density of states and fermi energy concepts. how do electrons and holes populate the bands? the...
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Density of Statesand
Fermi Energy Concepts
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How do Electrons and Holes Populate the Bands?
dEEgc )(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).
dEEgv )(
Density of States Concept
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 (k=8.61710-5eV/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.
kTEE feEf /)(
1
1)(
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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Fermi-Dirac Distribution
How do electrons and holes populate the bands?
Ef
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How do electrons and holes populate the bands?
Probability of Occupation (Fermi function) Concept
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.
kT = 0.0259eV @300KkTEE feEf /)(
1
1)(
21
21
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How do electrons and holes populate the bands?
Probability of Occupation (Fermi function) Concept
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 )].
kTEE feEf /)(
1
1)(
kT = 0.0259eV @300K
21
21
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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.
kTEEkTEE ff eEfe/)(/)(
)(and1
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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).
VC EfEf 1
kTEEC FCeEf /1
1
kTEEkTEEV VFFV eeEf // 1
1
1
111
kT
EE
kT
EE FVFC
2VC
F
EEE
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The density of electrons (or holes) occupying the states in energy between E and E + dE is:
How do electrons and holes populate the bands?
Probability of Occupation Concept
0 Otherwise
dEEfEgc )()(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).
dEEfEgv )()(
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Fermi function and Carrier Concentration
How do electrons and holes populate the bands?
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How do electrons and holes populate the bands?
Probability of Occupation Concept
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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?
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Typical band structures of Semiconductor
Ev
Ec
0
Ec+
EF
VB
CB
E
g(E)
g(E) (E–Ec)1/2
fE)
EF
E
Forelectrons
For holes
[1–f(E)]
Energy band diagram
Density of states Fermi-Diracprobability
function
probability of occupancy of a state
nE(E) or p
E(E)
E
nE(E)
pE(E)
Area = p
Area
Ec
Ev
ndEEn E )(
g(E) X f(E)Energy density of electrons in
the CB
number of electrons per unit energy per unit volumeThe area under nE(E) vs. E is the electron concentration.
number of states per unit energy per unit volume
How do electrons and holes populate the bands?
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Allowed electronic-energy-state systems for metal and semiconductors.
States marked with an X are filled; those unmarked are empty.
Metals vs. Semiconductors
Metal Semiconductor
Ef
Ef
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Metals vs. Semiconductors
Ef
Ef
Allowed electronic-energy states g(E)
Metal Semiconductor
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.
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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.
Fermi function and Carrier Concentration
How do electrons and holes populate the bands?
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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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IntrinsicEqual numberof electronsand holes
n-typeMore electrons than Holes
p-typeMore holes than electrons
How do electrons and holes populate the bands?
Additional Dopant States
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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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Energy band diagrams
np = ni2
Note that donor and acceptor energy levels are not shown.
Ec
Ev
Ef i
CB
Ef p
Ef n
Ec
Ev
Ec
EvVB
(a) intrinsic (c) p-type (b) n-type
Intrinsic, n-Type, p-Type Semiconductors
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CB
g(E)
E
Impurities forming ba
nds
EFp
Ev
Ec
EFn
Ev
Ec
CB
VB
Degenerated n-type semiconductorLarge number of donors form a band that overlaps the CB
Degenerated p-type semiconductor
How do electrons and holes populate the bands?
Heavily Doped Dopant States