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ECEE 302: Electronic Devices

7 October 2002. ECEE 302: Electronic Devices. Lecture 3. Physical Foundations of Solid State Physics, Semi-Conductor Energy Bands, and Charge Carriers. Outline (1 of 2). Atomic Bonding in Solids ionic covalent Energy Bands in a Solid Formation Band Theory of Solids

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ECEE 302: Electronic Devices

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  1. 7 October 2002 ECEE 302: Electronic Devices Lecture 3. Physical Foundations of Solid State Physics, Semi-Conductor Energy Bands, and Charge Carriers

  2. Outline (1 of 2) • Atomic Bonding in Solids • ionic • covalent • Energy Bands in a Solid • Formation • Band Theory of Solids • Direct and Indirect Transitions in Semiconductors • Variation of Energy Band Structure through alloying • Plane Waves • Charge Carriers • Electons and Holes • Effective Mass of the charge carrier • Intrinsic and Extrinsic Semi-conductors • Quantum Wells

  3. Outline (2 of 2) • Charge Carrier Concentrations (Density) • Fermi Level • Electron and Hole Concentrations at Equilibrium • n-Type and p-Type semi-conductors • Temperature Dependence • Compensation and Space Charge Neutrality • Carrier Drift in Electro-Magnetic Fields • Conductivity and Mobility • Resistivity (Theory of Resistance) • Effect of Temperature and Doping on Mobility • Hall Effect • Invariance of the Fermi Level at Equilibrium

  4. Ionic Bond • Ionic Bonding • exchange of an electron between two atoms so each acheives a closed shell • result is a positive (electron donor) and negative (electron acceptor) ion • ions attract forming a bond • Examples: NaCl, KCl, KFl, NaFl 3p6 3p5 2p6 3s1 Na+ looses electron to Cl Cl- gains electron from Na Na Valence = +1 Cl Valence = -1 Na+ Cl- Electrostatic Attraction forms basis for Atomic Bonding into Crystal Structure

  5. Valance Bond • Valance Bond: Bonding due to two atoms of complementary valance combining chemically to share electrons across the bond • Valance Band 4 (and 4): C, Si, Ge, SiC • Valance Band 3 and 5: GaAs, InP, • Valance Band 2 and 6: CdS, CdTe • Examples • Face Centered Cubic: Diamond (C), Silicon (Si)

  6. Energy Bands in a Solid • Formation of the Energy Bands Text Figure 3-3 • Band Theory of Solids Text Figure 3-4 • Insulator • Semi - Conductor • Conductor • Energy Gap Eg • Direct and Indirect Transitions in Solids Text Figure 3-5 • E=hn or n=E/h (the Einstein-Planck Relationship) • Plane Waves, Expectation Value, and Momentum • Variation of Energy Band Structure • AlloyingSze, Figure, p299 • Band Gap Behavior Text Figure 3-6

  7. Plane Waves Wave, Y(x,t) at time t x Wave, Y(x+Dx,t+Dt) at time t+Dt

  8. Expectation value of an operator • Operators represent measured quantities • The expectation value of an operator is the average value associated with the operator.

  9. Relationship between momentumand wave number (Textbook, p64)

  10. Relationship between momentumand wave number (Textbook, p64) (2 of 2)

  11. Charge Carriers in a Crystal • Electrons and Holes Text Figures 3-7 & 3-8 • Electron Hole Pairs - EHP • Effective Mass of the charge carriers Text Figure 3-9 • Derivation of effective mass formula • Realistic Band Structures Text Figure 3-10 • Intrinsic and Extrinsic Semi-conductors Text Figure 3-12 • donors ==yield electrons to the crystal (n-material) • acceptors == take electrons from the band and form holes (p-material) • intrinsic electron and hole concentrations ni • intrinsic semi-conductor electron-hole generation rates gi • intrinsic semi-conductor recombination rates ri • Use of Bohr model to calculate Binding Energy of an electron in the solid • Quantum Wells Text Figure 3-13

  12. Derivation of Effective Mass Formula • Effective Mass of electron or hole depends on structure (curvature) of the Energy Band Surface within the solid • We will determine the effective mass formula from consideration of a free electron

  13. Intrinsic Semi-Conductor Carrier Concentrations

  14. Use of Bohr Model to calculate Binding Energy of Electron in a Solid (1 of 2)

  15. Use of Bohr Model to calculate Binding Energy of Electron in a Solid (2 of 2)

  16. Example (1 of 3) • Calculate the approximate donor binding energy for GaAs (Textbook example 3-3)

  17. Example (2 of 3)

  18. Example (3 of 3)

  19. Charge Carrier Density (or Concentration) (1 of 2) • Fermi Level Text Figure 3-14 • Distributions • Maxwell Boltzman • Bose-Einstein • Fermi-Dirac • Properties of the Fermi-Dirac Distribution • The Fermi Level Energy is the energy level in the solid where there is a probability of 0.5=1/2 that an electron will be present in that state • Application of the Fermi distribution to the description of intrinsic semi-conductions Text Figure 3-15 • Density of States Text Appendix IV, page 525 • Electron and Hole Concentrations (densities) at Equilibrium (T=constant) Text Figure 3-16 • Electron and Hole Concentration Calculation as a function of temperature

  20. Quantum Statistics • Maxwell Boltzmann Distribution Function • Classical Distribution of particles • Multiple particles • Distinguishable particles • Bose-Einstein Distribution • Quantum Mechanical Distribution • Multiple, Indistinguishable particles, in same state • integer spin particles • Examples: photons, phonons, mesons • Fermi-Dirac Distribution (Quantum Mechanical, indistinguishable particles • Quantum Mechanical Distribution • Single, indistinguishable Particle, in same state • half-integer spin particles • Examples: electrons, protons, neutrons

  21. Comparison of the Distribution Functions

  22. Maxwell-Boltzmann (Classical) Distribution Law

  23. Bose-Einstein Distribution Law

  24. Physical Foundation of Planck’s Black Body Radiation Formula • Classical Description: All state share the same amount of energy. This results in the ultra-violet catastrophe • Quantum Mechanical Description: The states are weighted by the Bose-Einstein distribution. States of higher energy are less likely to occur. The energy is determined the overall thermal energy kT. Most photons (radiation) is of lower energy

  25. Fermi-Dirac Distribution Law

  26. Fermi Energy

  27. Properties of the Fermi-Dirac Distribution and the Fermi Energy, EF (1 of 3)

  28. Properties of the Fermi-Dirac Distribution and the Fermi Energy, EF (2 of 3)

  29. Properties of the Fermi-Dirac Distribution and the Fermi Energy, EF (3 of 3)

  30. Calculation of electron (hole) concentration (1 of 7) • Electron or Hole Concentration = number of electrons (holes) per unit volume that will be found at an energy level, E, within the conduction (valance) band of a solid at temperature T • The electron concentration in the conduction band depends on three factors • the fermi distribution, f(E,T)=probability an electron will be found with energy E within a solid at temperature T • density of states N(E)=number of available states at Energy E within the solid • E is an allowed energy (the electron cannot be at any energy within the band gap). For an electron to be a carrier it must be in the conduction band • The hole concentration in the valance band depends on three factors • the fermi distribution, [1- f(E,T)]=probability a hole will be found with energy E within a solid at temperature T • density of states N(E)=number of available states at Energy E within the solid • E is an allowed energy (the hole cannot be at any energy within the band gap). For a hole to be a carrier it must be in the conduction band

  31. Calculation of electron concentration (2 of 7)

  32. Calculation of hole concentration (3 of 7)

  33. Calculation of hole concentration (4 of 7)Density of States N(E)

  34. Calculation of hole concentration (5 of 7)Density of States N(E)

  35. Calculation of hole concentration (6 of 7)Density of States N(E)

  36. Calculation of electron & hole concentration (7 of 7)

  37. Calculation of electron & hole concentration for Semi-Conductors (1 of 2)

  38. Calculation of electron & hole concentration for Semi-Conductors (2 of 2)

  39. Relationship between intrinsic and doped Semi-conductors (1 of 2)

  40. Relationship between intrinsic and doped Semi-conductors (2 of 2)

  41. Example: Textbook, example 3-5, p88

  42. Charge Carrier Density (or Concentration) (2 of 2) • Temperature Dependence of Carrier Text Figure 3-17 & 18 Concentrations • Intrinsic Carrier Concentration as function of temperature • Space Charge • Compensation Text Figure 3-19 • Space Charge Neutrality

  43. Carrier Drift in Electro-Magnetic Fields • Conductivity and Mobility • Conductivity = property of solid to carry current • Mobility = ease with which an electron (or hole) can move in a solid • Atomic Model of Resistance (conductivity) • electrons collide (are scattered) by the atomic centers • mean free path = average length an electron (or hole) travels before it is scattered • conductivity effective mass • Resistance and Ohm’s Law • Effects of Temperature and Doping on Text Figure 3-22 & 23 Electron and Hole Mobility • Dependence of scattering on mobility • High Field Effects Text Figure 3-24 • Breakdown of Ohm’s Law • Hall Effect Text Figure 3-25 • Hall Voltage • Hall Coefficient • Method of determining charge carriers (electrons or holes)

  44. Conductivity Model

  45. Scattering of Electrons in a Solid (1 of 2)

  46. Scattering of Electrons in a Solid (2 of 2)

  47. Current Density and Conductivity

  48. Mobility (electrons and holes)

  49. Hall Effect (Figure 3-25 of Text) (1 of 2)

  50. Hall Effect (Figure 3-25 of Text) (2 of 2)

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