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Extracting the Eliashberg Function Print

A multitude of important chemical, physical, and biological phenomena are driven by violations of the Born-Oppenheimer approximation (BOA), which decouples electronic from nuclear motion in quantum calculations of solids. Recent advances in experimental techniques combined with ever-growing theoretical capabilities now hold the promise of presenting an unprecedented picture of these violations. By means of high-resolution angle-resolved photoemission at the ALS and theoretical calculations, a multi-institutional collaboration that includes researchers from Oak Ridge National Laboratory, the University of Tennessee, Stanford University, and the ALS has obtained the first high-resolution spectroscopic images of the specific vibrational modes that couple to a given electronic state.

Living with Vibrating Atoms

Theorists face a huge problem when trying to apply quantum mechanics to the calculation of the properties of solids, most of which are determined by the spectrum of energies characterizing the electrons as they zip around and between the framework (lattice) of atomic nuclei that make up the skeleton of the solid. As long as the atoms stay put, there is no problem, but the ever-present atomic (lattice) vibrations mean there is no static framework. Salvation comes from the Born-Oppenheimer approximation, which allows theorists to consider the slow-moving atoms to be frozen in space in their average positions while the more energetic electrons do their thing.

This "decoupling" of electronic from atomic motions often works well, but it turns out that many of the most interesting behaviors of solids result from so-called vibronic interactions that are due to coupling of electronic motion and lattice vibrations; that is, from the breakdown of the Born-Oppenheimer approximation. For example, electron–lattice interactions are the foundation for superconductors, metals that lose resistance to the flow of electricity at ultracold temperatures. It is interesting that they are also required for the biochemical reactions in the cells of our bodies that are catalyzed by enzymes. Shi et al. have combined a well-known soft x-ray spectroscopy technique with theoretical calculations in a new way to paint the first precise pictures of which vibrational modes couple to which electrons.

Breakdown in the BOA results from low-energy excitations of the electrons near the Fermi energy coupling with vibrational excitations of the solid. The resulting vibronic interactions are a necessary ingredient in any process that makes or breaks a covalent bond, such as conventional catalysis or enzymatically facilitated biological reactions. Conventional superconductivity, which is driven by the electron-lattice interaction, is another classic result. And many of the emergent properties of complex materials or artificially nanostructured materials result from coupling of the electronic and lattice motion in systems that are inherently anisotropic.

The distortion in the measured dispersion (energy ε vs momentum k) near the Fermi energy (εF) relative to that from a first-principles calculation for the electron bands in a solid without vibrational motion (blue) yields the real part of the self energy ReΣ(ε).

The manifestation of this coupling is a distortion in the measured dispersion (energy ε vs momentum k) near the Fermi surface relative to that from a first-principles calculation for the electron bands in a solid without vibrational motion (a frozen lattice) or, failing that, a simple parabolic (ε ~ k2) band. This distortion has been seen in several materials using high-resolution angle-resolved photoelectron spectroscopy (ARPES). The anisotropic nature of this coupling can be seen directly because the ARPES technique allows measurements to be made as a function of the magnitude and direction of the electron's speed.

The researchers chose to study the (1010 with bar) surface of beryllium (Be) because it is a light element (Z = 4), is strongly bonded (i.e., has large vibrational energies), and is known to have two-dimensional surface states. Previous measurements on Be(0001) had already revealed a breakdown in the BOA near the Fermi energy. The group has found a similar breakdown in their comparison of the measured and calculated energy vs momentum for a surface state on Be(1010 with bar).

Physicists call the difference between the first-principles calculation and the measured curve the real part of the self energy ReΣ(ε), where ε is the energy measured in eV with respect to the Fermi energy εF. The breakthrough in this work occurred when the researchers demonstrated that by means of a procedure called the maximum entropy method (MEM), the experimental data could be used to extract from ReΣ(ε) the spectroscopic function α2F(ω). Known as the Eliashberg function, this function is the product of the density (F) of vibrational modes with frequency ω at the surface and the coupling constant (α2) of the modes to the electrons. Hence, for the first time, one can see which vibrational modes are important and compare them with experimental and theoretical determinations of the energies and character of the vibrational modes.

By means of the maximum entropy method (MEM), the experimental data was used to extract from ReΣ(ε) (blue) the Eliashberg function (red) α2F(ω), allowing identification of the specific vibrational states that couple to a given electronic state.

In the 1960s, the most definitive signature for determining the mechanism in conventional superconductors was the measurement of the electron tunneling I-V characteristic and the concomitant inversion procedure to display the Eliashberg function. The procedure developed in this work can produce in unprecedented detail a spectroscopic picture of the direction-dependent nature of the coupling between the electrons and the lattice vibrations in anisotropic two-dimensional systems. The researchers are now exploring the application of MEM to extract the Eliashberg function from photoemission data on other materials, including high-temperature superconductors.

Research conducted by J. Shi (Oak Ridge National Laboratory); S.-J. Tang (University of Tennessee); B. Wu (Oak Ridge National Laboratory and University of Texas at Austin); P.T. Sprunger (Louisiana State University); W.L. Yang, V. Brouet, and X-J. Zhou (Stanford University and ALS); Z.-X. Shen (Stanford University); Z. Hussain (ALS); and Z. Zhang and E.W. Plummer (Oak Ridge National Laboratory and University of Tennessee).

Research funding: National Science Foundation, Office of Naval Research, and U.S. Department of Energy (Oak Ridge National Laboratory). Operation of the ALS is supported by the U.S. Department of Energy, Office of Basic Energy Sciences.

Publication about this research: J. Shi, S.-J. Tang, B. Wu, P.T. Sprunger, W.L. Yang, V. Brouet, X.J. Zhou, Z. Hussain, Z.-X. Shen, Z. Zhang, and E.W. Plummer, "Direct extraction of the Eliashberg function for electron-phonon coupling: A case study of
Be(1010with bar)," Phys. Rev. Lett. 92, 186401 (2004).