A New Approximation of Fermi-Dirac Integrals of Order 1/2 by Prony’s Method and Its Applications in Semiconductor Devices

Electronic devices are vital for our modern life. Semiconductor devices are at the core of them. Semiconductor devices are governed by the transport and behavior of electrons and holes which in turn are controlled by Fermi-Level or the Quasi-Fermi Level. The most frequently used approximation for the population of electrons and holes based on the Boltzmann approximation of Fermi-Dirac distribution. However when the Fermi-level is closer to the majority carrier band edge, by less than 3kT, it causes significant errors in the number of the carriers. This in turn causes errors in currents and other quantities of interest. In heavily doped semiconductors, it is desirable to use Fermi-Dirac Integral itself. However this is a tabulated function and therefore approximations are developed. Most of the approximation are mathematically cumbersome and complicated and they are not easily differentiable and integrable. Although several approximations have been developed, some with very high precision, these are not simple nor are they sufficiently useful in semiconductor device applications. In this thesis after exploring and critiquing these approximations, a new set of approximations is developed for the Fermi-Dirac integrals of the order 1/2. This analytical expression can be differentiated and integrated, still maintaining high accuracy. These new approximation is in the form of an exponential series with few terms using Prony’s method. Application of this approximation for semiconductor device calculations are discussed. Substantial errors in carrier densities and Einstein relation are shown when compared with Boltzmann approximation. The efficacy of the approximation in the calculation of Junctionless transistor quantities is demonstrated as an example

A new approximation of Fermi-Dirac integrals of order 1/2 for degenerate semiconductor devices

The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.spmi.2018.03.072 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/There had been tremendous growth in the field of Integrated circuits (ICs) in the past fifty years. Scaling laws mandated both lateral and vertical dimensions to be reduced and a steady increase in doping densities. Most of the modern semiconductor devices have invariably heavily doped regions where Fermi-Dirac Integrals are required. Several attempts have been devoted to developing analytical approximations for Fermi-Dirac Integrals since numerical computations of Fermi-Dirac Integrals are difficult to use in semiconductor devices, although there are several highly accurate tabulated functions available. Most of these analytical expressions are not sufficiently suitable to be employed in semiconductor device applications due to their poor accuracy, the requirement of complicated calculations, and difficulties in differentiating and integrating. A new approximation has been developed for the Fermi-Dirac integrals of the order 1/2 by using Prony's method and discussed in this paper. The approximation is accurate enough (Mean Absolute Error (MAE) = 0.38%) and easy enough to be used in semiconductor device equations. The new approximation of Fermi-Dirac Integrals is applied to a more generalized Einstein Relation which is an important relation in semiconductor devices

Glosarium Matematika

273 p.; 24 cm

Low dimensional supersymmetric field theories on the lattice

In der vorliegenden Arbeit werden verschiedene supersymmetrische Modelle in einer und zwei Raumzeit-Dimensionen untersucht, welche wesentliche Bestandteile von realistischeren Theorien, wie z.B. dem minimalen supersymmetrischen Standardmodell, beinhalten. Durch die separate Untersuchung der einzelnen Aspekte ist es möglich die Vor- und Nachteile der jeweils verwendeten Gittermethoden herauszuarbeiten. Zusätzlich erlaubt die niedrige Dimensionalität sehr präzise numerische Studien, welche konzeptuelle und technische Probleme bei der Behandlung von supersymmetrischen Theorien auf dem Gitter aufdecken können. Am Beginn der Untersuchung von supersymmetrischen Theorien auf dem Gitter steht das pädagogische Beispiel einer supersymmetrischen Quantenmechanik mit dynamisch gebrochener Supersymmetrie. Hieran wird die grundlegende Anwendbarkeit von Gittermethoden auf Theorien mit dynamisch gebrochener Supersymmetrie verifiziert. Am N=2 Wess-Zumino-Modell in 1+1 Dimensionen werden fünf verschiedene Gitterformulierungen verglichen, von denen drei eine explizite Realisierung eines Teils der vollen Supersymmetrie auf dem Gitter darstellen. Die Durchführung von hochpräzisen Messungen stellt selbst in zweidimensionalen Theorien eine große numerische Aufgabe dar. Daher werden die algorithmischen Verbesserungen, die im Verlaufe dieser Arbeit benutzt wurden, am Beispiel des N=2 Wess-Zumino-Modells exemplarisch dargestellt. Als Minimalversion einer supersymmetrischen Feldtheorie mit supersymmetriebrechendem Phasenübergang wird das N=1 Wess-Zumino-Modell in 1+1 Dimensionen analysiert. Die letzte Modellklasse dieser Arbeit bilden (supersymmetrische) nichtlineare Sigma-Modelle. Zunächst wird die Instantonen-Struktur von bosonischen nichtlinearen CP(N)-Sigma-Modellen mit getwisteten Randbedingungen konstruiert. Die Arbeit schließt mit einer Analyse des supersymmetrischen nichtlinearen O(3)-Sigma-Modells auf dem Gitter

Direct computation of elliptic singularities across anisotropic, multi-material edges

We characterise the singularities of elliptic div-grad operators at points or edges where several materials meet on a Dirichlet or Neumann part of the boundary of a two- or three-dimensional domain. Special emphasis is put on anisotropic coefficient matrices. The singularities can be computed as roots of a characteristic transcendental equation. We establish uniform bounds for the singular values for several classes of three- and four-material edges. These bounds can be used to prove optimal regularity results for elliptic div-grad operators on three-dimensional, heterogeneous, polyhedral domains with mixed boundary conditions. We demonstrate this for the benchmark L--shape problem

The Foundations of Infinite-Dimensional Spectral Computations

Spectral computations in infinite dimensions are ubiquitous in the sciences. However, their many applications and theoretical studies depend on computations which are infamously difficult. This thesis, therefore, addresses the broad question, “What is computationally possible within the field of spectral theory of separable Hilbert spaces?” The boundaries of what computers can achieve in computational spectral theory and mathematical physics are unknown, leaving many open questions that have been unsolved for decades. This thesis provides solutions to several such long-standing problems. To determine these boundaries, we use the Solvability Complexity Index (SCI) hierarchy, an idea which has its roots in Smale's comprehensive programme on the foundations of computational mathematics. The Smale programme led to a real-number counterpart of the Turing machine, yet left a substantial gap between theory and practice. The SCI hierarchy encompasses both these models and provides universal bounds on what is computationally possible. What makes spectral problems particularly delicate is that many of the problems can only be computed by using several limits, a phenomenon also shared in the foundations of polynomial root-finding as shown by McMullen. We develop and extend the SCI hierarchy to prove optimality of algorithms and construct a myriad of different methods for infinite-dimensional spectral problems, solving many computational spectral problems for the first time. For arguably almost any operator of applicable interest, we solve the long-standing computational spectral problem and construct algorithms that compute spectra with error control. This is done for partial differential operators with coefficients of locally bounded total variation and also for discrete infinite matrix operators. We also show how to compute spectral measures of normal operators (when the spectrum is a subset of a regular enough Jordan curve), including spectral measures of classes of self-adjoint operators with error control and the construction of high-order rational kernel methods. We classify the problems of computing measures, measure decompositions, types of spectra (pure point, absolutely continuous, singular continuous), functional calculus, and Radon--Nikodym derivatives in the SCI hierarchy. We construct algorithms for and classify; fractal dimensions of spectra, Lebesgue measures of spectra, spectral gaps, discrete spectra, eigenvalue multiplicities, capacity, different spectral radii and the problem of detecting algorithmic failure of previous methods (finite section method). The infinite-dimensional QR algorithm is also analysed, recovering extremal parts of spectra, corresponding eigenvectors, and invariant subspaces, with convergence rates and error control. Finally, we analyse pseudospectra of pseudoergodic operators (a generalisation of random operators) on vector-valued $l^p$ spaces. All of the algorithms developed in this thesis are sharp in the sense of the SCI hierarchy. In other words, we prove that they are optimal, realising the boundaries of what digital computers can achieve. They are also implementable and practical, and the majority are parallelisable. Extensive numerical examples are given throughout, demonstrating efficiency and tackling difficult problems taken from mathematics and also physical applications. In summary, this thesis allows scientists to rigorously and efficiently compute many spectral properties for the first time. The framework provided by this thesis also encompasses a vast number of areas in computational mathematics, including the classical problem of polynomial root-finding, as well as optimisation, neural networks, PDEs and computer-assisted proofs. This framework will be explored in the future work of the author within these settings

High-precision computation of uniform asymptotic expansions for special functions

In this dissertation, we investigate new methods to obtain uniform asymptotic expansions for the numerical evaluation of special functions to high-precision. We shall first present the theoretical and computational fundamental aspects required for the development and ultimately implementation of such methods. Applying some of these methods, we obtain efficient new convergent and uniform expansions for numerically evaluating the confluent hypergeometric functions and the Lerch transcendent at high-precision. In addition, we also investigate a new scheme of computation for the generalized exponential integral, obtaining on the fastest and most robust implementations in double-precision floating-point arithmetic. In this work, we aim to combine new developments in asymptotic analysis with fast and effective open-source implementations. These implementations are comparable and often faster than current open-source and commercial stateof-the-art software for the evaluation of special functions.Esta tesis presenta nuevos métodos para obtener expansiones uniformes asintóticas, para la evaluación numérica de funciones especiales en alta precisión. En primer lugar, se introducen fundamentos teóricos y de carácter computacional necesarios para el desarrollado y posterior implementación de tales métodos. Aplicando varios de dichos métodos, se obtienen nuevas expansiones uniformes convergentes para la evaluación numérica de las funciones hipergeométricas confluentes y de la función transcendental de Lerch. Por otro lado, se estudian nuevos esquemas de computo para evaluar la integral exponencial generalizada, desarrollando una de las implementaciones más eficientes y robustas en aritmética de punto flotante de doble precisión. En este trabajo, se combinan nuevos desarrollos en análisis asintótico con implementaciones rigurosas, distribuidas en código abierto. Las implementaciones resultantes son comparables, y en ocasiones superiores, a las soluciones comerciales y de código abierto actuales, que representan el estado de la técnica en el campo de la evaluación de funciones especiales