Abstract
We study tunneling of quasiparticles through potential barriers in a graphene monolayer with the mass gap using a semiclassical (WKB) approach. The main equations are derived in a way similar to the WKB theory for the Schr\odinger equation, which allows for explicit solutions at all orders. The analog of the classical action is used to distinguish types of possible stationary states in the system. The analysis focuses on the resonant scattering and the hole states localized in the vicinity of a barrier that are often overlooked. The scattering coefficients for the physically interesting limits are obtained by matching the WKB approximation with the known solutions at turning points. The localized states demonstrate unconventional properties and lead to alterations of the single particle density of states.
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