Two-dimensional (2D) high-temperature ferromagnetic materials are important for spintronic application. Fortunately, a highly-air-stable PdSe$_2$ monolayer semiconductor has been made through exfoliation from the layered bulk material. It is very highly desirable to realize robust ferromagnetism, even half-metallic ferromagnetism (100% spin polarization), in such excellent nonmagnetic monolayer semiconductors. Here, the first-principles investigation shows that the PdSe$_2$ monolayer can be made to attain Stoner ferromagnetism with the maximal Curie temperature reaching to 800K, and the hole concentration threshold for ferromagnetism decreases with applied uniaxial stress. Furthermore, half-metallicity can be achieved in some hole concentration regions. For the strain of 10% (uniaxial tensile stress of 4.4 N/m), the monolayer can attain half-metallic ferromagnetism up to 150 K. The magnetic anisotropic energy is suitable to not only stabilizing the 2D ferromagnetism but also realizing fast magnetization reversal. The magnetization can be also controlled by applying a transverse uniaxial stress. The highly-air-stable PdSe$_2$ monolayer, with these advantages, should be promising for spintronic applications.
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