# Density of States and Fermi Energy Concepts How do Electrons and

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```					   Density of States
and
Fermi Energy Concepts
How do Electrons and Holes Populate the Bands?

q Density of States Concept

The number of conduction band
states/cm3 lying in the energy
range between E and E + dE
(if E ³ Ec).

The number of valence band
states/cm3 lying in the energy
range between E and E + dE
(if E £ Ev).

General energy dependence of
gc (E) and gv (E) near the band edges.
How do Electrons and Holes Populate the Bands?

q 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?

q Probability of Occupation (Fermi Function) Concept
Ø Now that we know the number of available states at each energy,
then how do the electrons occupy these states?
Ø We need to know how the electrons are “distributed in energy”.
Ø Again, Quantum Mechanics tells us that the electrons follow the
“Fermi-distribution function”.

Ef ≡ Fermi energy (average energy in the crystal)
k ≡ Boltzmann constant (k=8.617´10-5eV/K)
T ≡Temperature in Kelvin (K)

v f(E) is the probability that a state at energy E is occupied.
v 1-f(E) is the probability that a state at energy E is unoccupied.

Ø 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?

q Fermi-Dirac Distribution

Ef
How do electrons and holes populate the bands?

q Probability of Occupation (Fermi function) Concept

kT = 0.0259eV @300K

v At T=0K, occupancy is “digital”: No occupation of states above Ef and
complete occupation of states below Ef .
v 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?

q Probability of Occupation (Fermi function) Concept

kT = 0.0259eV @300K

v At T=0K, occupancy is “digital”: No occupation of states above Ef and
complete occupation of states below Ef .
v At T>0K, occupation probability is reduced with increasing energy.
f(E=Ef ) = 1/2 regardless of temperature.

v 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?

q Probability of Occupation (Fermi function) Concept

Ø If E ³ Ef +3kT Ú
Ø 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).
How do electrons and holes populate the bands?

q Probability of Occupation Concept
The density of electrons (or holes) occupying the states
in energy between E and E + dE is:

Electrons/cm3 in the conduction
band between E and E + dE
(if E ³ Ec).

Holes/cm3 in the conduction
band between E and E + dE
(if E £ Ev).

0            Otherwise
How do electrons and holes populate the bands?

q Fermi function and Carrier Concentration
How do electrons and holes populate the bands?

q 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?

q Typical band structures of Semiconductor
g (E) µ (E–Ec)1/2        E
E                                                                    E
Ec+c                                                    [1–f(E)]

CB
For                                      Area
electrons
Ec                                                                                                nE(E)
Ec
number of                                                             number of electrons per unit
states per unit                            probability of             energy per unit volume
EF                     energy per unit     EF                     occupancy of               The area under nE(E) vs. E is the
volume                                     a state                    electron concentration.

Ev                                                                                 Ev            pE(E)

Area = p
For holes
VB

0
g(E)                           fE)                                nE(E) or pE(E)
Energy band   Density of states                Fermi-Dirac                     g(E) X f(E)
diagram                                      probability            Energy density of electrons in
function                         the CB
Metals vs. Semiconductors

Ef

Ef

Metal                           Semiconductor

Ø Allowed electronic-energy-state systems for metal and semiconductors.
Ø States marked with an X are filled; those unmarked are empty.
Metals vs. Semiconductors

q Allowed electronic-energy states g(E)

Fermi level Ef immersed in the   The Fermi level Ef is at an intermediate
continuum of allowed states.     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?

q Fermi function and Carrier Concentration

Ø 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)

Ø 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.

Ø In the conduction band at 0K, there are no electrons even
though there are plenty of available states, but the Fermi
function is zero.

Ø 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?

q Energy Band Occupation
How do electrons and holes populate the bands?

q Intrinsic Energy (or Intrinsic Level)

Ef is said to equal                       equal number of
Ei (intrinsic energy)                     electrons and holes.
when…
How do electrons and holes populate the bands?

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?

q Pure-crystal energy-band diagram
How do electrons and holes populate the bands?

q n-type material
How do electrons and holes populate the bands?

q p-type material
Intrinsic, n-Type, p-Type Semiconductors

q Energy band diagrams

CB

Ec                      Ec                             Ec
Ef n
Ef i
Ef p
Ev                      Ev                             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?

q Heavily Doped Dopant States

E
CB
CB       EFn
Impurities
forming                    Ec                   Ec
bands
g(E)                  Ev                   Ev
EFp
VB

Degenerated n-type semiconductor         Degenerated p-type
Large number of donors form a            semiconductor
band that overlaps the CB

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