  We have studied the electronic and magnetic states of Co and Mn atoms at the
interface of the Co$_\mathrm{2}$Mn$_{\beta}$Si (CMS)/MgO ($\beta$=0.69, 0.99,
1.15 and 1.29) magnetic tunnel junction (MTJ) by means of x-ray magnetic
circular dichroism. In particular, the Mn composition ($\beta$) dependences of
the Mn and Co magnetic moments were investigated. The experimental spin
magnetic moments of Mn, $m_\mathrm{spin}$(Mn), derived from XMCD weakly
decreased with increasing Mn composition $\beta$ in going from Mn-deficient to
Mn-rich CMS films. This behavior was explained by first-principles calculations
based on the antisite-based site-specific formula unit (SSFU) composition
model, which assumes the formation of only antisite defect, not vacancies, to
accommodate off-stoichiometry. Furthermore, the experimental spin magnetic
moments of Co, $m_\mathrm{spin}$(Co), also weakly decreased with increasing Mn
composition. This behavior was consistently explained by the antisite-based
SSFU model, in particular, by the decrease in the concentration of
Co$_\mathrm{Mn}$ antisites detrimental to the half-metallicity of CMS with
increasing $\beta$. This finding is consistent with the higher TMR ratios which
have been observed for CMS/MgO/CMS MTJs with Mn-rich CMS electrodes.
