  We report on first-principles spin-polarised quantum transport calculations
(from NEGF+DFT) in MgO-spaced magnetic tunnel junctions (MTJs) based on two
different Mn-based Heusler ferrimagnetic metals, namely Mn$_3$Al and Mn$_3$Ga
in their tetragonal DO$_{22}$ phase. The former is a fully compensated
half-metallic ferrimagnet, while the latter is a low-moment
high-spin-polarisation ferrimagnet, both with a small lattice mismatch from
MgO. In identical symmetric and asymmetric interface reconstructions across a
3-monolayer thick MgO barrier for both ferrimagets, the linear response
(low-voltage) spin-transfer torque (STT) and tunneling magneto-resistance (TMR)
effects are evaluated. A larger staggered in-plane STT is found in the Mn$_3$Ga
case, while the STT in Mn$_3$Al vanishes quickly away from the interface
(similarly to STT in ferromagnetic MTJs). The roles are reversed for the TMR,
which is practically 100\% in the half-metallic Mn$_3$Al-based MTJs (using the
conservative definition) as opposed to 60\% in the Mn$_3$Ga case. The weak
dependence on the exact interface reconstruction would suggest
Mn$_3$Ga-Mn$_3$Al solid solutions as a possible route towards optimal trade-off
of STT and TMR in the low-bias, low-temperature transport regime.
