4. ANALYSIS OF CARRIER CONCENTRATIONS
IN SEMICONDUCTORS
4.1. Objective of the test
Analysis of physical processes that influence carrier concentrations in semiconductors. Investigation how carrier concentration
depends on impurity, temperature and parameters of semiconductor.
4.2. Theory and the main formulae
4.2.1. In an intrinsic semiconductor concentration ni of conduction electrons equals concentration pi of holes:
ni  pi  N C NV exp  W / 2kT  .
(4.1)
Here


 22m kT / h 
NC  2 2mnkT / h2
NV
3/ 2
2 3/ 2
p
,
(4.2)
,
(4.3)
N C is the effective density of states in the conduction band, NV – the
effective density of states in the valence band, W – gap energy, k –
Boltzmann’s constant, T – temperature, h – Plank’s constant, mn –
effective mass of an electron, m p – effective mass of a hole.
4.2.2. The concentration nn of conduction electrons in a donortype semiconductor at low temperatures (less than Ts ) is given by
nn  N D N C exp  WD / 2kT  ,
(4.4)
is donor concentration, WD – ionisation energy of donor
where N D
atom.
At medium temperatures (in the extrinsic
concentration nn equals concentration of donors:
nn  N D .
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range),
the
(4.5)
At high temperatures (higher than Ti ) an n-type semiconductor
has properties of an intrinsic semiconductor; the concentration of intrinsic carriers exceeds nn .
The above mentioned temperatures Ts and Ti are given by
TS 
WD
k ln( NC / N D )
and
Ti 
W
.
k ln( N C NV / N D2 )
(4.6)
4.2.3. The concentration of holes in an acceptor-type semiconductor can be found using Eqs (4.1)–(4.6) with some changes: NV
must be used instead of N C , N A (acceptor concentration) – instead of
ND .
4.2.4. The concentration of minority carriers in a doped semiconductor is calculated using mass action law:
(4.7)
np  ni2  pi2 .
4.2.5. The Fermi level in an n-type semiconductor in the extrinsic
range is always over the middle of the forbidden band. Its distance
from the bottom of the conduction band is given by
Wc  WF  kT ln
Nc
.
ND
(4.8)
4.2.6. The Fermi level in a p-type semiconductor in the extrinsic
range is bellow the middle of the forbidden band. Its distance from the
top of the valence band is given by
WF  Wv  kT ln
Nv
.
NA
(4.9)
4.2.7 Some parameters of semiconductors and impurities are
presented in Tables 4.1 and 4.2.
Table 4.1
Semiconductor
Ge
Si
GaAs
InP
Crystal lattice
Diamond type
Diamond type
Zinc blend type
Zinc blend type
15
Gap energy, eV
0.67
1.12
1.43
1.29
Table 4.2
Material
Impurity
type
B
Al
Ga
In
P
As
Sb
Acceptor
Acceptor
Acceptor
Acceptor
Donor
Donor
Donor


1.
2.
3.
4.
5.
6.
7.
8.
9.
Ionisation energy in Ge,
eV
WA
WD
0.0104
–
0.0102
–
0.0108
–
0.0112
–
–
0.0120
–
0.0127
–
0.0096
Ionisation energy in Si,
eV
WA
WD
0.045
–
0.057
–
0.065
–
0.072
–
–
0.044
–
0.049
–
0.039
4.3. Preparing for the test:
Using lecture-notes and referenced literature [1, p. 53–72],
examine physical processes that influence carrier concentrations
in semiconductors. Investigate how carrier concentrations depend
on impurity, temperature and parameters of semiconductors.
Prepare to answer the questions:
What types of carriers take part in conductance in an intrinsic
semiconductor? What is their nature?
How does carrier concentration change in an intrinsic semiconductor while temperature is increasing?
What carriers appear when covalent bond is broken?
What carriers appear due to ionisation of donor impurity atoms?
Are donor levels occupied by electrons at medium temperatures?
What is the position of the Fermi level in a donor-type semiconductor at medium temperatures?
Are acceptor levels occupied by electrons at medium temperatures?
What is the position of the Fermi level in an acceptor semiconductor at medium temperatures?
On what and how does the concentration of majority carriers in a
doped semiconductor at medium temperatures depend?
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10. On what and how does the concentration of minority carriers in a
doped semiconductor at medium temperatures depend?
4.4. In laboratory:
1. Answer the test question.
2. According to specified data calculate:
a) intrinsic carrier concentrations in two semiconductors with
different gap energies at given temperatures;
b) majority and minority carrier concentrations in a semiconductor at given temperatures in cases of different impurity
concentrations;
c) majority and minority carrier concentrations at given temperatures in cases of different impurities with the same
concentration;
d) the distance between the bottom of the conduction band and
the Fermi level in an n-type semiconductor at given
temperatures in the extrinsic range;
e) the distance between the top of the valence band and the
Fermi level in a p-type semiconductor at given temperatures
in the extrinsic range.
3. After calculations plot graphs and examine the results.
4. Prepare the report.
1.
2.
3.
4.
5.
4.5. Contents of the report
Objectives.
Initial data.
Results of calculations.
Graphs.
Conclusions.
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