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,n_{t}) and (n_{t},T) for local behavior at isolated equilibrium and at points on individual trajectories, where n, n_{t} 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 T_{0}. Based on obtained calculation results of time sequences of the three variables n, n_{t} and T, phase differences among these variables have been determined for different values of T_{0}. It has been established that the change in sample temperature lags behind change in current and this lag increases with T_{0}. Clearly defined correlations among systems (n,T), (n,n_{t}) and (n_{t},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 T_{0}.
Published in | Mathematical Modelling and Applications (Volume 9, Issue 2) |
DOI | 10.11648/j.mma.20240902.12 |
Page(s) | 38-42 |
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 |
Dynamical System, Phase Trajectories, Thermo-Electrical Instabilities, Semiconductor, Deep Traps
T_{0}=77 K | T_{0}=137 K | T_{0}=197 K | |
---|---|---|---|
0.754 | 1.363 | 2.26 | (n, T) |
2.19 | 0.272 | 1.64 | (n, n_{t}) |
1.13 | 3.32 | 0.503 | (n_{t}, T) |
2D | System of Two Ordinary Differential Equations |
3D | System of Three Ordinary Differential Equations |
[1] | Schoell E. Nonequilibrium phase transitions in semiconduc-tors. Self organization induced by generation and recombi-nation. Germany: Springer-Verlag. 1987, pp 48-56. |
[2] | Blakemore J. S. Solid State physics. UK: Cambridge University press; Second edition, 1985, pp. 306, 321. |
[3] | 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. |
[4] | 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. |
[5] | Grimmeiss H. G, Skarstam B.. Physical Review B, Vol. 23, N. 4, 1981, pp1947-1960. |
[6] | Kireev P. S. Physics of semiconductors. Russian second edition. Moscow: Nauka; 1975, pp 417. |
[7] | Rees G. J., Grimmeiss H. G., Janzen E., Skarstam B. J. Phys. C: Solid St. Phys., 13, 1980, pp 6157-6165. |
[8] | 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. |
[9] | Smith R. A. Semiconductors. Second edition. Great Britain: Cambridge Press; 1978, pp 424. |
[10] | Sze S. M. Semiconductor Devices. Physics and Technology. Second edition. New Delhi: Wiley India, 2011, pp 67-80. |
[11] | Ashcroft N. W., Mermin N. D. Solid State Physics. Delhi: Cengage Learning, 2013, pp 556, 580. |
[12] | Blakemore J. S. Semiconductor statistics. New York: Dover Publications, 1987, pp 227. |
[13] | Blakemore J. S. Solid State Physics. Second edition. Cambridge UK: Cambridge University Press, 1985, pp 305-306, 321. |
[14] | Arzikulova M. Journal of Mathematical Modeling and Applications. Vol. 9, Issue 2, 2024, pp 32-37. |
[15] | Arzikulova M. Journal of Applied Mathematical Sciences, ISSN 1314-7552. Vol. 17, N. 13, 2023, pp 635-639. |
[16] | A. A. Andronov, E. A. Leontovich, L. I. Gordon, A. G. Maier. Qualitative theory of second order dynamical systems. Moscow: Nauka; 1966, pp 135-165. |
APA Style
Arzikulova, M. (2024). Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems. Mathematical Modelling and Applications, 9(2), 38-42. https://doi.org/10.11648/j.mma.20240902.12
ACS Style
Arzikulova, M. Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems. Math. Model. Appl. 2024, 9(2), 38-42. doi: 10.11648/j.mma.20240902.12
AMA Style
Arzikulova M. Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems. Math Model Appl. 2024;9(2):38-42. doi: 10.11648/j.mma.20240902.12
@article{10.11648/j.mma.20240902.12, author = {Mukaddas Arzikulova}, title = {Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems }, journal = {Mathematical Modelling and Applications}, volume = {9}, number = {2}, pages = {38-42}, doi = {10.11648/j.mma.20240902.12}, url = {https://doi.org/10.11648/j.mma.20240902.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20240902.12}, 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. }, year = {2024} }
TY - JOUR T1 - Investigation of Thermo-Electrical Instabilities in a Semiconductor as 2D Dynamical Systems AU - Mukaddas Arzikulova Y1 - 2024/05/30 PY - 2024 N1 - https://doi.org/10.11648/j.mma.20240902.12 DO - 10.11648/j.mma.20240902.12 T2 - Mathematical Modelling and Applications JF - Mathematical Modelling and Applications JO - Mathematical Modelling and Applications SP - 38 EP - 42 PB - Science Publishing Group SN - 2575-1794 UR - https://doi.org/10.11648/j.mma.20240902.12 AB - 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. VL - 9 IS - 2 ER -