1. Read: Chapter 2 (Section 2.4)
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)
electron kinetic energy Ec
Increasing hole energy
Increasing electron energy Ev
hole kinetic energy
Electrons and holes tend to seek lowest-energy positions
Electrons tend to fall
Holes tend to float up (like bubbles in water)
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
Sand particles
Dish
Vibrating Table
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)
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
Probability
of occupancy
Carrier
distribution
EE130 Lecture 4, Slide 11

12. Obtain p(E) by multiplying gv(E) and 1-f(E)
Energy band
diagram
Density of
States
Probability
of occupancy
Carrier
distribution
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 Probability
of occupancy
Carrier
distribution
EE130 Lecture 4, Slide 16

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

18. Dependence of EF on TemperatureEc 3 K K 4
EF for donor-doped
EF for acceptor-doped 4 K 3 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