Lui Chi Kong
Department of Physics, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong
This paper aims to focus on the study the Hall effect in thin copper and zinc foils and p- and n-type germanium and to determine the hall coefficients of the samples. The hall coefficients of copper and zinc were found to be respectively (-4.8 + 0.2) x 10-11 m3/As and (4.2 + 0.2) x 10-11 m3/As. The hall coefficients of n- and p-type germanium were found to be (8.61 + 0.01) x 10-3 m3/As and (7.76 + 0.06) x 10-3 m3/As respectively. The energy of band gap for n- and p-type germanium were found to be (0.592 + 0.001) eV and (0.705 + 0.005) eV respectively. Some other properties such as conductivity at room temperature and Hall mobility for n- and p-type germanium were also obtained.
When a current-carrying conductor is placed perpendicularly in an applied magnetic field, a voltage gradient which is transverse to both the current and the magnetic field is developed. This is known as the Hall effect and was discovered by Edwin H. Hall in 1879. [1] Important information about the nature of the conduction process in semiconductor and metals may be obtained through analysis of this effect.
Consider a slab of current-carrying conductor as shown in Figure 1. The forces experienced by the carriers in the conductor act at the right angles to the directions to magnetic field and the current, given by Fleming’s left-hand rule, and cause the charge carriers to be pushed sideways, thus increasing their concentration towards one side of the conductor. As a result, a potential difference (and an electric field) is produced across the conductor, which is the Hall effect. [2]
Let VH be the Hall voltage. The force acting on the charge, F is given by
Where E is the electric field, Q is the charge and l is the length of conduction.
Since the force is perpendicular to the magnetic field, angle between the directions of the force and magnetic field, angle between the directions of the force and magnetic field is 90˚. Therefore,
F = Q v B sin90˚ = Q v B (2)
where v is the velocity of the electron and B is the magnetic field strength.
But V is given by
where n is the number of charge carriers per unit volume, A is the cross section area and I is the current flowing through the conductor. Also,
A = d l (4)
where d is the thickness of the conductor.
Equating equations (1) and (2),
VH = B v l (5)
Substituting equation (3) into equation (5), the Hall voltage is given by
The expression shows the value of VH is large for n is small, such as semiconductor since those have a smaller number of charge carrier per unit volume than metal.
The Hall coefficient RH is given by
The values of Hall coefficient of copper and zinc given in the literature are respectively -5.5 x 10-11 m3/As and +3.3 x 10-11 m3/As respectively. Negative sign indicates electron conduction and positive sign indicates holes concentration. [3]
Experimental
Apparatus and Procedures
To investigate the Hall effect, the magnetic field was set up by connecting two coils of each 300 turns with the iron core to the power supply and the transverse current was set up by connecting the sample carrier board to the power supply. For Hall effect in copper and zinc, a measuring amplifier with amplification 104 was connected. For Hall effect in n- and p-type germanium, a current-control box was inserted.
For each sample, the transverse current was first kept constant. The magnetic field strength was varied and the Hall voltage was measured. Then the magnetic field was kept constant. The transverse current was varied and the Hall voltage was measured.
For n- and p-type germanium samples, the change in resistance with applied magnetic field was studied by measuring the sample resistances in the absence and presence of magnetic field and calculating the fractional change in resistance. Also, a heating current was applied and a computer program was used in investigating the temperature dependence. For constant current, the sample voltage and temperature were measured. For constant magnetic field strength, the hall voltage and temperature of the sample were measured.
Data
Analysis and Results
The thickness of the copper sample is 18 μm. With constant current 4.7A, a graph of Hall voltage versus magnetic field strength is plotted in Figure 2(a), and the slope of the graph is (-0.114 + 0.002) x 10-4 V/T. From equation (6), the Hall coefficient RH of copper is (-4.35 + 0.08) x 10-11 m3/As. With constant magnetic field strength 250mT, a graph of Hall voltage versus transverse current is plotted in Figure 2(b), with slope (-0.0066 + 0.0003) x 10-4 V/A. The Hall coefficient RH of copper is (-4.8 + 0.2) x 10-11 m3/As.
The thickness of the zinc sample is 25 μm. With constant current 4.32A, a graph of Hall voltage versus magnetic field strength is plotted in Figure 3(a), with slope (0.072 + 0.005) x 10-4 V/T. The Hall coefficient RH of zinc is (4.2 + 0.3) x 10-11 m3/As. With constant magnetic field strength 250mT, a graph of Hall voltage versus transverse current is plotted in Figure 3(b), and the slope of the graph is (0.0042 + 0.0002) x 10-4 V/A. The Hall coefficient RH of zinc is (4.2 + 0.2) x 10-11 m3/As.
For n-germanium, the thickness is 1 x 10-3 m. With constant current 30 mA, a graph of Hall voltage versus magnetic field strength is plotted in Figure 4(a), with slope (0.2504 + 0.0006) V/T. The Hall coefficient RH of n-germanium is (8.35 + 0.02) x 10-3 m3/As. With constant magnetic field strength 200mT, a graph of Hall voltage versus transverse current is plotted in Figure 4(b), and the slope of the graph is (1.722 + 0.003) V/A. The Hall coefficient RH of n-germanium is (8.61 + 0.01) x 10-3 m3/As.
The resistivity of n-germanium increases non-linearly (quadratic) with the applied magnetic field. The conductivity at room temperature is 42.67 Ω-1m-1. The Hall mobility of the charge carriers is 0.356 m3/Vs. The Holes concentration of n-doped sample is 7.40 x 1020 m-3. When the heating current is applied but the magnetic field switched off, the conductivity against reciprocal temperature was studied. The energy of the band gap is calculated to be (0.592 + 0.001) eV. When both the heating current and the magnetic field were applied, the Hall voltage decreases with the temperature of the sample.
For p-germanium, the thickness is 1 x 10-3 m. With constant current 30 mA, a graph of Hall voltage versus magnetic field strength is plotted in Figure 5(a), with slope (0.228 + 0.002) V/T. The Hall coefficient RH of p-germanium is (7.59 + 0.06) x 10-3 m3/As. With constant magnetic field strength 250mT, a graph of Hall voltage versus transverse current is plotted in Figure 5(b), and the slope of the graph is (1.94 + 0.02) V/A. The Hall coefficient RH of p-germanium is (7.76 + 0.06) x 10-3 m3/As.
The resistivity of p-germanium increases non-linearly (quadratic) with the applied magnetic field. The conductivity at room temperature is 38.81 Ω-1m-1. The Hall mobility of the charge carriers is 0.295 m3/Vs. The Holes concentration of p-doped sample is 8.22 x 1020 m-3. When the heating current is applied but the magnetic field switched off, the conductivity against reciprocal temperature was studied. The energy of the band gap is (0.705 + 0.005) eV. When both the heating current and the magnetic field were applied, the Hall voltage decreases with the sample temperature.
Discussion
From the results, the copper and zinc samples have opposite sign of Hall coefficient, because of different charge carriers in the conductor. Negative sign indicates electrons conduction while positive sign indicates holes conduction. If the electrons are the charge carriers, to produce the indicated current direction, they move from right to left under the external field. The Lorentz force acting on the electrons is therefore drifted towards the upper edge of the sample. This means that a transverse electric field is set up across the sample in the direction. If the charge carriers are holes, for the same primary current direction (holes move from left to right), the holes would also deflect toward the same terminal. But since the holes charge the upper terminal of the sample to positive potential, the polarity of the transverse field is opposite to that of the electrons. Hence the signs of the Hall voltage are different.
Since electrons and holes give different deflections in the sample and hence obtain different polarities for the potentials. The Hall effect can therefore be used as a measure of determining the presence of holes in a material. [4]
Consider the change in resistivity of the germanium samples with the applied magnetic field. This phenomenon is called magnetoresistance and is due to the fact that the drift velocity of all carriers is not the same. With the magnetic field on, the Hall voltage compensates exactly the Lorentz force for carriers with the average velocity. Slower carriers will be overcompensated, and faster ones undercompensated, resulting in trajectories that are not along the applied external field. This results in an effective decrease in mean free path and hence an increase in resistivity. [5]
The Hall voltage of the germanium samples decreases with increasing temperature. Since the measurement were made with constant current, it is to be assumed that this is attributed to an increase in the number of charge carriers (transition from extrinsic to intrinsic conduction) are the associated reduction in drift velocity.
Some sources of uncertainty include the fluctuation of the multimeter after amplification when measuring the Hall voltage for copper and zinc, presence of the impurities in the sample, and the uncertainty in the thickness of the samples not mentioned by the producer.
Conclusion
To summarize, the Hall coefficients of copper, zinc and germanium samples were obtained in the experiment. Negative sign of Hall coefficient is associated with electrons conduction whereas positive sign indicates holes conduction. It is also found that for the germanium samples, the resistance increases non-linearly with the applied magnetic field, and the Hall voltage decreases with increasing temperature. Various properties of germanium samples can be determined from the Hall effect, such as the conductivity at room temperature, holes concentration and energy of band gap.
References
[1] Halliday, D., et al, Fundamentals of Physics: Extended, 5th edition, New York: John Wiley & Sons Inc., 1997, pp.706
[2] Duncan, T., Advanced Physics for Hong Kong, Volume 1, London: John Murray Ltd, 1999, pp.182-183
[3] Leadstone, G.S., The discovery of the Hall Effect, Physics Education (Great Britain), Volume 14, Number 6, 1979, pp.374-379
[4] Ferandeci, A.M., Physical Foundations of Solid State and Electron Device, 1st Edition, Hightstown: McGraw-Hill Inc., 1991, pp.95-140
[5] Melissinos, A.C., Experiments in Modern Physics, 1st Edition, San Diego: Academic Press, 1966, pp.80-98
Figure
Captions
Figure 1: A slab of current-carrying conductor in a magnetic field. The transverse current is perpendicular to the applied magnetic field.
Figure 2: Hall effect in copper sample: (a) Hall voltage as a function of magnetic field strength at constant current; (b) Hall voltage as a function of current at constant magnetic field strength.
Figure 3: Hall effect in zinc sample: (a) Hall voltage as a function of magnetic field strength at constant current; (b) Hall voltage as a function of current at constant magnetic field strength.
Figure 4: Hall effect in n-germanium sample: (a) Hall voltage as a function of magnetic field strength at constant current; (b) Hall voltage as a function of current at constant magnetic field strength.
Figure 5: Hall effect in p-germanium sample: (a) Hall voltage as a function of magnetic field strength at constant current; (b) Hall voltage as a function of current at constant magnetic field strength.