Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1991 Pseudo arc-length continuation method for multiple solutions in one-dimensional steady state semiconductor device simulation Tai-Yih Tso Iowa State University Follow this and additional works at:https://lib.dr.iastate.edu/rtd Part of theElectrical and Computer Engineering Commons, and theMathematics Commons Recommended Citation Tso, Tai-Yih, "Pseudo arc-length continuation method for multiple solutions in one-dimensional steady state semiconductor device simulation " (1991).Retrospective Theses and Dissertations. 10078. https://lib.dr.iastate.edu/rtd/10078 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. 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Ann Arbor, MI 48106 Pseudo arc-length continuation method for multiple solutions in one-dimensional steady state semiconductor device simulation by Tai-Yih Tso A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Depart ment: M at hem ati cs Major: Applied Mathematics Approved: Signature was redacted for privacy. Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. For t;lif^C!ra(luate College Iowa State University Ames. Iowa 1991 Copyright © Tai-Yih Tso, 1991. All rights reserved. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS v 1. INTRODUCTION 1 1.1 Overview 1 1.2 Basic semiconductor device equations 3 1.3 Physical ba,9is T I. I Scaled form of the equations 11 1.5 Mathematical analysis II I.fi Numerical methods Ifi 1.7 Computation of current-voltage charcteristics in the one-dimensional thyristor model I!) 2. ANALYTICAL INVESTIGATIONS OF THE CHARACTER OF SOLUTIONS 21 2.1 Existence and uniqueness of solutions 21 2.2 The p-n diode at thermal eciuilibrium 29 2.3 Singular perturbation analysis 32 3. DISCRETIZATION AND MESH DESIGNING 38 3.1 Discretization of the basic semiconductor device equations 38 3.1.1 Discretization of Poisson equation 39 iii 3.1.2 Discretization of continuity equations 40 •3.2 Analysis for the discretizations 47 3.3 Mesh designing -Ij 4. NUMERICAL SCHEMES IN SEMICONDUCTOR DEVICE SIMULATION .33 4.1 Iterative methods .34 4.2 Continuation method oî) 4.3 Implementation and conditioning G"j 5. RESULTS AND CONCLUSION V i 0.1 Computation results 73 -3.2 Conclusion 7G BIBLIOGRAPHY <)1 iv LIST OF FIGURES Figure 1.1: Two transistors analogy 3 Figure 1.2: Box scheme 18 Figure 1.3: Current-voltage characteristics of a thyristor 20 Figure 2.1: An abrupt junction 31 Figure 5.1: Speed of convergence 78 Figure 5.2: Current-voltage characteristic of a diode at forward bias . . . 79 Figure 5.3: Current-voltage characteristic of a diode at reverse bias ... SO Figure 5.4: Junction layer widths of a diode 81 Figure 5.5: Current-voltage characteristic of a thyristor 82 Figure 5.6: Electtric fields of a thyristor at blocking state 83 Figure 5.7: Electrostatic potential at blocking state 84 Figure 5.8: Electrostatic potential at unstable state 85 Figure 5.9: Electrostatic potential at on-state 80 Figure 5.10: Current densities at on-state 87 Figure 5.11: Current densities at unstable state 88 Figure 5.12: Current densities at blocking state 89 Figure 5.13: Non-monotone current-voltage characteristic 90 ACKNOWLEDGEMENTS I wish to thank the expert instruction of Dr. Roger K. Alexander during iny graduate studies at Iowa State University. He introduced this research to n\e. Under his guidance, 1 learned a great deal and enjoyed the learning.