OUTLINE

1
Lecture 2

• OUTLINE
• Semiconductor Fundamentals (contd)
• Energy band model
• Band gap energy
• Density of states
• Doping
• Reading Pierret 2.2-2.3, 3.1.5 Hu 1.3-1.4,1.6,
  2.4

2
Potential Energy Profiles
1 atom
2 atoms
Discrete allowed energy levels
When two atoms are in close proximity, the upper
energy levels are shifted to bonding and
anti-bonding levels.
V(r)?1/r is mostly a coulombic potential btwn
the positive nucleus negative electrons.
N atoms
many bonding/anti-bonding levels
Lecture 2, Slide 2
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Si From Atom to Crystal
R.F. Pierret, Semiconductor Fundamentals, Figure
2.5
Energy states in Si atom ? energy bands in Si
crystal

• The highest nearly-filled band is the valence
  band
• The lowest nearly-empty band is the conduction
  band

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Energy Band Diagram
Ec
Ev

• Simplified version of energy band model, showing
  only the bottom edge of the conduction band (Ec)
  and the top edge of the valence band (Ev)
• Ec and Ev are separated by the band gap energy EG

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Electrons and Holes (Band Model)

• Conduction electron occupied state in the
  conduction band
• Hole empty state in the valence band
• Electrons holes tend to seek lowest-energy
  positions
• ? Electrons tend to fall and holes tend to float
  up (like bubbles in water)

Ec
Ec represents the electron potential energy.
Ev
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Electrostatic Potential, Vand Electric Field, E

• The potential energy of a particle with charge -q
  is related to the electrostatic potential V(x)

• Variation of Ec with position is called band
  bending.

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Measuring the Band Gap Energy

• EG can be determined from the minimum energy of
  photons that are absorbed by the semiconductor

Ec
photon hn gt EG
Ev
Band gap energies of selected semiconductors
Semiconductor Ge Si GaAs
Band gap energy (eV) 0.67 1.12 1.42
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Density of States
E
dE
Ec
Ec
density of states, g(E)
Ev
Ev
g(E)dE number of states per cm3 in the energy
range between E and EdE Near the band edges
Electron and hole density-of-states effective
masses
Si Ge GaAs
mn,DOS/mo 1.08 0.56 0.067
mp,DOS/mo 0.81 0.29 0.47
Lecture 2, Slide 8
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Effective Mass, m

• When an electron is moving inside a solid
  material, the potential field will affect its
  movement.
• For low kinetic energy where p
  is the crystal momentum
• i.e. a conduction electron behaves as a particle
  but with an effective mass m

Schrödinger equation E total energy Y
wave function h reduced Planck constant
Lecture 2, Slide 9
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EG and Material Classification
silicon
Ec
EG 1.12 eV
Ev

• Neither filled bands nor empty bands allow
  current flow
• Insulators have large EG
• Semiconductors have small EG
• Metals have no band gap (conduction band is
  partially filled)

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Doping

• By substituting a Si atom with a special impurity
  atom (Column V or Column III element), a
  conduction electron or hole is created.

ND ionized donor concentration (cm-3)
NA ionized acceptor concentration (cm-3)
http//inventors.about.com/library/inventors/blsol
ar5.htm
Lecture 2, Slide 11
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Doping Silicon with a Donor
Example Add arsenic (As) atom to the Si crystal
Si
Si
Si
As
Si
Si
Si
Si
Si
The loosely bound 5th valence electron of the As
atom breaks free and becomes a mobile electron
for current conduction.
Lecture 2, Slide 12
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Doping Silicon with an Acceptor
Example Add boron (B) atom to the Si crystal
Si
Si
Si
B
Si
Si
Si
Si
Si
The B atom accepts an electron from a neighboring
Si atom, resulting in a missing bonding electron,
or hole. The hole is free to roam around the
Si lattice, carrying current as a positive charge.
Lecture 2, Slide 13
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Solid Solubility of Dopants in Si
F. A. Trumbore, Bell Systems Technical Journal,
1960
Atoms per cubic centimeter
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Doping (Band Model)
Ec
Ev
Ionization energy of selected donors and
acceptors in silicon
Donors Donors Donors Acceptors Acceptors Acceptors
Dopant Sb P As B Al In
Ionization energy (meV) Ec-ED or EA-Ev 39 45 54 45 67 160
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Dopant Ionization
R.F. Pierret, Semiconductor Fundamentals, Figure
2.13
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Charge-Carrier Concentrations
Charge neutrality condition ND p NA n At
thermal equilibrium, np ni2 (Law of Mass
Action)
Note Carrier concentrations depend on net dopant
concentration!
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n-type Material (n gt p)
ND gt NA (more specifically, ND NA gtgt ni)
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p-type Material (p gt n)
NA gt ND (more specifically, NA ND gtgt ni)
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Carrier Concentration vs. Temperature
R.F. Pierret, Semiconductor Fundamentals, Figure
2.22
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Terminology

• donor impurity atom that increases n
• acceptor impurity atom that increases p
• n-type material contains more electrons than
  holes
• p-type material contains more holes than
  electrons
• majority carrier the most abundant carrier
• minority carrier the least abundant carrier
• intrinsic semiconductor n p ni
• extrinsic semiconductor doped semiconductor
• such that majority carrier concentration net
  dopant concentration

Lecture 2, Slide 21
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Summary

• Allowed electron energy levels in an atom give
  rise to bands of allowed electron energy levels
  in a crystal.
• The valence band is the highest nearly-filled
  band.
• The conduction band is the lowest nearly-empty
  band.
• The band gap energy is the energy required to
  free an electron from a covalent bond.
• EG for Si at 300 K 1.12 eV
• Insulators have large EG semiconductors have
  small EG

Lecture 2, Slide 22
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Summary (contd)

• Ec represents the electron potential energy
• Variation in Ec(x) ? variation in electric
  potential V
• Electric field
• E - Ec represents the electron kinetic energy

Lecture 2, Slide 23
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Summary (contd)

• Dopants in silicon
• Reside on lattice sites (substituting for Si)
• Have relatively low ionization energies (lt50 meV)
  
• ? ionized at room temperature
• Group-V elements contribute conduction electrons,
  and are called donors
• Group-III elements contribute holes, and are
  called acceptors

Dopant concentrations typically range from 1015
cm-3 to 1020 cm-3
Lecture 2, Slide 24
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