  We perform hybrid-density functional calculations to investigate the charged
defect formation energy of native point defects in $Mg_2 Si$, $Mg_2 Sn$, and
their solid solutions. The band gap correction by hybrid-density functional is
found to be critical to determine the charged defect density in these
materials. For $Mg_2 Si$, $Mg$ interstitials are dominant and provide
unintentional $n$-type conductivity. Additionally, as the $Mg$ vacancies can
dominate in $Mg$-poor $Mg_2 Sn$, $p$-type conductivity is possible for $Mg_2
Sn$. However, the existence of low formation energy defects such as
$Mg_{Sn}^{1+}$ and $I_{Mg}^{2+}$ in $Mg_2 Sn$ and their diffusion can cause
severe charge compensation of hole carriers resulting in low $p$-type doping
efficiency and thermal degradation. Our results indicate that, in addition to
the extrinsic doping strategy, alloying of $Mg_2 Si$ with $Mg_2 Sn$ under
$Mg$-poor conditions would be necessary to enhance the $p$-type conductivity
with less charge compensation.
