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Lab 4: the Classical Hall Effect

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Lab 4: the Classical Hall Effect Lab 4: The Classical Hall E®ect 2 Background A particle with charge q moving with a velocity ¡!v in ¡! ¡! a uniform magnetic ¯eld B will experience a force F , ¡! ¡! F = q(¡!v £ B ) (1) Consider the motion of free electrons in a conducting 1 Introduction wire at room temperature. In the absence of an elec- tric ¯eld, the electrons will move in random directions with a characteristic velocity distribution. The aver- Understanding the motion of charge carriers in mag- age velocity of the electrons is zero, and their most netic ¯elds has led to several interesting practical ap- probable speed is related to the thermal energy of the plications. The deflection of an electron beam in a electrons. This thermal energy is proportional to the magnetic ¯eld can be used to measure the charge to temperature of the conductor. If the conductor is con- nected to a battery, the electrons will experience a mass ratio of electrons and ions. Mass spectrometers ¡! that are used to measure the masses of isotopes ¯nd force due to the electric ¯eld EB inside the conduc- widespread applications in archaeology, geology, and tor. The electrons will be accelerated in a direction planetary science. The period of orbital motion of a opposite the ¯eld. However, collisions with the lat- charged particle in a uniform magnetic ¯eld is inde- tice of ions that make up the wire will dissipate the pendent of the velocity of the particle. This is the ba- increase in the kinetic energy of the electrons due to sis for particle accelerators such as cyclotrons. High- the electric ¯eld. As a result, the electrons acquire a ¡! energy collisions in particle accelerators and cloud drift velocity vd in addition to their random ther- chambers (used for studying cosmic rays) are analyzed mal velocity. The magnitude of the drift velocity is by photographing the curved trajectories of charged usually small in comparison with the most probable particles (produced during these collisions) in mag- speed associated with the random motion. netic ¯elds. The total number of charge carriers in a conductor of ¡! in¯nitesimal length d L and cross-sectional area A is The classical Hall e®ect (discovered in 1879 by E. H. ¡! Hall) is also related to the motion of charge carriers nA(d L ), where n is the number of charge carriers per in magnetic ¯elds. It led to some of the earliest ex- unit volume. periments that established the sign of charge carriers ¡! in conductors. It has been used to measure the num- Since a total charge Q = nA(d L )q flows through the ¡! ¡! ber density of charge carriers in conductors, as well length d L in time ¢t = d L , the current flowing as the drift velocity of charge carriers in the presence vd through the wire (I = Q ) is, of a uniform electric ¯eld in the conductor. Such an ¢t electric ¯eld can be established by connecting the ter- I = nAqvd (2) minals of a battery to the conductor. The classical If the conductor is placed in a uniform magnetic ¯eld Hall e®ect is commonly used in magnetic ¯eld sensors perpendicular to the drift velocity, the electrons will that are reliable, accurate, and relatively inexpensive. feel a force transverse to their motion as speci¯ed by equation 1. The electrons will therefore pile up on During this lab, you will carry out some simple ex- one side of the conductor as shown in ¯gure 1. The periments that will allow you to verify the Hall e®ect. other side of the conductor will have an excess posi- You will determine the sign of the charge carriers in tive charge. The separation of charges creates a uni- ¡! two kinds of semiconductors. The drift velocity of the form electrostatic ¯eld E inside the conductor that charge carriers in the sample and the number of charge opposes the magnetic ¯eld. During the charge pileup, carriers per unit volume can also be determined from the magnitude of the electrostatic ¯eld increases until your data. the force due to this ¯eld balances the force due to the magnetic ¯eld. This condition is expressed as, EXERCISES 1 AND 7 PERTAIN TO THE BACKGROUND CONCEPTS AND EXER- qE = qvdB (3) CISES 2-6 AND 8-10 PERTAIN TO THE EX- When this condition is satis¯ed, electrons will no PERIMENTAL SECTIONS. longer experience a deflection due to the magnetic 4.1 ¯eld. The electric potential di®erence between the two sides of the conductor is known as the Hall volt- age VH . If the width of the conductor is d, the rela- tionship between the electrostatic ¯eld and the Hall voltage is given by, V E = H (4) d Using equations 3 and 4, we can express the drift velocity as, V v = H (5) d Bd R Battery It is therefore possible to infer the drift velocity by measuring the Hall voltage and the magnetic ¯eld. B in x x x x x x x x Rearranging equation 2, the number of charge carriers per unit volume can be expressed as, x x x x x xF x x I d n = (6) qAvd vd q + + + + + + + + + + + + + + + + + + + + + It should be evident that a measurement of the current flowing through the conductor can be combined with x x x x x x x x the inferred drift velocity to ¯nd the value of n. a) Hall effect for negative charge carriers Exercise 1: A rectangular slab of silver (an excel- R Battery lent electrical conductor) has a width of 2.5 cm and thickness 2.0 mm. The slab is placed in a magnetic B ¯eld of magnitude 1.0 T directed perpendicular to the in x x x x x x x x slab. A Hall voltage of 0.107 ¹V is measured when a current of 2.0 A flows through the slab. + + + + + + + + + + + + + + + + + + + + + F x x x x x x x x d a) Assuming that the electron charge is q = ¡1:6 £ 10¡19 C, calculate the number of charge carriers vd q per unit volume. x x x x x x x x b) Compare the number of charge carriers n obtained in exercise 1a to the number of atoms per unit b) Hall effect for positive charge carriers volume in silver obtained from the density of silver 3 Figure 1: Hall e®ect (10.5 g/cm ) and its molar mass (108 g/mol). In a semi-conductor, the number of charge carriers per unit volume is much smaller than in an excellent electrical conductor like silver. From equation 6, it can be expected that the drift velocity must be higher in a semiconductor than in a good conductor assum- ing that both materials have the same area of cross section, and that the current flowing through each material is the same. Since the Hall voltage is pro- portional to the drift velocity (see equation 5), it can be expected that the Hall voltage will be larger, and therefore easier to measure in a semiconductor. 4.2 Semi-conductors have another interesting property. The charge carriers in a semi-conductor are not al- R1 ways electrons. In some cases, the charge carriers are positively charged and are referred to as holes. From ¯gure 1, it should be clear that the sign of the Power supply Hall voltage should depend on the type of charge car- A (a) rier. If the charge carriers are electrons, the material is referred to as an n-type semicon- R1 ductor. If the charge carriers are holes, the material is called a p-type semiconductor. V Power supply (b) 3 Suggested Reading Figure 2: Ammeter (a) and Voltmeter (b) setup Refer to the chapters on Current and Resistance and 5 Experiment I: Determining the sign the Magnetic Field, of the charge carriers R. Wolfson and J. Pasacho®, Physics with Mod- ern Physics (3rd Edition, Addison-Wesley Longman, Don Mills ON, 1999) In order to interpret the results one needs to deter- mine the direction of the current I, the direction of D. Halliday, R. Resnick and K. S. Krane, Physics the magnetic ¯eld B and the sign of the Hall voltage. (Volume 2, 5th Edition, John Wiley, 2002) The following discussion describes how the signs of the quantities can be determined. If an ammeter is connected to a power source as shown 4 Apparatus in ¯gure 2a, it is con¯gured to show a positive deflec- tion or reading. If a voltmeter is connected between two points on a Refer to Appendix E for photos of the appara- circuit, as shown in ¯gure 2b, the voltmeter will read tus a positive voltage. In other words, the voltmeter will show a positive deflection or reading if the positive terminal is at a higher potential than the negative ² Digital multimeter terminal. ² n and p type semiconductors The direction of the magnetic ¯eld can be deduced on the basis of magnetic ¯eld lines. The direction of the ² Circuit board with Potentiometer ¯eld between the poles of the magnet is indicated on the magnet. ² 4.5V power supply (3 batteries) To determine the sign of the charge carriers we will use ² Ammeter the apparatus shown below. Figure 4a shows the cir- cuit board that holds resistors and connection plugs. ² Permanent magnet The reversing switch can be used to change the direc- tion of current through the semiconductor wafer. The ² Magnetic ¯eld sensor current can be set to a convenient value by adjusting the variable resistor, or potentiometer. ² Banana wires Figure 4b is the circuit diagram for the experiment. ² Vernier calipers In the circuit diagram, when the switch is in position 4.3 1 Zero Dial Wafer Power Supply Plug 1 3 4 Potentiometer 2 Resistor Plug 7 8 2 Reversing Switch Figure 3: Diagram of the wafer board Wafer Plug 5 I the current flows from X to Y, on the other hand, Ammeter 3 connections when the switch is in position II the current flows from Y to X.
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