In this Letter, we investigate the effect of next-to-nearest atom hopping on Klein tunneling in graphene. An effective quantum dynamics equation is obtained based on an emergent generalized Dirac structure by analyzing the tight-binding model beyond the linear regime. We show that this structure has some interesting theoretical properties. First, it can be used to simplify quantum transport calculations used to characterize Klein tunneling; second, it is not chirally symmetric as hinted by previous work. Finally, it is reminiscent of theories on a space with a discrete topology. Exploiting these properties, we show that the discrete topology of the crystal lattice has an effect on the Klein tunneling, which can be experimentally probed by measuring the transmittance through $n\text{−}p\text{−}n$ junctions. We argue that this simulates some quantum gravity models using graphene and we propose an experiment to perform such measurements.

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• Quantum gravity
• Quantum simulation
• Transport phenomena
• Graphene