Mathematical Model for Thermo-Electrical Instabilities in Semiconductors

Mukaddas Arzikulova

doi:doi:10.11648/j.mma.20240902.11

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Mathematical Model for Thermo-Electrical Instabilities in Semiconductors

Mukaddas Arzikulova^*

Published in Mathematical Modelling and Applications (Volume 9, Issue 2)

Received: 11 April 2024 Accepted: 28 April 2024 Published: 17 May 2024

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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.

Published in	Mathematical Modelling and Applications (Volume 9, Issue 2)
DOI	10.11648/j.mma.20240902.11
Page(s)	32-37
Creative Commons	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.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Dynamical System, Initial Value Problem, Time Series, Thermo-Electrical Instabilities, Semiconductor, Deep Traps

1. Introduction

Thermo electrical instabilities in semiconductors immersed in cryogenic media (low frequency oscillations of current and temperature of a sample as a result of Joule heating) have been a point of interest for researchers for many years and have been studied experimentally in a variety of semiconducting materials

[1-3]

. In

[1]

mathematical model based on 3 types of trap levels that account for both electron and hole capture for a thin cylindrical sample of crystalline A^IIB^VI compound with uniform distribution of temperature, electric field and electron-hole pair generation across the sample. System comprised of 5 differential equations taking into consideration kinetics of generation-recombination processes is investigated at steady state, taking all time dependences of the variables equal to 0 for simplified cases. Authors calculated threshold values for electric field and frequency of current and temperature oscillations comparing them with available experimental data.

Detailed, comprehensive and thorough research of the instabilities in chalcogenide and vanadium dioxide thin films have been conducted in

[2]

, where thermo electrical instabilities are characterized in terms of negative differential conductivity observed in voltage current characteristics of solids. Heat flow in a semiconductor is compared with RC (resistor-capacitor) network, and effects of circuit design, presence of in homogeneities like metallic inclusions are analyzed. Steady state heat conduction equation together with coupled electrical and thermal processes in a sample are solved on grid in the finite difference approximation.

[3]

presents research of another model with differential equations for free electron concentration n, trap population n_t and sample temperature T for A²B⁶ compounds. Equations for n and T were solved numerically by means of Gear’s method for stiff systems of ordinary differential equations. Time sequences of T(t) and phase trajectories on phase plane (n, T) were obtained as a result, for ambient temperature of T₀=283 K and different values of applied electric field.

Since it is impossible experimentally to detect oscillations of sample temperature, research of mathematical equations that describe the phenomenon is necessary. So far both simplified and more complex model investigations have been limited to study of either null-isoclines of systems of differential equations

[1]

or as in

[3]

only a partial investigation of suggested model. Construction of a simplified new model for more detailed study of the system of nonlinear ordinary differential equations in order to attempt to answer the questions how change of current and temperature are related to each other, and how deep traps involvement into instability progresses is presented in the paper. The quantitative analysis of periodic solutions can give insight into current-temperature relationship as well as behavior of deep traps in such a system.

2. Model Details

To set up a system of equations that describe the dynamics of electrons at conduction zone E_c n and on deep traps n_t, which play an important part in thermo-electrical instabilities in semiconductors, the generation-recombination model by Schoell

[4]

was employed. The author

[4]

considered the two trap level model for analysis of instabilities of the current density and electric field that were classified in his book as first order phase transitions. Choice of the model is preferred because the right hand side expressions are polynomials with respect to position independent variables n and n_t that gives a clear understanding of the mechanism which drives the instability. Equations for the electron concentration at E_c and on deep traps, taking into account thermal emission rates, are considered for an n-type homogenous semiconductor with non-degenerate electron statistics. Along with the equation for change in sample temperature, well known as heat conduction equation, it completes the dynamical system (1):

\begin{matrix} \frac{d n}{dt} = n^{2} (- T_{1}^{S} - X_{1}^{*}) + (N_{D} - n - n_{t}) X_{th} - \\ - n (n_{t} (X_{1} - X_{1}^{*}) + T_{1}^{S} (N_{t} - N_{D}) - N_{D} X_{1}^{*}); \\ \frac{d n_{t}}{dt} = - (X_{op} + X_{th} n_{t} + T^{*} (N_{D} - n - n_{t}) - X_{1}); \\ \frac{d T}{dt} = (\frac{k}{cρ} \frac{d^{2} T}{d x^{2}} - \frac{T - T_{0}}{t_{c}} + \frac{neμ E^{2}}{cρ}) . \end{matrix}

(1)

The variables and constants in the system are: n=n/N_D, n_t=n_t/N_D, T=T/T₀, t_c, T₀– free electron concentration, electron concentration on traps, sample temperature, thermal relaxation time of a sample and temperature of the cooling media; N_d, E, c,

ρ

, k,

μ

– are effective donor concentration, applied electric field (constant), heat capacity of a sample, its density and heat conduction, and electron mobility. Temperature dependence of the physical parameters like heat capacity, heat conduction, thermal relaxation time, electron mobility, electron concentration in a temperature interval of T₀=(77-197) K is taken into consideration based on accessible data, both experimental as well as theoretical research from

[5-11]

. Physical constants taken for spatially homogeneous n-Si sample of size (0.8 x 0.5 x 0.5) cm with partially compensated donors N_A < N_D, where acceptors are assumed to be the ground state (E₁ =0.302 eV), and shallow donors to be the first excited state (E₂=0.011 eV) of the deep traps

[4, 9, 12]

. Figure 1 represents the two level model in detail.

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Figure 1. Two level generation-recombination model for first order phase transitions in semiconductors by Schoell

[4]

As a result of combination of the thermal

X_{th}

, optical

X_{op}

excitements and impact ionization

X_{1}

of the ground state E₁ in the presence of an applied constant electric field, electrons are transferred onto the first excited state E₂. From E₂ electrons are emitted onto the conductive zone with the rates

X_{1}^{S}

for thermal and

X_{1}^{*}

for field aided emission, or back to the ground state E₁. Temperature independent optical excitement

X_{op}

populates the ground states with a new supply of electrons to continuously support oscillations of current and temperature.

T_{1}^{S}

T^{*}

- recombination at the excited and ground states; two electron Auger recombination at the ground state

T_{1}

is combined with

T^{*}

in the equations. Direct transitions between conduction E_c and valence band E_v are ignored

[4]

Expressions for the generation rates

X_{1}^{S}

X_{th}

X_{1}^{*}

and

X_{1}

are given by:

X_{1}^{S} = X_{1}^{S 0} \exp (- \frac{E_{2}}{T})

;

X_{1}^{*} = X_{1}^{* 0} \exp (- \frac{E_{2}}{eλE})

X_{th} = e_{0} \exp (- \frac{E_{1}}{T})

;

X_{1} = X_{1}^{0} \exp (- \frac{E_{1}}{eλE})

Where

X_{1}^{S 0}

; e₀;

X_{1}^{* 0}

;

X_{1}^{0};

T_{1}^{S};

T^{*}

– are constants

[4]

;

λ

- electron mean free path; E – applied constant electric field; e - electron charge. Optical excitement rate

X_{op}

calculated from

[7]

Equation for electron dynamics n in the conduction zone E_c accounts for field impact ionization and recombination, as well as thermal generation of electrons from E₂, field impact ionization and recombination at E₁ and E₂. Dynamics of the deep trap population n_t accounts for optical, thermal, field impact ionization of electrons from E₁; recombination through E₂.

The dynamics of sample temperature T in time depends on applied constant electric field, heat dissipation through cooling of the sample at T₀, and change in the concentration of free electrons in E_c and their mobility. Variation of T over the length x of the sample was set as boundary value problem with boundary conditions of type 3

[12]

. Obtained solutions T(x) for different values of T₀ are substituted into the equation for followed investigation of nonlinear autonomous system (1).

3. Methods

System (1) was studied as a set of three planar systems: (n, T), (n, n_t), (n_t, T). Each of the planar systems was tested for the presence of a limit cycle based on Bendixon’s criteria at fixed values of generation rates

X_{1}^{S}

X_{th}

X_{1}^{*}

and

X_{1}

; optical excitement

X_{op}

, as well as parameter T₀ for determined range of values of appropriate variables, which form a “square” on the phase plane, before being cast into so called “canonical form” by means of linear nonsingular transformation

[13]

. The right hand side nonlinear expressions of the system (1) for an appropriate planar system are expanded into two variable Taylor series at an isolated equilibrium. The first order derivatives of the expansions were used to form a Jacobian matrix; firstly for determining that the equilibrium is isolated, as well as for obtaining quadratic characteristic equations with respect to characteristic number

λ

. For the case when

λ

is a complex conjugate (for periodical trajectories on the phase plane), transformation (3) was employed to express a planar system in form (2) when equilibrium is at origin. The transformation (3) provides real functions of real variables to produce values of real and not complex domain. Linear parts of the system (2) are then set as a Cauchy problem with sets of initial conditions:

\{\begin{matrix} \frac{du}{dt} = αu - βv + φ (u, v) \\ \frac{dy}{dt} = βu + αv + ψ (u, v) \end{matrix}

(2)

t = 0; u = u_{0}; v = v_{0}

u = (α - a) x + cy

;

v = βx

;(3)

u_{0} = (α - a) x_{0} - y_{0}

;

v_{0} = β x_{0}

Here x, y – variables and

x_{0}, y_{0}

- initial values of an appropriate planar system accordingly;

a, c

– elements of Jacobian at an isolated equilibrium;

λ_{1, 2} = α \pm βi

- roots of characteristic equation;

φ (u, v)

ψ (u, v)

– nonlinear parts (second and higher order terms) of two variable Taylor series expansions. The initial value problem (2) was solved with variation of constants method for inhomogeneous linear systems. Real parts of the solutions were separated and plotted against time.

4. Results

Figure 2 presents time sequences of obtained real parts of the solutions of (2) for the planar systems dependant on the single parameter T₀. Variations of n, n_t and T in time are overlaid in the graphs. At T₀ = 77 K, amplitudes of n and n_t are large, while that of T is small, relative to n and n_t. As T₀ grows the amplitude of all variables T, n and n_t decrease. With growing T₀, the amplitude of T become larger relative to those of n and n_t. At higher T₀, the amplitude of n_t is limited to smaller deviations in time. A decrease in the frequency of n_t with a growing T₀ is also observed.

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Figure 2. Time sequences for n-brown; n_t–orange; T-pink at different values of temperature of cooling media T₀: (a) – 77 K; (b) – 137 K; (c) – 197 K.

5. Discussion

At a low value of T₀, small amplitude of T shows rapid cooling of the sample Figure 2a; large amplitudes of n_t and n indicate lower carrier population at conduction band E_c, so the transfer of electrons onto E_c from traps is happening in larger volumes. With growing T₀ Figure 2b, c, electron transfer from traps is no longer possible at the same volume, because of the constant recombination rates and limited number of allowed states per unit volume at E_c. The lowering of the frequency of n_t also points to that. As a result, an adjustment through an increase of frequency is taking place for both n and T, since the sample’s cooling is taking a longer time; oscillations of T fall behind the changes of current in time.

Comparing the findings of the research with those of

[3]

would be appropriate here. Study of the system in

[3]

included as part of the research, time sequences of T(t) in transition from regular to chaotic oscillations, as well as accompanying phase trajectories of planar system (n, T). No further investigation of interrelationships of the variables n, n_t, T were conducted and yet time sequences of n, n_t can give information about the involvement of deep traps, interrelationship with other variables of the system through their own behavior in time. In Figure 2 time sequences of each of the variables were overlaid to show not only how these variables change in time, but also how their frequencies, amplitudes, and phase differences relate and evolve with respect to each other and a growing T₀.

The next step in the research of the dynamical system (1) is conducting analysis of the phase trajectories of the planar systems depending on the parameter T₀ which will give a clearer view of the interrelationship of the variables n, n_t and T.

6. Conclusions

The fact that alternating current passing through solids creates heat due to Joule heating has been known for a long time and has found far reaching applications in the modern world. Yet the relationship between current and temperature remained unknown, since it is impossible to detect variations of temperature in time, experimentally. Theoretical research of this relationship in solids, based on information accessible to the author, has never been conducted in full. Findings of this research give interesting insight into the delicately balanced system where change in one variable causes strong effects on the others, depending on parameter T₀. Change in the temperature of the cooling media forces the system to adjust to a new thermo dynamical state by changing the frequency and amplitude of oscillations, thus giving an idea of the level of density states occupation at the conduction zone; oscillations of sample temperature always follow those of current, and this delay is present throughout the interval of change of T₀.

Acknowledgments

I am grateful to God for His guidance and support; Mr. John A. Witcher for his invaluable help with manuscript editing, Monroe, Georgia Uncle Remus Public library management and the staff; University of Georgia Main library, UGA, Athens, Georgia, USA, for their assistance in accessing digital resource files.

Author Contributions

Mukaddas Arzikulova is the sole author. The author read and approved the final manuscript.

Funding

This work is not supported by any external funding.

Data Availability Statement

The data is available from the corresponding author upon reasonable request.

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	Kalashnikov S. G., Pustovoit V. I., Pado G. S., Tokarev E. F., Phys. St. Sol., vol. 34, N. 2, 1969, pp 1255.
[2]	Shaw M. P., Yildirim N. Thermal and electrothermal instabilities in semiconductors. Advances in electronics and electron physics. USA: Academic press, Vol. 60, 1983, pp 307-382. ISBN 0-12-014660-6.
[3]	Balkarey Yu. I., Golik L. L., Pakseev V. E., Rjanov Yu. A., Loskutov V. S., Elinson M. I., Experimental and numerical research of stochastic dynamics of thermo electrical instabilities in semiconductors. Physics and technics of Semiconductors. vol. 21, N. 8, 1987, pp 1369-1378.
[4]	Schoell E. Nonequilibrium phase transitions in semiconduc-tors. Self organization induced by generation and recombination. Germany: Springer-Verlag. 1987, pp 48-56.
[5]	Flubacher P., Leadbetter A. J., Morrison J. A. The heat Capacity of pure Silicon and Germanium and properties of their vibrational frequency spectra. Phil. Mag. 4, 39, 1959, pp 273-294. http://doi.org/10.1080/14786435908233340
[6]	Glassbrenner C. J., Slack Glen A. Thermal conductivity of Silicon and Germanium from 3K to the melting point. Phys. Rev. 134, 4A, 1964, pp A1058-A1069. http://doi.org/10.1103/PhysRev.134.A1058
[7]	Grimmeiss H. G, Skarstam B.. Physical Review B, Vol. 23, N. 4, 1981, pp1947-1960.
[8]	Kireev P. S. Physics of semiconductors. Russian second edition. Moscow: Nauka; 1975, pp 417.
[9]	Rees G. J., Grimmeiss H. G., Janzen E., Skarstam B. J. Phys. C: Solid St. Phys., 13, 1980, pp 6157-6165.
[10]	Reggiani S., Valdinoci M., Colalongo L., Rudan M., Baccarani G. An analytical, temperature dependent model for majority- and minority- carrier mobility in Silicon devices. VLSI Design. Vol. 10, No 4, 2000, pp 467-483.
[11]	Smith R. A. Semiconductors. Second edition. Great Britain: Cambridge Press; 1978, pp 424.
[12]	Arzikulova M. On integration of fractional expressions containing transcendental functions for one boundary value problem. Journal of Applied Mathematical Sciences, ISSN 1314-7552. Vol. 17, N. 13, 2023, pp 635-639. http://doi.org/1012988/ams.2023.918517
[13]	Andronov A. A., Leontovich E. A., Gordon I. I., Maier A. G. Qualitative theory of second order dynamic systems. Moscow, Nauka, 1966, pp 135-165.

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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

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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

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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

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@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}
}

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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  -

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Author Information

Mukaddas Arzikulova

Department of Physics, Samarkand State University, Samarkand, Uzbekistan

Contact Email

http://orcid.org/0009-0004-4498-6876

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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

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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

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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

Copy | Download

@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}
}

Copy | Download

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  -

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