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# Parameterization of the dielectric function of semiconductor nanocrystals

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... SE determines the complex dielectric function of ε = ε 1 + iε 2 , ε 1 and ε 2 being the real and imaginary parts, respectively, with a precision of ≈ 10 − 4 . Since ε 2 is proportional to the joint density of electron states, the method is very sensitive to the change of long-range order in the lattice, i.e. to the crystallinity of the material 33 . Due to the large amount of spectroscopic data, complex models can be built with parameters that describe the formation of structures in depth 26,34 . ...
... The baseline of the ellipsometric angles was taken from region I and it was used to determine the dielectric function of the c-Ge substrate. From the numerous possible parameterizations of the dielectric function 33 , the dispersion of the c-Ge wafer was described using the Johs-Herzinger generalized critical point model 51 . Only the oscillator parameters of Ge transitions at 2.1, 2.3, and 3.4 eV were tted in a c-Ge/GeO 2 model. ...
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Ion implantation has been a key technology in microelectronics and generally, for the controlled surface modification of materials for tribology, biocompatibility, corrosion resistance and many more. In this work in-situ spectroscopic ellipsometry was used for accurately tracking and on-line evaluating the accumulation of voids and damage in crystalline Ge during implantation of 200-keV Sb ⁺ ions at a total fluence of 10 ¹⁶ cm − 2 using an ion flux of 2.1 × 10 ¹² cm − 2 s − 1 . The phases of damage accumulation were identified using unique optical multi-layer models describing the layer structure and composition. The formation of initial partial disorder was followed by complete amorphization and void formation occurring at the fluence of 1 × 10 ¹⁵ cm − 2 , reaching a high volume fraction of voids and a layer thickness of ≈ 200 nm by the end of the process. This agrees with numerical simulations and results of complementary measurements including ion beam analysis and electron microscopy. The developed in-situ method for controlling the dynamics of structural damage accumulation is a versatile ion-implantation tool for avoiding adverse void formation and for controlled evolution of subsurface nanocavities or cellular surface texture alike.
... 2 spectroscopic ellipsometers previously accessible in research ellipsometers only. There are numerous dielectric function parameterizations which can be applied to various types of materials [4][5][6][7]. However, such abundance of available dispersion models would be the scholarly user's "a true paradise" but rather great perplexity and confusion for the "typical" ...
... Typical practice in the IC methodology is to select the "best" model from the set of candidate models by using not the "raw" AIC or BIC values but other associated measures such as IC differences (ΔAIC, ΔBIC), IC weights of evidence (ω(AIC), ω(BIC)), and evidence ratio The content is identical to the published paper but without the final typesetting by the publisher. 7 Evidence ratio (ER): ...
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In the field of optical metrology, the selection of the best model to fit experimental data is absolutely nontrivial problem. In practice, this is a very subjective and formidable task which highly depends on metrology expert opinion. In this paper, we propose a systematic approach to model selection in ellipsometric data analysis. We apply two well-established statistical methods for model selection, namely, the Akaike (AIC) and Bayesian (BIC) Information Criteria, to compare different dispersion models with various complexities and objectively determine the “best” one from a set of candidate models. The information criteria suggest the most optimal way to quantify the balance between goodness of fit and model complexity. In combination with screening-type parametric sensitivity analysis based on so-called “elementary effects” (the Morris method) this approach allows to compare and rate various models, identify key model parameters and significantly enhance process of ellipsometric measurements evaluation.
... The dependences of conductivity and the dielectric behavior on the frequency give us useful information about the conduction mechanism of the materials under investigation. The AC measurements can offer useful information from both the fundamental and technological points of view [14][15][16] . The most common results on semiconducting nanocrystallites are a significant increase in the energy band gap (E g ), which is an important parameter for semiconductors, and a drastic reduction in the dielectric constant ε when the crystallite size approaches a few nanometers [15,16] . ...
... The AC measurements can offer useful information from both the fundamental and technological points of view [14][15][16] . The most common results on semiconducting nanocrystallites are a significant increase in the energy band gap (E g ), which is an important parameter for semiconductors, and a drastic reduction in the dielectric constant ε when the crystallite size approaches a few nanometers [15,16] . Decreas-ing ε causes an increase in the Coulomb interaction energy between electrons or holes and ionized shallow impurities. ...
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In this work, PbS and PbTe nanomaterials with various morphologies were synthesized by a hydro-thermal method. The structural properties were investigated by using X-ray diffraction (XRD) and corresponding scanning electron microscopy together with their EDX analysis. Both the PbS and PbTe nanomaterials possess good polycrystalline structure. The crystallite size, determined from the XRD data, of PbS is 5 nm whereas the corresponding value of PbTe is 2.69 nm. SEM micrographs reveal that the prepared PbS nanomaterial has star-shaped structures, micro-flowers, some cubes, and semi-dendrites, whereas PbTe has semi-flower structures. Additionally , the dielectric properties have been studied in a broad frequency range from 0.1 Hz up to 1 MHz at temperatures from 298-423 K. The real and imaginary parts of the complex dielectric constant ε' and ε'' of PbTe are comparatively higher than those of PbS. Moreover, the dielectric data were analyzed on the basis of the electric modulus.
... This larger void fraction indicates lower optical and physical density. The intrinsic refractive index spectra of ITO EA and ITO AC (Fig. 10) were calculated by using a combination of the B-EMA [22,23] and the B-spline [24][25][26] parameterization of the dispersion. The layer thicknesses and volume fractions of the voids were set to the values calculated from the above B-EMA model, and the dispersion of the ITO component was parameterized by using the B-spline approach, which describes the dispersion with the aid of splines between nodes separated by 0.3 eV. ...
... For this reason, this type of flexible parameterization is well suited to describe novel and/or complex materials. [17][18][19] The B-spline parameterization represents the imaginary part of the dielectric function, e 2 , as a function of photon energy, E, by a linear sum of B-spline basis polynomials of degree 3 over a given number of knots, m: 20 ...
Article
ZnSiP2 is a wide band gap material that is lattice matched with Si, offering the potential for Si-based optoelectronic materials and devices, including multijunction photovoltaics. We present a carbon-free chemical vapor deposition process for the growth of both epitaxial and amorphous thin films of ZnSiP2–Si alloys with tunable Si content on Si substrates. Si alloy content is widely tunable across the full composition space in amorphous films. Optical absorption of these films reveals relatively little variation with Si content, despite the fact that ZnSiP2 has a much wider band gap of 2.1 eV. Post-growth crystallization of Si-rich films resulted in epitaxial alignment, as measured by X-ray diffraction and transmission electron microscopy. These films have an optical absorption onset near 1.1 eV, suggesting the possibility of band gap tuning with Si content in crystalline films. The optical absorption is comparably strong to pure ZnSiP2, suggesting a more direct transition than in pure Si.
... Moreover, the material optical properties might be unknown a priori and need to be determined during characterization. In that situation, when the optical constants are not really "constant", i.e., fixed and invariable, and/or not established beforehand, modeling or parameterization of the material n&k's becomes an absolute necessity [6][7][8] and for this purpose many analytical physics-based and Kramers-Kronig consistent expressions (models) have been developed. These models have been used to describe various types of materialsamorphous and crystalline semiconductors and dielectrics, metals, organic films, optical metamaterials, etc. ...
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We reviewed studies reporting the applications of the Tauc–Lorentz (TL) parameterization for the complex dielectric function in spectroscopic ellipsometry. Since this model became very popular for representation of the dielectric functions of amorphous semiconductors and dielectrics, there is a need for an improved multi-oscillator TL model with a constraint on the band-gap parameter Eg, which forces it to be common for all TL oscillators, and possibility to represent weak absorption features below the bandgap which previously very often were associated with the exponential Urbach tail. In this paper we propose an extended TL model by inclusion of additional unbounded Lorentz and/or Gaussian oscillators with transition energies located below the bandgap. We conclude that the proposed model is the most appropriate for the characterization of various materials with sub-band absorption features and gives meaningful value for the energy bandgap. A few examples to illustrate practical use of modified model have been provided.
... Moreover, the material optical properties might be unknown a priori and need to be determined during characterization. In that situation, when the optical constants are not really "constant", i.e., fixed and invariable, and/or not established beforehand, modeling or parameterization of the material n&k's becomes an absolute necessity [11][12][13] and for this purpose many analytical physics-based and Kramers-Kronig consistent expressions (models) have been developed which describe various types of materials-amorphous and crystalline semiconductors and dielectrics, metals, organic films, optical metamaterials, etc. ...
Article
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Analytic representations of the complex dielectric function, which describe various types of materials, are needed for the analysis of optical measurements, in particularly, ellipsometric data. Here, we examine an improved multi-oscillator Tauc-Lorentz (TL) model with a constraint on the band-gap parameter Eg, which forces it to be common for all TL oscillators, and possibility to represent reasonably weak absorption features below the bandgap by inclusion of additional unbounded Lorentz and/or Gaussian oscillators with transition energies located below Eg. We conclude that the proposed model is the most appropriate for the characterization of various materials with sub-band absorption features and provides meaningful value for the energy bandgap. A few examples to illustrate the use of modified model have been provided.
... 196-203]). For the purpose of modelling or parameterization many analytical physics-based and Kramers-Kronig consistent expressions (models) have been developed which describe various types of materialsamorphous and crystalline semiconductors and dielectrics, metals, organic films, optical metamaterials, etc. [12,13]. Parameterization should be accompanied by sensitivity analyses [14][15][16]. ...
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Ellipsometry as an indirect optical measurement method requires the use of optical modelling which include model parameterization. In practice, there are many ways to select a model and its parameters to fit the experimental data. Very often this fact leads to ad hoc decisions, i.e., based on experience or subjective opinion, instead use of some systematic approaches which provide predictive capability. In this paper we use the Akaike and Bayesian information criteria to perform optical model selection and its best parameterization to fit a particular set of ellipsometric data. We demonstrate that this approach accompanied by post hoc study of the inter-parameter correlations can significantly enhance optical modelling, in particularly, the process of model selection and data interpretation and improve the characterization of multilayered thin-film structures.
... The Tauc-Lorentz model has been proven to be useful for modeling (with perfect Kramer-Kronig consistency) semimetals [44,45], semiconductors and dielectrics, in particular oxides [46,47]. It has shown to be specially successful for the modeling of the response of amorphous and nanocrystalline materials [48,49]. In our case, the Tauc-Lorentz method fits perfectly the ellipsometric measurements values as shown in Figure 3 a) b). ...
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Nanocrystalline textured EuO thin films are prepared by an oxygen loss process from a pure Eu2O3 bulk ceramic target through pulsed laser deposition in vacuum at room temperature. X-ray diffraction spectra evidence a well-defined diffraction peak corresponding to the EuO phase textured along the (110) direction. Analysis of the XRD peak profile indicates that the films are nanocrystalline (average crystallite size of 11 nm) with a compressive residual strain. The formation of stoichiometric EuO is further confirmed by a strong signal from Eu2+ in the X-ray photoelectron spectra. The complex refractive index in the near infrared has been determined by spectroscopic ellipsometry and shows that the EuO films have a high transparency (k < 10-3) and a refractive index of 2.1. A band-gap shift of 0.25 eV is found with respect to the EuO bulk. These films, deposited by an accessible and efficient method, open a new route to produce EuO films with optical quality, suitable for NIR optoelectronic components.
... The importance of proper modeling of the real and imaginary parts of the complex dielectric function ε = ε 1 − iε 2 or the complex index of refraction N = n − ik (here we follow the traditional 1968 Nebraska ellipsometric convention which defines the imaginary parts with a "minus" sign 1 ) in spectroscopic ellipsometry data analysis is pretty much impossible to underestimate. At present, multiple well-established analytical models for the optical properties of materials as functions of photon energy (dispersion models) have been developed [1][2][3] . Typically, a model dielectric function can be established for the imaginary part ε 2 (or k) and then the real part ε 1 (or n) can be found by Kramers-Kronig integration. ...
... Effective medium methods are the most widely used to describe the optical function for such a composite structure, assuming that distinct phases are smaller than the wavelength of incident light. 11,12 In this work, we applied the Bruggeman effective medium approximation (BEMA), which is the most widely used description of the surface roughness in ellipsometry (we assumed 50% of voids in the roughness layer of SBN). ...
Article
Complex optical function e(E) ¼ e1(E) þ ie2(E) components parallel ekðEÞ and perpendicular e?(E) to the optic axis are determined in the photon energy range E ¼ 2–10 eV for SrxBa1–xNb2O6 single crystals with the composition parameters x ¼ 0.40, 0.50, 0.61, 0.65, and 0.75. The spectra are obtained from the evaluation of the ellipsometric data using a relevant optical model which takes into account the optical anisotropy of the crystal and surface imperfection of the measured specimens. We report the parameters necessary to construct the optical function for each studied composition. Additionally, the energies of electronic inter-band transitions are obtained from the analysis in terms of standard analytical line shapes.
... The Tauc-Lorentz model has been proven to be useful for modeling (with perfect Kramer-Kronig consistency) semimetals [44,45], semiconductors and dielectrics, in particular oxides [46,47]. It has shown to be specially successful for the modeling of the response of amorphous and nanocrystalline materials [48,49]. In our case, the Tauc-Lorentz method fits perfectly the ellipsometric measurements values as shown in Fig. 3(a) and (b). ...
Article
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Nanocrystalline textured EuO thin films are prepared by an oxygen loss process from a pure Eu2O3 bulk ceramic target through pulsed laser deposition in vacuum at room temperature. X-ray diffraction spectra evidence a well-defined diffraction peak corresponding to the EuO phase textured along the (1 1 0) direction. Analysis of the XRD peak profile indicates that the films are nanocrystalline (average crystallite size of 11 nm) with a compressive residual strain. The formation of stoichiometric EuO is further confirmed by a strong signal from Eu²⁺ in the X-ray photoelectron spectra. The complex refractive index in the near infrared has been determined by spectroscopic ellipsometry and shows that the EuO films have a high transparency (k < 10⁻³) and a refractive index of 2.1. A band-gap shift of 0.25 eV is found with respect to the EuO bulk. These films, deposited by an accessible and efficient method, open a new route to produce EuO films with optical quality, suitable for NIR optoelectronic components.
... At present, multiple well-established analytical models for the optical properties of materials as functions of photon energy (dispersion models) have been developed. [1][2][3] Typically, a model dielectric function can be established for the imaginary part ε 2 (or k) and then the real part ε 1 (or n) can be found by Kramers-Kronig integration. Instead of analytical model representation, one can also use "point-by-point data fit" (also called "exact numerical inversion" 2 ), when the real and imaginary parts of the complex dielectric function or the complex index of refraction are calculated at each wavelength independently. ...
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About ten years ago Johs and Hale developed the Kramers-Kronig consistent B-spline formulation for the dielectric function modeling in spectroscopic ellipsometry data analysis. In this article we use popular Akaike, corrected Akaike and Bayesian Information Criteria (AIC, AICc and BIC, respectively) to determine an optimal number of knots for B-spline model. These criteria allow finding a compromise between under-and overfitting of experimental data since they penalize for increasing number of knots and select representation which achieves the best fit with minimal number of knots. Proposed approach provides objective and practical guidance, as opposite to empirically driven or "gut feeling" decisions, for selecting the right number of knots for B-spline models in spectroscopic ellipsometry. AIC, AICc and BIC selection criteria work remarkably well as we demonstrated in several real-data applications. This approach formalizes selection of the optimal knot number and may be useful in practical perspective of spectroscopic ellipsometry data analysis.
... 26 However, it has been successfully applied to crystalline material. 27,28 To meet our main objective of detecting the phase transition from temperature dependent SE measurements, we found that the use of TL dispersion is adequate to fit the ellipsometric data of the NaNbO 3 film. The obtained film thickness of t ¼ (43.5 6 0.4) nm is in good agreement with the xray data (t ¼ 42.5 6 0.5 nm), and a surface roughness of about (0.40 6 0.04) nm could be determined. ...
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We have investigated high temperature phase transitions in NaNbO 3 thin films epitaxially grown under tensile lattice strain on (110) DyScO 3 substrates using metal-organic vapor phase epitaxy. At room temperature, a very regular stripe domain pattern consisting of the monoclinic a 1 a 2 ferroelectric phase was observed. Temperature-dependent studies of the refractive index and the optical bandgap as well as in situ high-resolution x-ray diffraction measurements prove a ferroelectric–ferroelectric phase transition in the range between 250 and 300 °C. The experimental results strongly suggest that the high-temperature phase exhibits a distorted orthorhombic a 1 /a 2 crystal symmetry, with the electric polarization vector lying exclusively in the plane. A second phase transition was observed at about 500 °C, which presumably signifies the transition to the paraelectric phase. Both phase transitions show a pronounced temperature-dependent hysteresis, indicating first-order phase transitions.
... To minimize the number of fit parameters the dependence of the data on time and wavelength should be parameterized, of which only the latter has been utilized frequently (mainly for semiconductors [92]). Therefore, there is a large potential in the development of data evaluation methods and strategies. ...
Article
Understanding interface processes has been gaining crucial importance in many applications of biology, chemistry, and physics. The boundaries of those disciplines had been quickly vanishing in the last decade, as metrologies and the knowledge gained based on their use improved and increased rapidly. Optical techniques such as microscopy, waveguide sensing, or ellipsometry are significant and widely used means of studying solid‐liquid interfaces because the applicability of ions, electrons, or X‐ray radiation is strongly limited for this purpose due to the high absorption in aqueous ambient. Light does not only provide access to the interface making the measurement possible, but utilizing the phase information and the large amount of spectroscopic data, the ellipsometric characterization is also highly sensitive and robust. This article focuses on ellipsometry of biomaterials in the visible wavelength range. The authors discuss the main challenges of measuring thickness and optical properties of ultra‐thin films such as biomolecules. The authors give an overview on different kinds of flow cells from conventional through internal reflection to combined methods. They emphasize that surface nanostructures and evaluation strategies are also crucial parts of in situ bioellipsometry and summarize some of the recent trends showing examples mainly from their research.
... E p for the CL model of a-SiC is fixed at 0.77 eV. whereas SE is mostly sensitive to their long-range order and properties in larger clusters forming crystals vs. amorphous materials which primarily influences how collective electrons form bands [38]. Optical spectra such as those plotted in Fig. 3 are fingerprints of the joint density of electron states in solids that change substantially as the longrange order of atoms are changing in the solids [39]. ...
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Protective SiC-rich nano layer was created by ion beam mixing of Si/C multilayers. The transformation of the Si and C layers into a homogeneous SiC layer was analyzed using complementary depth profiling by spectroscopic ellipsometry (SE) and Auger electron spectroscopy (AES). The distribution of elements and their chemical states was revealed by AES, whereas SE measured the accurate thicknesses, density, crystallinity, and the distribution of different phases. The optical properties of the created SiC layer determined by SE were almost identical to that of ion implantation-amorphized SiC. Amorphous Si and void formations were revealed by SE. The void profile determined by SE correlated well with the Xe distribution measured by AES. The complementary capabilities of SE and AES for a detailed chemical investigation were pointed out, such as the atomic (AES), structural (AES and SE) and non-destructive depth profiling (SE) features. It was shown that besides the synergism that the two methods reveal, SE is capable of a quick, sensitive and non-destructive testing of such layer structures and materials.
... For example, some vertical inhomogeneity would create wavelength-dependent measurement artifacts in ellipsometry because the penetration depth of the illuminating light is different for each of the constituent wavelengths and the sample optical properties are integrated over different depths. 57 Our obtained refractive index values between the Semilab and Horiba measurements are very similar (Δn < 5 × 10 −3 in the transparent region), which is of the order of the experimental uncertainty, and they were reproducible for measurements taken at different spots in each sample. Therefore, we propose that the average value of these results for the refractive index of MBE-grown AlxGa (1−x) As may be used with a high degree of confidence in the future work. ...
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A series of AlxGa(1−x)As ternary alloys were grown by molecular beam epitaxy (MBE) at the technologically relevant composition range, x < 0.45, and characterized using spectroscopic ellipsometry to provide accurate refractive index values in the wavelength region below the bandgap. Particular attention is given to O-band and C-band telecommunication wavelengths around 1.3 µm and 1.55 µm, as well as at 825 nm. MBE gave a very high accuracy for grown layer thicknesses, and the alloys’ precise compositions and bandgap values were confirmed using high-resolution x-ray diffraction and photoluminescence, to improve the refractive index model fitting accuracy. This work is the first systematic study for MBE-grown AlxGa(1−x)As across a wide spectral range. In addition, we employed a very rigorous measurement-fitting procedure, which we present in detail.
... 26 However, it has been successfully applied to crystalline material. 27,28 To meet our main objective of detecting the phase transition from temperature dependent SE measurements, we found that the use of TL dispersion is adequate to fit the ellipsometric data of the NaNbO 3 film. The obtained film thickness of t ¼ (43.5 6 0.4) nm is in good agreement with the xray data (t ¼ 42.5 6 0.5 nm), and a surface roughness of about (0.40 6 0.04) nm could be determined. ...
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We provide a combined theoretical and experimental study of the electronic structure and the optical absorption edge of the orthorhombic perovskite LaInO3, employing density functional theory and many-body perturbation theory. We find the lowest-energy excitation at 0.2 eV below the fundamental gap (5 eV), reflecting a sizable electron-hole attraction. Since the transition from the valence band maximum (Γ point) is, however, dipole forbidden, the onset is characterized by weak excitations from transitions around it. The first intense excitation appears about 0.32 eV above. Interestingly, this value coincides with an experimental value obtained by ellipsometry (4.80 eV) which is higher than the onset from optical absorption spectroscopy (4.35 eV). The latter discrepancy is attributed to the fact that the weak transitions that define the optical gap are not well enough resolved by the ellipsometry measurement. Through temperature-dependent measurements of the optical gap, we assess renormalization effects by electron-phonon coupling, enhancing the quantitative comparison between theoretical and experimental results.
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The research on solar cells based on photonic, plasmonic and various nanostructured materials has been increasing in the recent years. A wide range of nanomaterial approaches are applied from photonic crystals to plasmonics, to trap light and enhance the absorption as well as the efficiency of solar cells. The first part of this chapter presents examples on applications that utilize nanostructured materials for photovoltaics. In the second part, ellipsometry related metrology issues are discussed briefly, dividing the topic in two major parts: effective medium and scatterometry approaches.
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Ion implantation has been a key technology for the controlled surface modification of materials in microelectronics and generally, for tribology, biocompatibility, corrosion resistance and many more. To form shallow junctions in Ge is a challenging task. In this work the formation and accumulation of shallow damage profiles was studied by in-situ spectroscopic ellipsometry (SE) for the accurate tracking and evaluation of void and damage fractions in crystalline Ge during implantation of 200-keV Sb ⁺ ions with a total fluence up to 10¹⁶ cm⁻² and an ion flux of 2.1 × 10¹² cm⁻²s⁻¹. The consecutive stages of damage accumulation were identified using optical multi-layer models with quantitative parameters of the thickness of modified layers as well as the volume fractions of amorphized material and voids. The effective size of damaged zones formed from ion tracks initiated by individual bombarding ions can be estimated by numerical simulation compared with the dynamics of damage profiles measured by ion beam analysis and ellipsometry. According to our observations, the formation of initial partial disorder was followed by complete amorphization and void formation occurring at the fluence of about 1 × 10¹⁵ cm⁻², leading to a high volume fraction of voids and a modified layer thickness of ≈200 nm by the end of the irradiation process. This agrees with the results of numerical simulations and complementary scanning electron microscopy (SEM) measurements. In addition, we found a quasi-periodic time dependent behavior of amorphization and void formation represented by alternating accelerations and decelerations of different reorganization processes, respectively. For the understanding and prevention of adverse void formation and for controlled evolution of subsurface nanocavities or cellular surface texture the in-situ monitoring of the dynamics of structural damage accumulation by the developed SE method is essential.
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Porous silicon layers were prepared by electrochemical etching of p-type single-crystal Si (c-Si) of varying dopant concentration resulting in gradually changing morphology and nanocrystal (wall) sizes in the range of 2-25 nm. We used the model dielectric function (MDF) of Adachi to characterize these porous silicon thin films of systematically changing nanocrystal size. In the optical model both the surface and interface roughnesses have to be taken into account, and the E0, E1, and E2 critical point (CP) features are all described by a combination of several lineshapes (two-dimensional CP, excitonic, damped harmonic oscillator). This results in using numerous parameters, so the number of fitted parameters were reduced by parameter coupling and neglecting insensitive parameters. Because of the large number of fitted parameters, cross correlations have to be investigated thoroughly. The broadening parameters of the interband transitions in the measured photon energy range correlate with the long-range order in the crystal. The advantage of this method over the robust and simple effective medium approximation (EMA) using a composition of voids and c-Si with a nanocrystalline Si reference [Petrik et al., Appl. Surf. Sci. 253, 200 (2006)] is that the combined EMA+MDF multilayer method of this work provides a more detailed description of the material and layer structure.
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The aim of a joint research activity in the FP6-ANNA project (http://www.i3-anna.org) is to develop and improve metrologies for the measurement of nanocrystal properties. Within the framework of this cooperation we optimized the sample preparation techniques to obtain a range of structures containing nanocrystals. Based on these samples we have optimized our characterization methods. In this work we focus on ellipsometry and X-ray diffraction measurements for the characterization of nanocrystal sizes in silicon rich oxide and porous silicon. We demonstrate the capabilities of dielectric function parametrizations in the ellipsometric evaluations, revealing the correlation between the broadening parameters of the critical point features and the nanocrystal size.
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Polysilicon layers with thicknesses between 8 and 600 nm deposited by low-pressure chemical vapor deposition at temperatures ranging from 560 to 640 °C were characterized by spectroscopic ellipsometry (SE) to determine the layer thicknesses and compositions using multilayer optical models and the Bruggeman effective-medium approximation. The dependence of the structural parameters on the layer thickness and deposition temperature have been investigated. A better characterization of the polysilicon layer is achieved by using the reference data of fine-grained polysilicon in the optical model. The amount of voids in the polysilicon layer was independently measured by Rutherford backscattering spectrometry (RBS). The SE and RBS results show a good correlation. The comparison of the surface roughness measured by SE and atomic force microscopy (AFM) shows that independently of the AFM window sizes, a good correlation of the roughness determined by SE and AFM was obtained. © 2000 American Institute of Physics.
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We have developed a Kramers-Kronig consistent analytical expression to fit the measured optical functions of hydrogenated amorphous silicon (a-Si:H) based alloys, i.e., the real and imaginary parts of the dielectric function (epsilon(1),epsilon(2)) (or the index of refraction n and absorption coefficient alpha) versus photon energy E for the alloys. The alloys of interest include amorphous silicon-germanium (a-Si1-xGex:H) and silicon-carbon (a-Si1-xCx:H), with band gaps ranging continuously from similar to1.30 to 1.95 eV. The analytical expression incorporates the minimum number of physically meaningful, E independent parameters required to fit (epsilon(1),epsilon(2)) versus E. The fit is performed simultaneously throughout the following three regions: (i) the below-band gap (or Urbach tail) region where alpha increases exponentially with E, (ii) the near-band gap region where transitions are assumed to occur between parabolic bands with constant dipole matrix element, and (iii) the above-band gap region where (epsilon(1),epsilon(2)) can be simulated assuming a single Lorentz oscillator. The expression developed here provides an improved description of epsilon(2) (or alpha) in the below-band gap and near-band gap regions compared with previous approaches. Although the expression is more complicated analytically, it has numerous applications in the analysis and simulation of thin film a-Si:H based p-i-n and n-i-p multilayer photovoltaic devices. First, we describe an approach whereby, from a single accessible measure of the optical band gap, the optical functions can be generated over the full solar spectrum for a sample set consisting of the highest quality intrinsic a-Si:H based alloys prepared by plasma-enhanced chemical vapor deposition using the principle of maximal H-2 dilution. Second, we describe quantitatively how such an approach can be modified for sample sets consisting of lower quality alloy materials. Finally, we demonstrate how the generated optical functions can be used in simulations of the absorption, reflection, and quantum efficiency spectra of a-Si:H based single-junction and multijunction solar cells. (C) 2002 American Institute of Physics.
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An empirical study was carried out on the porous silicon layer (PSL) formation process by stain etching on p- and n-type silicon wafers having different doping concentrations. PSL formation differs from the electrochemical etching process, since the top surface of the porous layers is continuously etched during formation. A porosity gradient is developed in the porous layers formed on p-, p+-, n- and n+-type silicon because stain etching, as a wet chemical etching method attacks the pore walls. This complex process can result in limited thickness for PSLs depending on the doping type and concentration of the substrate. The total mass of the silicon dissolved from the top surface and from the pores, was measured by gravimetry. The porosity and thickness values extracted from spectroscopic ellipsometrical (SE) measurements and with the measured mass of dissolved silicon are used for studying the etching process. The structure of the layers was characterized by backscattering spectrometry (BS) and cross-sectional transmission electron microscopy (XTEM). The PSLs exhibit amorphous structure on p-type silicon, while p+ layers have crystalline structure according to ion beam channeling experiments, and XTEM images.
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The complex dielectric function ε(omega) of GaAs was measured from 20 to 750 K with a scanning rotating-analyzer ellipsometer. The structures observed in the 1.3-5.5-eV photon-energy region, attributed to transitions near the Gamma point of the Brillouin zone (E0, E0+Delta0, E'0), along the Lambda direction (E1, E1+Delta1), and near the X point (E2), are analyzed by fitting the second-derivative spectrum d2ε(omega)/domega2 to analytic critical-point line shapes. The E'0 and E2 critical points are best fitted in the whole temperature region by a two-dimensional line shape, whereas the E1 and E1+Delta1 transitions are best fitted up to room temperature by a Lorentzian interacting with a continuum of interband transitions (Fano line shape). The excitonic character of the E1 and E1+Delta1 transitions is discussed within several theoretical approaches. The experiments indicate that up to room temperature the localized Lorentzian interacting with the continuum is dominant, whereas at higher temperatures the modification of the two-dimensional Van Hove singularity due to the electron-hole attractive perturbation is a better description of the measurements. For all critical points, the energy decreases with increasing temperature while the broadening increases. This dependence on temperature is analyzed in terms of averaged phonon frequencies which cause a renormalization of the energies and a broadening of the band gaps.
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The complex dielectric function ε(omega) of Si was measured ellipsometrically in the 1.7-5.7-eV photon-energy range at temperatures between 30 and 820 K. The observed structures are analyzed by fitting the second-derivative spectrum d2ε/domega2 with analytic critical-point line shapes. Results for the temperature dependence of the parameters of these critical points, labeled E'0, E1, E2, and E'1, are presented. The data show good agreement with microscopic calculations for the energy shift and the broadening of interband transitions with temperature based on the electron-phonon interaction. The character of the E1 transitions in semiconductors is analyzed. We find that for Si and light III-V or II-VI compounds an excitonic line shape represents best the experimental data, whereas for Ge, alpha-Sn, and heavy III-V or II-VI compounds a two-dimensional critical point yields the best representation.
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In the past years spectroscopic ellipsometry (SE) was applied to materials science problems as an optical technique for non destructive depth profiling and characterization of multilayer structures and interfaces with considerable success. The measured optical response of the multicomponent and/or multilayer structure under investigation can only be related to actual material properties by a model calculation. The successful application of ellipsometry is not only determined by the quality of the measurements, but more importantly by the quality of the optical model. Several examples for the different application of SE are reviewed. Two recent examples of multilayer analysis illustrate possibilities: in the first example damage created by ion implantation in single-crystalline silicon and in silicon carbide was characterized using ellipsometry and Rutherford Backscattering Spectrometry (RBS) in combination with channeling. In the second example electrochemically prepared porous silicon layers (PSL) were investigated by SE.
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Polysilicon layers prepared by low pressure chemical vapor deposition (LP-CVD) on oxidized silicon were measured by spectroscopic ellipsometry (SE), atomic force microscopy (AFM), and transmission electron microscopy (TEM). SE was used to determine layer thicknesses and compositions using multi-layer optical models. The measured spectra were simulated and fitted using a linear regression algorithm (LRA). The dielectric function of composite materials was calculated by the Bruggeman effective medium approximation (B-EMA). The dependence of the surface roughness and layer structure on the deposition temperature was studied. The interface layer between the buried oxide and the polysilicon layer, which represents the initial phase of growth, was modeled with a thin layer having polycrystalline silicon and voids. The precision of the SE layer thickness measurements was determined by a comparison with AFM and TEM results taking into account the 95% confidence limits of the LRA. The root mean square (RMS) roughness values measured by AFM using different scan sizes were compared to the thicknesses of the top layer in the SE model simulating the surface roughness. It was shown that the correlation between the SE and the AFM surface roughness results are affected by the scan size of AFM and the surface characteristics.
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Dielectric function of disorder in single-crystalline silicon (c-Si) implanted by He with energy of 40keV and fluences from 1×1016 to 1×1017cm−2 were determined around the E1 and E2 critical points (CPs) by spectroscopic ellipsometry. The implanted material was modeled by an effective medium composition of c-Si and damaged Si. The dielectric function of damaged Si was calculated using the model dielectric function of Adachi to fit the E1 and E2 CP parameters of the MDF. The penetration depth of light in the photon energy range of 3–5eV is less than 100nm, which allows a simple layer structure of (surface oxide)/(surface amorphous layer)/(c-Si+damaged Si as a substrate). The oscillator energies and strengths decrease, while the broadening parameters increase with increasing fluence. Rutherford backscattering spectrometry was used for cross-checking of the surface disorder.
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Porous silicon layers (PSLs) were prepared by electrochemical etching of p-type single-crystalline silicon (c-Si) wafers having different dopant concentrations to obtain systematically changing sizes of nanocrystals (walls). The microstructure of the porous material was characterized using spectroscopic ellipsometry with multi-layer effective medium approximation (EMA) models. The dielectric function of PSL is conventionally calculated using EMA mixtures of c-Si and voids. The porosity is described by the concentration of voids. Some PSL structures can be described only by adding fine-grained polycrystalline silicon (nc-Si) reference material to the EMA model. Modified model dielectric functions (MDF) of c-Si have been shown to fit composite materials containing nanocrystalline regions, either by fitting only the broadening parameter or also other parameters of the parametric oscillator in MDF. The broadening parameter correlates with the long-range order in the crystalline material, and, as a consequence, with the size of nanocrystals. EMA and MDF models were used to describe systematically changing nanostructure of PSLs. Volume fraction of nc-Si in EMA and broadening parameter in MDF provide information on the nanocrystal size. The longer-term goal of this work is to provide a method for the quantitative characterization of nanocrystal size using quick, sensitive and non-destructive optical techniques.
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The development of an optical constant library for Hg1−xCdxTe as a function of composition (x=0–0.5) and temperature (T=0–250°C) which is suitable for precise composition control by spectroscopic ellipsometry (SE) during MBE growth is described. An efficient methodology for acquiring in situ optical constants as a function of composition and temperature is first presented. Optical constants extracted from these in situ measurements, as well as literature data from room temperature values, were used to obtain internally Kramers-Kronig consistent parametric optical constant models at discrete compositions and temperatures. Then a global data analysis over temperature T' and composition x' was performed in which the internal parameters of the optical constant model were fitted as polynomials in T and x. This parametric model was developed to replace, without compromising the quality of ellipsometric data fits, the usual tabulated optical constant lists while using a reasonably small set of adjustable parameters. The model is flexible enough to describe the complicated critical point structures of semiconductors, yet stable enough to generate optical constants as a function of composition and temperature and permit limited extrapolation outside the measured range.
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Using a simple model that describes the decrease of the amplitudes of optical structures in ion-implanted crystals, projected areas of several valence and core excitons in GaAs are determined. The last remnant of crystal-related optical structure vanishes for crystallite areas less than (16 A)Â².
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Four-parameter fitting of multiple-angle-of-incidence (MAI) ellipsometry data is developed to characterize near-surface layers on semiconductors damaged by implantation. We used coupled half-Gaussians to describe the damage depth profiles. The method was tested on Ge-implanted silicon layers (at a wavelength of 632.8 nm) and was cross-checked with high depth resolution RBS and channeling.
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ZnO thin films doped by Ga and In as well as multilayer structures of ZnO/Al2O3 have been investigated by X-ray fluorescence, Raman spectrometry, spectroscopic ellipsometry and vacuum ultra violet reflectometry. Systematic changes in the optical properties have been revealed even for Ga concentrations below 1%. The Raman active phonon mode of Ga doping at 580 cm−1 shows a correlation with the Ga concentration. Optical models with surface nanoroughness correction and different parameterizations of the dielectric function have been investigated. There was a good agreement between the dielectric functions determined by the Herzinger–Johs polynomial parameterization and by direct inversion. It has been shown that the correction of the nanoroughness significantly influences the accuracy of the determination of the layer properties. The band gap and peak amplitude of the imaginary part of the dielectric function corresponding to the excitonic transition changes systematically with the Ga-content and with annealing even for low concentrations.
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Aluminum doped Zinc Oxide (AZO) and Lithium doped Zinc Oxide (LZO) thin films are obtained by Pulsed Laser Deposition (PLD) method. These films are characterized by using Spectroscopic Ellipsometry (SE), X-ray Diffraction (XRD) and Photoluminescence (PL). By modeling the ellipsometry spectra we get the dielectric functions, the optical band gap E(g), and the electrical properties. Our results show the influence of the processing parameters on the optical and structural properties of doped ZnO thin films. The post-annealing treatment applied to AZO thin films, changes strongly the optical properties, by lowering the resistivity and red-shifting the band gap. (C) 2009 Elsevier B.V. All rights reserved.
Article
We propose to analyze ellipsometry data by using effective medium approximation (EMA) models. Thanks to EMA, having nanocrystalline reference dielectric functions and generalized critical point (GCP) model the physical parameters of two series of samples containing silicon nanocrystals, i.e. silicon rich oxide (SRO) superlattices and porous silicon layers (PSL), have been determined. The superlattices, consisting of ten SRO/SiO2 layer pairs, have been prepared using plasma enhanced chemical vapor deposition. The porous silicon layers have been prepared using short monopulses of anodization current in the transition regime between porous silicon formation and electropolishing, in a mixture of hydrofluoric acid and ethanol. The optical modeling of both structures is similar. The effective dielectric function of the layer is calculated by EMA using nanocrystalline components (nc-Si and GCP) in a dielectric matrix (SRO) or voids (PSL). We discuss the two major problems occurring when modeling such structures: (1) the modeling of the vertically non-uniform layer structures (including the interface properties like nanoroughness at the layer boundaries) and (2) the parameterization of the dielectric function of nanocrystals. We used several techniques to reduce the large number of fit parameters of the GCP models. The obtained results are in good agreement with those obtained by X-ray diffraction and electron microscopy. We investigated the correlation of the broadening parameter and characteristic EMA components with the nanocrystal size and the sample preparation conditions, such as the annealing temperatures of the SRO superlattices and the anodization current density of the porous silicon samples. We found that the broadening parameter is a sensitive measure of the nanocrystallinity of the samples, even in cases, where the nanocrystals are too small to be visible for X-ray scattering. Major processes like sintering, phase separation, and intermixing have been revealed as a function of annealing of the SRO superlattices.
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In a previous paper, the authors proposed a model for the optical dielectric function of zinc-blende semiconductors. It was found to be more generally valid than previous models. In this paper, it is used to obtain an analytic expression for the dielectric function of the alloy series AlxGa1-xAs as a function of ω and x, which is compared with spectroscopic ellipsometry data between 1.5 and 6.0 eV. The model enables us to determine accurately the critical point energies and linewidths of AlxGa1-xAs as a function of x. Also, it leads us to model the optical dielectric function of these alloys better than any previous model in that (1) it covers the entire photon energy range between 1.5 and 6.0 eV as well as the entire alloy composition range between 0.0 and 1.0, (2) it calculates the optical properties of AlxGa1-xAs as a function of ω and x with the highest accuracy, and (3) it allows one to accurately calculate the values of the refractive indices below 1.5 eV as a function of ω and x.
Article
Optical properties of P+ ion-implanted Si(100) wafers have been studied using spectroscopic ellipsometry (SE). The P+ ions are implanted at 150 keV with fluences ranging from 1×1014 to 2×1015 cm−2 at room temperature. An effective-medium-approximation analysis suggests that the ion-implanted layer can be explained by a physical mixture of microcrystalline and amorphous silicon. The ϵ(E) spectrum of the microcrystalline component is found to differ appreciably from that of single-crystalline silicon, especially in the vicinity of the sharp critical-point features. This difference in ϵ(E) can be successfully interpreted by increasing the broadening parameter at each critical point. Considering these and previous data, we obtain an expression, A=(5.13×1011/EacM)1.872, which enables us to estimate the amorphization-threshold fluence A for silicon implanted with optional ion species of mass number M at energy Eac in keV. No clear change in the original structure of silicon surface after P+ ion implantation has been observed by atomic force microscopy. SE has been proven to be an easy, fast, and nondestructive technique which can be used to assess important ion-implantation parameters. © 2002 American Institute of Physics.
Article
Real time spectroscopic ellipsometry (RTSE) has been applied to characterize compositionally graded amorphous silicon-carbon alloy (a-Si1−xCx:H) thin films, prepared using continuous variations in the flow ratio z = [CH4]/{[SiH4]+[CH4]} during rf plasma-enhanced chemical vapor deposition. Triangular variations in z versus time, yielding 50–130-Å-thick a-Si1−xCx:H graded layers with 0.02⩽x⩽0.24, were applied in order to assess the performance of the RTSE data analysis procedure. This procedure employs a four-medium virtual interface approximation, and returns the time evolution of (i) the near-surface C content, (ii) the instantaneous growth rate, and (iii) the surface roughness layer thickness. In the depth profiles of the graded structures, an apparent resolution of ∼10 Å is obtained with a composition uncertainty of ±0.004. © 1997 American Institute of Physics.
Article
The dielectric functions ε of polycrystalline CdS and CdTe thin films sputter deposited onto Si wafers were measured from 0.75 to 6.5 eV by in situ spectroscopic ellipsometry. Differences in ε due to processing variations are well understood using an excited carrier scattering model. For each sample, a carrier mean free path λ is defined that is found to be inversely proportional to the broadening of each of the band structure critical points (CPs) deduced from ε. The rate at which broadening occurs with λ−1 is different for each CP, enabling a carrier group speed υg to be identified for the CP. With the database for υg, ε can be analyzed to evaluate the quality of materials used in CdS/CdTe photovoltaic heterojunctions.
Article
Accurate dielectric function values are essential for spectroscopic ellipsometry data analysis by traditional optical model-based analysis techniques. In this paper, we show that B-spline basis functions offer many advantages for param- eterizing dielectric functions. A Kramers–Kronig consistent B-spline formulation, based on the standard B-spline recursion relation, is derived. B-spline representations of typical semiconductor and metal dielectric functions are also presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Article
Bulk single-crystal CdTe, sputter-deposited polycrystalline CdTe films of about 2.5 micron thickness, and single-crystal Si (c-Si) have been ion implanted using 350 keV Xe at fluences ranging from 1×1013 to 16×1013 cm–2 so as to create disorder in a controlled way from fully single-crystalline to fully amorphous material. The general purpose of the investigations is to seek a parameterization of the critical point structures and establish a database for fitting the optical properties of CdTe films having different unknown grain sizes whereby the grain size will be described in terms of an effective defect density. The polycrystalline CdTe samples were magnetron sputtered onto c-Si followed by CdCl2 and Br2-methanol treatment to improve properties in terms of grain size and surface smoothness, respectively. The fluences for use in the ion implantation of CdTe were estimated using the SRIM (Stopping and Range of Ions in Matter) software, and cross-checked by simultaneous implantation of bulk c-Si samples. The optical properties were characterized by second derivative analysis and by a generalized critical point model. Although the damage created by 350 keV Xe in the simultaneously implanted c-Si samples, as measured by both spectroscopic ellipsometry and Rutherford backscattering/ channeling spectrometry (RBS/C), agrees well with the expectations based on the SRIM simulation, the damage created in CdTe remains at a very low level even for doses several times higher than the amorphization level estimated by simulation. The character of the dechanneling of the RBS/C spectra indicates extended defects (presumably dislocation loops). This effect was similar in both single-crystal and thin film polycrystalline CdTe, although less pronounced in thin film samples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Article
The results of multiparameter fitting of spectroscopic ellipsometric (SE) spectra on porous silicon layers (PSL) were connected with the processing parameters (oxidation, etching time, porosity, argon implantation dose).Two optically different types of silicon forms, a bulk-type silicon (c-Si) and polycrystalline-like silicon with enhanced absorption in the grain boundaries (p-Si) needed to be mixed with voids in the appropriate ratio, and the PSL had to be divided in depth in several different sections in order to obtain the best fit. The sectioning reflects the effect of upper and lower interfaces or inhomogeneity in depth. The effective porosity and the sublayer thicknesses are determined with high precision.In the case of argon implantation, we used the dielectric function of c-Si and implanted amorphous silicon (a-Si) mixed with voids. The regression analysis of SE spectra clearly shows the effect of different doses of implantation. The sectioning reflects that to the full range of argon ions the porous silicon became amorphous and denser. The overall thickness of the originally porous layer also significantly reduced (from 670 nm to 320 nm) due to the argon implantation.
Article
Spectroscopic ellipsometry (SE) has been used to routinely characterize amorphous silicon nitride and diamond thin films. Since SE measurements do not yield quantities of interest directly, the SE data must first be fit to a model to obtain useful parameters such as film thickness and optical functions. The Tauc–Lorentz (TL) model for the optical functions of amorphous materials [Appl. Phys. Lett. 69, 371/373, 2137 (1996)] has been shown to be very useful in interpreting these SE results. A four-parameter model is usually sufficient to describe the optical functions of the thin film to the accuracy of the ellipsometer. One of these parameters, the band gap Eg, correlates with other mechanical and chemical properties of the film, such as the silicon-to-nitrogen ratio in silicon nitride films, and to the sp3-bonded carbon fraction and the hardness of amorphous carbon films.
Article
The optical properties of ion implantation induced disorder in SiC have been investigated in the photon energy range of 5–9 eV using spectroscopic ellipsometry (SE). The most characteristic interband transitions of SiC are located between 5 and 8 eV. This photon energy region is extremely important for the sensitive characterization of lattice order in SiC. The dielectric function of the disordered layer has been calculated taking into account the surface overlayer consisting of oxide and roughness using complementary characterization tools. The dielectric function of the damaged region has been analyzed using different techniques like second derivative analysis and effective medium approximation (EMA) based on reference dielectric functions. The disorder determined by SE has been verified by Rutherford backscattering spectrometry combined with channelling (RBS/C). Using derivative lineshape analysis combined with simulations, the track size can be estimated. The results can give insight into the effect of the decreasing characteristic size of the unchanged crystalline regions on the optical properties. We created near-surface damage using heavy ions, since the penetration depth of light at photon energies around the direct interband transitions is very small (in the range of 10 nm). We used 100-keV Xe at fluences ranging from a slight damage to full amorphization (between 2.0 × 1013 cm− 2 and 1.6 × 1014 cm− 2).
Article
An ellipsometer utilizing the polarizer-sample-retarder-rotating analyser configuration measures all four elements of the Stokes vector. Consequently the ellipsometric parameters ψ and Δ as well as the polarization transfer factor D of the sample can be measured simultaneously as a function of the photon energy E. The measurement of all four elements of the Stokes vector is shown to be well suited for the ellipsometric characterization of non-ideal samples, e.g. laterally inhomogeneous or rough samples, and it may be used to improve the precision of high lateral resolution (“microspot”) measurements. Formulae for the error spectra δψ(E), δΔ(E), and δD(E) were derived by analysing the influence of the random errors of the measurement. These error spectra provide the weights in minimization procedures for the determination of the geometrical and strutural sample parameters and results, via the analysis of the hessian matrix, in the errors of the sample parameters adjusted to the spectra. The effect of systematic alignment errors is discussed and fast Kramers-Kronig-consistent spectral inversion is shown to be a useful tool for the reduction of systematic model errors.
Article
CdZnO thin films with different ratio of CdO and ZnO (3:1, 1:1, and 1:3) were grown on glass substrate using sol-gel spin coating method. The morphology of the CdZnO films depends on the amount of ZnO and CdO in the films. The optical band gap of the CdZnO films depends on the compositions of CdO and ZnO. Films having higher amount of CdO shows the presence of grains along with the fiber nature of ZnO, whereas the film with lower percentage of CdO shows fiber nature of the film very similar to pure ZnO film. The optical bandgap of CdZnO (3:1), CdZnO (1:1), and CdZnO (1:3) films was calculated to be 2.80, 2.49, and 2.52 eV, respectively. Other optical properties such as refractive index, extinction coefficient, and dielectric constants were calculated using the optical data. The volume and surface energy loss functions were also calculated and observed to increase with increase in the photon energy.
Article
A several‐parameter fitting of spectroscopic ellipsometry data is developed to characterize near‐surface layers in semiconductors damaged by implantation. The damage depth profiles are described by either rectangular, trapezoid‐type, or coupled half‐Gaussian (realistic) optical models. The rectangular model has three parameters: the average damage level, the effective thickness of the implanted layer, and the thickness of the native oxide. The trapezoid‐type model is enhanced with a fourth parameter, the width of the amorphous/crystalline interface. The realistic optical model consists of a stack of layers with fixed and equal thicknesses. The damage levels are determined by a depth profile function (presently coupled half‐Gaussians). Five parameters are used: the position of the maximum, the height, and two standard deviations of the profile, plus the thickness of the native oxide. The complex refractive index of each layer is calculated from the actual damage level by the Bruggeman effective medium approximation. The optical models were tested on Ge‐implanted silicon samples and cross checked with high‐depth‐resolution Rutherford backscattering spectrometry and channeling.
Article
A parameterization of the optical functions of amorphous semiconductors and insulators is presented in which the imaginary part of the dielectric function ϵ 2 is determined by multiplying the Tauc joint density of states by the ϵ 2 obtained from the Lorentz oscillator model. The real part of the dielectric function ϵ 1 is calculated from ϵ 2 using Kramers–Kronig integration. The parameters of this model are fit to n and k data for amorphous Si (2 data sets), SiO, As 2 S 3 , and Si 3 N 4 . Comparative fits are made with a similar parameterization presented earlier by Forouhi and Bloomer [Phys. Rev. B 34, 7018 (1986)]. In all cases, the new parameterization fits the data better. © 1996 American Institute of Physics.
Article
The optical properties of undoped and P‐doped silicon prepared by low‐pressure chemical vapor deposition were measured by spectroscopic ellipsometry over the energy range 3.0–6.0 eV. A marked effect of material microstructure is observed. Approximate values of the density deficit and of the volume fractions of crystalline and amorphous material are estimated as components of the microstructure by comparing measured spectra to those synthesized from constituent spectra in the Bruggeman effective‐medium approximation.
Article
The model of Forouhi and Bloomer (FB) for the optical properties of amorphous semiconductors is modified in order to describe more accurately the dispersion of the optical constants observed for amorphous carbon (a-C) and amorphous hydrogenated carbon (a-C:H) thin films. The FB model represents the optical absorption as the product of a lineshape function and a joint density of states function, which is derived by assuming the condition and valence bands to be parabolic and separated by an energy gap within which there are no allowed electronic states. Two modifications to this model are discussed to address the cases of non-parabolic bands and/or electron energy levels in the energy gap. These modified parametric models are then fit to a large number of a-C and a-C:H film optical constant spectra, and results are presented which indicate that non-parabolicity of the conduction and valence bands is the most important correction to the standard FB model required to describe a-C:H thin films. The modified model incorporating non-parabolic bands is shown to fit a broad range of both a-C and a-C:H spectra very well, and provides useful information about the optical absorption process and physical properties of the films.
Article
In this paper we discuss the connection between the microstructure of a heterogeneous thin film and its macroscopic dielectric response ε. Effective medium theory is developed from a solution of the Clausius-Mossotti problem from basic principles. The solution is generalized to obtain the Lorentz-Lorenz. Maxwell Garnett and Bruggeman expressions. The connection between microstructure and absolute limits to the allowed values of the dielectric response of two-phase composites is reviewed. The form of these limits for two-phase composites of known composition and two- or three-dimensional isotropy can be used to derive simple expressions for ε and also for the average fields within each phase. These results are used to analyze dielectric function spectra of semiconductor films for information about density, polycrystallinity and surface roughness. Examples illustrating the detection of unwanted overlayers and the real-time determination of nucleation growth are also given.
Article
Titanium dioxide films with the anatase and rutile single phase were formed on Si substrates by rf sputtering through a precise control of critical parameters. The structure of the films was studied by X-ray diffraction (XRD) and transmission electron microscopy (TEM), and the optical properties were evaluated with spectroscopic ellipsometry (SE). Lattice distortion was found in both anatase and rutile films from TEM observation. The obtained refractive indices n exhibit higher values than those reported for thin films due presumably to the density structure of the sputtered films. Optical band gaps were calculated by Tauc plot using the obtained extinction coefficient separately for anatase and rutile, with values larger than those reported for bulk materials. The reasons for the larger band gap might be due to the strain from lattice distortion.
Article
We extend earlier calculations of the temperature dependence of the electronic states at the Γ point of Si and Ge to other points of the Brillouin zone. Thus we are able to calculate the temperature dependence of gaps and critical-point energies: the indirect gap and the E1 and E2 critical points, as well as the E0 gap of Si and the E0’ gap of Ge. Both the Fan self-energy and the Debye-Waller terms of the electron-phonon interaction are included. The theoretical results, corrected for the contribution of thermal expansion to the temperature shifts, show satisfactory agreement with experimental data.
Article
A new method is described for calculation of the real and imaginary parts of the dielectric function of semiconductors at energies below and above the lowest band gaps, in which the model is based on the Kramers-Kronig transformation and strongly connected with the electronic energy-band structures of the medium. This model reveals distinct structures at energies of the E0, E0+Delta0, E1, E1+Delta1, and E2 critical points. Analyses are presented for GaP, GaAs, GaSb, InP, InAs, and InSb, and results are in satisfactory agreement with the experimental information over the entire range of energies. The model is able to properly give the optical constants, such as the refractive indices and the absorption coefficients, which are important for a variety of optoelectronic device applications.
Article
A method is described for calculation of the real (ε1) and imaginary parts (ε2) of the dielectric function of Si and Ge at energies below and above the fundamental absorption edge, in which the model is based on the Kramers-Kronig transformation and strongly connected with the electronic energy-band structure of the medium. A complete set of the critical points (CP's) are considered in this study. This model reveals distinct structures at energies of the E0, E0+Delta0 [three-dimensional (3D) M0], E1, E1+Delta1 (3D M1 or 2D M0), E2 [a mixture of damped harmonic oscillator (DHO) and 2D M2], E'1, and E'0 (triplet) CP's (DHO). The indirect-band-gap transitions also play an important part in the spectral dependence of ε2 of Si. Results are in satisfactory agreement with the experimental information over the entire range of photon energies. The strength and broadening parameters at energies of each CP are obtained and discussed.
Article
We have succeeded in fabricating the mostly crystallized Si:H materials having a wide optical band gap of up to 2.4 eV by means of a reactive sputtering technique with a low substrate temperature of $\sim${}100 K. The structural analysis showed that the materials consist of small crystalline silicon particles surrounded by hydrogen atoms, whose diameters are 20\char21{}30 A\r{}. The widening of the optical band gap can be explained by a three-dimensional quantum-well effect in the small particles.
Article
We have carried out reflectivity measurements, for photon energies from 2.0 to 5.6 eV in the electronic interband regime, for a series of unannealed ion-implanted GaAs samples which had been exposed to 45-keV ${\mathrm{Be}}^{+}$ ions at various fluences up to 5\ifmmode\times\else\texttimes\fi{}${10}^{14}$ ions/${\mathrm{cm}}^{2}$. The microstructure of the near-surface implantation-induced damage layer in these samples is known (from previous Raman work) to consist of a fine-grain mixture of amorphous GaAs and GaAs microcrystals, with the characteristic microcrystal size L of the microcrystalline component decreasing with increasing fluence (L=55 A\r{} at 5\ifmmode\times\else\texttimes\fi{}${10}^{14}$ ${\mathrm{cm}}^{\mathrm{$-${}}2}$). The optical dielectric function of each sample's damage layer has been derived from the observed reflectivity spectrum by Lorentz-oscillator analysis. Then, by inverting the effective-medium approximation, we have extracted the dielectric function of the microcrystalline component ($\mu${}-GaAs) within the damage layer.
Article
A model is proposed for the line shape of the optical dielectric function of zinc-blende semiconductors. For comparison with previously proposed models, this model is used primarily with spectroscopic ellipsometry data (but also transmission data below 1.5 eV) to obtain an analytic room-temperature dielectric function for GaAs. It is found to be more generally valid than the harmonic-oscillator model, the critical-point (CP) model, or the model of Adachi. It is applicable over the entire range of photon energies, below and above the lowest band gaps, incorporates the electronic band structure of the medium, and exactly satisfies the Kramers-Kronig transformation. It goes beyond the CP parabolic-band approximation in that it correctly takes into account the full analytic form of the electronic density of states and thus does not require the use of arbitrary cutoff energies. Also, it allows one to go beyond the usual approximation of Lorentzian broadening, which is known to be incorrect for elements and compounds above very low temperatures. For these reasons, it results in excellent quantitative agreement with experimental results for the dielectric function and for its derivatives with respect to photon energy, much better than that given by earlier models. Finally, the parameters of the model are physically significant and are easily determined as functions of composition for semiconductor alloys. Application of the model to the fitting of spectroscopic data on GaAs strongly suggests that spectroscopic ellipsometry does not measure the true bulk dielectric function. It also supports the conclusion that the line-shape broadening in GaAs at room temperature is more nearly Gaussian than Lorentzian.
Article
We have performed real-time spectroscopic ellipsometry (SE) measurements over the photon-energy range from 1.5 to 4.0 eV during the growth of microcrystalline silicon (muc-Si:H) by plasma-enhanced chemical-vapor deposition on chromium at 250 °C. We focus on the regime when the film consists of isolated microcrystallites and intervening void volume. In this regime, the observed three-dimensional growth behavior allows us to associate the crystallite size with the physical thickness of the film. The SE measurements are self-contained in that they provide not only microstructural information, including film thickness and void-volume fraction, but also the effective optical functions of the film. From this combination of results, the optical functions of the Si crystallites, themselves, can be deduced by mathematically extracting the influence of the void-volume fraction on the effective optical functions. A critical-point (CP) analysis of the E1 transitions visible near 3.3 eV in the crystallite optical functions provides information on the electronic properties as a continuous function of crystallite size. Over the physical thickness range accessible in these experiments (~200-250 Å), the transition energy and phase deduced in the CP analysis are constant at the single-crystal values, within experimental error, while the broadening parameter decreases with increasing thickness. The latter behavior is consistent with a finite-size effect in which electron scattering at surfaces modifies the optical response of the microcrystallites.
Article
We present models for the optical functions of 11 metals used as mirrors and contacts in optoelectronic and optical devices: noble metals (Ag, Au, Cu), aluminum, beryllium, and transition metals (Cr, Ni, Pd, Pt, Ti, W). We used two simple phenomenological models, the Lorentz-Drude (LD) and the Brendel-Bormann (BB), to interpret both the free-electron and the interband parts of the dielectric response of metals in a wide spectral range from 0.1 to 6 eV. Our results show that the BB model was needed to describe appropriately the interband absorption in noble metals, while for Al, Be, and the transition metals both models exhibit good agreement with the experimental data. A comparison with measurements on surface normal structures confirmed that the reflectance and the phase change on reflection from semiconductor-metal interfaces (including the case of metallic multilayers) can be accurately described by use of the proposed models for the optical functions of metallic films and the matrix method for multilayer calculations.
• A S Ferlauto
• G M Ferreira
• J M Pearce
• C R Wronski
• R W Collins
A.S. Ferlauto, G.M. Ferreira, J.M. Pearce, C.R. Wronski, R.W. Collins, J,. Appl. Phys. 92 (2002) 2424.
• A R Forouhi
• I Bloomer
A.R. Forouhi, I. Bloomer, Phys. Rev. B 34 (1986) 225.
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