On the Hall Effect in Germanium Crystals
Shane Duane
ID: 08764522
JS Theoretical Physics
21st November 2010
Abstract
The Hall effect in two different Ge crystals was investigated using
electro- and rotating permanent magnets. It was shown that an electromagnet has no unique value for its remnant field Br . The Hall coefficient
RH was shown to be an intrinsic property of a sample, not dependent on
the current I passed through it. The samples were shown to have vastly
different RH values of (1.22 ± 0.01) × 10−4 C −1 and (1.184 ± 0.005) ×
10−2 C −1 . also, the charge carrier mobilities were shown to differ by a
factor of 105 , while their conductivities differed by a factor of 10.
1
Theory & Introduction
~ H , known as the
The Hall effect is the occurrence of a special electric field E
Hall field, in a metal or semiconductor. This field is created transversely to a
current density J~ passed through the sample when it is placed in a magnetic
~
field B.
The Hall field is given by
~ H = RH B
~ × J,
~
E
(1)
where RH is known as the Hall coefficient.
Let our sample have dimensions l, w, t as in Fig 1. Suppose the charge carri~ . Since they are charged, they will experience
ers in the sample have velocity V
the Lorentz force
~ ×B
~
F~ = q V
which will deflect them to one side of the sample. This will lead to a build up
of charge on one side, and an equal quantity of opposite charge will be created
on the opposite side by charge conservation. Hence a transverse electric field
~ H will be created, and will reach a value to balance the Lorentz force, allowing
E
the current to flow normally. Therefore
~ H = −q V
~ ×B
~
qE
~H = B
~ ×V
~.
⇒E
~ , where N is the carrier density for carriers of charge q, thus
Now J~ ≡ N q V
~ × J,
~
~H = 1 B
E
Nq
1
(2)
and so we are led to define the Hall coefficient as
RH ≡
1
.
Nq
(3)
The Hall field has an associated Hall voltage VH = EH w. Now suppose the
~ is applied perpendicular to J~ as in Fig 1. Since the current is
magnetic field B
I = wtJ we find
VH = RH BI/t.
(4)
Defining αH ≡ RH /t we find that VH ∝ BI, namely
VH = αH BI.
(5)
In Fig 1 we see the setup if the charge carriers are assumed to be positive
holes. If, however, the carriers are negative electrons, then the same equations
~ E
~ H , F~ change direction
hold. This is true since in the case of electrons all of J,
and hence lead to
~H = B
~ ×V
~,
E
the same dynamical equation as for the case of holes.
Knowing RH allows us to find the carrier density N and also the carrier
mobility, µ, if the electrical conductivity, σ, is also known, since for one type of
carrier only, we have
σ = N qµ.
(6)
The conductivity is defined as σ ≡ 1/ρ where the resistivity ρ is related to the
sample resistance R between terminals 1 & 2 by
R=
ρl
.
wt
(7)
This experiment consisted of three sections. In section I the properties of an
electromagnet were investigated. Section II dealt with the measurement of the
Hall voltage VH and conductivity σ using the electromagnet. In section III a
magic cylinder magnet was used, and the properties of a lock-in amplifier were
supposed to be investigated, yet there was not enough time available at the end
to do this
2
Experimental Method
Section I Properties of an Electromagnet
In this section the characteristics of a two-coil electromagnet were investi~ between the coils as a function of Ic , the current between
gated by measuring B
the coils. The remnant field magnitude, Br , was also investigated.
Here B was measured using a Hall Probe Gaussmeter for values of Ic between
±2.5A. The data was plotted in Graph 1(Fig 3 below).
Section II Hall Effect using Electromagnet
In section II the voltage VH in the first germanium sample was measured as
a function of B and I using the two-coil electromagnet from section I. Then the
2
conductivity of the sample was found by measuring V12 across terminals 1 & 2
and I.
The apparatus was set up as in Fig 2 using the first Ge sample. The voltage
V34 between terminals 3 & 4 was measured, along with the value of B. This was
done for a range of B values for each of three I values. The results are plotted
in Graphs 2, 3 & 4 (see Figs 4, 5, 6).
~
The sign of RH was obtained by using a compass to find the direction of B
and then noting the sign of V34 . When both are the same sign, then the current
goes from 1 to 2 and the carriers are holes. Otherwise they are electrons.
The voltage V12 across terminals 1 & 2 was measured along with I in order
to obtain σ.
Section III Hall Effect using Magic Cylinder
Here VH was measured using a ”magic cylinder” permanent cylindrical magnet as in Fig 3 for a second germanium sample. This was done with cylinder
fixed, and then rotating. The properties of a lock-in-amplifier were not investigated due to time constraints.
The apparatus was set up with the magic cylinder here, instead of the electromagnet. Firstly V34 was measured for I between 5 and 25 mA and B = ±170mT
with positive and negative sign by rotating the cylinder. Then VH , V0 were calculated using
V34 = VH + V0
and
1
(V34 (B, I) − V34 (−B, I))
2
1
V0 = (V34 (B, I) + V34 (−B, I)).
2
VH =
(8)
(9)
The results for this are tabulated in Table 1.
Then the cylinder was set to rotate continuously by electrical power. Here an
oscilloscope was used to measure the peak-to-peak difference ∆V ≡ V34 (B, I) −
V34 (−B, I) and from this RH , N, σ & µ were calculated. The results of this are
tabulated in Table 2.
3
Results & Analysis
NOTE: Unless Noted otherwise, all measurements have an error of ±
1 in their last significant figure.
Section I Properties of an Electromagnet
A linear relationship for B vs Ic was found from Graph 1. The remnant field
Br takes values between ±8.7 mT, and has no unique value. It can be seen from
the graph that its strength depends on the previous maximum Ic value.
Section II Hall Effect using Electromagnet
~ and V34 were found to be positive at the same time, hence RH had
Both B
positive sign, and thus the charge carriers in the first Ge sample were holes.
We obtained a value of RH = (1.22 ± 0.01) × 10−4 C −1 for the Hall coefficient.
3
The density N was found to be 5.02 × 1022 m−3 and the carrier mobility was
µ = (2.24 ± 0.03) × 10−4 (C · Ω · m)−1 .
Section III Hall Effect using Magic Cylinder From the manual rotation
of the magic cylinder we found a value of RH = 0.0115 ± 0.0005 C −1 . This led
to values of σ = 17.9 ± 0.5 (Ω · m)−1 and µ = 0.206 ± 0.001(Ω · C · m)−1 .
The continuously rotating cylinder gave a value of RH = 0.01184±0.00005 C −1 .
From this we obtained values of σ = 17.6±0.3 (Ω·m)−1 and µ = 0.208±0.004 (Ω·
C · m)−1 . Using the continuously rotating cylinder it was not possible to find
~ for a
the sign of RH , since it is not possible to determine the alignment of B
given peak/trough on the oscilloscope.
4
Conclusions
From investigating the electromagnet we found that there is no unique B value
for a given Ic and hence we had to measure B directly using a Gaussmeter for
Section II.
From Section II we see that RH is a characteristic property of a material,
insomuch as it does not depend on I.
In section III it was seen that the manual and rotating cylinder methods
agree well on measurements of V34 . The continuous method is probably more
accurate however, since the peaks are clearly defined and easily measurable on
an oscilloscope. With the manual method it can be more difficult to align the
cylinder so as to achieve maximum field strength at the sample.
It was found that the carriers in the second Ge sample were 105 times more
mobile and also the conductivity was an order of magnitude greater than for
the first sample. Clearly these two samples have quite different properties.
There was not enough time left after performing the aforementioned experiments to carry out an investigation into the lock in amplifier, and so it has been
left out of this report.
References
1. ”Hall Effect”, http://en.wikipedia.org/wiki/Hall effect
4
Appendix: Figures & Graphs
Figure 1: Sample schematic
Figure 2: Circuit Diagram for Section II
5
Figure 3: Graph for Section I
Figure 4: Graph for Section II
VH (mV)
8
20.35
30.65
40.95
51.15
V0 (mV)
12
22.65
36.35
51.45
67.25
V0 /V12
0.0220
0.0204
0.0218
0.0229
0.0238
RH (1/C)
0.0095
0.0120
0.0120
0.0120
0.0120
Table 1: Results for Section III, manual rotation
6
Figure 5: Graph for Section II
Figure 6: Graph for Section II
I (mA)
5.02
10.04
15.17
20.2
25.1
∆V (mV)
20
40
61.6
81.6
102
V12 (V)
0.562
1.126
1.78
2.28
2.84
Table 2: Results for Section III, continuous rotation
7