Scattering by Quasi-Circular Potentials in Two-Dimensional Dirac Materials

We investigate electron scattering by quasi-circular potential regions in two-dimensional Dirac materials. While the case of a perfectly circular potential step admits an exact analytical solution based on pseudospin angular-momentum conservation, small deviations from circular symmetry require a more general treatment. We develop a perturbative method for boundaries whose radius varies smoothly with angle, expanding the boundary conditions in Fourier and Taylor series to obtain analytic corrections to the scattering and transmission amplitudes. Using these results, we calculate the differential cross section for plane-wave incidence and analyze how angular deformations modify the scattering patterns. Comparison between the two inequivalent valleys shows that the broken symmetry leads to valley-dependent reflection, suggesting a possible mechanism for valley filtering. This approach provides a compact analytical framework to describe non-circular scattering in Dirac systems.