Impurity scattering and Friedel oscillations in altermagnets

Altermagnetism is a newly predicted type of magnetism with spin-polarized electron bands but zero net magnetization. Here, we investigate the impurity scattering effect in an altermagnetic metal by using the $T\text{\ensuremath{-}}\mathrm{matrix}$ approach. We first calculate the impurity-induced local density of states (LDOS) in momentum space for a single scalar/magnetic impurity. In the absence (presence) of spin-orbit coupling and Zeeman field, the LDOS exhibits (does not exhibit) ${C}_{4}$ symmetry due to the $d\text{\ensuremath{-}}\mathrm{wave}$ nature of the Fermi surfaces. In real space, we obtain the analytical expressions for the Friedel oscillation of LDOS based on the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ Hamiltonian. The behavior of LDOS oscillations without spin-orbit coupling and Zeeman field also shows ${C}_{4}$ symmetry. The amplitude of the LDOS decreases with ${r}^{\ensuremath{-}1}$ in all directions at long distances, which is the same as that in conventional metals since they both have parabolic dispersion. However, the period of Friedel oscillations in an altermagnet is strongly anisotropic and depends on the direction of observation. Specifically, along the direction with (without) the spin splitting of the Fermi surface, the oscillation periods of the spin-up and spin-down LDOS are different (identical), resulting in an unfixed (a fixed) period at a given energy for the total LDOS. The difference between the two periods is proportional to the magnitude of the spin splitting of the Fermi surface. Our results reveal unique and universal features of the Fermi surfaces of an altermagnet that can be tested by spin-polarized scanning tunneling microscopy.