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# Effective medium approximation of ellipsometric response from random surface roughness simulated by finite-element method

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... 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. ...
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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. ...
Article
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.
... 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: ...
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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 ($$\varepsilon_{1}$$) is negative almost in the whole spectrum region indicating that the Au films are continuous. For the imaginary part of dielectric function ($$\varepsilon_{2}$$), 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 $$\varepsilon_{1}$$ and $$\varepsilon_{2}$$ 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.
... 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. ...
Article
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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. ...
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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. ...
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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). ...
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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.
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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.
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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.
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Chapter
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.
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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.
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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.
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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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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.
Chapter
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.
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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.
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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.
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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.
Article
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.
Article
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).
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