Near-field spectroscopy of silicon dioxide thin films

Scattering scanning near-field optical microscopy (s-SNOM) Keilmann and Hillenbrand (2004); Novotny and Hecht (2006); Keilmann and Hillenbrand (2009) is a powerful tool for probing local electromagnetic response of diverse materials. The s-SNOM achieves spatial resolution of $10$–$20\,\text{nm}$, which is especially valuable in the physically interesting infrared region Basov et al. (2011); Basov and Chubukov (2011) where the resolution of conventional spectroscopy is fundamentally limited by a rather large wavelength $\lambda\sim 5$–$500\,\mu\text{m}$. The s-SNOM techniques have been rapidly advancing, Amarie and Keilmann (2011); Huth et al. (2011) which enabled their applications to imaging spectroscopy of complex oxides Qazilbash et al. (2007); Zhan et al. (2007); Qazilbash et al. (2009); Frenzel et al. (2009); Lai et al. (2010); Qazilbash et al. (2011) and graphene. Fei et al. (ress)

The s-SNOM utilizes scattering of incident light by the tip of an atomic force microscope (AFM) positioned next to the probed sample (Fig. 1). The tip couples to the sample via evanescent waves of large in-plane momenta $q\sim 1/a$, where $a$ is the tip radius of curvature (a few tens of nm). This is why the lateral resolution of the s-SNOM is determined primarily by $a$ rather than $\lambda$. Lai et al. (2007); Huber et al. (2008); Olmon et al. (2008)

One of the interesting open questions is the depth ($z$-coordinate) resolution of the s-SNOM probes. Previous experiments suggested that it is comparable to the lateral resolution $\sim a$, based on imaging of small sub-surface particles. Taubner et al. (2005) Surprisingly, our recently near-field measurements of SiO₂ thin films have demonstrated that films as thick as several hundred nm have a response clearly different from that of the bulk material. Andreev et al. Thus, if instead of particles one has layers, then the s-SNOM is able to detect them at much larger depths.

In this paper these experimental results are re-analyzed and compared with two theoretical models, the conventional point-dipole approximation Keilmann and Hillenbrand (2004); Hillenbrand et al. (2002) and the spheroidal model. The former is very simple to implement but is also very crude. Predictably, it yields a bulk-like response of the s-SNOM signal as soon as the SiO₂ film thickness exceeds the tip radius, in disagreement with the experiment. A plausible reason for shortcomings of the point-dipole model is its failure to account for the strongly elongated shape of the tip. Such a tip acts as an optical antenna Keilmann and Hillenbrand (2004); Novotny and Hecht (2006); Keilmann and Hillenbrand (2009) that greatly enhances the electric field inside the tip-sample nanogap. Unfortunately, analytical models, Cvitkovic et al. (2007); Moon et al. (2011) that attempt to treat elongated tips do not apply to layered substrates. This compels us to study the problem numerically.

To make the calculations tractable, we follow examples in the literature Porto et al. (2003); Renger et al. (2005); Esteban et al. (2009) and model the tip as a metallic spheroid of total length $2L\gg a$, see Fig. 1. As shown below, this gives results in a much better agreement with the experiment in terms of both the frequency and the thickness dependence of the near-field signal. We attribute the origin of the more gradual film-thickness dependence in the spheroidal model to the aforementioned “antenna effect.” The magnitude of this effect is determined by the material response over length scales ranging from $a$ to $2L$, and so it truly saturates only when the film thickness becomes much larger than $2L$.

The remainder of the paper is organized as follows. In Sec. II we summarize the experimental procedures and results. In Secs. III and IV we discuss the two theoretical models and compare their predictions with the measurements. Concluding remarks are given in Sec. V.

To make the paper self-contained we summarize the results of our recent experiments Andreev et al. in this Section. We investigated commercially available calibration gratings, which contain strips or islands of SiO₂ thermally grown on Si. The manufacturer specified thicknesses of the SiO₂ layer spanned the range $d_{1}=2$, $18$, $22$, $108$, and $300\,\text{nm}$. A combination of CO₂ and tunable quantum cascade lasers (Daylight Solutions) allowed us to cover the frequency range between 890 $\text{cm}^{-1}$ and 1250 $\text{cm}^{-1}$. The near-field data were collected using a Neaspec system.

The measured s-SNOM signal represents the electromagnetic field backscattered by the probe and the scanned sample. The complex amplitude $s(\omega,t)$ of the backscattered field varies periodically with the tapping frequency $\Omega\sim 40\,\text{kHz}$ as the distance $z_{\text{tip}}$ between the sample and the nearest point of the tip undergoes harmonic oscillations

where $\Delta z=50\,\text{nm}$ typically. In order to suppress unwanted background and isolate the part of the signal scattered by the probe tip, the signal is demodulated. Namely, we extracted the absolute values $s_{n}(\omega)$ and phases $\phi_{n}(\omega)$ at tapping harmonics

The experimental results for the spectra are shown in Fig. 2(a). These spectra were intended to be taken in the tapping mode, i.e., for zero $z_{0}$. However, experimentally $z_{0}$ can be determined only up to an additive constant $\sim 1\,\text{nm}$. Therefore, we measured $z_{0}$-dependence of $s_{3}$ (the approach curves) shown in Fig. 3(a) and selected the largest observed $s_{3}$. Our results are in a qualitative agreement with previous experimental study, Taubner et al. (2005) which reported approach curves for SiO₂ at a few discrete frequencies and film thicknesses.

The data points in Fig. 2(a) represent the normalized amplitude $s_{3}(\text{SiO}_{2})/s_{3}(\text{Si})$, where $s_{3}(\text{SiO}_{2})$ and $s_{3}(\text{Si})$ are the raw third-order demodulation signals averaged over the entire SiO₂ and Si areas, respectively. The statistical uncertainty of these averaged data traces is about $2\%$.

For each thickness studied, the normalized amplitude $s_{3}(\text{SiO}_{2})/s_{3}(\text{Si})$ exhibits several maxima. The main peak is situated at $\omega\approx 1130\,\text{cm}^{-1}$. The key aspect of the data is a rapid decrease in the normalized amplitude of this peak as the thickness is reduced. A trace of this resonance can be reliably identified even for the $2$-nm thick SiO₂ film. Another notable feature is the growing strength and frequency shift of the secondary peaks on the high-$\omega$ side of the main peak as $d_{1}$ is decreased.

Since the response of Si is frequency independent in our experimental range, the frequency dependence of the spectra in Fig. 2(a) originates from that of SiO₂. We attribute the maxima of $s_{3}(\text{SiO}_{2})/s_{3}(\text{Si})$ to the phonon modes localized at the air-SiO₂ interface. Amarie and Keilmann (2011) These resonances occur in the frequency region between the bulk transverse and longitudinal modes of SiO₂ (the outer dashed lines in Fig. 4 below).

The results of our theoretical calculations for the normalized scattering amplitude are presented in the remaining panels of Figs. 2 and 3. They are discussed and compared with the experimental findings in the following Sections.

The sample is modeled as a two-layer system. The first layer with dielectric function $\epsilon_{1}(\omega)$ occupies the slab $-d_{1}<z<0$. The second layer with dielectric function $\epsilon_{2}(\omega)$ occupies the half-space $z<-d_{1}$. The half-space $z>0$ (“layer 0”) is filled with air (dielectric constant $\epsilon_{0}=1$). The fundamental response functions of the system are the reflection coefficients $r_{X}(q,\omega)$, which are functions of in-plane momentum $q$, frequency $\omega$, and polarization $X=S$ or $P$. The domain of definition of $r_{X}(q,\omega)$ is understood to include nonradiative modes $q>\sqrt{\epsilon_{0}}\,\omega/c$. It is known from previous studies that the s-SNOM signal is dominated by the $P$-polarized waves. In our two-layer model their reflection coefficient is given by a Fresnel-like formula

where $z$-axis momenta $k^{z}_{j}$ are defined by

Equation (3) is valid for arbitrary $q$. In the near-field case where $q$ is large and $k^{z}_{j}\simeq iq$, it simplifies to

Assuming all $\epsilon_{j}$ are $q$-independent, the effective dielectric function $\epsilon_{*}(q,\omega)$ depends on $q$ only via the product $qd_{1}$ in this limit. Therefore, $r_{{}_{\text{P}}}(q,\omega)$ for one thickness $d_{1}$ can be obtained from another by rescaling $q$. As discussed in Sec. I and shown in more detail below, the most important momenta are $q\sim 1/a$ where $a\sim 30\,\text{nm}$ is the tip radius. Therefore, we can get an approximate understanding of the system response by examining the behavior of $r_{{}_{\text{P}}}(q,\omega)$ as a function of $\omega$ at fixed $qd_{1}\sim d_{1}/a$. This behavior is dictated by the spectrum of surface collective modes, as follows.

In general, surface modes correspond to poles of the response functions $r_{X}$. Function $r_{{}_{\text{P}}}$ given by Eq. (6) can have up to two poles at each $qd_{1}$, see, e.g., Ref. Prade et al., 1991. They are defined by the following condition on $\epsilon_{1}(\omega)$:

At large $qd_{1}$, where $\tanh qd_{1}=1$, this condition yields $\epsilon_{1}(\omega)=-\epsilon_{0}$ or $\epsilon_{1}(\omega)=-\epsilon_{2}$, which correspond to modes localized at the upper $0$–$1$ and the lower $1$–$2$ interfaces, respectively. Actually, the latter “pole” has vanishingly small residue because evanescent waves do not reach the lower interface at $qd_{1}=\infty$. There is no $q$-dispersion and no coupling of the two modes in this limit. The dispersion appears at finite $qd_{1}$, where the two modes become mixed. In particular, we find

at $qd_{1}\ll 1$. At finite $q$, both interfaces participate in generating these excitations. The labels “0–1” and “1–2” are for convenience: they indicate at which interface a given dispersion branch is ultimately localized as $q$ increases. At $qd_{1}=0$, the “0–1” and “1–2” branches are characterized by $\epsilon_{1}(\omega)=0$ and $\epsilon_{1}(\omega)=-\infty$, which correspond, respectively, to the bulk longitudinal and transverse phonon frequencies $\omega_{{}_{\text{LO}}}$ and $\omega_{{}_{\text{TO}}}$.

If we try to apply this formalism to real materials, we face the problem that Eq. (7) has no solutions for real $\omega$ because the dielectric functions have finite imaginary parts. This is why in practice the collective mode spectra are usually defined differently. They are identified with the maxima of dissipation, i.e., $\text{Im}\,r_{{}_{\text{P}}}$. The number of these maxima can be fewer than the total allowed number of the modes because some of them can be overdamped. Similarly, we define $\omega_{{}_{\text{LO}}}$ and $\omega_{{}_{\text{TO}}}$ as the frequencies that correspond to the maxima of $-\text{Im}\,\epsilon_{1}^{-1}(\omega)$ and $\text{Im}\,\epsilon_{1}(\omega)$.

To see what kind of spectra are realized in our system, we use our ellipsometry data for $\epsilon_{1}(\omega)$ [Fig. 4(a)] and Eq. (6) to compute $r_{{}_{\text{P}}}$ for several values of $qd_{1}$. The plot of these quantities as a function of $\omega$ is presented in Fig. 4(c). Three maxima on each curve in the region of primary interest $\omega>1000\,\text{cm}^{-1}$ are apparent. They exist already at $qd_{1}=\infty$, and so all of them belong to the upper (air-SiO₂) interface. In fact, we do not expect sharp modes at the lower (SiO₂-Si) interface because the dielectric function of Si is quite large $\epsilon_{2}\approx 11.7$ in the studied range of $\omega$. The lowest value of $\text{Re}\,\epsilon_{1}\approx-5.0$ is not sufficient to compensate $\epsilon_{2}$ and generate “1-2” modes, cf. Eq. (7).

The main peak of $\text{Im}\,r_{{}_{\text{P}}}$ at $qd_{1}=\infty$ defines the surface phonon frequency of SiO₂ $\omega_{{}_{\text{SP}}}\approx 1164\,\text{cm}^{-1}$. There also exist secondary peaks at $\omega\approx 1100\,\text{cm}^{-1}$ and $\omega\approx 1220\,\text{cm}^{-1}$. Their evolution as a function of $qd_{1}$ comply with the general scheme outlined above. As $qd_{1}$ decreases, all the three peaks loose strength, as expected, because the amount of SiO₂ diminishes. The lower-$\omega$ secondary peak redshifts, moving towards $\omega_{{}_{\text{TO}}}$, and then quickly disappears. This agrees with the SiO₂-Si resonance being highly damped. The higher-$\omega$ secondary peak becomes dominant at $qd_{1}<0.5$ and demonstrates a systematic shift towards $\omega_{{}_{\text{LO}}}$, see Fig. 4(c).

A notable feature of Fig. 4(b) is the clustering of the crossing points of the different curves near $\omega=1036\,\text{cm}^{-1}$. This is the frequency where the dielectric function of SiO₂ is the closest to that of Si, $\epsilon_{2}\approx 11.7$. As a result, the two layers act almost as one bulk material, so that $r_{{}_{\text{P}}}(\omega)$ is approximately thickness-independent.

There is a qualitative correspondence between the features displayed by the reflection coefficient $r_{{}_{\text{P}}}$ and the observed near-field signal $s_{3}(\text{SiO}_{2})/s_{3}(\text{Si})$, cf. Figs. 2(a) and 4(b),(c). However, the relation between $r_{{}_{\text{P}}}(q,\omega)$ and the measured s-SNOM signal is nontrivial. For example, the frequency positions of the maxima in $\text{Im}\,r_{{}_{\text{P}}}(q,\omega)$ and those in $s_{3}(\text{SiO}_{2})/s_{3}(\text{Si})$ differ by as much as $40\,\text{cm}^{-1}$. We also suspect that there may be some slight differences between the optical constants of thick films we assume in our calculations and those of the small SiO₂ structures we probe by the s-SNOM. This is the likely reason why the crossing point of the experimental curves occurs near $1060\,\text{cm}^{-1}$ rather than $1036\,\text{cm}^{-1}$ predicted by both our models, cf. Fig. 2.

Developing a reliable procedure for inferring $r_{{}_{\text{P}}}(q,\omega)$ from $s_{3}$ remains a challenge for the theory. The next section presents our current approach towards this ultimate goal.

Both radiative and nonradiative waves may play significant roles in the s-SNOM experiment. Porto et al. (2003) The radiative modes magnify the signal by a certain far-field factor (FFF) $F(q_{s},\omega)$, where $q_{s}=(\omega/c)\sin\theta$ is the momentum of these modes for the angle of incidence $\theta$. The nonradiative modes influence the effective polarizability $\chi(\omega,z_{\text{tip}})$ of the tip, i.e., the ratio of its dipole moment $p^{z}$ and the external electric field $E_{\text{ext}}^{z}$. Altogether the demodulated s-SNOM signal $s_{n}e^{i\phi_{n}}$ can be written as

Below we discuss the FFF and the tip polarizability separately.

In this paper we analyzed the results of experimental study of amorphous SiO₂ films on Si obtained by scanning near-field optical spectroscopy. Andreev et al. We discussed the collective mode spectra of such structures and compared measurements with two theoretical calculations. The first is based on a conventional approximation in which the tip of the scanned probe is modeled as a point dipole. In the second the tip is treated as an elongated spheroid, significantly improving agreement with the experiment.

We explain the qualitative difference between the two models as follows. An important physical ingredient missing in the point-dipole model is the enhancement of the electric field near the apex of the tip — the antenna effect. This phenomenon is well-known from classical electrostatics. The enhancement of the field is controlled primarily by the ratio of the total length of the tip $2L$ (actually, the smaller of $2L$ and $\lambda$) and the apex radius of curvature $\sim a$. The point-dipole model has been successful in the past without this enhancement factor only on account of the normalization procedure. Instead of absolute $s_{n}$, one usually reports $s_{n}$ normalized to some reference material such as Au or in our case, Si. This way, one eliminates any possible frequency dependence of the source radiation, but at the same time cancels the part of the signal scaling with tip size. For a stratified sample this cancellation is imperfect because the the field enhancement depends also on the dielectric response of the sample, which is a function of momentum $q$. For a tip of length $2L$, harmonics relevant for the field enhancement have momenta ranging from $q\sim 1/a$ down to $q\sim 1/L$. Therefore, one may expect that the dependence of the s-SNOM signal on the thickness $d_{1}$ of the top layer would saturate only when $d_{1}\sim L$. Our simulations provide direct evidence for this claim. Therefore, we think that the spheroid model holds a great promise as an analysis tool for near-field experiments. It captures a lot of physics relevant to the near-field interaction while remaining computationally fast.

The strong experimentally observed thickness dependence of the near-field signal Andreev et al. indicates that s-SNOM is capable of not only high lateral resolution but can also probe the system in the third dimension. However, the response of a layered system is different from those containing small subsurface particles Taubner et al. (2005). We hope that experimental and theoretical approaches presented in this paper may be of use for accurate depth profiling of various dielectric and metallic nanostructures.

The work at UCSD is supported by ONR, AFOSR, NASA, and UCOP. AHCN and LMZ acknowledge DOE grant DE-FG02-08ER46512 and ONR grant MURI N00014-09-1-1063. We thank F. Keilmann and R. Hillenbrand for illuminating discussions.