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EEE 352: Lecture 16
Intrinsic Semiconductors—Last time we pointed out
that we still need to determine “
How many electrons
and holes are present at temperature T?
”
To do this,
we need to determine:
* Density of states
⇒
Electrons
⇒
Holes
* Electron and hole carrier densities
Intrinsic Semiconductors
±
By
INTRINSIC
semiconductors, we mean:
¾
There are
NO IMPURITIES
¾
That is, the semiconductor is ideally
PURE
±
Hence, we study the properties of the thermally excited electrons
and holes.
±
We recall that Si would be an insulator, but has a small band gap
(only 1.1 eV) so that thermal excitation across the gap gives free
carriers that make it a very poor insulator (but a good
semiconductor).
±
Previously we saw that
SEMICONDUCTORS
are materials with a
SMALL
energy gap between empty and filled energy bands
±
Electrons can be
EXCITED
across this gap at higher temperatures thus
INCREASING
the conductivity
Density of States
AT FINITE TEMPERATURES
CURRENT IS CARRIED BY
HOLES
IN THE
VALENCE
BAND
AND
ELECTRONS
IN THE
CONDUCTION
BAND
FULL
VALENCE
BAND
EMPTY
CONDUCTION
BAND
ABSOLUTE ZERO
T = 0 K
ENERGY
GAP E
g
E
F
P(E)
FINITE TEMPERATURE
T > 0 K
E
F
P(E)
The THERMAL excitation of electrons
across the GAP creates electrons in the
conduction band and holes in the
valence band.
HOW MANY ARE THERE?
Density of States
n
(
E),p(E)
ENERGY
GAP
E
•
THE
UPPER
BRANCH IS THE
ELECTRON
DENSITY
OF STATES
•
THE
LOWER
BRANCH IS THE
HOLE
DENSITY OF
STATES
•
THE ELECTRON AND HOLE DENSITY OF STATES
VANISH
AT
CONDUCTION AND VALENCE
BAND
EDGES
, RESPECTIVELY
•
Previously we say that the density of states
VANISHES
for energies in the gap
* In semiconductors we can then define
TWO
types of density of states:
⇒
One for
ELECTRONS
excited
ABOVE
the energy gap
⇒
And one more for
HOLES
left
BELOW
the energy gap
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Definition of Energy Levels
This is a plot of the energy levels in
the momentum space.
We desire to
discuss them in the REAL SPACE.
E
c
 the lowest conduction band state
E
v
 the highest valence band state
The DENSITY OF STATES counts
the NUMBER os states at any
energy E.
Band Structure
ρ
(
E
)
E
This is our real space plot:
±
NEAR
the gap we assume that the density of states takes the form we derived
earlier Remembering too that
INCREASING
hole energy corresponds
⇒
To moving
DOWN
the valence band
±
We can then compute the
EFFECTIVE
number of electrons and holes available as a
function of
TEMPERATURE
Density of States
[]
G
v
c
v
h
c
e
E
E
E
E
E
m
E
p
Holes
E
E
m
E
n
Electrons
=
−
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
2
/
1
2
/
3
2
*
2
2
/
1
2
/
3
2
*
2
2
2
1
)
(
:
2
2
1
)
(
:
h
h
π
THE ELECTRON AND HOLE DENSITY OF STATES
ARE DEFINED
IN THE FOLLOWING MANNER:
n
(
E),p(E)
ENERGY
GAP
E
E
c
E
v
•
To compute the
EFFECTIVE
number of electrons and holes at any temperature:
* We need to know where the Fermi level lies
⇒
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 Spring '08
 Ferry

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