Semiclassical theory of potential scattering for massless Dirac fermions

• A careful semiclassical description of scattering for Dirac fermions is constructed.
• Uniform formulas for reflection and transmission coefficients are obtained.
• Above-barrier scattering is treated as virtual Klein tunneling.
• Good agreement with numerical results is achieved.

In this paper we study scattering of two-dimensional massless Dirac fermions by a potential that depends on a single Cartesian variable. Depending on the energy of the incoming particle and its angle of incidence, there are three different regimes of scattering. To find the reflection and transmission coefficients in these regimes, we apply the Wentzel–Kramers–Brillouin (WKB), also called semiclassical, approximation. We use the method of comparison equations to extend our prediction to nearly normal incidence, where the conventional WKB method should be modified due to the degeneracy of turning points. We compare our results to numerical calculations and find good agreement.