Kronig Analysis Research Articles

Optical properties of popular dielectric substrate materials in a wide spectral range from far-infrared to ultraviolet

We investigated the optical properties of 13 different dielectric materials (slide glass, quartz, Al2O3 (c-cut), DyScO3 (110), KTaO3 (001), LaAlO3 (001), (LaAlO3)0.3-(Sr2AlTaO6)0.7 (001) (LSAT), MgF2 (100), MgO (100), SiC, SrTiO3 (001), TbScO3 (110), and TiO2). The single-bounce reflectance spectra of the bulk samples were measured using Fourier transform infrared (FTIR) and monochromatic spectrometers across a wide spectral range, from far infrared to ultraviolet (80–50,000 cm−1). Using the Kramers–Kronig analysis, we obtained the optical conductivity and dielectric function of the dielectric materials from their measured reflectance spectra. Moreover, we measured the transmittance spectra of the materials to obtain their bandgaps. We fitted the measured reflectance spectra using the Lorentz model to obtain phononic structures. Each dielectric material exhibits unique phononic structures and optical bandgaps, associated with the composition and crystal structure of the material. The observed optical properties of these dielectric materials provide valuable information for the optical analysis of thin films grown on them.

Giant ab‐Plane Birefringence in Quasi‐1D Fibrous Red Phosphorus

AbstractQuasi‐one‐dimensional (quasi‐1D) van der Waals crystal fibrous red phosphorus (RP) exhibits pronounced in‐plane optical anisotropy, positioning it as a potential candidate for polarization‐related micro‐nano devices. Unfortunately, a comprehensive investigation into the complex refractive index of fibrous RP and the structure–activity relationship connecting the distinctive quasi‐1D structure with optical anisotropy is currently deficient. Herein, we have collectively determined the complex refractive index of the fibrous RP flakes within the ab‐plane through Kramers–Kronig (KK) analysis and theoretical calculation. Notably, the maximum birefringence of fibrous RP reaches 0.642@475 nm with an absolute extinction coefficient of only 0.08, superior to the reported traditional optical crystals and the emerging low‐dimensional materials as well. The remarkable birefringence can be attributed to the synergistic influence of the large electronic dipole polarizability, anisotropic electron density distribution and the distortion of stereochemically active lone pair (SCALP). This work demonstrates the potential of fibrous RP for polarization‐sensitive devices, illuminating possibilities to exploit novel giant birefringent crystals based on the structure–activity relationship.

Estimation of dielectric response functions for NIR-SWIR absorbing dyes by inverse analysis of diffuse reflectance

This study describes the estimation of dielectric response functions for NIR-SWIR absorbing dyes by inverse analysis of diffuse reflectance. The inverse analysis procedure is based on the Kramers–Kronig relations, the Kubelka–Munk model of diffuse reflectance, and the background subtraction of substrate spectra. The calculated response functions provide estimates of the dielectric response characteristics for NIR-SWIR absorbing dyes, as well as their sensitivity to material-background environments. This study demonstrates that Kramers–Kronig analysis can be adopted for calculating estimates of dielectric functions, given any reasonable estimate of absorbance. Specifically, that Kramers–Kronig analysis provides filtering of dielectric response structure with respect to spectrum artifacts due to inverse-analysis and spectroscopic-measurement procedures. This can significantly reduce the experimentation necessary to achieve desired dielectric response characteristics for NIR-SWIR dye and substrate combinations. Furthermore, these calculated response functions can support the construction of approximate effective medium models capable of estimating reflectance features for dyed fabrics.

Effect of B2O3 substitution for P2O5 on the refractive index and optical absorption in pyro SnO−P2O5−B2O3 glasses

We investigated the impact of substituting B2O3 for P2O5 on a refractive index and optical absorption of pyro SnO−P2O5−B2O3 glasses. To date, we have reported that thermal properties and glass structure have a characteristic composition dependence when B2O3 replaces P2O5. In this paper, we report that optical properties such as optical absorption and refractive index also have a characteristic composition dependence. We observed a consistent increase in the refractive index with increasing B2O3 substitution, categorized into three regions, which is consistent with the trend in density. In addition, we observed that the optical absorption edge of the glass is shifted to a longer wavelength with increasing B2O3 concentration. To gain further insight into the optical properties of the glasses, we utilized Kramers–Kronig analysis to experimentally correlate the observed changes in refractive index and optical absorption. These changes in density and refractive index were predicted by elucidating the short-range structures of the glass, which were quantitatively provided by determining the proportions of phosphate and borate units. This information was then used to calculate the ion packing ratio.

Examining the anisotropic behavior of the excitons in anatase TiO2 by angle-resolved electron energy-loss spectroscopy

The behavior of excitons excited in anatase crystalline materials affects the photocatalytic performance of the materials. Thus, investigating the properties of such excitons is essential for understanding the origin of their influence on photocatalytic performance. Here, angle-resolved electron energy-loss spectroscopy was applied to evaluate the size of the spatial spread of excitons in anatase TiO2. Three kinds of excitons (I–III), which were reported by polarized light absorption experiments, were experimentally identified in the electron energy-loss spectroscopy (EELS) spectra. Exciton size was evaluated from absorption intensity, which was derived by the Kramers–Kronig analysis of the EELS spectra, depending on the momentum transfer (q). The sizes of excitons I, II, and III were evaluated to be 8, 5, and 6 nm, respectively. The larger size of exciton I than that of III was with the same tendency as in the theoretical results. Exciton II, which was evaluated as a delocalized one, was evaluated to have a finite size in this experiment. The largest size of exciton I, approximately 8 nm, is the same order of the exciton diffusion lengths of the material. Therefore, exciton I should significantly influence the photocatalytic activity of anatase.

Complex dielectric function and opto-electronic characterization using VEELS for the lead-free BCZT electro-ceramic perovskite

The current work presents the complex dielectric function and the opto-electronic properties of lead-free Ba0.8Ca0.2Ti0.9Zr0.1O3 (BCZT) electro-ceramic, derived from valence electron energy loss spectroscopy, in transmission electron microscopy (VEELS–TEM).A single tetragonal perovskite phase, with P4mm space group, was determined by Rietveld refinement of the x-ray diffraction pattern. The VEELS–TEM experiment scanned the energy interval from 0–50 eV. The spectroscopic analysis started with the chemical identification of the atoms that conforms the BCZT solid-solution. Bulk and surface plasmons were located at 27.2 eV and 12.9 eV, respectively in the energy loss function. Complex dielectric function was obtained using Kramers–Kronig analysis from the Gatan Microscopy Suite software. Dielectric constant was calculated from the real part of the complex dielectric function, while the inter-band transitions were identified in the joint density of states function. The refraction index n and the extinction coefficient k, as a function of energy, were obtained from the complex dielectric function. The bandgap energy was determined using a polynomial fit in the optical absorption coefficient plot with an Eg = 3.2 eV.

Comment on the paper “Improving Poor Man’s Kramers-Kronig analysis and Kramers-Kronig constrained variational analysis”

The title paper [Spectrochim. Acta A213 (2019): 391–396] reports an improvement of the ”Poor Man’s Kramers–Kronig analysis” and of the ”Kramers–Kronig constrained variational analysis” thanks to an ad hoc modification of some analytical formulas existing in the literature. This ad hoc modification is not based on mathematical grounds. In this comment we show that no ad hoc modification is required but a correction of the analytical formula used by the authors of the title paper [Spectrochim. Acta A213 (2019): 391–396].

Anisotropic Complex Refractive Indices of Atomically Thin Materials: Determination of the Optical Constants of Few-Layer Black Phosphorus.

In this work, studies of the optical constants of monolayer transition metal dichalcogenides and few-layer black phosphorus are briefly reviewed, with particular emphasis on the complex dielectric function and refractive index. Specifically, an estimate of the complex index of refraction of phosphorene and few-layer black phosphorus is given. The complex index of refraction of this material was extracted from differential reflectance data reported in the literature by employing a constrained Kramers–Kronig analysis combined with the transfer matrix method. The reflectance contrast of 1–3 layers of black phosphorus on a silicon dioxide/silicon substrate was then calculated using the extracted complex indices of refraction.

Identification of Plastic Type and Surface Roughness of Film-Type Plastics in Water Using Kramers–Kronig Analysis

The knowledge of the plastic type, thickness, and the nature of the surface is important towards the monitoring of microplastic pollution in water bodies, especially when vis-NIR spectroscopy is utilized. Factors such as complex environment and surface roughness induced-light scattering of the probing light limit the optical detection of these parameters in in-situ measurements, however. In this paper, a novel application of Kramers–Kronig analysis was exploited to identify both smooth and rough film-type macroplastics with unknown thickness. This method is particularly useful in the in-situ identification of unknown film-like macroplastics; although the sample is large, the ratio function is detected from an area that corresponds to the size of a MP. Therefore, it can be applied for the case of large size MPs. The validity of the method was demonstrated using transmittance data for smooth and roughened plastics given in Kanyathare et al., 2020.

Optical dielectric function of two-dimensional WS2 on epitaxial graphene

Heterostacks composed of two-dimensional WS2 and graphene exhibit interesting opto-electronic properties, including ultrafast charge transfer and fast, efficient photodetection. The optical properties of WS2 are heavily influenced by the presence of graphene in the heterostack, yet, so far, their characterization in the whole visible range is missing. In this work we report the complex dielectric function of two-dimensional WS2 flakes on epitaxial graphene on silicon carbide, obtained by means of spectroscopic ellipsometry (SE). The so-called A, B and C excitonic features are precisely identified, and significant differences with respect to literature data of WS2 on fused silica are highlighted. Since this investigation is based on SE, the complex dielectric function of WS2 is retrieved without performing Kramers Kronig analysis, which is instead necessary when employing reflectance or transmittance spectroscopy. Furthermore, the approach described in this work can be used in principle to characterize any two-dimensional flakes, both in terms of complex dielectric function and percentage of surface coverage.

Polarization, thin-film optics, ellipsometry, and polarimetry: Retrospective

As plenary speaker at the 2019 ICSE-8 Conference in Barcelona, Spain, the author was invited to review several of his published contributions to polarization optics of interfaces and thin films, ellipsometry, and polarimetry over the past 50 years. This paper is an abridged account of his oral presentation. The author reconsiders the reflection of p- and s-polarized and unpolarized light at dielectric-dielectric and dielectric-conductor interfaces and obtains interesting results. For example, the author shows how the retrieval of the reflection phase shift of s-polarized light from p and s intensity reflectance data enables the determination of the complex dielectric function of an absorbing medium on a wavelength-by-wavelength basis without ellipsometry or Kramers–Kronig analysis. Novel achromatic and angle-insensitive single- and four-reflection wave retarders are presented that are particularly suited for the near IR. The author also shows how binary-thickness thin-film optical coatings on a high-index substrate can generate binary spatial and temporal polarization modulation by reflection. Finally, O'Bryan's 1936 vacuum ellipsometer is revisited and the author shows how it can be adapted for the measurement of the optical properties of absorbing liquids using attenuated total reflection spectroscopic ellipsometry.

Design and application of a relativistic Kramers–Kronig analysis algorithm

Low-loss electron energy loss spectroscopy (EELS) in the scanning transmission electron microscope probes the valence electron density and relevant optoelectronic properties such as band gap energies and other band structure transitions. The measured spectra can be formulated in a dielectric theory framework, comparable to optical spectroscopies and ab-initio simulations. Moreover, Kramers–Kronig analysis (KKA), an inverse algorithm based on the same name relations, can be employed for the retrieval of the complex dielectric function. However, spurious contributions traditionally not considered in this framework typically impact low-loss EELS modifying the spectral shapes and precluding the correct measurement and retrieval of the dielectric information. A relativistic KKA algorithm is able to account for the bulk and surface radiative-loss contributions to low-loss EELS, revealing the correct dielectric properties. Using a synthetic low-loss EELS model, we propose some modifications on the naive implementation of this algorithm that broadens its range of application. The robustness of the algorithm is improved by regularization, applying previous knowledge about the shape and smoothness of the correction term. Additionally, our efficient numerical integration methodology allows processing hyperspectral datasets in a reasonable amount of time. Harnessing these abilities, we show how simultaneous relativistic KKA processing of several spectra can share information to produce an improved result.

Near-Infrared Reflectivity of Superconducting FeSe Thin Films

We present an investigation on the intrinsic optical properties of superconducting FeSe thin films deposited on CaF2 substrates. It is found that the samples respond strongly to the near-infrared (NIR) irradiation in different temperature regimes. Thus, we are able to measure the optical reflectivity of the in-plane (ab-plane) thin films from room temperature to 5 K. A critical behavior near the superconducting transition temperature ( T c∼10 K) can be observed optically in the temperature-dependent reflectivity spectrum. We carry out the Kramers–Kronig analysis of the experimental data and discuss the basic optical properties of superconducting FeSe thin films in different structural phases above and below the transition temperature T s∼100 K. By fitting experimental data with the Drude–Smith and Drude–Lorentz formulas, we obtain the key sample parameters such as the electronic relaxation time and dc conductivity. The interesting and important findings obtained from this study not only shed new light on the pairing mechanism in FeSe but also demonstrate that the NIR reflection experiment is a powerful and convenient optical technique for contactless characterization and investigation of superconducting thin film materials.

Model-Based Analyses of Confined Polymer Electrolyte Nanothin Films Experimentally Probed by Polarized ATR–FTIR Spectroscopy

Orientational ordering within nanoscale (70–8 nm) thickness fluorinated ionomer films on Si substrates was investigated through the use of attenuated total reflection Fourier transform infrared (ATR–FTIR) spectroscopy in conjunction with electromagnetic field calculations. A spectral model was developed for Nafion thin films across the 1400–950 cm–1 region from frequency-dependent, isotropic optical constants derived from Kramers–Kronig analysis of ionomer transmission infrared spectra. The model considered infrared light propagation within the parallel boundary regions between the Ge ATR crystal, the ionomer film, and the Si substrate supporting the film. The calculations reproduced overall polymer thickness-dependent changes in peak frequencies and band shapes observed in experimental spectra recorded with p- and s-polarized light. General trends were traceable to effects of anomalous dispersion and electric field enhancement within the nanoscale gap separating the Ge and Si phases. However, optical eff...

Spectral refractive index assessment of turbid samples by combining spatial frequency near-infrared spectroscopy with Kramers–Kronig analysis

A practical algorithm for estimating the wavelength-dependent refractive index (RI) of a turbid sample in the spatial frequency domain with the aid of Kramers-Kronig (KK) relations is presented. In it, phase-shifted sinusoidal patterns (structured illumination) are serially projected at a high spatial frequency onto the sample surface (mouse scalp) at different near-infrared wavelengths while a camera mounted normally to the sample surface captures the reflected diffuse light. In the offline analysis pipeline, recorded images at each wavelength are converted to spatial absorption maps by logarithmic function, and once the absorption coefficient information is obtained, the imaginary part (k) of the complex RI (CRI), based on Maxell's equations, can be calculated. Using the data represented by k, the real part of the CRI (n) is then resolved by KK analysis. The wavelength dependence of n ( λ ) is then fitted separately using four standard dispersion models: Cornu, Cauchy, Conrady, and Sellmeier. In addition, three-dimensional surface-profile distribution of n is provided based on phase profilometry principles and a phase-unwrapping-based phase-derivative-variance algorithm. Experimental results demonstrate the capability of the proposed idea for sample's determination of a biological sample's RI value.

Synthesis, sintering, transport and thermal properties of Na2Fe2Ti6O16 freudenbergite

Na2Fe2Ti6O16 freudenbergite has been synthesized using a simple solid-state route and conventionally sintered to reach 94% density ceramics. High temperature electrical and thermal transport properties have been investigated for the first time. In the temperature range of 468 K–1000 K, the electrical conductivity of Na2Fe2Ti6O16 can be well fitted with small polaron hopping model. For a further verification of the model, we measured the room-temperature optical reflectivity and derived the optical conductivity (σ(ω)) with Kramers–Kronig analysis. A mid-infrared peak in σ(ω) confirms the presence of small polarons. From 468 K to 1000 K, Na2Fe2Ti6O16 is an n-type oxide and exhibits a temperature-independent Seebeck coefficient (∼−590 μV K−1), in agreement with the small polaron hopping mechanism. The thermal conductivity decreases with increasing temperature from 16.5 W m−1.K−1 (373 K) to 3.4 W m−1.K−1 (1000 K). Na2Fe2Ti6O16 shows a maximum figure of merit (ZT) ∼ 3 × 10−4 at 1000 K.

Crystal structure characteristics, dielecric properties and vibrational spectra of Nb-rich non-stoichiometric Ba[(Zn1/3Nb2/3)1−xNbx]O3 ceramics

Nb-Rich non-stoichiometric Ba(Zn1/3Nb2/3)1−xNbxO3 (BZNN) (x = 0.01, 0.02, 0.03, 0.04) ceramics were synthesized using solid-state reaction method. Crystal structures and morphologies were analyzed by X-ray diffraction (XRD) and scanning electron microscopy methods. Vibrational modes were obtained by both Raman spectroscopy and Fourier transform far-infrared reflection (FTIR) spectroscopy. The relationship among crystal structures, dielectric properties, and phonon modes were analyzed in detail. XRD results showed that the main phase of BZNN is Ba(Zn1/3Nb2/3)O3, and the cell volume and lattice parameter reach a minimum value where x = 0.02. The Raman results displayed that two new modes appeared at about 200 cm− 1 where x > 0.01, which showed that structures of BZNN transform from 1:1 to 1:2 ordering with increasing Nb contents. FTIR results demonstrated that great changes took place for the shape of the bands below 180 cm−1. The FTIR spectra were subjected to the Kramers–Kronig analysis.

Optical transmittance and reflectance studies and evidence of weak electron–phonon interaction in Type-I Ge clathrate Ba8Ga16Ge30

The optical properties of ternary type-I Ge clathrate Ba8Ga16Ge30 are investigated by optical reflectance and transmittance measurements. The refractive index and extinction coefficient are calculated from the reflectance spectrum via the modified Kramers–Kronig analysis method between 0.5 and 3.2 eV. The photon energy dependence of the optical absorption coefficient reveals that Ba8Ga16Ge30 is an indirect band gap semiconductor with a gap energy Eg of 0.69 eV at 5.7 K and 0.66 eV at 285 K. The temperature dependence of Eg can be satisfactorily described by an equation based on the Bose–Einstein statistics model. When compared with those of common elemental, III–V, and II–VI semiconductors, the small temperature coefficient dEg/dT of the Ba8Ga16Ge30 can be considered to represent the weak electron–phonon interaction in the Ba8Ga16Ge30 clathrate.

Semiconductor-to-metal transition in the bulk of WSe2 upon potassium intercalation

We present electron energy-loss spectroscopic measurements of potassium (K) intercalated tungsten diselenide (WSe2). After exposure of pristine WSe2 films to potassium, we observe a charge carrier plasmon excitation at about 0.97 eV, which indicates a semiconductor-to-metal transition. Our data reveal the formation of one particular doped K-WSe2 phase. A Kramers–Kronig analysis allows the determination of the dielectric function and the estimation of the composition of K0.6WSe2. Momentum dependent measurements reveal a substantial plasmon dispersion to higher energies.

High-resolution monochromated electron energy-loss spectroscopy of organic photovoltaic materials

Advances in electron monochromator technology are providing opportunities for high energy resolution (10 – 200meV) electron energy-loss spectroscopy (EELS) to be performed in the scanning transmission electron microscope (STEM). The energy-loss near-edge structure in core-loss spectroscopy is often limited by core-hole lifetimes rather than the energy spread of the incident illumination. However, in the valence-loss region, the reduced width of the zero loss peak makes it possible to resolve clearly and unambiguously spectral features at very low energy-losses (<3eV). In this contribution, high-resolution EELS was used to investigate four materials commonly used in organic photovoltaics (OPVs): poly(3-hexlythiophene) (P3HT), [6,6] phenyl-C61 butyric acid methyl ester (PCBM), copper phthalocyanine (CuPc), and fullerene (C60). Data was collected on two different monochromated instruments – a Nion UltraSTEM 100 MC ‘HERMES’ and a FEI Titan3 60–300 Image-Corrected S/TEM – using energy resolutions (as defined by the zero loss peak full-width at half-maximum) of 35meV and 175meV, respectively. The data was acquired to allow deconvolution of plural scattering, and Kramers–Kronig analysis was utilized to extract the complex dielectric functions. The real and imaginary parts of the complex dielectric functions obtained from the two instruments were compared to evaluate if the enhanced resolution in the Nion provides new opto-electronic information for these organic materials. The differences between the spectra are discussed, and the implications for STEM-EELS studies of advanced materials are considered.