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

TL;DR

This study investigates the infrared optical properties of Mg$_{1-x}$Al$_x$(B$_{1-y}$C$_{y}$)$_2$ single crystals, revealing doping-induced changes in scattering rates, interband excitations, and superconducting gaps, primarily driven by disorder effects.

Contribution

It provides experimental and theoretical insights into how doping affects the electronic structure and superconductivity in MgB$_2$-based crystals, especially the interband excitations and gap signatures.

Findings

Doping increases scattering rates in $c$ and $c$ bands, more so with C doping.

The $c$-band plasma frequency changes with doping, $c$-band remains nearly unchanged.

Interband excitation shifts to lower energy with doping, explained by Fermi energy increase.

Abstract

The reflectivity $R (\omega)$ 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 $1300<\omega<17000$ cm$^{-1}$. An increase with doping of the scattering rates in the $\pi$ and $\sigma$ bands is observed, being more pronounced in the C doped crystals. The $\sigma$-band plasma frequency also changes with doping due to the electron doping, while the $\pi$-band one is almost unchanged. Moreover, a $\sigma\to\sigma$ interband excitation, predicted by theory, is observed at $\omega_{IB} \simeq 0.47$ eV in the undoped sample, and shifts to lower energies with doping. By performing theoretical calculation of the doping dependence $\omega_{IB}$, the experimental observations can be explained with the increase with electron doping of the Fermi energy of the holes in the $\sigma$-band. On the other hand, the $\sigma$ 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 $30<\omega<150$ cm$^{-1}$ in one pure and one C-doped sample. In the undoped sample ($T_c$ = 38.5 K) a signature of the $\pi$-gap only is observed. At $y$ = 0.08 ($T_c$ = 31.9 K), the presence of the contribution of the $\sigma$-gap indicates dirty-limit superconductivity in both bands.