Theory of Spatial Coherence in Near-Field Raman Scattering

A theoretical study describing the coherence properties of near-field Raman scattering in two- and one-dimensional systems is presented. The model is applied to the Raman modes of pristine graphene and graphene edges. Our analysis is based on the tip-enhanced Raman scheme, in which a sharp metal tip located near the sample surface acts as a broadband optical antenna that transfers the information contained in the spatially correlated (but nonpropagating) near field to the far field. The dependence of the scattered signal on the tip-sample separation is explored, and the theory predicts that the signal enhancement depends on the particular symmetry of a vibrational mode. The model can be applied to extract the correlation length $L_{\mathrm{c}}$ of optical phonons from experimentally recorded near-field Raman measurements. The coherence properties of optical phonons have been broadly explored in the time and frequency domains, and the spatially resolved approach presented here provides a complementary methodology for the study of local material properties at the nanoscale.

Popular Summary

The importance of spatial coherence in optics is well established, but so far it has been neglected in Raman spectroscopy targeting vibrational modes. Raman scattering, which is an inelastic process, has been broadly treated in papers and classical textbooks as spatially incoherent, but we show that inelastic light scattering in the near-field regime is a partially coherent process that can be used to measure nanoscale correlation lengths in various material systems.

We theoretically investigate the Raman modes of pristine monolayer graphene to determine how the scattered signal depends on the distance between the sample and a laser-irradiated gold tip; the tip acts as a broadband optical antenna to transmit information from the near field to the far field. As the correlation length increases, we find increasingly different behaviors for the strengths of various bands present in the Raman spectrum of graphene. We note that the characteristic correlation lengths are nearly an order of magnitude smaller than optical wavelengths. As a result of coherence, we find that the Raman intensities on the nanoscale depend strongly on phonon symmetry and spatial confinement.

Our work presents a theoretical breakthrough in our understanding of inelastic scattering and defines a new paradigm for studying correlation properties in emerging material systems.

Figure 1

Illustration of two individual scattering paths associated with a scatterer $\mathbb{D}$ irradiated by the field $\mathbf{E}(\omega )$. On length scales $|{\mathbf{r}}_1-{\mathbf{r}}_2|$ smaller than the phonon correlation length $L_{\mathrm{c}}$ of a vibrational mode $\gamma$, the partial fields $\overleftrightarrow{\mathrm{G}}({\mathbf{r}}_0,{\mathbf{r}}_1;{\omega }_{\mathrm{s}}){\mathbf{p}}^{\gamma }({\mathbf{r}}_1;{\omega }_{\mathrm{s}},\omega )$ and $\overleftrightarrow{\mathrm{G}}({\mathbf{r}}_0,{\mathbf{r}}_2;{\omega }_{\mathrm{s}}){\mathbf{p}}^{\gamma }({\mathbf{r}}_2;{\omega }_{\mathrm{s}},\omega )$ add coherently at location ${\mathbf{r}}_0$ of the detector. On the other hand, for length scales larger than $L_{\mathrm{c}}$, there is no phase correlation between the scattering events; hence, the partial fields at the detector add incoherently.

Figure 2

Illustration of spatially resolved near-field Raman scattering of a graphene sample. The electric field $\mathbf{E}$ confined to the apex of a laser-irradiated gold tip interacts locally with the graphene lattice characterized by the Raman polarizability ${\overleftrightarrow{\alpha }}^{\gamma }$, where $\gamma$ denotes a specific phonon mode ($\gamma \in [\mathrm{G},\mathrm{D},{\mathrm{G}}^{\prime }])$.

Figure 3

Interaction series in TERS. Red arrows represent induced dipoles; black arrows indicate electromagnetic interactions.

Figure 4

Comparison of original function (top panels) and approximation (bottom panels) according to Eq. (31). The left panels correspond to $n=j$ ($=x$ in this case), while the right panels correspond to $n\ne j$. In both cases, the following fitting constants were used: $a_0=0.74$, $b_0=4.0$, $c_0=0.08$, and $d_0=1.5$.

Figure 5

Distance dependence of the $\mathbf{TST}$ signal for the G and ${\mathrm{G}}^{\prime }$ bands. Panels (a-d) are evaluated for different correlation lengths $L_{\mathrm{c}}$: 0, 15, 30, and 45 nm, respectively. In all cases, we used ${\tilde{f}}_{\mathrm{e}}=3$. The signal $S({\mathbf{r}}_0)$ is normalized to 1 for $z_{\min }=20\mathrm{nm}$, which corresponds to the closest tip-sample distance for $r_{\text{tip}}=15\mathrm{nm}$, considering that the minimal distance between the tip apex and the sample is about 5 nm. For $L_{\mathrm{c}}=0$, the curves for G and ${\mathrm{G}}^{\prime }$ coincide. This case corresponds to an incoherent addition of the Raman response in different points of the graphene lattice. As $L_{\mathrm{c}}$ increases, interferences between neighboring lattice points which are symmetric with respect to the lateral location of the tip give rise to a distance dependence that is different for the G and ${\mathrm{G}}^{\prime }$ bands.

Figure 6

Symmetries of near-field Raman scattering in the $\mathbf{TST}$ configuration. The yellow spots represent the top view of the tip (axis along the $z$ direction) with the graphene lattice underneath (lying in the $xy$ plane). In (a), the $\otimes$ symbol represents the tip dipole induced by the incident field. The green arrows represent the $x$ and $y$ in-plane components of the electric field generated by the tip dipole at the graphene lattice. The respective induced Raman dipoles in the graphene plane are represented by the red arrows in (b-d) for the modes with ${\mathrm{A}}_1$, ${\mathrm{E}}_{2\mathrm{g}1}$, and ${\mathrm{E}}_{2\mathrm{g}2}$ symmetries, respectively. By considering the fully coherent case ($L_{\mathrm{c}}\to \infty$), the scattered fields generated by the ${\mathrm{A}}_1$ Raman dipoles add constructively at the tip apex, generating a strong induced dipole at the tip [represented by the $\otimes$ symbol in (b)]. On the other hand, the field generated by the Raman dipoles interfere destructively at the tip apex for the ${\mathrm{E}}_{2\mathrm{g}1}$ and ${\mathrm{E}}_{2\mathrm{g}2}$ symmetries [(c) and (d), respectively].

Figure 7

Comparison of original function (top panel) and approximation (bottom panel) according to Eq. (43), for $j=x$. The following fitting constants were used: $a_0^{\prime }=0.78$, $b_0^{\prime }=2.4$, $c_0^{\prime }=0.18$, and $d_0^{\prime }=0.56$.

Figure 8

Distance dependence of the $\mathbf{ST}$ signal for the G and ${\mathrm{G}}^{\prime }$ bands. Panels (a-d) account for different values of the correlation length $L_{\mathrm{c}}$: 0, 15, 30, and 45 nm, respectively, as indicated in the graphics. In all cases, we used ${\tilde{f}}_{\mathrm{e}}=3$ and $r_{\text{tip}}=15\mathrm{nm}$. The signal $S({\mathbf{r}}_0)$ is normalized to 1 at $z_{\min }=20\mathrm{nm}$. Compared to the $\mathbf{TST}$ signal shown in Fig. 5, the $\mathbf{ST}$ signal presents a weaker decay and therefore contributes to the measured signal only for larger tip-sample distances.

Figure 9

Dependence of the $\mathbf{TST}$ of the D band on tip-sample separation $z$. Panels (a-d) account for different values of $L_{\mathrm{c}}$: 0, 15, 30, and 45 nm, respectively, as indicated in the graphics. In all cases, we used ${\tilde{f}}_{\mathrm{e}}=3$ and $r_{\text{tip}}=15\mathrm{nm}$. The signal $S({\mathbf{r}}_0)$ is normalized to 1 at $z_{\min }=20\mathrm{nm}$.

Figure 10

Distance dependence of the $\mathbf{ST}$ signal for the D band. Panels (a-d) account for different values of $L_{\mathrm{c}}$: 0, 15, 30, and 45 nm, respectively, as indicated in the graphics. In all cases, we set ${\tilde{f}}_{\mathrm{e}}=3$ and $r_{\text{tip}}=15\mathrm{nm}$. The signal $S({\mathbf{r}}_0)$ is normalized to 1 at $z_{\min }=20\mathrm{nm}$.

Figure 11

Distance dependence of the $\mathbf{TST}+\mathbf{ST}$ signals for the D, G, and ${\mathrm{G}}^{\prime }$ bands. Panels (a-d) account for different values of $L_{\mathrm{c}}$: 0, 15, 30, and 45 nm, respectively, as indicated in the graphics. In all cases, we used ${\tilde{f}}_{\mathrm{e}}=3$ and $r_{\text{tip}}=15\mathrm{nm}$. The signal $S({\mathbf{r}}_0)$ is normalized to 1 at $z_{\min }=20\mathrm{nm}$.