1. Introduction

When the real part of the dielectric function Re[ε] of a deep-subwavelength film approaches zero while its imaginary part remains small, the out-of-plane component of the electric field E_⊥ is both enhanced and strongly confined in the film via continuity of εE_⊥ at the interfaces [1]. In the infrared (IR) region, Re[ε] = 0 is a characteristic feature of polaritonic materials, e.g. inorganic oxides or doped semiconductors [2]. Thin films composed of such materials support two confined polaritons near Re[ε] = 0: the Berreman mode (BE) and the so-called epsilon-near-zero mode (ENZ) [1,3]. Their dispersions are separated by the light line via k_||(BE) < ω/c < k_||(ENZ) where ω is the angular frequency, k_|| the in-plane wave vector, and c the speed of light. The BE mode is a radiative mode [4] that can be excited from free space using E_⊥ provided by p-polarized light in thin films with thicknesses ≤ 200 nm [5]. The ENZ mode is a non-radiative mode [1] that can be excited locally in ultrathin films with thicknesses < 20 nm [3] via coupling to near-field E_⊥ [6] of localized plasmon resonances (LPR) [7]. Spectrally, this corresponds to ω(BE) > ω(ENZ) [1,4].

BE modes and locally excited ENZ modes allow strong manipulation of IR light–matter interactions at the nanoscale via spatial and spectral distribution of enhanced and confined E_⊥(r,ω) near Re[ε(ω)] = 0. Therefore, optical properties of thin films such as IR absorption per unit volume given by

However, accessing IR absorption [Eq. (1)] near Re[ε] = 0 by using far-field IR polarimetric techniques remains challenging since both reflection and transmission measurements of BE and ENZ modes are needed. An alternative near-field IR approach by scattering-type scanning near-field optical microscopy (s-SNOM) provides limited sensitivity to E_⊥ strongly confined in the film near Re[ε] = 0 [14], and relies on accurate modeling of nanoscale tip–sample interactions and geometric optimization. We propose a novel mid-infrared (MIR) nanospectroscopic approach for unambiguous identification of BE and ENZ effects by atomic force microscope infrared spectroscopy (AFM-IR) [15]. This technique provides direct information on absorption of IR light by thin films via their photothermal expansion probed by the AFM cantilever. We apply a commercially available setup in standard top side p-polarized illumination configuration [16] with a fixed 70° angle of incidence (AOI) accessing out-of-plane optical properties of thin films with high sensitivity.

AFM-IR has been widely used for analyzing IR absorption [Eq. (1)] in organic films where Im[ε] is large and E is typically small. This work establishes the broad applicability of AFM-IR studies to the opposite case near Re[ε] = 0 where thin film photothermal expansion is due to large E_⊥ of BE and ENZ modes. Here, we focus on homogeneous SiO₂ films on Si. The MIR Re[ε] = 0 properties of thin SiO₂ films have promising applications in BE mediated directive emission [17] and ENZ induced scattering suppression [6]. Characterization of ultrathin native SiO₂ films plays an important role in Si oxidation level control via BE mode [18] and Si-based plasmonic nanostructure design via ENZ local field confinement [19].

We first apply AFM-IR to non-invasively probe BE enhanced absorption in a thermally formed 100 nm SiO₂ film and compare the data to a 100 nm Si₃N₄ film on Si. Then, strong ENZ absorption in a 2 nm native SiO₂ film is characterized by AFM-IR via coupling to the LPR of the tip. We show that the physical origin of BE and ENZ absorption can be extracted from AFM-IR spectra without the need of complex modeling, contrary to s-SNOM. Finally, we apply our results to the far-field analysis of nanoscale plasmonic ENZ gratings on a 2 nm native SiO₂ layer on Si that exhibit both polaritonic effects. This is done by polarization-dependent IR microscopy [20].

2. The Berreman effect

Vanishing Re[ε] of SiO₂ in the MIR region is a direct consequence of the strong stretching vibration ν(SiO₂) that corresponds to the maximum of Im[ε] shown in Fig. 1(a). The dielectric function was extracted from electrochemically formed 8 nm SiO₂ film on Si [21] where Re[ε] = 0 is at 1211 cm⁻¹. To demonstrate excitation and thus the radiative nature of the BE mode via coupling to E_⊥ of incident p-polarized light, we performed a far-field IR polarimetric experiment [Fig. 1(b)] of a thermally formed 100 nm SiO₂ film on Si at an angle of incidence of 70° (AOI of AFM-IR). This was achieved using a spectroscopic IR polarimeter externally attached to a Bruker IFS 55 FT-IR [20]. In the p-polarized reflectance spectrum in Fig. 1(c), the BE peak is stronger than the ν(SiO₂) peak indicating enhancement of E_⊥ in the film [Eq. (1)]. Here, the 4 cm⁻¹ modulation of the spectrum is due to interference caused by backside reflections of the MIR transparent Si substrate.

 

Fig. 1 (a) Dielectric function of electrochemically grown SiO₂ in MIR region with marked 1211 cm⁻¹ position where Re[ε] = 0 and Im[ε] = 0.73. (b) Schematic of far-field IR polarimetry at AOI = 70°. (c) P-polarized reflectance R_p(70°) showing silicon dioxide stretching vibration ν(SiO₂) and BE mode of thermally grown 100 nm SiO₂ on Si.

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We now demonstrate the physical origin of the BE effect in the 100 nm SiO₂ film on Si. The theoretical background is provided by calculations of Maxwell’s equations using the commercial optical simulation package RSoft. In this work, we use the built-in finite-difference time-domain (FDTD) and rigorous coupled-wave analysis (RCWA) algorithms for near- and far-field simulations, respectively. The dielectric function of the transparent Si substrate was set to ε(Si) = 11.71.

Using ε(SiO₂) from Fig. 1(a) leads to a BE absorption maximum at 1234 cm⁻¹. The corresponding simulation of the Berreman mode at a fixed point in time is presented in Fig. 2. It is evident that the BE effect is a result of periodic enhancement of confined E_⊥ near Re[ε] = 0 created by the incident wave fronts seen in Fig. 2(a). Therefore, the BE mode radiates power back to the far-field as it propagates along the SiO₂ film, which is accompanied by E_⊥ induced resonant absorption [Eq. (1)] shown in Fig. 2(b).

 

Fig. 2 Simulated E_⊥ (a) and absorption (b) in a 100 nm SiO₂ film on Si under p-polarized illumination at AOI = 70° showing propagating BE mode at 1234 cm⁻¹ at a fixed point in time.

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The BE mode can be considered a macroscopic confined polariton since its wavelength λ(BE) diverges as the AOI approaches zero leading to an electrostatic-like response [2]. As the AOI approaches 90°, the light cone serves as the lower bound of λ(BE) [1,4]. The wavelength of the BE mode in Fig. 2 is given by

3. AFM-IR: BE mode at the nanoscale

We use a commercially available AFM-IR setup by Anasys Instruments (nanoIR2-FS) equipped with a tunable p-polarized MIR quantum cascade laser (QCL) by Daylight Solutions (MIRcat) [16]. The setup uses gold coated silicon AFM cantilevers where the radius of the tip apex is around 30 nm. The measured photothermal expansion is proportial to IR absorption and is typically in pm region in thin films [22]. To increase the sensitivity of the method, mechanical signal enhancement mechanism has been applied by tuning QCL pulse rate to a cantilever bending mode in contact with the sample [15]. AFM-IR spectra are collected by detecting the QCL wavelength-dependent resonant cantilever oscillation amplitude in the 900–1350 cm⁻¹ region. The setup allows measurements of BE and ENZ modes with a spectral resolution of 1 cm⁻¹ at 20 cm⁻¹/s sweep rate in ambient conditions on arbitrary substrates.

Incident p-polarized QCL light at AOI = 70° induces a dipole at the tip apex with out-of-plane orientation that provides tip-enhancement of the measured signal. Using a simple model of AFM-IR, we take tip induced E_⊥ into account by simulating a conical gold nanostructure on the SiO₂ film resembling a cantilever tip apex where r_tip = 30 nm. In 100 nm SiO₂ films, the short-range LPR of the tip cannot couple to the ENZ mode [3] so that the time-averaged BE mode is not affected by the AFM cantilever, as demonstrated in Fig. 3(a,b). Here, ε(Au) was taken from [23], and the SiO₂ film thickness was reduced to 50 nm for clarity.

 

Fig. 3 Time-averaged simulated ν(SiO₂) (a) and homogeneous 1234 cm⁻¹ BE (b) absorption in a 50 nm SiO₂ film on Si under p-polarized illumination at AOI = 70° in the presence of gold AFM tip apex. Corresponding simulations without the tip are shown in (c) and (d), respectively.

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Figure 3(a) clearly shows that ν(SiO₂) absorption is primarily a tip induced effect due to strong plasmonic enhancement of Im[ε] of SiO₂. BE absorption in Fig. 3(b), on the other hand, shows a homogeneous spatial distribution at 1234 cm⁻¹ that does not change when removing the tip in Fig. 3(d), contrary to ν(SiO₂) in Fig. 3(c). Therefore, the resulting photothermal expansion is due to the BE mode excited by the incident QCL light [Eq. (2)] and not due to a local feature induced by AFM-IR. This indicates that the method probes material-specific BE modes in a non-invasive fashion, i.e. without disturbing the propagation of the confined polaritons.

An AFM-IR spectrum of a thermally formed 100 nm SiO₂ film on Si is presented in Fig. 4(a). Here, tip induced ν(SiO₂) is the dominating effect, which is consistent with Fig. 3(a). The non-invasively measured BE absorption is a weaker effect [Fig. 3(b)] found at 1241 cm⁻¹. The dielectric function of thermally grown thin SiO₂ films [20] is strongly dependent on sample preparation parameters leading to spectral shifts of the BE mode. The method is highly sensitive to changes in the dielectric function [15,16] which could explain the 7 cm⁻¹ blueshift of the measured BE absorption in Fig. 4(a).

 

Fig. 4 Normalized AFM-IR spectra of thermally grown 100 nm SiO₂ (a) and Si₃N₄ (b) films on Si indicating strong tip induced absorption at ν(SiO₂) and ν(Si₃N₄), and the non-invasively probed BE absorption. The insets show zoomed-in BE regions. The noise is due to MIR transparency of Si.

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To show that BE modes can also be probed by AFM-IR in other thin Re[ε] = 0 films, we measured a 100 nm Si₃N₄ film on Si in Fig. 4(b). Here, a similar behavior of the AFM-IR amplitude can be observed. The dominating tip-enhanced ν(Si₃N₄) peak is much broader, and only the right shoulder of the resonance lies within the measured spectral region. This is consistent with [24]. The weaker BE absorption of Si₃N₄ is found at 1151 cm⁻¹.

When measuring propagating confined polaritons such as Berreman modes, tip-enhancement is an unwanted effect since it enhances ν(SiO₂) absorption. However, it is essential in order to locally excite the ENZ mode in 2 nm native SiO₂. The ε(SiO₂) data in Fig. 1(a) were found to be very similar to the one of ν(SiO₂) vibrations in native oxide layers [20,21] so that the ENZ effect could be simulated spectrally and directly compared to AFM-IR.

4. AFM-IR: ENZ local field confinement

We use the tip apex model shown in Fig. 3(a,b) to demonstrate the physical origin of ENZ local field confinement in ultrathin SiO₂ films on Si. The simulations of ν(SiO₂) and ENZ absorption are presented in Figs. 5(a) and 5(b), respectively. They indicate that both effects are tip induced, and that ENZ absorption dominates, in contrast to Fig. 3(a,b). As opposed to the radiative BE mode calculated at 1234 cm⁻¹ [Eq. (2)], the non-radiative ENZ mode calculated at 1229 cm⁻¹ [Fig. 5(b)] is largely independent of k_||(ENZ) [3]. This leads to resonant plasmon–ENZ coupling and confinement of short range near-field E_⊥ provided by the out-of-plane LPR of the tip, rendering the far-field E_⊥ induced BE mode [Fig. 2(a)] a minor effect in ultrathin SiO₂ films. For clarity, the thickness in Fig. 5(a,b) was set to 5 nm.

 

Fig. 5 Simulated tip induced ν(SiO₂) (a) and 1229 cm⁻¹ ENZ (b) absorption in a 5 nm SiO₂ film on Si under p-polarized illumination at AOI = 70° according to Fig. 3(a,b). (c) Schematic of AFM-IR of a 2 nm native SiO₂ layer on Si with indicated dimensions of the simulated absorption monitor. (d) Corresponding cascaded and normalized simulated absorption (top) and AFM-IR amplitude spectra (raw and Savitzky–Golay filtered, bottom) showing dominating ENZ local field confinement.

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The locally enhanced ENZ absorption in a 2 nm native SiO₂ layer on Si is unambiguously identified by AFM-IR where the cantilever is used as both ENZ confined polariton exciter and detector. To simulate the AFM-IR signal, we monitored absorption in a 2 nm SiO₂ film under the tip by integrating Eq. (1) over a domain with dimensions 2r_tip × 2r_tip × 2 nm³ as shown in Fig. 5(c). Both results are presented together in Fig. 5(d) where raw measured data were Savitzky–Golay filtered for clarity. The good agreement between AFM-IR and simulation shows that the method is applicable to ENZ materials without complex modeling of measurement procedure and geometric optimization. Here, measured and simulated peak positions of dominating ENZ local field confinement at 1229 cm⁻¹ coincide, proving the similarity between ε(SiO₂) in Fig. 1(a) and the one of thin native oxide layers. The AFM-IR amplitude noise level is due to small absorption volume and fast heat dissipation reducing photothermal deflection of the cantilever.

The inversion of ν(SiO₂)/ENZ and ν(SiO₂)/BE peak ratios as well as spectral positions of the two confined polaritons ω(ENZ) < ω(BE) [1,4] measured by AFM-IR in Figs. 5(d) and 4(a), respectively, allow fast nanospectroscopic diagnostics of Re[ε] = 0 induced effects in thin films. Furthermore, AFM-IR analyses allow drawing conclusions about far-field optical responses of plasmonic ENZ materials where IR absorption is generated primarily by the enhanced near-field in the Re[ε] = 0 layer [Fig. 5(a,b)], making the technique highly usable for their nanoscale design. To demonstrate far-field applicability of the polaritonic AFM-IR nanospectroscopy, we performed an IR microscopic characterization of ENZ and BE modes of a gold grating on 2 nm native SiO₂ on Si.

5. Plasmonic ENZ grating as a far-field application

The grating was fabricated by nanoimprint lithography [25] with a period (P) of 800 nm and width of the stripes (L) of 330 nm as shown in the scanning electron microscope (SEM) image in Fig. 6(a). The height of the grating stripes was 58 nm. Here, 54 nm Au was deposited on a 4 nm Ti adhesive layer on Si with native SiO₂. In the measured 900–1675 cm⁻¹ region, the grating is non-diffractive. In contrast to AFM-IR where the tip dipole is oriented out-of-plane, the ENZ mode in gratings couples to near-field E_⊥ provided by the in-plane dipolar LPR of the stripes at 90° azimuth [6]. This resonance can be excited by s-polarized light, so that ENZ is not sensitive to AOI, out-of-plane polarization, and grating periodicity in absence of diffraction effects. The BE mode, on the other hand, is excited between the stripes via coupling to incident p-polarized light at 0° azimuth [Eq. (2)].

 

Fig. 6 (a) SEM image of gold grating on Si with 2 nm native SiO₂ layer. The dimensions are P = 800 nm, L = 330 nm. In-plane polarization indications and azimuth angles are marked. (b) Schematic of homogeneous modification of the grating with 12 nm NB layer. (c) Offset-corrected azimuthally (a) anisotropic reflectance by polarization-dependent IR microscopy. ENZ and BE modes of native SiO₂ and the near-field enhanced NO₂ modes of NB are marked.

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This in-plane anisotropy separating ENZ local field confinement and BE mode is accessible by polarization-dependent far-field IR microscopy [20] via azimuthal rotation of the grating. Since only the in-plane polarization direction needs to be defined as indicated in Fig. 6(a), mixed p- and s-polarization states can be used allowing focusing optics. We applied a Bruker Hyperion 3000 FT-IR microscope equipped with a 15 × Cassegrain objective providing an average AOI of about 17°. The setup includes both polarizer and analyzer with parallel alignment, and measures MIR reflectance of a 160 × 160 µm² spatial region with 1 cm⁻¹ spectral resolution.

To demonstrate the local character and in-plane plasmonic excitation of the ENZ mode in the 2 nm native SiO₂ film on Si, we functionalized the grating with a homogeneous 12 nm nitrobenzene (NB) layer by electrochemical grafting [26] as shown in Fig. 6(b). The resulting azimuth-dependent reflectance pseudocolored according to Fig. 6(a) is presented in Fig. 6(c). For clarity, we applied an offset correction for an isotropic response at 1675 cm⁻¹. At 90° azimuth, ENZ local field confinement dominates according to AFM-IR in Fig. 5(d). Here, enhanced ν_sym(NO₂) and ν_asym(NO₂) vibrations of NB are present, proving the LPR near-field E_⊥ origin of ENZ [6,10]. These modes are visualized by DFT simulations of a NB monomer. At 0° azimuth where Drude behavior is evident [7], E_⊥ is provided solely by the incident IR light. Here, the ENZ mode is absent together with the NB modes, and only the BE mode as minor far-field Re[ε] = 0 effect in plasmonic ENZ materials is clearly seen.

We simulated total absorption at AOI = 17° by integrating Eq. (1) over the unit cell of the grating. Here, ε(Ti) was taken from [23]. Near-field induced ENZ absorption in Fig. 7(a) is consistent with AFM-IR and IR microscopy results in Figs. 5(d) and 6(c), respectively. ENZ and BE absorption maxima in the 2 nm native SiO₂ film are found at 1217 and 1234 cm⁻¹, respectively. The BE/ENZ peak ratio is about 0.0125 so that ENZ absorption is a dominating effect in accordance with AFM-IR simulation in Fig. 5(b).

 

Fig. 7 (a) Simulated p-polarization-dependent total absorption of the grating at AOI = 17° where ENZ and BE are at 1217 cm⁻¹ and 1234 cm⁻¹, respectively. (b) Simulation of in-plane plasmon induced E_⊥ at AOI = 0° showing unit cell distribution of dipolar ENZ local field confinement in SiO₂.

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Compared to AFM-IR where the ENZ maximum is at 1229 cm⁻¹, the 12 cm⁻¹ redshift of the grating ENZ towards the Re[ε] = 0 position [Fig. 1(a)] can be explained by spatial dispersion of differently oriented tip and grating LPRs resulting in stronger coupling of plasmonic near-field E_⊥ in the case of in-plane orientation. Figure 7(b) presents simulated E_⊥ in the unit cell of the grating at normal incidence where the thickness of SiO₂ layer was increased to 5 nm for clarity. The dipolar ENZ mode shows larger spatial distribution of the confined E_⊥ field, indicating stronger coupling compared to AFM-IR in Fig. 5(d). In addition to the AFM-IR characterization of bare ultrathin films, spatial dispersion of the nanostructures needs to be carefully taken into account when designing and optimizing plasmonic ENZ materials. This combined approach could open up new routes to LPR-based IR sensing since ENZ modes are indicative of near-field E_⊥ [Eq. (1)], and may be applied to quantification of analytes [Fig. 6(c)].

6. Conclusion

Two fundamental polaritonic effects in thin films near Re[ε] = 0, Berreman mode and epsilon-near-zero local field confinement, were unambiguously identified at the nanoscale by AFM-IR. The results provide direct information on the enhanced light–matter interactions in homogeneous SiO₂ films via IR absorption of these resonant phenomena that can be understood in terms of FDTD calculations. This study renders AFM-IR a promising technique for high-sensitivity thin film analysis for novel Re[ε] = 0 based IR photonic materials with tailored optical and thermal response. We believe that the method could allow precise characterizations of heterogeneous and multilayered films exhibiting Re[ε(r,ω)] = 0 contrast [27], and doped semiconductor films with voltage controlled dielectric function [12] where AFM-IR is expected to provide time-resolved BE and ENZ mode studies of active Re[ε] = 0 materials.