  Structure-property relationships have always been guiding principles in
discovering new materials. Here we explore the relationships to discover novel
two-dimensional (2D) materials with the goal of identifying 2D magnetic
semiconductors for spintronics applications. In particular, we report a density
functional theory + $U$ study of single-layer antiferromagnetic (AFM)
semiconductor CoS$_2$ with the pentagonal structure forming the so-called Cairo
Tessellation. We find that this single-layer magnet exhibits an indirect
bandgap of 1.06 eV with light electron and hole effective masses of 0.03 and
0.10 $m_0$, respectively, which may lead to high carrier mobilities. The hybrid
density functional theory calculations correct the bandgap to 2.24 eV. We also
compute the magnetocrystalline anisotropy energy (MAE), showing that the easy
axis of the AFM ordering is out of plane with a sizable MAE of 153 $\mu$eV per
Co ion. We further calculate the magnon frequencies at different spin-spiral
vectors, based on which we estimate the N$\acute{e}$el temperatures to be 20.4
and 13.3 K using the mean field and random phase approximations, respectively.
We then apply biaxial strains to tune the bandgap of single-layer pentagonal
CoS$_2$. We find that the energy difference between the ferromagnetic and AFM
structures strongly depends on the biaxial strain, but the ground state remains
the AFM ordering. Although the low critical temperature prohibits the magnetic
applications of single-layer pentagonal CoS$_2$ at room temperature, the
excellent electrical properties may find this novel single-layer semiconductor
applications in optoelectronic nanodevices.
