Nonadiabatic effects in the phonon dispersion of ${\mathrm{Mg}}_{1-x}{\mathrm{Al}}_x{\mathrm{B}}_2$

Superconducting ${\mathrm{MgB}}_2$ shows an $E_{2g}$ zone center phonon, as measured by Raman spectroscopy, that is very broad in energy and temperature dependent. The Raman shift and lifetime show large differences with the values elsewhere in the Brillouin zone measured by inelastic x-ray scattering (IXS), where its dispersion can be accounted for by standard harmonic phonon theory, adding only a moderate electron-phonon coupling. Here we show that the effects rapidly disappear when electron-phonon coupling is switched off by Al substitution on the Mg sites. Moreover, using IXS with very high wave-vector resolution in ${\mathrm{MgB}}_2$, we can follow the dispersion connecting the Raman and the IXS signal, in agreement with a theory using only electron-phonon coupling but without strong anharmonic terms. The observation is important in order to understand the effects of electron-phonon coupling on zone center phonon modes in ${\mathrm{MgB}}_2$, and also in all metals characterized by a small Fermi velocity in a particular direction, typical for layered compounds.

Figure 1

Left panel: IXS spectra (circle) and fit (lines) for ${\mathrm{Mg}}_{1-x}{\mathrm{Al}}_x{\mathrm{B}}_2$ at the reduced wave vector $q=$(0 0 0.2), with colors coding the content $x$. Error bars can be inferred from the data dispersion. Lines are models of the data, with pseudo-Voigt fits (dashed, as in Ref. [5]) or simple Gaussian (continuous lines) profiles. Right panel: Energy shift for the phonon at $q=$(0 0 0.2) as a function of the Al content $x$, from the fit of the data in the left panel (error bars from fit, mostly smaller than symbols), as well as from calculations in the virtual crystal approximation (VCA).

Figure 2

Calculated and measured dispersion of ${\mathrm{Mg}}_{0.6}{\mathrm{Al}}_{0.4}{\mathrm{B}}_2$. $q$ is given in Brillouin zone (BZ) reciprocal lattice units (r.l.u.).

Figure 3

Left panel: ${\mathrm{Mg}}_{1-x}{\mathrm{Al}}_x{\mathrm{B}}_2$ phonon dispersion along $\mathrm{\Gamma }$-A and $\mathrm{\Gamma }$-M directions: IXS data for $x=0$ (blue diamonds symbols) and 0.4 (white filled diamonds); Raman data for $x=0$ (filled squares, cyan, average values for $T\le 100$ K, and yellow, at ambient temperature, from Ref. [5]) and $x=0.4$ (black square). Error bars from the fit are smaller or of the same order as the symbol size. DFT calculation (red lines) for $x=0$ only (taken from Ref. [5]). The black line represents an empirical renormalization of the DFT calculation (see text). Right panel: zoom on the $E_{2g}$ dispersion along the $\mathrm{\Gamma }$-M direction. The dashed red line corresponds to the DFT calculation (red continuous line in Fig. 3). The black continuous line represents the nonadiabatic correction to the DFT calculations as explained in the text. We approximately indicate empirical adiabatic (pale yellow), and nonadiabatic (pale blue) region, separated by a crossover region in pale green, estimated from the difference between the two calculated dispersion curves.