Impact of Ge, Ga, and Al doping on the mechanical and electronic properties of Cr$_3$Si: insights from first-principles calculations

Abstract

This study systematically investigates the effects of Ge, Ga, and Al doping on the mechanical and electronic properties of cubic Cr$_3$Si using first-principles density functional theory (DFT). Doping increases lattice constants from 4.50 {\AA} for undoped Cr$_3$Si to 4.51-4.53 {\AA} (Ge), 4.52-4.54 {\AA} (Ga), and 4.51-4.54 {\AA} (Al) as doping concentrations increase from 12.5 $\%$ to 50 $\%$. Negative formation enthalpies across all configurations confirm thermodynamic stability, with values ranging from -0.35 eV/atom for undoped Cr$_3$Si to -0.33 eV/atom (Ge), -0.31 eV/atom (Al), and -0.25 eV/atom (Ga) at 50 $\%$ doping. Mechanical properties exhibit significant degradation with increased doping: bulk modulus decreases from 248.7 GPa for undoped Cr$_3$Si to 241 GPa (12.5 $\%$), 238 GPa (25 $\%$), 235 GPa (37.5 $\%$), and 231 GPa (50 $\%$) for Ge doping, with similar trends for Ga (230 GPa at 50 $\%$) and Al (232 GPa at 50 $\%$). Shear modulus and Young's modulus follow similar reductions, with shear modulus going from 158.9 GPa to 147 GPa (Ge), 145 GPa (Ga), and 147 GPa (Al) at 50 $\%$ doping. Elastic anisotropy increases notably with Al and Ga doping, while Ge maintains a relatively isotropic behavior. The wave velocities and Debye temperatures decrease for all dopants, with Debye temperature dropping from 720 K for undoped Cr$_3$Si to 700 K (Ge), 685 K (Ga), and 690 K (Al) at 50 $\%$ doping, reflecting a softer lattice and diminished thermal conductivity. While Al and Ga doping introduce higher anisotropy and reduce mechanical rigidity, Ge doping preserves isotropic mechanical behavior, making it the most suitable dopant for applications requiring balanced mechanical and thermal properties. These findings offer critical insights into tailoring Cr$_3$Si-based alloys for high-performance applications, highlighting trade-offs between stiffness, anisotropy, and thermal performance.