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ECSE

6230
Semiconductor Devices and Models I
Lecture 4
Prof. Shayla Sawyer
Bldg. CII, Rooms 8225
Rensselaer Polytechnic Institute
Troy, NY 12180

3590
Tel. (518)276

2164
Fax. (518)276

2990
e

mail: sawyes@rpi.edu
1
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Outline
•
Carrier Concentration at Thermal Equilibrium
–
Introduction
–
Fermi Dirac Statistics
•
Donors and Acceptors
•
Determination of Fermi Level
•
Dopant Compensation
2
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Carrier Concentration Introduction
•
One of most important properties of a
semiconductor is that it can be doped with
different types and concentrations of
impurities
•
Intrinsic material

no impurities or lattice
defects
•
Extrinsic

doping, purposely adding impurities
–
N

type mostly electrons
–
P

type mostly holes
3
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Carrier Concentration Introduction
•
To calculate semiconductor electrical
properties, you must know the number of
charge carriers per cm
3
of the material
•
Must investigate distribution of carriers over
the available energy states
•
Statistics are needed to do so
Fermi

Dirac statistics
•
Distribution of electrons over a range of
allowed energy levels at thermal equilibrium
4
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Fermi

Dirac Distribution
Probability that an available energy
state at E will be occupied by an
electron at absolute temperature T
Mathematically,
E
F
(Fermi Energy) is the energy at
which f(E) = 1/2
The transition region in (E

E
F
)
from f(E) =1 to f(E) = 0 is within
3 k T.
When T
0,
E is discontinous at E = E
F
.
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Fermi

Dirac Distribution
•
To apply the Fermi

Dirac distribution, we must recall
that f(E) is the probability of occupancy of an
available
state
at E.
•
Where can we find available states?
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Carrier Concentration
At Thermal Equilbrium
•
Number of electrons (occupied conduction band
levels) given by:
•
Density of states g(E) can be approximated by
the
density near the bottom of the conduction band
where
M
C
is the number of equiv. minima
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Carrier Concentration
At Thermal Equlibrium
•
The integral can be evaluated as
Where N
C
is the effective
density of states in the
conduction band given by:
For the valence band, consider light and
heavy holes for the density of states
effective mass for holes (m
dh
)
and use similar equation
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Carrier Concentration
At Thermal Equlibrium: Intrinsic
•
For intrinsic material lies at some intrinsic level E
i
near the middle of the band gap, electron and hole
concentrations are
•
Law of mass action: product of maj. and min.
carriers is fixed
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Donors and Acceptors
•
Doping by substituting Si atoms with Column III or V of
the Periodic Table.
•
Very dilute doping level, typical 10
14
to 10
18
cm

3
, results
in discrete energy levels.
Donor level is neutral if filled with e

,
positively charged if empty.
e.g., P, As, and Sb in Si.
Acceptor level is neutral if empty,
negatively charged if filled with e

.
e.g., B and Al in Si.
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Donors and Acceptors
“Hydrogen

like” Model to describe
dopant atom ionization.
Hydrogen Atom
•
Ground state (n=1) ionization energy
of hydrogen is 13.6 eV.
•
To
estimate
ionization energy of
donors, replace m
0
with m* and
0
and
S
(e.g., 11.7
0
for Si).
E
D
= (
0
/
S
)
2
( m*/ m
0
) E
H
~
0.006 eV for Ge,
0.025 eV for Si,
0.007 eV for GaAs
E
A
~ 0.015 eV for Ge,
0.05 eV for Si,
0.05 eV for GaAs
http://gemologyproject.com/wiki/index.php?title=
The_Chemistry_of_Gemstones
kT~0.026eV
Comparable to thermal energies so ionization
is complete at room temperature
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Donors and Acceptor Levels
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Determination of Fermi Level
Intrinsic
Semiconductor

E
F
~ E
g
/ 2
Extrinsic
Semiconductor

E
F
adjusted to preserve
space charge neutrality
Space Charge Neutrality
n
0
+ N
A

= N
D
+
+ p
0
Total Neg. Charges = Total Positive Charges
electrons and ionized acceptors=holes and ionized donors
100% ionization assumed.
Ionized Concentration of Donors
When impurities are introduced:
where
g
D
is the ground state degeneracy of donor impurity
g
D
= 2 (i) electrons with either spin
(ii) no electrons at all
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Determination of Fermi Level
Ionized Acceptors
where
g
A
is the ground state degeneracy of acceptor impurity
g
A
= 4 for Ge, Si, and GaAs because
(i) Acceptor levels can receive electrons with either spin and
(ii) Valence band double degeneracy.
Space Charge Neutrality
N

type Semiconductor is assumed.
n=N
D
+
+p ~ N
D
+
therefore
Charge Neutrality
•
Since the material must balance
electrostatically, the Fermi level must adjust
such that charge neutrality remains.
•
The Fermi level therefore can be calculated for
a set given N
D
, E
D
, N
C
, and T
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Dopant Compensation
When both n

and p

type (donor and acceptor) impurities are present,
the space charge neutrality condition
n
0
+ N
A

= N
D
+
+ p
0
holds, even when the impurities are deep levels.
In an n

type semiconductor where N
D
>>>N
A
Fermi level can be obtained from
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Dopant Compensation
In an p

type semiconductor where N
A
>>>N
D
Fermi level can be obtained from
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Example Problem
A hypothetical semiconductor has an intrinsic carrier
concentration of 1.0 x 10
10
cm

3
at 300 K, it has a
conduction and valence band effective density of
states N
C
and N
V
both equal to 10
19
cm

3
.
a)
What is the band gap Eg?
b)
If the semiconductor is doped with N
d
= 1x10
16
donors/cm
3
, what are the equilibrium electron and
hole concentrations at 300K?
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Example Problem
A hypothetical semiconductor has an intrinsic carrier
concentration of 1.0 x 10
10
cm

3
at 300 K, it has a
conduction and valence band effective density of
states N
C
and N
V
both equal to 10
19
cm

3
.
c) If the same piece of semiconductor, already having N
d
= 1x10
16
donors/cm
3,
is also doped with N
a
= 2x10
16
acceptors/cm
3
, what are the new equiliblrium
electron and hole concentrations at 300 K?
a)
Consistent with your answer to part (c), what is the
Fermi level position with respect to the intrinsic
Fermi level, E
F
–
E
i
?
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