Title: Engineering the electronic and magnetic properties of monolayer TiS$_2$ through systematic transition-metal doping

Abstract: Layered materials that exhibit magnetic ordering in their pristine form are very rare. Several standard approaches, such as adsorption of atoms, introduction of point defects, and edge engineering, have been developed to induce magnetism in two-dimensional materials. In this way, we investigate the electronic and magnetic properties of monolayer TiS$_2$ doped with 3$d$ transition metals (TMs) atoms in both octahedral 1T and trigonal prismatic 1H structures using first-principles calculations. In its pristine form, TiS$_2$ is a non-magnetic semiconductor. The bands near the Fermi energy primarily exhibit $d$ orbital characters, and due to the presence of ideal octahedral and trigonal arrangements, they are well separated from other bands with $p$ character. Upon substituting 3$d$-TM atoms in both structures, a variety of electronic and magnetic phases emerge, including magnetic semiconductor, magnetic half-metal, non-magnetic metal, and magnetic metal. Chromium exhibits the largest magnetic moment in both the 1T and 1H structures. The 1T structure shows a slightly higher magnetic moment of 3.419 $\mu_B$ compared to the 1H structure 3.138 $\mu_B$, attributed to the distorted octahedral structure of the 1T structure. Unlike pristine TiS$_2$, the deficiency in saturation of neighboring S atoms in the presence of impurities leads to the proximity of energy levels of $d$ and $p$ states, resulting in unexpectedly sizable magnetic moments. Another interesting case is Cobalt, which leads to a magnetic moment of approximately 0.805 $\mu_B$ in the 1H structure, while the Co exhibits a non-magnetic state in the 1H structure. These materials demonstrate a high degree of tunability and can be optimized for various magnetic applications.