Download Fractals - TCD Maths

The Hall Effect
Declan Cockburn: 02038633. Lab partner: William Walshe. November 2004
Abstract:
This experiment involves the investigation into the Hall effect, properties of a magnetic
coil, examination of the properties of germanium in a magnetic field and the learning of
and how to use the lock in amplifier.
We found that the charge carriers in germanium was (unlike most other semi-conductors)
actually electrons. Thus the sign of R was negative.
For the first sample:
3
3
RH The hall coefficient was found to be: 12.2  10  0.2  10 Tm
VA
20
20
3
The carrier density was found to be: 5.12  10  .08  10 m
The conductance was found to be 2.14  103  0.06  103  1.m 1
For the second sample
3
3
1 1
RH The hall coefficient was found to be: 11.4  10  .2  10 TmV A
The carrier density was found to be: 5.48  1020  .01  1020 m 3
The conductance was found to be: 7.6  103  .8  103  1.m 1
Introduction:
In 1879 E. H. Hall discovered what is today called the Hall effect.
It is a rather simple phenomenon whereby a conducting plate has a DC current and a
transverse magnetic field going through it as in Fig. 1.
What happens is: there is a Lorentz force  ev  B
caused by the magnetic field, which tends to deflect
the moving charge carriers to the left hand side in the
case of Fig.1. Due to this a build up of charge occurs
the sides of the plate, which creates EH a transverse
electric field that quickly balances out the Lorentz
force. This means that the carriers can now return to
traveling straight.
This is all pretty fundamental, the interesting thing
about the Hall effect however, is the complete lack of
Fig.1
the free electron theory’s ability to predict the polarity of the electric field in some
conductors. Why is this? Well in the majority of cases it is the electrons which are the
charge carriers. In some cases though it is positive holes that carry charge. A stationary
magnetic field affects all moving charges, regardless of the sign of the charge. Then the
(+) holes are pushed to the same side as the (-) electrons would be if they were the
carriers, even though the conventional current remains the same!
The magnitude of the electric field is given by:
 
E H  RH B  j
Where:

j =I and is the current
RH is called the Hall coefficient. It is defined as being

1
Nq
Where N is the number of charge carriers per unit
density and q is the charge on said carrier.
But the Hall voltage is given by: VH  E H w 
RH
BI   H BI
t
Another term of interest is the carrier mobility given by:  

Nq
As can be seen it is proportional to the conductivity which is merely the reciprocal of the
resistivity.
Experimental Details:
(i)
The aim of this experiment was to measure the magnetic field between magnetic coils as
a function of the current applied to the coils. This was very straightforward and required
us to slowly and consistently increment a measurable a current -2.5A  +2.5A and back
again while tabulating the readings with the associated field between the coils
(ii)
The aim of this experiment was to show that the Hall voltage was proportional to the
product of the magnetic field and the current between contact points 1&2.
Secondly to determine the sign of RH
(and hence know what are the charge
carriers) and lastly to measure the
conductivity of the sample.
The circuit was setup as in Fig. 2. With the
magnetic field B perpendicular to the
page. There also existed a 10 kΩ
potentiometer not shown that was used to
remove the reading of V0 (extra
independent voltage caused by the
misalignment of contacts) as much as
possible. V0 is caused by a slight
misalignment between the contacts of
3&4.
Readings were taken of V34 against B for
three different values of I (10, 20, 30 mA)
and plotted.
Also I was measured against the voltage
drop between contacts 1&2
Fig.2
(iii)
The aims of this experiment were to determine the carrier type/sign of RH . To measure
conductivity and obtain carrier mobility. To estimate the misalignment of two transverse
contacts. To learn about the characteristics and operation of a Lock In Amplifier (LIA)
and finally to determine RH using a rotating field.
This experiment is quite a long one and involves the use of a magic cylinder, which far
from being magic, uses 8 permanently magnetic segments. Their poles are pointed in
such directions as to have a transverse magnetic field through it’s core. See Figure 3.
Also a very “interesting” and sensitive instrument was used called an LIA.
For the first part of this experiment, using a DMM: V34 was measured against I = 5, 10,
15, 20, 25 mA with B = 170mT and repeated for B = -170mT.
Next the above experiment was repeated except with the cylinder rotating, and the
measurements being taken by an oscilloscope.
For the next part of the experiment the LIA was required. A lot of reading, trial and error
was required to figure out how to operate this unfamiliar tool. Once our understanding of
it was brought to fruition we began by hooking it up to detect V34. Set the cylinder
rotating, set the gain to achieve a suitable deflection and finally chose a suitable time
constant.
The phase setting was then investigated where the phase was measured against the
voltage given on the LIA (which is the rms value of V34). A maximum was found.
For I = 5, 10, 15, 20 mA, V34 was measured at the max value of the phase setting which
was found each time.
As a result of the LIA’s noise reduction capability, it can give a much better readings at
low voltages than the oscilloscope. To examine this V34 was measured for I= 1, 0.5 and
0.1 mA. Again the phase setting was corrected to be at its maximum each time.
The results were then tabulated.
Results and Analysis:
(i) To investigate the properties of an electromagnet:
Investigating the properties of an electromagnet:
Magnetic Field vs. Current Through Coils
200
150
100
B (mT)
50
0
-3
-2
-1
0
1
2
3
-50
-100
-150
-200
Ic (Amps)
Fig. 4
As can be seen in Fig. 4, the magnetic field, though proportional to the current, has
another factor affecting it: what state it was in just previously. Even though the magnet
we were using has a soft iron core, this shows that nevertheless it is affected by a residual
magnetic field. This is significant as it shows that there is no unique value of B verses Ic.
This means that B must be measured directly for each value of V34.
(ii) Measurements of VH using the electromagnet:
V43 vs B at 10mA
y = 0.1211x + 5.6185
30
25
20
V34 (mV)
15
10
5
0
-200
-150
-100
-50
-5 0
50
-10
-15
-20
B (mT)
C
m
Value
5.62
0.1211
Fig. 5.1
Error
0.07
0.0006
V43   H BI  I
m  0.1211  0.0006 T
V
 H I
I  10  .1mA
t  1  .02mm
RH 1   H t 
mt
 12.1  10 3  .4  10 3 Tm
VA
I
100
150
200
250
V43 vs B at 20mA
y = 0.2454x + 10.476
60
50
40
V34 (mV)
30
20
10
0
-200
-150
-100
-50
-10 0
50
-20
-30
-40
B (mT)
C
M
Value
10.5
0.245
Error
Fig. 5.2
0.1
0.001
V43   H BI  I
m  0.245  0.001 T
V
 H I
I  20  .1mA
t  1  .02mm
RH 2   H t 
mt
 12.3  10 3  .4  10 3 Tm
VA
I
100
150
200
V43 vs B at 30mA
y = 0.3682x + 14.123
100
80
60
V34 (mV)
40
20
0
-200
-150
-100
-50
-20
0
50
100
150
200
-40
-60
B (mT)
Fig. 5.3
C
M
Value
14.1
0.368
Error
0.2
0.002
V43   H BI  I
m  0.368  0.002 T
V
 H I
I  30  .1mA
t  1  .02mm
RH 3   H t 
mt
 12.3  10 3  .3  10 3 Tm
VA
I
As can be seen from the three graphs above (Figs. 5.1, 5.2 and 5.3) it is quite plain to see
that V43 (and therefore VH) is directly proportional to B. Also it can be seen from the
associated calculations that RH is the same for each (and thus independent of) I.
RH 
Since
1
Nq then:
N 
1
RH q where
RH 
RH 1  RH 2  RH 3
3
Thus:
12.1  12.3  12.3
.4 2  .4 2  .32
RH  

3
3


3
3
3
  10  12.2  10  0.2  10 TmVA

And:
N
1
 5.12  1020  .08  1020 m 3
3
19
(12.2  10  0.2  10 )(1.6  10 )
3
Where |q| was taken to be |e| = 1.6x10-19C
Determining the sign of RH requires more information than above. It requires knowledge
of the direction of the magnetic field using a compass.
So for a value of V34 say +40mV at I = 30mA.
The direction of B was as shown in Fig.6
Fig. 6
Thus it is shown in Fig. 7 according to the
Lorentz force that the charge carriers are
obviously electrons, which means that the
sign of RH in this case is negative.
Fig. 7
The conductivity of the sample is given in this case by:
Where resistance
R
Vdrop
I

l
Rwt
Voltage drop vs. I
Voltage drop (mV))
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
I (mA)
Fig. 8
Value
-0.18
0.933
C
m
Error
0.06
0.003
Fig 8. allows us to get an accurate measurement of R the resistance across the sample.
Where:
R  m  0.933  0.003

(10  10 3  .02  10 3 )
 2.14  103  0.06  103  1 .m 1
3
3
3
3
(.933  .003)(1  10  .02  10 )( 5  10  .02  10 )
The carrier mobility  can now be determined:
Since (assuming):
Then:



Nq
2.14  103  0.06  103
 26  1m 2 1C 1
20
20
19
(5.12  10  .08  10 )(1.6  10 )
(iii) Measurements of VH using a “magic cylinder” permanent magnet:
This part of the experiment uses the magic cylinder (see Fig.3 for apparatus) which here
is set to it’s arbitrary +/- 170mT. The results are tabulated below in Fig. 9 with “a”
denoting +B and “b” denoting a -B.
VH and V0 were found by:
1
VH  {V34 ( B, I )  V34 (  B, I )}
2
1
V0  {V34 ( B, I )  V34 (  B, I )}
2
I (mA)
5
10
15
20
25
error=.1
V34a (mV)
22.4
44
64.4
83.8
102.6
error=.1
V12a (mV)
1.09
2.53
4.04
5.47
6.86
error=.01
V34b (mV)
3
4.8
5.9
6.5
7.4
error=.1
V12 (mV)
1.12
2.55
4.08
5.45
6.86
error=.01
VH (mV)
9.7
19.6
29.25
38.65
47.6
error=.1
V0 (mV)
12.7
24.4
35.15
45.15
55
error=.1
Fig. 9
RH and its associated error was found from the following equation for each of the above
values.
RH 
VH t
, but because of the direction of the magnetic field for VH, RH will be negative
BI and thus charge carriers electrons.
RH
0.0114
0.0115
0.0115
0.0114
0.0112
RH
0.0006
0.0004
0.0003
0.0003
0.0003
The mean turning out to be:
N
V0/V12
11.6
9.64
8.7
8.25
8.02
(V0/V12)
0.2
0.09
0.05
0.04
0.03
RH  11.4  103  .2  103 TmV 1 A1
1
 5.48  1020  .01  10 20 m 3
RH q
R turns out to be .26 +/- .02Ω
 
l
 7.6  10 3  .8  10 3  1.m 1
Rwt
And the carrier mobility:


Nq

7.6  103  .8  103
 87  9m 2  1C 1
(5.48  1020  .01  1020 )(1.6  1019 )
Next for the same values of I, V34 was measured again except this time with the digital
oscilloscope and with the cylinder rotating. The oscillator has to be DC coupled because
of the two components in V34: the Hall voltage which varies with time and the constant V0.
If it wasn’t DC coupled then we would be unable to measure V0.
RH cannot be determined.
The following values were obtained
I (mA)
5
10
15
20
25
error=.1
V34 (mV)
20
40
60
85
98
error=1
V12 (mA)
1.09
2.42
3.92
5.33
6.6
error=.01
These correlate well with the above values of V34.
Now the LIA is used to measure a value of V vs. the phase setting at I=10mA to
investigate the maximum.
Output voltage vs Phase setting (g)
Vout (centivolts)
1
0.5
0
0
50
100
150
200
-0.5
-1
-1.5
Phase (degrees)
Fig. 10
250
300
350
From Fig 9. it is obvious that if we were to have continued taking readings they would
have formed a sine-like function. Fig.10 also shows the max value of V to be 12 +/- .5
mV which approx corresponds to the rms value of the input signal which is VH.
The values obtained were:
Vrms (mV)
6
12
17.5
23
I (mA)
5
10
15
20
Vrms
0.2
0.5
0.5
1
B (mT)
170
170
170
170
error=.1
Thus:
VH (mV)
8.5
17
24.7
32
And since:
RH
0.01
0.01
0.0096
0.0094
VH
0.3
0.7
0.7
1
RH 
VH t
BI
RH
0.0008
0.0007
0.0005
0.0005
Thus the mean is:
10  10  9.6  9.4
.82  .7 2  .52  .52 
3
3
3
RH  

  10  9.8  10  0.3  10 TmVA
4
4


Thus:
N
1
 6.4  1020  .2  1020 m 3
3
19
(9.8  10  0.3  10 )(1.6  10 )
3
The sign of RH cannot be determined.
Finally, we investigate the accuracy of the LIA. The rms (as in ) value of VH is measured
for small values of current to investigate the relationship. The values are as such:
Vrms (mA)
I (mA)
VH (mA)
VH/I
1
1.25
1.77
1.77
0.5
0.68
0.96
1.92
0.1
0.36
0.51
5.1
The most startling thing is that VH/I is not a constant. This is because it is very difficult to
get precise measurements of VH at such low currents in-spite of the LIA’s ability to
practically detect the brainwave of an amoeba.
Conclusions:
Properties of an electromagnet were investigated.
VH was shown to be proportional to
For the first sample:
RH The hall coefficient was found to be:
12.2  103  0.2  103 Tm
VA
3
The carrier density was found to be: 5.12  10  .08  10 m
The conductance was found to be 2.14  103  0.06  103  1.m 1
And the mobility was found to be 26  1m 2  1C 1
20
20
The second sample
RH The hall coefficient was found to be: 11.4  10
3
 .2  103 TmV 1 A1
Using the LIA the hall coefficient was found to be:  9.8  10
3
 0.3  103 Tm
VA
3
The carrier density was found to be: 5.48  10  .01  10 m
Using the LIA the carrier density was found to be  6.4  1020  .2  1020 m 3
The conductance was found to be: 7.6  103  .8  103  1.m 1
And the mobility was found to be: 87  9m 2 1C 1
20
20
For the second sample the LIA’s N and RH do not agree with the conventional method. I
would trust the conventional method more as I am not entirely confident with my
evaluation of VH using LIA.