Panoply of Ni-Doping-Induced Reconstructions, Electronic Phases, and Ferroelectricity in 1T-MoS$_2$

The distorted phases of monolayer 1T-MoS$_2$ have distinct electronic properties, with potential applications in optoelectronics, catalysis, and batteries. We theoretically investigate the use of Ni-doping to generate distorted 1T phases, and find not only the ones usually reported but also two further phases (${3} \times {3}$ and ${4} \times {4}$), depending on the concentration and the substitutional or adatom doping site. Corresponding pristine phases are stable after dopant removal, which might offer a potential route to experimental synthesis. We find large ferroelectric polarizations, most notably in ${3} \times {3}$ which -- compared to the recently measured 1T$''$ -- has 100 times greater ferroelectric polarization, a lower energy, and a larger bandgap. Doped phases include exotic multiferroic semimetals, ferromagnetic polar metals, and improper ferroelectrics with only in-plane polarization switchable. The pristine phases have unusual multiple gaps in the conduction bands, with possible applications for intermediate band solar cells, transparent conductors, and nonlinear optics.