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We investigate the thermoelectric properties of CaAl$_2$Si$_2$-type Zintlphase compounds $AB_2X_2$ ($A$ = Mg, Ca, Sr, Ba, $B$ = Mg, Zn, Cd, and $X$ = P,As, Sb) using first principles band calculations within the Boltzmann transporttheory assuming the constant relaxation time approximation. We introduce theeffective degree of valley degeneracy $n_{TE}$ to focus on the relationshipbetween the thermoelectric properties and the multivalley character of theelectronic band structure around the Fermi level. We also introduce a quantity$\gamma_{TE}$, which takes into account $n_{TE}$ and anisotropy of the valleystructure, and it is found that $\gamma_{TE}$ enables us to well understand theoverall trend of the material dependence of the power factor. We finallysuggest promising thermoelectric materials, e.g. BaMg$_2$P$_2$ for PF $\sim20\mu$W/cmK$^2$ and $ZT > 0.2$ at 300K and SrZn$_2$As$_2$ for PF $\sim35\mu$W/cmK$^2$ and $ZT > 0.35$ at 300K assuming a relaxation time of 10 fs anda lattice thermal conductivity value of 2 W/mK.

First principles study on the thermoelectric performance of CaAl$_2$Si$_2$-type Zintl phase compounds