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SEMICONDUCTORS - PROBLEMS
electron mass = 9.11 x 10–31kg,
electron charge = 1.60 x 10–16C,
Boltzmann constant = 1.38 x 10–23J/K, Planck’s constant = 6.63 x 10–34J.s
1. Calculate the total number of energy states in silicon between the lowest level
in the conduction band and a level kT above this level, at T = 300 K. k =1.38
x10-23. J/K. electron mass = 9.11x 10-31 kg., effective mass of electron in the
conduction band is 1.08 times that of free electron, Planck’s constant = 6.63 x
10-34 Js.
2. Determine the total number of energy states per unit volume in silicon
between the highest level in the valence band and a level k T below this level,
at T = 300K. the effective mass of the hole in the valence band is 0.56 times
the mass of free electron.
3. Calcualte the probability that an energy level (a) k T (b) 3 kT (c) 10 k T above
the Fermi-level is occupied by an electron.
4. The fermil-evel in a semiconductor is o.30 e V below the conduction band. (a)
what is the probability of occupation of an energy state at the bottom of the
concduction band at (i) 300 K (ii) 400 K (b) what is the probability of
occupation of an energy state at a level k T above the bottom of the
conduction band, at (i) 300 k (ii) 400 k.
5. The fermilavel in a semiconductor is 0.35 e V above the valence band (a)
what is the probability of non – occupation of an energy stateat the top of the
valence band at (i) 300 K (ii) 400 K (b) what is the probability of nonoccupation of an energy state at a level k T below the top pof valence at (i)
300 K (ii) 400 K
6. Calculate the temperature at which there is a 1 % probanbility of nonoccupation of an energy state at a level 0.30 e V below the fermilevel.
7. Calculate the temperature at which there is a 10-6 probability of occupation of
an energy state at a level 0.55 e V above the fermilvel.
8. An intrinsic semiconductor has energy gap of (a) 0.7 e V (b) 0.4 e V.
Calculate the probability of occupation of the lowest level in the conduction
band at (i) 0C (ii) 50C (ii) 100C.
9. The energy band gap in an intrinsic semiconductor is 0.67 eV. What is the
temperature at which the probability of occupation of an energy state by an
electron, at the bottom of the conduction band, is three times that at 16C.
(Ans: 42C)
10. The effective mass of the conduction electron in Si is 0.31 times the free
electron mass. Find the conduction electron density at 300 K, assuming that
the fermi-level lies exactly at the centre of the energy band gap (= 1.11 eV).
ANSWER: 2.1 x 1015 /m3
11. The effective mass of hole and electron in GaAs are respectively 0.48 and
0.067 times the free electron mass. The band gap energy is 1.43 eV. How
much above is its fermi-level from the top of the valence band at 300 K?
(Ans: 0.75 eV = 1.2 x 10–19J).
12. Find the conductivity of Si at 300 K in (a) intrinsic condition (b) with
donor impurity of 1 in 108 (c) with acceptor impurity of 1 in 5 x 107
and (d) with both the above impurities present simultaneously. The
intrinsic carrier density in Si is 1.5 x 1016 /m3. The mobility of electrons
is 0.13 m2/ V.s. The mobility of holes is 0.05 m2/ V.s. The density of
Si atoms is 5 x 1028/m3.
(Answer: 4.32 x 10–4 S/m, 10.4 S/m, 8.0 S/m, 4.0 S/m)
13. In intrinsic GaAs, the electron and hole mobilities are 0.85 and 0.04
m2/ V.s respectively and the effective masses of electron and hole
respectively are 0.068 and 0.50 times the electron mass. The energy
band gap is 1.43 eV. Determine the intrinsic carrier density and
conductivity at 300K.
(Answer: 1.98 x 1012/m3, 2.82 x 10–7 S/m)
15. The resistivity of an intrinsic semiconductor is 4.5 Ωm at 20 C and 2.0 Ωm at
32 C. What is the band gap energy? Assume the mobilities of electrons and
holes to be equal and the same at both the temperatures.
(Answer: 0.96 eV [1.54 x 10–19 J] )
16. The intrinsic charge carrier density in a semiconductor is 1.1 x 1016/m3.
Electron mobility is 0.17 m2/V.s, Hole mobility is 0.035 m2/V.s. Calculate the
resistivity.
(Answer: 2800 Ωm)
17. A sample of silicon at room temperature has an intrinsic resistivity of 2.5 x 103
Ωm. The sample is doped with 4 x 1016 donor atoms/m3 and 1016 acceptor
atoms/m3. Find the total current density if an electric field of 400 V/m is
applied across the sample. Electron mobility is 0.125 m2/V.s, Hole mobility is
0.0475 m2/V.s.
(Answer: 0.305 A/m2.)
20. A silicon sample has to be converted into an n-type semiconductor with a
conductivity of 0.022 S/m at room temperature. What should be the donor
atom density? Electron mobility is 0.125 m2/V.s
(Answer: 1.1 x 1018/m3)
21. The conductivity of an extrinsic Si semiconductor is 3.0 x 104 S/m. What are
the densities of electrons and holes if it is (a) n-type? (b) p-type? Intrinsic
carrier density is 1.5 x 1016/m3. Electron mobility is 0.125 m2/V.s, Hole
mobility is 0.0475 m2/V.s.
(Answer: 1.44 x 1024/m3, 1.56 x 108/m3, 3.8 x 1024/m3, 6.0 x107/m3.)
22. The band gap energy of an intrinsic semiconductor is 1.2 eV. What is the
ratio of its conductivity at 600 K and that at 300 K? Assume that the
mobilities do not change with temperature.
(Answer: 3.1 x 105)
23. A sample of pure Ge has an intrinsic charge carrier density of 2.5 x 1019/m3 at
300 K. It is doped with donor impurity of 1 in every 106 Ge-atoms. (a) What
is the resistivity of the doped-Ge? Electron mobility is 0.38 m2/V.s. Ge-atom
density is 4.2 x 1028/m3. (b) If this Ge-bar is 5.0 mm long and 25 x 10–12 m2
in cross-sectional area, what is its resistance? What is the voltage drop
across the Ge-bar for a current of 1A?
(Answer: 3.9 x 10–4Ωm, 78 kΩ, 78 mV)
24. The conductivity of a n-type semiconductor is 10.0 S/m and its electron
mobility is 0.050 m2/V.s. Determine the electron density.
(Answer: 12.5 x 1020/m3.)
25. An n-type germanium has a donor atom density of 1021/m3. It is arranged in a
hall-effect experiment where the magnetic induction is 0.5 T and current
density is 500 A/m2. What is the hall-voltage if the specimen is 3 mm thick?
(Answer: 4.7 mV)
26. A n-type semiconductor has a hall coefficient of 2.0 x 10–4m3/C and and its
conductivity is 1.0 S/m. Find its electron mobility.
(Answer: 0.020 m2/V.s)
27. The hall coefficient of a doped silicon sample is found to be
+3.66  10–4m3/c. The resistivity of the specimen is 8.93  10–3 Ωm.
Find the mobility and density of the charge carrier, assuming single
carrier conduction.
(Answer: 0.041 m2/V.s, 1.7 x 1022/m3)
28. A rectangular plate of a semiconductor has dimensions 2.0 cm along ydirection, 1.0 mm along z-direction. Hall probes are attached on its two
surfaces parallel to xz-plane and a magnetic field of 1.0 tesla is applied along
z-direction. A current of 3.0 mA is is set up along the x-direction. Calculate
the hall voltage measured by the probes, if the hall coefficient of the material
is 3.66 x 10–4m3/C. Also, calculate the charge carrier concentration.
(Answer: 1.1 mV, 1.7 x 1022/m3)
29. The hall-coefficient of a p-type semiconductor is 1.00 x 10–4m3/C. Its
conductivity is 500 S/m. Calculate the resistivity of the semiconductor and the
mobility of the holes in it.
(Answer: 0.0020 Ωm, 0.050m2/V.s)
32. The conductivity of intrinsic silicon is 4.17 x10–5/Ωm and 4.00 x 10–4Ωm, at
0 C and 27 C respectively. Determine the band gap energy of silicon.
(Answer: 1.11 eV)
33.The density of conduction electrons in pure silicon at room temperature is
1016/m3. (a) What fraction of silicon atoms must be replaced by phosphorous
atoms to increase the electron-density by a factor of 106? (b) What mass of
phosphorous would be needed to dope a 1.0 g sample of silicon to this
extent? (c) On average, how far apart are these phosphorous atoms in the
sample?
(d) What is the resistivity of this phosphorous-silicon sample?
(e) What hall voltage would you expect in a sample 100 m thick when a
current of 1 mA is passed perpendicular to a magnetic field of 0.10 Tesla?
The molar mass of silicon = 28.1 g/mol
The molar mass of phosphorous = 31.0 g/mol
The density of silicon = 2330 kg/m3
The density of phosphorous = 1820 kg/m3
Avogadro constant = 6.02  1023 / mol.
The mobility of electrons in phosphorous-silicon = 0.07 m2 / V.s.
(Answer: 10–6 , 2.2 x 10–7 g, 49 nm, 8.9 x 10–3 Ωm, 0.63 mV)
37.Calculate the contact potential in a pn-junction at 300 K with 1.0 x 1024
donor/m3 on p-side and 1.0 x 1021 acceptor/m3 on the n-side. The intrinsic
carrier density is 1.5 x 1016 /m3.
(Answer: 0.75 V)
38. Calculate the contact potential in a silicon pn-junction at 300 K with 1.0 x 1022
donor/m3 on p-side and 5.0 x 1023 acceptor/m3 on the n-side. The intrinsic
carrier density is 1.5 x 1016 /m3.
(Answer: 0.795 V)
39. Calculate the contact potential in a silicon pn-junction at 300 K with 2.0 x 1022
donor/m3 on p-side and 1.0 x 1021 acceptor/m3 on the n-side. The intrinsic
carrier density is 1.5 x 1016 /m3.
(Answer: 0.673 V)
40. The resistivities of the p-region and the n-region of a germanium pn-junction
are 0.060 m and 0.040 m, respectively. (a) Calculate the contact potential
and the potential energy barrier. (b) If the doping densities of both the p- and
n-regions are doubled, what are the new values of the contact potential and
the potential energy barrier? The intrinsic carrier density in germanium is 2.5
x 1019 / m3. The mobility of electrons = 0.38 m2/V.s, The mobility of holes =
0.18 m2/V.s. kT/e = 0.026 V at 300 K.
(Answer: (a) 0.154 V, 0,154 eV, (b) 0.190 V, 0.190 eV)
41.Calculate the space-charge width in a silicon pn-junction at 300 K, with the
doping densities of 1.0 x 1022 acceptor/m3 on p-side and 1.0 x 1021 donor/m3.
The intrinsic carrier density is 1.5 x 1016 /m3. The relative pemittivity of Si is
11.7. The permittivity of vacuum is 8.854 x 10–12 F/m.
(Answer: 0.952 m)
42.Calculate the junction capacitance and the total capacitance of a pn-junction
at 300 K with a reverse bias of 5.0 V. The doping densities are 1.0 x 10 22
acceptor/m3 on p-side and 1.0 x 1021 donor/m3. The intrinsic carrier density
is 1.5 x 1016 /m3. The relative pemittivity of the semiconductor is 11.7. The
permittivity of vacuum is 8.854 x 10–12 F/m. The cross-sectional area of the
pn-junction is 1.0 x 10–8 m2.
(Answer: 3.66 x 10–5 F/m2, 0.366 pF)
43.When a reverse bias is applied to a germanium pn-junction, the reverse
saturation current at room temperature is 0.30 A. Find the current through
the diode when a forward bias of 0.15 V is applied at room temperature.
Assume ideal behaviour for the germanium diode.
(Answer: 98.5 A)
44.The equilibrium current across an unbiased pn-junction is 10 A at 300 K.
Calculate the current when the junction is (a) forward biased by 0.1 volt, (b)
reverse biased by 0.1 volt at 300 K.
(Answer: 467 A, 10 A)