We report a systematic investigation on the magnetization relaxation properties of iron-based half-metallic Heusler alloy Fe$_{2}$Cr$_{1-x}$Co_${x}$Si (FCCS) thin films using broadband angular-resolved ferromagnetic resonance. Band structure engineering through Co doping (x) demonstrated by first-principles calculations is shown to tune the intrinsic magnetic damping over an order of magnitude, namely 0.01-0.0008. Notably, the intrinsic damping constants for samples with high Co concentration are among the lowest reported for Heusler alloys and even comparable to magnetic insulator yttrium iron garnet. Furthermore, a significant reduction of both isotropic and anisotropic contributions of extrinsic damping of the FCCS alloys was found in the FCCS films with x=0.5-0.75, which is of particular importance for applications. These results demonstrate a practical recipe to tailor functional magnetization for Heusler alloy-based spintronics at room temperature
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