Density of States
and
Fermi Energy Concepts
How do Electrons and Holes Populate the Bands?
 Density of States Concept
gc ( E ) dE
gv ( E ) dE
General energy dependence of
gc (E) and gv (E) near the band edges.
The number of conduction band
states/cm3 lying in the energy
range between E and E + dE
(if E  Ec).
The number of valence band
states/cm3 lying in the energy
range between E and E + dE
(if E  Ev).
How do Electrons and Holes Populate the Bands?
 Density of States Concept
Quantum Mechanics tells us that the number of available states
in a cm3 per unit of energy, the density of states, is given by:
Density of States
in Conduction Band
Density of States
in Valence Band
How do electrons and holes populate the bands?
 Probability of Occupation (Fermi Function) Concept
 Now that we know the number of available states at each energy,
hen how do the electrons occupy these states?
t
 We need to know how the electrons are “distributed in energy”.
 Again, Quantum Mechanics tells us that the electrons follow the
“Fermi-distribution function”.
f (E) 
1
1 e
( E  E f ) / kT
Ef ≡ Fermi energy (average energy in the crystal)
k ≡ Boltzmann constant (k=8.61710-5eV/K)
T ≡Temperature in Kelvin (K)
 f(E) is the probability that a state at energy E is occupied.
 1-f(E) is the probability that a state at energy E is unoccupied.
 Fermi function applies only under equilibrium conditions, however, is
universal in the sense that it applies with all materials-insulators,
semiconductors, and metals.
How do electrons and holes populate the bands?
 Fermi-Dirac Distribution
Ef
How do electrons and holes populate the bands?
 Probability of Occupation (Fermi function) Concept
1
2
f (E) 
1
2
1
1 e
( E  E f ) / kT
kT = 0.0259eV @300K
 At T=0K, occupancy is “digital”: No occupation of states above Ef and
complete occupation of states below Ef .
 At T>0K, occupation probability is reduced with increasing energy.
f(E=Ef ) = 1/2 regardless of temperature.
How do electrons and holes populate the bands?
 Probability of Occupation (Fermi function) Concept
1
2
f (E) 
1
2
1
1 e
( E  E f ) / kT
kT = 0.0259eV @300K
 At T=0K, occupancy is “digital”: No occupation of states above Ef and
complete occupation of states below Ef .
 At T>0K, occupation probability is reduced with increasing energy.
f(E=Ef ) = 1/2 regardless of temperature.
 At higher temperatures, higher energy states can be occupied, leaving
more lower energy states unoccupied [1 - f(Ef )].
How do electrons and holes populate the bands?
 Probability of Occupation (Fermi function) Concept
 If E  Ef +3kT  e
( E  E f ) / kT
 1 and f ( E )  e
 ( E  E f ) / kT
 Consequently, above Ef +3kT the Fermi function or filled-state
probability decays exponentially to zero with increasing energy.
How do electrons and holes populate the bands?
Example 2.2
The probability that a state is filled at the conduction band edge (Ec)
is precisely equal to the probability that a state is empty at the
valence band edge (Ev).
Where is the Fermi energy locate?
Solution
The Fermi function, f(E), specifies the probability of electron occupying
states at a given energy E.
The probability that a state is empty (not filled) at a given energy E is equal
to 1- f(E).
f EC   1  f EV 
f  EC  
EC  EF
kT
1
1  e  EC  EF / kT

EV  E F
kT
1  f EV   1 
EF 
EC  EV
2
1
1  e  EV  EF / kT

1
1  e  EF  EV / kT
How do electrons and holes populate the bands?
 Probability of Occupation Concept
The density of electrons (or holes) occupying the states
in energy between E and E + dE is:
gc ( E ) f ( E ) dE
Electrons/cm3 in the conduction
band between E and E + dE
(if E  Ec).
gv ( E ) f ( E ) dE
Holes/cm3 in the conduction
band between E and E + dE
(if E  Ev).
0
Otherwise
How do electrons and holes populate the bands?
 Fermi function and Carrier Concentration
How do electrons and holes populate the bands?
 Probability of Occupation Concept
How do electrons and holes populate the bands?
Fermi-Dirac distribution function
describing the probability that an
allowed state at energy E is occupied
by an electron.
The density of allowed states for a
semiconductor as a function of
energy; note that g(E) is zero in the
forbidden gap between Ev and Ec.
The product of the distribution
function and the density-of-states
function
How do electrons and holes populate the bands?
 Typical band structures of Semiconductor
E
g (E)  (E–Ec)1/2
E
Ec+c
E
[1–f(E)]
CB
Area   n E ( E ) dE  n
For
electrons
Ec
number of
states per unit
energy per unit
volume
EF
Ec
probability of
occupancy of
a state
EF
Ev
Ev
For holes
nE(E)
number of electrons per unit
energy per unit volume
The area under nE(E) vs. E is the
electron concentration.
pE(E)
Area = p
VB
0
g(E)
Energy band
diagram
Density of states
fE)
nE(E) or pE(E)
Fermi-Dirac
probability
function
g(E) X f(E)
Energy density of electrons in
the CB
Metals vs. Semiconductors
Ef
Ef
Metal
Semiconductor
 Allowed electronic-energy-state systems for metal and semiconductors.
 States marked with an X are filled; those unmarked are empty.
Metals vs. Semiconductors
 Allowed electronic-energy states g(E)
Fermi level Ef immersed in the
continuum of allowed states.
The Fermi level Ef is at an intermediate
energy between that of the conduction band
edge and that of the valence band edge.
Ef
Ef
Metal
Semiconductor
How do electrons and holes populate the bands?
 Fermi function and Carrier Concentration
 Note that although the Fermi function has a finite value in the
gap, there is no electron population at those energies.
(that's what you mean by a gap)
 The population depends upon the product of the Fermi
function and the electron density of states. So in the gap
there are no electrons because the density of states is zero.
 In the conduction band at 0K, there are no electrons even
though there are plenty of available states, but the Fermi
function is zero.
 At high temperatures, both the density of states and the
Fermi function have finite values in the conduction band, so
there is a finite conducting population.
How do electrons and holes populate the bands?
 Energy Band Occupation
How do electrons and holes populate the bands?
 Intrinsic Energy (or Intrinsic Level)
Ef is said to equal
Ei (intrinsic energy)
when…
equal number of
electrons and holes.
How do electrons and holes populate the bands?
 Additional Dopant States
Intrinsic
Equal number
of electrons
and holes
n-type
More electrons
than Holes
p-type
More holes
than electrons
How do electrons and holes populate the bands?
 Pure-crystal energy-band diagram
How do electrons and holes populate the bands?
 n-type material
How do electrons and holes populate the bands?
 p-type material
Intrinsic, n-Type, p-Type Semiconductors
 Energy band diagrams
CB
Ec
Ef i
Ev
Ec
Ef n
Ec
Ev
Ef p
Ev
VB
(a) intrinsic
(b) n-type
(c) p-type
np = ni2
Note that donor and acceptor energy levels are not shown.
How do electrons and holes populate the bands?
 Heavily Doped Dopant States
E
Impurities
forming
bands
g(E)
CB
EFn
Ec
Ev
Degenerated n-type semiconductor
Large number of donors form a
band that overlaps the CB
CB
Ec
Ev
EFp
VB
Degenerated p-type
semiconductor