Infrared properties of Mg$_{1-x}$Al$_x($B$_{1-y}$C$_{y}$)$_2$ single crystals in the normal and superconducting state

Abstract

The reflectivity $R (ω)$ of $ab$-oriented Mg$_{1-x}$Al$_x$(B$_{1-y }$C$_y$)$_2$ single crystals has been measured by means of infrared microspectroscopy for $1300<ω<17000$ cm$^{-1}$. An increase with doping of the scattering rates in the $π$ and $σ$ bands is observed, being more pronounced in the C doped crystals. The $σ$-band plasma frequency also changes with doping due to the electron doping, while the $π$-band one is almost unchanged. Moreover, a $σ\toσ$ interband excitation, predicted by theory, is observed at $ω_{IB} \simeq 0.47$ eV in the undoped sample, and shifts to lower energies with doping. By performing theoretical calculation of the doping dependence $ω_{IB}$, the experimental observations can be explained with the increase with electron doping of the Fermi energy of the holes in the $σ$-band. On the other hand, the $σ$ band density of states seems not to change substantially. This points towards a $T_c$ reduction driven mainly by disorder, at least for the doping level studied here. The superconducting state has been also probed by infrared synchrotron radiation for $30<ω<150$ cm$^{-1}$ in one pure and one C-doped sample. In the undoped sample ($T_c$ = 38.5 K) a signature of the $π$-gap only is observed. At $y$ = 0.08 ($T_c$ = 31.9 K), the presence of the contribution of the $σ$-gap indicates dirty-limit superconductivity in both bands.