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Effective medium approximation of ellipsometric response from random surface roughness simulated by finite-element method
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... For example, Chang et al [25] reported that the effective thickness of a SiO 2 layer on a Si crystal obtained via ellipsometry and EMA is qualitatively consistent with that measured using other methods, such as mediumenergy ion scattering spectroscopy and transmission electron microscopy. Numerical simulations were also conducted by Fodor et al [39] using the finite-element method and by Liu et al [40] using the finite-difference time-domain (FDTD) method; they observed a proportional relationship between the surface roughness parameters of a Si substrate and the effective optical thickness modelled using EMA. However, because of technical limitations, i.e. the measurement accuracies and computational costs, both experimental and numerical investigations regarding ultra-thin film problems have not been fully studied. ...
... In this regard, we conducted FDTD simulations of the optical characteristics of a dielectric mixture layer on a Si substrate to investigate the validity of using EMA layers in XRCD measurements, as preliminarily reported in a previous proceeding [43]. Because the surfaces of the Si spheres were cleaned using a detergent and water with ethanol [1] and the surface layer of the Si spheres might be a mixture layer rather than a rough surface, which was assumed in [39,40], we computed the complex reflection coefficients of Si substrates with randomly distributed thin mixture films composed of water/SiO 2 and water/carbonaceous materials at both normal and oblique incidences of plane electromagnetic waves using a dedicated in-house FDTD code. The results were compared with those obtained for ideally layered surfaces to evaluate the effects of the mixture layers on the effective optical characteristics when the ideally layered surface model is hypothetically used for the mixture layer surface. ...
This paper reports on a numerical simulation of the optical characteristics of a dielectric mixture layer formed on a Si substrate using the finite-difference time-domain (FDTD) method. This study investigated the validity of using effective medium approximation (EMA) layers in thin mixture film problems with the optical constants of bulk materials. The complex reflection coefficients of Si substrates with randomly distributed thin mixture films fabricated from water/SiO2 and water/carbonaceous materials were numerically evaluated at normal and oblique incidences of plane electromagnetic waves via the FDTD principle, and the possible effects of the mixture layers on the effective optical characteristics were investigated. We observed that Bruggemann's model based on EMA provides a good prediction of the behaviour of the plane wave reflected by the Si surface with randomly distributed mixture layers. The results also suggested that mixture layers have effects under oblique incidence, but they have no significant impact under normal incidence.
... The film thickness is in good agreement with the X-ray data (39 ± 1 nm) while the surface roughness is slightly overestimated compared to the AFM rms surface roughness of 0.15 nm. Despite the deviation in the surface roughness (which can be attributed to the specific measurements conditions and evaluation of AFM and SE data [34,35], the result basically confirms our model assumption. ...
The dielectric function and interband critical points of compressively strained ferroelectric K0.85Na0.15NbO3 thin film grown by metal-organic vapor phase epitaxy (MOVPE) are studied in broad spectral and temperature ranges by spectroscopic ellipsometry (SE). The temperature dependence of the measured pseudodielectric functions is strongly affected by a structural phase transition from the monoclinic Mc-phase to the orthorhombic c-phase at about 428 K. Using a parametric optical constant model, the corresponding dielectric functions as well as the interband optical transitions of the film are determined in the spectral range of 0.73–6.00 eV. Standard critical point (SCP) analysis of the 2nd derivatives of the dielectric functions identified three and four critical points for monoclinic and orthorhombic symmetries, respectively. A systematic redshift of the threshold energies with increasing temperatures was observed.
... The effective medium approximation (EMA) is a prevalent theoretical method in spectroscopic ellipsometry for modeling optical properties of thin films with including their surface roughness. Essentially, EMA streamlines the intricate, microscopically rough surface of a sample into a flat layer described by an effective dielectric function [38]. EMA excels in converting the intricate, microscopic surface irregularities into a more manageable and analyzable form. ...
This study examines the optical properties of thin Cu (Ag)-layered structures covered with protective layers based on graphene, titanium (TiO2), or aluminium (Al2O3) oxides. The objective is to investigate the impact of these coatings on the optical behaviors of underlying metallic layers, specifically in the spectral range of excitation of surface plasmon resonances. Combining the methods of spectroreflectometry and spectro-ellipsometry was used to analyze the optical characteristics of the hybrid metal-oxide-graphene films. The study shows that graphene, due to its exceptional electrical conductivity and unique optoelectronic properties, significantly modifies the optical behavior of investigated structures. It includes notable changes in refractive and absorption indices, and optical conductivity indicating potential for enhancing light-matter interactions in plasmonic-graphene layered structures with the aim to apply as biosensor. It is important that addition of TiO2 and Al2O3 layers has also strong effects on the optical properties, which are relevant to their respective applications in the fields of optoelectronics and microelectronics. Employing the effective medium approximation and the Tauc–Lorentz model promotes deeper understanding the interplay between interband and intraband electronic transitions at the nanoscale level. It was revealed that the layer thickness of constituted materials and their individual dielectric functions together with addition of a graphene monolayer commit the significance for altering the optical properties of hybrid layered structures. The obtained results are important for the fields of plasmonics and nanotechnology, providing insights for designing sensors and devices with improved optical characteristics.
... The effective medium approximation (EMA) model, where the surface roughness layer is replaced by a single homogeneous and flat layer, is usually studied and applied by Sensors 2024, 24, 1242 2 of 12 researchers to correct for the influence of rough surfaces in ellipsometry [12][13][14]. The root mean square (rms) roughness σ and the correlation length τ are used to characterize the morphologies of randomly rough surfaces. ...
In this work, we discuss the precision of the effective medium approximation (EMA) model in the data analysis of spectroscopic ellipsometry (SE) for solid materials with micro-rough surfaces by drawing the regime map. The SE parameters ψ (amplitude ratio) and Δ (phase difference) of the EMA model were solved by rigorous coupled-wave analysis. The electromagnetic response of the actual surfaces with micro roughness was simulated by the finite-difference time-domain method, which was validated by the experimental results. The regime maps associated with the SE parameters and optical constants n (refractive index) and k (extinction coefficient) of the EMA model were drawn by a comparison of the actual values with the model values. We find that using EMA to model micro-rough surfaces with high absorption can result in a higher precision of the amplitude ratio and extinction coefficient. The precisions of ψ, Δ, n and k increase as the relative roughness σ/λ (σ: the root mean square roughness, λ: the incident wavelength) decreases. The precision of ψ has an influence on the precision of k and the precision of Δ affects the precision of n. Changing σ alone has little effect on the regime maps of the relative errors of SE parameters and optical constants. A superior advantage of drawing the regime map is that it enables the clear determination as to whether EMA is able to model the rough surfaces or not.
... Several methods have been employed to solve ISPs, such as library search [11], non-linear regression [12], and heuristic optimization algorithms [13], [14]. However, these methods require numerical electromagnetic (EM) solvers, such as rigorous coupled wave analysis [15] (RCWA) method, finite-difference time-domain method [14], and finite element method [16], which are time-and resource-consuming. With the rapid development of the semiconductor industry, nanostructures in electronic devices are becoming increasingly complex [17], necessitating a highly efficient scatterometry technique for in-situ measurement. ...
As a non-destructive and rapid technique, optical scatterometry has gained widespread use in the measurement of film thickness and optical constants. The recent advances in deep learning have presented new and powerful approaches to the resolution of inverse scattering problems. However, the application of deep-neural-network-assisted optical scatterometry for nanostructures still faces significant challenges, including poor stability, limited functionalities, and high equipment requirements. In this paper, a novel characterization method is proposed, which employs deep-neural-network-assisted ellipsometry to address these challenges. The method processes ellipsometric angles, which are measured by basic ellipsometers, as functional signals. A comprehensive model is developed to profile nano-gratings fabricated by diverse techniques, by incorporating rounded corners, residual layers, and optical constants into an existing model. The stability of the model is enhanced by implementing several measures, including multiple sets of initial values and azimuth-resolved measurements. A simple compensation algorithm is also introduced to improve accuracy without compromising efficiency. Experimental results demonstrate that the proposed method can rapidly and accurately characterize nano-gratings fabricated by various methods, with relative errors of both geometric and optical parameters well controlled under 5 %. Thus, the method holds great promise to serve as an alternative to conventional characterization techniques for in-situ measurement.
... This approach was first shown in [8], where Aspnes et al. split the roughness layer into sublayers and developed the corresponding effective medium models. This method has also been used in analyzing spectroscopic ellipsometry measurements [15][16][17][18] of rough surfaces. However, in all cases, the EMT was static, without the flexibility to be adjusted to different shapes and slopes. ...
Spectroscopic reflectance is a versatile optical methodology for the characterization of transparent and semi-transparent thin films in terms of thickness and refractive index. The Fresnel equations are used to interpret the measurements, but their accuracy is limited when surface roughness is present. Nanoroughness can be modelled through a discretized multi-slice and effective medium approach, but to date, this offered mostly qualitative and not quantitative accuracy. Here we introduce an adaptive and nonlocal effective medium approach, which considers the relative size and environment of each discretized slice. We develop our model using finite-difference time-domain simulation results and demonstrate its ability to predict nanoroughness size and shape with relative errors < 3% in a variety of test systems. The accuracy of the model is directly compared to the prediction capabilities of the Bruggeman and Maxwell–Garnett models, highlighting its superiority. Our model is fully parametrized and ready to use for exploring the effects of roughness on reflectance without the need for costly 3D simulations and to be integrated into the Fresnel simulator of spectroscopic reflectance tools.
... In contrast, effective medium approximations (EMAs) offer a more computationally inexpensive solution to the problem while maintaining a reasonable level of accuracy. 27,28 Among two of the widely popular EMAs, i.e., Maxwell-Garnet 29 and Bruggeman effective medium approximation (BEMA), 30 the latter one, when combined with the anisotropic description of the constituents [termed as anisotropic Bruggeman effective medium approximation (ABEMA)], is found to be very effective in characterizing highly ordered roughness of slanted columnar thin films. 31,32 Though ABEMA was developed originally for these ordered surface features, it is found very useful in describing the optical features of gratings with randomly rough edges. ...
Any degree of surface roughness could play a significant role in determining the optical properties of ultra-thin films required for epsilon-near-zero (ENZ) applications. In this report, we have provided a systematic analysis of the evolution of an ENZ mode with increasing surface roughness values and established both experimentally and theoretically that roughness acts as a supporting mechanism for achieving a strong ENZ plasmon resonance response in randomly rough indium tin oxide thin films. For pulsed laser deposited indium tin oxide thin films, ENZ plasmon-mediated absorption is enhanced monotonically with the increasing surface roughness. A value of 99.75%, depicting near-perfect absorption, at a wavelength of 1335 nm for the incidence angle of 50° is demonstrated experimentally via Kretschmann–Raether configuration for the film with the highest surface roughness. A modified transfer matrix method based on the anisotropic Bruggemann effective medium approximation is being used to effectively simulate the experimental spectra, and based on this analysis, an even higher absorption is predicted at lower angles outside the experimentally viable domain. Such a high value of absorption just above the ENZ wavelength is due to the strong electric field enhancement inside the film layer, while in terms of absorption loss, surface roughness leads the way and contributes immensely toward the occurrence of perfect absorption in the collective media. Modification of the ENZ mode dispersion in the presence of a surface roughness layer is also discussed, and observed perfect absorption is recognized as the outcome of the crossover between the internal damping and radiation damping terms.
... Ellipsometry is a promising metrology tool for the next generation of semiconductor devices [1], and a major tool in new material characterization [2][3][4]. In addition, the technology has been applied in biomedical detection [5,6] and surface roughness estimation [7]. The working principles of ellipsometry are as follows. ...
Ellipsometry is an important metrology tool in a plethora of industries. The measurement accuracy can be significantly affected by the existence of Poisson-Gaussian mixed noise. This paper quantifies the induced error on normalized Mueller matrix measurements through statistical analysis. A method is then proposed to mitigate the effects of Poisson-Gaussian noise in spectroscopic ellipsometry signal demodulation, based on maximum likelihood estimation. The noise is characterized through experiments on an in-house setup. The improved performance of dimension reconstruction from the proposed method is demonstrated through simulations.
... In order to obtain the thickness and dielectric function of the Au film, a fivelayer optical model consisting of surface roughness layer/Au layer/Cr layer/SiO 2 layer/Si substrate is used in the fitting process as shown in Fig. 1a. The surface roughness layer can be depicted by the effective medium approximation (EMA) theory [19]. The Drude and critical points (DCP) [20] model is utilized to obtain the thickness and dielectric function of Au films, which can be expressed as: ...
A smooth gold (Au) film with thickness below 10 nanometer (nm) is hard to fabricate as well as to accurately measure its thickness and the corresponding dielectric function. Here, we report 5.4, 6.6 and 7.5 nm thick continuous Au films prepared on Chromium (Cr) seed layer. The thickness and dielectric function of the Au films are obtained using spectroscopic ellipsometry and first principles calculation. From the fitting results of the ellipsometric parameters, the value of the real part of dielectric function () is negative almost in the whole spectrum region indicating that the Au films are continuous. For the imaginary part of dielectric function (), it decreases with increasing of the Au film thickness because the surface electrons scattering decreases. Moreover, the calculated and measured results of 5.4, 6.6 and 7.5 nm thick Au films present a good agreement in the wavelength range from 400 to 1600 nm. From the results of the first principles calculation, both and decrease with increasing of the Au film thickness. These precise measurement and calculation results of dielectric function are beneficial to nano-photoelectronic devices design with sub-10 nm Au films involved.
... it is well known that the relation between EMA effective roughness layer and S q should be quadratic [43], but voxel material averaging gives a contribution linear in S q . ...
Surface and volume imperfections can significantly affect the performance of nanoscale or microscale devices used in photonics, optoelectronics or scientific instrumentation. In this article we present an open source software package for Finite-Difference Time-Domain electromagnetic field calculations suitable for calculations on graphics cards. Its special features include handling realistic models of imperfect nanoscale objects, such as treatment of arbitrary geometries including addition of random roughness to any geometrical object. The method is compared to conventional optical approach represented by Rayleigh-Rice theory. Practical applicability is demonstrated using a calculation of variation of field enhancement at proximity of a rough nanoscale antenna and rough particle scattering. It is shown that such approach can be namely useful in the areas where many repeated calculations are necessary, e.g. when studying how the optical response of nanoscale objects can vary when they are rough.
Program summary
Program Title: GSvit
CPC Library link to program files: https://doi.org/10.17632/k424zbsxnk.1
Licensing provisions: GPLv2
Programming language: C
Nature of problem: If we want to analyse impact of random imperfections, in particular surface roughness, on optical response of nanoscale and microscale objects, we need to run many calculations with different random realisations. GSvit is a general electromagnetic field solver optimized for running such calculations, via fast computing on graphics cards and algorithms for loading of arbitrary data and modification of their geometry to construct randomly rough surfaces and interfaces.
Solution method: Finite-Difference Time-Domain method implemented on computer processor and on graphics card, with advanced pre-processing in order to add realistic roughness to 3D objects.
Additional comments including restrictions and unusual features: The current version of software, including the documentation and downloadable examples can be found on http://gsvit.net/.
... For rectangular gratings, the topographical parameters become the depth h, the line width b, and the period Λ. When the quantities of those variables are known, Ψ and Δ can be obtained from the first-principles calculation methods of electromagnetic scattering, such as the rigorous coupled-wave analysis (RCWA) [32,33], the finiteelement method (FEM) [34], and the finite-difference time-domain (FDTD) method [23,35]. In this work, the FDTD method was employed to simulate the interaction of the electromagnetic wave with Gaussian distributed randomly micro-rough surfaces, while RCWA was used to calculate the electromagnetic response of rectangular gratings. ...
The effective medium approximation (EMA) model may cause a large deviation in the data analysis of spectroscopic ellipsometry (SE) for solid materials with randomly micro-rough surfaces since it ignores the influence of the lateral irregularities of the rough surfaces on the electromagnetic scattering. In this work, a novel inversion framework is developed to extract optical constants from the SE parameters for solid materials with randomly micro-rough surfaces. Our approach enables the integration of the Levenberg-Marquardt optimization algorithm and the first-principles calculations of electromagnetic scattering. In each iterative step, the electromagnetic interactions with rough surfaces are accurately obtained from first-principles calculations without using the EMA model for rough estimation, which significantly guarantees the precision and wide applicability of our method for actual surfaces without a perfectly Gaussian height distribution. Furthermore, a superior advantage of our approach is that its error can be feasibly evaluated from the instrumental errors of the surface morphology detectors and the SE.
... In many cases the boundary roughness cannot be neglected and the inclusion of this defect into the structural models used in the optical characterization is necessary. The effective medium approximation (EMA) [1][2][3][4][5][6], the Rayleigh-Rice theory (RRT) [7][8][9][10][11][12][13][14] and the scalar diffraction theory (SDT) [15][16][17][18][19][20][21][22] are often used for this purpose. The EMA is applicable if the heights and lateral dimensions of the roughness are small, the RRT is usable for the roughness with small slopes and the heights of irregularities much smaller than the wavelength of light, the SDT can be utilized if the roughness can be viewed as locally smooth. ...
An efficient and numerically stable method for calculating the optical quantities of multi-layer systems with slightly rough boundaries using the second order Rayleigh-Rice theory is developed. It is assumed that the mean planes of the boundaries are parallel and all the media forming the system are nonmagnetic, isotropic and homogeneous. The perturbation series is formulated using the four-dimensional formalism inspired by the Yeh matrix formalism, but the final result is written using the two-dimensional formalism which is more efficient for the numerical calculations. The final formulae, which are expressed using an arbitrary power spectral density function (PSDF), include the mixing between the p and s polarizations occurring for anisotropic roughness. Although in the general case the calculation of optical quantities requires evaluation of double integrals, it is shown that for the PSDF given by the isotropic Gaussian function some integrals can be calculated analytically and only single integrals have to be evaluated numerically. The random roughness of boundaries is a defect that occurs frequently in practice, and it must be taken into account in the optical characterization and synthesis of thin film systems exhibiting this defect. The presented method is suitable for these purposes, since both of the mentioned applications require methods that are very fast.
... Article μm 2 scans for the 4−40 and 123 min growth duration samples, respectively, are plotted in the second panel of Figure 8. Due to the nature of the effective medium approximation, SE is most sensitive to roughness between the atomic scale and a tenth of the wavelength of the probing light, 27 with finite element method simulations showing the breakdown of the approximation as the roughness increases. 27,59 In comparison, AFM offers greater sensitivity to larger undulations of the order of a micron and reduces the surface texture down to a single parameter. 27,60 Correlating the two characterization methods, SE and AFM studies on hydrogenated amorphous Si produced with plasma-enhanced CVD have demonstrated a relationship between the thickness of the roughness layer deduced with SE (t SE ) and the RMS roughness obtained with AFM (R RMS ) of the form t SE ≈ 1.5R RMS + 0.4 nm for 1 ≤ t SE < 10 nm. ...
With the retention of many of the unrivaled properties of bulk diamond but in thin-film form, nanocrystalline diamond (NCD) has applications ranging from micro-/nano-electromechanical systems to tribological coatings. However, with Young’s modulus, transparency, and thermal conductivity of films all dependent on the grain size and nondiamond content, compositional and structural analysis of the initial stages of diamond growth is required to optimize growth. Spectroscopic ellipsometry (SE) has therefore been applied to the characterization of 25–75 nm thick NCD samples atop nanodiamond-seeded silicon with a clear distinction between the nucleation and bulk growth regimes discernable. The resulting presence of an interfacial carbide and peak in nondiamond carbon content upon coalescence is correlated with Raman spectroscopy, whereas the surface roughness and microstructure are in accordance with values provided by atomic force microscopy. As such, SE is demonstrated to be a powerful technique for the characterization of the initial stages of growth and hence the optimization of seeding and nucleation within films to yield high-quality NCD.
... This method models the sample as a stratified media, incorporating plane wave propagation and multiple reflections and refractions at the layer interfaces (described by the Fresnel equations or for the more general anisotropic case by the 4 × 4 matrix method of Berreman) [23]. Other simulation approaches, like the finiteelement method [24][25][26], the finite-difference time-domain method [27] or the rigorous coupled-wave analysis [26,28] are yet not very wide spread in the ellipsometric community, due to extreme requirements in computational power or limited software implementations (i.e. different specific problems would require different specific, highly complex, model approaches). ...
Columnar mesoporous Si thin films and dense nanowire (SiNW) carpets were investigated by spectroscopic ellipsometry in the visible-near-infrared wavelength range. Porous Si layers were formed by electrochemical etching while structural anisotropy was controlled by the applied current. Layers of highly oriented SiNWs, with length up to 4.1 μm were synthesized by metal-assisted chemical etching. Ellipsometric spectra were fitted with different multi-layered, effective medium approximation-based (EMA) models. Isotropic, in-depth graded, anisotropic and hybrid EMA models were investigated with the help of the root mean square errors obtained from the fits. Ellipsometric-fitted layer thicknesses were also cross-checked by scanning electron microscopy showing an excellent agreement. Furthermore, in the case of mesoporous silicon, characterization also revealed that, at low current densities (<100 mA/cm²), in-depth inhomogeneity shows a more important feature in the ellipsometric spectra than anisotropy. On the other hand, at high current densities (>100 mA/cm²) this behaviour turns around, and anisotropy becomes the dominant feature describing the spectra. Characterization of SiNW layers showed a very high geometrical anisotropy. However, the highest fitted geometrical anisotropy was obtained for the layer composed of ∼1 μm long SiNWs indicating that for thicker layers, collapse of the nanowires occurs.
The chemical structure was analyzed by spectrophotometer measurements and models were developed based on the Swanepoel method using transmittance, absorbance, and reflectance spectra. The Swanepoel model obtained the optical constants (n – refractive index, and k – extinction coefficient) and the thickness (d).KeywordsOptical and chemical propertiesUV-VIS spectroscopySpectral transmissionReflections spectraSwanepoel methodRefractive indexExtinction coefficientBand-gapUrbach energyAbsorption coefficientDielectric constantOpenFiltersAntireflection coatingOptical conductivitySpectro-ellipsometry methodInfrared spectroscopy (FT-IR)Raman spectroscopy
This paper focuses on one specific alloy among the wide varied family of semiconductors. From III-N group we chose to add scandium percentages to aluminum nitride, and to study the influence of scandium element on the variation of parameters such as permittivity (epsilon) the idea is to use it as a thin film prepared by co-sputtering and characterized by ellipsometry spectroscopy and XRD.
Moreover, in order to show the effect of scandium ratio on the relative permittivity we use effective medium approximation more specifically plan layer effective medium and Maxwell garnet method that is the most adapted to be used with ellipsometry spectroscopy.
Results of the optical characterization of randomly rough silicon surfaces covered with native oxide layers based on processing experimental data obtained by ellipsometry and reflectometry are presented. It is shown that the Rayleigh‐Rice theory is suitable theoretical approach for characterizing micro‐rough surfaces in contrast to effective medium approximation. Combination of the Rayleigh‐Rice theory and scalar diffraction theory is efficient and reliable approach for characterizing rougher surfaces with the rms values of heights larger than 10 nm. Thickness of native oxide layers and roughness parameters, ie, the rms values of heights and autocorrelation lengths, are determined for micro‐rough and rougher surfaces using the corresponding theoretical approaches.
The modification of the effective medium approximation for randomly microrough surfaces covered by very thin overlayers based on inhomogeneous fictitious layers is formulated. The numerical analysis of this modification is performed using simulated ellipsometric data calculated using the Rayleigh–Rice theory. The system used to perform this numerical analysis consists of a randomly microrough silicon single crystal surface covered with a SiO2 overlayer. A comparison to the effective medium approximation based on homogeneous fictitious layers is carried out within this numerical analysis. For ellipsometry of the system mentioned above the possibilities and limitations of both the effective medium approximation approaches are discussed. The results obtained by means of the numerical analysis are confirmed by the ellipsometric characterization of two randomly microrough silicon single crystal substrates covered with native oxide overlayers. It is shown that the effective medium approximation approaches for this system exhibit strong deficiencies compared to the Rayleigh–Rice theory. The practical consequences implied by these results are presented. The results concerning the random microroughness are verified by means of measurements performed using atomic force microscopy.
A Monte Carlo calculation for light intensities scattered from a random Gaussian-correlated surface is presented for the first time. It is shown that small randomness on a grating surface can considerably change the intensities and, in particular, the surface polariton resonances. These results should be used to check perturbation-theory calculations.
An overview on recent applications of the finite-element method
Maxwell-solver JCMsuite to simulation tasks in nanooptics is given. Numerical
achievements in the fields of optical metamaterials, plasmonics, photonic
crystal fibers, light emitting devices, solar cells, optical lithography,
optical metrology, integrated optics, and photonic crystals are summarized.
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.
Optical constant spectra for silicon and thermally grown silicon dioxide have been simultaneously determined using variable angle of incidence spectroscopic ellipsometry from 0.75 to 6.5 eV. Spectroscopic ellipsometric data sets acquired at multiple angles of incidence from seven samples with oxide thicknesses from 2 to 350 nm were analyzed using a self-contained multi-sample technique to obtain Kramers–Kronig consistent optical constant spectra. The investigation used a systematic approach utilizing optical models of increasing complexity in order to investigate the need for fitting the thermal SiO2 optical constants and including an interface layer between the silicon and SiO2 in modeling the data. A detailed study was made of parameter correlation effects involving the optical constants used for the interface layer. The resulting thermal silicon dioxide optical constants were shown to be independent of the precise substrate model used, and were found to be approximately 0.4% higher in index than published values for bulk glasseous SiO2. The resulting silicon optical constants are comparable to previous ellipsometric measurements in the regions of overlap, and are in agreement with long wavelength prism measurements and transmission measurements near the band gap. © 1998 American Institute of Physics.
In this theoretical paper, formulae for important optical quantities of single layers with slightly randomly rough boundaries are derived by means of a generalized Rayleigh–Rice theory. Thus the formulae for the specular reflectances and ellipsometric parameters of the layers mentioned are presented. The theoretical results are illustrated by a numerical analysis. Practical features implied by this analysis to be relevant from the experimental point of view are introduced as well. Moreover, relations expressing the flux of scattered light are presented.
In the framework of a recent EU funded research project (EUROSEISRISK Seismic hazard assessment, site effects s soil-structure interaction studies in an instrumented basin), a bridge pier model was constructed, instrumented and tested in the EUROSEISTEST experimental site (http://euroseis.civil.auth.gr), located close to Thessaloniki in Greece. The prior aim is the experimental investigation of the dynamic characteristics of the model, the study of the soil-structure-interaction effects, and in particular the wave fields emanating from the oscillating structure to the surrounding ground; to accomplish this task a well-designed set of free-vibration tests were conducted. Experimental results were compared with careful 3D numerical simulations of the soil-foundation structure system, in the frequency and time domain. Several fundamental aspects of SSI are discussed and the available analytical impedance expressions are compared with the experimental and numerical results of the present study. The study of the dynamic behavior of a simple SDOF system consisting of a model bridge pier with surface foundation in real soft soil conditions and the numerical FE modeling of the experiments, enable us to enhance our knowledge on various soil-structure interaction aspects.
Ellipsometry, in general, is one of the most sensitive spectroscopic techniques nowadays in both macro-and nano-scale research. Advantages like absolute measurements (without need for references), high precision, non-destructive character, easy real-time monitoring and many possible applications make Imaging Ellipsometry and Spectroscopic Ellipsometry unavoidable tools in modern research practice. Still, one of themain disadvantages of ellipsometry concerns the fact that an optical model is required in order to define constituents, the structure and morphology of each sample. In most cases, a good optical model that properly describes the structure is not possible to make without additional information like the number of layers, their structure, surface roughness or type of interfaces. For this reason we need additional measurements that corroborate ellipsometric ones in order to construct a valid optical model.We call thesemeasurements CorrelationMeasurements and they are the topic of this communication. We describe the most frequently used ones like SEM, AFM, STM, Raman or FTS, and we also mention new trends that combine spectroscopic or imaging ellipsometers with one or more correlation techniques in the same instrument.
Real-time spectroscopic ellipsometry (SE) has been applied to
investigate the nucleation and grain growth processes in
microcrystalline silicon (μc-Si:H) thin films deposited by a
conventional plasma-enhanced chemical vapor deposition using hydrogen
dilution of silane source gas. Real-time SE results revealed the
μc-Si:H nucleation from hydrogenated amorphous silicon (a-Si:H)
phase, followed by the coalescence of isolated μc-Si:H grains exposed
on growing surfaces. In the μc-Si:H grain growth process, the
μc-Si:H shows an enhanced surface roughening. The onset of the
μc-Si:H grain growth and the coalescence of μc-Si:H grains were
readily characterized by monitoring surface roughness evolution. We
found that a μc-Si:H nuclei density increases significantly as the
hydrogen dilution ratio R=([H2]/[SiH4]) increases.
In contrast, a film thickness at which most of the surface is covered
with the μc-Si:H, gradually reduces with increasing R. The real-time
SE results described above showed remarkable agreement with those
estimated by transmission electron microscopy and atomic force
microscopy. For the a-Si:H/μc-Si:H mixed-phase surface formed during
the phase transition, however, the SE results showed relatively large
errors in the analyses. Such difficulties in the real-time SE analysis
for the μc-Si:H thin film are discussed.
Real-time spectroscopic ellipsometry (SE) data collected during the
nucleation and growth of hydrogenated amorphous silicon (a-Si:H) thin
films have been analyzed by applying one and two layer optical models
incorporating different effective medium theories (EMT's). The purpose
of the EMT's is to simulate the dielectric functions of the
microscopically inhomogeneous nucleating and surface roughness layers
used in the models. Five one-parameter EMT's have been considered in
this study for the characterization of three classes of microscopically
inhomogeneous layers, including (i) 5-20 Å-thick nucleating layers
consisting of isolated a-Si:H clusters on the underlying substrate, (ii)
10-15 Å-thick nucleation-induced surface roughness layers on very
thin (<200 Å) a-Si:H films, and (iii) 40-80 Å-thick
substrate-induced surface roughness layers on thicker (>2500 Å)
a-Si:H films. In all three applications, the Bruggeman effective medium
approximation (EMA) provides the best overall fits to the time evolution
of the SE data, and complexities beyond the simple one-parameter EMA
cannot be justified in view of existing experimental limitations.
Furthermore, many of the general features of nucleation, coalescence,
and bulk layer growth deduced in the SE analysis and used in previous
studies to understand and optimize materials and device fabrication, are
found to be essentially independent of the EMT used in the analysis.
Polysilicon layers prepared by low-pressure chemical vapor deposition at 560°C, 620°C, 660°C, and 700°C were measured by Atomic Force Microscopy (AFM) and Spectroscopic Ellipsometry (SE). Morphology, cross-sectional profile, roughness spectral density, and roughness of the surfaces were investigated by AFM using window sizes of 1×1 μm2, 10×10 μm2, and 50×50 μm2. The layer structure and the surface roughness were determined by SE using the Bruggemann-Effective Medium Approximation (B-EMA). The Root Mean Square (RMS) and mean square (Ra) roughness values measured by AFM were compared to the thickness of the top layer of the SE model describing the surface roughness. Although AFM results depend on the used window size, good correlation was found between the roughness values determined by AFM and SE for each window sizes. The results show that SE calibrated with AFM could be used for quantitative surface roughness determination.
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.
Measurements of Si surface roughness by atomic force microscopy and ellipsometry have been performed over a wide range of conditions. Advanced methods of data analysis have been applied to both techniques leading to a quantitative comparison of root-mean-square (rms) roughness to the ellipsometric paramter Δ. Differences in Δ are observed for surfaces with the same rms roughness, but different roughness spectral densities, as expected from theory.
The modeling of surface and interface roughness is a key issue in the interpretation of ellipsometric measurements. Materials properties are often extracted from ellipsometry measurements in an indirect way by modeling the optical response of the material. Since roughness is known to affect the scattering of light on an interface, how roughness is incorporated into these models can affect the outcome of the fitting procedure.
Using Rayleigh-Rice scattering theory we have studied the influence of surface morphology on the optical response of self-affine surfaces. We have established a mathematical relationship between the surface roughness (d) as determined by spectroscopic ellipsometry (SE) using the effective medium approximation (EMA) and the parameters controlling the morphology of the surface: root-mean-square roughness (w), correlation length (ξ), and roughness (Hurst) exponent (α). These three parameters affect the roughness value measured by ellipsometry. However, when the correlation length is smaller than the wavelength, the dependence is contained in a single parameter wδ that is proportional to the product of the surface roughness and the local slope δ=w/wξα ξα. The fact that the local slope of a surface increases only very slowly during growth explains the linear dependence experimentally found between w as measured by scanning-probe microscopy and the vertical roughness determined by the effective medium approach.
Using measured dielectric function data from 2.1 to 5.5 eV for chemical-vapor-deposition—grown smooth amorphous (a-Si) and microscopically rough fine-grained polycrystalline (p-Si) films, we show that the dielectric properties of microscopically rough layers of thicknesses 100-500 ÅA are accurately modeled in the effective-medium approximation. These microscopically rough layers show essentially no macroscopic light scattering, and thus are inaccessible to measurement by usual scattering techniques. The unambiguous identification of microscopic roughness, as opposed to, e.g., an overlying oxide, is shown to require a spectroscopic capability. Statistical-analysis techniques are introduced to determine model parameters systematically and objectively, and also to establish correlations and confidence limits that show which parameters are defined by the data and which are statistically indeterminate. A best-fit five-parameter model for the sample with the thickest surface region shows that the density profile is characteristic of hemispherical, not pyramidal, irregularities. This indicates that surface roughness arises from a three-dimensional nucleation and growth process in these samples. In a comparison of the three one-parameter effective-medium models, Bruggeman and Maxwell Garnett(2) theories are found to adequately represent the data, while the Lorentz-Lorenz model, previously used exclusively to model roughness in single-wavelength applications, predicts only qualitatively the spectral dependence and gives poor results.
We investigate the evolution of surface morphology during hot-wire chemical vapor deposition of amorphous silicon films onto rough substrates. Using in situ spectroscopic ellipsometry, we find that the surface smoothens as the film grows. However, postdeposition atomic force microscopy reveals that the roughness is actually increasing linearly. We resolve this discrepancy by examining the power spectrum densities of the atomic force images, which indicate that the growth surface experiences both short-range smoothening and global roughening. The ellipsometry data are consistent with the short-range atomic force microscopy data, but they exclude information about the long-range components of roughness. The slope of the power spectrum density indicates surface diffusion is the dominant smoothening mechanism; the linear increase in roughness is consistent with columnar growth caused by self-shadowing.
In this paper, the theoretical analysis of the correctness of applying the effective medium approximation (EMA) at the ellipsometric studies of rough surfaces is presented. Within this analysis the Rayleigh–Rice theory (RRT) is used to calculate the simulated ellipsometric data of various slightly randomly rough surfaces. This simulated data are treated using the least-squares method for finding the values of the parameters characterizing the rough surfaces within the EMA, i.e., the values of thickness and packing density factor describing a fictitious (effective) thin films replacing these surfaces within the EMA. It is shown that the EMA could be used to interprete the ellipsometric data of the rough surfaces in a correct way if their roughness only contains the high spatial frequencies. For the low spatial frequencies it is shown that the influence of the surface roughness on the ellipsometric quantities is small in contrast to its influence on the reflectance. Moreover, it is shown that the EMA must be used carefully for optical characterizing rough surfaces and that it is more reasonable to use the RRT for this purpose. It is also presented that in general it is impossible to expect a good agreement between the thickness values determined using EMA within the ellipsometric analysis and the rms values of the heights of the irregularities evaluated by atomic force microscopy (AFM) for the same rough surfaces.
In this work, we investigated static and dynamic aspects of the rms local surface slope "ρ" for self-affine random surfaces. The rms local slope is expressed as a function of the rms roughness amplitude σ, the in-plane correlation length ξ, and the roughness exponent H (0<H<1), as well as is shown to scale as ρ∼σξ-H. Application to room temperature heteroepitaxial silver films shows the rms local slope to be closely time invariant in the thickness range 10<h<1000 nm with an asymptotic value ρ≈0.7. However, discrepancies in deposition details could alter the mode of film growth leading to a power law growth of the local slope as a function of the film thickness h; ρ∝hc (c>0).
Relationship between surface morphology and efhttp
- A Yanguas-Gil
- H Wormeester
A. Yanguas-Gil, H. Wormeester, Relationship between surface morphology and efhttp://dx.doi.org/10.1007/978-3-642-33956-1.
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