Influence of domain walls in the incommensurate charge density wave state of Cu intercalated 1$T$-TiSe$_2$

We report a low-temperature scanning tunneling microscopy study of the charge density wave (CDW) order in 1$T$-TiSe$_2$ and Cu$_{0.08}$TiSe$_2$. In pristine 1$T$-TiSe$_2$ we observe a long-range coherent commensurate CDW (C-CDW) order. In contrast, Cu$_{0.08}$TiSe$_{2}$ displays an incommensurate CDW (I-CDW) phase with localized C-CDW domains separated by domain walls. Density of states measurements indicate that the domain walls host an extra population of fermions near the Fermi level which may play a role in the emergence of superconductivity in this system. Fourier transform scanning tunneling spectroscopy studies suggest that the dominant mechanism for CDW formation in the I-CDW phase may be electron-phonon coupling.