The Hall Effect
Introduction
In an atom,
electrons are tightly bound to the positive nucleus, and have well confined quantum
mechanical wave functions. In a crystalline solid, many atoms are arranged in a
closely spaced periodic array, with the result that quantum mechanical electron
waves are influenced by the entire crystal of ions, and their behavior changes
dramatically from that of a simple collection of atoms. In an individual atom,
the electron has sharp quantized energy levels, but in a crystalline solid,
electrons may have energies within particular bands, and the bands are
separated by gaps where extended electron states cannot exist. Because of the
Pauli exclusion principle, only two electrons (spin up and spin down) may fill
any of the states in a band, so that a crystal with many electrons must fill up
the bands. The nature of the constituent atoms determines how the bands get filled,
and this determines how the electrons can behave (e.g., move) in the crystal
[1-3]. If a crystal has a partially filled band (a “conduction” band), then
there are empty states into which the electrons may move, and the crystal will
be an electrical conductor. If the electrons completely fill a band (a “valence
band”), then they must be excited across a gap to find available states; if the
gap is large, the crystal will be an insulator, and if the gap is small, the
crystal will be an intrinsic semiconductor. If a semiconductor is doped with
impurities, then states close to band edges may be induced, and electrons may
be thermally excited into or out of
these impurity
states. An interesting property of semiconductors is that when electrons are excited
out of lower energy states, they leave behind a vacancy, which itself can act
like a mobile positive charge; such vacancy charge carriers are called “holes.”
Because of the role of thermal excitation in populating the various states,
some materials may have properties with interesting temperature dependence.
Needless to say, understanding such materials requires a means for measuring
the properties of the charge carriers; a method for accomplishing this involves
a phenomenon called the Hall effect [2-4].
The situation for
the Hall effect is illustrated in Fig. 1. The sample of solid material is a rectangular
parallelepiped (often referred to as a “Hall bar”), illustrated with its edges parallel
to the x, y, and z-axes of a Cartesian
coordinate system. Electrical leads are connected to the sample faces which are
normal to the x-axis (indicated with
the letter “I”), at points near the centers of the faces which are normal to
the y-axis (indicated as V1
and V2), and at one additional point near
the end (indicated
as V3).
Figure 1. Situation for the Hall effect.
An electrical
current of magnitude I established by
an external current source, is maintained by the charge carriers drifting in
the x-direction; note that the end
faces of the sample should be covered with a good conductor so that the current
inside the sample is uniform. If the material has electrical resistance, then
by Ohm's Law, an electrical potential will be present between V1 and
V3; and an electrical resistance will be given by (V1 - V3)
/ I.
Question: Why
isn't the resistance determined by measuring the voltages at the ends of the sample,
using the same leads which are supplying the current I? [Look up “four-terminal
resistance measurement.”]
Task: Let l, w,
and t represent the dimensions of the
sample in the x, y, and z-directions, respectively.
Assuming that the distance between the V1 and V3 contacts
is l /2; show that the electrical
resistivity of the material is given by (V1 - V3) l / (2twI).
If a magnetic
field is applied to the sample in the positive z-direction, then the charge carriers will experience a Lorentz
force perpendicular to their velocity, and they will drift parallel to the y-axis until they are stopped at the
sides of the Hall bar. The charge at the sides of the sample will build up,
establishing a transverse electrical potential (in the y-direction), and steady state will be reached when the force due
to the transverse potential just cancels the Lorentz force due the magnetic field.
The transverse potential, called the Hall voltage, has a magnitude given by |V2
-V1|; the sign of the Hall voltage can be used to determined the sign
of the charge carriers. The density of the charge carriers may be determined
using a theoretical model for the resistivity of the material.
Experimental
Apparatus
Electromagnet and power supplies
Closed cycle refrigerator, with compressor
Multimeters
Current source
Computer
Hall samples
Procedure
In the Hall
Effect experiment, you will measure the properties of the charge carriers in
various Hall bar samples. The major experimental equipment includes a closed cycle
refrigerator (for cooling samples to determine temperature dependence) and a
large electromagnet for producing the magnetic field. Other equipment includes
a sample holder within the refrigerator, a temperature sensor and regulation
system, a probe for measuring the strength of the magnetic field, a current
source, meters for monitoring various voltages and currents, and a computer for
logging and analyzing data.
Separate
manuals are available for learning the particulars of the closed cycle
refrigerator, the temperature sensor and regulation system, and the magnetic field
probe. The temperature regulation system uses a feedback heater, and you should
learn the intricacies of feedback systems. Not discussed in the refrigerator
manual is the sample mount system inside the refrigerator, since it was custom
made. The details of the mount are illustrated in Fig. 2. In this figure, the
magnetic field is perpendicular to the sample mount platform.
Figure 2. Sample
mount system inside the closed cycle refrigerator.
The sample mount
is located on the end of a “cold finger,” which, like the sample mount itself,
is made of metal with high thermal conductivity so as to establish good thermal
contact between the sample and the cold head of the refrigerator. The base of
the sample mount contains the temperature sensor and feedback heater. The
sample mount platform contains an integrated circuit socket and a metal “cold
plate” which is raised about one millimeter above the surface of the sample
mount platform. The purpose of the cold plate is to facilitate making good
thermal contact with the Hall bar sample. The Hall bar sample itself is
attached to the underside of an integrated circuit “chip carrier;” electrical
leads attached to the Hall bar, as discussed with Fig. 1, are connected to pins
of the chip carrier; these connections will be shown in more detail later. When
the chip carrier is installed in the socket in the sample mount platform, the
Hall bar sample is pressed against the cold plate, making good thermal contact.
To prevent the electrically conducting cold plate from shorting out the Hall
bar sample and its leads, a thin (~7 mm) Mylar film is situated between the Hall
bar sample and the cold plate. Although the Mylar is thermally insulating as
well as electrically insulating, the use of a thin film maintains electrical
isolation but permits good thermal contact. It should be noted that the
particular Mylar film used has a metallic coating on one side, and it is
important that this side face the cold plate, and not the Hall bar sample. To further
enhance the thermal contact, the cold plate, Mylar film, and sample are coated
with Apiezon N grease; the grease provides a more uniform contact between the
solid surfaces and has no undesirable consequences (e.g., cracking) when it
solidifies at low temperatures. The final step in ensuring good thermal contact
with the cold plate is to increase the pressure on the sample. This is
accomplished by tightly wrapping many turns of dental floss around the sample
chip carrier and the sample mount platform. Dental floss is used because it can
maintain tension at low temperatures. It should be noted the the wiring on the
side of the sample platform opposite the sample must be protected from damage
by the dental floss; a cylindrical shell is used for this purpose, as
illustrated in Fig. 2.
Figure 3
illustrates how the Hall bar sample is wired to the chip carrier; the
illustration is a top view, looking down on the chip carrier, the Hall bar
sample, and the sample platform socket. Electrical leads from the sample platform
socket travel down the cold finger and ultimately lead to a multi-pin connector
located at the cold head of the refrigerator; the letters in Fig. 3 refer to the
pin designations on the multi-pin connector. A mating multi-pin plug connects
the leads to a junction box with five insulated-shield BNC connectors. The
center sockets and shields of the BNC connectors are designated with letters on
the junction box; these letters indicate the connections to the sample socket,
coinciding with the letters in Fig. 3.
Figure 3.
Electrical connections from the Hall bar sample to the chip carrier and sample
platform socket. The view is from the top, and the letters refer to pin
designations on a multi-pin connector and junction box used in the Hall effect
measurements.
Connections
for the Hall bar sample, the junction box, and the equipment used in the Hall effect
measurement are illustrated in Fig. 4. The letters are as discussed earlier. To
avoid ground loops, the entire system is grounded at only one point, which is
the ground on the current source; the case of the junction box, which is metal,
is connected to the ground point with a short length of heavy braided copper
ground strap. As already mentioned, the shields of the BNC connectors on the
junction box are isolated from the case; this is to permit measurements without
having unwanted currents and voltage drops in connecting coax cable shields.
[You should be able to explain this.] However, in this experiment measurements
are made with differential voltmeters using only the center conductors of different
connections; thus the shields should be grounded rather than isolated. In order
to have the choice of isolated or grounded shields, the junction box contains a
switch which can be used to connect or disconnect the BNC shields from the
junction box case. It should be noted that this option applies to only four of
the BNC connectors; the fifth has a permanently grounded shield.
Figure 4.
Electrical connections for the Hall bar sample, junction box, and measuring
equipment for the Hall effect measurement.
Connections
with the junction box and other equipment are made with BNC cables with only
one end of the shields connected to ground, as indicated in Fig. 4. Connections
without the shield are made with adapters having a BNC socket on one end and a
banana plug on the other end. Note that sample resistivity measurements are
made by connecting leads G and E to the differential voltmeter, and Hall
voltage measurements are made by connecting leads C and E to the differential
voltmeter. To determine the resistivity, measure the voltage between leads G
and E as a function of the current through the sample. The minimum current
should be large enough to yield voltages above the noise, and the maximum
current should produce negligible Joule heating in the sample (for reference,
see how much power the feedback heater is using). A plot of voltage versus
current should be a straight line whose slope is the sample resistance. If
there is a non-zero intercept (a voltage between G and E when there is no
current), how would you explain this?
The limited
number of electrical leads through the cold head results in a complication.
When current is flowing through the Hall bar, there is a continuous voltage
change down its length. If the Hall voltage connections V1 and V2
at the sample are not at exactly the same distance along the sample length,
then a voltage will be measured even when there is no applied magnetic field.
This voltage offset may be eliminated by making measurements with the magnetic field
first in one direction, and then in the opposite direction; one-half the difference
in the two data sets (the anti-symmetric dependence on magnetic field) will be
the Hall voltage without the offset. The average of the two data sets (the
symmetric dependence on magnetic field) will give the magnetoresistance of the
material (after subtracting the voltage offset at zero magnetic field).
The magnetic
field for the experiment is obtained with an electromagnet (with 10 cm diameter
pole pieces) which involves large currents and Joule heating. The magnet must
be cooled with flowing water; an interlock prevents the use of the magnet power
supply without the flow of cooling water. To avoid eddy current heating, the
current to the magnet should be changed slowly. The magnet power supply has a
reversing switch for changing the direction of the field; the field must be
reduced to zero before the reversing switch is used.
The
measurements you should make include the Hall voltage as a function of magnetic
field (both directions) at various temperatures from 10K to 300K, and the
resistivity as a function of temperature. From your data you should be able to
determine the electron mobility and energy gap of the semiconductor.
The current Hall
samples have properties as follows:
Hall bar sample
#1: Undoped germanium, dimensions (l, w, t) 25.4 ´ 3.0 ´ 2.0 mm.
Hall bar sample
#2: Doped germanium, dimensions (l, w, t)
18.5 ´ 2.7 ´ 0.8 mm.
References
[1] R. A. Serway,
C. J. Moses, and C. A. Moyer, Modern Physics,
lishing,
[2] C. Kittel, Introduction
to
[3] C. Kittel, Quantum
Theory of Solids, John Wiley & Sons,
[4] E. H. Putley, The
Hall Effect and Semi-conductor Physics