N I S T Center for Neutron Research

Accomplishments and Opportunities 2001

Giant Anharmonicity and Electron-Phonon Mediated Superconductivity in MgB_{2} at 39 K

The recent surprise discovery of superconductivity in MgB_{2} 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 T_{c} in MgB_{2}: an electron-phonon or other exotic mechanism? To answer
this fundamental question, we calculated T_{c}, its pressure dependence, and its isotope effect from
the electronic band structure and lattice dynamics of MgB_{2} 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 E_{2g} 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 (E_{2g}) = 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 p_{xy} 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 E_{2g} phonons (Refer to Figure 3). Note that
the other bands are not affected by the E_{2g} phonons.

Graphics Caption FIGURE 1. The crystal structure of MgB_{2} consisting of B and Mg
hexagonal layers. The in-plane boron modes (shown by arrows) are strongly coupled to the
boron p_{x,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 E_{2g} 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 E_{2g} phonons (u_{B} ≈ 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 T_{c}. Bottom: Pressure
dependence of T_{c} 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 T_{c} with ω (E_{2g}) and taking a typical value for the Coulomb
repulsion µ* = 0.15, we obtain a T_{c} 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 (E_{2g}) = 291.8 M^{-0.575} and λ = 0.615l M^{0.169},
which yields the T_{c} -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 T_{c} 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 (E_{2g}) 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 T_{c} shown in Figure 4.

For isotropic pressure T_{c} 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 T_{c}-P curve around P ≈ 20 GPa.

A similar cusp was recently observed experimentally. Our calculations indicate that
T_{c} 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
MgB_{2} become available, measurements of the a b - and c - compression dependence
of T_{c} should provide a critical test of our theory.

MgB_{2} may be the ultimate B C S s - wave superconductor, with parameters controlling
T_{c} fully optimized to yield the highest possible T_{c}. However, even if
T_{c} cannot be increased further, the low cost, light mass, easy fabrication, nearly
isotropic high conductivity of MgB_{2} , 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