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)

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

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

Carrier Concentration vs. Temperature EE130 Lecture 4, Slide 4

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

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

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

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

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

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

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

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

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

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

Intrinsic Carrier Concentration EE130 Lecture 4, Slide 15

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

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

Dependence of EF on Temperature Ec 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

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

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