We investigate the impact of mechanical strains and a perpendicular electric field on the electronic and magnetic ground-state properties of two-dimensional monolayer CrI$_3$ using density functional theory. We propose a minimal spin model Hamiltonian, consisting of symmetric isotropic exchange interactions, magnetic anisotropy energy, and Dzyaloshinskii-Moriya (DM) interactions, to capture most pertinent magnetic properties of the system. We compute the mechanical strain and electric field dependence of various spin-spin interactions. Our results show that both the amplitudes and signs of the exchange interactions can be engineered by means of strain, while the electric field affects only their amplitudes. However, strain and electric fields affect both the directions and amplitudes of the DM vectors. The amplitude of the magnetic anisotropy energy can also be substantially modified by an applied strain. We show that in comparison with an electric field, strain can be more efficiently used to manipulate the magnetic and electronic properties of the system. Notably, such systematic tuning of the spin interactions is essential for the engineering of room-temperature spintronic nanodevices.
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