Subsurface chemical nanoidentification by nano-FTIR spectroscopy

Nano-FTIR spectroscopy based on Fourier transform infrared near-field spectroscopy allows for label-free chemical nanocharacterization of organic and inorganic composite surfaces. The potential capability for subsurface material analysis, however, is largely unexplored terrain. Here, we demonstrate nano-FTIR spectroscopy of subsurface organic layers, revealing that nano-FTIR spectra from thin surface layers differ from that of subsurface layers of the same organic material. Further, we study the correlation of various nano-FTIR peak characteristics and establish a simple and robust method for distinguishing surface from subsurface layers without the need of theoretical modeling or simulations (provided that chemically induced spectral modifications are not present). Our experimental findings are confirmed and explained by a semi-analytical model for calculating nano-FTIR spectra of multilayered organic samples. Our results are critically important for the interpretation of nano-FTIR spectra of multilayer samples, particularly to avoid that geometry-induced spectral peak shifts are explained by chemical effects.


Finite Dipole Model
The finite dipole model (FDM) 1,2 is explained and illustrated (Fig. 4) in the main text in a simplified way. With this Supplementary Method section, we provide a more detailed description and illustration (Supplementary Figure 1) of the model. Nano-FTIR signals are described by calculating the scattering coefficient = scat / 0 . The tip is approximated as a prolate spheroid with apex radius R and major half-axis length (here we use = 30 nm and = 200 nm). The electric field 0 of the external illumination is incident on the tip directly and indirectly via reflection from at the sample surface with the far-field reflection coefficient . The local electric field at the tip, loc = (1 + ) 0 , induces an electric dipole 0 = 2 0 along the tip-axis.
In the FDM the tip-sample interaction is mediated only via the charge 0 , which is located at a distance 0 = 1.31 +2 ≈ from the tip apex. 1 being the quasi-electrostatic Fresnel reflection coefficient for semiinfinite samples. The distance of 0 ′ to the sample surface is the same as the height of 0 above the sample: 0 = + 0 (method of image charges 3 ). 0 ′ acts back on the tip by inducing the additional charge 1 at a distance 1 = /2 from the tip apex, and its counter charge − 1 at the spheroid center. Self-consistent treatment of the problem yields where are geometry factors depending on the tip apex radius , spheroid major half-axis length and tip-sample distance , and given by The -factor is a model parameter that describes the amount of induced charge still relevant for the near-field interaction. It is empirically found to be ≈ 0.7 ± 0.1, 2 here we use = 0.65.
The charges ± 1 form the electric dipole 1 = 1 , yielding for the total electric dipole moment induced in the tip = 0 + 1 . By using Supplementary Equation (1) and the expressions for 0 and 1 , we rewrite where we have used that 0 is proportional to the local electric field at the tip loc . The effective polarizability of the coupled tip-sample system is defined via = eff loc and thus in the FDM it is given by (proportional to) The scattered (far) field of this dipole is measured directly and indirectly via reflection from the sample, yielding scat = (1 + ) and thus the tip scattering coefficient

Reflected electric monopole field
As explained in the previous Supplementary Methods section, the FDM describes the tipsample interaction in a simple image charge model: The tip produces an electric field distribution similar to that of a charge 0 , which induces a mirror charge 0 ′ in the sample (i.e. for bulk samples 0 ′ = − 0 ), which then acts back onto the tip (Supplementary Equation (1)).
In order to derive an effective ̅ , such that the image charge of 0 in multilayered samples is described by 0 ′ = − ̅ 0 , we analyse and compare the electric field distributions produced by 0 and 0 ′ , the latter corresponding to the field of 0 after reflection at the sample surface (method of image charges 3 ).
We express in the following (i) the electric field of the monopole (in the absence of a sample) in the angular spectrum representation 4 and (ii) the electric field reflected from the multilayered sample, refl , from which we derive (iii) the effective (momentum-integrated) near-field reflection coefficient ̅ that is used in the main text. Such treatment fully accounts for the evanescent part of the plane wave spectrum necessary for proper description of the nearfield tip-sample interaction.

(i) Electric monopole field
The electric field of a point charge a e −i (located in the origin of a coordinate system) is given at an arbitrary point = ( , , ) in space by 5 where Φ is the electric potential of the point charge, the nabla operator is given by = ( , , ) and the oscillation frequency is related to the electromagnetic wave momentum = / . We obtain the angular spectrum representation of Supplementary Equation (6) by using the Weyl identity, 4 yielding for the electric field of the monopole field For simplicity, we restrict ourselves to analysing the -component of the electric field along the -axis ( = 0, = 0), which we justify by the elongated shape of the probing tip (providing near fields below the tip apex that are essentially polarized along the -direction) and the rotational symmetry of the problem: The FDM models the nano-FTIR probing tip as a prolate spheroid of length 2 , which is much shorter than the wavelength of infrared radiation used in our experiments. Thus, we further simplify Supplementary Equation (9) by taking the electrostatic limit (which leads to the condition = i√ 2 + 2 ): We obtain our final expression for ( ) after coordinate transformation from cartesian coordinates ( , , ) to cylindrical coordinates ( = √ 2 + 2 , , ) and integration over : As consistency check, we perform the integration over and reproduce the typical −2 dependence for the electric monopole field: We now derive the field after reflection from the (multilayer) sample surface, starting from Supplementary Equation (11). The monopole field is purely -polarized (due to the rotational symmetry of the problem), which allows us to express all reflections at the sample surface by the quasi-electrostatic Fresnel reflection coefficient for -polarized light ( ) which is given in Equation (5) Note that Supplementary Equation (14) is independent of the momentum (after integration).
In principle, Supplementary Equation (14) can be evaluated at arbitrary , however, in order to describe the tip-sample interaction, we make an approach similar to Aizpurua et. al. 6 and Fei et. al. 7 and evaluate the reflected field at the position of the charge a itself, ,refl ( = a ), which we compare with the (incident) monopole field ( ′ = 2 a ), ensuring that we evaluate the incident and reflected field at the same distance from the respective charges (mirror charge in case of the reflected field). We thus define the effective near-field reflection coefficient (which is valid at the position of a ) as We note that the integral in the nominator of Supplementary Equation (15) contains a coupling weight function (Equation (7) in the main text) that is proportional to , similar to expressions found in the work of Hauer et. al. 8 On the other hand, a 2 -dependency is found in similar momentum-integrals contained in the point dipole model for multi-layered samples 6 and the lightning rod model 9,10 , as they are derived from a reflected dipole field, rather than a reflected monopole field.

Supplementary Figures
Supplementary Figure 1: Detailed illustration of the finite dipole model for bulk samples. The nano-FTIR tip is modelled as a prolate spheroid of length 2L and apex radius R, which is located at height H(t) above a bulk sample with permittivity and electrostatic reflection coefficient = ( − 1)/( + 1). The incident electric field E0 induces the primary electric dipole p0, which consists of the point charges ± 0 which are located at distances 0 from the tip apexes. The point charge Q0 creates a mirror charge 0 ′ in the sample, which yields an additional (near-field induced) dipole p1 in the tip. The dipole 1 consists of the point charges 1 at a distance 1 from the sample-near tip apex and − 1 in the spheroid center. The model accounts for far-field illumination and detection of the tip-scattered field Escat via reflection at the sample surface, with Fresnel reflection coefficient r (indicated by red arrows). Figure 2: Calculated nano-FTIR phase spectra of (a,b) thin PMMA layers of varying thickness 1 on silicon and of (c,d) a 2 = 59 nm-thin PMMA layer silicon buried at different depths below PS, calculated with a spectral resolution of Δ = 0.4 cm −1 . As in the experiment, the spectral baseline (non-absorbing frequency range; in calculated spectra around = 1770 to 1780 cm −1 ) has been subtracted before finding the maxima, defined as points with the highest . The black dots indicate the calculated maxima used in Fig. 3 of the main text.  (1) of the main text, which accounts for far-field illumination and detection via the sample surface. Vertical gray lines indicate peak positions. The figure shows that the factor (1+r)² yields an additional red-shift of the peak (panel a) and an increasingly pronounced dispersive line shape with increasing depth of the subsurface layer. Note that in experiment this effect appears already at smaller depth d2. Figure 4: Momentum-dependent spectral peak shifts in far-field infrared reflection spectroscopy. Calculated reflected power | | 2 of -polarized infrared far-field radiation, reflected from the surface of a 10 nm-thin PMMA layer on a silicon substrate (as illustrated in the inset), for (a) varying in-plane momenta q which correspond to varying angles of incidence via = 0 ⋅ sin ( ) for a fixed frequency = 1737 cm −1 near the C = O vibrational mode of PMMA, (b) varying frequency for normal-incidence (0°, red) and grazing-incidence (80°, blue). (c) Same as panel b, but shown after baseline subtraction and normalization for better visibility of the angle-dependent spectral peak-shift Δ ≈ 10 cm −1 . max and (c) peak height ratios C of PMMA surface (black symbols) and PMMA subsurface (red symbols) layers are plotted versus the corresponding peak heights 3 max , for the vibrational modes I-IV of PMMA. Arrows indicate decreasing PMMA surface layer thickness t1 (black) and increasing PMMA subsurface layer depth d2 (red). Subsurface PMMA layer thickness is t2 = 59 nm. Green areas in (c) indicate the data spaces that correspond to subsurface material. The figure shows for various (partially spectrally overlapping) vibrational modes of PMMA that: (1) nano-FTIR peak positions shift to lower frequencies (red-shift) when the thickness t1 of a surface layer decreases or when the depth d2 of a subsurface layer increases. (2) The red-shift is stronger for subsurface layers as compared to surface layers. (3) Most interesting and important, the peak height ratios C = 4 max / 3 max observed for all molecular vibrations of PMMA behave nearly the same as that of the C=O peak (Fig. 6), and thus can be considered as a rather robust criterium for distinguishing surface and subsurface layers.