Download The Hall Effect - Ryerson Department of Physics

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THE HALL EFFECT
Objective
The objective of this experiment is to determine, for a germanium sample, the following:
a)
b)
c)
sign of the charge carriers,
charge carrier concentration,
mobility of the charge carriers.
Theory
An electric charge moving perpendicular to a magnetic field B will experience a
magnetic force. This also applies to the current flowing in a conductor placed in a magnetic
field. The magnetic force will tend to move the charge carriers to one edge of the conductor
leaving a deficiency of charge carriers on the opposite edge. Thus one edge will have a positive
charge and the other a negative charge, resulting in an electric field being set up within the
conductor. Equilibrium will be reached when the electric force exactly balances the magnetic
force exerted on a charge carrier.
The resulting electric field within the conductor produces a potential difference across the
material called the Hall voltage VH. The polarity of the Hall voltage depends on the current
direction. This relationship can be used to determine the sign of the charge carriers present in
any conducting material.
It can be shown that for electrons moving in a semiconductor sample, the Hall voltage is
equal to:
where n
e
B
i
t
is the density of charge carriers (number per m3)
is the magnitude of the charge on an electron
is the magnetic field perpendicular to the sample
is the current in the sample
is the thickness of the sample in the direction of B
The charge carriers in some materials are more mobile than those in other materials and
will thus contribute to a greater current flow. A parameter called "mobility" is defined to
measure this effect. Mobility is the ratio of drift velocity to the electric field parallel to the drift
velocity.
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The mobilities of the carriers can be found in the following equation:
where µ
is the mobility of the charge carriers,
L
is the length of the sample in the direction of current flow,
w
is the width of the sample in the direction of VH,
Vx
is the voltage drop along the sample in the direction of L (i.e., a
measure of the resistance of the sample) when carrying current ix.
A rectangular wafer of germanium is provided on a circuit board. Handle the board
carefully as the germanium wafer is delicate. The dimensions of the germanium wafer are:
length, L
width, w
thickness, t
10.0 mm
5.0 mm
1.0 mm
A box containing a power supply and two VOM meters is also provided. The power
supply is used to drive a current through the length of the wafer and one meter is used to measure
the current. The other meter is used to measure either the Hall voltage developed across the
width of the wafer or the resistive voltage developed along the length of the wafer.
The magnet has pole pieces shaped to provide a reasonably uniform magnetic field. Be
careful not to allow any steel pieces to crash into the magnet (it has a very strong pull) as this
causes demagnetization. When inserting the germanium wafer, try to centre the wafer between
the pole pieces and over the centre of the pole pieces. There is a scratch along the plexiglass that
indicates the centre of the magnetic line up with the copper line parallel to the length of the
sample.
Procedure
1.
Connect the power supply and a milliammeter to the circuit board so that current will
flow along the length of the wafer (follow the connections given in Figure 1 on the next page).
With the wafer away from the magnet, turn on the power supply and adjust the current to 50 mA
(do not exceed 50mA at any time during the experiment). Connect the other meter across the
width of the wafer to measure mV D.C. It is difficult to make two permanent connections on
either side of the wafer exactly opposite one another so a "zero pot" is provided which allows the
lead on that side to be adjusted so that it is "electrically" opposite the other side. Adjust that
potentiometer now for a zero mV reading.
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2.
Using a gaussmeter, measure the magnitude and direction of the magnetic field
generated by the permanent magnet. The wafer will be inserted in this field, so note the
variation of the field over the region that the wafer will occupy. This may be one source of
uncertainty in the value of B.
3.
Insert the wafer between the pole pieces of the magnet so that the field is perpendicular to
the face of the wafer and the wafer is centred as described above. Set the current to 5 mA and
measure the voltage developed across the width of the wafer. This will be the Hall voltage (VH).
Take voltage measurements every 5 mA up to a maximum of 50 mA. Note the relative
directions of the magnetic field, the current through the sample and the Hall voltage polarity.
This information will be needed to determine whether the charge carriers are holes or electrons.
When trying to determine the sign of your charge carriers note that the charges will always be
pushed to the same side of the wafer irrespective of their charge. You should determine which
side of the wafer it is (using the direction of the current and the magnetic field) and then predict
which type of charge will result in a positive or negative hall voltage (which will depend on how
you are measuring your voltage). Having done this you can now say whether your charge
carriers are negative or positive by comparing your predictions to the actual sign of the Hall
voltage that was measured.
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4.
Remove the wafer from the magnetic field. Disconnect the meter used to measure the
Hall voltage and reconnect it to measure the voltage drop along the length of the wafer (Vx).
Measure Vx for ix (current passing through length of sample) set to 5 mA. Increase ix in steps of
5 mA until the maximum of 50 mA is reached, recording Vx at each step.
Calculations
1.
Determine whether the majority carriers are holes or electrons and explain how this was
determined, with reference to a sketch of the sample in the magnetic field.
2.
Plot the Hall voltage vs current and use the slope of the graph (VH/I) to calculate the
charge carrier density "n".
3.
Plot Vx vs ix and use the slope of the graph to calculate the mobility "µ" of the charge
carriers.