Lecture #4

1. Lecture #4 OUTLINE Energy band model (revisited) Thermal equilibrium Fermi-Dirac distribution Boltzmann approximation Relationship between EF and n, p Read: Chapter 2 (Section 2.4)

2. Important Constants • Electronic charge, q = 1.610-19 C • Permittivity of free space, eo = 8.85410-14 F/cm • Boltzmann constant, k = 8.6210-5 eV/K • Planck constant, h = 4.1410-15 eVs • Free electron mass, mo = 9.110-31 kg • Thermal voltage kT/q = 26 mV EE130 Lecture 4, Slide 2

3. Dopant Ionization (Band Model) EE130 Lecture 4, Slide 3

4. Carrier Concentration vs. Temperature EE130 Lecture 4, Slide 4

5. Electrons and Holes (Band Model) • Electrons and holes tend to seek lowest-energy positions • Electrons tend to fall • Holes tend to float up (like bubbles in water) electron kinetic energy Ec Increasing hole energy Increasing electron energy Ev hole kinetic energy EE130 Lecture 4, Slide 5

6. Thermal Equilibrium • No external forces are applied: • electric field = 0, magnetic field = 0 • mechanical stress = 0 • no light • Dynamic situation in which every process is balanced by its inverse process • Electron-hole pair (EHP) generation rate = EHP recombination rate • Thermal agitation  electrons and holes exchange energy with the crystal lattice and each other  Every energy state in the conduction band and valence band has a certain probability of being occupied by an electron EE130 Lecture 4, Slide 6

7. Analogy for Thermal Equilibrium • There is a certain probability for the electrons in the conduction band to occupy high-energy states under the agitation of thermal energy (vibrating atoms) Sand particles Dish Vibrating Table EE130 Lecture 4, Slide 7

8. Fermi Function • Probability that an available state at energy E is occupied: • EF is called the Fermi energy or the Fermi level There is only one Fermi level in a system at equilibrium. If E >> EF : If E << EF : If E = EF : EE130 Lecture 4, Slide 8

9. Effect of Temperature on f(E) EE130 Lecture 4, Slide 9

10. Boltzmann Approximation Probability that a state is empty (occupied by a hole): EE130 Lecture 4, Slide 10

11. Equilibrium Distribution of Carriers • Obtain n(E) by multiplying gc(E) and f(E) Energy band diagram Density of States Carrier distribution Probability of occupancy EE130 Lecture 4, Slide 11

12. Obtain p(E) by multiplying gv(E) and 1-f(E) Energy band diagram Density of States Carrier distribution Probability of occupancy EE130 Lecture 4, Slide 12

13. Equilibrium Carrier Concentrations • Integrate n(E) over all the energies in the conduction band to obtain n: • By using the Boltzmann approximation, and extending the integration limit to , we obtain EE130 Lecture 4, Slide 13

14. Integrate p(E) over all the energies in the valence band to obtain p: • By using the Boltzmann approximation, and extending the integration limit to -, we obtain EE130 Lecture 4, Slide 14

15. Intrinsic Carrier Concentration EE130 Lecture 4, Slide 15

16. N-type Material Energy band diagram Density of States Carrier distribution Probability of occupancy EE130 Lecture 4, Slide 16

17. P-type Material Energy band diagram Density of States Carrier distribution Probability of occupancy EE130 Lecture 4, Slide 17

18. Dependence of EF on Temperature Ec K 0 0 3 K 0 0 4 EF for donor-doped EF for acceptor-doped 4 0 0 K 3 0 0 K Ev 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 Net Dopant Concentration (cm-3) EE130 Lecture 4, Slide 18

19. Summary • Thermal equilibrium: • Balance between internal processes with no external stimulus (no electric field, no light, etc.) • Fermi function • Probability that a state at energy E is filled with an electron, under equilibrium conditions. • Boltzmann approximation: For high E, i.e.E – EF > 3kT: For low E, i.e.EF– E > 3kT: EE130 Lecture 4, Slide 19

20. Relationship between EF and n, p : • Intrinsic carrier concentration : EE130 Lecture 4, Slide 20