  We present a theoretical study on the Landau levels (LLs) and magneto-optical
absorption of a two-dimensional semi-Dirac electron system under a
perpendicular magnetic field. Based on an effective k.p Hamiltonian, we find
that the LLs are proportional to the two-thirds power law of the magnetic field
and level index, which can be understood as a hybridization of the LL of
Schrodinger and Dirac electrons with new features. With the help of Kubo
formula, we find the selection rule for interband (intraband) magneto-optical
transition is anisotropic (isotropic). Specifically, the selection rules for
interband magneto-optical transitions are $\Delta n$=0, $\pm2$ ($\pm2$, $\pm4$)
for linearly polarized light along the linear (parabolic) dispersion direction,
while the selection rules for the intraband transition are $\Delta n$=$\pm1$,
$\pm3$ regardless of the polarization direction of the light. Further, the
magneto-optical conductivity for interband (intraband) transition excited by
linearly polarized light along the linear dispersion direction is two (one)
orders of magnitude larger than that along the parabolic dispersion direction.
This anisotropic magneto-optical absorption spectrum clearly reflects the
structure of the LLs, and results in a strong linear dichroism. Importantly, a
perfect linear dichroism with magnetic-field tunable wavelength can be achieved
by using the interband transition between the two lowest LLs, i.e, from Ev0 to
Ec0. Our results shed light on the magneto-optical property of the two
dimensional semi-Dirac electron systems and pave the way to design magnetically
controlled optical devices.
