Regular beating feature of Friedel oscillations induced by imperfections in a Mexican-hat dispersion material

The Mexican-hat dispersion exhibits a special concentric contour in its Fermi surface, giving rise to novel features for scattering from imperfections. Here, we study the mechanisms of point-impurity and line-defect scatterings in a material with a Mexican-hat dispersion. By adopting Green's function combined the T-matrix approximation, we calculate the local density of states (LDOS) in the momentum space (also called FT-LDOS) and real space near the point impurity. We find that the pattern of the FT-LDOS effectively reflects the shape of the Fermi surface. Notably, inside the Mexican hat, three scattering processes occur with three characteristic wave vectors, i.e., the inter-surface scattering between the inner and outer Fermi surfaces, and the intra-surface scattering on both Fermi surfaces, which leads to the LDOS in real space exhibiting a regular beating feature. The LDOS for each scattering exhibits an asymptotic behavior with decay index x^-1, similar to the parabolic dispersion, although the Mexican-hat dispersion has a quartic form. By using the stationary phase approximation, we calculate the LDOS oscillation in real space near the line defect. It is observed that two scatterings occur between two pairs of stationary phase points on the Fermi surfaces with two characteristic wave vectors, also leading to the beating feature for the LDOS. The LDOS for each scattering exhibits an asymptotic behavior with decay index x^-1/2, also similar to the parabolic dispersion. Remarkably, the emergence of regular beating features for scattering from a point impurity or line defect is different from what is observed for 2D electron-gas and Dirac-electron systems. Our results underscore a distinctive aspect of the Friedel oscillation in a Mexican-hat dispersion material, and reveal unique features of the Fermi surfaces, which can be tested using a scanning tunneling microscope.