  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 constant
($K_\mathrm{u}$) 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 Bruno's 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.
