The Co-based full-Heusler alloys such as Co2FeA1 may find important applications in magnetic random access memory (MRAM) devices employing the spin transfer torque (STT) effect [1].These alloys normally exhibit very small Gilbert damping α as a result of their half metallic nature [2], so their implementation in MRAM has the advantage of low current threshold required for the magnetization switching.When epitaxially grown as a single crystalline thin film on the GaAs (001) or MgO (001) substrate, the Co-based Heusler alloy often displays a pronounced lattice strain-induced magneto-crystalline anisotropy (MCA) with its strength comparable or even larger than that of the intrinsic bulk MCA [3,4].Considering that the strain-dependent MCA, i.e.magneto-elastic anisotropy (MEA), originates from the alteration of spin orbit coupling (SOC) due to the change of the distance between the magnetic atoms, and the intrinsic α is strongly correlated with the SOC, the possible impacts of the strain and the resultant MEA on α need to be examined before designing the appropriate STT-based structures.