To accurately analyze an arbitrary semiconductor structure which is intended as a seif contained device under various operating conditions, a mathematieal model has to be given. The equations which form this mathematieal model are commonly called the basic semiconductor equations. They can be derived from Maxwell’s equations (2-1), (2-2), (2-3) and (2-4), several relations obtained from solid-state physics knowledge about semiconductors and various — sometimes overly simplistic — assumptions.
$$\text{rot}\,\vec H = \vec J + \frac{{\partial \vec D}} {{\partial t}}$$
$$\text{rot}\,\vec E = - \frac{{\partial \vec B}} {{\partial t}}$$
$$\text{div }\vec D = \rho$$
$$\text{div }\vec B = 0$$
\(\vec E\) and \(\vec D\)are the electric field and displacement vector; \(\vec H\)and \(\vec B\)are the magnetic field and induction vector, respectively. \(\vec J\)denotes the conduction current density, and ρ is the electric charge density.


Carrier Concentration Boltzmann Equation Bipolar Transistor Impurity Band Electric Field Vector 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [2.1]
    Adler, M. S.: Accurate Calculations of the Forward Drop and Power Dissipation in Thyristors. IEEE Trans. Electron Devices ED-25, No. 1, 16–22 (1978).CrossRefGoogle Scholar
  2. [2.2]
    Adler, M. S.: An Operational Method to Model Carrier Degeneracy and Band Gap Narrowing. Solid-State Electron. 26, No. 5, 387–396 (1983).CrossRefGoogle Scholar
  3. [2.3]
    Ankri, D., Eastman, L. F.: GaAlAs-GaAs Ballistic Hetero-Junction Bipolar Transistor. Electronics Lett. 18, No. 17, 750–751 (1982).CrossRefGoogle Scholar
  4. [2.4]
    Anselm, A. I.: Einführung in die Halbleitertheorie. Berlin: Akademie-Verlag 1964.Google Scholar
  5. [2.5]
    Antoniadis, D. A., Dutton, R. W.: Models for Computer Simulation of Complete IC Fabrication Process. IEEE J. Solid State Circuits SC-14, No. 2, 412–422 (1979).CrossRefGoogle Scholar
  6. [2.6]
    Awano, Y., Tomizawa, K., Hashizume, N., Kawashima, M.: Monte Carlo Particle Simulation of a GaAs Submicron n+-i-n+ Diode. Electronics Lett. 18, No. 3, 133–135 (1982).CrossRefGoogle Scholar
  7. [2.7]
    Awano, Y., Tomizawa, K., Hashizume, N., Kawashima, M.: Monte Carlo Particle Simulation of a GaAs Short-Channel MESFET. Electronics Lett. 19, No. 1, 20–21 (1983).CrossRefGoogle Scholar
  8. [2.8]
    Aymerich-Humett, X., Serra-Mestres, F., Millan, J.: An Analytical Approximation for the Fermi-Dirac Integral F3/2(x). Solid-State Electron. 24, No. 10, 981–982 (1981).CrossRefGoogle Scholar
  9. [2.9]
    Baccarani, G., Mazzone, A. M.: On the Diffusion Current in Heavily Doped Silicon. Solid-State Electron. 18, 469–470 (1975).CrossRefGoogle Scholar
  10. [2.10]
    Baccarani, G.: Physics of Submicron Devices. Proc. VLSI Process and Device Modeling, pp. 1–23. Katholieke Universiteit Leuven, 1983.Google Scholar
  11. [2.11]
    Bennett, H. S.: Improved Concepts for Predicting the Electrical Behavior of Bipolar Structures in Silicon. IEEE Trans. Electron Devices ED-30, No. 8, 920–927 (1983).CrossRefGoogle Scholar
  12. [2.12]
    Blakemore, J. S.: Approximations for Fermi-Dirac Integrals, especially the Function Fl/2(x) Used to Describe Electron Density in a Semiconductor. Solid-State Electron. 25, No. 11, 1067–1076 (1982).CrossRefGoogle Scholar
  13. [2.13]
    Blatt, F. J.: Physics of Electronic Conduction in Solids. New York: McGraw-Hill 1968.Google Scholar
  14. [2.14]
    Bltekjaer, K.: Transport Equations for Electrons in Two-Valley Semiconductors. IEEE Trans. Electron Devices ED-17, No. 1, 38–47 (1970).CrossRefGoogle Scholar
  15. [2.15]
    Bonch-Bruevich, V. L.: On the Theory of Heavily Doped Semiconductors. Soviet Physics Solid State 4, No. 10, 1953–1962 (1963).Google Scholar
  16. [2.16]
    Buot, F. A., Frey, J.: Effects of Velocity Overshoot on Performance of GaAs Devices, with Design Information. Solid-State Electron. 26, No.7, 617–632 (1983).CrossRefGoogle Scholar
  17. [2.17]
    Capasso, F., Pearsall, T. P., Thornber, K. K.: The Effect of Collisional Broadening on Monte Carlo Simulations ofHigh-Field Transport in Semiconductor Devices. IEEE Electron Dev. Lett. EDL-2, 295 (1981).CrossRefGoogle Scholar
  18. [2.18]
    Chryssafis, A., Love, W.: A Computer-Aided Analysis of One-Dimensional Thermal Transients in n-p-n Power Transistors. Solid-State Electron. 22, 249–256 (1979).CrossRefGoogle Scholar
  19. [2.19]
    Cody, W. J., Thacher, H. C.: Rational Chebyshev Approximations for Fermi-Dirac Integrals of Orders-1/2, 1/2 and 3/2. Math. Comp. 21, 30–40 (1967).Google Scholar
  20. [2.20]
    Cohen, M. L., Bergstresser, T. K.: Band Structures and Pseudopotential Form Factors for Fourteen Semiconductors of the Diamond and Zinc-blende Structures. Physical Review 141, 789–796 (1966).CrossRefGoogle Scholar
  21. [2.21]
    Conwell, E. M.: High Field Transport in Semiconductors. New York: Academic Press 1967.Google Scholar
  22. [2.22]
    Cook, R. K., Frey, J.: Two-Dimensional Numerical Simulation of Energy Transport Effects in Si and GaAs MESFET’s. IEEE Trans. Electron Devices ED-29, No. 6, 970–977 (1982).CrossRefGoogle Scholar
  23. [2.23]
    Curtice, W. R.: Direct Comparison of the Electron Temperature Model with the Particle Mesh (Monte-Carlo) Model for the GaAs MESFET. IEEE Trans. Electron Devices ED-29, No. 12, 1942–1943 (1982).CrossRefGoogle Scholar
  24. [2.24]
    DeMan, H. J. J.: The Influence of Heavy Doping on the Emitter Efficiency of a Bipolar Transistor. IEEE Trans. Electron Devices ED-18, No. 10, 833–835 (1971).CrossRefGoogle Scholar
  25. [2.25]
    DelAlamo, J. A., Swanson, R. M., Lietoila, A.: Photovoltaic Measurement of Bandgap Narrowing in Moderately Doped Silicon. Solid-State Electron. 26, No. 5, 483–489 (1983).CrossRefGoogle Scholar
  26. [2.26]
    Dhariwal, S. R., Ojha, V. N.: Band Gap Narrowing in Heavily Doped Silicon. Solid-State Electron. 25, No. 9, 909–911 (1982).CrossRefGoogle Scholar
  27. [2.27]
    Dorkel, J. M.: On Electrical Transport in Non-Isothermal Semiconductors. Solid-State Electron. 26, No. 8, 819–821 (1983).CrossRefGoogle Scholar
  28. [2.28]
    Engl, W. L., Dirks, H. K., Meinerzhagen, B.: Device Modeling. Proc. IEEE 71, No. 1, 10–33 (1983).CrossRefGoogle Scholar
  29. [2.29]
    Frey, J.: Transport Physics for VLSI. In: Introduction to the Numerical Analysis of Semiconductor Devices and Integrated Circuits, pp. 51–57. Dublin: Boole Press 1981.Google Scholar
  30. [2.30]
    Frey, J.: Physics Problems in VLSI Devices. In: Introduction to the Numerical Analysis of Semiconductor Devices and Integrated Circuits, pp. 47–50. Dublin: Boole Press 1981.Google Scholar
  31. [2.31]
    Froelich, R. K., Blakey, P. A.: Energy and Momentum Conserving Simulation of Millimeter Wave Impatt Diodes. Proc. NASECODE II Conf., pp. 208–212. Dublin: Boole Press 1981.Google Scholar
  32. [2.32]
    Gaensslen, F. H., Jaeger, R. C., Walker, J. J.: Low-Temperature Threshold Behavior of Depletion Mode Devices — Characterization and Simulation. Proc. Int. Electron Devices Meeting, pp. 520–524 (1976).Google Scholar
  33. [2.33]
    Gaensslen, F. H., Rideout, V. L., Walker, E. J., Walker, J. J.: Very Small MOSFET’s for Low Temperature Operation. IEEE Trans. Electron Devices ED-24, No. 3, 218–229 (1977).CrossRefGoogle Scholar
  34. [2.34]
    Gaensslen, F. H., Jaeger, R. C.: Temperature Dependent Threshold Behaviour of Depletion Mode MOSFET’s. Solid-State Electron. 22, 423–430 (1979).CrossRefGoogle Scholar
  35. [2.35]
    Gaur, S. P., Navon, D. H.: Two-Dimensional Carrier Flow in a Transistor Structure under Nonisothermal Conditions. IEEE Trans. Electron Devices ED-23, 50–57 (1976).CrossRefGoogle Scholar
  36. [2.36]
    Gnädinger, A. P., Talley, H. E.: Quantum Mechanical Caleulations of the Carrier Distribution and the Thickness of the Inversion Layer of a MOS Field-Effect Transistor. Solid-State Electron. 13, 1301–1309 (1970).CrossRefGoogle Scholar
  37. [2.37]
    Grondin, R. O., Lugli, P., Ferry, D. K.: Ballistic Transport in Semiconductors. IEEE Electron Device Lett. EDL-3, No. 12, 373–375 (1982).CrossRefGoogle Scholar
  38. [2.38]
    Halperin, B. I., Lax, M.: Impurity Band Tails in the High Density Limit. I. Minimum Counting Method. Physical Review 148, 722–740 (1966).CrossRefGoogle Scholar
  39. [2.39]
    Heasell, E. L.: A Self-Consistent Calculation of Effective Intrinsie Concentration in Heavily Doped Silicon. Int. J. Electronics 38, No. 1, 127–135 (1975).CrossRefGoogle Scholar
  40. [2.40]
    Heasell, E. L.: On the Role of Degeneracy in the “Heavy Doping” Phenomenon. Solid-State Electron. 23, 183 (1980).CrossRefGoogle Scholar
  41. [2.41]
    Hess, K.: Ballistic Electron Transport in Semiconductors. IEEE Trans. Electron Devices ED-28, 937–940 (1981).CrossRefGoogle Scholar
  42. [2.42]
    Heywang, W., Pötzl, H. W.: Bandstruktur und Stromtransport. Berlin-Heidelberg-New York: Springer 1976.Google Scholar
  43. [2.43]
    Hillbrand, H.: Untersuchungen des Transportverhaltens von III-V Halbleitern bei hohen elektrischen Feldern mit Monte-Carlo-Methoden. Dissertation, Technische Hochschule Wien, 1974.Google Scholar
  44. [2.44]
    Hofmann, H.: Das elektromagnetische Feld, 2. Aufl. Wien-New York: Springer 1982.Google Scholar
  45. [2.45]
    Holden, A. J., Debney, B. T.: Improved Theory of Ballistic Transport in One Dimension. Electronics Lett. 18, No. 13, 558–559 (1982).CrossRefGoogle Scholar
  46. [2.46]
    Jüngling, W.: Hochdotierungseffekte in Silizium. Diplomarbeit, Technische Universität Wien, 1983.Google Scholar
  47. [2.47]
    Jüngling, W., Guerrero, E., Selberherr, S.: On Modeling the Intrinsie Number and Fermi Levels for Device and Process Simulation. Proc. NASECODE III Conf., pp. 144–149. Dublin: Boole Press 1983.Google Scholar
  48. [2.48]
    Kane, E. O.: Thomas Fermi Approach to Impure Semiconductor Band Structure. Physical Review 131, No. 1, 79–88 (1963).CrossRefGoogle Scholar
  49. [2.49]
    Kireev, P. S.: Semiconductor Physics. Moscow: MIR Publishers 1978.Google Scholar
  50. [2.50]
    Kittel, C.: Introduction to Solid-State Physics. New York: Wiley 1967.Google Scholar
  51. [2.51]
    Kleppinger, D. D., Lindholm, F. A.: Impurity Concentration Dependent Density of States and Resulting Fermi Level for Silicon. Solid-State Electron. 14, 407–416 (1971).CrossRefGoogle Scholar
  52. [2.52]
    Landsberg, P. T., Hope, S. A.: Diffusion Currents in Semiconductors. Solid-State Electron. 19, 173–174 (1976).CrossRefGoogle Scholar
  53. [2.53]
    Landsberg, P. T., Hope, S. A.: Two Formulations of Semiconductor Transport Equations. Solid-State Electron. 20, 421–429 (1977).CrossRefGoogle Scholar
  54. [2.54]
    Langer, E.: Numerische Simulation der Halbleiterdiode. Diplomarbeit, Technische Universität Wien, 1980.Google Scholar
  55. [2.55]
    Lanyon, H. P. D., Tuft, R. A.: Bandgap Narrowing in Heavily Doped Silicon. Proc. International Electron Device Meeting, pp. 316–319 (1978).Google Scholar
  56. [2.56]
    Lanyon, H. P. D., Tuft, R. A.: Bandgap Narrowing in Moderately to Heavily Doped Silicon. IEEE Trans. Electron Devices ED-26, No.7, 1014–1018 (1979).CrossRefGoogle Scholar
  57. [2.57]
    Lee, D. S., Fossum, J. G.: Energy Band Distortion in Highly Doped Silicon. IEEE Trans. Electron Devices ED-30, No. 6, 626–634 (1983).Google Scholar
  58. [2.58]
    Lue, J. T.: Theory of Schottky Barrier Heights of Amorphous MIS Solar Cells. Solid-State Electron. 25, No. 9, 869–874 (1982).CrossRefGoogle Scholar
  59. [2.59]
    Lundstrom, M. S., Schwartz, R. J., Gray, J. L.: Transport Equations for the Analysis of Heavily Doped Semiconductor Devices. Solid-State Electron. 24, 195–202 (1981).CrossRefGoogle Scholar
  60. [2.60]
    Lundstrom, M. S., Schuelke, R. J.: Modeling Semiconductor Heterojunctions in Equilibrium. Solid-State Electron. 25, No. 8, 683–691 (1982).CrossRefGoogle Scholar
  61. [2.61]
    Mertens, R. P., Van Meerenbergen, J. L., Nijs, J. F., Van Overstraeten, R. J.: Measurement of the Minority-Carrier Transport Parameters in Heavily Doped Silicon. IEEE Trans. Electron Devices ED-27, 949–955 (1980).CrossRefGoogle Scholar
  62. [2.62]
    Mertens, R. P.: Modeling of Heavy Doping Effects. Proc. VLSI Process and Device Modeling, pp. 1–29. Katholieke Universiteit Leuven, 1983.Google Scholar
  63. [2.63]
    Mock, M. S.: Transport Equations in Heavily Doped Silicon, and the Current Gain of a Bipolar Transistor. Solid-State Electron. 16, 1251–1259 (1973).CrossRefGoogle Scholar
  64. [2.64]
    Moglestue, C., Beard, S. J.: A Particle Model Simulation of Field Effect Transistors. Proc. NASECODE I Conf., pp. 232–236. Dublin: Boole Press 1979.Google Scholar
  65. [2.65]
    Moglestue, C.: A Monte-Carlo Particle Model Study of the Influence of the Doping Profiles on the Characteristics of Field-Effect Transistors. Proc. NASECODE II Conf., pp. 244–249. Dublin: Boole Press 1981.Google Scholar
  66. [2.66]
    Morgan, T. N.: Broadening of Impurity Bands in Heavily Doped Semiconductors. Physical Review. 139, No. 1A, A 343-A 348 (1965).Google Scholar
  67. [2.67]
    Nag, B. R.: Diffusion Equation for Hot Electrons. Phys. Rev. BU, No. 8, 3031–3036 (1974).Google Scholar
  68. [2.68]
    Nag, B. R., Chakravarti, A. N.: Comments on the Generalized Einstein Relation for Semiconductors. Solid-State Electron. 18, 109–110 (1975).CrossRefGoogle Scholar
  69. [2.69]
    Nag, B. R.: Parallel Diffusion Constant of Hot Electrons in Silicon. Appl. Phys. Lett. 28, No. 9, 550–551 (1976).CrossRefGoogle Scholar
  70. [2.70]
    Nakagawa, A.: One-Dimensional Device Model of the npn Bipolar Transistor Including Heavy Doping Effects under Fermi Statistics. Solid-State Electron. 22, 943–949 (1979).CrossRefGoogle Scholar
  71. [2.71]
    Paul, R.: Halbleiterphysik. Heidelberg: Hüthig-Verlag 1975.Google Scholar
  72. [2.72]
    Phillips, A.: On Modeling the Bipolar Transistor with Realistic Impurity Distributions, Heavy Doping Physics. Technical Report TR22.2045, IBM (1976).Google Scholar
  73. [2.73]
    Polsky, B. S., Rimshans, J. S.: Two-Dimensional Numerical Simulation of Bipolar Semiconductor Devices Taking into Account Heavy Doping Effects and Fermi Statistics. Solid-State Electron. 26, No.4, 275–279 (1983).CrossRefGoogle Scholar
  74. [2.74]
    Raychaudhuri, D., Chattopadhyay, D.: Harmonie Generation Due to Ballistic Electron Transport in GaAs. Proc. IEEE 71, No. 3, 440–441 (1983).CrossRefGoogle Scholar
  75. [2.75]
    Robinson, J. E., Rodriguez, S.: Ionized Impurity Scattering in Degenerate Many-Valley Semiconductors. Physical Review 135, No. 3A, A779-A784 (1964).CrossRefGoogle Scholar
  76. [2.76]
    Seeger, K.: Semiconductor Physics. Wien-New York: Springer 1973.Google Scholar
  77. [2.77]
    Shibib, M. A.: Inclusion of Degeneracy in the Analysis of Heavily Doped Regions in Silicon Solar Cells and Other Semiconductor Devices. Solar Cells 3, 81–85 (1981).CrossRefGoogle Scholar
  78. [2.78]
    Shur, M. S., Eastman, L. F.: Ballistic Transport in Semiconductor at Low Temperature for Low-Power High-Speed Logic. IEEE Trans. Electron Devices ED-26, No. 11, 1677–1683 (1979).CrossRefGoogle Scholar
  79. [2.79]
    Slotboom, J. W., DeGraaff, H. C.: Measurements of Bandgap Narrowing in Si Bipolar Transistors. Solid-State Electron. 19, 857–862 (1976).CrossRefGoogle Scholar
  80. [2.80]
    Slotboom, J. W., DeGraaff, H. C.: Bandgap Narrowing in Silicon Bipolar Transistors. IEEE Trans. Electron Devices ED-24, No. 8, 1123–1125 (1977).CrossRefGoogle Scholar
  81. [2.81]
    Slotboom, J. W.: The pn-Product in Silicon. Solid-State Electron. 20, 279–283 (1977).CrossRefGoogle Scholar
  82. [2.82]
    Smith, R. A.: Semiconductors. Cambridge: Cambridge University Press 1978.Google Scholar
  83. [2.83]
    Stern, F.: Effect of Band Tails on Stimulated Emission of Light in Semiconductors. Physical Review 148, No. 1, 186–193 (1966).CrossRefGoogle Scholar
  84. [2.84]
    Stern, F.: Optical Absorption Edge of Compensated Germanium. Physical Review B3, No. 10, 3559–3560 (1971).Google Scholar
  85. [2.85]
    Stratton, R.: Semiconductor Current-Flow Equations (Diffusion and Degeneracy). IEEE Trans. Electron Devices ED-19, No. 12, 1288–1292 (1972).CrossRefGoogle Scholar
  86. [2.86]
    Sze, S. M.: Physics of Semiconductor Devices. New York: Wiley 1969.Google Scholar
  87. [2.87]
    Teitel, S. L., Wilkins, J. W.: Ballistic Transport and Velocity Overshoot in Semiconductors: Part I — Uniform Field Effects. IEEE Trans. Electron Devices ED-30, No. 2, 150–153 (1983).CrossRefGoogle Scholar
  88. [2.88]
    Thoraber, K. K.: Current Equations for Velocity Overshoot. IEEE Electron Device Lett. EDL-3, No. 3, 69–70 (1982).CrossRefGoogle Scholar
  89. [2.89]
    Tihanyi, J.: Integrated Power Devices. Proc. International Electron Devices Meeting, pp. 6–10 (1982).Google Scholar
  90. [2.90]
    Van Overstraeten, R. J., DeMan, H. J., Mertens, R. P.: Transport Equations in Heavy Doped Silicon. IEEE Trans. Electron Devices ED-20, 290–298 (1973).CrossRefGoogle Scholar
  91. [2.91]
    Van Vliet, K. M.: The Shockley-Like Equations for the Carrier Densities and the Current Flows in Materials with a Nonuniform Composition. Solid-State Electron. 23, 49–53 (1980).CrossRefGoogle Scholar
  92. [2.92]
    Wieder, A. W.: Emitter Effects in Shallow Bipolar Devices: Measurements and Consequences. IEEE Trans. Electron Devices ED-27, No. 8, 1402–1417 (1980).CrossRefGoogle Scholar
  93. [2.93]
    Williams, C. K., Glisson, T. H., Littlejohn, M. A., Hauser, J. R.: Ballistic Transport in GaAs. IEEE Electron Device Lett. EDL-4, No. 6, 161–163 (1983).CrossRefGoogle Scholar
  94. [2.94]
    Wulms, H. E. J.: Base Current of I2L Transistors. IEEE J. Solid-State Circuits SC-12, No. 2, 143–150 (1977).CrossRefGoogle Scholar
  95. [2.95]
    Ziman, J. M.: Electrons and Phonons. London: Clarendon Press 1963.Google Scholar
  96. [2.96]
    Zimmerl, O.: Iterative Lösung der Boltzmanngleichung für heiße Elektronen in InSb. Dissertation, Technische Hochschule Wien, 1972.Google Scholar

Copyright information

© Springer-Verlag/Wien 1984

Authors and Affiliations

  • Siegfried Selberherr
    • 1
  1. 1.Institut für Allgemeine Elektrotechnik und ElektronikTechnische Universität WienAustria

Personalised recommendations