EXP 5
Hall Effect in n-type and p-type Germanium
5.1. Objectives
• You will measure the Hall voltage UH as a function of the current at constant magnetic
field B.
• You will measure the Hall voltage as a function of the the magnetic field B at constant
current: determination of the Hall coefficient RH together with the Hall mobility mH and
the carrier concentration ( n: n-type, p: p-type ).
• You will measure the Hall voltage as a function of temperature: Investigation of the transition from extrinsic to intrinsic conductivity.
5.2. Related Concepts
Semiconductor
Meyer-Neldel Rule
valence band
magnetoresistance
Hall coefficient
band theory
intrinsic conduction
conduction band
mobility
forbidden zone
extrinsic conduction
Lorentz force
conductivity
83
5.3. THEORY
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
5.3. Theory
Figure 5.1: Hall effect in sample of rectangular section.
If a current I flows through a conducting strip of rectangular section and if the strip is traversed
by a magnetic field at right angles to the direction of the current, a voltage − the so called Hall
voltage − is produced between two superposed points on opposite sides of the strip.
This phenomenon arises from the Lorentz force : the charge carriers giving rise to the current
→
−
flowing through the sample are deflected in the magnetic field B as a function of their sign and
−
their velocity →
v :
→
−
→
−
−
F = q(→
v × B)
(5.1)
→
−
( F = force acting on charge carriers, q = quantity of charge).
Since negative and positive charge carriers in semiconductors move in opposite directions, they
are deflected in the same direction.
The type of charge carrier causing the flow of current can therefore be determined from the
polarity of the Hall voltage, knowing the direction of the current and that of the magnetic field.
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5.3. THEORY
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
5.3.1. Hall voltage as a function of current
There is a linear relationship between the current Ip and the Hall voltage UH :
UH = α · Ip
(5.2)
where α = proportional factor.
5.3.2. Hall voltage as a function of magnetic field
• Hall coefficient
The Hall coefficient RH is then given by
RH =
UH d
·
B Ip
(5.3)
where d = sample thickness.
• Conductivity
The conductivity is calculated from the sample length w, the sample cross-section A, and
the sample resistance R as follows:
σ=
w
R·A
(5.4)
• Hall mobility
The Hall mobility µH of the charge carriers can now be determined from
µH = RH · σ
(5.5)
• Carrier concentration
The electron concentration n of n-doped samples (the hole concentration p of p-doped
samples)is calculated from
n=
1
e · RH
(5.6)
5.3.3. Hall voltage as a function of temperature
The Hall voltage decreases with increasing temperature. Since the experiment was performed
with a constant current, it can be assumed that the increase of charge carriers (transition from
extrinsic to intrinsic conduction) with the associated reduction of the drift velocity v is responsible for this.
85
5.3. THEORY
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
The same current for a higher number of charge carriers means a lower drift velocity. The drift
velocity is in turn related to the Hall voltage by the Lorentz force.
86
5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
5.4. Experiment
5.4.1. Equipment
Description
Manufacturer
Connecting cord
Amount
7
Hall effect module
PHYWE 11801-01
1
Hall effect, n-Ge, carrier board
PHYWE 11802-01
1
Hall effect, p-Ge, carrier board
PHYWE 11805-01
1
Coil, 600 turns
PHYWE 06514-01
2
Iron core, U-shaped, laminated
PHYWE 06501-00
1
Pole pieces, Plane, 30 × 30 × 48 mm, 1 pair
PHYWE 06489-00
1
Teslameter, digital
PHYWE 13610-93
1
Hall probe, tangent., prot. cap
PHYWE 13610-02
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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5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
p-Ge, n-Ge, Carrier board
Figure 5.2: The carrier boards with the doped germanium sample.
• Sample dimensions: (20 × 10 × 1) mm3
• Resistivity:
n-Ge
p-Ge
(2 ∼ 2.5) Ω·cm
(2.5 ∼ 3) Ω·cm
• Temperature probe: Pt 100
88
5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE 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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5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
• Function elements and operating elements (Figure 5.3)
Figure 5.3: 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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5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
5.4.2. Set-up & procedure
Figure 5.4: Experimental setup.
The experimental setup is shown in Figure 5.4.
• 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 plate must be brought up to the magnet very carefully, so as not to damage the
crystal. In particular, avoid bending the plate.
• The Hall voltage is measured with a multimeter. Therefore, use the sockets on the front-side
of the module.
• The current and temperature can be easily read on the integrated display of the module.
Calibration of the magnetic field
• The magnetic field has to be measured with the teslameter via a hall probe, which can be
directly put into the groove in the module as shown in Figure 5.4. So you can be sure that
the magnetic flux is measured directly on the Ge-sample.
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5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
Procedure
Compensation of the Hall voltage
• It is possible that the Hall contacts do not lie directly opposite each other because of
production reasons.
• In this case, a fault voltage will be measurable at sockets 5 when current passes through
the sample and there is no magnetic field.
• Use rotary knob 8 to compensate for this voltage at each sample current intensity.
• Hall voltage as a function of current.
– Set the magnetic field to a value of 250 mT by changing the voltage and current on
the power supply.
– Connect the multimeter to the sockets of the hall voltage (UH ) on the front side of
the module.
– Set the display on the module into the “current mode”.
– Determine the hall voltage as a function of the current from −30 mA up to 30 mA
in steps of nearly 5 mA.
• Hall voltage as a function of magnetic field
– Set the current to a value of 30 mA.
– Start with -300 mT by changing the polarity of the coil current and increase the
magnetic field in steps of nearly 20 mT. At zero point, you have to change the polarity.
– Determine the Hall voltage as a function of the magnetic field.
• Hall voltage as a function of temperature
– Set the current to 30 mA and the magnetic field to 300 mT.
– Set the display in the temperature mode.
– Start the measurement by activating the heating coil with the “on/off” knob on the
backside of the module.
– It is recommended that a control measurement be carried out during the cooling
phase.
– Determine the Hall voltage as a function of the temperature.
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5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
• While doing this experiment the current should be kept under 30 mA and
the temperature under 150 ◦ C.
• 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.
93
5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
APPENDIX A
Measuring Example
• Hall voltage as a function of current
Figure 5.5: Hall voltage as a function of current: (left) n-Ge, T = 300 K, B = 300 mT / (right)
p-Ge, T = 300 K, B = 250 mT.
• Hall voltage as a function of magnetic field
Figure 5.6: Hall voltage as a function of magnetic field: (left) n-Ge, T = 300 K, I = 30 mA /
(right) p-Ge, T = 300 K, I = 30 mA.
Figure 5.6 shows a linear relation between Hall voltage UH and magnetic field B. The
regression line with the formula
UH = U0 + b · B
For n-Ge, a slope
b = 0.144 V/T (sb = ±0.004 V/T)
For p-Ge, a slope
b = 0.125 V/T (sb = ±0.003 V/T)
– Hall coefficient
94
(5.7)
5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
RH =
d
UH d
=b·
·
B Ip
Ip
(5.8)
Thus, if the thickness of specimen d = 1 × 10−3 m and Ip = 0.030 A,
then for n-Ge
RH = 4.8 × 10−3
m3 /A · s,
sRH = ±0.2 × 10−3
m3 /A · s,
sRH = ±0.08 × 10−3
m3 /A · s
then for p-Ge
RH = 4.17 × 10−3
– Conductivity
With the measured values at room temperature
l = 0.02m, A = 1 × 10−5 m2 ,
for n-Ge
R = 37.3Ω,
σ = 53.6Ω−1 · m−1
R = 35.0Ω,
σ = 57.14Ω−1 · m−1
for p-Ge
– Hall mobility
For n-Ge
µH = 0.257 ± 0.005
m2
V·s
µH = 0.238 ± 0.005
m2
V·s
for p-Ge
– Carrier concentration
For n-Ge
n = 13.0 × 1020 m−3 .
For p-Ge
p = 14.9 × 1020 m−3 .
95
m3 /A · s
5.4. EXPERIMENT
CHAPTER 5. HALL EFFECT IN N-TYPE AND P-TYPE GERMANIUM
• Hall voltage as a function of temperature
Figure 5.7: Hall voltage as a function of temperature: (left) n-Ge, B = 300 mT, I = 30 mA /
(right) p-Ge, B = 300 mT, I = 30 mA.
Figure 5.7(right) shows accordingly the reversal of sign of the Hall voltage, typical of p-type
materials, above a particular temperature.
96
BIBLIOGRAPHY
BIBLIOGRAPHY
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