At the Frontier of Physics and Chemistry  

Improved computational capability introduced a decisive change in the theoretical picture of superconductivity. In January 1986, Georg Bednorz and Karl Müller found that a novel copperoxide compound chilled to 30 degrees above absolute zero allowed electricity to flow without resistance. "The Superconductivity Revolution" announced the cover of Time magazine in 1987 after the two IBM scientists won the Nobel Prize. Their discovery brought an exotic, quantum phenomenon into public consciousness and awakened a dream of technological nirvana — roomtemperature superconductivity. To transmit electrical current without the slightest loss of energy is magic without trickery, perhaps the closest thing to a free lunch Mother Nature offers. A material that is superconducting at room temperature would likely lead to highspeed trains that levitate on superconducting magnets, practical electric cars and superfast networks and computer chips. Although roomtemperature superconductivity remains an elusive quest, the 1986 breakthrough jumpstarted research in superconductivity around the globe that continues today. Within a few years, scientists found other copperoxide materials and soon pushed the critical temperature, T_{c}, where resistance drops, well above 100 degrees Kelvin (100 K). Though still very cold — absolute zero, 0 K, is minus 273 degrees Celsius — it's warmer than liquid nitrogen, 77 K, which has enabled some useful applications. A few urban utility companies have tripled their capacity to carry power simply by replacing their existing underground cables with liquidnitrogen cooled superconducting cables, and cellular telephone towers have extended their reception range and callhandling ability with superconducting signal filters. Along with furious laboratory efforts to find ever higher T_{c} materials, the 1986 breakthrough stirred intensive theoretical work, and while engineers have developed uses for highT_{c} materials, physicists still can't explain why they are superconducting. One leader in the field, David Pines, staff scientist at Los Alamos National Laboratory, says that understanding hightemperature superconductivity is "arguably the major problem in physics today" with thousands of published papers a year contributing to the effort. "If we can arrive at a complete theoretical explanation of hightemperature superconductivity," says solidstate physicist Mark Jarrell of the University of Cincinnati, "then we should be able to design and synthesize a roomtemperature superconductor, which would have tremendous technological implications." Superconductivity is a quantum phenomenon in the solid state, and theoretical formulations to describe it depend on highperformance computing to solve the equations. The solid state, which includes metals, semiconductors and insulators, is a densely packed, regularly spaced lattice of atoms with electrons moving among them. The electrons and electron states that must be accounted for are, like fish in the sea, essentially infinite, and it's not possible, therefore, even with the most powerful supercomputers, to exactly calculate all the interactions that bear on the electronic properties of a solidstate material. The theoretical challenge is to develop computational approaches that can reasonably approximate the complex physics and produce reliable predictions. Within the past few years, Jarrell has developed an original approach, called the Dynamical Cluster Approximation, that extends and overcomes a serious limitation of another approach. Using the prototype Terascale Computing System at Pittsburgh Supercomputing Center and a massively parallel implementation of this new approach, he and his colleague, postdoctoral fellow Thomas Maier, carried out computations with a theoretical model, the twodimensional Hubbard model, that has gained general acceptance as a theoretical framework for highT_{c} materials. Because the highT_{c} materials are structurally a series of copperoxide planes, with apparently almost no interactions between the planes, they can be modeled as 2D systems. Jarrell's computations for the first time indicate clearly, nevertheless, that the 2D Hubbard model is incomplete as a description of hightemperature superconductivity. "Up until now," says Jarrell, "nobody has been able to address the question as to whether this model describes highT_{c} superconductivity." 


ShortLived SuccessSuperconductivity first revealed itself in 1911. Dutch physicist Heike KamerlinghOnnes chilled mercury to superlow temperatures and found that electrical resistance vanished at 4.2 K. He called this strange physical state superconductivity and won the 1913 Nobel Prize in physics for his work. In subsequent years, other superconducting materials were discovered, with T_{c} reaching to almost 20 K by the time of Bednorz and Müller's discovery. Due to the underlying physics of these pre1986 materials, now called lowtemperature or conventional superconductors, many physicists were convinced that superconductivity couldn't exist above 23 K, and part of the shockeffect of the 1986 breakthrough was the demolition of this mental barrier. For many years after 1911, there was no way to explain superconductivity. It just existed. As quantum theory came into being in the 1920s and 30s, however, a few physicists realized that this bizarre electronic behavior fell into the rare category of a quantum effect manifested on a macroscopic scale. By the mid1950s, John Bardeen, Leon Cooper and John Schrieffer had worked out the mathematical details of what has come to be called BCS theory. Under normal conditions, an electrical current flows in metal as a stream of electrons. Resistance occurs as the moving electrons bang around in the lattice of atoms and lose energy, which creates microscopic vibrations that spread and dissipate as heat. Being negatively charged, electrons usually repel each other, but at low enough temperatures, says BCS theory, they form pairs — called Cooper pairs — that exist as a single quantum entity with the seemingly magical ability to defy conventional physics. Like the Wuxia masters in "Crouching Tiger, Hidden Dragon" who float effortlessly through space, Cooper pairs flow through the metal lattice unimpeded. Electrons pair up despite their mutual repulsion, says BCS theory, through the mediation of quantum vibrations. By their interactions with positively charged ions in the lattice, electrons generate packets of vibrational energy, dubbed phonons, and according to BCS theory, low temperatures allow phonons to operate as matchmakers that facilitate a telepathiclike link between electrons. At higher temperatures, stronger vibrations break up the phononinduced Cooper pairs, and superconductivity goes away. The proof of the pudding for BCS theory is that it fits comprehensively with the phenomenology of conventional superconductivity, successfully explaining all the measured data and predicting some new effects. This theoretical success story, however, reached the end of its road in 1986. 


Something Happening HereBCS theory won a 1972 Nobel Prize for its creators, but it doesn't work for hightemperature superconductivity. Some version of Cooper pairs still appears to be the joy juice of the new superconductivity, but the phononinduced mechanism of BCS theory isn't strong enough to hold electrons together at the higher temperatures. In the words of a 60s song, "There's somethin' happening here. What it is ain't exactly clear." Fifteen years of prodigious work on highT_{c} materials has established that they constitute a new realm of solidstate physics. In virtually every respect, their normal state — behavior above T_{c} — differs markedly from conventional superconductors. "What's weird about hightemperature superconductors," explains Jarrell, "is they're a pathetically bad metal." Unlike metals, the copperoxide materials, also known as cuprates, are flaky in consistency. Along with being devilishly tricky to prepare, they're almost more like insulators than metals at room temperature, with poor ability to conduct electricity. Also unlike conventional superconductors, their electrical and magnetic properties are highly directional (anisotropic). And a range of quantum properties in the normal state — things like characteristic spinexcitation energy — differ dramatically from lowT_{c} materials. The big job of finding a theory that pulls this exotic new solidstate world into a coherent picture has gone in many directions, but the most widely accepted starting place has been the 2D Hubbard model. A decade ago, Jarrell's University of Cincinnati colleague, Fuchun Zhang, showed that this model, a mathematical expression formulated 40 years ago to describe magnetism, captures the essential physics of the cuprates. Part of the beauty of the model is its twodimensionality, which means reduced mathematical complexity from the full realism of three dimensions. "It's the simplest possible model you could construct," says Jarrell. One term describes electronelectron interactions and another accounts for electrons hopping sitetosite in the lattice, a minimal number of parameters that, despite its relative simplicity, has shown an ability to accurately calculate many of the strange properties associated with highT_{c} superconductors, at least in the normal state. The main problem has come in finding a way to solve the Hubbard model under conditions that replicate the transition to superconductivity. Solution of the Hubbard model for the infinite number of electrons in a solidstate lattice requires an approximation scheme. An approach called the Dynamical Mean Field has proven useful in many calculations, but is inherently inadequate for the highT_{c} transition because it's "localized." The DMF approximation accounts for interactions between electrons at one atomic site, while other sites in the lattice are in effect averaged as a mean field. Studies have shown, however, that a fundamental characteristic of hightemperature superconductivity is that the pairing interactions are nonlocalized — electrons from neighboring atoms, rather than the same atom, interact strongly. 

Redrawing the MapA good deal of activity has gone toward developing nonlocalized extensions to the DMF approximation. In the last few years, Jarrell developed a sophisticated approach, the Dynamical Cluster Approximation, that incorporates nonlocal corrections to the DMF approximation by mapping the problem onto a cluster of sites, which is itself embedded within the mean field. In 2000, he used the DCA approach to solve the 2D Hubbard model on a CRAY T3E at Ohio Supercomputer Center. With a cluster of four sites, the smallest possible, his results showed properties in good general agreement with hightemperature superconducting materials, including transition to the superconducting state. "Initially, everything looked very promising," says Jarrell. The poor electrical conductivity of the normal state was replicated, and also the transition to superconductivity. "For small clusters, we found essentially the phase diagrams of a cuprate." The scale of the computation increases dramatically with expanded cluster size, and Jarrell was temporarily precluded from looking at the effect of larger cluster sizes. In spring 2001, however, Jarrell and Maier gained access to the prototype Terascale Computing System, facilitating a series of calculations expanding cluster size up to 16. The increase in computational capability introduced a decisive change in the theoretical picture. "As we systematically increased cluster size," says Jarrell, "superconductivity systematically went away. That tells you something fundamental. We believe that the 2D Hubbard model is not sufficient by itself. Something new has to be introduced." Among several possibilities, Jarrell notes that a fully accurate model of superconductivity may need to add coupling between copperoxide planes. Initial explorations of this showed restoration of superconductivity, but even with the prototype TCS it wasn't possible to systematically explore the effect of coupling. Another possibility is "chemical disorder" of the cuprates, variation in the number of oxygen atoms from region to region. It's impossible to predict how much computing it will take to thoroughly explore these questions, says Jarrell, but the fullscale TCS will allow him to get started. "We don't understand hightemperature superconductivity in an analytic fashion. If we put in chemical disorder, we don't know what the effects are going to be. This is research driven by tremendous curiosity and ignorance." Sooner or later, answers will come. When solidstate physicists arrive at the full story of how paired electrons stay paired at high temperatures, they'll be a big step closer to knowing how to create a roomtemperature superconductor. When that happens, our use of electricity will never be the same.
