  When complex mechanisms are involved, pinpointing high-performance materials
within large databases is a major challenge in materials discovery. We focus
here on phonon-limited conductivities, and study 2D semiconductors doped by
field effects. Using state-of-the-art density-functional perturbation theory
and Boltzmann transport equation, we discuss 11 monolayers with outstanding
transport properties. These materials are selected from a computational
database of exfoliable materials providing monolayers that are dynamically
stable and that do not have more than 6 atoms per unit cell. We first analyse
electron-phonon scattering in two well-known systems: electron-doped InSe and
hole-doped phosphorene. Both are single-valley systems with weak
electron-phonon interactions, but they represent two distinct pathways to fast
transport: a steep and deep isotropic valley for the former and strongly
anisotropic electron-phonon physics for the latter. We identify similar
features in the database and compute the conductivities of the relevant
monolayers. This process yields several high-conductivity materials, some of
them only very recently emerging in the literature (GaSe, Bi$_2$SeTe$_2$,
Bi$_2$Se$_3$, Sb$_2$SeTe$_2$), others never discussed in this context
(AlLiTe$_2$, BiClTe, ClGaTe, AuI). Comparing these 11 monolayers in detail, we
discuss how the strength and angular dependency of the electron-phonon
scattering drives key differences in the transport performance of materials
despite similar valley structure. We also discuss the high conductivity of
hole-doped WSe$_2$, and how this case study shows the limitations of a
selection process that would be based on band properties alone.
