lecture #4 outline energy band model (revisited) thermal equilibrium fermi-dirac distribution –...
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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)
EE130 Lecture 4, Slide 2Spring 2007
Important Constants
• Electronic charge, q = 1.610-19 C
• Permittivity of free space, o = 8.85410-14 F/cm
• Boltzmann constant, k = 8.6210-5 eV/K
• Planck constant, h = 4.1410-15 eVs
• Free electron mass, mo = 9.110-31 kg
• Thermal voltage kT/q = 26 mV
EE130 Lecture 4, Slide 5Spring 2007
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
hole kinetic energy
Incr
easi
ng
ele
ctro
n e
ner
gy
Incr
easi
ng
ho
le e
ner
gy
Ec
Ev
EE130 Lecture 4, Slide 6Spring 2007
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 7Spring 2007
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)
Dish
Vibrating Table
Sand particles
EE130 Lecture 4, Slide 8Spring 2007
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 :
kTEE FeEf /)(1
1)(
EE130 Lecture 4, Slide 10Spring 2007
Boltzmann Approximation
Probability that a state is empty (occupied by a hole):
kTEEF
FeEfkTEE /)()( ,3 If
kTEEF
FeEfkTEE /)(1)( ,3 If
kTEEkTEE FF eeEf /)(/)()(1
EE130 Lecture 4, Slide 11Spring 2007
Equilibrium Distribution of Carriers
• Obtain n(E) by multiplying gc(E) and f(E)
Energy banddiagram
Density ofStates
Probabilityof occupancy
Carrier distribution
EE130 Lecture 4, Slide 12Spring 2007
• Obtain p(E) by multiplying gv(E) and 1-f(E)
Energy banddiagram
Density ofStates
Probabilityof occupancy
Carrier distribution
EE130 Lecture 4, Slide 13Spring 2007
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
band conduction of top
cE
c(E)f(E)dEgn
2/3
2
* /)( 2
2 where
h
kTmNeNn nc
kTEEc
Fc
EE130 Lecture 4, Slide 14Spring 2007
• 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
1band valenceof bottom vE
v dEf(E)(E)gp
2/3
2
* /)( 2
2 where
h
kTmNeNp pv
kTEEv
vF
EE130 Lecture 4, Slide 15Spring 2007
Intrinsic Carrier Concentration
2
/ /)(
/)( /)(
i
kTEvc
kTEEvc
kTEEv
kTEEc
n
eNNeNN
eNeNnpGvc
vFFc
2/ kTEvci
GeNNn
EE130 Lecture 4, Slide 16Spring 2007
N-type Material
Energy banddiagram
Density ofStates
Probabilityof occupancy
Carrier distribution
EE130 Lecture 4, Slide 17Spring 2007
P-type Material
Energy banddiagram
Density ofStates
Probabilityof occupancy
Carrier distribution
EE130 Lecture 4, Slide 18Spring 2007
Dependence of EF on Temperature
1013 1014 1015 1016 1017 1018 1019 1020
300K
400K
400K300K
kTEE
cFceNn /)( nNkTEE ccF ln
Net Dopant Concentration (cm-3)
Ec
Ev
EF for donor-doped
EF for acceptor-doped
EE130 Lecture 4, Slide 19Spring 2007
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:
kTEE FeEf /)(1
1)(
kTEE FeEf /)()(
kTEEFeEf /)()(1