N I S T Center for Neutron Research
Accomplishments and Opportunities 2001
Giant Anharmonicity and Electron-Phonon Mediated Superconductivity in MgB2 at 39 K
The recent surprise discovery of superconductivity in MgB2 at 39 K has stimulated a great deal of research on this intercalated graphite-like system (Refer to Figure 1). Sparked by this discovery, we set out to unlock the structural secrets and, in particular, to reveal the origin of the high Tc in MgB2: an electron-phonon or other exotic mechanism? To answer this fundamental question, we calculated Tc, its pressure dependence, and its isotope effect from the electronic band structure and lattice dynamics of MgB2 using density functional theory within the generalized gradient approximation (Refer to References 1 through 3).
Figure 2 shows that the features in the calculated phonon density of states (D O S) are in excellent agreement with the neutron data (G D O S), giving confidence that the calculations provide a sound description of the physical properties of the system. The D O S consists of two bands of phonons, one below 40 meV corresponding primarily to acoustic phonon modes, and one above 50 meV that mostly involves the boron motions. Inspection of the calculated phonon-dispersion curves that make up the high-energy band in the D O S reveals that the in-plane boron phonons (as depicted in the inset to Figure 3) are very anharmonic. To demonstrate this, in Figure 3 we plot the total energy as boron atoms move according to one of these zone-center in-plane phonons with E2g symmetry. The potential indicates a very strong anharmonic term. Numerically solving the Schrödinger equation for this anharmonic potential yields a phonon energy of h ω (E2g) = 74.5 meV, a 25 % enhancement over the harmonic value of 60.3 meV. This value is in good agreement with recent Raman measurements. The giant anharmonicity revealed gives the first hint that the in-plane modes are strongly coupled to the pxy σ bonding orbitals of boron, as shown schematically in Figure 1. This coupling is also evident from the splitting of the boron σ bands (red lines) with the E2g phonons (Refer to Figure 3). Note that the other bands are not affected by the E2g phonons.
Graphics Caption FIGURE 1. The crystal structure of MgB2 consisting of B and Mg hexagonal layers. The in-plane boron modes (shown by arrows) are strongly coupled to the boron px,y σ bands shown as the green contour and isosurface plots.
Graphics Caption FIGURE 2. Generalized (top) and the calculated (bottom) phonon density of states. The intensities of the peaks do not agree well because in D O S B and Mg ions contribute equally while in G D O S they are weighted by neutron cross-sections and inverse masses.
Graphics Caption FIGURE 3. Top: Total energy curve as a function of boron displacement for E2g mode (shown in the inset), indicating a large anharmonic term in the potential. Bottom: Band structure of the undistorted (left) and distorted structures (right) by E2g phonons (uB ≈ 0.06 Å). See Reference 1 for the animation of the zone center phonons and their coupling with the B σ bands.
Graphics Caption FIGURE 4. Top: Boron mass dependence of the Tc. Bottom: Pressure dependence of Tc as a function of uniform ab-axes and c-axis compression, respectively.
The splitting of the boron σ bands, when averaged over the Fermi surface, gives an electron-phonon (E P) coupling constant λ ≈ 1. Using this value in the McMillan expression for Tc with ω (E2g) and taking a typical value for the Coulomb repulsion µ* = 0.15, we obtain a Tc of 39.4 K, in excellent agreement with experiments. We also solved the Schrödinger equation for the potential shown in Figure 3 for different boron masses and obtained ω (E2g) = 291.8 M-0.575 and λ = 0.615l M0.169, which yields the Tc -M curve shown in Figure 4 and a boron isotope effect α = 0.21, in good agreement with the experimental value of 0.26 ± 0.03.
Since the pressure dependence of Tc puts a stringent test on any theory of superconductivity, we repeated the calculations of phonons and electronic band structures discussed above for isotropic, uniaxial (along c-axis), and biaxial (in the ab-plane) pressures (Refer to Reference 3). We find that while ω (E2g) increases with increasing pressure, the density of states at the Fermi energy decreases. The E P constant λ also shows significant changes with pressure. Inserting all these competitive effects into the McMillan formula yields the pressure dependence of Tc shown in Figure 4.
For isotropic pressure Tc decreases with increasing pressure almost linearly at a rate of ≈ -1.0 K/GPa, in excellent agreement with the experimental value of -1.1 K/GPa. We also predict a cusp in the Tc-P curve around P ≈ 20 GPa.
A similar cusp was recently observed experimentally. Our calculations indicate that Tc should increase first and then decrease with increasing c - compression, while it should decrease rapidly with a b - compression. Hence, when single crystal samples of MgB2 become available, measurements of the a b - and c - compression dependence of Tc should provide a critical test of our theory.
MgB2 may be the ultimate B C S s - wave superconductor, with parameters controlling Tc fully optimized to yield the highest possible Tc. However, even if Tc cannot be increased further, the low cost, light mass, easy fabrication, nearly isotropic high conductivity of MgB2 , which also has a large critical current, will no doubt find many important technological applications in the near future.
References
[1] For full details of this work, see the website: h t t p : / / w w w . n c n r . n i s t . g o v / s t a f f / t a n e r / m g b 2.
[2] T. Yildirim, O. Gulseren, J. F. Lynn, C. M. Brown, T. J. Udovic, Q. Huang, N. Rogado, A. Regan, M. A. Hayward, J. S. Slusky, T. He, M. K. Haas, P. Khalifah, K. Inumaru, and R. J. Cava, Phys. Rev. Lett. 87, 37001 (2001).
[3] T. Yildirim and O. Gulseren, J. Phys.Chem. Solids (in press, 2002).
Authors
T. Yildirim
N I S T Center for Neutron Research
National Institute of Standards and Technology
Gaithersburg, MD 20899-8562
O. Gulseren
N I S T Center for Neutron Research
National Institute of Standards and Technology
Gaithersburg, MD 20899-8562
and
Department of Materials Science and Engineering
University of Pennsylvania
Philadelphia, PA 19104-6202