Additional applications of the Lambert W function to solid state physics

IF 0.6 4区教育学 Q4 EDUCATION, SCIENTIFIC DISCIPLINES

European Journal of Physics Pub Date : 2024-09-03 DOI:10.1088/1361-6404/ad6cb3

Ahmed Houari

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引用次数: 0

Abstract

Analytical solutions are always desirable in physics for the sake of mathematical exactness and physical insight. In this study, I obtain closed-form analytical expressions for various physical quantities taken from solid state physics. More precisely, I derive analytical expressions in terms of the Lambert W function for the work function and the field-enhancement factor of a field emitting material from the Fowler–Nordheim equation. Additionally, I derive similar analytical expressions for the localization length and the density of states in amorphous semiconductors from the Mott hopping conductivity. Similarly, I also derive analytical formulae based on the Lambert W function to compute the extrinsic-intrinsic transition temperature in a partially compensated semiconductor and the Kondo exchange coupling constant. All the obtained results are exact and explicit. Moreover, some of them allow a direct determination of some physical quantities of interest compared to their indirect determination from semi-logarithmic experimental plots. The findings of this paper are accessible and suitable for students enrolled in graduate solid state physics courses.

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兰伯特 W 函数在固态物理学中的其他应用

在物理学中，为了数学的精确性和物理洞察力，分析解总是令人向往的。在本研究中，我从固体物理学中获得了各种物理量的闭式分析表达式。更确切地说，我从 Fowler-Nordheim 方程推导出了朗伯 W 函数的分析表达式，即场发射材料的功函数和场增强因子。此外，我还从莫特跳跃电导率推导出了非晶半导体中局部化长度和状态密度的类似分析表达式。同样，我还根据兰伯特 W 函数推导出分析公式，计算出部分补偿半导体的本征-外征转变温度和近藤交换耦合常数。所有得到的结果都是精确和明确的。此外，与从半对数实验图中间接得出的结果相比，其中一些结果还能直接确定一些感兴趣的物理量。本文的研究结果易于理解，适合修读固体物理研究生课程的学生。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

European Journal of Physics 物理-物理：综合

CiteScore

1.70

自引率

28.60%

发文量

128

审稿时长

3-8 weeks

期刊介绍： European Journal of Physics is a journal of the European Physical Society and its primary mission is to assist in maintaining and improving the standard of taught physics in universities and other institutes of higher education. Authors submitting articles must indicate the usefulness of their material to physics education and make clear the level of readership (undergraduate or graduate) for which the article is intended. Submissions that omit this information or which, in the publisher''s opinion, do not contribute to the above mission will not be considered for publication. To this end, we welcome articles that provide original insights and aim to enhance learning in one or more areas of physics. They should normally include at least one of the following: Explanations of how contemporary research can inform the understanding of physics at university level: for example, a survey of a research field at a level accessible to students, explaining how it illustrates some general principles. Original insights into the derivation of results. These should be of some general interest, consisting of more than corrections to textbooks. Descriptions of novel laboratory exercises illustrating new techniques of general interest. Those based on relatively inexpensive equipment are especially welcome. Articles of a scholarly or reflective nature that are aimed to be of interest to, and at a level appropriate for, physics students or recent graduates. Descriptions of successful and original student projects, experimental, theoretical or computational. Discussions of the history, philosophy and epistemology of physics, at a level accessible to physics students and teachers. Reports of new developments in physics curricula and the techniques for teaching physics. Physics Education Research reports: articles that provide original experimental and/or theoretical research contributions that directly relate to the teaching and learning of university-level physics.