  Monolayer transition-metal dichalcogenides possess a pair of degenerate
helical valleys in the band structure that exhibit fascinating optical valley
polarization. Optical valley polarization, however, is limited by carrier
lifetimes of these materials. Lifting the valley degeneracy is therefore an
attractive route for achieving valley polarization. It is very challenging to
achieve appreciable valley degeneracy splitting with applied magnetic field. We
propose a strategy to create giant splitting of the valley degeneracy by
proximity-induced Zeeman effect. As a demonstration, our first principles
calculations of monolayer MoTe$_2$ on a EuO substrate show that valley
splitting over 300 meV can be generated. The proximity coupling also makes
interband transition energies valley dependent, enabling valley selection by
optical frequency tuning in addition to circular polarization. The valley
splitting in the heterostructure is also continuously tunable by rotating
substrate magnetization. The giant and tunable valley splitting adds a readily
accessible dimension to the valley-spin physics with rich and interesting
experimental consequences, and offers a practical avenue for exploring device
paradigms based on the intrinsic degrees of freedom of electrons.
