Surface integral

In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration. Surface integrals have applications in physics, particularly with the theories of classical electromagnetism. Surface integrals of scalar fields Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) va

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Optimization of high efficiency silicon heterojunction solar cells using silane-plasma diagnostics

Christophe Ballif, Loris Barraud, Priscille Marguerite Bôle Rothen, Stefaan De Wolf, Antoine Descoeudres

In silicon heterojunction solar cells, the passivation of the crystalline silicon wafer surfaces and fabrication of emitter and back surface field are all performed by intrinsic and doped amorphous silicon thin layers, usually deposited by plasma-enhanced chemical vapor deposition (PECVD). Since the properties of materials deposited by PEVCD are directly linked to the plasma properties, plasma diagnostics are very useful tools to optimize such devices. A novel diagnostic has been developed to measure in-situ the molecular silane depletion fraction in the plasma during deposition. It is found that the silane depletion strongly depends on the process parameters, and appears to be a relevant parameter for the quality of the passivating layers. Good passivation is indeed obtained from highly depleted silane plasmas. Based on this, layers deposited in a large-area PECVD reactor working at very high frequency (40.68 MHz) were optimized for heterojunction solar cells. All other fabrication steps were also fully industry compatible, using sputtering for transparent conductive oxide layers and screenprinting for the front grid. The best 2 x 2 cm2 cell shows a high open-circuit voltage of 717 mV, yielding a conversion efficiency of 20.3% (aperture area). Keywords: Heterojunction, PECVD, High-Efficiency

2010

Silane plasma diagnostics for high-efficiency silicon heterojunction solar cells

Christophe Ballif, Loris Barraud, Priscille Marguerite Bôle Rothen, Stefaan De Wolf, Antoine Descoeudres

2011

Random Surface Covariance Estimation by Shifted Partial Tracing

Tomas Masák, Victor Panaretos

The problem of covariance estimation for replicated surface-valued processes is examined from the functional data analysis perspective. Considerations of statistical and computational efficiency often compel the use of separability of the covariance, even though the assumption may fail in practice. We consider a setting where the covariance structure may fail to be separable locally-either due to noise contamination or due to the presence of a nonseparable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a nonseparable but banded component. We introduce nonparametric estimators hinging on the novel concept of shifted partial tracing, enabling computationally efficient estimation of the model under dense observation. Due to the denoising properties of shifted partial tracing, our methods are shown to yield consistent estimators even under noisy discrete observation, without the need for smoothing. Further to deriving the convergence rates and limit theorems, we also show that the implementation of our estimators, including prediction, comes at no computational overhead relative to a separable model. Finally, we demonstrate empirical performance and computational feasibility of our methods in an extensive simulation study and on a real dataset. Supplementary materials for this article are available online.

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