Carrier distribution and negative compressibility in graphene-MoS$_{2}$ heterostructures*

We report the investigation of electrical properties and magnetotransport in monolayer graphene - multilayered MoS$_{2}$ heterostructures. The devices are fabricated by dry transfer of graphene layers onto exfoliated MoS$_{2}$. The conductivity dependence on the back-gate bias shows the ambipolar behavior characteristic of graphene, along with a marked saturation of the conductivity on the electron branch. Magnetotransport measurements reveal that the conductivity saturation is the result of electrons populating the lower mobility MoS$_{2}$ layer at a positive, threshold back-gate bias. Experimental data from heterostructures with different thicknesses allow the extraction of the band offset between the MoS$_{2}$ conduction band and the graphene charge neutrality point. Surprisingly, the carrier density in graphene reveals a marked decrease as a function of gate bias near the MoS$_{2}$ population threshold, an observation which implies that electrons in MoS$_{2}$ have negative compressibility at low carrier density.