Model for quantitative tip-enhanced spectroscopy and the extraction of nanoscale-resolved optical constants

Near-field infrared spectroscopy by elastic scattering of light from a probe tip resolves optical contrasts in materials at dramatically subwavelength scales across a broad energy range, with the demonstrated capacity for chemical identification at the nanoscale. However, current models of probe-sample near-field interactions still cannot provide a sufficiently quantitatively interpretation of measured near-field contrasts, especially in the case of materials supporting strong surface phonons. We present a model of near-field spectroscopy derived from basic principles and verified by finite-element simulations, demonstrating superb predictive agreement both with tunable quantum cascade laser near-field spectroscopy of ${\mathrm{SiO}}_2$ thin films and with newly presented nanoscale Fourier transform infrared (nanoFTIR) spectroscopy of crystalline SiC. We discuss the role of probe geometry, field retardation, and surface mode dispersion in shaping the measured near-field response. This treatment enables a route to quantitatively determine nanoresolved optical constants, as we demonstrate by inverting newly presented nanoFTIR spectra of an ${\mathrm{SiO}}_2$ thin film into the frequency dependent dielectric function of its mid-infrared optical phonon. Our formalism further enables tip-enhanced spectroscopy as a potent diagnostic tool for quantitative nanoscale spectroscopy.

Figure 1

(a) Scanning electron micrograph of a typical commercial near-field probe exhibiting a conical geometric profile and characteristic length scales (probe length and tip radius) separated by nearly three orders of magnitude. The surface profile (blue) is considered in Sec. 6. (b) Schematic description of the probe-sample near-field interaction, involving emission of cylindrical evanescent fields from charge elements in the probe and their reflection by the sample. (c) Probe response function $\Lambda (q,z)$ (defined in the main text) computed by the boundary element method (Appendix pp3) for evanescent fields ${\vec{E}}_q$ of increasing momenta $q$. Dashed curves indicate the geometric profile of the probe, and surface charge distribution profiles are normalized by their minimum values for viewing purposes.

Figure 2

Spectral near-field contrast between the $1130 \mathrm{cm}{}^{-1}$ surface phonon polariton of ${\mathrm{SiO}}_2$ and silicon (providing normalization) as predicted by the lightning rod model for an ellipsoidal probe in the quasistatic approximation. Contrast increases monotonically beyond experimentally observed levels as the probe length is increased.

Figure 3

(a) Scattered field of a realistic near-field probe geometry under plane wave illumination (incident along the viewing angle) as computed by the finite-element method. Oscillatory fields near the tip apex are associated with standing-wave-like surface charge densities resulting from field retardation. (b) (Bottom) Field enhancement at the tip apex computed quasistatically (QS) and electrodynamically (ED) for two probe geometries of varying size illuminated perpendicular to their principle axes. (Top) Near-field $S_3$ contrast between ${\mathrm{SiO}}_2$ at the surface optical phonon resonance $\left({\omega }_{}\mathrm{SO}\right)$ and silicon simulated by the fully retarded lightning rod model. The vertical dashed line indicates the length of a typical near-field probe, $L\approx 19\mu \mathrm{m}$.

Figure 4

(a) Near-field response of ${\mathrm{SiO}}_2$ thin films etched to varying thicknesses on a silicon substrate measured by tunable QCL spectroscopy and normalized to silicon [40] (see text). The faint curves are provided as guides to the eye. (Inset) Sample near-field signal $S_3$ at $\omega =1130 \mathrm{cm}{}^{-1}$ overlaid on simultaneously acquired AFM topography. (b) Near-field $S_3$ spectra predicted by the lightning rod model using optical constants from literature [44]. Data points from the 300-nm film are superimposed for point of comparison. Our model captures the key features of the data; we infer that discrepancies with ultrathin film data result from substantial variations in optical properties.

Figure 5

(a) Example of strong surface optical phonon dispersion for a 100-nm-thick ${\mathrm{SiO}}_2$ film on silicon computed by the Fresnel reflection coefficient $r_p(q,\omega )$ [Eq. (21)]. (b) The momentum-dependent distribution of electric fields at the sample surface (dashed line) calculated by the lightning rod model at the tip-sample phonon resonance for a conical tip in full contact.

Figure 6

(a) Amplitude $S_3$ and phase ${\varphi }_3$ of the backscattered near-field signal from a 6H SiC crystal, measured in the vicinity of the surface optical phonon and referenced to a surface-deposited gold film, as obtained in a single acquisition by nanoFTIR. (Left inset) Visible light image (width $60 \mu \mathrm{m}$) above the near-field probe at the SiC/gold interface. (Right inset) PSHet near-field $S_3$ image (width $1 \mu \mathrm{m}$) of the interface with sample and reference nanoFTIR locations indicated. (b) Lightning rod model predictions for near-field signal from SiC using optical constants measured by in-house ellipsometry. While insensitive to details of probe geometry (see text), fully retarded (Ret.) calculations provide superior agreement to the experimental spectra than the quasistatic (QS) approximation.

Figure 7

Amplitude $S$ and phase $\varphi$ of the backscattered near-field signal from a 300-nm ${\mathrm{SiO}}_2$ film, measured in the vicinity of the surface optical phonon and referenced to the silicon substrate, as obtained in a single acquisition by nanoFTIR.

Figure 8

(a) Typical variation in optical constants among thermal oxide thin films taken from literature ellipsometry. Pairs of red and blue curves with identical line style are associated with distinct thin film samples [44]. (b) Optical constants of a 300-nm ${\mathrm{SiO}}_2$ film extracted from near-field spectra $S_2\left(\omega \right)$ and $S_3\left(\omega \right)$ following the method of Eq. (24).

Figure 9

(Left) The field radiated from an axisymmetric body to an observation point at inclination angle $\theta$ is constructed from the contributions of currents (shown in red) through infinitesimal surface annuli. (Right) The two independent polarizations composing any annular axisymmetric current distribution, with associated angular radiation profiles [Eq. (E1)] shown schematically in blue.