lecture 2 outline semiconductor fundamentals (cont’d) – energy band model – band gap energy...
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Lecture 2
OUTLINE• Semiconductor Fundamentals (cont’d)
– 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
Potential Energy Profiles
Discrete allowed energy levels
When two atoms are in close proximity, the upper energy levels are shifted to bonding and anti-bonding levels.
Lecture 2, Slide 2EE130/230A Fall 2013
V(r)1/r is mostly a coulombic potential btwn the positive nucleus & negative electrons.
N atoms
1 atom 2 atoms
many bonding/anti-bonding levels
Energy states in Si atom energy bands in Si crystal
Si: From Atom to Crystal
• The highest nearly-filled band is the valence band• The lowest nearly-empty band is the conduction band
Lecture 2, Slide 3EE130/230A Fall 2013
R.F. Pierret, Semiconductor Fundamentals, Figure 2.5
Energy Band Diagram
• 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
Ec
Ev
ele
ctro
n e
ne
rgy
distance
Lecture 2, Slide 4EE130/230A Fall 2013
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)
Lecture 2, Slide 5
Incr
easi
ng h
ole
ener
gy
Incr
easi
ng e
lect
ron
ener
gy
Ec
Ev
electron kinetic energy
hole kinetic energyreferencecP.E. EE
Ec represents the electron potential energy.
EE130/230A Fall 2013
Electrostatic Potential, Vand Electric Field, E
• The potential energy of a particle with charge -q is related to the electrostatic potential V(x):
)(1
creference EEq
V
qVP.E.
Lecture 2, Slide 6
dx
dE
qdx
dV c1
• Variation of Ec with position is called “band bending.”
0.7 eV
EE130/230A Fall 2013
• EG can be determined from the minimum energy of photons that are absorbed by the semiconductor
Measuring the Band Gap Energy
Band gap energies of selected semiconductors
Lecture 2, Slide 7
Semiconductor Ge Si GaAsBand gap energy (eV) 0.67 1.12 1.42
Ec
Ev
photonh > EG
EE130/230A Fall 2013
g(E)dE = number of states per cm3 in the energy range between E and E+dE
Near the band edges:
Density of States
28)(
2/3*,3 cDOSnc EEm
hEg
for E Ec
Lecture 2, Slide 8
Ec
Ev
dE
E
density of states, g(E)Ec
Ev
for E Ev 28
)(2/3*
,3EEm
hEg vDOSpv
Si Ge GaAs
mn,DOS*/mo 1.08 0.56 0.067
mp,DOS*/mo 0.81 0.29 0.47
Electron and hole density-of-states effective masses
EE130/230A Fall 2013
• 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 : wave functionħ : reduced Planck constant
Effective Mass, m*
Vm
E 2
0
2
2
*2
2
m
pE
Lecture 2, Slide 9EE130/230A Fall 2013
EG and Material Classification
• 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)
silicon
Lecture 2, Slide 10
EG = 1.12 eVEG = ~ 9 eV
silicon dioxideEc
Ev
Ec
Ev
metal
Ev
Ec
EE130/230A Fall 2013
• By substituting a Si atom with a special impurity atom (Column V or Column III element), a conduction electron or hole is created.
Doping
Donors: P, As, Sb
ND ≡ ionized donor concentration (cm-3)
Lecture 2, Slide 11
Acceptors: B, Al, Ga, In
NA ≡ ionized acceptor concentration (cm-3)
EE130/230A Fall 2013http://inventors.about.com/library/inventors/blsolar5.htm
Doping Silicon with a DonorExample: Add arsenic (As) atom to the Si crystal
The loosely bound 5th valence electron of the As atom “breaks free” and becomes a mobile electron for current conduction.
Lecture 2, Slide 12EE130/230A Fall 2013
As
Si
Si SiSi
SiSi
SiSi
Doping Silicon with an Acceptor
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
Example: Add boron (B) atom to the Si crystal
EE130/230A Fall 2013
B
Si
Si SiSi
SiSi
SiSi
Solid Solubility of Dopants in Si
ATOMS PER CUBIC CENTIMETER
F. A. Trumbore, Bell Systems Technical Journal, 1960
Lecture 2, Slide 14EE130/230A Fall 2013
Doping (Band Model)
Ionization energy of selected donors and acceptors in silicon
Lecture 2, Slide 15
Donors AcceptorsDopant Sb P As B Al InIonization energy (meV)Ec-ED or EA-Ev
39 45 54 45 67 160
Ec
Ev
Donor ionization energy
ED
EA
Acceptor ionization energy
EE130/230A Fall 2013
Dopant Ionization
Lecture 2, Slide 16EE130/230A Fall 2013
R.F. Pierret, Semiconductor Fundamentals, Figure 2.13
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!
Lecture 2, Slide 17EE130/230A Fall 2013
n-type Material (n > p)
ND > NA (more specifically, ND – NA >> ni):
Lecture 2, Slide 18EE130/230A Fall 2013
p-type Material (p > n)
Lecture 2, Slide 19
NA > ND (more specifically, NA – ND >> ni):
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Carrier Concentration vs. Temperature
Lecture 2, Slide 20EE130/230A Fall 2013
R.F. Pierret, Semiconductor Fundamentals, Figure 2.22
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 semiconductorsuch that majority carrier concentration = net dopant concentration
Lecture 2, Slide 21EE130/230A Fall 2013
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 22EE130/230A Fall 2013
Summary (cont’d)
• Ec represents the electron potential energy
Variation in Ec(x) variation in electric potential V
Electric field
• E - Ec represents the electron kinetic energy
dx
dE
dx
dE vc
Lecture 2, Slide 23EE130/230A Fall 2013
Summary (cont’d)
• Dopants in silicon:– Reside on lattice sites (substituting for Si)
– Have relatively low ionization energies (<50 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 24EE130/230A Fall 2013
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