Transition rules and optical properties of armchair graphene nanoribbons embedded in hexagonal boron nitride lattices are studied for the first time. Based on tight binding calculations considering first and second nearest neighbors, we show that the optical transition rules of such structures are completely different from that of conventional graphene nanoribbons. These rules are explained by the symmetry properties of the subband wave functions. The optical spectrum, the quantum efficiency, and the photoresponsivity of different nanoribbons are evaluated and their application in photodetector devices is investigated. The results are verified with first principles calculations.

In this work, we present analytical solutions for the wave functions and energy dispersion of zigzag graphene nanoribbons. A nearest neighbor tight-binding model is employed to describe the electronic band structure of graphene nanoribbons. However, an exact analytical solution for the dispersion relation and the wave functions of zigzag nanoribbons cannot be obtained. We propose two approximations of the discrete energies, which are valid for a wide range of nanoribbon indices. Employing these models, selection rules for optical transitions and optical properties of zigzag graphene nanoribbons are studied.

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