  The recent discovery of high thermoelectric performance in Mg$_3$Sb$_2$ has
been critically enabled by the success in $n$-type doping of this material,
which is achieved under Mg-rich growth conditions, typically with chalcogens
(Se, Te) as extrinsic dopants. Using first-principles defect calculations, we
previously predicted that higher electron concentrations ($\sim10^{20}$
cm$^{-3}$) can be achieved in Mg$_3$Sb$_2$ by doping with La instead of Se or
Te. Subsequent experiments showed that free electron concentration in La-doped
Mg$_3$Sb$_{2-x}$Bi$_x$ indeed exceeds those in the Te-doped material. Herein,
we further investigate $n$-type doping of Mg$_3$Sb$_2$ and predict that, in
addition to La, other group-3 elements (Sc, Y) are also effective as $n$-type
dopants; Y is as good as La while Sc slightly less. Overall, we find that
doping with any group-3 elements should lead to higher free electron
concentrations than doping with chalcogens.
