We propose the design of low strained and energetically favourable mono and bilayer graphene overlayer on anatase TiO$_2$ (001) surface and examined the electronic structure of the interface with the aid of first principle calculations. In the absence of hybridization between surface TiO$_2$ and graphene states, dipolar fluctuations govern the minor charge transfer across the interface. As a result, both the substrate and the overlayer retain their pristine electronic structure. The interface with the monolayer graphene retains its gapless linear band dispersion irrespective of the induced epitaxial strain. The potential gradient opens up a few meV bandgap in the case of Bernal stacking and strengthens the interpenetration of the Dirac cones in the case of hexagonal stacking of the bilayer graphene. The difference between the macroscopic average potential of the TiO$_2$ and graphene layer(s) in the heterostructure lies in the range 3 to 3.13 eV, which is very close to the TiO$_2$ bandgap ($sim$ 3.2 eV). Therefore, the proposed heterostructure will exhibit enhanced photo-induced charge transfer and the graphene component will serve as a visible light sensitizer.
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