  Here we use low-temperature scanning tunneling microscopy and spectroscopy to
study the polar surfaces of PdCoO2. On the CoO2-terminated polar surface, we
detect the quasiparticle interference pattern originating from the Rashba
spin-split surface states. On the well-ordered Pd-terminated polar surface, we
observe a regular lattice which has larger lattice constant than the atomic
lattice of PdCoO2. In comparison with the shape of the hexagonal bulk Fermi
surface, we identify this regular lattice as a fully two-dimensional
incommensurate charge modulation that is driven by the Fermi surface nesting.
More interestingly, we also find the moir\'e pattern induced by the
interference between the two-dimensional incommensurate charge modulation in
the Pd layer and the underlying CoO2 lattice. Our results not only show a new
charge modulation on the Pd surface of PdCoO2, but also pave the way for fully
understanding the novel electronic properties of this material.
