Inclusion of the sample-tip interaction term in the theory of tip-enhanced Raman spectroscopy

The scattering process in tip-enhanced Raman spectroscopy (TERS) can occur in distinct ways, depending on the number of interaction events and their respective time sequences. These events are fundamentally described by the interaction of the radiation field with the plasmonic nano-structure that generates local enhancement or with the scattering medium itself. The interactions are historically denominated as T, in reference to field-tip interaction, and S, referencing field-sample interaction. The TERS intensity was analytically derived for the ST and TST terms in Phys. Rev. X 4, 031054 (2014). Here we provide further development on the TERS theory by presenting a comprehensive description of the physical picture and the mathematical steps for the obtention of analytical expressions that account for the TERS intensity related to the TS term for one- and two-dimensional samples. Finally, we performed a tip-approach TERS experiment in graphene with considerably higher accuracy than what has been reported in the literature and, with the inclusion of the TS term, the phonon coherence length of graphene, as measured by TERS, is revised to $\ensuremath{\approx}40$ nm.