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We report a low-temperature scanning tunneling microscopy study of the chargedensity wave (CDW) order in 1$T$-TiSe$_2$ and Cu$_{0.08}$TiSe$_2$. In pristine1$T$-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) phasewith localized C-CDW domains separated by domain walls. Density of statesmeasurements indicate that the domain walls host an extra population offermions near the Fermi level which may play a role in the emergence ofsuperconductivity in this system. Fourier transform scanning tunnelingspectroscopy studies suggest that the dominant mechanism for CDW formation inthe I-CDW phase may be electron-phonon coupling.

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