  Two-dimensional gapped semi-Dirac (GSD) materials are systems with a finite
band gap that their charge carriers behave relativistically in one direction
and Schr\"odinger-like in the other. In the present work, we show that besides
the two well-known energy bands features (curvature and chirality), the
band-gap parameter also play a crucial role in the index- and magnetic
field-dependence of the Landau levels (LLs) in a GSD system. We take the
monolayer phosphorene as a GSD representative example to explicitly provide
physical insights into the role of this parameter in determining the index- and
magnetic field-dependence of LLs. We derive an effective one-dimensional
Schr\"odinger equation for charge carriers in the presence of a perpendicular
magnetic field and argue that the form of its effective potential is clearly
sensitive to a dimensionless band-gap that is tunable by structural parameters.
The theoretical magnitude of this effective gap and its interplay with oval
shape $k$-space cyclotron orbits resolve the seeming contradiction in
determining the type of the quantum Hall effect in the pristine monolayer
phosphorene. Our results strongly confirm that the dependence of LLs on the
magnetic field in this GSD material is as conventional two-dimensional
semiconductor electron gases up to a very high field regime. Using the
strain-induced gap modification scheme, we show the field dependence of the LLs
continuously evolves into $B^{2/3}$ behavior, which holds for a gapless
semi-Dirac system. The highlighted role of the band-gap parameter may affect
the consequences of the band anisotropy in the physical properties of a GSD
material, including magnetotransport, optical conductivity, dielectric
function, and thermoelectric performance.

Mesoscale and Nanoscale Physics