We performed calculations of electronic, optical and transport properties of graphene on hBN with realistic moire patterns. The latter are produced by structural relaxation using a fully atomistic model. This relaxation turns out to be crucially impo
rtant for electronic properties. We describe experimentally observed features such as additional Dirac points and the Hofstadter butterfly structure of energy levels in a magnetic field. We find that the electronic structure is sensitive to many-body renormalization of the local energy gap.
Electronic analogue of generalized Goos-H{a}nchen shifts is investigated in the monolayer graphene superlattice with one-dimensional periodic potentials of square barriers. It is found that the lateral shifts for the electron beam transmitted through
the monolayer graphene superlattice can be negative as well as positive near the band edges of zero-$bar{k}$ gap, which are different from those near the band edges of Bragg gap. These negative and positive beam shifts have close relation to the Dirac point. When the condition $q_A d_A= -q_B d_B= m pi$ ($m=1,2,3...$) is satisfied, the beam shifts can be controlled from negative to positive when the incident energy is above the Dirac point, and vice versa. In addition, the beam shifts can be greatly enhanced by the defect mode inside the zero-$bar{k}$ gap. These intriguing phenomena can be verified in a relatively simple optical setup, and have potential applications in the graphene-based electron wave devices.
