Introduction

The mid-infrared (IR) spectral range, pertaining to frequencies within the range 400–2500 cm−1, is host to a broad range of applications in renewable energy1,2,3,4, molecular sensing5,6, IR spectroscopy7 and thermal camouflage8. In the search of materials relevant for mid-IR photonics, emerging layered van der Waals (vdW) materials hold a prominent role, because they support several intriguing phenomena9, for instance negative refraction10,11, negative reflection12, topological transitions of polariton dispersion13,14,15, and deeply subwavelength light-matter interactions16,17,18. Often, these phenomena rely on the strong coupling of light to lattice vibrations (phonons) in polar materials, yielding long-lived evanescent excitations termed surface phonon polaritons. These occur at mid-IR frequencies19,20, when the dielectric function (ϵ(ω)) of the material is resonant and becomes negative for a narrow spectral range called the Reststrahlen band (see Fig. 1a)9,20. VdW materials like hexagonal boron nitride (hBN)17,21 and α-MoO322,23 support such phonon polaritons and additionally exhibit strong optical anisotropies, yielding a hyperbolic dispersion24,25.

Fig. 1: Comparison of extraction methods.
figure 1

a The complex dielectric function ϵ(ω) in black (left axis) and the complex refractive index in red (right axis) of a polaritonic material with parameters: \({\epsilon }_{\inf }=5\), ωTO = 1360 cm−1, ωLO = 1610 cm−1, and γ = 7 cm−1 (similar to values reported for hBN44). b The far-field reflectance spectrum of a d = 0.3 μm-thick layer of an isotropic material with dielectric function that of panel (a), on a gold substrate. c The complex wavenumber of phonon polaritons, qx(ω), for a d = 0.120 μm-thick layer of the material with dielectric function in panel (a), in air.

Full size image

The highest crystal quality of such vdW materials occurs when they are mechanically exfoliated from bulk crystals26. Although several larger-scale crystal fabrication techniques exist, such as chemical vapor deposition27 and vdW epitaxy28, the quality and optical properties of mechanically exfoliated vdW flakes continue to be superior29. Additionally, the extreme anisotropy of various vdW materials, e.g. hBN and α-MoO3, is preserved best when they are exfoliated30. Unfortunately, the lateral dimensions of exfoliated flakes do not typically exceed some tens of micrometers31, or hundreds of micrometers in exceptional cases. The small dimensions of vdW exfoliated flakes pose significant challenges in mid-IR experimentation and characterization, due to the large cross-sectional area of a mid-IR beam that exceeds that of a flake.

To obtain the complex dielectric function of a material, especially near a resonance (e.g. Reststrahlen band), where both the real, Re(ϵ), and imaginary parts, Im(ϵ), are large, at least two observable quantities ought to be measured. The conventional approach for estimating ϵ(ω) is spectroscopic ellipsometry32, which relies on measurements of the reflection coefficient at oblique incidence for two linear polarizations, detecting changes in the amplitude (Ψ) and phase (Δ) of their ratio. The cross-sectional area of an IR beam at oblique incidence, however, can reach the millimeter scale. Evidently, this makes the extraction of the dielectric function of mechanically exfoliated vdW flakes with small lateral dimensions a challenging task33. To overcome this challenge, previous approaches relied on exciting evanescent modes, for example, surface phonon polaritons, and estimating the dielectric function from an experimentally-obtained dispersion diagram ω(q), where q is the polariton’s wavenumber. Most often, the experimental observables are the polariton’s wavelength \({\lambda }_{{\rm{p}}}=2\pi /{\rm{Re}}(q)\) and decay length \({L}_{{\rm{p}}}=1/{\rm{Im}}(q)\)21,22.

The excitation of evanescent and polaritonic modes, however, requires expensive and delicate near-field instrumentation such as tip-based scanning probes34,35, as well as precise nano-imaging for the accurate extraction of a dispersion diagram of a mode36. Such nano-scale imaging involves reconstructing hyperspectral data, which can be cumbersome37. Additionally, correlating these measurements with the intrinsic dielectric function of a material is convoluted, requiring considerable numerical fitting and theoretical modeling38, especially for materials with strongly anisotropic properties39, spatial dispersion40, or when the sample dimensions (e.g. film’s thickness, lateral dimensions) are unknown. Near-field tip-based measurements are generally time-consuming as well, require a tunable narrow-band laser for probing an extended frequency range, and cannot be easily adapted to non-standard experimental conditions such as low temperatures (which requires state-of-the-art cryogenic microscopy) or high pressures, relevant in the research of vdW materials.

There have been efforts to utilize more conventional approaches for the extraction of the dielectric function of small-sized flakes17,41,42,43 that rely on far-field excitation with Fourier Transform Infrared (FTIR) micro-spectroscopy. Alvarez-Perez et al. refined the dielectric function obtained by FTIR reflectance (R(ω)) measurements by probing the dispersion and damping rate of phonon polaritons, revealed by a near-field scanning approach41. FTIR reflectance measurements have also been used to extract the dielectric function of naturally abundant17 and isotopically enriched hBN42,44. In the aforementioned approaches, however, the main challenge of applying reflectance-fitting to exfoliated flakes is their small lateral size. When the flake dimensions approach the wavelength of light, as is often the case for vdW flakes at mid-IR wavelengths, the approximation of having an film with infinite lateral dimensions break down, leading to discrepancies between simulated and measured spectra.

In this work, we present a simple, empirical method to experimentally extract the complex dielectric function of vdW materials near a phonon resonance that relies on measurements of reflectance spectra taken with conventional far-field FTIR micro-spectrometry. This method is applicable to exfoliated flakes with small lateral dimensions. The smallest cross-sectional area of a flake for which the method remains robust depends on the resolution of the microscope used and is typically on the order of few tens of microns. The method is free from the aforementioned challenges of near-field-based approaches and can be readily applied by anyone possessing a microscope and a spectrometer in a variety of experimental conditions (e.g. dynamic temperatures changes, high-pressure). Importantly, this method does not require elaborate fitting algorithms or an a priori estimation of the dielectric function of the measured material, but rather relies on fundamental Kramers-Kronig relations upon identifying the positions of minima in R(ω). In particular, via simple Fabry-Pérot analysis, it is straightforward to show that, as long as the flake is atop a reflecting substrate, the frequency where reflectance minima occur (ωd) directly yields the real part of the refractive index of the material (\({\rm{Re}}(\sqrt{\epsilon (\omega )})\)). Unlike conventional approaches where one sample suffices, here, multiple flakes of varying thicknesses are required in order to extract the dielectric function of a material, over an extended spectral range, since the position of a Fabry-Pérot resonance depends on the flake's thickness, d. This, however, does not impose a considerable limitation in the case of exfoliated samples, as it is typical that at least tens of flakes with varying thicknesses are usually exfoliated onto a substrate45, and these can be easily measured with an FTIR microscope. As a benchmark, we demonstrate the accuracy of our method by applying it to exfoliated flakes of hBN and α-MoO3, as cases of in-plane isotropic and in-plane anisotropic vdW materials, respectively, at frequencies near their phonon resonances. These materials have been widely explored in recent literature on mid-IR photonics12,13,20,25,46,47, and their dielectric properties have been extensively characterized by complementary near- and far-field techniques, making them suitable reference materials for validation of our method.

Results

Let us model the dielectric function ϵ(ω) of a polar material near a phonon resonance using a Lorentz oscillator48:

$$\epsilon (\omega )={\epsilon }_{\inf }\left[1+\frac{{\omega }_{{\rm{LO}}}^{2}-{\omega }_{{\rm{TO}}}^{2}}{{\omega }_{{\rm{TO}}}^{2}-{\omega }^{2}-i\gamma \omega }\right]$$
(1)

where \({\epsilon }_{\inf }\) is the high-frequency permittivity, ωTO and ωLO are the transverse and longitudinal optical phonon frequencies, respectively, and γ is the phonon inverse lifetime (damping rate). Within the Reststrahlen band, between ωTO and ωLO, Re(ϵ) can be negative, while Im(ϵ) exhibits a resonance as shown in Fig. 1a for \({\epsilon }_{\inf }=5\), ωTO = 1360 cm−1, ωLO = 1610 cm−1, and γ = 7 cm−1. These values are similar to values reported for the dielectric function of hBN44. Extracting ϵ(ω) requires determining the four dielectric parameters \({\epsilon }_{\inf }\), ωTO, ωLO, and γ of Eq. (1).

Extracting ϵ(ω) via near-field excitation of polaritons

As discussed above, previous approaches for experimentally extracting ϵ(ω) of vdW flakes via near-field nano-imaging probe the surface phonon polariton wavelength λp(ω) and decay length Lp(ω)21,41. Measuring λp(ω) and Lp(ω) allows recovering the in-plane wavenumber q(ω) of the polariton, which, in turn, contains the components of the dielectric function. In the general case of an anisotropic material with \(\overleftrightarrow{\epsilon }={\rm{diag}}\{{\epsilon }_{{\rm{x}}}(\omega ),{\epsilon }_{{\rm{y}}}(\omega ),{\epsilon }_{{\rm{z}}}(\omega )\}\), where ϵx and ϵy are the dielectric functions along the x- and y-coordinate directions in the plane of the crystal, ϵz is the out-of-plane dielectric function, the dispersion relation for a thin film of thickness d is21,41:

$$q(\omega )=\frac{\rho (\omega )}{d}\left[\arctan \left\{\frac{\rho (\omega )}{{\epsilon }_{{\rm{z}}}(\omega )}\right\}+\arctan \left\{\frac{\rho (\omega ){\epsilon }_{{\rm{s}}}}{{\epsilon }_{{\rm{z}}}(\omega )}\right\}\right]$$
(2)

where \(\rho (\omega )=i\sqrt{{\epsilon }_{{\rm{z}}}(\omega )/({\epsilon }_{{\rm{x}}}(\omega ){\cos }^{2}\psi +{\epsilon }_{{\rm{y}}}(\omega ){\sin }^{2}\psi )}\) with ψ being the angle between the x axis and the in-plane wavevector q, and ϵs is the dielectric function of the substrate, which is considered isotropic. A conventional polaritonic dispersion relation q(ω) (with ψ = 0) for an isotropic material with the ϵ(ω) given by Eq. (1) (Fig. 1a) and a thickness of d = 120 nm, which is a typical thickness of a flake of a van der Waals material upon exfoliation41, is shown in Fig. 1c. In order to evaluate q(ω) in Eq. (2), it is assumed that d ≪ λ, with λ being the wavelength of light in free space. This condition is easily met at mid-IR frequencies where λ is on the order of tens of micrometers. Once the complex q(ω) is obtained from the experimentally measured parameters λp(ω) and Lp(ω) via nano-imaging, in the case of isotropic properties with ϵx = ϵy = ϵz = ϵ, both the real and imaginary part of ϵ can be extracted simultaneously.

Near-field methods are particularly well suited for extracting the dielectric function, as they can be applied to flakes with lateral dimensions of a few-tens-of-μm. Because the phonon-polariton decay lengths are much shorter than the flake dimensions, the measured Lp(ω) and λp(ω) remain largely unaffected by finite-size effects, making near-field approaches more reliable than far-field methods for retrieving optical constants. However, due to the convoluted nature of Eq. (2), extracting ϵ(ω) requires significant numerical fitting. In the presence of anisotropy, as is often the case with vdW materials, one can probe simultaneously either ϵx and ϵz by setting ψ = 0 or ϵy and ϵz by setting ψ = π/2. These two distinct values of ψ pertain to incident light along the x-z plane and along the y-z plane, respectively. For both measurements, however, four parameters (real and imaginary parts of ϵz(ω) and ϵx/y(ω)) are fitted simultaneously to only two experimental observables: λp(ω) and Lp(ω), making the extraction problem under-determined and ill-posed39. To address this, fitting techniques and inverse problem algorithms are required to regularize the solution, adding complexity38,39.

Extracting ϵ(ω) based on far-field reflectance spectra

As mentioned in the previous section, the prerequisite for obtaining the dielectric function ϵ(ω) of a vdW flake via near-field nano-imaging is the excitation of a phonon polaritonic mode at the surface of the material. For a phonon polariton to exist, the dielectric function ought to be negative49. Thereby, near-field-based extraction approaches rely on the spectral region within the Reststrahlen band (see Fig. 1a, c).

In the spectral region outside of the Reststrahlen band, the far-field reflectance spectrum of a polaritonic thin-film has a characteristic signature. This is shown in Fig. 1b for the hypothetical material with ϵ(ω) of Fig. 1a on a gold substrate, assuming a thickness of d = 1 μm. In the spectral region highlighted with orange-shaded color in Fig. 1a, b, the polaritonic material behaves like a dielectric (ϵ(ω) > 0). The dip in reflectance, at the frequency indicated as ωd in Fig. 1b corresponds to a Fabry-Pérot resonance and depends on the thickness of the film, d, whereas the smaller dip at higher frequencies closer to ωTO arises from the intrinsic resonance of the material. The reflectance, R, is given by the absolute square of the reflection coefficient, R(ω) = ∣r(ω)∣2. Without loss of generality, for linearly polarized incident light with an electric field along the x-direction at normal incidence, the reflection coefficient, rx(ω), of a thin film can be expressed analytically as (see Supplementary Information, Section 1.1.):

$${r}_{{\rm{x}}}(\omega )=\frac{\left(1-\frac{1}{{\aleph }_{{\rm{s}}}}\right)+i\left(\frac{1}{{\aleph }_{{\rm{x}}}(\omega )}-\frac{{\aleph }_{{\rm{x}}}(\omega )}{{\aleph }_{{\rm{s}}}}\right){T}_{{\rm{ex}}}(\omega )}{\left(1+\frac{1}{{\aleph }_{{\rm{s}}}}\right)-i\left(\frac{1}{{\aleph }_{{\rm{x}}}(\omega )}+\frac{{\aleph }_{{\rm{x}}}(\omega )}{{\aleph }_{{\rm{s}}}}\right){T}_{{\rm{ex}}}(\omega )}$$
(3)

where \({\aleph }_{{\rm{x}}}(\omega )=\sqrt{{\epsilon }_{{\rm{x}}}(\omega )}={n}_{{\rm{x}}}(\omega )+i{k}_{{\rm{x}}}(\omega )\) and \({\aleph }_{{\rm{s}}}=\sqrt{{\epsilon }_{{\rm{s}}}}={n}_{{\rm{s}}}+i{k}_{{\rm{s}}}\) are the complex refractive indices of the flake and the substrate, respectively, and \({T}_{{\rm{ex}}}(\omega )=\tan (2\pi d{\aleph }_{{\rm{x}}}(\omega )/\lambda )\). The same equation holds for reflection along the y axis by exchanging the subscript x ↔ y.

The dielectric permittivity of materials can also been extracted from reflectance spectra17,42,43,44,50,51, via fitting to an a priori known model calculated with the Fresnel coefficients, in other words the transfer matrix approach52. However, a single experimental observable, in this case R(ω), is not sufficient for the extraction of both real and imaginary parts of ϵx/y(ω), especially within a highly dispersive spectral range and for flakes with lateral sizes less than 100 μm. This can be understood by comparing the measured and calculated (Eq. (3)) reflectance spectra for two α-MoO3 flakes with different lateral dimensions (Fig. S5 of Supplementary Information). The mismatch between the Fresnel coefficient-based transfer matrix model (Eq. (3)) and the measured spectra systematically increases as the lateral dimensions of a flake decrease.

To circumvent this issue, in previous approaches, measurements of R(ω) at multiple angles of oblique incidence were used43,53,54. Nonetheless, reflectance measurements at oblique incidence are particularly challenging for exfoliated flakes in the mid-IR, since their small size (few-tens-of-μm) necessitates the use of reflective objectives, which inherently restrict angle-resolved measurements.

In addition, once oblique incidence is considered, Eq. (3) must be modified to include the effect of ϵz, the out-of-plane dielectric function, whose influence becomes significant at large angles of incidence. This further convolutes the extraction of the permittivity in the case of anisotropic materials, since one ought to extract four parameters out of a single experimental observable, similar to the near-field-based approaches discussed above. Some previous approaches utilized far-field reflectance measurements at normal incidence to overcome these challenges, however, this required supplementary near-field measurements in order to extract the full dielectric tensor of vdW and other emerging materials17,41,55.

Extraction of ϵ(ω) based on reflectance minima

In this work, we introduce an empirical approach for extracting the mid-IR permittivity of small-sized exfoliated flakes, using reflectance measurements at normal incidence, without any additional near-field scanning probe. With the introduced approach, the dielectric function is obtained along both in-plane orthogonal directions (x and y). Unlike near-field approaches, that require the excitation of polaritonic modes, the dispersion of which unavoidably depends on both ϵx and ϵz (for ψ = 0), or ϵy and ϵz (for ψ = π/2) as shown in Eq. (2), thereby introducing complexity in the extraction of these parameters independently, the expression of rx/y (Eq. (3)) only depends on ϵx/y. Thereby, by aligning the polarization of the incident electric field along a certain coordinate direction (x or y), one can extract unambiguously the dielectric permittivity along that direction. The experimental quantity that we consider in this extraction approach is not the reflectance spectrum directly; instead, upon measuring this spectrum, we carefully detect the frequency ωd at which this spectrum reaches a minimum due to a Fabry-Perot resonance.

Direct extraction of \({\rm{Re}}(\sqrt{\epsilon (\omega )})\) outside the Reststrahlen band

In deriving ωd from Eq. (3), we observe in Fig. 1a that \({\rm{Im}}(\epsilon )\) rapidly decreases for frequencies outside the Reststrahlen band. By considering \({\rm{Im}}({\epsilon }_{{\rm{x}}/{\rm{y}}}(\omega ))\,\ll\, {\rm{Re}}({\epsilon }_{{\rm{x}}/{\rm{y}}}(\omega ))\), where the indices x/y indicate the dielectric function along the x/y-direction, from the first and second derivatives of Eq. (3), we can write56:

$${\omega }_{{\rm{d,x/y}}}=\frac{1}{4d}\left[\frac{1}{{n}_{{\rm{x}}/{\rm{y}}}({\omega }_{{\rm{d}}})}-\frac{1}{\pi {n}_{{\rm{x}}/{\rm{y}}}({\omega }_{{\rm{d}}})}\arctan \left\{\frac{-2{n}_{{\rm{x}}/{\rm{y}}}({\omega }_{{\rm{d}}}){k}_{{\rm{s}}}}{{n}_{{\rm{x}}/{\rm{y}}}^{2}({\omega }_{{\rm{d}}})-{n}_{{\rm{s}}}^{2}-{k}_{{\rm{s}}}^{2}}\right\}\right]$$
(4)

where nx/y is the real part of the refractive index (\(\sqrt{{\epsilon }_{{\rm{x}}/{\rm{y}}}}\)) of the flake along the x/y-direction, and ns and ks are the real and imaginary parts of the refractive index of the substrate. Importantly, kx/y does not appear in Eq. (4). Therefore, the experimental measurement of ωd,x/y directly and unambiguously yields nx/y(ω = ωd), assuming that the thickness of the film and the optical properties of the substrate are known. So far, we have established that one can obtain nx/y(ωd,x/y) from measurements of the normal-incidence reflectance spectra of exfoliated flakes on a reflective substrate by aligning the polarization of the incident electric field along either the x or y axis (see Fig. 2a), as long as ωd lies within a spectral range where the material exhibits low losses.

Fig. 2: Extraction of nx(ωd) of hBN outside the Reststrahlen band.
figure 2

a Schematic of the experimental setup. The sample is represented by an AFM scan of an exfoliated flake, for example and without loss of generality, an in-plane anisotropic flake for which ϵx ≠ ϵy, e.g. α-MoO3, on a metallic substrate. Reflectance spectra were measured using a conventional FTIR microscope equipped with two polarizers to select the polarization direction of the light. b Microscope image of exfoliated hBN flakes on a gold substrate, with the thicknesses of four representative flakes labeled. c Simulated reflectance spectra for varying flake thicknesses (left axis), overlaid with analytically calculated reflectance minima frequencies, ωd (dashed lines), obtained using Eq. (4) under the low-loss assumption. The ϵx(ω) used for these calculations is shown in Fig. 1a. The ratio nx(ω)/kx(ω) is plotted as a black dashed line (right axis), and the low-loss approximation is considered valid in the frequency range where nx(ω) ≳ 10kx(ω) (shaded in yellow).

Full size image

Let us define more specifically this spectral region, termed “low-loss” spectral range henceforth, as the region where nx/y(ω) ≳ 10kx/y(ω). The location of this spectral region depends on the thickness of the flake, d (see Eq. (4)). For conventional vdW materials, the low-loss condition is satisfied for a large range of thicknesses, as long as they are not ultra-thin. In Fig. 2b, we present a microscope image of flakes of hBN that are simultaneously exfoliated from the same crystal onto a gold substrate. hBN is in-plane isotropic (ϵx = ϵy). The thickness of each flake, measured by atomic force microscopy (AFM), is indicated.

The corresponding reflectance spectra are shown in Fig. 2c. As shown in the yellow-highlighted region, the low-loss condition (nx ≳ 10kx) is satisfied for the majority of flakes, where the optical properties of hBN were taken from ref. 44. As long as the flakes are thicker than ~ 100 nm, ωd lies within the low-loss spectral range of hBN, where our method can be applied accurately. Precise determination of flake thickness is critical for our approach, and for all flakes used in this work, the uncertainty in thickness measurements is below 3% (see tabulated data in Supplementary Information Section 2). Since the flakes used here exceed 100 nm in thickness, they lie outside the range where dielectric properties exhibit noticeable thickness dependence57.

Thereby, by taking measurements of reflectance spectra on different exfoliated flakes of various thicknesses, di, on the same substrate, one obtains \({\omega }_{{{\rm{d}}}_{{\rm{i}}}}\), from which, via Eq. (4), we can obtain \({n}_{{\rm{x}}}({\omega }_{{{\rm{d}}}_{{\rm{i}}}})\), for i = 1, 2,... hence constructing the frequency dependence of nx(ω). The imaginary part of the refractive index of the flakes, kx is then easily estimated. In particular, kx(ω) can be obtained either via Kramers-Kronig relations50, or by fitting nx(ω) to Eq. (1), from which all parameters of the Lorentz oscillator can be determined.

So far, we have discussed the extraction of the dielectric function along both in-plane coordinate directions assuming the low-loss condition (nx/y(ω) ≳ 10kx/y(ω)). We note that this can be applied not only for ω < ωTO (left highlighted region in Fig. 1a) but also for ω≥ωLO (see Fig. 1a) as we shall show below. To obtain the dielectric function along the x- and y-coordinate directions, one ought to align the incident electric field with either the x or y axis of the crystal, and to take reflectance spectra along that direction for frequencies outside the Reststrahlen band. By fitting Eq. (1) to the experimentally obtained values of n(ω) obtained via Eq. (4), all dielectric parameters of Eq. (1) are obtained, and one can thereby estimate the dielectric function of the material within the Reststrahlen band as well.

Numerical extraction of \({\rm{Im}}(\sqrt{\epsilon(\omega)})\) within the Reststrahlen band for hyperbolic materials

Due to the low γ of many polaritonic materials, like α-MoO341, hBN17, and LiV2O555, their dielectric function varies very rapidly with ω within the Reststrahlen band. Thereby, it is also useful to be able to verify the accuracy of the dielectric function obtained as outlined previously, in the low-loss regime, within the Reststrahlen band. It is possible to obtain this information for either in-plane coordinate direction (x and y), as long as the material behaves as a dielectric and satisfies the low-loss condition (ny/x(ω) ≳ 10ky/x(ω)) along the orthogonal direction (y or x). This is the case for a family of vdW materials, namely hyperbolic ones, for which \({\rm{Re}}({\epsilon }_{{\rm{x}}}(\omega )){\rm{Re}}({\epsilon }_{{\rm{y}}}(\omega )) < 0\)41,55.

In particular, when incident light is polarized at an angle ψ = 45° relative to the x-axis (inset shown in Fig. 3a), the effective reflection coefficient is given by (see Supplementary Information, Section 1.2):

$$r(\omega ,\psi =4{5}^{\circ })=\frac{{r}_{{\rm{x}}}(\omega )+{r}_{{\rm{y}}}(\omega )}{2}$$
(5)

where rx(ω) and ry(ω) can be calculated from Eq. (3). The reflectance spectra, R(ω, ψ = 45°), shown in Fig. 3, are calculated for a material with the dielectric function of Fig. 1a along the x-axis and a frequency-independent dielectric function (ϵy = 20) along the y-axis. We see that within the Reststrahlen band along the x-direction, R(ω, ψ = 45°) exhibits distinct minima when the incident electric field has both x and y components, occurring at frequencies labeled as ωd,RB. It is very common in the case of hyperbolic materials like hBN and α-MoO3 to exhibit such a dielectric response, having a Reststrahlen band along one coordinate direction and a positive dielectric permittivity along an orthogonal direction. Assuming that the dielectric response along the y-axis is obtained as outlined in the previous section, it becomes possible to obtain the dielectric function within the Reststrahlen band along the x-direction (and vice-versa).

Fig. 3: Numerical extraction of kx(ωd) in the Reststrahlen band for in-plane anisotropic materials.
figure 3

Simulated reflectance spectra at ψ = 45° (the incident electric field is schematically represented in the inset) for various flake thicknesses (left axis), calculated using Eq. (5). Reflectance dip frequencies (ωd,RB) within the Reststrahlen band are indicated by vertical dashed lines. The dielectric function ϵx(ω) corresponds to that used in Fig. 1 along the x-direction, while a constant real-valued ϵy = 20 is considered along the y-direction. The ratio kx(ω)/nx(ω) is shown as a black dashed line (right axis). The method enables extraction of kx(ωd,RB) within the frequency range where kx(ω) ≳ 10nx(ω) (shaded in yellow).

Full size image

In particular, from Fig. 1a, within the Reststrahlen band, kx(ω) ≫ nx(ω). In fact, as shown in Fig. 3, it holds that kx(ω) ≳ 10nx(ω) within the Reststrahlen band, thereby the reflection coefficient rx(ω) (Eq. (3)) becomes nearly independent of nx(ω). Consequently, Eq. (5) is primarily governed by kx(ω) (since ny is known), allowing kx(ωd) to be extracted numerically from the observed reflectance minima ωd,RB. Upon applying this approach to flakes of different thicknesses and thus obtaining kx(ω), via Eq. (1) or via Kramers-Kronig relations, one can directly obtain nx(ω) within the Reststrahlen band as well (see Supplementary Information Section 2.1 and 2.2).

To summarize, we have demonstrated two approaches for obtaining the dielectric function of a polaritonic material within the Reststrahlen band. In the previous section, we discussed how to indirectly obtain this information by extrapolating from the dielectric function obtained experimentally outside the Reststrahlen band. In parallel, in this subsection, we discussed how to numerically recover the dielectric function within the Reststrahlen band via numerical fitting of Eqs. (3), (5). In the following sections, we discuss the basics of extraction of the dielectric response of exfoliated flakes in the case of strong anisotropy, and then we apply our method to hBN and α-MoO3.

Extraction of the dielectric function of hBN

In this section, we utilize the method described above to extract the dielectric response of hBN, which is isotropic in-plane (ϵx = ϵy), and has a dielectric function that follows the Lorentz oscillator model of Eq. (1). We measure the reflectance spectrum R(ω) of eleven hBN flakes of varying thickness ranging from 117 nm to 320 nm, exfoliated onto a gold substrate. For these measurements, we use a Bruker Hyperion 2000 microscope coupled to a Bruker Tensor FTIR spectrometer, equipped with a Mercury-Cadmium-Telluride detector which has a spectral range of (600−8000) cm−1 (1.5 μm−16.6 μm). Figure 4a presents microscope images of three representative hBN flakes while their corresponding thicknesses and reflectance spectra shown in Fig. 4b, c, respectively. The reflectance dips in Fig. 4c occur in two separate spectral ranges, which are both outside the Reststrahlen band of hBN. One spectral region pertains to frequencies for which ω < ωTO whereas the other pertains to ω > ωLO. At the frequency of each dip, corresponding to a Fabry-Pérot resonance, we recover the corresponding refractive index nx(ωd) using Eq. (4). The extracted values of nx(ωd) are plotted as circles in Fig. 4d (circles), and their absolute values are presented in Table S1 of Supplementary Information Section 2.3. We note that, in total, eleven reflectance spectra were taken, one for each flake. For the sake of conciseness, we present only three spectra in Fig. 4c pertaining to the three flakes in Fig. 4a, b. In Section 2.3 of the Supplementary Information, we present all spectra combined. Details of the error estimation for n(ωd) are provided in the Methods “Estimating errors”.

Fig. 4: Extracting ϵ(ω) of hBN.
figure 4

a Microscope images of three hBN flakes of different thicknesses. (b) Thickness profiles measured by AFM of the three flakes shown in panel a. The extracted thicknesses d are 285 nm, 220 nm, and 160 nm, and these were measured along the paths marked with the dashed lines in panel (a). c FTIR-measured reflectance spectra for the three flakes. d Extracted real part of the refractive index, nx(ω) as a function of frequency. The solid circles are values extracted directly from Eq. (4), whereas the solid curves are the result of fitting Eq. (1) to the data points depicted as solid circles. Results are color-coded according to the three different flakes shown in panel (a). The estimation error on n(ωd) is shown in panel (d).

Full size image

By fitting Eq. (1) to the experimentally retrieved nx(ωd), we extract the frequency-dependent refractive index, which is shown in Fig. 4d with the solid curve. From this fitting, we also extract the four dielectric parameters of Eq. (1) along with their respective standard errors: \({\epsilon }_{\inf }=4.75\pm 0.37\), ωTO = 1362 ± 1.7 cm−1, ωLO = 1635 ± 30 cm−1 and γ = 8.75 ± 1.9 cm−1. The global root-mean-square error (RMSE) of the numerical fitting is 0.25. The method used to calculate the standard errors is detailed in the Methods “Estimating errors”. These extracted parameters closely match previously reported values, as reported by Li et al.37, Giles et al.44 and Caldwell et al.17, as can be seen in Table 1, where the standard errors of our method are presented as a percentage.

Table 1 Extracted parameters for the in-plane ϵ(ω) of hBN according to Eq. (1)
Full size table

The following observations can be made about the estimation of each parameter (ωTO, ωLO, γ, \({\epsilon }_{\inf }\)); the reflectance dips at frequencies below the Reststrahlen band (ω < ωTO) are very pronounced and are thereby easily detectable spectroscopically. Consequently, the extracted values of ωTO and ωLO have relatively low uncertainties. In contrast, the reflectance dips at frequencies ω > ωLO are broader and less pronounced, resulting in a less precise estimation of the frequencies ωd in this spectral range. This yields a larger σE for \({\epsilon }_{\inf }\). Accurate estimation of the damping factor γ requires utilizing experimental measurements within the Reststrahlen band and around the TO and LO phonon frequencies. However, due to the in-plane isotropy of hBN, the method described previously cannot be applied to evaluate k(ωd) in this region. Moreover, since the low-loss approximation fails in the vicinity of ωTO (Fig. 2c), n(ωTO) cannot be extracted from the measurements. As a result, the extracted value of γ has the highest uncertainty among the parameters, although the associated σE remains within acceptable limits for practical use.

Extraction of the dielectric function of α-MoO3

The crystal structure of α-MoO3 is orthorhombic, which leads to strong in-plane anisotropy23. The dielectric functions along the crystal’s principal axes [100], [001], and [010], labeled as ϵx, ϵy, and ϵz, respectively, are radically different from each other. Here, we use our method to extract ϵx and ϵy. Figure 5a shows a representative α-MoO3 flake, exfoliated from a bulk crystal onto a gold substrate, along with its AFM thickness map (panel b). As shown, exfoliated flakes typically have a rectangular cross-sectional area with edges aligned to the [100] and [001] directions. In total, we measured the reflectance of thirteen flakes with varying thicknesses ranging from 350 nm to 2200 nm.

Fig. 5: Extracting ϵ(ω) of α-MoO3 along the x-direction.
figure 5

a Microscope image of an α-MoO3 flake on gold with in-plane crystal axes (x, y) labeled. b The AFM-measured thickness map of the flake with extracted thickness being d = 920 nm. c FTIR-measured reflectance spectra for polarizer angle ψ = 0° (x) for the flake. d Extracted real part of the refractive index, nx(ω) as a function of frequency. The solid circles are values extracted directly from Eq. (4), whereas the solid curves are the result of fitting Eq. (6) to the data points. The data point corresponding to the flake shown in panel (a) is marked as red circles whereas the rest data points correspond to flakes of other thicknesses. The error bars indicate the estimation error on nx(ωd).

Full size image

To extract ϵx(ω), we measured the reflectance spectra of the flakes using a polarizer aligned along the x-direction (ψ = 0 with respect to Fig. 5a), and the results are shown in Fig. 5c. We note that all thirteen exfoliated flakes satisfy the low-loss condition discussed in “Direct extraction of \({\rm{Re}}(\sqrt{\epsilon (\omega )})\) outside the Reststrahlen band”. For the sake of conciseness, we present only one spectrum in Fig. 5c, pertaining to the flake in Fig. 5a, b. In Section 2.4 of the Supplementary Information, we present all spectra combined. For each flake and each reflectance dip, we estimated nx(ωd) using Eq. (4), similar to the case of hBN (see “Extraction of the dielectric function of hBN”). The results are plotted as solid dots in Fig. 5d. These data are tabulated in Table S2 of Supplementary Information Section 2.4. Despite having only thirteen flakes to measure, there are more than thirteen data points in Fig. 5d. This is because, as explained in “Direct extraction of \({\rm{Re}}(\sqrt{\epsilon (\omega )})\) outside the Reststrahlen band”, each reflectance spectrum can, in principle, yield more than one reflectance dip, namely one below ωTO and one above ωLO, and each of these dips yields a value nx(ωd) (Eq. (4)) at a different frequency.

We then fit a model for the dielectric permittivity of α-MoO3 to the data points shown in Fig. 5d, in order to obtain a continuous curve that describes nx(ω). To do this, based on previous theoretical58 and experimental studies23,59, we take into account that α-MoO3 actually displays two distinct Lorentzian-shaped resonances in its dielectric response along the x-axis. Thereby, we consider the following oscillator model:

$${\epsilon }_{{\rm{x}}}(\omega )={\epsilon }_{\inf }\left[1+\frac{{\omega }_{{{\rm{LO}}}_{1}}^{2}-{\omega }_{{{\rm{TO}}}_{1}}^{2}}{{\omega }_{{{\rm{TO}}}_{1}}^{2}-{\omega }^{2}-i{\gamma }_{1}\omega }\right]\left[1+\frac{{\omega }_{{{\rm{LO}}}_{{\rm{2}}}}^{2}-{\omega }_{{{\rm{TO}}}_{{\rm{2}}}}^{2}}{{\omega }_{{{\rm{TO}}}_{{\rm{2}}}}^{2}-{\omega }^{2}-i{\gamma }_{{\rm{2}}}\omega }\right].$$
(6)

Fitting Eq. (6) to the retrieved values of nx(ωd) (Fig. 5d) yields the following Lorentz model parameters and respective standard errors: \({\epsilon }_{\inf }=4.42\pm 0.26\); \({\omega }_{{{\rm{TO}}}_{1}}=(589.5\pm 36)\,{{\rm{cm}}}^{-1}\), \({\omega }_{{{\rm{LO}}}_{1}}=(606.9\pm 25)\,{{\rm{cm}}}^{-1}\), γ1 = (51.5 ± 22) cm−1; \({\omega }_{{{\rm{TO}}}_{{\rm{2}}}}=(812\pm 2)\,{{\rm{cm}}}^{-1}\), \({\omega }_{{{\rm{LO}}}_{{\rm{2}}}}=(997\pm 12)\,{{\rm{cm}}}^{-1}\), γ2 = (17.5 ± 1.6) cm−1. The global RMSE of the numerical fitting is 0.16. The extracted parameters are summarized in Table 2, where they can be compared with previous results. In the same table, we present the standard errors of each parameter, in percentage. Since the FTIR measurement range is limited to (600–8000) cm−1, the first oscillator’s resonance (\({\omega }_{{{\rm{TO}}}_{1}}\)) lies outside of the spectral window of the FTIR. As a result, parameter extraction for this oscillator requires extrapolation, leading to higher uncertainty (6% for \({\omega }_{{{\rm{TO}}}_{1}}\) as compared to 0.2% for ωTO,2). In contrast, the parameters of the second oscillator fall within the measured range of the FTIR, hence they are estimated with high precision 9% uncertainty for γ2 as compared to 42% for γ1). We note for reference that, as discussed in ref. 60, the reported MSE values for each individual parameter \({\epsilon }_{\inf }\), ωTO, ωLO in the method of ref. 41 for the x-direction of α-MoO3 are 4%, 0.4%, and 0.6%, respectively, considering only far-field extraction of the dielectric permittivity. These values, however, cannot be directly compared with the standard errors in Table 2, which refer to differential errors (see Methods “Estimating errors” for details).

Table 2 Extracted parameters for ϵx(ω) of α-MoO3 along the x-direction according to Eq. (6)
Full size table

Next, we consider the dielectric function along the y-coordinate direction, ϵy(ω), which we model with a single Lorentz oscillator (Eq. (1)), based on previous results41. We take reflectance spectra when the polarizer is oriented along ψ = 90° (Fig. 3), thereby by aligning the electric field with the y-coordinate direction. According to23,41, the TO phonon frequency ωTO along y-direction lies outside the spectral window of the FTIR (see Table 3), thereby we cannot observe prominent reflectance dips for frequencies ω < ωTO along the y-direction. The dip in the reflectance spectrum shown with the solid red curve in Fig. 6a for ψ = 90° pertains to frequencies higher than ωLO,y, when the material satisfies the low-loss condition introduced in “Direct extraction of \({\rm{Re}}(\sqrt{\epsilon (\omega )})\) outside the Reststrahlen band”, and corresponds to the same flake that is shown in Fig. 5a, which has a thickness of 920 nm. We note that the low-loss condition is satisfied for nine out of the thirteen flakes that we exfoliated in total (see Supplementary Information Section Table S3 of Section 2.4). Based on this dip and similar ones for the other flakes considered, as discussed in “Direct extraction of \({\rm{Re}}(\sqrt{\epsilon (\omega )})\) outside the Reststrahlen band”, we extract values for ny(ωd) that are shown in Fig. 6b with solid dots. By fitting Eq. (1) to the data points of ny(ωd), we obtain the solid curve ny(ω) for frequencies outside the Reststrahlen band along the y-direction.

Table 3 Extracted parameters for ϵy(ω) of α-MoO3 along the y-direction according to Eq. (1)
Full size table
Fig. 6: Extracting ϵ(ω) of α-MoO3 along the y-direction.
figure 6

a FTIR-measured reflectance spectra for polarizer angles ψ = 90° (y-axis, red curve) and ψ = 45° with respect to the x-axis (blue curve), for the flake shown in Fig. 5a. b,c Extracted real and imaginary parts of the refractive index, ny(ω) and ky(ω) respectively as a function of frequency. The data points corresponding to the flake shown in Fig. 5a are marked as red and blue circles whereas the black data points correspond to flakes of other thicknesses. The error bars in panels (b) and (c) indicate the estimation error on ny(ωd) and ky(ωd,RB) respectively.

Full size image

Subsequently, we apply the method outlined in “Numerical extraction of \({\rm{lm}}(\sqrt{\epsilon(\omega)})\) within the Reststrahlen band for hyperbolic materials”, since the spectral range of the Reststrahlen band α-MoO3 along the y-coordinate direction overlaps spectrally with the range where the material behaves as a lossless dielectric along the x-direction41. In this hyperbolic region, the conditions ky(ω) ≳ 10ny(ω) and nx(ω) ≳ 10kx/y(ω) hold (see Fig. 5d), allowing extraction of ky(ωd,RB) within the Reststrahlen band. We apply Eq. (5) to estimate ky(ωd,RB) using previously extracted values of nx(ω) and kx(ω) (Table 2). Figure 6a (blue curve) shows the reflectance spectrum of the same flake as that of Fig. 5a, measured at ψ = 45°. The extracted values of ky(ωd,RB) from five flakes for which the frequencies ωd,RB are within the Reststrahlen band are shown in Fig. 6c as black circles and are tabulated in Table S3 of Supplementary Information Section 2.4. By fitting Eq. (1) simultaneously to the extracted values of ny(ωd) (Fig. 6b) and ky(ωd,RB) (Fig. 6c), we obtain the solid curves ny(ω) and ky(ω) for frequencies within and outside the Reststarhlen band along the y-direction. The best-fit parameters are: \({\epsilon }_{\inf }=4.9\pm 0.26\), ωTO = (534 ± 9) cm−1 and ωLO = (859 ± 11) cm−1. Accurate estimation of the damping factor γ requires data points close to ωTO, which is, unfortunately, outside the spectral range accessible with the FTIR used. Consequently, we fix γ = 9.5 cm−1, as reported in Alvarez-Pérez et al.41, for the purpose of fitting. The global RMSE of the fit is 0.17. We compare the extracted parameters we have obtained with previous literature in Table 3, where the errors of our method are presented as percentages. To conclude, as shown from Tables 2, 3, our method yields good agreement with previous literature results along both in-plane axes of the α-MoO3 crystal.

Discussion

We presented an empirical method for extracting the complex dielectric function ϵ(ω) of vdW flakes in the vicinity of their phonon resonances, using far-field FTIR microspectroscopy in contrast to previous approaches that have largely utilized near-field nano-imaging of the propagation characteristics of surface phonon polaritons23,34,37,38,39,41,55 and thus required complicated instrumentation.

Our approach enables reliable extraction of ϵ(ω) from far-field reflectance spectra by analyzing the frequencies of the reflectance minima, ωd, arising from Fabry-Pérot resonances, and applies as long as the material being measured is atop a reflective substrate. Our analysis reveals that the frequencies where reflectance minima occur, ωd, are highly sensitive to either the real or imaginary part of the refractive index, depending on the spectral region. This key feature allows for direct extraction of \(\sqrt{\epsilon ({\omega }_{{\rm{d}}})}\), per frequency. By measuring multiple exfoliated flakes of varying thicknesses, we obtain a frequency-dependent dielectric response.

For an accurate extraction of the dielectric response, as considered throughout this article, the reflectance measurements ought to correspond to normally incident light. However, the very weak angular dependence of Fabry-Pérot resonances as can be calculated with Fresnel coefficients suggests that the dielectric response can be extracted reliably even for oblique angles of incidence (see Supplementary Information Fig. S2). This becomes relevant and useful in the case of FTIR micro-spectroscopy, since microscope objectives collimate the IR beam. In our experimental setup, the central angle of incidence was 22°. An analysis over the validity of our method for small but non-zero angles of incidence is discussed in Supplementary Information Section 1.

Furthermore, we note that our approach is currently limited to extraction of the dielectric function only along the in-plane directions (see Eqs. (3), (4), (5)). In principle, the framework can be extended to extract the out-of-plane permittivity ϵz(ω) by applying Eqs. (3), (5) to oblique incidence, as Fabry-Pérot dips in this geometry depend explicitly on ϵz(ω) (Eq. S6 of Supplementary Information). Such dips are observed in α-MoO3 near ω ≈ 1000 cm−1 (Figs. 5c and 6a). Combining normal-incidence data with angle-dependent reflectance from flakes of different thicknesses could thus yield ϵz(ω).

While Fabry-Pérot resonances have previously been used to characterize weakly dispersive or low-loss materials61,62,63, our method remains effective and is most useful in regimes of strong frequency dispersion, where conventional approaches typically fail. In contrast to previous far-field techniques64,65, which are limited in the estimation of the damping factor, γ, and the high-frequency permittivity, \({\epsilon }_{\inf }\), our approach overcomes these limitations with a significantly simplified experimental framework.

We apply our method to the case of the highly dispersive Reststrahlen bands of hBN and α-MoO3, as materials in the classes of in-plane isotropic and in-plane anisotropic polaritonic materials at mid-IR frequencies. We extract the complete set of Lorentz oscillator parameters that describe them (Tables 1, 2, 3). We demonstrate that the extracted dielectric function in either case is in good agreement with results obtained using near-field probes41 and FTIR micro-spectroscopy17. We further note that for hBN, our results are directly validated against measurements on samples from the same source reported in the literature17,37,44, while for α-MoO3, numerical simulations using the extracted dielectric function reproduce experimental observations with good fidelity66.

Although demonstrated for known materials, the method is also applicable to materials for which no prior model for the dielectric function exists. In that case, the dielectric function is first estimated by measuring the reflectance spectra of flakes with a wide range of thicknesses and identifying the corresponding dip frequencies ωd to extract nd(ωd) and fit an initial permittivity model, yielding approximate Reststrahlen band frequencies (ωTO and ωLO). The analysis can then be refined by retaining only the data satisfying the low-loss approximation, with additional measurements near the estimated Reststrahlen band included if necessary. Our approach is broadly applicable to lossy, frequency-dispersive materials and offers a practical and accurate alternative to near-field techniques. By enabling far-field optical characterization of small-area flakes with simple FTIR spectroscopy, we hope to simplify the infrared characterization of emerging vdW materials with high precision at minimal experimental overhead.

Methods

Exfoliated samples and characterization

A glass substrate was used as the base layer, onto which a 150 nm gold film was deposited via thermal evaporation. The dielectric function of the gold-on-glass structure was characterized using a spectroscopic ellipsometer (IR-VASE Mark II, J.A. Woollam) over the spectral range of (333–8000) cm−1. The complex refractive index of the gold layer was extracted using the CompleteEASE software (J.A. Woollam).

Hexagonal boron nitride (hBN) and α-MoO3 flakes were mechanically exfoliated and transferred onto the gold-coated glass substrates (150nm gold thickness) using a polydimethylsiloxane (PDMS)-based exfoliation and transfer method (X0 retention, DGL type, Gelpak) at 90 °C. The thickness of the exfoliated flakes was measured using an atomic force microscope (Park NX20) operated in non-contact mode with an AC160TS cantilever (Olympus).

The reflectance spectra were obtained with a spectral resolution of 2 cm−1 using Bruker Hyperion 2000 microscope coupled to a Bruker Tensor FTIR spectrometer, equipped with a Mercury-Cadmium-Telluride detector which has a spectral range of (600–8000) cm−1 (1.5 –16.6 μm). The light was focused on the sample and reflected light was collected by a reflective objective (× 36, angular spread: 15° − 30°, numerical aperture: 0.5). For the reflectance from α-MoO3, two linear polarizers (ZnSe), parallel to each other were placed in the FTIR microscope, one for the incident light and other for the reflected light.

The internal aperture of the microscope was adjusted to fit the part of the crystal where the reflectance was to be measured. All the reflectance spectra from the flakes were normalized to a reference gold mirror.

The acquired spectra were filtered using cubic smoothing spline interpolation. Then, the frequencies of the local reflectance minima (ωd) were identified manually. The data are tabulated in the Supplementary Information Section 2. The fitting to extract the Lorentz model parameters was done by the least square method, with the model equation being \(n={\rm{Re}}(\sqrt{\epsilon })\) and/or \(k={\rm{Im}}(\sqrt{\epsilon })\), with ϵ being Eq. (1) or Eq. (6) as applicable.

Estimating errors

For each flake, the error on the measured thickness (δd) and on ωd (δωd) were evaluated. The contribution of the two on the estimation of n(ωd) can then be calculated from Eq. (4) neglecting the second term as δnd = δd/(4d2ωd) and \(\delta {n}_{\omega }=\delta {\omega }_{{\rm{d}}}/(4d{\omega }_{{\rm{d}}}^{2})\) respectively. The precision on the evaluation of n(ωd) was then calculated as \(\sqrt{{\delta {n}_{{\rm{d}}}}^{2}+{\delta {n}_{\omega }}^{2}}\).

For the errors on k(ωd,RB), Eq. (5) was minimized considering the errors δd and δωd,RB and the error on evaluating k(ωd,RB) for each contribution was determined as δkd and δkω respectively. The precision on the evaluation of k(ωd,RB) was then calculated as \(\sqrt{{\delta {k}_{{\rm{d}}}}^{2}+{\delta {k}_{\omega }}^{2}}\).

To evaluate the dielectric parameters, we performed a linear least square fitting. The 95% confidence intervals (CI) for each of the parameters were calculated from the inverse R factor (of the QR decomposition of the Jacobian), the degrees of freedom for error, and the RMSE. The Standard Error is then σE = CI/t, where t is the inverse cumulative distribution function and is ≈ 1.96 for 20 data points in the case of hBN, ≈ 2.03 for 24 data points for α-MoO3 along x and ≈ 1.8 for 14 data points for α-MoO3 along y.