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e/k of a trans istor
Experiment-419
S
DETERMINATION OF RATIO OF
ELECTRONIC CHARGE TO
BOLTZMANN CONSTANT (e/k) USING
A SILICON TRANSISTOR
Jeethendra Kumar P K
KamalJeeth Instrumentation & Service Unit, Tata Nagar, Bengaluru-560092, INDIA
Email: [email protected]
Abstract
Using a silicon transistor (SL100) the collector current variation with emitter
base voltage is studied at fixed temperature and ratio of e/k is calculated and
compared with standard value.
Introduction
The transistor invented by Bardeen, Brattain and Shockley is one of revolutionary
inventions in Physics. It is mainly responsible for ushering in the present day electronic
revolution. The theoretical aspects of semiconductor physics offer a means to determine
various universal constants in lab experiments. It provides a means to experimentally
determine various physical constants in Physics. One such experiment is the
determination of ratio of e/k using the transistor equation, as described below.
Theory
Neglecting the reverse current in the base region, which is very small in a silicon
transistor compared to a germanium transistor, the collector current of a transistor is
given by [1,2]
౛౒
IC = ICO (eηౡ౐ − 1)
…1
where IC is collector current,
ICO is collector current with its base open which gives the reverse saturation
current,
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KAMALJEETH INSTRUMENTS
e/k of a trans istor
T is temperature of the device,
e is electronic charge (1.6x10-19 Coulomb)
k is Boltzmann constant (1.38x10 -23 m2 kg s-2 K-1),
V is the voltage across the depletion region, and
η is known as the ideality factor.
B
E
C
JE
JC
Depletion regions
Figure-1: The two depletion regions in an npn silicon transistor
In Equation-1 the collector current is an exponential function of ICO, which is in turn a
function of temperature; the value of the current doubles for every ten degree increase
in the temperature. Figure-1 shows an npn silicon transistor with two depletion regions.
In the normal use of a transistor, the emitter-base junction (JE) is forward biased and
collector- base (JC) junction is reverse biased. In this mode the transistor conducts and
collector current flows through the transistor. The emitter –base junction (JE) is forward
biased and as a result the depletion region becomes very thin. The voltage across JE is
VBE, hence replacing ‘V’ in Equation-1 by VBE; the collector current can be written as
౛౒ాు
IC = ICO (e ηౡ౐ -1)
…2
Taking natural logarithmic of both the sides of Equation-2, we can write
୚
ాు
ln (IC) = ln (ICO) + ln [e η୩୘
] +ln (-1)
ଵ
ୣ
log IC = log ICO + ଶ.ଷ଴ଷ ୘ η୩ VBE
…3
…4
Equation-4 represents a straight line in log IC versus VBE graph such that
Y = log IC
X = VBE
Y-intercept C = log ICO, and
Slope =
ଵ
ୣ
ଶ.ଷ଴ଷ ୘ η୩
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KAMALJEETH INSTRUMENTS
e/k of a trans istor
Hence from the slope of the straight line represented by log IC versus VBE, one can
determine the value of e/k.
ୣ
୩
= slope 2.303 ηT
…5
Equation-5 shows that e/k is dependent on the temperature of the device. At different
temperatures the slope of the straight line is different such that the value of e/k is
constant.
Ideality factor (η)
The ideality factor (η) of a transistor is not constant but depends on the voltage across
the depletion region and the current flowing through it. The value of η is given by
ୣ
୚ ି୚భ
మష ୍భ )
η = ୩୘ [୪୬ሺ୍మ
…6
where V1, I1 and V2, I2 are voltages across the junctions, and currents passing through
the junctions 1 and 2 at a given temperature. From Equation-6, η is seen to be inversely
proportional to temperature. Hence as seen from Equation-5, when temperature T is
varied, both slope and η vary. Hence to get accurate value for e/k, one should know the
value of η at the given temperature [2].
Figure-2 shows the variation of η with temperature for the silicon transistor SL100. In
the temperature range of 27-80°C, the variation is linear, as shown in Figure-2. At room
temperature (27°C), the ideality factor is 1.7 which matches with the value calculated
from Equation-6.
Ideality factor (η)
2
1.5
1
0.5
0
0
10
20
30
40
50
60
70
80
90
Temperature (0C)
Figure-2: Variation of ideality factor with temperature
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KAMALJEETH INSTRUMENTS
e/k of a trans istor
Apparatus used
The experimental set-up consists of: a regulated power supply (0-10V), digital DC milliammeter (0-200mA), and digital DC voltmeter (0-2V).
Experimental procedure
I
I
0-10V
V
C
V
BE
330
Figure-3: Circuit connections for determination of e/k
The transistor is connected to power supply and milli-ammeter as shown in Figure-3. In
this mode, the emitter base junction is forward biased and collector base junction is
reverse biased.
1. The room temperature is noted using a digital themometer
T =27.1°C
2. The transistor SL100 is mounted on its socket and circuit connections are made
as shown in Figure-3.
3. The emitter-base voltage (VBE) is varied by varying the power supply voltage
control knob and the correponding collector current is noted
VBE = 0.572V and IC =2.4 mA
The readings obtained are tabulated in Table-1.
4. The experiment is repeated by varying the emitter-base voltage in suitable steps
up to the maximum value of 0.65V and noting the correponding collector
current. The corresponding collector base volatge is recorded in Table-1.
Table-1: IC variation with VBE at 27°C
VBE(V) IC(mA) Log (IC)
-0.572
-2.4
-2.619
-0.582
-3.0
-2.522
-0.592
-3.8
-2.420
-0.602
-4.5
-2.346
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KAMALJEETH INSTRUMENTS
e/k of a trans istor
-0.612
-0.622
-0.632
-0.642
-5.6
-7.5
-9.6
-11.5
-2.251
-2.124
-2.017
-1.939
5. A graph is drawn taking log IC along Y-axis and VEB along X-axis. The slope of
the straight line and Y-intercept are noted by extrapolating the line to meet the
Y-axis as shown in Figure-4.
0
Log IC
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-4
-6
-8
-10
VBE (V)
Figure-4: Variation of log IC versus VBE at 27°C
Slope =9.842
Y-intercept= -8.25 = log ICO
IC0= 5.6x10 -9 =5.6nA
ୣ
= Slope 2.303 ηT = 9.842 x2.303x1.7x300= 11560
୩
6. The experiment is repeated on a sunny day with room temperature =31°C. The
IC variation with VBE variation is tabulated in Table-2. From the graph in Figure2, the ideality factor at 31°C is found as
η31 = 1.7= η27
Table-2: IC variation with VBE at 31°C
VBE(V) IC(mA) Log (IC)
-0.572
-2.4
-2.619
-0.582
-3.0
-2.522
-0.592
-3.8
-2.420
-0.602
-4.5
-2.346
-0.612
-5.6
-2.251
-0.622
-7.5
-2.124
-0.632
-9.6
-2.017
-0.642
-11.5
-1.939
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KAMALJEETH INSTRUMENTS
e/k of a trans istor
0
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
-2
Log IC
-4
-6
-8
-10
VBE(V)
Figure-5: Variation of log IC versus VBE at 31°C
Slope =9.896
Y-intercept= -8.10 = log ICO
IC0= 7.94x10-9 =7.94nA
From Figure-2, the η value (1.7) for 27° and 31° temperatures are almost the
same, hence the slope remained also most same at 31°C, hence e/k.
ୣ
୩
= Slope 2.303 ηT = 9.896 x2.303x1.7x300= 11623
Results
The average value of
ୣ
୩
obtained 11591 is found to tally with the standard value of
11594.
The reverse saturation current of SL100 = 5.6nA at 27°C and it increased to 7.94nA at
30°C.
To observe the effect of η on e/k one can repeate the experiment at higher temperature
using a digitally controlled oven. The transistor is now placed in side the oven and
connected to the set-up. In this digitally controlled oven one can set the temperature
from room temperature to 150°C and experiment can be repeated. The temperature
controlled oven is shown in Figure-6.
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e/k of a trans istor
Figure-6:Microcontroller based oven for temperature variation study of
seminconductor devices
Reference
[1]
John D Ryder, Electronic fundamenatls and applications, 5th Edition,PrenticeHall of India Private Ltd, Page-56, 1978
[2]
S Sankararaman and Miss Jaiby Joseph, Variation of ideality factor, LE,Vol-4, No3, Sept.-2004, Page-174
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KAMALJEETH INSTRUMENTS