Download band gap

Department of Physics
University of Jaffna
Experimental Report of
Band Gap
D.D.A.D.J. Prabodha
S 9493
Band Gap
PHY301M4
1.0 Abstract
This experiment was designed and conducted on behalf of finding the band gap of a given semiconductor. Here it has been given an n-Ge test piece. Practically it was obtained as
0.67698eV at room temperature (303K) and this value was deviated 4.15% from the theoretical
value 0.65eV.
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Band Gap
PHY301M4
2.0 INTRODUCTION
2.1 Energy band and Fermi level
The Bohr model of the atom tells us that electrons can only have certain discrete amounts of
energy corresponding to certain electron orbits around the atomic nucleus. The electrons have
a minimum energy state and only certain discrete higher energy states are allowed. A fixed
amount of energy is needed to jump the electron up into a higher energy state and when the
electron falls back into a lower energy state that energy is given up as electromagnetic
radiation. The highest filled band is called the valence band. The next higher band is the
conduction band, which is separated from the valence band by an energy gap, also called a
band gap. This band gap represents the energy required to knock electrons out of atoms into
the conduction band. In any material, for conduction to occur, there must be electrons
available in the highest energy band. The energy required for an electron to jump the band
gap can be provided by heat or some form of radiation or from an electric field and is usually
expressed in electron Volts (eV) where 1 eV is equivalent to 1.6 X 10-19 Joules (J).
In a metal the minimum energy needed to liberate an electron from the surface of the metal
is called the "work function".
CB
Conduction
Band (C B)
CB
Forbidden gap
Over
lap
Valence
Band (V B)
VB
Fermi level
VB
Conductor
Fermi
Level
Fermi level
Insulator
17
Figure2.1:- Energy band picture of materials
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Gap
(1eV)
(Several eV)
2
Semiconductor
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Band Gap
PHY301M4
2.2 Conductor Energy Bands
Already have free electrons in the conduction band which are available to carry the current
and a small band gap which makes it easy to jump more electrons into the conduction band.
In the conductors valence band & conduction band are overlap. Metals have a positive
coefficient of resistance since the thermal agitation of the electrons increases with temperature
and impedes electron flow.
2.3 Insulator Energy Bands
Have a wide band gap typically higher than 5 eV (electron Volts) with no electrons in the
conduction band, at ordinary temperatures no electron can reach the conduction band.
Conduction can occur if a high enough field is used to force electrons into the conduction
band but this usually results in the breakdown of the insulating material.
2.4 Semiconductor Energy Band
Have few electrons in the conduction band making them poor conductors but their relatively
low band gap (0.67 eV for Germanium and 1.12 eV for Silicon) permits their conductivity to
be increased by using an external stimulus to raise the energy level of the electrons. When
this occurs the higher energy electron breaks free from the covalent bond between the
semiconductor atoms creating an electron-hole pair of new charge carriers thus increasing the
conductivity of the semiconductor material. Adding small numbers of new current carriers by
doping allows dramatic changes to the conductivity of the semiconductor.
According to that there are two types of semiconductors such as intrinsic semiconductor and
extrinsic semiconductor. Technically, there are no intrinsic or extrinsic conductors. Intrinsic
and extrinsic conduction are terms that are applied to semiconductors. These terms refer to
methods of enhancing the conductivity of semiconductors.
The conductivity of
semiconductors arising from thermal excitation is called intrinsic conductivity. Another way
to increase the conductivity of a semiconductor is by doping (the addition of impurities). After
doping, the resulting conductivity is called extrinsic conductivity.
In doped semiconductors, extra energy levels are added.
The increase in conductivity with temperature can be modeled in terms of the Fermi function,
which allows one to calculate the population of the conduction band
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Band Gap
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At finite temperatures, the number of electrons which reach the conduction band and
contribute to current can be modeled by the Fermi function. That current is small compared
to that in doped semiconductors under the same conditions.
More free electrons
Free electrons
Donor level
Acceptor level
(Impurity
level)
(Impurity level)
Holes
Less holes
(a) Intrinsic semiconductor
(b) n-type semiconductor
More
Lessholes
free
electrons
(c) p-type semiconductor
Figure 2.2:- Energy band diagram of semiconductors.
2.5 Germanium Energy Bands
Germanium was the material on which early transistors were based. It has a low melting point
and is easy to work with and a low resistivity which helps in achieving high frequency
response. Unfortunately it also has a low maximum working temperature of 175 °C and
suffers from inherent high leakage currents due to its low band gap of only 0.67 eV.
For intrinsic semiconductors like silicon and germanium, the Fermi level is essentially
halfway between the valence and conduction bands. Although no conduction occurs at 0 K,
at higher temperatures a finite number of electrons can reach the conduction band and provide
some current.
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Band Gap
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0K (No electrons in conduction band)
Fermi level
300K
0.74eV
Figure 2.3:- Energy band diagram of Germanium semiconductor
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Band Gap
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3.0 THEORY
The conductivity of semiconductors is characteristically a function of temperature.
There ranges can be distinguished at low temperatures we have extrinsic conduction. As the temperature rises
charge carriers are activated from the impurities. At moderate temperatures
We talk of impurity depletion (saturation range). Since a further temperature rises. At high temperatures it is
intrinsic conduction.
Figure 3.1:- Conductivity of semi-conductor as a
function of the reciprocal of the
temperature.
Figure 3.2:- Conductivity of intrinsic range
At high temperatures it is intrinsic conduction which finally predominates. In this instance charge carriers are
additionally transferred by thermal excitation from the valence band to the conduction band. The temperature
dependence is in this case essentially described by an exponential function.

Eg 

 2 KT 
   0 exp 
………………………... (1)
Where,
𝐸𝑔 - Energy gap.
K - Boltzmann’s constant.
T - Absolute temperature.
Here,
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T 
Where,
UT

 T0
UT - Voltage across the thermocouple

40V
K
T0 – Room temperature
But, Resistivity
Conductivity


RA
l
1
l

 RA
……………………... (2)
From Ohm’s law
V  IR
……………………………… (3)
By using equations (2) and (3),
From equation (1)
 Eg  1
  ln( 0 )
ln( )  
 2K  T
y = -mx + c
From the above linear equation y = - m x+ c gives, gradient m
 Eg
m  
 2K
So, Energy gap
Where,



E g  2Km
K = 8.625x10-5 eV/K
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Band Gap
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4.0 PROCEDURE
Figure 4.1:- Experimental Set Up
As shown in the figure 4.1, the apparatus was arranged according to the setup diagram that has been given in
the procedure card. A millivolt meter was connected to the output terminals which come out from the
thermocouple that is placed behind the n-Ge test piece. An ac voltage source was applied to the heating coil
at the back of the test piece and the applied voltage has a capability of varying. Constant DC voltage value
was given n-Ge test piece and the given potential must not be exceeded the maximum allowable current
between the test piece all over the processing time. If it does, it has to be decreased the DC value immediately.
A miliampere meter and a voltage meter were connected accordingly on behalf of measuring the current and
voltage through the test piece. 100 Ω resistor was connected as series, right next to the test piece for protecting
from over current.
First, the alternating voltage was set to 2V. Checked the motionless temperature. Again the alternating voltage
was increased to 4V and notice the motionless temperature. Same method was done for 6V by keeping mind
that the maximum temperature of test piece would not be exceeded. Then the test piece was allowed to cool
and the current, voltage variations were recorded as a function of temperature in the steps of 5K. Steps of
temperature were calculated from the millivolt meter readings that of 200µV was to 5K as the equation given
for thermocouple, 40µV per Kelvin.
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Band Gap
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5.0 RESULTS
Graph 5.1:- Logarithm of conductivity of Germanium test piece versus reciprocal of absolute temperature
for graph
m
= 3924.50623 K
Practical value of Band gap of Ge
= 2 × 3924.50623 K × 8.625 × 10−5 eVK −1
= 0.67698eV
Theoretical value of band gap of Ge
= 0.65eV
Deviation percentage of band gap
= 4.15%
Dimension of Ge sample
= 20mm×10mm×1mm
Room temperature
= 303 K
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Band Gap
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6.0 CONCLUSION
This practical was arranged to determine the band gap of n-Ge test piece. Logarithmic of conductivity versus
reciprocal of absolute temperature graph was plotted. Gradient of the graph gave the opportunity to derive the
band gap or 𝐸𝑔 of test piece and it was determined as 0.67698eV at room temperature (303K). The obtained
value was deviated 4.15% from the theoretical value 0.65eV.
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Band Gap
PHY301M4
7.0 DISCUSSION
Here we are dealing with a test piece which has a small energy gap and hence the values must be
so accurate for a better answer. But the instruments which we used here were not ideal and even
the connectors and connecting wires have a considerable resistance. This facts directly affect to
the final result.
As the readings; voltage across the test piece and the current flow through the test piece must be
taken simultaneously for different voltages of thermocouple. It was too difficult take values
simultaneously as they had been changing rapidly all over the practical. Values taking from a
detector that is changing rapidly can be given an error values even at the noticing down. It will be
affected to the calculations.
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Band Gap
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8.0 REFERENCES
1. https://www.gopracticals.com/physics/physics-determine-band-gap-semiconductor
2. http://vlabs.iitb.ac.in/vlabs-dev/labs/physics-basics/labs/energy-band-gap-iitk/simulation
3. Novak, G.S., Conversion of units of measurement. IEEE Transactions on Software Engineering,
1995. 21(8): p. 651-661.
4. Kittel, C., Introduction to solid state physics. 2005: Wiley.
5. https://www.hiram.edu/wp-content/uploads/2016/12/exp-band-gap.pdf.
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9.0 Appendix
Table 9.1(Readings of heating coil voltage 2v &4v & 6v)
Temperature Temperature (T)
(C)
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
(K)
403
398
393
388
383
378
373
368
363
358
353
348
343
338
333
328
323
318
313
308
303
4000
3800
3600
3400
3200
3000
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
Current
Voltage
1
T
(mA)
29.5
28.7
28
27.5
26.6
25.5
24.5
23.4
22.1
20.5
19.4
18.1
16.4
15
13.8
12
10.5
9.1
7.4
6.3
5.2
(V)
0.72
0.78
0.89
1
1.04
1.12
1.24
1.34
1.45
1.65
1.76
1.9
2.07
2.25
2.41
2.55
2.72
2.86
3.03
3.11
3.21
−1
0.0024814
0.0025126
0.0025445
0.0025773
0.002611
0.0026455
0.002681
0.0027174
0.0027548
0.0027933
0.0028329
0.0028736
0.0029155
0.0029586
0.003003
0.0030488
0.003096
0.0031447
0.0031949
0.0032468
0.0033003
σ
−1
Ln( σ)
−1
20.48611111
18.3974359
15.73033708
13.75
12.78846154
11.38392857
9.879032258
8.731343284
7.620689655
6.212121212
5.511363636
4.763157895
3.961352657
3.333333333
2.863070539
2.352941176
1.930147059
1.590909091
1.221122112
1.012861736
0.809968847
3.01974715
2.912211302
2.755591146
2.621038824
2.548543322
2.432202586
2.290414557
2.166919228
2.030866872
1.826502418
1.706812076
1.560910872
1.376585547
1.203972804
1.051894664
0.85566611
0.657596196
0.464305608
0.1997702
0.012779727
-0.210759492
Table 9.1:- Readings of heating coil voltage 2v &4v & 6v.
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