Semiconductor Physics GATE Problems (Part - II)
1. Consider two energy levels: E1, E eV above Fermi level and E2, E eV below
the Fermi level. P1 and P2 are respectively the Probabilities of E1 being
occupied by an electron and E2 being empty. Then
(a) P1 > P2
(b) P1 = P2
(c) P1 < P2
(d) P1 and P2 depend on number of free electrons
[GATE 1987: 2 Marks]
Soln. Given
P1 โ Prob. of E1 being occupied by electron
P2 โ Prob. of E2 being empty (not occupied)
๐ธ1
๐ธ๐น
๐ธ2
E
E
Probability that state at energy E1 is occupied is
๐(๐ฌ๐) =๐
๐+๐(๐ฌ๐โ๐ฌ๐ญ)/๐๐ป
Probability that state is empty is given by ๐ โ ๐(๐ฌ๐)
From the plot of Fermi Prob. function vs energy, it is observed that
probability of state being occupied above EF is relatively very low.
While the state not occupying below EF (top portion of plot) can be
written as
๐ โ ๐(๐ฌ)
Which is much larger
๐ = 0 ๐พ
๐1 ๐2
๐ธ๐น ๐ธ
๐2 > ๐1
๐(๐ธ)
1
1/2
(Fermi Prob. Function vs energy for different temps.)
Note that probability of empty state is holes in the valence band.
So, ๐ท๐ > ๐ท๐
Option (c)
2. In an intrinsic Semiconductor the free electron concentration depends on
(a) Effective mass of electrons only
(b) Effective mass of holes only
(c) Temperature of the Semiconductor.
(d) Width of the forbidden energy hand of the semiconductor.
[GATE 1987: 2 Marks]
Soln. There is relationship between intrinsic concentration and density of
states in conduction and valence band
๐๐๐ = ๐ต๐ช๐ต๐ฝ . ๐
โ๐ฌ๐
๐๐ป
Or, ๐๐ โ ๐จ . ๐ป๐
๐โ . ๐โ๐ฌ๐
๐๐๐ปโ
Where
NC โ Density of states in conduction band
NV โ Density of states is valence band
Note, both NC and NV vary as ๐ป๐ ๐โ
So, ๐๐๐ โ ๐ป๐ ๐๐ ๐๐ โ ๐ป๐ ๐โ ๐๐ ๐ โ ๐ป๐ ๐โ
Option (c)
3. According to the Einstein relation, for any semiconductor the ratio of
diffusion constant to mobility of carriers
(a) Depends upon the temperature of the semiconductor.
(b) Depends upon the type of the semiconductor.
(c) Varies with life time of the semiconductor.
(d) Is a universal constant.
[GATE 1987: 2 Marks]
Soln. The Einstein relation is given by
๐ซ
๐=
๐ซ๐
๐๐=
๐ซ๐
๐๐=
๐๐ป
๐=
๐ป๐๐ฒ
๐๐, ๐๐๐
From the above relation we notice that ๐ซ/๐ depends on temperature.
Option (a)
4. Direct band gap semiconductors
(a) Exhibit short carrier life time and they are used for fabricating BJTโs
(b) Exhibit long carrier life time and they are used for fabricating BJTโs
(c) Exhibit short carrier life time and they are used for fabricating Lasers.
(d) Exhibit long carrier life time and they are used for fabricating BJTโs
[GATE 1987: 2 Marks]
Soln. Direct band gap Semiconductors. (DBG):
In such semiconductors the transition from max point of valence band
to minimum of conduction band takes place without change in
momentum.
Ex. GaAs
They exhibit short carrier life time. Thus used for fabricating lasers.
In DBG semiconductor during the recombination the energy is released
in the form of light.
Option (c)
5. Due to illumination by light, the electron and hole Concentrations in a
heavily doped N type semiconductor increases by โ ๐ ๐๐๐ โ ๐ respectively
if ni is the intrinsic concentration then ,
(a) โ ๐ < โ ๐
(b) โ ๐ > โ ๐
(c) โ ๐ = โ ๐
(d) โ ๐ ร โ ๐
[GATE 1989: 2 Marks]
Soln. Due to light illumination electron hole pair generation occurs
So, โ๐= โ๐
Where, โ๐ is increase in electron concentration due to illumination of
light.
โ๐ is increase in hole concentration due to illumination by light.
Option (c)
6. A Silicon Sample is uniformly doped with 1016 phosphorus atoms / cm3 and
2 ร 1016 boron atoms / cm3. If all the dopants are fully ionized the material
is
(a) n โ type with carrier concentration of 1016/๐๐3
(b) p โ type with carrier concentration of 1016/๐๐3
(c) p โ type with carrier concentration of 2 ร 1016/๐๐3
(d) n โ type with carrier concentration of 2 ร 1016/๐๐3
[GATE 1991: 2 Marks]
Soln. Given,
Phosphorus atom of 1016 will make n โ type semiconductor. Since
dopants are fully ionized the number of electrons will be equal to donor
atoms i.e. ๐ต๐ซ = ๐ = ๐๐๐๐/๐๐๐
Similarly,
๐ต๐จ = ๐ = ๐ฉ๐๐๐๐ ๐๐๐๐๐ = ๐ ร ๐๐๐๐/๐๐๐
Note NA > ND
The resultant maternal will be p โ type semiconductor
Carrier concertation
= ๐ต๐จ โ ๐ต๐ซ
= ๐ ร ๐๐๐ โ ๐๐๐๐
= ๐๐๐๐/๐๐๐
Option (b)
7. The intrinsic carrier density at 300 0K is 1.5 ร 1010/๐๐3, in silicon for
n โ type silicon doped to 2.25 ร 1015 ๐๐ก๐๐๐ / ๐๐3 the equilibrium electron
and hole densities are
(a) ๐ = 1.5 ร 1015/๐๐3, ๐ = 1.5 ร 1010/๐๐3
(b) ๐ = 1.5 ร 1010/๐๐3, ๐ = 2.25 ร 1015/๐๐3
(c) ๐ = 2.25 ร 1015/๐๐3, ๐ = 1.0 ร 105/๐๐3
(d) ๐ = 1.5 ร 1010/๐๐3, ๐ = 1.5 ร 1010/๐๐3
[GATE 1997: 2 Marks]
Soln. Given,
For n type semiconductor electron density
= ๐ = ๐ต๐ซ = ๐. ๐๐ ร ๐๐๐๐ ๐๐๐๐๐/๐๐๐
Intrinsic carrier concentration ๐๐ = ๐. ๐ ร ๐๐๐๐ / ๐๐๐
Note, here ๐ โซ ๐๐
Using Mass Action Law
๐. ๐ = ๐๐๐
๐ =๐๐
๐
๐=
(๐.๐ร๐๐๐๐)๐
๐.๐๐ร๐๐๐๐
= ๐๐๐๐๐๐๐๐/๐๐๐
Option (c)
8. The electron concentration in a sample of uniformly doped n โ type silicon
at 300 0K varies linearly from1017/๐๐3 ๐๐ก ๐ฅ = 0 ๐ก๐ 6 ร 1016/
๐๐3 ๐๐ก ๐ฅ = 2๐๐. Assume a situation that electrons are supplied to keep
this concentration gradient constant with time. If electronic charge is 1.6 ร
1019 coulomb and the diffusion constant ๐ท๐ = 35 ๐๐2/๐, the current
density in the silicon, if no electric field is present is
(a) Zero
(b) 120 A/cm2
(c) +1120 A/cm2
(d) โ1120 A/cm2
[GATE 1997: 2 Marks]
Soln. Since sample is of n โ type total electron current density is given by
๐ฑ๐ = ๐ ๐๐ . ๐ ๐ฌ + ๐ ๐ซ๐ ๐ ๐
๐ ๐
Given E = 0
So, there will be only diffusion current
๐ฑ๐ = ๐ซ๐. ๐ .๐ ๐
๐ ๐
n โ type
1017 ๐๐3โ 6 ร 1016 ๐๐3โ
๐ฅ = 0 ๐ฅ = 2๐๐
๐ ๐
๐ ๐=
๐ ร ๐๐๐๐ โ ๐๐๐๐
๐ ร ๐๐โ๐ โ ๐=
๐ ร ๐๐๐๐ โ ๐๐ ร ๐๐๐๐
๐ ร ๐๐โ๐
=โ๐ ร ๐๐๐๐
๐ ร ๐๐โ๐= โ๐ ร ๐๐๐๐
๐ฑ๐ = ๐ซ๐ . ๐ .๐ ๐
๐ ๐
= ๐๐ ร ๐. ๐ ร ๐๐โ๐๐ ร (โ๐ ร ๐๐๐๐)
= โ๐๐๐๐ ๐จ/๐๐๐
Option (d)
9. An n โ type silicon bar 0.1 cm long and 100 ยตm2 in cross โ sectional area
has a majority carrier concentration of 5 ร 1020/๐3 and the carrier mobility
is 0.13๐2/๐ฃ โ ๐ at 300 oK. If the charge of electron is 1.6 ร 10โ19
coulomb, then the resistance of the bar is
(a) 106ฮฉ
(b) 104ฮฉ
(c) 10โ1ฮฉ
(d) 10โ4ฮฉ
[GATE 1997: 2 Marks]
Soln. Given,
Si bar = 0.1 cm = 10-3 m.
Convert it into meters since all other parameters are in meters. Cross
sectional area (๐จ) = ๐๐๐ ๐๐๐ = ๐๐๐ ร ๐๐โ๐๐๐๐
๐ = ๐ ร ๐๐๐๐/๐๐
๐๐ = ๐. ๐๐ ๐๐/๐ โ ๐
๐ = ๐. ๐ ร ๐๐โ๐๐ ๐๐๐๐๐๐๐
Since semiconductor is of n โ type
๐๐ = ๐๐ ๐๐
๐น๐๐๐๐๐๐๐๐๐๐(๐) =๐
๐๐=
๐
๐๐๐๐
Also, Resistance =๐๐
๐จ
๐
๐๐จ=
๐๐โ๐
๐ ร ๐๐๐๐ ร (๐. ๐ ร ๐๐โ๐๐)(๐. ๐๐) ร (๐๐๐ ร ๐๐โ๐๐)
๐น = ๐๐๐๐
Option (a)
10. The resistivity of a uniformly doped n โ type silicon sample is 0.5 ฮฉ โ ๐๐.
If the Electron mobility (๐๐) is 1250 ๐๐2/๐ โ ๐ ๐๐ and the charge of an
electron is 1.6 ร 10โ19 coulomb, the donor impurity concentration (๐๐ท) in
the sample is
(a) 2 ร 1016/๐๐3
(b) 1 ร 1016/๐๐3
(c) 2.5 ร 1015/๐๐3
(d) 2 ร 1015/๐๐3
[GATE 2003: 2 Marks]
Soln. Given,
Resistivity of n โ type Silicon sample = ๐. ๐๐ โ ๐๐
๐๐ = ๐๐๐๐ ๐๐๐/๐ โ ๐
Resistivity for n โ type sample (๐) =๐
๐๐๐๐
Assuming complete ionization
๐ = ๐ต๐ซ
๐บ๐, ๐ต๐ซ =๐
๐ ๐๐ ๐
=๐
๐.๐ร๐๐โ๐๐ ร ๐๐๐๐ ร ๐.๐
= ๐๐๐๐/๐๐๐
Option (b)
11. A silicon sample โAโ is doped with 1018 atoms/cm3 of Boron. Another
sample โBโ of identical dimensions is doped with 1018 atoms/cm3 of
phosphorus. The ratio of Electron to hole mobility is 1/3. The ratio
conductivity of the sample A to B is
(a) 3
(b) 1/3
(c) 2/3
(d) 3/2
[GATE 2004: 2 Marks]
Soln. Sample โAโ doped with 1018 atoms / cm2 of Boron (trivalent) so P โ type
semiconductor
Sample โBโ doped with same no. of phosphorus (N - type)
๐๐
๐๐=
๐
๐
Note ๐๐ = ๐ ๐ ๐๐
๐๐ = ๐๐ ๐๐
๐๐
๐๐=
๐๐
๐๐=
๐
๐
Option (b)
12. A heavily doped n โ type semiconductor has the following data.
Hole โ electron mobility ratio: 0.4
Doping concentration: 4.2 ร 108 ๐๐ก๐๐๐ /๐3
Intrinsic concentration: 1.5 ร 104 ๐๐ก๐๐๐ /๐3
The ratio of conductance of the n โ type semiconductor to that of intrinsic
semiconductor of same material and at the same temperature is given by
(a) 0.00005
(b) 2,000
(c) 10,000
(d) 20,000
[GATE 2005: 2 Marks]
Soln. Given,
Heavily doped n โ type semiconductor
๐๐
๐๐= ๐. ๐
๐ = ๐. ๐ ร ๐๐๐ ๐๐๐๐๐/๐๐
๐๐ = ๐. ๐ ร ๐๐๐ ๐๐๐๐๐/๐๐
We have to find
๐๐
๐๐
Conductivity of n โ type semiconductor
๐๐ = ๐๐ ๐๐
Total conductivity
๐ = ๐(๐ ๐๐ + ๐ ๐๐)
For intrinsic semiconductor
๐๐ = ๐๐ ๐ (๐๐ + ๐๐)
So, ๐๐
๐๐=
๐ ๐ ๐๐
๐๐ ๐ (๐๐+๐๐)=
๐ ๐๐
๐๐ ๐๐ (๐+๐๐
๐๐)
=๐
๐๐ (๐+๐๐
๐๐)=
๐.๐ร๐๐๐
๐.๐ร๐๐๐(๐+๐.๐)
= ๐ ร ๐๐๐
Option (d)
13. The majority carriers in an n โ type semiconductor have an average drift
velocity โVโ in a direction perpendicular to a uniform magnetic field โBโ.
The electric field โEโ induced due to Hall effect acts in the direction.
(a) ๐ ร ๐ต
(b) ๐ต ร ๐
(c) Along โVโ
(d) Opposite to โVโ
[GATE 2005: 2 Marks]
Soln. As per the problem the
Block of n โ type semiconductor. Majority carriers will be electrons.
Electron motion in x โ direction. Magnetic field in z โ direction
๐ต๐ง
๐ฆ
๐ฅ
๐โ
๐ธ๐ฆ ๐โ
๐ง
Using Lorentz force equation.
๏ฟฝฬ ๏ฟฝ = ๐ . ๏ฟฝโโ๏ฟฝ ร ๏ฟฝโโ๏ฟฝ
= (โ๐) . ๐ . ๐ฉ in (โ๐) direction.
So, direction of force will be in y โ direction
So, electrons will get collected at the face shown
Option (b)
14. Silicon is doped with boron to a concentration of 4 ร 1017 ๐๐ก๐๐๐ /๐๐.
Assume the intrinsic carrier concentration of silicon to be 1.5 ร 1010/๐๐3
and the value of ๐พ๐
๐ to be 25mV at 300 K. Compared to un - doped silicon,
the Fermi level of doped silicon.
(a) Goes down by 0.13 eV
(b) Goes up by 0.13 eV
(c) Goes down by 0.427 eV
(d) Goes up by 0.427 eV
[GATE 2007: 2 Marks]
Soln. ๐บ๐ is doped with Boron i.e. P type semiconductor i.e. P โ type
semiconductor
๐ต๐จ = ๐ ร ๐๐๐๐ ๐๐๐๐๐/๐๐๐
๐๐ = ๐. ๐ ร ๐๐๐๐ / ๐๐๐
๐๐ป
๐= ๐๐ ๐๐
Since the semiconductor is of p โ type, the Fermi level will go down
with respect to intrinsic level.
The expression for quasi Fermi level for p โ type semiconductor in
terms of Acceptor concentration
๐ฌ๐ญ๐ โ ๐ฌ๐ญ๐ = ๐น๐ป ๐ฅ๐ง (๐ต๐จ
๐๐)
๐ฌ๐ญ๐ โ ๐ ๐๐ซ๐ฆ๐ข ๐ฅ๐๐ฏ๐๐ฅ ๐๐จ๐ซ ๐ข๐ง๐ญ๐ซ๐ข๐ง๐ฌ๐ข๐ ๐ฌ๐๐ฆ๐ข๐๐จ๐ง๐๐ฎ๐๐ญ๐จ๐ซ
๐ฌ๐ญ๐ โ ๐ ๐๐ซ๐ฆ๐ข ๐ฅ๐๐ฏ๐๐ฅ ๐๐จ๐ซ ๐๐จ๐ซ ๐ฉ โ ๐ญ๐ฒ๐ฉ๐ Semiconductor
So, ๐ฌ๐ญ๐ โ ๐ฌ๐ญ๐ = ๐๐ ร ๐๐โ๐ ๐ฅ๐ง (๐ร๐๐๐๐
๐.๐ร๐๐๐๐)
= ๐๐ ร ๐๐โ๐ ๐ฅ๐ง(๐. ๐๐ ร ๐๐๐)
= 0.425 ev
Option (c)
Common Data for Q.18 & Q.19
The silicon sample with unit cross-sectional area shown below is in
thermal equilibrium. The following information is given: T = 300oK,
electronic charge = 1.6 ร 10โ19๐ถ, thermal voltage = 26mV and electron
mobility = 1350 cm2/v-s
15. The magnitude of the electric field at ๐ฅ = 0.5๐๐ is
(a) 1KV/cm
(b) 5 KV/cm
(c) 10 KV/cm
(d) 25 KV/cm
[GATE 2010: 2 Marks]
Soln. Given silicon sample is in thermal equilibrium i.e. steady state is
reached.
๐ป = ๐๐๐ ๐ฒ
Electronics charge = ๐. ๐ ร ๐๐โ๐๐ ๐ช
Thermal voltage = 26 m V
๐๐ = ๐๐๐๐ ๐๐๐/๐ฝ โ ๐
๐ต๐ซ = ๐๐๐๐ / ๐๐๐
So, the electric filed in the given sample is
๐ฌ =๐ ๐ฝ
๐ ๐๐=
๐
๐ ร ๐๐โ๐= ๐๐๐๐ฝ/ ๐
Or, ๐ฌ = ๐๐ ๐ฒ๐ฝ / ๐๐
Note, the electric filed will be same throughout the sample
Option (c)
16. The magnitude of the electric drift current density x = 0 ยตm is
(a) 2.16 ร 104๐ด/๐๐2
(b) 1.08 ร 104๐ด/๐๐2
(c) 4.32 ร 103๐ด/๐๐2
(d) 6.48 ร 102๐ด/๐๐2
[GATE 2010: 2 Marks]
Soln. Electron drift current density is given by
๐ฑ = ๐๐ฌ = ๐ต๐ซ ๐ ๐๐ ๐ฌ
= ๐๐๐๐ ร ๐. ๐ ร ๐๐โ๐๐ ร ๐๐๐๐ ร ๐๐ ร ๐๐๐
= ๐. ๐ ร ๐๐๐๐ ร ๐๐
๐ฑ = ๐. ๐๐ ร ๐๐๐ ๐จ / ๐๐๐
Note, the drift current density will same throughout the sample, so it
will be same at ๐ = ๐. ๐
Option (a)
17. Assume electronic charge ๐ = 1.6 ร 10โ19 ๐ถ,๐๐
๐= 25 ๐๐ and electron
mobility ๐๐ = 1000 ๐๐2/๐ โ ๐ . If the concentration gradient of electrons
injected into a P โ type silicon sample is 1 ร 1021/๐๐4, the magnitude of
electron diffusion current density (in A/cm2) is __________.
[GATE 2014: 2 Marks]
Soln. Given
๐ = ๐. ๐ ร ๐๐โ๐๐ ๐ช
๐๐ป
๐= ๐๐ ๐๐ฝ
๐๐ = ๐๐๐๐ ๐๐๐/ ๐ฝ
Concertation gradient of electrons is ๐ ร ๐๐๐๐ / ๐๐๐
Diffusion current density is given by
๐ฑ๐(๐) = ๐ . ๐ซ๐
๐ ๐
๐ ๐
As per the Einstein relation which relates diffusivity and mobility
๐ซ๐ = (๐๐ป
๐) ๐๐ = ๐ฝ๐ป ๐๐
= ๐๐ ร ๐๐โ๐ ร ๐๐๐๐
๐ซ๐ = ๐๐
So, ๐ฑ๐(๐) = ๐ ๐ซ๐ ๐ ๐
๐ ๐
= ๐. ๐ ร ๐๐โ๐๐ ร ๐๐ ร ๐ ร ๐๐๐๐
= ๐๐ ร ๐๐๐ = ๐๐๐๐ ๐จ/๐๐๐
Ans. 4000 A / cm2
18. Consider a silicon sample doped with ๐๐ท = 1 ร 1015/๐๐3 donor atoms.
Assume that the intrinsic carrier concentration ๐๐ = 1.5 ร 1010/๐๐3. If the
sample is additionally doped with ๐๐ด = 1 ร 1018/๐๐3 acceptor atoms, the
approximate number of electrons/cm3 in the sample, at T =300K, will be ___
[GATE 2014: 2 Marks]
Soln. When the semiconductor is doped by both type of impurities then it is
called compensated semiconductor.
In the present problem semiconductor is doped by donor atoms
๐ต๐ซ = ๐ ร ๐๐๐๐ / ๐๐๐
And also doped by Acceptor atoms
๐ต๐จ = ๐ ร ๐๐๐๐ / ๐๐๐
Since the acceptor atoms are more so the semiconductor will be p โ
type. As per the charge neutrality equation
๐ต๐ซ+ + ๐ = ๐ต๐จ
โ + ๐
Or, ๐ท = (๐ต๐จโ โ ๐ต๐ซ
+) + ๐
๐ท โ (๐ต๐จ โ ๐ต๐ซ)
Also,
๐ =๐๐
๐
๐ท=
๐๐๐
(๐ต๐จ โ ๐ต๐ซ)
=๐. ๐๐ ร ๐๐๐๐
(๐๐๐๐ โ ๐๐๐๐)=
๐. ๐๐ ร ๐๐๐๐
(๐๐๐๐ ร ๐๐๐๐ โ ๐๐๐๐)
=๐. ๐๐ ร ๐๐๐๐
๐๐๐ ร ๐๐๐๐= ๐. ๐๐๐ ร ๐๐๐
= 225.2 Answer
19. An N โ type semiconductor having uniform doping is biased as shown in
the figure
N โ type semiconductor
V
If Ec is the lowest energy level of the conduction band, EV is the highest
energy level of the valance band and EF is the Fermi level, which one of the
following represents the energy band diagram for the biased N โ type
semiconductor?
(a) (b)
(c) (d)
๐ธ๐ถ
๐ธ๐น
๐ธ๐ถ
๐ธ๐น
๐ธ๐ ๐ธ๐
๐ธ๐ถ
๐ธ๐น ๐ธ๐ถ
๐ธ๐น
๐ธ๐
๐ธ๐
Soln.
๐ฌ(๐) =๐
๐ .๐ ๐ฌ๐
๐ ๐ ๐๐๐ ๐๐๐๐๐๐๐๐๐
Diagram indicates electron energies. One has to note that slope in the
bands must be such that electrons drift downhill in the field. Thus E
points uphill in the band diagram (Above equation)
Option (d)๐ธ๐ถ
๐ธ๐น
๐ธ๐
E+-
20. A dc voltage of 10 V is applied across an n โ type silicon bar having a
rectangular cross โ section and length of 1 cm as shown in figure. The donor
doping concentration ND and mobility of electrons ๐๐ are 1016 cm-3 and
1000 ๐๐2 ๐โ1 ๐ โ1, respectively. The average time (๐๐ ๐๐ ) taken by the
electrons to move from one end of the bar to other end is ______
10 V
๐ โ ๐บ๐
1 cm
[GATE 2015: 2 Marks]
Soln. Length of bar = 1 cm
๐ต๐ซ = ๐๐๐๐ / ๐๐๐
๐๐ = ๐๐๐๐ ๐๐๐ / ๐ฝ โ ๐บ
Electric filed (๐ฌ) = ๐๐ ๐ฝ ๐๐โ
Drift velocity of electrons
๐ฝ๐ = ๐๐ ๐ฌ = ๐๐๐๐ ร ๐๐
= ๐๐๐ ๐๐ ๐๐๐โ
Time taken by electrons to move from one end of the bar to the other
end
๐๐๐๐ ๐๐๐๐๐ = ๐ณ
๐ฝ๐ =
๐
๐๐๐
= ๐๐โ๐ ๐๐๐ = ๐๐๐ ๐ ๐๐๐ Answer