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
2 atoms
1 atom
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
EE130/230A Fall 2013
Lecture 2, Slide 2
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
EE130/230A Fall 2013
Lecture 2, Slide 3
Energy Band Diagram
electron energy
Ec
Ev
distance
• 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
EE130/230A Fall 2013
Lecture 2, Slide 4
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
Increasing electron energy
Increasing hole energy
 Electrons tend to fall and holes tend to float up (like bubbles in water)
EE130/230A Fall 2013
electron kinetic energy
hole kinetic energy
Lecture 2, Slide 5
Ec
Ev
Ec represents the
electron potential
energy.
P.E.  Ec  Ereference
Electrostatic Potential, V
and Electric Field, E
• The potential energy of a
particle with charge -q is
related to the electrostatic
potential V(x):
P.E.  qV
1
 V  ( Ereference  Ec )
q

0.7 eV
EE130/230A Fall 2013
dV 1 dEc


dx q dx
• Variation of Ec with position is
called “band bending.”
Lecture 2, Slide 6
Measuring the Band Gap Energy
• EG can be determined from the minimum energy of
photons that are absorbed by the semiconductor
Ec
photon
hn > EG
Ev
Band gap energies of selected semiconductors
Semiconductor
Ge
Si
GaAs
Band gap energy (eV)
EE130/230A Fall 2013
0.67
Lecture 2, Slide 7
1.12
1.42
Density of States
E
Ec
dE
Ec
density of states, g(E)
Ev
Ev
g(E)dE = number of states per cm3 in the energy range between E and E+dE
Near the band edges:
gc (E) 
gv (E) 
3/ 2
8 2 *


m
n , DOS
h3
3/ 2
8 2 *


m
p , DOS
h3
EE130/230A Fall 2013
E  Ec  for E  Ec
Ev  E  for E  Ev
Lecture 2, Slide 8
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
Effective Mass, m*
• When an electron is moving inside a solid material, the potential
field will affect its movement.
Schrödinger equation:
2 2
E  
   V
2m0
E : total energy
Y : wave function
ħ : reduced Planck constant
p2
• For low kinetic energy E 
where p is the crystal momentum
2m *
i.e. a conduction electron behaves as a particle but with an
effective mass m*
EE130/230A Fall 2013
Lecture 2, Slide 9
EG and Material Classification
silicon dioxide
silicon
metal
Ec
Ec
EG = 1.12 eV
EG = ~ 9 eV
Ev
Ev
Ec
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)
EE130/230A Fall 2013
Lecture 2, Slide 10
Doping
• By substituting a Si atom with a special impurity atom (Column V
or Column III element), a conduction electron or hole is created.
Donors: P, As, Sb
Acceptors: B, Al, Ga, In
ND ≡ ionized donor concentration (cm-3)
NA ≡ ionized acceptor concentration (cm-3)
EE130/230A Fall 2013
Lecture 2, Slide 11
http://inventors.about.com/library/inventors/blsolar5.htm
Doping Silicon with a Donor
Example: Add arsenic (As) atom to the Si crystal
Si
Si
Si
Si
As
Si
Si
Si
Si
The loosely bound 5th valence electron of the As atom “breaks free” and
becomes a mobile electron for current conduction.
EE130/230A Fall 2013
Lecture 2, Slide 12
Doping Silicon with an Acceptor
Example: Add boron (B) atom to the Si crystal
Si
Si
Si
Si
B
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.
EE130/230A Fall 2013
Lecture 2, Slide 13
Solid Solubility of Dopants in Si
F. A. Trumbore, Bell Systems Technical Journal, 1960
EE130/230A Fall 2013
Lecture 2, Slide 14
ATOMS PER CUBIC CENTIMETER
Doping (Band Model)
Donor ionization energy
Ec
ED
EA
Ev
Acceptor ionization energy
Ionization energy of selected donors and acceptors in silicon
Donors
Sb P As
Dopant
Ionization energy (meV)
39
Ec-ED or EA-Ev
EE130/230A Fall 2013
45
Lecture 2, Slide 15
54
Acceptors
B
Al
In
45
67
160
Dopant Ionization
R.F. Pierret, Semiconductor Fundamentals, Figure 2.13
EE130/230A Fall 2013
Lecture 2, Slide 16
Charge-Carrier Concentrations
Charge neutrality condition:
N D + p = NA + n
At thermal equilibrium, np = ni2 (“Law of Mass Action”)
Note: Carrier concentrations depend on net dopant concentration!
EE130/230A Fall 2013
Lecture 2, Slide 17
n-type Material (n > p)
ND > NA (more specifically, ND – NA >> ni):
EE130/230A Fall 2013
Lecture 2, Slide 18
p-type Material (p > n)
NA > ND (more specifically, NA – ND >> ni):
EE130/230A Fall 2013
Lecture 2, Slide 19
Carrier Concentration vs. Temperature
R.F. Pierret, Semiconductor Fundamentals, Figure 2.22
EE130/230A Fall 2013
Lecture 2, Slide 20
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
EE130/230A Fall 2013
Lecture 2, Slide 21
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
EE130/230A Fall 2013
Lecture 2, Slide 22
Summary (cont’d)
• Ec represents the electron potential energy
Variation in Ec(x)  variation in electric potential V
Electric field

1 dEc 1 dEv


q dx q dx
• E - Ec represents the electron kinetic energy
EE130/230A Fall 2013
Lecture 2, Slide 23
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
EE130/230A Fall 2013
Lecture 2, Slide 24