A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region
Abstract
:1. Introduction
- Facility of implementation in circuit simulators through analog electrical components with the system of equations developed as a solution of the ADE.
- The mathematical procedure to solve the spatio-temporal second-order differential ADE equation is not complex in comparison to the numerical solutions proposed in the previous reported references.
- The calculation variables used in each expression are a function of time and space, ensuring an adequate approximation with experimental values.
- The developed model calculates the low doping zone width during the space charge accumulation, allowing the excess of the carriers’ injected calculation on the N-region.
- A three-dimensional variable analysis of the carriers’ behavior is presented.
- The results show that the proposed solution is robust for its integration in commercial electrical circuit simulators.
- The obtained solution ensures a balance between mathematical calculations and accuracy results that are appropriate for its use by electronic designers.
- A graphical and numerical estimated error is presented to validate the accuracy of the obtained results.
2. Ambipolar Diffusion Equation (ADE)
2.1. Carrier Transport
2.1.1. Drift Mechanisms
2.1.2. Diffusion
2.2. Carrier Control Model
3. The Empirical Solution of the ADE
3.1. Static Phase
3.2. Conduction Phase
3.2.1. Activation
3.2.2. Blocking
4. Simulation Results
4.1. Static Phase
4.2. Conduction Phase
4.3. Reverse Blocking Phase
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
A | Active area |
ADE | Ambipolar diffusion equation |
B = µn/µp | Mobility relation |
BJT | Bipolar Junction Transistor |
D | Ambipolar diffusion constant. |
Dn | Ambipolar diffusion constant for electrons |
Dp | Ambipolar diffusion constant for holes |
E | Electric field |
Eg | Bandgap |
error(n) | Quadratic error |
Exp(n) | Sampled Experimental data |
Gn,p | Generation rate of electrons and holes |
IGBT | Insulated-Gate Bipolar Transistor |
In(0,t) | Electron current injected in the union P+N-. |
In_drift | Electron drift current |
In(WB,t) | Electron current injected in the union N-N+. |
In_difusion | Electron diffusion current |
Ip(0,t) | Hole current injected in the union P+N- |
Ip_drif | Hole drift current |
Ip(WB,t) | Hole current injected in union N-N+ |
Ip_difusion | Hole diffusion current |
ITD(x,t) | Total injected current in the diode |
IT_drift | Total drift current in the diode |
Lon | Ambipolar diffusion lengthen in turn-on phase |
Loff | Ambipolar diffusion length in turn-off phase |
LS | Ambipolar diffusion length in static phase |
MSE | Mean Square Error |
ni | Intrinsic concentration. |
nP+ | Electron concentration in the region P+ |
n(x,t) | Electron concentration in the region N- in function of space and time |
Px=0 | Initial concentration close to the union P+N- |
Px=WB | Initial concentration close to the union N-N+ |
p(x,t) | Hole concentration in the region N- in function of space and time |
Sim(n) | Simulated sampled data |
WB | Low doping region width |
Vn | Electron drift velocity |
Vp | Hole drift velocity |
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Current [A] | P0 [Charges/cm3] | PW [Charges/cm3] |
---|---|---|
0.5 | 8.043 × 1015 | 1.384 × 1015 |
1 | 1.310 × 1016 | 2.571 × 1015 |
1.5 | 1.712 × 1016 | 3.627 × 1015 |
2 | 2.056 × 1016 | 4.588 × 1015 |
2.5 | 2.362 × 1016 | 5.476 × 1015 |
3 | 2.640 × 1016 | 6.305 × 1015 |
3.5 | 2.897 × 1016 | 7.085 × 1015 |
4 | 3.137 × 1016 | 7.824 × 1015 |
4.5 | 3.362 × 1016 | 8.529 × 1015 |
5 | 3.576 × 1016 | 9.203 × 1015 |
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Hernandez-Gonzalez, L.; Ramirez-Hernandez, J.; Ulises Juarez-Sandoval, O.; Olivares-Robles, M.A.; Sanchez, R.B.; Gibert Delgado, R.d.P. A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region. Mathematics 2021, 9, 458. https://doi.org/10.3390/math9050458
Hernandez-Gonzalez L, Ramirez-Hernandez J, Ulises Juarez-Sandoval O, Olivares-Robles MA, Sanchez RB, Gibert Delgado RdP. A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region. Mathematics. 2021; 9(5):458. https://doi.org/10.3390/math9050458
Chicago/Turabian StyleHernandez-Gonzalez, Leobardo, Jazmin Ramirez-Hernandez, Oswaldo Ulises Juarez-Sandoval, Miguel Angel Olivares-Robles, Ramon Blanco Sanchez, and Rosario del Pilar Gibert Delgado. 2021. "A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region" Mathematics 9, no. 5: 458. https://doi.org/10.3390/math9050458
APA StyleHernandez-Gonzalez, L., Ramirez-Hernandez, J., Ulises Juarez-Sandoval, O., Olivares-Robles, M. A., Sanchez, R. B., & Gibert Delgado, R. d. P. (2021). A New Approach for Approximate Solution of ADE: Physical-Based Modeling of Carriers in Doping Region. Mathematics, 9(5), 458. https://doi.org/10.3390/math9050458