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Beyond band theory
STANFORD -- Band theory explains how electrons move in solids. It is a simple model that has been phenomenally successful. In fact, it is the theoretical base upon which Silicon Valley and the entire microelectronics industry is built. But it may have met its Waterloo in a class of materials that includes the new copper-oxide superconductors discovered unexpectedly in 1986.
Despite a number of proposed modifications, band theory has failed to explain electron conduction, metallic behavior and other physical properties in these new superconductors that can carry electricity without resistance at much higher temperatures than previously had been possible.
Discovery of the new superconductors created tremendous excitement both among basic scientists and those interested in potential applications such as more efficient electrical generators, lossless powerlines and magnetically levitated vehicles. Slower than anticipated progress toward the goal of room temperature superconductivity has dampened the excitement of those looking for short-term applications. But interest has remained high among basic researchers, who argue that improved understanding of how these materials work is the most likely way to achieve this difficult and elusive goal.
Now a group of scientists at Stanford and the Massachusetts Institute of Technology has published the results of a series of experiments on six high temperature superconductors. They hope that their data will provide an empirical basis for developing a more general theory of electron transport that can describe not only how electrons move through materials where band theory works extremely well but also how they move within materials like high temperature superconductors, where the current theory breaks down.
“There are things going on in these materials that band theory can't explain,” says Zhi-Xun Shen, assistant professor of applied physics at Stanford, who headed the research effort. “That is one of the reasons that they have such surprising properties.”
The paper reports the results of the most detailed studies of high temperature superconductors yet performed by a demanding experimental technique called photoemission spectroscopy, which provides direct information about electron movement within such materials. Its authors are Shen; William E. Spicer, Stanford professor emeritus of electrical engineering; Stanford graduate student David M. King; Stanford Linear Accelerator Center physicist Daniel S. Dessau; and postdoctoral fellow Barry O. Wells of the Massachusetts Institute of Technology.
(A similar analysis of a single high temperature superconductor published two years ago in Physical Review Letters by this group was recently ranked by the Institute for Scientific Information as one of the 10 most highly cited physics papers last year.)
"For a number of years, we have known that band theory doesn't explain electron transport in certain materials, such as transition metal oxides like nickel oxide, but this has largely been ignored because the theory has proven so successful on commercially important materials like silicon," says Spicer, who helped pioneer photoemission spectroscopy.
Band theory, more properly called one-electron band theory, provides a conceptually simple explanation for electron transport. Instead of dealing individually with the millions of electrons that flow through a solid, the theory starts with an individual electron. It then approximates the complex interplay of forces acting upon that electron with a series of wave functions that represent the average conditions that an electron experiences. This approximation has proven remarkably accurate for estimating electron flow in silicon chips and many other materials.
"In materials like the new superconductors, however, the electrical and magnetic interactions between electrons appear to be so strong that the one-electron approximation of band theory no longer works," says Shen.
Instead, he and his colleagues argue that a fundamentally different approach will be necessary. This different approach is called many-body theory, because it explicitly takes into account the electrical interactions between large numbers of individual electrons. Band theory is delocalized, that is, the exact location of the electron is unimportant. In fact, there is no restriction against two electrons occupying the same site - something that cannot happen in real life because of the strong electrical repulsion between electrons.
"In many-body theory, however, two electrons can only occupy the same site at great energy cost. The repulsion between individual electrons is important. If I'm an electron, and I sit in one spot, and you are an electron, you don't want to get too close. You must find an empty chair, or site, to sit in," explains Shen.
Electron interactions are directly related to the phenomenon of superconductivity. The only known way that electrons can move through a material without colliding either with the atoms or each other - and so achieve electrical movement without resistive losses - is by moving in pairs.
In conventional superconductors, which operate only at temperatures under minus 220 degrees Celsius, the force that causes electron pairing is created by vibrations in the material's lattice. At the higher temperatures - slightly under minus 100 degrees Celsius - at which the new materials superconduct, however, lattice vibrations are not strong enough to hold the electrons together. So a stronger mechanism is required.
According to the researchers, photoemission spectroscopy can obtain direct measurements of this critical binding force between the electron pairs. It does so by shooting photons into the material. A certain number of the photons knock electrons loose and some of these electrons fly out from the surface, where the scientists measure their direction and speed. This information can tell the scientists a great deal about the conditions within the material.
"The process is so fast that it is almost as if you freeze the electrons and remove them with all the information intact," Shen says.
However, the process is also very painstaking. It requires very high vacuum and the samples must be extraordinarily pure or the results can be misleading, Spicer says. "Even with considerable help from Stanford's KGB, it took us over six months just to prepare the samples," he says. (KGB is a nickname for the collaboration among applied physicists Aharon Kapitulnik, Theodore H. Geballe and Malcolm Beasley, who have made major contributions to the development of new superconducting materials.)
To measure the binding force between the superconducting electron pairs, the researchers looked at the differences in the energy and direction of the electrons emitted when a material is in a normal and a superconducting state.
In all the high temperature superconductors that they tested, the scientists found that this binding force is stronger in one direction, relative to the material's lattice, than in another. This provides an important clue to the origin of this force. In conventional superconductors, the binding force is the same in all directions. But theories that propose a magnetic source for the superconducting electron binding force predict that it will be stronger in some directions than in others and are consistent with these observations.
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