Absence of inter-layer tunnel coupling of $K$-valley electrons in bilayer MoS$_{2}$

In Bernal stacked bilayer graphene interlayer coupling significantly affects the electronic bandstructure compared to monolayer graphene. Here we present magnetotransport experiments on high-quality $n$-doped bilayer MoS$_{2}$. By measuring the evolution of the Landau levels as a function of electron density and applied magnetic field we are able to investigate the occupation of conduction band states, the interlayer coupling in pristine bilayer MoS$_{2}$, and how these effects are governed by electron-electron interactions. We find that the two layers of the bilayer MoS$_{2}$ behave as two independent electronic systems where a two-fold Landau level's degeneracy is observed for each MoS$_{2}$ layer. At the onset of the population of the bottom MoS$_{2}$ layer we observe a large negative compressibility caused by the exchange interaction. These observations, enabled by the high electronic quality of our samples, demonstrate weak interlayer tunnel coupling but strong interlayer electrostatic coupling in pristine bilayer MoS$_{2}$. The conclusions from the experiments may be relevant also to other transition metal dichalcogenide materials.