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Magnetoelastic coupling induced magnetic anisotropy in Co$_2$(Fe/Mn)Si thin films

The influence of epitaxial strain on uniaxial magnetic anisotropy of Co$_{2}$FeSi (CFS) and Co$_{2}$MnSi (CMS) Heusler alloy thin films grown on (001) SrTiO$_3$ (STO) and MgO is reported. The in-plane biaxial strain is susceptible to tune by varying the thickness of the films on STO, while on MgO the films show in-plane easy axis for magnetization (\overrightarrow{M}) irrespective of their thickness. A variational analysis of magnetic free energy functional within the Stoner-Wohlfarth coherent rotation model with out-of-plane uniaxial anisotropy for the films on STO showed the presence of magnetoelastic anisotropy with magnetostriction constant $\approx$ (12.22$\pm$0.07)$\times 10^{-6}$ and (2.02$\pm$0.06)$\times 10^{-6}$, in addition to intrinsic magnetocrystalline anisotropy $\approx$ -1.72$\times 10^{6}$ erg/cm$^{3}$ and -3.94$\times 10^{6}$ erg/cm$^{3}$ for CFS and CMS, respectively. The single-domain phase diagram reveals a gradual transition from in-plane to out-of-plane orientation of magnetization with the decreasing film thickness. A maximum canting angle of 41.5$^{\circ}$ with respect to film plane is predicted for the magnetization of the thinnest (12 nm) CFS film on STO. The distinct behaviour of \overrightarrow{M} in the films with lower thickness on STO is attributed to strain-induced tetragonal distortion.