Crystalline semiconductors under specific conditions, with an applied electric field, switch or oscillate between two conductive states, thus producing low frequency oscillations of electric current flowing through the sample and as a result of Joule heating oscillations of sample temperature. These phenomena are recognized to be thermo - electrical instabilities. Although current oscillations can be detected and registered experimentally, there is no device that can detect, register and allow us to study the sample temperature change in time. The purpose of this study is to learn about the relationship of electric current and sample temperature coupled with deep traps that play an important part in supporting the phenomenon. This can be done only by setting up a mathematical model that describes the phenomenon in detail. The equations that make up the model are continuity equations for free electron and deep traps carrier populations, as well as a heat conduction equation – a set of ordinary nonlinear inhomogeneous differential equations. The system is transformed into a so called “canonical form” as a result of linearization of the system at isolated equilibrium. It is achieved by expansion of the right hand sides of the equations into two variable Taylor series at isolated equilibrium involving linear non-singular transformation. The mathematical model for thermo-electrical instabilities in an n-type semiconductor with non-degenerate electron statistics has been studied as 3D dynamical system. The system of differential equations is broken down into component planar systems, each of them being tested for existence of limit cycles on a determined phase plane, followed by quantitative investigation of their local behavior at isolated equilibrium and at points on individual trajectories on phase plane dependant on single parameter T_{0}. Solutions of sets of initial value problems as time series of the variables: free electron concentration; sample temperature; deep trap population is presented. The investigation results show that oscillations of sample temperature follow those of current. Change in T_{0} forces the system to adjust to new thermodynamical state by changing frequency and amplitude of the oscillations as well as dynamics of deep trap population.
Published in | Mathematical Modelling and Applications (Volume 9, Issue 2) |
DOI | 10.11648/j.mma.20240902.11 |
Page(s) | 32-37 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Dynamical System, Initial Value Problem, Time Series, Thermo-Electrical Instabilities, Semiconductor, Deep Traps
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APA Style
Arzikulova, M. (2024). Mathematical Model for Thermo-Electrical Instabilities in Semiconductors. Mathematical Modelling and Applications, 9(2), 32-37. https://doi.org/10.11648/j.mma.20240902.11
ACS Style
Arzikulova, M. Mathematical Model for Thermo-Electrical Instabilities in Semiconductors. Math. Model. Appl. 2024, 9(2), 32-37. doi: 10.11648/j.mma.20240902.11
AMA Style
Arzikulova M. Mathematical Model for Thermo-Electrical Instabilities in Semiconductors. Math Model Appl. 2024;9(2):32-37. doi: 10.11648/j.mma.20240902.11
@article{10.11648/j.mma.20240902.11, author = {Mukaddas Arzikulova}, title = {Mathematical Model for Thermo-Electrical Instabilities in Semiconductors }, journal = {Mathematical Modelling and Applications}, volume = {9}, number = {2}, pages = {32-37}, doi = {10.11648/j.mma.20240902.11}, url = {https://doi.org/10.11648/j.mma.20240902.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20240902.11}, abstract = {Crystalline semiconductors under specific conditions, with an applied electric field, switch or oscillate between two conductive states, thus producing low frequency oscillations of electric current flowing through the sample and as a result of Joule heating oscillations of sample temperature. These phenomena are recognized to be thermo - electrical instabilities. Although current oscillations can be detected and registered experimentally, there is no device that can detect, register and allow us to study the sample temperature change in time. The purpose of this study is to learn about the relationship of electric current and sample temperature coupled with deep traps that play an important part in supporting the phenomenon. This can be done only by setting up a mathematical model that describes the phenomenon in detail. The equations that make up the model are continuity equations for free electron and deep traps carrier populations, as well as a heat conduction equation – a set of ordinary nonlinear inhomogeneous differential equations. The system is transformed into a so called “canonical form” as a result of linearization of the system at isolated equilibrium. It is achieved by expansion of the right hand sides of the equations into two variable Taylor series at isolated equilibrium involving linear non-singular transformation. The mathematical model for thermo-electrical instabilities in an n-type semiconductor with non-degenerate electron statistics has been studied as 3D dynamical system. The system of differential equations is broken down into component planar systems, each of them being tested for existence of limit cycles on a determined phase plane, followed by quantitative investigation of their local behavior at isolated equilibrium and at points on individual trajectories on phase plane dependant on single parameter T0. Solutions of sets of initial value problems as time series of the variables: free electron concentration; sample temperature; deep trap population is presented. The investigation results show that oscillations of sample temperature follow those of current. Change in T0 forces the system to adjust to new thermodynamical state by changing frequency and amplitude of the oscillations as well as dynamics of deep trap population. }, year = {2024} }
TY - JOUR T1 - Mathematical Model for Thermo-Electrical Instabilities in Semiconductors AU - Mukaddas Arzikulova Y1 - 2024/05/17 PY - 2024 N1 - https://doi.org/10.11648/j.mma.20240902.11 DO - 10.11648/j.mma.20240902.11 T2 - Mathematical Modelling and Applications JF - Mathematical Modelling and Applications JO - Mathematical Modelling and Applications SP - 32 EP - 37 PB - Science Publishing Group SN - 2575-1794 UR - https://doi.org/10.11648/j.mma.20240902.11 AB - Crystalline semiconductors under specific conditions, with an applied electric field, switch or oscillate between two conductive states, thus producing low frequency oscillations of electric current flowing through the sample and as a result of Joule heating oscillations of sample temperature. These phenomena are recognized to be thermo - electrical instabilities. Although current oscillations can be detected and registered experimentally, there is no device that can detect, register and allow us to study the sample temperature change in time. The purpose of this study is to learn about the relationship of electric current and sample temperature coupled with deep traps that play an important part in supporting the phenomenon. This can be done only by setting up a mathematical model that describes the phenomenon in detail. The equations that make up the model are continuity equations for free electron and deep traps carrier populations, as well as a heat conduction equation – a set of ordinary nonlinear inhomogeneous differential equations. The system is transformed into a so called “canonical form” as a result of linearization of the system at isolated equilibrium. It is achieved by expansion of the right hand sides of the equations into two variable Taylor series at isolated equilibrium involving linear non-singular transformation. The mathematical model for thermo-electrical instabilities in an n-type semiconductor with non-degenerate electron statistics has been studied as 3D dynamical system. The system of differential equations is broken down into component planar systems, each of them being tested for existence of limit cycles on a determined phase plane, followed by quantitative investigation of their local behavior at isolated equilibrium and at points on individual trajectories on phase plane dependant on single parameter T0. Solutions of sets of initial value problems as time series of the variables: free electron concentration; sample temperature; deep trap population is presented. The investigation results show that oscillations of sample temperature follow those of current. Change in T0 forces the system to adjust to new thermodynamical state by changing frequency and amplitude of the oscillations as well as dynamics of deep trap population. VL - 9 IS - 2 ER -