  The electronic structures and thermoelectric properties of semiconducting
transition-metal dichalcogenide monolayers $\mathrm{MX_2}$ (M=Zr, Hf, Mo, W and
Pt; X=S, Se and Te) are investigated by combining first-principles and
Boltzmann transport theory, including spin-orbital coupling (SOC). It is found
that the gap decrease increases from S to Te in each cation group, when the SOC
is opened. The spin-orbital splitting has the same trend with gap reducing.
Calculated results show that SOC has noteworthy detrimental effect on p-type
power factor, while has a negligible influence in n-type doping except W cation
group, which can be understood by considering the effects of SOC on the valence
and conduction bands. For $\mathrm{WX_2}$ (X=S, Se and Te), the SOC leads to
observably enhanced power factor in n-type doping, which can be explained by
SOC-induced band degeneracy, namely bands converge. Among all cation groups, Pt
cation group shows the highest Seebeck coefficient, which leads to best power
factor, if we assume scattering time is fixed. Calculated results show that
$\mathrm{MS_2}$ (M=Zr, Hf, Mo, W and Pt) have best p-type power factor for all
cation groups, and that $\mathrm{MSe_2}$ (M=Zr and Hf), $\mathrm{WS_2}$ and
$\mathrm{MTe_2}$ (M=Mo and Pt) have more wonderful n-type power factor in
respective cation group. Therefore, these results may be useful for further
theoretical prediction or experimental search of excellent thermoelectric
materials from semiconducting transition-metal dichalcogenide monolayers.
