Kohn Anomaly in ${\mathrm{MgB}}_2$ by Inelastic X-Ray Scattering

We study phonons in ${\mathrm{MgB}}_2$ using inelastic x-ray scattering (1.6 and 6 meV resolution). We clearly observe the softening and broadening of the crucial $E_{2g}$ mode through the Kohn anomaly along $\Gamma M$, in excellent agreement with ab initio calculations. Low temperature measurements (just above and below $T_c$) show negligible changes for the momentum transfers investigated and no change in the $E_{2g}$ mode at $A$ between room temperature and 16 K. We report the presence of a longitudinal mode along $\Gamma A$ near in energy to the $E_{2g}$ mode that is not predicted by theory.

Figure 1

Spectra along $\Gamma M$. Room temperature (299 K) and 6 meV resolution. Note the shift and increasing linewidth of the $E_{2g}$ mode (indicated by arrows). Solid lines are fits to the data (see text).

Figure 2

Phonon dispersion. Points show measured values while lines are calculations of 9. Dashed lines show branches not investigated in this work due to polarization selection rules or choice of scan range. Open symbols are measured using high resolution, while filled symbols are measured using a high flux setup. Circles (green, online) are room temperature measurements, squares (red, online) at 44 or 43 K, and triangles (blue, online) at 16 K. Points approximately at the calculated $E_{2g}$ and $B_{1g}$ energies along $\Gamma A$ are anomalous longitudinal modes. See text for discussion.

Figure 3

Measured and calculated $E_{2g}$ linewidth along $\Gamma M$ ($M$ at $1.17{\AA }^{-1}$). See text for details. The horizontal bars on the figure show the diameters of the sigma Fermi surfaces projected into the plane perpendicular to the $\Gamma A$ axis (from 18) giving a naive indication of the region where the Kohn anomaly should occur.

Figure 4

Temperature dependence of the transverse modes at $A$ ($210.5$). 6 meV resolution, offset for clarity in the main figure, but overlapped in the inset. See text for discussion.