Abstract
We measure phonon dispersion and linewidth in a single crystal of
MgB$_2$ along the $\Gamma-A$, $\Gamma-M$, and $A-L$ directions using
inelastic x-ray scattering. We use density functional theory to compute
the effect of both electron-phonon coupling and anharmonicity on
the linewidth, obtaining excellent agreement with experiment. Anomalous
broadening of the E2g phonon mode is found all along $\Gamma-A$.
The dominant contribution to the linewidth is always the electron-phonon
coupling.
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