Perpendicular magnetization is essential for high-density memory application using magnetic materials. High-spin polarization of conduction electrons is also required for realizing large electric signals from spin-dependent transport phenomena. Heusler alloy is a well-known material class showing the half-metallic electronic structure. However, its cubic lattice nature favors in-plane magnetization and thus minimizes the perpendicular magnetic anisotropy (PMA), in general. This study focuses on an inverse-type Heusler alloy, Mn$_{2-delta}$CoGa$_{1+delta}$ (MCG) with a small off-stoichiometry ($delta$) , which is expected to be a half-metallic material. We observed relatively large uniaxial magnetocrystalline anisotropy energy ($K_mathrm{u}$) of the order of 10$^5$ J/m$^3$ at room temperature in MCG films with a small tetragonal distortion of a few percent. A positive correlation was confirmed between the $c/a$ ratio of lattice constants and $K_mathrm{u}$. Imaging of magnetic domains using Kerr microscopy clearly demonstrated a change in the domain patterns along with $K_mathrm{u}$. X-ray magnetic circular dichroism (XMCD) was employed using synchrotron radiation soft x-ray beam to get insight into the origin for PMA. Negligible angular variation of orbital magnetic moment ($Delta m_mathrm{orb}$) evaluated using the XMCD spectra suggested a minor role of the so-called Brunos term to $K_mathrm{u}$. Our first principles calculation reasonably explained the small $Delta m_mathrm{orb}$ and the positive correlation between the $c/a$ ratio and $K_mathrm{u}$. The origin of the magnetocrystalline anisotropy was discussed based on the second-order perturbation theory in terms of the spin-orbit coupling, claiming that the mixing of the occupied $uparrow$- and the unoccupied $downarrow$-spin states is responsible for the PMA of the MCG films.
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