  Short-period super-Earth-sized planets are common. Explaining how they form
near their present orbits requires understanding the structure of the inner
regions of protoplanetary discs. Previous studies have argued that the hot
inner protoplanetary disc is unstable to the magneto-rotational instability
(MRI) due to thermal ionization of potassium, and that a local gas pressure
maximum forms at the outer edge of this MRI-active zone. Here we present a
steady-state model for inner discs accreting viscously, primarily due to the
MRI. The structure and MRI-viscosity of the inner disc are fully coupled in our
model; moreover, we account for many processes omitted in previous such models,
including disc heating by both accretion and stellar irradiation, vertical
energy transport, realistic dust opacities, dust effects on disc ionization and
non-thermal sources of ionization. For a disc around a solar-mass star with a
standard gas accretion rate ($\dot{M}$$\sim$$10^{-8}$M$_\odot$yr$^{-1}$) and
small dust grains, we find that the inner disc is optically thick, and the
accretion heat is primarily released near the midplane. As a result, both the
disc midplane temperature and the location of the pressure maximum are only
marginally affected by stellar irradiation, and the inner disc is also
convectively unstable. As previously suggested, the inner disc is primarily
ionized through thermionic and potassium ion emission from dust grains, which,
at high temperatures, counteract adsorption of free charges onto grains. Our
results show that the location of the pressure maximum is determined by the
threshold temperature above which thermionic and ion emission become efficient.
