Research Article
Mathematical Model for Thermo-Electrical Instabilities in Semiconductors
Mukaddas Arzikulova*
Issue:
Volume 9, Issue 2, June 2024
Pages:
32-37
Received:
11 April 2024
Accepted:
28 April 2024
Published:
17 May 2024
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.
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 - electrica...
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Research Article
Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems
Mukaddas Arzikulova*
Issue:
Volume 9, Issue 2, June 2024
Pages:
38-42
Received:
30 April 2024
Accepted:
20 May 2024
Published:
30 May 2024
Abstract: A semiconducting sample placed in cryogenic media with applied electric field generates low frequency oscillations of electric current and sample temperature and known to be thermo-electrical instabilities. Although observation of current oscillations on oscilloscope is possible, change of sample temperature cannot be detected experimentally. Description of the phenomenon through mathematical equations helps to understand relationship of the two variables as well as their connection to deep trap behavior that are involved in supporting the instability. Mathematical model for thermo-electrical instabilities in an n type semiconductor based on the two deep trap level model with non-degenerate electron statistics has been introduced in order to investigate the unique relationship between the change in time of both electric current flowing through a semiconductor sample and the sample temperature. The 3D dynamical system of nonlinear inhomogeneous ordinary differential equations has been investigated as component 2D dynamical systems (n,T), (n,nt) and (nt,T) for local behavior at isolated equilibrium and at points on individual trajectories, where n, nt and T are free electron concentration at conduction band, electron concentration at deep traps and temperature of a semiconductor sample accordingly. Each of the planar systems is expressed in canonical form and investigated as a Cauchy problem with a set of appropriate initial values. This paper presents investigation results of phase trajectories of the planar systems depending on a single parameter – the temperature of cooling media T0. Based on obtained calculation results of time sequences of the three variables n, nt and T, phase differences among these variables have been determined for different values of T0. It has been established that the change in sample temperature lags behind change in current and this lag increases with T0. Clearly defined correlations among systems (n,T), (n,nt) and (nt,T) are seen, being the result of balance between field aided and thermal ionization mechanisms for charge carrier generation and recombination processes. Thermal and field assisted generation mechanisms compete with one another in achieving steady non equilibrium state in the system depending on temperature of cooling media T0.
Abstract: A semiconducting sample placed in cryogenic media with applied electric field generates low frequency oscillations of electric current and sample temperature and known to be thermo-electrical instabilities. Although observation of current oscillations on oscilloscope is possible, change of sample temperature cannot be detected experimentally. Descr...
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