Theory for tip-enhanced Raman spectroscopy of two-dimensional materials including out-of-plane Raman response
Tip-Enhanced Raman Spectroscopy (TERS) can be used to make nanoscale spatial measurements of 2D materials, such as graphene and transition metal dichalcogenides (TMDs). The TERS theory introduced in [L. Cançado et al., Phys. Rev. X 4, 031054 (2014)], however, was tailored for graphene, whose out-of-plane Raman response is neglected. In the present work, we include the out-of-plane response in the TERS theory. In doing so, we provide an exact analytical expression for the field propagation between the tip and the sample, and show that the contribution to the TERS signal that scatters first at the sample, then at the tip (sample-tip, or TS) is important only when the out-of-plane response is significant. We extensively study the variation of TERS experimental measurements when varying physical parameters of the system, like the tip radius, the out-of-plane response, the TERS coherence length, and others. It becomes evident that the TERS enhancement is very sensitive to the out-of-plane Raman response of the phonon mode, while normalized tip-approach measurements are more sensitive to the coherence length, and we show that the medium refractive index leads to an effective tip enhancement factor $f_e$. Our results lead to the conclusion that, in general, a strong TERS enhancement is a necessary condition for investigating the physics discussed here, which here means surveying the difference in TERS signals between different Raman modes. We use our model to analyze some graphene TERS experiments, showing that they are consistent with a negligible out-of-plane Raman response and a nonzero TERS coherence length in the fitting.