Magnetic Order and Magneto-Elasticity in the Electronic Excitations of Gd-$i$-MAX

We report the investigation of electronic collective modes in rare-earth-based magnets (Mo$_{2/3}$RE$_{1/3}$)$_2$AlC (also known as RE-$i$-MAX phases), where RE=Gd, Yb, and Dy, using single crystal samples. A detailed investigation of the Raman spectra of Gd-$i$-MAX samples at low temperatures, with a focus on the phonon behavior in relation to the antiferromagnetic (AFM) phase transition at 26 K is presented. Significant shifts in the central frequencies of several low-frequency phonon modes were observed below 25 K, correlating with the N\'{e}el transition. Integrated Raman intensity measurements indicated a reduction in the electronic background below the AFM transition temperature, suggesting the opening of a magnetic gap. Our analysis showed no new phonon modes. Therefore, we do not see any indication of a Brillouin zone folding of phonon mode to the $\Gamma$-point in our measurement. However, the hardening of all phonon modes at low temperatures points to a strong spin-phonon coupling effect. Using a temperature-dependent model of phonon frequency, we determined the spin-phonon coupling constant $\lambda$ to be less than 0.1 cm$^{-1}$ for all frequencies, which is of the same order of magnitude as found in other antiferromagnetic materials such as MnF$_{2}$ and FeF$_{2}$ with $T_N=68~K$ and $T_N=78~K$, respectively, but significantly lower than that of $CuO$ with $T_N=213~K$.