Band Population in Semiconductors
Lecture 6
Density of States , Fermi-Dirac
Function and Charge carriers
Recap
Electron and hole concentration has been calculated at
thermal equilibrium and their product is calculated.
N (E) is proportional to E
1/2
so the density of states in
conduction band increases with electron energy.
Fermi function becomes extremely small for large energies.
Electrons are accumulated at the bottom of the conduction
band edge and holes are accumulated at the top of the valence
band.
How do electrons and holes populate the bands?
E
f
≡ Fermi energy (average energy in the crystal)
k
≡ Boltzmann constant (
k
=8.617
10
-5
eV/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
.
Metals vs. Semiconductors
E
f
E
f
Allowed electronic-energy states
g
(E) or N (E )
Metal
Semiconductor
The Fermi level
E
f
is at an intermediate energy
between that of the conduction band edge and that
of the valence band edge.
Fermi level
E
f
immersed in the
continuum of allowed states.
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
.
Density of states
Fermi Dirac function
n and p
Energy band diagrams
n
0
p
0
= n
i
2
E
c
E
v
E
f
i
C
B
E
f
p
E
f
n
E
c
E
v
E
c
E
v
V
B
(
a
) intrinsic
(
c
)
p
-type
(
b
)
n
-type
Intrinsic, n-Type, p-Type Semiconductors
How do Electrons and Holes Populate the Bands?
The number of conduction band states/cm
3
lying in the energy range between
E
and
E
+
dE .
(if
E
E
c
).
The number of valence band states/cm
3
lying
in the energy range between
E
and
E
+
dE
(if
E
E
v
).
Density of States Concept
General energy dependence of
g
c
(
E
) and
g
v
(
E
) near the band edges.
Density of states in conduction band
increases in upward direction as a function
of square root of energy while density of
states in valence band increases in
downward direction as a function of square
root of energy .
Typical band structures of
Semiconductor
Energy band
diagram
Density of states
Fermi-Dirac
probability
function
probability of
occupancy of a
state
g(E)
X
f(E)
E
nergy density of electrons in the CB
number of electrons per unit energy
per unit
v
olume
The area under
n
E
(E)
vs.
E
is
the
electron concentration.
number of states
per unit energy per
unit volume
How do electrons and holes populate the bands?
Pure-crystal energy-band diagram
In an intrinsic semiconductor, n = p. So there are equal number of states in
equal-sized energy bands at the edges of the conduction and valence bands,
n=p implies that there is an equal chance of finding an electron at the
conduction band edge as there is of finding a hole at the valence band edge
:
f(EC) = 1−f(EV)
For an n-typesemiconductor, there are more electrons in the conduction band than
there are holes in the valence band. This also implies that the probability of finding an
electron near the conduction band edge is larger than the probability of finding a hole
at the valence band edge. Therefore, the Fermi level is closer to the conduction band
in an n-type semiconductor:
n-type material
p-type material
For a p-type semiconductor, there are more holes in the valence band than there
are electrons in the conduction band. This also implies that the probability of
finding an electron near the conduction band edge is smaller than the probability
of finding a hole at the valence band edge. Therefore, the Fermi level is closer to
the valence band in an p-type semiconductor
Summary
Density of states in conduction band increases in upward direction as a
function of square root of energy while density of states in valence band
increases in downward direction as a function of square root of energy .
Electrons accumulate at the edge of the conduction band in case of n-type
semiconductor and showing more concentration of electron than holes and in
case of p-type semiconductor holes accumulated at top edge of the valence
band showing more concentration of holes .
Distribution functions are nothing but the probability density functions used
to describe the probability with which a particular particle can occupy a
particular energy level. In n-type Fermi-Dirac function exaggerated towards
conduction band above E
c
and in case of p-type it is exaggerated to valence
band below E
v
.
Exploring the concepts of electron and hole population in semiconductor bands through the Fermi-Dirac function, density of states, and Fermi energy. Learn how the Fermi function influences carrier concentration, the difference between metals and semiconductors in band structure, and the behavior of electrons and holes at various energy levels based on temperature conditions.
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Presentation Transcript
Lecture 6 Density of States , Fermi-Dirac Function and Charge carriers
Recap Electron and hole concentration has been calculated at thermal equilibrium and their product is calculated. N (E) is proportional to E1/2 so the density of states in conduction band increases with electron energy. Fermi function becomes extremely small for large energies. Electrons are accumulated at the bottom of the conduction band edge and holes are accumulated at the top of the valence band.
How do electrons and holes populate the bands? 1 Ef Fermi energy (average energy in the crystal) k Boltzmann constant (k=8.617 10-5eV/K) T Temperature in Kelvin (K) = = f E ( ) E E kT ( / ) + + e 1 f f(E) is the probability that a state at energyE 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.
Metals vs. Semiconductors Allowed electronic-energy states g(E) or N (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
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.
Intrinsic, n-Type, p-Type Semiconductors Energy band diagrams CB Ec Ec Ec Ef n Ef i Ef p Ev Ev VB Ev (c) p-type (a) intrinsic (b) n-type n0p0= ni2
How do Electrons and Holes Populate the Bands? Density of States Concept gc The number of conduction band states/cm3 lying in the energy range between E and E + dE .(if E Ec). dE E gv ) ( E dE ( ) The number of valence band states/cm3 lying in the energy range between E and E + dE (if E Ev). Density of states in conduction band increases in upward direction as a function of square root of energy while density of states in valence downward direction as a function of square root of energy . General energy dependence of gc (E) and gv (E) near the band edges. band increases in
How do electrons and holes populate the bands? Typical band structures of Semiconductor g(E) (E Ec)1/2 E E E Ec+ [1 f(E)] CB For electrons = = = = Area nE E dE n ( ) nE(E) Ec Ec number of states per unit energy per unit volume number of electrons per unit energy per unit volume The area under nE(E) vs. E is the electron concentration. probability of occupancy of a state EF EF Ev pE(E) Ev Area = p For holes VB 0 fE) nE(E) or pE(E) g(E) X f(E) g(E) Energy band diagram Density of states Fermi-Dirac probability function Energy density of electrons in the CB
Pure-crystal energy-band diagram In an intrinsic semiconductor, n = p. So there are equal number of states in equal-sized energy bands at the edges of the conduction and valence bands, n=p implies that there is an equal chance of finding an electron at the conduction band edge as there is of finding a hole at the valence band edge : f(EC) = 1 f(EV)
n-type material For an n-typesemiconductor, there are more electrons in the conduction band than there are holes in the valence band. This also implies that the probability of finding an electron near the conduction band edge is larger than the probability of finding a hole at the valence band edge. Therefore, the Fermi level is closer to the conduction band in an n-type semiconductor:
p-type material For a p-type semiconductor, there are more holes in the valence band than there are electrons in the conduction band. This also implies that the probability of finding an electron near the conduction band edge is smaller than the probability of finding a hole at the valence band edge. Therefore, the Fermi level is closer to the valence band in an p-type semiconductor
Summary Density of states in conduction band increases in upward direction as a function of square root of energy while density of states in valence band increases in downward direction as a function of square root of energy . Electrons accumulate at the edge of the conduction band in case of n-type semiconductor and showing more concentration of electron than holes and in case of p-type semiconductor holes accumulated at top edge of the valence band showing more concentration of holes . Distribution functions are nothing but the probability density functions used to describe the probability with which a particular particle can occupy a particular energy level. In n-type Fermi-Dirac function exaggerated towards conduction band above Ec and in case of p-type it is exaggerated to valence band below Ev.
