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
Spring 2007
EE130 Lecture 4, Slide 2
Dopant Ionization (Band Model)
Spring 2007
EE130 Lecture 4, Slide 3
Carrier Concentration vs. Temperature
Spring 2007
EE130 Lecture 4, Slide 4
Increasing electron energy
Increasing hole energy
Electrons and Holes (Band Model)
electron kinetic energy
Ec
hole kinetic energy
Ev
• Electrons and holes tend to seek lowest-energy positions
– Electrons tend to fall
– Holes tend to float up (like bubbles in water)
Spring 2007
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
Spring 2007
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)
Spring 2007
EE130 Lecture 4, Slide 7
Fermi Function
• Probability that an available state at energy E is occupied:
f (E) 
1
1 e
( E  E F ) / kT
• 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 :
Spring 2007
EE130 Lecture 4, Slide 8
Effect of Temperature on f(E)
Spring 2007
EE130 Lecture 4, Slide 9
Boltzmann Approximation
If E  EF  3kT , f ( E )  e
 ( E  EF ) / kT
If EF  E  3kT , f ( E )  1  e
( E  EF ) / kT
Probability that a state is empty (occupied by a hole):
1  f (E)  e
Spring 2007
( E  EF ) / kT
e
EE130 Lecture 4, Slide 10
 ( EF  E ) / kT
Equilibrium Distribution of Carriers
• Obtain n(E) by multiplying gc(E) and f(E)
Energy band
diagram
Spring 2007
Density of
States
Probability
of occupancy
EE130 Lecture 4, Slide 11
Carrier
distribution
• Obtain p(E) by multiplying gv(E) and 1-f(E)
Energy band
diagram
Spring 2007
Density of
States
Probability
of occupancy
EE130 Lecture 4, Slide 12
Carrier
distribution
Equilibrium Carrier Concentrations
• Integrate n(E) over all the energies in the
conduction band to obtain n:
n
top of conductionband
Ec
g c(E)f(E)dE
• By using the Boltzmann approximation, and
extending the integration limit to , we obtain
n  Nce
Spring 2007
 ( Ec  E F ) / kT
 2m kT 

where N c  2
 h

EE130 Lecture 4, Slide 13
*
n
2
3/ 2
• Integrate p(E) over all the energies in the
valence band to obtain p:
p
Ev
bottomof valence band
gv(E)1  f(E)dE
• By using the Boltzmann approximation, and
extending the integration limit to -, we obtain
p  Nve
Spring 2007
 ( E F  Ev ) / kT
 2m kT 

where N v  2
 h



EE130 Lecture 4, Slide 14
*
p
2
3/ 2
Intrinsic Carrier Concentration

np  N c e
 ( Ec  E F ) / kT
 Nc Nve
N e
 ( E F  Ev ) / kT
v
 ( Ec  Ev ) / kT
 Nc Nve
n
2
i
ni  N c N v e
Spring 2007
 EG / 2 kT
EE130 Lecture 4, Slide 15

 EG / kT
N-type Material
Energy band
diagram
Spring 2007
Density of
States
Probability
of occupancy
EE130 Lecture 4, Slide 16
Carrier
distribution
P-type Material
Energy band
diagram
Spring 2007
Density of
States
Probability
of occupancy
EE130 Lecture 4, Slide 17
Carrier
distribution
Dependence of EF on Temperature
Ec
n  Nce
 ( Ec  E F ) / kT
EF  Ec  kT ln N c n 
EF for donor-doped
EF for acceptor-doped
Ev
1013
1014
1015
1016
1017
1018
Net Dopant Concentration (cm-3)
Spring 2007
EE130 Lecture 4, Slide 18
1019
1020
Summary
• Thermal equilibrium:
– Balance between internal processes with no
external stimulus (no electric field, no light, etc.)
– Fermi function
f (E) 
1
1  e ( E  EF ) / kT
• Probability that a state at energy E is filled with an
electron, under equilibrium conditions.
• Boltzmann approximation:
 ( E  EF ) / kT
For high E, i.e. E – EF > 3kT: f ( E )  e
For low E, i.e. EF – E > 3kT:
Spring 2007
EE130 Lecture 4, Slide 19
1  f (E)  e
 ( EF  E ) / kT
• Relationship between EF and n, p :
n  Nce
 ( Ec  E F ) / kT
p  Nve
 ( E F  Ev ) / kT
• Intrinsic carrier concentration :
ni  N c N v e
Spring 2007
 EG / 2 kT
EE130 Lecture 4, Slide 20