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

An efficient magnetic tight-binding method for transition metals and alloys

International audienceAn efficient parameterized self-consistent tight-binding model for transition metals usings, p and d valence atomic orbitals as a basis set is presented. The parameters of our tight-binding model for pure elements are determined from a fit to bulk ab-initiocalculations. Avery simple procedure that does not necessitate any further fitting is proposed to deal with systems made of several chemical elements. This model is extended to spin (and orbital) polarized materials by adding Stoner-like and spin–orbit interactions. Collinear and non-collinear magnetism as well as spin-spirals are considered. Finally the electron–electron intra-atomic interactions are taken into account in the Hartree–Fock approximation. This leads to an orbital dependence of these interactions, which is of a great importance for low-dimensional systems and for a quantitative description of orbital polarization and magneto-crystalline anisotropy. Several examples are discussed.Nous présentons un modèle de liaisons fortes paramétré et auto-cohérent utilisant une base d’orbitales atomiques s, p, et d pour décrire les électrons de valence des métaux de transition. Les paramètres du modèle sont déterminés à partir d’un ajustement non linéaire sur des résultats de calculs ab initio d’éléments purs en volume. Notre procédure ne nécessite aucun paramètre ni ajustement supplémentaire pour l’étendre aux systèmes avec plusieurs atomes de natures chimiques différentes. Nous avons généralisé notre modèle aux matériaux présentant une polarisation de spin et orbitale à l’aide de termes de Stoner et de couplage spin–orbite. Nous traitons aussi bien le magnétisme colinéaire que non colinéaire ainsi que les spirales de spin. Enfin nous montrons comment prendre en compte l’interaction électron–électron intra-atomique dans l’approximation de Hartree–Fock. Cela introduit une dépendance orbitale des interactions qui peut s’avérer importante dans les systèmes de basse dimensionalité et pour décrire correctement l’anisotropie magnéto- cristalline et la polarisation orbitale. Nous illustrons notre propos à l’aide de plusieurs exemples

Optimal plug-in Gaussian processes for modelling derivatives

Derivatives are a key nonparametric functional in wide-ranging applications where the rate of change of an unknown function is of interest. In the Bayesian paradigm, Gaussian processes (GPs) are routinely used as a flexible prior for unknown functions, and are arguably one of the most popular tools in many areas. However, little is known about the optimal modelling strategy and theoretical properties when using GPs for derivatives. In this article, we study a plug-in strategy by differentiating the posterior distribution with GP priors for derivatives of any order. This practically appealing plug-in GP method has been previously perceived as suboptimal and degraded, but this is not necessarily the case. We provide posterior contraction rates for plug-in GPs and establish that they remarkably adapt to derivative orders. We show that the posterior measure of the regression function and its derivatives, with the same choice of hyperparameter that does not depend on the order of derivatives, converges at the minimax optimal rate up to a logarithmic factor for functions in certain classes. We analyze a data-driven hyperparameter tuning method based on empirical Bayes, and show that it satisfies the optimal rate condition while maintaining computational efficiency. This article to the best of our knowledge provides the first positive result for plug-in GPs in the context of inferring derivative functionals, and leads to a practically simple nonparametric Bayesian method with optimal and adaptive hyperparameter tuning for simultaneously estimating the regression function and its derivatives. Simulations show competitive finite sample performance of the plug-in GP method. A climate change application for analyzing the global sea-level rise is discussed.Comment: This paper supersedes the second part of the technical report available at arXiv:2011.13967v1. That technical report has been split: The first part on equivalence theory will be extended and become 2011.13967v2. The results on Bayesian inference for function derivatives have evolved into this paper. arXiv admin note: text overlap with arXiv:2011.1396

A rigorous proof of the Bohr-van Leeuwen theorem in the semiclassical limit

The original formulation of the Bohr-van Leeuwen (BvL) theorem states that, in a uniform magnetic field and in thermal equilibrium, the magnetization of an electron gas in the classical Drude-Lorentz model vanishes identically. This stems from classical statistics which assign the canonical momenta all values ranging from $-\infty$ to $\infty$ what makes the free energy density magnetic-field-independent. When considering a classical (Maxwell-Boltzmann) interacting electron gas, it is usually admitted that the BvL theorem holds upon condition that the potentials modeling the interactions are particle-velocities-independent and do not cause the system to rotate after turning on the magnetic field. From a rigorous viewpoint, when treating large macroscopic systems one expects the BvL theorem to hold provided the thermodynamiclimit of the free energy density exists (and the equivalence of ensemble holds). This requires suitable assumptions on the many-body interactions potential and on the possible external potentials to prevent the system from collapsing or flying apart. Starting from quantum statistical mechanics, the purpose of this article is to give, within the linear-response theory, a proof of the BvL theorem in the semiclassical limit when considering a dilute electron gas in the canonical conditions subjected to a class of translational invariant external potentials.Comment: 50 pages. Revised version. Accepted for publication in R.M.

Anisotropy of thermoelectric power in bismuth telluride

"January 15, 1961." "This report is identical with a thesis submitted to the Department of Electrical Engineering, M.I.T., in partial fulfillment of the requirements for the degree of Doctor of Philosophy."Bibliography: p. 52.Army Signal Corps Contract No. DA36-039-sc-78108. Dept. of the Army Task 3-99-20-001 and Project 3-99-00-000.Jane Hodgson Dennis

Electrical properties of polycrystalline semiconductor films

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