Surface states and quasiparticle interference in Bernal and rhombohedral graphite with and without trigonal warping

We use an exact analytical technique [Phys. Rev. B \textbf{101}, 115405 (2020), Phys. Rev. B \textbf{102}, 165117 (2020)] to recover the surface Green's functions for Bernal (ABA) and rhombohedral (ABC) graphite. For rhombohedral graphite we recover the predicted surface flat bands. For Bernal graphite we find that the surface state spectral function is similar to the bilayer one, but the trigonal warping effects are enhanced, and the surface quasiparticles have a much shorter lifetime. We subsequently use the T-matrix formalism to study the quasiparticle interference patterns generated on the surface of semi-infinite ABA and ABC graphite in the presence of impurity scattering. We compare our predictions to experimental STM data of impurity-localized states on the surface of Bernal graphite which appear to be in a good agreement with our calculations.