Sensitivity analysis for indirect measurement in scatterometry and the reconstruction of periodic grating structures

In this work, we discuss some aspects of numerical algorithms for the determination of periodic surface structures (gratings) from light diffraction patterns. With decreasing structure details of lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method are applied to simple periodic line structures in order to evaluate the quality of the manufacturing process. Using scatterometry, geometrical parameters of periodic structures including period (pitch), side-wall angles, heights, top and bottom widths of trapezoid shaped bridges can be determined. The mathematical model for the scattering is based on the time-harmonic Maxwell's equations and reduces in case of grating structures to the Helmholtz equation. For the numerical simulation, e.g. finite element methods can be applied to solve the corresponding boundary value problems. More challenging is the inverse problem, where the grating geometry is to be reconstructed from the measured diffraction patterns. Restricting the class of gratings and the set of measurements, the inverse problem can be reformulated as a non-linear operator equation in Euclidean spaces. The operator maps the parameters describing the grating to special efficiencies of plane wave modes diffracted by the grating. We employ a Newton type iterative method to solve this operator equation. The reconstruction properties and the convergence of the numerical algorithm, however, is controlled by the local conditioning of the non-linear mapping, i.e. by the condition numbers of its Jacobian matrix. To improve the convergence of the iteration and the accuracy of the reconstruction, we determine optimal sets of efficiencies for the measurements by optimizing the condition numbers of the corresponding Jacobians. Numerical examples for a chrome-glass mask and for an inspecting light of wave length 632.8 nm confirm that an optimization of the measurement data results in better solutions

Profile reconstruction in EUV scatterometry: Modeling and uncertainty estimates

Scatterometry as a non-imaging indirect optical method in wafer metrology is also relevant to lithography masks designed for Extreme Ultraviolet Lithography, where light with wavelengths in the range of 13 nm is applied. The solution of the inverse problem, i.e. the determination of periodic surface structures regarding critical dimensions (CD) and other profile properties from light diffraction patterns, is incomplete without knowledge of the uncertainties associated with the reconstructed parameters. With decreasing feature sizes of lithography masks, increasing demands on metrology techniques and their uncertainties arise. The numerical simulation of the diffraction process for periodic 2D structures can be realized by the finite element solution of the two-dimensional Helmholtz equation. For typical EUV masks the ratio period over wave length is so large, that a generalized finite element method has to be used to ensure reliable results with reasonable computational costs. The inverse problem can be formulated as a non-linear operator equation in Euclidean spaces. The operator maps the sought mask parameters to the efficiencies of diffracted plane wave modes. We employ a Gau{\ss}-Newton type iterative method to solve this operator equation and end up minimizing the deviation of the measured efficiency or phase shift values from the calculated ones. We apply our reconstruction algorithm for the measurement of a typical EUV mask composed of TaN absorber lines of about 80 nm height, a period of 420 nm resp.~720 nm, and with an underlying MoSi-multilayer stack of 300 nm thickness. Clearly, the uncertainties of the reconstructed geometric parameters essentially depend on the uncertainties of the input data and can be estimated by various methods. We apply a Monte Carlo procedure and an approximative covariance method to evaluate the reconstruction algorithm. Finally, we analyze the influence of uncertainties in the widths of the multilayer stack by the Monte Carlo method

Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry

We investigate the impact of line edge and line width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The 2D rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method (FEM) providing a numerical solution of the two-dimensional Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using the LER/LWR perturbed efficiency datasets for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values

Got milk? Breastfeeding and milk analysis of a mother on chronic hemodialysis

Purpose: Women on dialysis rarely become pregnant. However, the overall rate of successful pregnancies is increasing in this patient population and breastfeeding becomes an option for mothers on dialysis. In this study we performed a systematic breast milk composition analysis of a mother on chronic hemodialysis (HD). Methods: Specimens of breast milk and blood were collected in regular intervals before and after HD from a 39-year old woman starting on day 10 postpartum. Samples were analyzed for electrolytes, retention solutes, nutrients and other laboratory measurements. Breast milk samples from low-risk mothers matched for postpartum age were used as controls. Results: Significantly higher levels of creatinine and urea were found in pre-HD breast milk when compared to post-HD. A similar post-dialytic decrease was only found for uric acid but not for any other investigated parameter. Conversely, sodium and chloride were significantly increased in post-HD samples. Compared to controls creatinine and urea were significantly higher in pre-HD samples while the difference remained only significant for post-HD creatinine. Phosphate was significantly lower in pre- and post-HD breast milk when compared to controls, whereas calcium showed no significant differences. In terms of nutrient components glucose levels showed a strong trend for a decrease, whereas protein, triglycerides and cholesterol did not differ. Similarly, no significant differences were found in iron, potassium and magnesium content. Conclusion: To the best of our knowledge this is the first report on a breastfeeding mother on chronic dialysis. Although we found differences in creatinine, urea, sodium, chloride and phosphate, our general analysis showed high similarity of our patient’s breast milk to samples from low-risk control mothers. Significant variations in breast milk composition between pre- and post-HD samples suggest that breastfeeding might be preferably performed after dialysis treatment. In summary, our findings indicate that breastfeeding can be considered a viable option for newborns of mothers on dialysis

Got milk? Breastfeeding and milk analysis of a mother on chronic hemodialysis

Purpose: Women on dialysis rarely become pregnant. However, the overall rate of successful pregnancies is increasing in this patient population and breastfeeding becomes an option for mothers on dialysis. In this study we performed a systematic breast milk composition analysis of a mother on chronic hemodialysis (HD). Methods: Specimens of breast milk and blood were collected in regular intervals before and after HD from a 39-year old woman starting on day 10 postpartum. Samples were analyzed for electrolytes, retention solutes, nutrients and other laboratory measurements. Breast milk samples from low-risk mothers matched for postpartum age were used as controls. Results: Significantly higher levels of creatinine and urea were found in pre-HD breast milk when compared to post-HD. A similar post-dialytic decrease was only found for uric acid but not for any other investigated parameter. Conversely, sodium and chloride were significantly increased in post-HD samples. Compared to controls creatinine and urea were significantly higher in pre-HD samples while the difference remained only significant for post-HD creatinine. Phosphate was significantly lower in pre- and post-HD breast milk when compared to controls, whereas calcium showed no significant differences. In terms of nutrient components glucose levels showed a strong trend for a decrease, whereas protein, triglycerides and cholesterol did not differ. Similarly, no significant differences were found in iron, potassium and magnesium content. Conclusion: To the best of our knowledge this is the first report on a breastfeeding mother on chronic dialysis. Although we found differences in creatinine, urea, sodium, chloride and phosphate, our general analysis showed high similarity of our patient’s breast milk to samples from low-risk control mothers. Significant variations in breast milk composition between pre- and post-HD samples suggest that breastfeeding might be preferably performed after dialysis treatment. In summary, our findings indicate that breastfeeding can be considered a viable option for newborns of mothers on dialysis

Magnetic screening in the hot gluon system

The gluon transverse self-energy of the pure Yang-Mills system at high-temperature is analysed in the static limit and at fourth order in the coupling. Possible contributions to this function are collected, seen to be gauge-fixing independent subsets and shown to vanish all, except those which are either regulators or constituents of the self-energy of Euklidean 3D Yang-Mills theory at zero temperature. The latter self-energy, in turn, is known from the non-perturbative analysis by Karabali and Nair.Comment: 11 pages, latex, correction of prefactor in (24

CFD analysis for solar chimney power plants

Paper presented at the 9th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Malta, 16-18 July, 2012.Computational Fluid Dynamics (CFD) was employed for investigating Solar Chimney Power Plants (SCPP). The effect of the geometric dimensions on the fluid dynamics and heat transfer was investigated. The thermal efficiency of the collector was found to improve with increasing scale, due to an increase of the heat transfer coefficient. The spread in relevant Reynolds numbers for the collector and chimney was four orders of magnitude from the smallest to the largest scale. Parametric studies were also performed to determine the effect of the distance of the collector from the ground on the power output. An optimum distance was determined for two different scales.dc201