Raman-active $E_{2g}$ phonon in ${\mathrm{MgB}}_2$: Electron-phonon interaction and anharmonicity

Measurements of the Raman spectra in ${\mathrm{MgB}}_2$ up to 750 K are presented. The width of the $E_{2g}$ phonon line shows a pronounced maximum in the range of 350–400 K. In the same temperature range the frequency exhibits a sharp softening. The $E_{2g}$ phonon line is superimposed on the electronic background, whose frequency also strongly depends on temperature. The phonon self-energies and spectra of electronic Raman scattering were calculated, taking into account a renormalization of ab initio electronic structure due to an electron-phonon scattering. We obtained a satisfactory description of the temperature dependence of the phonon linewidth and proved that it is mainly determined by nonadiabatic effects. Since the wave vector of the $E_{2g}$ phonon is in the nonadiabatic range, the contribution of electrons to phonon damping dominates and determines its nonmonotonic behavior as a function of temperature. When the anharmonic contributions to the phonon frequency are taken into account, the total shift of the $E_{2g}$ phonon energy agrees with the experiment at 300 K, but the estimates of the anharmonic terms in the high-temperature range need further clarification.

Figure 1

The magnetic moment vs the temperature for our sample. Micrograph of rectangular (top) and hexagonal (bottom) ${\mathrm{MgB}}_2$ crystallites in the used sample, as seen in an optical microscope.

Figure 2

Raman susceptibility of ${\mathrm{MgB}}_2$ taken at different temperatures with the linear polarizations using 532 nm excitation.

Figure 3

Raman susceptibility of ${\mathrm{MgB}}_2$ taken at different temperatures with circular polarizations using 532 nm excitation (left panel). Fitted phonon profiles at different temperatures obtained after the subtraction of the electronic background (right panel).

Figure 4

Experimental and calculated $E_{2g}$ phonon frequency (a) and linewidth (b) vs temperature for $q\parallel c$. Dotted line in (a): thermal expansion contribution; dashed line: electron-phonon contribution; solid line: total calculated frequency. Lines in (b): electron-phonon induced linewidth. The inset shows ${\alpha }^{2}F\left(\mathrm{\Omega }\right)$ used in our calculations.

Figure 5

The $E_{2g}$ phonon spectral functions for 40, 140, 240, 340, 440, 540, 640, and 740 K, calculated for $q\parallel c$.

Figure 6

$Q$ dependences of the renormalized ${\omega }_{E_{2g}}$ and imaginary part of the phonon self-energy ${\mathrm{\Pi }}^{″}\left(\omega \right)$. The used probed wave vector distribution $U\left(q\right)$ and position of its maximum are plotted by dotted lines.

Figure 7

The Raman spectra of ${\mathrm{MgB}}_2$ combined from calculated phonon profiles superimposed on the electronic component of the Raman scattering at different temperatures. Spectra were measured (solid lines) and calculated (dashed lines: electronic Raman scattering; dotted lines: total spectra) for $q\parallel c$; the dash-dotted line shows the fit of the electronic Raman scattering spectrum at 370 K by Eq. (6).