Density%20of%20States%20and%20Fermi%20Energy%20Concepts

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Density of StatesandFermi Energy Concepts
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How do Electrons and Holes Populate the Bands?

• Density of States Concept

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).
General energy dependence of gc (E) and gv (E)
near the band edges.
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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
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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, then how do the electrons
  occupy these states?
• We need to know how the electrons are
  distributed in energy.
• Again, Quantum Mechanics tells us that the
  electrons follow the Fermi-distribution
  function.

Ef Fermi energy (average energy in the
crystal) k Boltzmann constant
(k8.617?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.

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How do electrons and holes populate the bands?

• Fermi-Dirac Distribution

Ef
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How do electrons and holes populate the bands?

• Probability of Occupation (Fermi function) Concept

kT 0.0259eV _at_300K

• At T0K, occupancy is digital No occupation
  of states above Ef and complete occupation of
  states below Ef .
• At Tgt0K, occupation probability is reduced with
  increasing energy.
• f(EEf ) 1/2 regardless of temperature.

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How do electrons and holes populate the bands?

• Probability of Occupation (Fermi function) Concept

kT 0.0259eV _at_300K

• At T0K, occupancy is digital No occupation
  of states above Ef and complete occupation of
  states below Ef .
• At Tgt0K, occupation probability is reduced with
  increasing energy.
• f(EEf ) 1/2 regardless of temperature.
• At higher temperatures, higher energy states can
  be occupied, leaving more lower energy states
  unoccupied 1 - f(Ef ).

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How do electrons and holes populate the bands?

• Probability of Occupation (Fermi function) Concept

• If E ? Ef 3kT ?
• Consequently, above Ef 3kT the Fermi function or
  filled-state probability decays exponentially to
  zero with increasing energy.

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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).
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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
Electrons/cm3 in the conduction band between E
and E dE (if E ? Ec).
Holes/cm3 in the conduction band between E and E
dE (if E ? Ev).
0 Otherwise
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How do electrons and holes populate the bands?

• Fermi function and Carrier Concentration

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How do electrons and holes populate the bands?

• Probability of Occupation Concept

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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
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How do electrons and holes populate the bands?

• Typical band structures of Semiconductor

number of electrons per unit energy per unit
volume The area under nE(E) vs. E is the electron
concentration.
number of states per unit energy per unit volume
probability of occupancy of a state
g(E) X f(E) Energy density of electrons in the CB
Energy band diagram
Density of states
Fermi-Dirac probability function
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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.

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Metals vs. Semiconductors

• Allowed electronic-energy states g(E)

The Fermi level Ef is at an intermediate energy
between that of the conduction band edge and that
of the valence band edge.
Fermi level Ef immersed in the continuum of
allowed states.
Ef
Ef
Metal
Semiconductor
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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.

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How do electrons and holes populate the bands?

• Energy Band Occupation

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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.
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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
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How do electrons and holes populate the bands?

• Pure-crystal energy-band diagram

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How do electrons and holes populate the bands?

• n-type material

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How do electrons and holes populate the bands?

• p-type material

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Intrinsic, n-Type, p-Type Semiconductors

• Energy band diagrams

C
B
E
E
E
c
c
c
E
f
n
E
f
i
E
f
p
E
E
E
v
v
v
V
B
(
c
) p-type
(
a
) intrinsic
(
b
) n-type
np ni2
Note that donor and acceptor energy levels are
not shown.
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How do electrons and holes populate the bands?

• Heavily Doped Dopant States

E
C
B
E
C
B
n
F
Impurities forming bands
E
E
c
c
g
(
E
)
E
E
v
v
E
F
p
V
B
Degenerated n-type semiconductor Large number of
donors form a band that overlaps the CB
Degenerated p-type semiconductor