1. Introduction

In the fields of solid-state physics, the complex dielectric function is a fundamental cornerstone for exploring light-matter interactions in emerging materials [1], as well as for the rationally designing various photonic and optoelectronic devices, such as photonic chips [2,3], biosensing [4,5], photochemistry [6,7], etc.

Recent decades have evidenced great advancements in commercial spectroscopic ellipsometry (SE) techniques, leading to a significant expansion of the artificial material database. High-quality materials with low optical loss or poor ambient tolerance, including two-dimensional materials [8,9], perovskites [10,11], and other related organic and inorganic materials, are noted for their highly desired physical properties. However, most of these materials often necessitate in-situ vacuum-based SE measurement integrated with the fabrication processes, significantly raising infrastructure costs and operational complexity, thus limiting practical applications. An alternative strategy to manage reactive materials outside the vacuum involves applying stable encapsulating films to the targets. However, this introduces additional fitting parameters, complicating the SE optical models and making the fitting of the imaginary part of dielectric functions rather challenging, particularly for materials with low absorption coefficients or losses. Consequently, there is an urgent demand for ambient available alternative methods to traditional SE, especially for research involving low-loss and highly sensitive materials.

Specifically, as for plasmonic materials, the integration of various optical fitting models of SE with well-developed metallic samples, from polycrystalline and single crystalline films [12] to high-quality alloys [13,14], shows that conventional noble metal-based plasmonics are almost approaching the inherent optical loss limit predicted by theories [15,16]. Recently, Yang Wang and coworkers utilized the thermo-assisted spin coating method to overcome the fabrication bottleneck, creating high-quality sodium films encapsulated by quartz substrates [17]. This experimental breakthrough led to a record-low-loss plasmonic material, surpassing noble metal silver in the near-infrared and achieving an optical damping rate of 10 meV through standard multi-parameter SE modeling, which serves as crucial evidence and new referencing systems for the ongoing explorations such as low threshold nanolasors [18], integrated optics [19] and single particle strong coupling [20], etc. However, applying standard SE measurements to bulk dispersive encapsulated systems is still challenging, which will introduce significant measuring errors or even unphysical fitting solutions [17] (see Supplement 1 for more fitting details). Therefore, there is an urgent need for a precise and convenient SE technique that does not rely on fitting models, which is crucial for advancing the study of encapsulated material systems and thus of significant interest across various branches of solid-state physics.

Here, we demonstrate an analytical ellipsometry method (AEM) for the measurement of dielectric functions of encapsulated materials, which effectively eliminates fitting errors and avoids nonphysical results. Taking the highly pursued low-loss plasmonic materials, such as sodium films, as an example, which are prepared by the thermo-assisted spin-coating method proposed in our previous work, with a thickness of hundreds of microns approximately. After encapsulation with fused silica and side packaging with ultraviolet curing glue, the sodium films can be transferred freely out of glove boxes for in-air SE measurements by RC2 ellipsometer (J.A. Woollam Corporation). We measured the dielectric functions of the sodium films across a broad spectral range from 0.50 to 4.97 eV (250 nm to 2.5 µm). In addition, by fitting the Drude model to the dielectric functions of more than 200 sodium samples, the statistical Drude parameters of sodium (ɛ_∞, ħω_p, ħγ) can be obtained, which provides direct experimental evidence that sodium is an ideal candidate with a low damping rate beyond noble metals. Finally, the measurement errors of the proposed method can be analytically evaluated to be less than 1.7% for the real part and 4.4% for the imaginary part of the dielectric function. In conclusion, our method provides a simple and precise alternative to traditional methods for low-loss and highly sensitive materials, beneficial for both precise optical designs in photonics and fundamental studies in solid-state physics.

2. Analytical ellipsometry method (AEM)

To establish an analytic formula of the dielectric functions of sodium as a function of the SE raw data ρ_m(ω), the dielectric functions of encapsulated metals ${\varepsilon _m}(\omega )$ can be expressed as the following formula by a simple formulation from Snell's Law and Fresnel Formula:

More detailed derivations are available in Supplement 1, Sec. 2. As depicted in Fig. 1, α is the incident angle, β is the refractive angle on silica obtained by $sin\beta = \frac{{sin\alpha }}{{\sqrt {{\varepsilon _{sub}}(\omega )} }}$, and the parameter T can be derived as the following formula:

 

Fig. 1. The schematic diagram of the in-air ellipsometry measurement on the encapsulated surface. The light beams depicted in dark red are the measurement light beams needed to collect.

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${\varepsilon _{sub}}(\omega )$, the dielectric function of the isotropic quartz, can be obtained by the following formula:

To minimize the depolarization effect due to the fused silica substrate, there physical phenomena that generate partially polarized light upon light reflection, including surface light scattering, multiple reflections, and incident angle variation, are taken into account in our experimental design and analysis. Eventually, a 1.5-mm-thick fused silica with a surfaces roughness of ∼100 pm is used as the substrate (more details of thickness choice are available in Supplement 1, Sec. 3). Besides, to evaluate the validity of the proposed AEM, a pre-evaluation procedure of the proposed AEM is conducted on conventional plasmonic metals, such as gold and silver, before the measurements on sodium (more experiment details are available in Supplement 1, Sec. 4). Figs. 2(a) and (b) illustrate the measured ε₁ and ε₂ of the gold film and silver film respectively, obtained by analytically measuring without the fused silica (solid line) and the AEM with the fused silica (dotted line). It is depicted that, the dielectric functions of noble metals measured with fused silica coincide well with the corresponding standard data, evidencing the validity of the AEM. Furthermore, measurements at multiple angles of incidence are also conducted on sodium samples (see Supplement 5), which also show the validity of the AEM.

 

Fig. 2. Evaluation the validity of the AEM. The dielectric function of (a) Au and (b) Ag measured with and without covering fused silica respectively.

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3. Results and discussion

The dielectric function of materials reflects the intrinsic light-matter interactions. Specifically, it includes electromagnetic field confinement and intrinsic optical loss for plasmonic metals. Fig. 3(a) shows ${\varepsilon _1}$ of the sodium film across a spectral range of 0.50–4.97 eV in red circles, while Figs. 3(b) and (c) show ${\varepsilon _2}$ in the energy range of 1.77–4.97 eV and 0.50–1.77 eV in yellow and blue circles, respectively. For comparison, data from Wang (left-triangle) [17], Hodgson (rhombus) [21], Smith (up-triangle) [22], Palmer (circles) [23], Monin (down-triangle) [24], and Inagaki (squares) [25] are plotted. The dielectric functions are sensitive to sample quality and measurement methods, which may induce differences among data from our work and previous literature.

 

Fig. 3. (a) The ${\varepsilon _1}$ and (b)-(c) ${\varepsilon _2}$ of the dielectric functions of sodium from 0.50 eV to 4.97 eV (250-2500 nm). The measured data of the dielectric functions of sodium reported in the literature are shown for comparison. The statistical data of Drude model parameters (d) ${\varepsilon _\infty }$, (e) $\hbar {\omega _p}$, and (f) $\hbar \gamma $ are obtained from over 200 sample points.

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The ${\varepsilon _1}$ is related to the capability of field confinement nearby the sodium/quartz interface, which is determined by the density of free carriers theoretically. Since the free electron density of sodium is generally up to 3 × 10²⁸ /m³ [26], the ${\varepsilon _1}$ of sodium fabricated and measured by different groups are close to each other. However, the imaginary parts measured by different research groups show distinct diversity. The ${\varepsilon _2}$ is associated with intrinsic optical loss, including interband and intraband absorption. In Fig. 3(b), the ${\varepsilon _2}$ shows an inconspicuous Lorentz-like peak around 2.8 eV basically consistent with the literature, corresponding to the weak interband transition in sodium. Unlike the interband absorption, the intraband absorption of free carriers within the conduction band mainly determines the ${\varepsilon _2}$ in the infrared spectral range, which is of great importance in the performance of surface plasmon polariton devices. As shown in Fig. 3(c), the ${\varepsilon _2}$ in the infrared spectral range is smaller than the data of other works, indicating the superiority of our fabrication method and high sample quality.

Based on the near free electron gas behaviors of sodium in the near-infrared range, the measured dielectric functions can be described by Drude model:

The parameter ${\varepsilon _\infty }$ accounts for the net contribution from the positive ion cores. The volume plasma frequency ${\omega _p}$ is related to the effective mass of the electron m* and the electron density N through ${\omega _p} = \sqrt {N{e^2}/{\varepsilon _0}{m^\mathrm{\ast }}} $. And the electron relaxation rate $\gamma $ describes the effective electron scattering rate regarded as a figure of merit for optical loss, with a corresponding relaxation time τ = 1/$\gamma $. The dielectric function ${\varepsilon _m}(\omega )$ in Eq. (4) can be decomposed into real ${\varepsilon _1}(\omega )$ and imaginary part ${\varepsilon _2}(\omega )$, which can be simplified as follows under the condition of $\mathrm{\omega } \gg \mathrm{\gamma }$ but below the onset of the interband response,

Equations (5) and (6) allow for the direct calculations of ${\varepsilon _\infty },\textrm{}{\omega _p}$, and $\mathrm{\gamma }$ from ${\varepsilon _1}(\omega )$ and ${\varepsilon _2}(\omega )$. Using this approach in the region of 800 nm to 2.5 µm, it could be obtained that the fitting parameters of sodium are ${\varepsilon _\infty }$ = 1.18 ± 0.23, $\hbar {\omega _p}$ = 5.79 ± 0.08 eV, and $\hbar \mathrm{\gamma }$ = 13.5 ± 7.5 meV. Fig. 3(d-f) demonstrate the statistical data of three parameters from over 200 sample points. Each parameter has a distribution close to the Gauss profile. The peak width is originating from sample quality variations and random errors.

The intraband absorption of free carriers provides information about the electron scattering rate γ, contributed by electron-phonon scattering, electron-electron scattering, surface roughness scattering, and other possible scattering mechanisms. The average $\hbar \mathrm{\gamma }$ extracted from the Drude model is 13.5 meV, and the lowest $\hbar \mathrm{\gamma }$ is merely 6 meV. As a result, the maximum electron relaxation time τ of these sodium samples is ∼ 109 fs. Furthermore, the maximum free path of the electron of sodium is ∼ 116 nm, which can be calculated by $l = {v_f}\tau $, where ${v_f} = $ 1.07 × 10⁸cm/s is the Fermi velocity of sodium. For comparison, the electron scattering rate of sodium is lower than that of gold (35 ∼ 80 meV) [27–29] and silver (15 ∼ 50 meV) [28,30] in the optical region, which means that sodium metal is a more promising plasmonic candidate than noble metals. In addition, since the smaller ${m^\mathrm{\ast }}$ corresponds to more smooth metal surfaces with the same electron density N [29], the fact that the plasma frequency $\hbar {\omega _p}$ of our sodium sample is slightly higher than that of literature (5.2 ∼ 5.8 eV) [21–25] demonstrates the high optical quality of our sodium samples among previous works as well. Another parameter of the Drude model is the infinite dielectric constant ${\varepsilon _\infty }$, which is related to the conformity between the metal and free electron structure. The ${\varepsilon _\infty }$ of our sodium samples is close to 1, which agrees well with the previous literature (1.0 ∼ 1.3) [21–25] and means that sodium is very close to the free electron structure.

To evaluate the effectiveness of the measured data, we finally perform the error analysis of the proposed AEM measurements. One of errors is due to a surface layer between sodium films and substrate as shown in Fig. 4(a), since sodium is chemically reactive and easy to react with oxygen. In our sample preparation method, with the experiment condition of O₂ ∼ 0.1 ppm, the thickness of the oxide layer is only ∼ 0.004 nm, which would be applied to estimate the errors in the dielectric function of sodium. Besides, referring to [31] , the refractive index of oxidation layer is assumed to be from 1.4 to 1.6. As Fig. 4(b) and (c) show, thanks to the efficiency of the fabrication method in removing oxide layers, the error related to interface oxide layer in the real part of the dielectric function is within 0.004%, and that in the imaginary part is within 0.006% (more details are available in Supplement 1, Sec. 6).

 

Fig. 4. Errors of the AEM. (a) The schematic diagram of the error due to a surface layer. (b) The relative error of ε₁ and (c) the relative error of ε₂ of sodium samples covered with an 0.004-nm oxidation layer with a refractive index of 1.4, 1.5, and 1.6. (d) The schematic diagram of the error due to surface roughness-induced depolarization effect. (e) The relative error of ε₁ and (f) the relative error of ε₂ of sodium samples encapsulated by quartz substrates with a surface roughness of ∼100 pm. (g) The schematic diagram of the error due to depolarization effect resulting from substrate dispersion and focusing beam. The light beam depicted in dark red is a beam of a certain wavelength with an incident angle α, while the light beam indicated in light red is a beam of other wavelength with an incident angle α+θ. (h) The relative error of ε₁ and (i) the relative error of ε₂ of sodium samples caused by substrate dispersion and focusing beam.

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The second error here is related to surface roughness-induced depolarization effect as shown in Fig. 4(d), which is inevitable but can be minimized by decreasing the surface roughness of quartz substrates. To quantitatively evaluate this error, a simplified optical model with an equivalent rough surface layer is proposed (more details are available in Supplement 1, Sec. 7), where the dielectric function of this layer is calculated by effective medium theories. As shown in Fig. 4(e) and (f), the relative error related to surface roughness (∼100 pm) in the real part of the dielectric function is < 0.7%, and that in the imaginary part is < 1.0%, which suggests that the impact of surface roughness can be considered negligible in our work.

Another potential error is due to depolarization effect resulting from the substrate dispersion and focusing beam. For the encapsulated measuring system, to eliminate this error, a large-area (∼ 4 mm) and high-quality sample surface together with a well-defined ultrathick substrate (see Supplement 1, Fig. S12 in Sec. 8) is ideally needed to fulfill the non-fusing measuring requirement, which is too heavy for the spin coater and raises significant challenges in the sample fabrication process. As an alternative scheme, this depolarization effect can be minimized by carefully optimizing the thickness of the bulky substrate. In this case, the angle of incident light at certain wavelengths detected is actually α+θ due to the substrate dispersion as shown in Fig. 4(g), but the incident angle is considered to be α. Thus, the relative error could be calculated with the analytic expression in Eq. (7), reading as

Fig. 4(h) and (i) show, as the thickness of the fused silica substrate is 1.5 mm and the incident angle is 65°, the relative error of the ε₁ (ε₂) is < 1.0% (< 3.4%). Note that these relative errors would increase as the substrate thickness increases as predicted in Supplement 4. That is why the 1.5-mm-thick fused silica was chosen as the spin-coating substrate and encapsulation layer.

According to the error analysis, the total relative error of the ε₁ and ε₂ of sodium, is less than 1.7% and 4.4%, respectively. Furthermore, for film materials that can be grown on thick substrate and formed in a large area with high quality, the total relative error of the ε₁ (ε₂) by this method could be decreased to < 0.7% (< 1.0%). Therefore, compared to the traditional fitting method, the measured dielectric function of encapsulated materials, especially the imaginary part, is more reliable and precise.

4. Conclusion

Our work introduces a novel analytical ellipsometry method (AEM) for accurately measuring dielectric functions of chemically reactive materials under bulk encapsulation without the multi-parameter fitting errors and the issue of potentially unphysical results in traditional spectroscopic ellipsometry methodologies. The successful application of this method to sodium films, analyzing critical issues like surface overlayer effects and depolarization effects, demonstrates substantial improvements in precision. Specifically, our method reduced measurement errors to less than 1.7% for the real part and 4.4% for the imaginary part of the dielectric function, showcasing a significant advancement in the precision of optical measurements. Furthermore, by fitting the Drude model to the dielectric functions of more than 200 sodium samples, we observed a remarkably low damping rate in sodium, with Drude model parameters indicating an average electron relaxation rate (ℏγ) of 13.5 ± 7.5 meV and the lowest reaching merely 6 meV, pointing to sodium's exceptional potential as a low-loss plasmonic material beyond conventional noble metals. In conclusion, our results provide a generalized and convenient ellipsometry measurement strategy for highly sensitive materials under bulk encapsulation, particularly beneficial for precise optical designs in photonics and fundamental studies in solid-state physics.