EXP 4
Energy Gap of Germanium
4.1. Objectives
• You will measure the voltage drop at an undoped Ge crystal as a function of the temperature when the current through the crystal is constant, and calculate the conductivity
σ.
• You will determine the band gap Eg of germanium.
4.2. Related Concepts
band theory
conduction electron
valence band
semiconductor
impurity depletion
Fermi level
hole
conduction band
intrinsic conduction
Pauli exclusion principle
charge carrier
energy gap
extrinsic conduction
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4.3. THEORY
CHAPTER 4.
ENERGY GAP OF GERMANIUM
4.3. Theory
4.3.1. Band theory
Classification of solids using band theory
Figure 4.1: Simple picture of energy bands in solids.
When an electric field is applied to a solid, free electrons are accelerated. They lose or decrease
their kinetic energy by collisions with the atoms of the lattice. The current that results is proportional to the average electron velocity, which is determined by the intensity of the applied
electric field and the collision frequency. Only electrons with energy near the Fermi level may be
accelerated, as the other electrons inhabit states that are surrounded by full (occupied) states,
and are consequently forbidden to accelerate (change state) by the Pauli Exclusion Principle.
If the valence band is full and does not overlap empty bands, the lack of adjacent empty states
severely limits conduction. This is the accepted model for insulators and semiconductors.
Resitivity
Electrons may move through an ideal crystal without resistance, but in real crystals electrons
collide with phonons, dislocations, vacancies, impurities, and any other lattice imperfections. The
resistivity due to impurities and imperfections is called the residual resistivity and is usually
independent of temperature. The total resistivity is the sum of the residual resistivity and
thermal contributions.
Semiconductor: Intrinsic, Extrinsic
Semiconducting materials are comparatively poor conductors of electricity. They are characterized by a nearly full valence band, which is separated from a nearly empty conduction band by
an energy gap of 2 eV or less. Thermal energy can promote electrons to the conduction band
and leave vacant states, called holes, in the valence band. Both conduction electrons and holes
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4.3. THEORY
CHAPTER 4.
ENERGY GAP OF GERMANIUM
serve as charge carriers.
When only valence and conduction states are involved, the semiconductor is said to be intrinsic.
Substitutional impurity atoms of higher or lower valence may contribute electrons to the conduction band or holes to the valance band. If they make a dominant contribution to the conductivity,
the material is said to be an extrinsic semiconductor, either n-type or p-type, respectively.
Electrical conductivity
If the gap is a few electron volts or more, very high fields are required to surmount it. At ordinary
voltages, very little current flows, so the material behaves as an insulator. There are, however,
other ways for electrons to cross the energy gap. The energy of thermal motion will allow a
small fraction of the electrons in the valance band to enter the conduction band. This fraction
increases with increasing temperature.
In the case of extrinsic semiconductors, impurities and imperfections contribute to the conductivity by creating new energy levels, and this becomes predominant at low temperatures.
69
4.3. THEORY
CHAPTER 4.
ENERGY GAP OF GERMANIUM
4.3.2. Temperature dependence of conductivity in semiconductors
Figure 4.2: Plot of ln G vs 1/T for an intrinsic semiconductor. The slope of the curve gives the
band gap Eg .
The conductance G is G = 1/R = F σ
G=
1
= Fσ
R
(4.1)
where F is a geometrical factor and σ is the conductivity
Eg
σ = C · e− 2kT
Then
ln G = ln(F C) −
and the slope of a graph of ln G versus
1
1
T
(4.2)
Eg
2kT
(4.3)
E
yields − 2kg 1 as illustrated in Figure 4.2 [1].
E (0)
− g2k , where Eg (0): energy gap at 0 K. The temperature dependence of the energy gap was measured by
optical absorption by Macfarlane et al (1957, 1958) [2] [3].
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4.3. THEORY
CHAPTER 4.
ENERGY GAP OF GERMANIUM
Figure 4.3: Plot of ln G vs 1/T for an extrinsic semiconductor. In the extrinsic region, the slope
of the curve also contains the energy Ed of the donor level below the conduction band edge.
If the germanium is not sufficiently pure, it will be necessary to go to a temperature considerably above room temperature before the thermally-generated carriers are numerous enough to
dominate the carrier concentration. This type of behavior is most commonly seen in extrinsic
semiconductors, as shown in Figure 4.3.
Note that if carriers from ionized impurities are dominant the carrier concentration will be
relatively independent of T .
71
4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
4.4. Experiment
4.4.1. Equipment
Description
Manufacturer
Connecting cord
Amount
4
Hall effect module
PHYWE 11801-01
1
Intrinsic conductor, Ge, carrier board
PHYWE 11807-01
1
Tripod base
PHYWE 02002-55
1
Support rod
PHYWE 02025-55
1
Right angle clamp, square
PHYWE 02040-55
1
Digital multimeter
FLUKE 189
2
Power supply
1
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4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
Intrinsic conductor, Ge, carrier board
Figure 4.4: The carrier boards with the non-doped germanium sample.
• Sample dimensions: (20 × 10 × 1) mm3
• Resistance: ≈ 50 Ω
• Temperature probe: Pt 100
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4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
Hall Effect Module
• Description
– The Hall effect module must be supplied with a 12 V alternating voltage.
– The module creates from this an adjustable and controlled sample direct current of
each sign, a fault voltage compensator and the heating power for the meandering
heating path on a carrier board.
– Via the temperature sensor on the carrier board the sample temperature is controlled.
Thus an exceeding of the max. allowed temperature of T = 170 ◦ C is avoided. This
safety function avoids overheating, which would cause the soldering tin at the semiconductor sample contacts to be melted off.
– You can select whether the sample current or the sample temperature is to be displayed by the 3-place LED display.
– The module has 4 mm safety sockets for feeding in the supply voltage and for the
determination of the Hall and sample voltages.
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4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
• Function elements and operating elements (Figure 5.3)
Figure 4.5: Hall effect module, front (left) / back (right).
Function elements at the front of the Hall module:
1
2
3
4
5
6
7
8
9
10
Rotary knob for the sample current Ip
Digital display, displays either sample current Ip or sample temperature Tp as selected
Threaded socket for screwing in the holding rod supplied
Series of LEDs which indicate the operating mode of the sample heating, and whether
the digital display shows sample current Ip or sample temperature Tp
Pair of 4 mm safety sockets for pick up of the Hall voltage UH
Positioning bore hole for a tangential magnetic field probe
Press switch for selection of the display of sample current Ip or sample temperature Tp
Rotary knob for compensation of the Hall voltage UH for fault voltage
Shaft for acceptance of the sample board with contact strip
4 mm safety sockets for pick up of the sample voltage Up
Function elements at the back of the Hall module:
11
12
Pair of 4 mm safety sockets for connection of the supply voltage
Press switch for heating to be “On” or “Off”
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4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
4.4.2. Set-up & procedure
Figure 4.6: Experimental setup for the determination of the band gap of germanium.
The experimental setup is shown in Figure 4.6.
• The test piece on the board has to be put into the hall effect module via the guide groove.
• The module is directly connected with the 12 V∼output of the power unit over the AC
input on the backside of the module.
• The voltage across the sample is measured with a multimeter. Therefore, use the two lower
sockets on the front side of the module.
• The current and temperature can be easily read on the integrated display of the module.
Be sure, that the display works in the temperature mode during the measurement. You
can change the mode with the “Display” knob.
Procedure
• At the beginning, set the current to a value of 5 mA. The current remains nearly constant
during the measurement, but the voltage changes according to a change in temperature.
• Set the display in the temperature mode, now.
• Start the measurement by activating the heating coil with the “on/off” knob on the backside
of the module.
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4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
• Determine the change in voltage dependent on the change in temperature for a temperature
range of room temperature to a maximum of 170 ◦ C.
• While doing this experiment the current should be kept under 30 mA and
the temperature under 150 ◦ C.
• Avoid sample currents above 5 mA, as these would lead to self-heating,
which would exert a falsifying effect on the measured values.
• The exchangeable carrier board can get very hot during operation. There
is a danger of burns to hands. Do not handle the board until the module
has been switched off and an appropriate cooling-down time has elapsed.
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4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
APPENDIX C
Electrical Conductivity: Conductor, Insulator
• Electrical Resistivity of conductors
The total resistivity of a conductor is given by the sum of the thermal component ρth and
the residual component ρr in an equation known as Matthiessen’s Rule.
ρ = ρth + ρr
(4.4)
Above the Debye temperature, the thermal component of the resistivity is approximately
linear in conductors. A simple way to estimate the overall purity and perfection of a conductor is to therefore measure the ratio of the resistivities at room temperature and at
liquid helium temperature (4.2 K), where ρ = ρr at 4.2 K, because the thermal component
is effectively zero at such low temperature. Likewise, for an ultrapure element, ρ = ρth at
all temperatures because the residual resistivity is effectively zero due to the absence of
appreciable impurity scattering.
To find ρr from measurements of resistance, R, at different temperatures write Matthiessen’s
rule at two temperatures, say T1 and T2
ρ(T1 ) = ρth (T1 ) + ρr
(4.5)
ρ(T2 ) = ρth (T2 ) + ρr
(4.6)
Define
R(T1 )
R(T2 )
(4.7)
ρ(T1 )
ρth (T1 )
=
ρ(T2 )
ρth (T2 )
(4.8)
α=
and for a pure element,
κ=
where R is the resistance you measure and ρth (T ) can be obtained from the literature. For
pure copper, ρth (273 K) = 1.55 micro-ohm-cm and ρth (77 K) = 0.195 micro-ohm-cm, and
ρr =
κ−α
ρth (T2 )
α−1
78
(4.9)
4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
• Insulators
Electrical conductivity in ionic solids arises from the migration of ions under the influence
of an electric field. In alkali halides the positive ions are much more mobile than the
negative ions.
Consequently, most of the current is carried by positive ions and we will neglect the
contribution of the negative ions. The mobility of ions in a perfect lattice is extremely
low. This is because of the high energy barrier preventing interchange of atoms. Hence,
the presence of some lattice defects is essential for the occurrence of ionic conductivity.
The concentration of a pair of positive and negative ion vacancies is given by
W
n = c · e− 2kT
(4.10)
where n is the number of anion-cation vacancy pairs (“Schottky” defects), W is the formation energy of such a pair, and c is a constant depending on the material. The positive
Figure 4.7: Schematic ionic conductivity vs temperature curve for NaCl. The change to a steeper
slope at higher temperature(lower 1/T ) comes from the creation of more Schottky defects, which
contribute to the conductivity.
ions surrounding a positive ion vacancy will have a finite probability of jumping into it.
As a result, the vacancy will move through the crystal by virtue of positive ion-vacancy
exchange, and diffusion becomes possible. The expression of the conductivity, σ, derived
on this basis has the following form
Em +
σ = A · exp −
kT
79
W
2
!
(4.11)
4.4. EXPERIMENT
CHAPTER 4.
ENERGY GAP OF GERMANIUM
where A is a constant for a particular crystal, and Em is the activation energy for migration. Therefore, from a plot of ln σ vs (1/T ) you can calculate the value of (Em + W/2).
Plots of this sort for real alkali halides typically have the form of Figure 4.7.
There is a break observed somewhere in the temperature range between 400 ◦ C and 700 ◦ C
depending upon the purity of the crystal. Below a critical temperature essentially all the
vacancies arise because of the impurities, and the number of vacancies becomes temperature independent. In that case the temperature dependence of the conductivity is controlled
only by the mobility energy Em . Above the critical temperature the number of thermally
produced vacancies predominates over the number due to the impurity atoms and consequently the conductivity depends on the energy W , specifically on (Em + W/2). This
means that the slope in Fig.9 should be −Em /2k and −(Em + W/2)/2k respectively in
the two ranges of temperature. By determining these two slopes, the values of Em and W
can be calculated.
80
BIBLIOGRAPHY
BIBLIOGRAPHY
Bibliography
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(1957).
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(1958).
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900–905 (1951).
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[11] University of California at Santa Cruz, EE-145L; Properties of Materials Laboratory, Lab
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[12] Queen Mary, University of London, PHYSICS LABORATORY (PHY-201), Hall effect in
germanium.
[13] National Institute of Science Education and Research, PL-343 & PL-344 :Advanced
Physics Laboratory Course, Band Gap of a semiconductor: A Germanium test-piece.
[14] San Jose State University, MatE 25; Materials Engineering 25 Laboratory, Lab 10: Temperature Dependence of Conductivity in Metals and Semiconductors.
[15] University of California, Berkeley, Engineering 45; Properties of Materials Laboratory, Lab
6: Electrical Conductivity.
81
BIBLIOGRAPHY
BIBLIOGRAPHY
[16] Online Experiment Showing Temperature Effects on Semiconductor Resistivity, http:
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82