Experiment 12 1 The Hall Effect Physics 2150 Experiment No. 12 University of Colorado Introduction The Hall Effect can be used to illustrate the effect of a magnetic field on a moving charge to investigate various phenomena of electric currents in conductors and especially semi-­‐conductors. When a current-­‐carrying conductor is placed in a magnetic field of magnitude B that has its direction perpendicular to the current, a voltage difference will appear magnetic. This voltage is called the Hall voltage and is the discovery of E.H. Hall in 1879. This Hall voltage is proportional to the product of the current and component of 𝐵 normal to the current. More recently, the Hall Effect is widely employed throughout industry in modern Hall Effect gauss-­‐meters. Consider the simplified case shown in Fig. 1, where the current 𝐼 through the sample is in the negative x-­‐direction. The sample has dimensions a, b, b’, and c. Dimension b’ is the distance between the probe tips in the y-­‐direction Fig. 1 and is not labeled therein. The magnetic field 𝐵 is perpendicular to 𝐼 and is in the positive z-­‐direction. Figure 1: Simplified Diagram of Hall Effect Apparatus The three probe resistance network is shown. Experiment 12 2 Hall Coefficient for Positive Carriers If the carriers are positively charged, they are then moving in the negative x-­‐
direction in Fig. 1. The magnetic field exerts a force on them that will be in the positive y-­‐
direction of magnitude given by 𝐹! = 𝐹! = −𝑞 𝑣! 𝐵 (1) where 𝑣! is the average velocity of the carriers in the x-­‐direction. These carriers will thus be forced toward the top edge of the slab, which will then develop a higher potential than the bottom edge. An electric field EH will grow until the force on a charge carrier due to the magnetic field is just canceled out, preventing further buildup of charge. This electric field will be given by 𝐸! = 𝐸! = 𝑞 𝑣! 𝐵/𝑞 = 𝑣! 𝐵 . (2) The current density in the x-­‐direction is given by 𝑗! = 𝑛 𝑞 𝑣! (3) where 𝑛 is the average density of the carriers. Thus, the Hall field is given by 𝐸! = 𝑗! 𝐵/𝑛 𝑞. (4) This is customarily written 𝐸! = 𝑅! 𝑗! 𝐵 (5) and the “Hall Coefficient” 𝑅! is thus given by the positive number 𝑅! = 1/𝑛 𝑞. (6) Typical units for 𝑅! are cubic meters per Coulomb. The Hall Coefficient for Negative Carriers The force on minus charges due to the 𝐵 field is still in the positive y-­‐direction, as negative carriers have to be traveling in a direction opposite to the current. In this case, negative carriers will be forced upward and the upper edge of the slab will develop a lower potential than the top. An electric field pointing in the positive y-­‐
Experiment 12 3 direction will grow until the electric field’s upward force just balances the downward force due to the magnetic field. This electric field will be given by: 𝐸! = 𝐸! = − 𝑞𝑣! 𝐵 𝑞 = − 𝑣! 𝐵. (7) Since the current density is given by 𝑗! = 𝑛 𝑞𝑣! , (8) the Hall electric field will be ! !
𝐸! = − !!! = 𝑅! 𝑗! 𝐵. (9) In this case, the Hall coefficient is negative: !
RH = − ! ! . Thus, for carriers of either sign, the Hall coefficient is !
𝑅! = !" . (10) (11) An experimental measurement of the Hall coefficient thus allows one to determine both the sign and the density of the charge carriers in a material. Historically, it was this effect which first conclusively established that current is carried in metallic conductors by negatively charged particles. In materials which possess positive and negative charge carriers, the expression for the Hall coefficient is considerably more complicated. In metals, which have a large density of conduction electrons, 𝑅! is small and negative. In a semiconducting material with a small excess of positive charge carriers over negative charge carriers, the Hall coefficient can be large and positive. For copper, a typical value is -­‐6 x 10-­‐11 m3/C, and for the semiconductor bismuth, -­‐4 x 10-­‐7 m3/C, nearly 10,000 times larger. (This is not the accepted value for this particular sample.) Note that when a steady state has been reached, the charge carriers drift in the x-­‐
direction, although the total electric field makes some angle 𝜃, called the Hall angle, with respect to the x-­‐axis. This angle is usually small and can be written as 𝜃 = tan 𝜃 = 𝐸! /𝐸! . (12) From Fig. 1 it can be seen that if two probe contacts are placed oppositve each other at the top and bottom faces of the slab and the voltage between them is measured, the Hall field can be found from the relation Experiment 12 !
! !
4 !
𝐸! = 𝐸! = − !"# !!!"##"$ = − !!! (13) where 𝑏 ! is the distance between the pointed contacts, i.e. the effective width of the Hall probe. Also, from Fig. 1, if 𝐼 is the total current through the sample, in the x-­‐direction, the current density 𝑗! will be !
𝑗! = !" . (14) Thus, 𝑅! =
!!
!! !
= −
!!!
!
!"
!!
, (15) where the measured Hall voltage, 𝑉! , is corrected by the ratio of the strip width (𝑏) to the transverse distance between the probe tips (𝑏 ! ). These dimensions are given on the apparatus table. This experiment is difficult because the Hall voltage is usually very small, typically a few hundred microvolts in the present apparatus. Therefore, special precautions have to be taken to minimize stray voltages. One problem is that, due to the large current 𝐼 which flows along the sample in the x-­‐direction, there is a large potential gradient in the x-­‐
direction. If the probes on opposite sides of the sample are not exactly opposite each other, the voltage measured between the probes will be due in part to this potential gradient, rather than due to the Hall effect. In the present apparatus a resistance network has been incorporated to cancel out this effect. The apparatus is diagrammed in Fig. 2. Figure 2: Three-­‐Probe Network Attached to the Sample On one side of the sample there are mounted two probes, P2 and P3, at distances of about 1 cm on either side of the point, which is truly opposite the single probe P1. Between P2 and P3 is connected a potentiometer, R1. By adjusting the potentiometer wiper arm, the Experiment 12 5 potential of the arm may be made equal to that of any point between P2 and P3. In practice, to set the wiper arm to the potential of that point which is exactly opposite P1, a sensitive galvanometer is connected between P1 and the arm, the current 𝐼 is set flowing through the sample, which is removed from the magnetic field so that B = 0, and the arm is adjusted until no deflection is seen on the galvanometer. In principle, even if the current 𝐼 is changed, no further adjustment of the wiper arm should be necessary. Another source of inconsistent results is the presence of temperature differences between the wires used in making the electrical connections in this apparatus. Such temperature differences can cause voltages of thermoelectric origin big enough to mask the Hall voltage at lower magnetic fields. Banana plugs or clip leads should not be allowed to touch cold metal tabletops and should not be touched with warm fingers. This warning should be followed particularly for that portion the circuit containing the sample, probes, and galvanometer. Incorrect readings of the Hall voltage would also result if a current were allowed to flow through the contract P1, for this would result in an additional potential drop across the sample. The apparatus has therefore been designed so that measurements are taken when no current flows, by matching the Hall voltage against another measurable voltage from a standard reference source. The apparatus for these measurements is shown in Fig. 1. A standard reference cell of about 1.5V emf is connected in series with a 3000-­‐ohm resistor and a 2-­‐ohm resistor. Across the 2-­‐ohm resistor is a 10-­‐turn potentiometer R2, whose wiper arm can pick off any voltage from 0 to about 900 𝜇V. A sensitive microvoltmeter V, which draws essentially no current is conducted to the arm of this potentiometer; this microvoltmeter is used for actually taking the voltage readings. The galvanometer G is extremely sensitive and is used only for detecting the null, which is the position of the wiper arm at which no current flows in the galvanometer and at which the wiper arm must be at a potential equal to the Hall voltage. Procedure The sample consists of a thin strip of bismuth to which three probes are attached. It is encased in Plexiglas for protection. The dimensions of the strip are given on page 12.6 in the supplementary notes. Slide the sample and its holder out from the gap between the pole pieces of the magnet. The sample is extremely delicate and fragile. Vibration and handling may cause changes in contact resistance internally, so move it slowly and by the stand on which it mounts. Be sure that magnet power is off. On the strip current power supply, set the voltage knob to 25 volts, full scale, and the current control to 1,000 milliamperes, full scale. Turn on this power supply and set the voltage so that 0.5 amperes is read on the D.C. ammeter that rests alongside the galvanometer. Set R2 to zero. Turn the galvanometer on; set the sensitivity switch to either “Direct” or “1”. Set the potentiometer R1 to about midrange. The light spot on the scale of the null-­‐
detecting galvanometer should now be visible. If not, readjust the potentiometer until the Experiment 12 6 galvanometer reads zero. The position of the wiper arm should now correspond exactly to the point on the sample opposite the single probe P1. (This can be tested by changing the current 𝐼 ; however, when actually taking measurements, 𝐼 should never be changed.) Now insert the sample into the exact center of the magnet gap. Note that you may now have a noticeable deflection on the null-­‐detecting galvanometer. This is due to remnant magnetization in the magnet iron. Turn on the magnet power supply and increase the current. Note the effect on the galvanometer. Pull the probe from the magnet and rotate the probe 180°. Then reinsert it into the magnetic field. Does the galvanometer react as you would expect? Increase the magnet current and record the Hall voltage as a function of 𝐵 by adjusting the potentiometer R2 until null readings are obtained, taking voltage measurements directly from the microvoltmeter V. Plot the Hall voltage versus magnetic field. Can you fit a straight line through the points? Should the Hall coefficient vary with changes in 𝐵? Using Linfit or an equivalent linear regression program, determine the best value of the Hall coefficient of the sample. From this, determine the density of charge carriers in this particular sample of Bi, and compare it with the number of atoms per unit volume of bismuth. The Hall angle, which is the angle between the direction of the current in the sample and the Hall electric field, varies as the magnetic field varies. Calculate the Hall angle in the sample, in both radians and degrees, given a Hall voltage of 500 microvolts. For this calculation, in Eq. (12), set 𝐸! equal to (0.81 ± 0.02) volts/meter at 𝐼 = 0.5A. There is extensive literature on the Hall Effect in bismuth that indicates strong dependences of the coefficient on temperature, impurity concentration, impurity type, and direction of 𝐵 field with respect to crystal axes. Both positive and negative coefficients are reported for very low concentrations of impurities (e.g. 0.001 %) at room temperature. The polarity of charge carriers in this particular sample can be determined by applying the rationale of Eq. (1). First determine the direction of the force deflecting the charges, then examine the nulling circuit polarity to determine the sign of the charge carriers. Experiment 12 Supplementary Notes Dimensions of the Bismuth Strip: Length Width Distance between probe tips in the Y direction Thickness a (6.8 ± 0.2) * 10-­‐2 meters b (6.41 ± 0.03) * 10-­‐3 meters b’ (2.84 ± 0.03) * 10-­‐3 meters c (1.40 ± 0.01) * 10-­‐4 meters 7