At the Frontier of Physics and Chemistry


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Photo: Mark Jarrell Mark Jarrell, University of Cincinnati.




Photo: Mark Jarrell Thomas Maier, University of Cincinnati.

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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 copper-oxide 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 — room-temperature 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 high-speed trains that levitate on superconducting magnets, practical electric cars and superfast networks and computer chips.

Although room-temperature superconductivity remains an elusive quest, the 1986 breakthrough jump-started research in superconductivity around the globe that continues today. Within a few years, scientists found other copper-oxide materials and soon pushed the critical temperature, Tc, 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 liquid-nitrogen cooled superconducting cables, and cellular telephone towers have extended their reception range and call-handling ability with superconducting signal filters.

Along with furious laboratory efforts to find ever higher Tc materials, the 1986 breakthrough stirred intensive theoretical work, and while engineers have developed uses for high-Tc 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 high-temperature 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 high-temperature superconductivity," says solid-state physicist Mark Jarrell of the University of Cincinnati, "then we should be able to design and synthesize a room-temperature superconductor, which would have tremendous technological implications."

Superconductivity is a quantum phenomenon in the solid state, and theoretical formulations to describe it depend on high-performance 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 solid-state 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, post-doctoral fellow Thomas Maier, carried out computations with a theoretical model, the two-dimensional Hubbard model, that has gained general acceptance as a theoretical framework for high-Tc materials. Because the high-Tc materials are structurally a series of copper-oxide planes, with apparently almost no interactions between the planes, they can be modeled as 2-D systems.

Jarrell's computations for the first time indicate clearly, nevertheless, that the 2-D Hubbard model is incomplete as a description of high-temperature superconductivity. "Up until now," says Jarrell, "nobody has been able to address the question as to whether this model describes high-Tc superconductivity."





Short-Lived Success

Superconductivity first revealed itself in 1911. Dutch physicist Heike Kamerlingh-Onnes 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 Tc reaching to almost 20 K by the time of Bednorz and Mller's discovery. Due to the underlying physics of these pre-1986 materials, now called low-temperature or conventional superconductors, many physicists were convinced that superconductivity couldn't exist above 23 K, and part of the shock-effect 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 mid-1950s, 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 telepathic-like link between electrons. At higher temperatures, stronger vibrations break up the phonon-induced 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.





Model Neuromuscular Junction Model Neuromuscular Junction

Something Happening Here

BCS theory won a 1972 Nobel Prize for its creators, but it doesn't work for high-temperature superconductivity. Some version of Cooper pairs still appears to be the joy juice of the new superconductivity, but the phonon-induced 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 high-Tc materials has established that they constitute a new realm of solid-state physics. In virtually every respect, their normal state — behavior above Tc — differs markedly from conventional superconductors.

"What's weird about high-temperature superconductors," explains Jarrell, "is they're a pathetically bad metal." Unlike metals, the copper-oxide 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 spin-excitation energy — differ dramatically from low-Tc materials.

The big job of finding a theory that pulls this exotic new solid-state world into a coherent picture has gone in many directions, but the most widely accepted starting place has been the 2-D 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 two-dimensionality, 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 electron-electron interactions and another accounts for electrons hopping site-to-site 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 high-Tc 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 solid-state lattice requires an approximation scheme. An approach called the Dynamical Mean Field has proven useful in many calculations, but is inherently inadequate for the high-Tc 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 high-temperature superconductivity is that the pairing interactions are non-localized — electrons from neighboring atoms, rather than the same atom, interact strongly.

Spectral data of a cuprate superconductor in the normal (non-superconducting) state as calculated from the 2-D Hubbard model.
These pictures from Jarrell's computations agree well with angle-resolved photo emission studies, an experimental method that measures electron distribution and maps the "Fermi surface" — a geometrical representation of a solid's electrical conductivity. Peaks in the data (blue-violet) show where there are many electrons at the Fermi energy. Agreement with ARPES results exemplifies that although the 2-D Hubbard model alone may not adequately describe the cuprate superconducting state, it describes many unusual properties of the normal state. (Thomas Pruschke of the University of Augsburg produced these figures using software he and Jarrell developed.) The electronic properties of cuprates are strongly sensitive to the addition of impurities, called doping. At low doping (pointer) [n=.95], the volume of the Fermi surface is larger than it should be for a metal and is shifted away from the axis (black line) of the Fermi surface predicted for a normal metal. At high doping (top) [n = 0.80], however, the volume decreases and the surface lines up with prediction. "Among the mysteries of the weakly doped cuprates," says Jarrell, "is that the Fermi surface is too large and not centered around the origin. Many theories are proposed for these anomalies, but the simplicity of the 2-D Hubbard model leads to only one interpretation. Short-ranged magnetic correlations are responsible for both the anomalous volume and center of the Fermi surface."

Download a larger version of these images.

Redrawing the Map

A good deal of activity has gone toward developing non-localized extensions to the DMF approximation. In the last few years, Jarrell developed a sophisticated approach, the Dynamical Cluster Approximation, that incorporates non-local 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 2-D 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 high-temperature 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 2-D 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 copper-oxide 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 full-scale TCS will allow him to get started. "We don't understand high-temperature 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 solid-state 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 room-temperature superconductor. When that happens, our use of electricity will never be the same.

Download PDF Electronic Nirvana
from Projects in Scientific Computing 2001
Researchers Mark Jarrell, University of Cincinnati
Thomas Maier, University of Cincinnati
Hardware TCSini
Software User-developed code.
Related Material
on the Web
BCS Theory of Superconductivity,Georgia State University
References M. Jarrell, S. Mouskouri & Th. Maier, "A Quantum Monte Carlo algorithm for non-local corrections to the Dynamical Mean-Field Approximation," Physical Review B (2001), forthcoming.

M. Jarrell, Th. Maier, M. H. Hettler & A. N. Tahvildarzadeh, "Phase Diagram of the Hubbard Model: Beyond the Dynamical Mean Field," European Physics Letters (2001), forthcoming.
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