  By using a first-principles approach, monolayer PbI$_2$ is found to have
great potential in thermoelectric applications. The linear Boltzmann transport
equation is applied to obtain the perturbation to the electron distribution by
different scattering mechanisms. The mobility is mainly limited by the
deformation-potential interaction with long-wavelength acoustic vibrations at
low carrier concentrations. At high concentrations, ionized impurity scattering
becomes stronger. The electrical conductivity and Seebeck coefficient are
calculated accurately over various ranges of temperature and carrier
concentration. The lattice thermal conductivity of PbI$_2$, 0.065 W/mK at 300
K, is the lowest among other 2D thermoelectric materials. Such ultralow thermal
conductivity is attributed to large atomic mass, weak interatomic bonding,
strong anharmonicity, and localized vibrations in which the vast majority of
heat is trapped. These electrical and phonon transport properties enable high
thermoelectric figure of merit over 1 for both p-type and n-type doping from
300 K to 900 K. A maximum $zT$ of 4.9 is achieved at 900 K with an electron
concentration of 1.9$\times$10$^{12}$ cm$^{-2}$. Our work shows exceptionally
good thermoelectric energy conversion efficiency in monolayer PbI$_2$, which
can be integrated to the existing photovoltaic devices.
