Physics Experiment

Physics Experiment

By: David Long (03457885) Partner: Matthew Slack

Title: The Hall Effect

Abstract:

In this experiment, it was found, using an electromagnet, that VH, the Hall Voltage, is proportional to BI, the applied magnetic field times the applied current. For a Ge crystal, RH, along with the carrier concentration, mobility, and the electrical conductivity of the sample were measured and found to be:RH = 0.01348 +/- 0.00019Ω, the carriers are holes, N = 4.638 x 1020 +/- 6.5 x 1018 carriers, , and σ = 1.79 x 105 +/- 1.72 x 103 Ω-1 m-1. For the permanent magnet “magic cylinder”, the same values of a different Ge crystal were measured and found to be: RH = 1.23 x 10-02 +/- 8.99 x 10-05Ω, N = 5.07 x 1020 +/- 3.71 x 1018 carriers, σ = 428.247 +/- 14.5 Ω-1m-1, μ = 5.28 +/- 0.21 m-1C-1Ω-1carrier-1. The carriers were found to be electrons. This experiment was a successful attempt to examine the Hall Effect, and tabulate results.

Introduction:

The Aim of this experiment was to examine the Hall Effect for a Ge crystal in a static magnetic field, and in a rotating magnetic field. This was done by determining the characteristics of an electromagnet, and then use the electromagnet to measure the Hall Voltage VH, and hence the magnitude of the Hall Coefficient for the crystal in a static magnetic field. The process was then repeated for a crystal in a rotating magnetic field (a “Magic Cylinder”). In each case, the carrier type and concentration, along with the electrical conductivity of the sample and the carrier mobility were also measured.

The experiment is in three main parts. In the first section the characteristics of an electromagnet are examined. This is necessary so that any fluctuations in the magnetic field are found before attempting to measure the Hall Voltage. Here, the magnetic field between the poles of the magnet was measured as a function of the current in the coils, for values of IC from 0A to 2.5A to 0A to -2.5A and back again to 0A. From this it was found that the magnetic field is not independent of the previous value of the current IC, and so needs to be measured each time.

In the second part of the experiment, it was shown that VH is proportional to BI. Hence the magnitude and sign of RH, as well as the concentration and type of charge carriers in the crystal were measured. How this was done is outlined in the Experimental Details.

Finally in the third part of the experiment, VH was measured using a “Magic Cylinder” magnet. In this case, the germanium crystal was mounted inside the rotating cylinder of permanent magnets. Firstly V34 was measured using the stationary cylinder, with the magnetic field aligned perpendicular to the sample for I = 5, 10, 15, 20, 25mA. From this it was then possible to measure VH and V0. V34 was then measured for the same values of I, but with the Cylinder continuously rotating, and using an oscillator. The oscillator is DC coupled as this ensures that there is no noise contaminating the reading. It is also essential that neither of the terminals of the sample current power supply are earthed as this ensures that the sample current is of the same form as the measured current. Next, a lock-in amplifier was used, as this gives a better reading than the oscillator by filtering out the noise in the signal. VH was again determined for I = 5, 10, 15, 20 mA, hence giving RH, and N.

The equations given for finding the Number of charge carriers work the same for both electrons and holes as they can be regarded as effectively the same thing – Electrons are the Negative charge carriers, while holes are the Positive charge carriers. They both have the same characteristics, except that they have opposite sign. Therefore, the equations can apply equally well to both.

Part I:

Experimental details:

The apparatus was set up in the Lab already. It consisted of an electromagnet with a Hall probe gaussmeter between the two poles. The probe was oriented until it gave the largest reading, at which position it was used in the experiment. B, the magnetic Field was then measured for the current through the coils (IC) from 0A to 2.5A to 0A to -2.5A to 0A. From this plot, the remnant field Br was determined.

Results and Analysis:

The results of this part of the experiment are shown below, along with their graphical representation:

Ic (A) / B (+/- 0.1) (mT)
0 / 0.632
0.5 / 30.7
1 / 63.7
1.5 / 97.4
2 / 130.6
2.5 / 163.4
2 / 135
1.5 / 103.8
1 / 70.7
0.5 / 36.5
0 / 0.701
-0.5 / -35.9
-1 / -72.5
-1.5 / -109.2
-2 / -143.8
-2.5 / -177.3
-2 / -149.5
-1.5 / -118.8
-1 / -85.9
-0.5 / -51.9
0 / -15.92

As can be seen, the magnetic field depends on the previous maximum of the current through the coil. In figure 1 above, the lower line shows the line of increasing current, while the upper line shows the line of decreasing current. It can be shown that the remnant field of the electromagnet is -15.219 T. This is the difference between the field when coming from a current of 2.5A, and coming from a current of -2.5A. It can easily be noticed that the values of B for IC increasing differ from those with IC. This is due to the effect of the electric field on the “soft” iron of the electromagnet. The fact that the electromagnet is made of “soft” iron means that some of the dipoles in the iron remain aligned even after the external magnetic field has been removed. This gives rise to a remnant field Br. As expected this remnant field depends on the previous maximum value of the current – where the current was at a positive maximum, the field will tend to be slightly more positive, and where the current was at a negative maximum, the field will tend to be slightly more negative. The same shape of loop would be followed in the maximum values of IC were changed, but the magnitude of the loop would depend on the maximum values of IC.

Part II:

Experimental Details:

In this part of the experiment, the electromagnet was used again, with the exception that a Germanium sample mounted on a circuit board between the poles of the magnet was used. Firstly some of the voltage difference between the terminals 1 and 2 (V0) was cancelled out to ensure that only V34 (the Potential difference between terminals 3 & 4) was measured. This was done by setting I = 10mA and B = 0. The potentiometer attached to the circuit board was then adjusted to get V34 approximately 0. [I in this case refers to the current through the sample, as opposed to IC which is the current through the coils of the electromagnet. V34 describes the voltage drop between terminals 3 & 4, V0 refers to the voltage drop between terminals 1 & 2, while VH is the Hall Voltage (V34 = VH + V0).] V34 was then measured versus B (for IC from -2.5A to 2.5A as in part I) for three values of I (10, 20, 30mA). V34 was then plotted against B for each value of I, to show that VH is proportional to B. RH was determined from the slope of each plot, and then used to determine the carrier concentration N. By picking one pair of values of B & I, the sign of RH was then determined. The conductivity of the sample was measured by measuring I vs. the voltage drop V12. This was then combined with N to evaluate the carrier mobility.

Results and Analysis:

The table of results is shown below, along with the graphs of each set. As can be seen, V34 was measured against B for I = 10, 20, 30 mA. As can be seen in the Graphs, VH is proportional to B, as required.

I=10mA / I=20mA / I=30mA
Ic / B (mT) / V34 (mV) / Ic / B / V34 (mV) / Ic / B / V34 (mV)
-2.5 / -167.5 / 20.4 / -2.5 / -170.4 / 42.6 / -2.5 / -168.6 / 61.8
-2 / -141 / 17.2 / -2 / -142.4 / 36.2 / -2 / -141.8 / 51.9
-1.5 / -111.7 / 13.5 / -1.5 / -113.1 / 29 / -1.5 / -113 / 41.1
-1 / -81.1 / 9.5 / -1 / -82.4 / 21.2 / -1 / -81.5 / 28.9
-0.5 / -49 / 5.1 / -0.5 / -49.7 / 12.5 / -0.5 / -49.1 / 16
0 / -15.52 / 0.5 / 0 / -15.63 / 3.4 / 0 / -15.54 / 2.2
0.5 / 19.23 / -4.3 / 0.5 / 19.35 / -6.2 / 0.5 / 18.62 / -11.8
1 / 52.1 / -9 / 1 / 52.9 / -15.7 / 1 / 53.9 / -26.1
1.5 / 85.2 / -13.7 / 1.5 / 87.9 / -25.1 / 1.5 / 86.1 / -40
2 / 118.1 / -18.3 / 2 / 119.6 / -33.7 / 2 / 118.2 / -53.2
2.5 / 149.6 / -22.6 / 2.5 / 152 / -42.2 / 2.5 / 150.3 / -66.1

The Equation of the graph is: V34 = αHBI +βI. When compared with the equations above, it can be seen that αHI is equal to the slope of the Graph. αH is equivalent to RH/t (T = 1mm – thickness of the sample). Therefore, for Graph 1;

RH = 0.01368 Ω

Similarly for Graph 2, just substitute in the value of the slope. Doing this, and working out gives a value of RH = 0.01331Ω. For Graph 3, the value is RH = 0.01346Ω.

It can be seen that RH is approximately the same for each value of I. The difference between the values is due to experimental error. The average of RH = 0.01348 +/- 0.00019Ω. The Carrier concentration N can be determined as follows:

(q is the magnitude of the charge carrier)

N = 4.638 x 1020 +/- 6.5 x 1018 carriers

Using the average of the values of RH found, the average number of charge carriers can be found as shown above.

The sign of RH was determined by placing a compass between the poles of the electromagnet. The compass was observed to point from the left coil to the right coil, whereas the current flow was into the right coil, and out of the left coil. As the compass points in the direction of B, it was found that the flow of holes was responsible for the movement of charge. Therefore, RH is positive.

The Conductivity of the sample was then measured by measuring the current I vs. the voltage drop V12. The results and graph are illustrated below:

I (mA) / V12 (V)
5 / 0.436
10 / 0.873
15 / 1.314
20 / 1.766
25 / 2.21
30 / 2.67
35 / 3.13
40 / 3.59
45 / 4.08
48 / 4.38

As can be seen, there is a linear relationship between I and V12. The equation of the graph is:. Therefore, the slope of the graph is 1/R. The c-component of the graph is merely the residual voltage of the system. The electrical conductivity of the sample is found from the formula , where l, w, and t are the length, width and thickness of the sample respectively (l = 10mm, w = 5mm, t = 1mm). Therefore, σ = 1.79 x 105+/- 1.72 x 103Ω-1 m-1. From this then, the carrier mobility (μ) can be found to be:

The equations used in this part of the experiment were all given in the experiment hand-out.

Part III:

Experimental Details (i):

For this part of the experiment, a “Magic Cylinder” rotating permanent magnet was used to examine the Hall Effect in a rotating magnetic field. Firstly, the apparatus was set up to give B perpendicular to the sample. This was done using the engraved arrows on the tops of the permanent magnets making up the magic cylinder, and the diagram in the experimental handout. V34 was measured for both directions of B. Hence VH and V0 were measured using the formulae:

These measurements were then repeated for I = 5, 10, 15, 20, 25, mA. V12 was also measured for each I. The results for VH, V0, V0/V12, RH, and R were then tabulated. Using these results, N, the sign of RH, the carrier type, sample conductivity and mobility were all found. The misalignment of the contacts 3 & 4 was also calculated.

Results and Conclusions (i):

The results received from the first half of part III of the experiment are tabulated below:

I (A) / V34 (+B) mV / V34 (-B) mV / VH / V0 / V12 (V) / V0/V12 / RH / R
5.00E-03 / 2.22E-02 / 1.10E-03 / 0.01055 / 0.01165 / 1.11 / 0.010495 / 1.24E-02 / 4.44E+00
1.00E-02 / 4.57E-02 / 3.50E-03 / 0.0211 / 0.0246 / 2.11 / 0.011659 / 1.24E-02 / 4.57E+00
1.50E-02 / 7.03E-02 / 7.50E-03 / 0.0314 / 0.0389 / 3.05 / 0.012754 / 1.23E-02 / 4.69E+00
2.00E-02 / 9.61E-02 / 1.27E-02 / 0.0417 / 0.0544 / 3.98 / 0.013668 / 1.23E-02 / 4.81E+00
2.50E-02 / 1.22E-01 / 1.84E-02 / 0.05175 / 0.07015 / 4.9 / 0.014316 / 1.22E-02 / 4.88E+00

The I value is fixed for each set of measurements. V34 is then measured for B positive and negative (this is done by rotating the magic cylinder so that the B-field is perpendicular to the sample and facing in either the positive or negative directions). VH is found using the formula given above, similarly for V0. V12 is measured. V0/V12 is found by dividing V0 by V12. RH is found using the formula , where t is the thickness of the sample. RH (average) was found to be: RH = 1.23 x 10-02 +/- 8.99 x 10-05Ω. The resistance R is found using the formula . Using the above values and measurements, N, the sign of RH, along with the carrier type, the sample conductivity and mobility may all be found. These values are tabulated below:

N / rho (resistivity) / conductivity / mobility
5.04E+20 / 0.00222 / 450.4504505 / 5.59E+00
5.04E+20 / 0.002285 / 437.6367615 / 5.43E+00
5.08E+20 / 0.002343333 / 426.742532 / 5.25E+00
5.10E+20 / 0.0024025 / 416.2330905 / 5.10E+00
5.13E+20 / 0.002438 / 410.1722724 / 4.99E+00

N is found using the given equation:. N (average) was therefore found to be, N = 5.07 x 1020 +/- 3.71 x 1018 carriers. The conductivity is given by: , where l, w, and t are the dimensions of the sample, and R is the resistance (given above). σ(average) was therefore found to be, σ = 428.247 +/- 14.5 Ω-1m-1. The carrier mobility is then found using: . In the table above, the resistivity is equal to 1/σ. μ (average) was found to be, μ = 5.28 +/- 0.21 m-1C-1Ω-1carrier-1. The sign of RH is found by examining the sample in the magic cylinder. V34 is at a maximum when B is perpendicular to the sample. This occurs when the direction of the field is into the sample, meaning that the voltage V34 is from terminal 4 to terminal 3, as B is at right angles to the direction of V34. Therefore, the movement of carriers is from 4 to 3, i.e.: from negative to positive. Therefore, electrons are the carriers in this sample. In this case, therefore, RH is negative.

Experimental Details (ii):

In the second half of this part, the cylinder was set continuously rotating, and an oscilloscope was connected to it. V34 was then measured using the oscilloscope output for values of B = +170mT and -170mT, for each of the previous values of I. The results were tabulated, and RH, σ, and μ were calculated (see results below). Next, a lock-in amplifier was used to investigate the effect of the phase setting. Terminals 3 & 4 were connected to input A of the LIA. The reference input of the LIA was connected to the square wave provided by the rotation unit. I was set to 10mA and the rotation started. The gain was then slowly increased until a good deflection was observed. The effect of the phase setting was examined by measuring and plotting the output voltage vs. the phase over a range of 180o. From this, a value for the phase at which the output was a max was determined. Then, for I = 5, 10, 15, 20 mA, VH was determined, and used to calculate RH and N.

Results and Analysis (ii):

The first set of results, (for the cylinder continuously rotating) are as follows:

I (A) / V34 (+B) mV / V34 (-B) mV / VH / V0 / V12 (V) / V0/V12 / RH / R
5.00E-03 / 1.12E-02 / -1.12E-02 / 0.0112 / 0 / 1.13E+00 / 0 / 1.32E-02 / 2.24E+00
1.00E-02 / 2.16E-02 / -2.16E-02 / 0.0216 / 0 / 2.19E+00 / 0 / 1.27E-02 / 2.16E+00
1.50E-02 / 3.20E-02 / -3.20E-02 / 0.032 / 0 / 3.2 / 0 / 1.25E-02 / 2.13E+00
2.00E-02 / 4.16E-02 / -4.16E-02 / 0.0416 / 0 / 4.09 / 0 / 1.22E-02 / 2.08E+00
2.50E-02 / 5.12E-02 / -5.12E-02 / 0.0512 / 0 / 5.03E+00 / 0 / 1.20E-02 / 2.05E+00

The I value was fixed for each measurement. The value of VH was then measured using the oscilloscope, by measuring the peak and trough values on the screen of the oscilloscope – these values refer to the maximum and minimum values of B respectively. Using the same formulae and methods as described above in the first half of this part, the number of carriers, the type of carrier, along with the sample conductivity and mobility were calculated. These results are outlined in the table below:

N / rho (resistivity) / mu (conductivity) / mobility
4.74E+20 / 0.00112 / 892.8571429 / 1.18E+01
4.92E+20 / 0.00108 / 925.9259259 / 1.18E+01
4.98E+20 / 0.001066667 / 937.5 / 1.18E+01
5.11E+20 / 0.00104 / 961.5384615 / 1.18E+01
5.19E+20 / 0.001024 / 976.5625 / 1.18E+01

Again, by the method illustrated above, the carriers were found to be electrons, meaning that RH is negative.

phase (Degrees) / Vout (V)
0.00E+00 / 6.00E+00
5.00E+00 / 5.98E+00
1.00E+01 / 5.93E+00
1.50E+01 / 5.83E+00
2.00E+01 / 5.69E+00
2.50E+01 / 5.48E+00
3.00E+01 / 5.25E+00
3.50E+01 / 4.98E+00
4.00E+01 / 4.66E+00
4.50E+01 / 4.29E+00
5.00E+01 / 3.92E+00
5.50E+01 / 3.52E+00
6.00E+01 / 3.11E+00
6.50E+01 / 2.62E+00
7.00E+01 / 2.15E+00
7.50E+01 / 1.66E+00
8.00E+01 / 1.16E+00
8.50E+01 / 6.20E-01
9.00E+01 / 1.10E-01
9.50E+01 / -4.20E-01
1.00E+02 / -9.60E-01
1.05E+02 / -1.49E+00
1.10E+02 / -2.00E+00
1.15E+02 / -2.51E+00
1.20E+02 / -2.95E+00
1.25E+02 / -3.38E+00
1.30E+02 / -3.80E+00
1.35E+02 / -4.18E+00
1.40E+02 / -4.52E+00
1.45E+02 / -4.85E+00
1.50E+02 / -5.15E+00
1.55E+02 / -5.37E+00
1.60E+02 / -5.60E+00
1.65E+02 / -5.75E+00
1.70E+02 / -5.86E+00
1.75E+02 / -5.96E+00
1.80E+02 / -6.00E+00

For the next part of the experiment, a Lock-in Amplifier was used. In this case, the output voltage was measured as a function of the phase setting of the Lock-in amplifier. These results are tabulated on the left hand side. A plot of these results can be found below. The phase setting value at which the output voltage is a maximum can be seen to be 0o. This value gives a maximum voltage of 6V. This value is the RMS value of the input signal.

VH was then determined for I = 5, 10, 15, 20 mA, hence giving RH and N. These results are tabulated below:

I (A) / VH (V) / RH / N
5.00E-03 / 2.99E+00 / 3.52E+00 / 4.54849E-20
1.00E-02 / 6.03E+00 / 3.55E+00 / 4.51078E-20
1.50E-02 / 9.00E+00 / 3.53E+00 / 4.53333E-20
2.00E-02 / 1.20E+01 / 3.53E+00 / 4.53711E-20
1.00E-03 / 6.40E-01 / 3.76E+00 / 4.25E-20
5.00E-04 / 3.10E-01 / 3.65E+00 / 4.3871E-20
1.00E-04 / 6.00E-02 / 3.53E+00 / 4.53333E-20

These results were found again using the methods outlined above. Again, RH was found to be negative.

Conclusions:

In conclusion, the overall aim of this experiment was to examine the Hall Effect through its effect on the voltage across a Germanium wafer. This was done using a stationary electromagnet, and using a stationary, and rotating, permanent magnetic cylinder. It was found that the types of charge carriers could be worked out quite easily using measurable information. Not only could the types of charge carriers be worked out, but also the mobility of the carriers, as well as the conductivity of the sample.

It was also necessary, as part of the experiment, to familiarize oneself with the characteristics of a soft iron electromagnet, as well as a Lock-in amplifier. Both these instruments were examined, and used successfully in finding results and answers to the questions asked.