STM). While none of the experiments provides any definitive answers, each one pushes the experimental boundaries and provides some new insights that have intensified the ongoing debate.

One of the STM groups, led by Ali Yazdani of Princeton University, found ² that superconducting energy gaps nucleate locally in puddlelike nanoscale patches of the copper–oxygen plane above T _c, with those patches proliferating as the temperature is lowered through T _c. A second STM team, ³ headed by Eric Hudson of MIT, reports evidence for a second, spatially homogeneous gap that opens below T _c.

A third STM group of experimenters, under Séamus Davis of Cornell University, studied an underdoped sample well below T _c by looking at the behavior of electrons in real and in momentum space simultaneously. As Davis and coworkers have reported at recent conferences, they see quasiparticles suggestive of nonlocal superconductivity at low excitation energies and, at higher energies, holes localized in an inhomogeneous stripe pattern. ⁴

Scanning tunneling microscopy

Scanning tunneling microscopes probe with high resolution the spatial distribution of the density of electronic states of the copper oxide compounds. ⁵ The basic measurement is the current that flows as electrons tunnel between the probe’s tip and the underlying sample. The derivative of that current with respect to the tip-to-surface bias voltage—the differential conductance—is a measure of the density of states. Typical V-shaped spectra measured at energies above and below T _c are shown in figure 2(a). The width of the dip is taken to be the energy gap. The spectra measured below T _c typically manifest sharp peaks; above T _c, the peaks vanish, leaving indistinct humps.

Past STM studies have shown that energy gaps in the density of states vary widely from point to point in the copper–oxygen plane where the superconductivity occurs. ⁶ The average gap in the superconducting spectra appears to evolve continuously with temperature from the superconducting state into the pseudogap region.

Recently, the STM groups led by Yazdani and by Hudson both managed to follow the evolution of the gap at each local position as the temperature is raised. Tracking these local regions required the researchers to account for thermal expansion as the sample was heated and find a way to reposition the tip in its original location with atomic-scale resolution. The two groups explored different aspects of the data.

Yazdani, together with colleagues from Princeton and from the Central Research Institute of Electric Power Industry in Tokyo, ² varied the temperature from 20 K to 140 K. The collaborators wanted to find the temperature at which each local gap closed. In studies of overdoped, optimally doped, and underdoped crystals of a bismuth strontium calcium copper oxide compound known as Bi-2212, they first measured the distribution of gap sizes at a low temperature. They found that the larger the gap in a given nanoscale region, the higher the temperature at which the gap in that region closed. Many local regions remain gapped to well above T _c (see figure 1(b) and the gap maps in figures 2(b) and 2(d)).

For the overdoped and optimally doped samples, the local gap size Δ scaled linearly with the gap-closing temperature T _p according to the relation $2\Delta /k_{\text{B}}{T}_{\text{P}}=\mathrm{8.0.}$ . Conventional Bardeen-Cooper-Schrieffer superconductors follow a similar scaling law but with a smaller constant, 3.5, and a gap that is spatially uniform. Still, the scaling suggests to Yazdani that the pseudogap is connected with the superconducting pairing gap, at least in the optimal and overdoped regions.

The situation is more complex for the underdoped samples, for which the simple scaling between gap size and gap-closing temperature T _p breaks down. For underdoped samples, the Princeton–Tokyo experimenters find kinks in the tunneling spectra below T _c in 30% of the regions. Could those kinks be the superconducting gap, with the larger gap then being attributed to a pseudogap? Yazdani says that he is not confident in making such a conclusion but will continue to explore the underdoped region.

The Princeton–Tokyo researchers relate their results to the region of fluctuating superconductivity that has been uncovered by Nai Phuan Ong and colleagues from Princeton, together with collaborators from Boston College and Tokyo’s Central Research Institute of Electric Power Industry. ⁷ In cuprates above T _c, Ong’s group detected the presence of freely moving vortices, which constitute a kind of “vortex liquid.” The existence of such vortices implies that superconducting pairs still exist in the region above T _c but have only short-range phase coherence there.

The vortex motion was detected through extensive measurements of both diamagnetism and the Nernst effect (the flow of transverse electrical current in response to the simultaneous application of a thermal gradient and a magnetic field) in Bi-2212 and other cuprates. Ong and his team found that the region of fluctuating superconductivity extends to surprisingly high temperatures, as shown in figure 1(a), but does not go as high as T*.

Yazdani and his coworkers note that their results provide a microscopic basis for understanding the region of fluctuating superconductivity. The temperature at which the roughly 50% of the local gaps have closed, as determined by STM measurements, corresponds to the upper limit for detection of a vortex liquid in the bulk magnetic measurements (see the T ₀ line in figure 1(b)). Ong surmises that an adequate number of pairs must be present in the samples before the vortex response is large enough to measure.

Andrew Millis of Columbia University comments that the link between the STM experiment and the vortex-liquid measurements suggests that the gap seen in STM is indeed associated with superconductivity.

Evidence for a second gap?

In the other recent STM experiment that tracked local gaps as a function of temperature, Hudson joined with collaborators from MIT and Nagoya University in Japan ³ to study an overdoped sample of a bismuth lead strontium copper oxide compound called Bi-2201. Their measurements extended up to 20 K, which is 5 K above the material’s T _c.

The MIT–Nagoya experimenters found that the local gaps evolved smoothly as the temperature was raised through T _c. To bring out any details in the spectra that might be seen only in the superconducting state, the researchers divided each spectrum by a spectrum measured at the same position but at a temperature above T _c. The spectra normalized in that way reveal a small gap that coexists with the larger gap below T _c but that vanishes at T _c. Unlike the larger gap, which is seen to vary spatially, the small gap has the same value throughout the sample.

The results from Hudson’s group echo the findings from some angle-resolved photoemission spectroscopy (ARPES) experiments ^8,9 and from Raman scattering measurements, ¹⁰ all of which have reported evidence for a second gap below T _c that’s smaller than the pseudogap. At the same time, other ARPES groups claim to see just one gap. ^11,12

Looking in momentum space

A third STM group, led by Davis, has been studying the behavior of underdoped cuprates well below T _c in both real ⁴ and momentum space. ¹³ At recent conferences, Davis has reported finding different behavior depending on the energy of the tunneling electrons. For low excitation energies (corresponding to low voltage bias), Davis and his collaborators find evidence for a superconducting condensate. For excitation energies of 35 mV and higher, they find only static spatial patterns, indicating that the electrons are partially localized.

Davis’s collaborators hail from Cornell, Brookhaven National Laboratory, the University of Colorado in Boulder, the University of Sherbrooke in Canada, and, in Japan, the University of Tokyo, RIKEN’s Discovery Research Institute in Wako, Kyoto University, and the National Institute of Advanced Industrial Science and Technology in Tsukuba.

Traditionally, STM has complemented ARPES, with STM peering into real space and ARPES probing k space. A number of years ago, theorists noted that one can infer k-space information from STM. Scanning tunneling microscopes can measure the spatial interference patterns, such as that seen in figure 3(a), made by long-lived quasiparticles—electrons from broken Cooper pairs—as they scatter off impurities or other defects. Fourier transforming such patterns yields the k-space scattering vectors.

Last year, Davis and his colleagues explored a new way to look at STM data in real space, exploiting an asymmetry of the tunneling spectrum of high-T _c superconductors ¹⁴ that was noted back in 1995. The asymmetry in STM reflects the strong electron interactions, which make it easier to extract an electron from the superconductor than it is to inject one. ¹⁵ Taking the ratio of current at negative bias to that at positive bias can yield a measure of the hole density.

The Cornell collaboration measured that tunneling asymmetry in very underdoped samples ¹² of Bi-2212 and of a particular compound of calcium, sodium, copper, oxygen, and chlorine atoms—called Na-CCOC. In both materials, the experimenters found regions where the holes appeared to be localized and ordered in a stripelike pattern. The orientation and location of such regions were random, as shown in figure 3(b).

For the work reported at recent conferences, Davis and his coworkers exploited both the space- and momentum-tracking capabilities of STM. They presented evidence for nonlocal states at low-bias voltages and partially localized states at higher bias.

ARPES studies of underdoped cuprates have also uncovered interesting changes in behavior as a function of direction in k space. Below T _c, ARPES measurements confirm that the gap has a cosine variation in momentum space, as expected for electrons that pair in a d-wave state. Thus, the gap function has four nodal points on the Fermi surface where the gap is zero, and four antinodal points, where the gap is maximal.

Above T _c, ARPES and other techniques find that the simple d-wave structure is distorted. Many points along the Fermi surface have no superconducting gap. The locus of those gapless points along a section of the Fermi surface forms an arc centered on the nodal point, while the gap remains open in the antinodal regions. Some theorists have speculated that the nodal region is governed by superconductivity and the antinodal region by a different mechanism; other theorists disagree. (See Physics Today, July 2007, page 26 .)

In the STM measurements, Davis’s group sees the long-lived (nonlocal) quasiparticles near the nodal region of k space and the partially localized states near the antinodal regions. The experimenters extrapolate an energy scale for the superconductor that increases with decreasing doping, as does the pseudogap measured by other experiments.

While the new STM experiments—and others—don’t settle any questions, the rich new details they provide are confronting theory in a region where many investigators believe they may find the key to high-T _c superconductivity.