Lab 2 - Hall Effect - F23

Toronto Metropolitan University - PCS224 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited The Hall Effect Physics Topics If necessary, review the following topics and relevant textbook sections from Serway / Jewett “Physics for Scientists and Engineers”, 10th Ed. • Electric Force (Serway 22.3) • Potential Difference in a Uniform Field (Serway, 24.2) • Electric Current (Serway, 26.1) • Analysis Model: Particle in a Field (Magnetic) (Serway, 28.1) • The Hall Effect (Serway, 28.6) Introduction Current is carried by charges. If a current carrying material is placed in a magnetic field, the charges within it will feel a magnetic force ⃗ F = q⃗v × ⃗ B and will move in response to it. The magnetic force will push the charge carriers to one side of the sample (a lack of charges on the opposite side causes it to become oppositely charged). This separation of charges creates an electric field ⃗ E H and an associated electric potential ∆ V H which can be measured with a voltmeter. This effect was first observed by Edwin Hall in 1879. The central concept relevant for the Hall Effect is a competition between electric and mag- netic forces. Initially the charge carriers only feel a magnetic force, and so they get pushed to one side of the sample. However, as the concentration of charges on that side builds up, they will tend to electrically repel other charge carriers away. As long as the magnetic force on the charge carriers is larger than the electrical repulsion, the charge will continue to build up. Once the electric repulsion exactly balances the magnetic force, the system comes to equilibrium, the buildup of charge stops, and the Hall Voltage becomes constant. This occurs when the electric force has an equal magnitude as the magnetic force: | q || ⃗ E H | = | q || ⃗ v d || ⃗ B | (1) Throughout this lab, we will assume the current carrying sample is a rectan- gular solid with current flowing along the length L direction, and the other two dimensions denoted as width W and thickness t . If the magnetic field ⃗ B points along the thickness direction ( t ), and the current flows along the length direction ( L ), the Hall Electric field points along the “width” ( W ) of the sample. Thus the associated potential difference is | ∆ V H | = | ⃗ E H | W . Combining this with above, we have | ∆ V H | = W | ⃗v d || ⃗ B | (2) Page 1 of 7

Toronto Metropolitan University - PCS224 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited You may wonder how we know that electrons (negatively charged particles) are the current carriers in a metal. Conventional experiments which do not use the Hall Effect can only determine the direction of the current, but not the sign of the carriers. On the other hand, the sign of the Hall Voltage is sensitive to the sign of the charge carriers. The Hall Effect is also one of the main tools we have for studying the electrical properties of a semiconductor. In fact, the Hall Effect can tell us • The sign of the charge carrier • The density n of charge carriers • The mobility µ of charge carriers. (Recall that mobility is defined as the ratio of the speed of the charge carriers to the applied electric field magnitude | ⃗v | = µ | ⃗ E | .) Using (2) as a starting point, the Pre-Lab Questions will guide you through derivation of formulas which allow you to determine n , µ and the sign of the charge carriers. Pre-Lab Questions Please complete the following questions prior to coming to lab. They will help you prepare for both the lab and the pre-lab quiz (Found on D2L). 1.) Read through the entire lab writeup before beginning 2.) What is the specific goal of this lab? Exactly what question(s) are you trying to answer? Be as specific as possible. (“To learn about topic X...” is not specific!) 3.) What specific measurements or observations will you make in order to answer this question? 4.) The following set of questions will help you determine the sign of the charge car- riers from the sign of the Hall Voltage. Shown below are two possible cases: one with the positive charge carriers, and one with negative charge carriers. A few charge carriers are shown in each figure. In both figures, the conventional current flows to the left , and the magnetic field points along the thickness direction (into the page). (a) On each figure, draw the velocity vectors for the charges. Remember that con- ventional current is defined as the flow of positive charge. (b) On each figure, draw the direction of the magnetic force on the charges. Page 2 of 7

Toronto Metropolitan University - PCS224 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited A rectangular wafer of germanium (a semiconductor) is provided on a circuit board. Handle the board carefully as the wafer is delicate. The dimensions of the germanium wafer are L × W × t =10.0mm × 5.0mm × 1.0mm. Figure 1: Circuit Board Connections for Hall Effect Lab A DC power supply is used to drive a current along the length of the wafer. CAUTION: do NOT exceed 50mA of current at any time during the experiment. A multimeter can be used to measure the voltage drop ∆ V L across the length ( L ) of the wafer. The multimeter can also be used to measure the (Hall) Voltage drop along the width ( W ), ∆ V H . A strong permanent magnet has poles shaped to provide a reasonably uniform magnetic field. Be careful not to allow any steel pieces to crash into the magnet (it has a very strong pull), as that will cause demagnetization. Procedure 1.) The next two steps will allow you to measure the magnitude and direction of the magnetic field generated by the permanent magnet. Note that gauss meter probes are delicate; do not bend the probe. Place the tip of the probe into the zero-Gauss Page 4 of 7

Toronto Metropolitan University - PCS224 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited chamber (a small cylindrical hole in a small black piece of plastic). Push the “zero” button on the gauss meter to zero the probe. 2.) Insert the gaussmeter in between the poles of the permanent magnet. With the lettering on the blue part of the probe facing up, a ⃗ B -field in the down direction will give a positive reading on the meter. Record your measurement of the magnitude and direction of the magnetic field in between the poles of the magnet. Try moving the probe around in between the poles in order to get a sense of how much the field varies in between the poles. Use your observations to record an estimate of uncertainty in the magnetic field. 3.) Connect the power supply to the wafer board so that current will flow in the same direction as shown in Fig. 1. Connect the multimeter across the width of the wafer as shown in the figure and set it to measure DC volts. 4.) With the wafer far away from the magnet, turn on the power supply and adjust the current to 50mA [do NOT exceed 50mA at any time during this experiment!]. 5.) Adjust the zero potentiometer (“zero pot” as shown in the figure) so that your multi- meter reads 0V when the wafer is far away from the magnet. 6.) Insert the wafer between the poles of the magnet so that the field is perpendicular to the face of the wafer, and the wafer is centered. There is a scratch along the plexiglass which indicates the center of the poles; try to line up your wafer so that the current flows parallel to this scratch. 7.) Note the relative directions of the current, magnetic field, and how your have hooked up your multimeter. Draw a diagram in your notebook for future reference. 8.) Vary the current from 0mA to 50mA and record the Hall voltage at each measurement (make sure to take note of the sign). As always, record your data with an estimate of uncertainty. 9.) Remove the wafer from the magnetic field. Disconnect the multimeter from the wafer and reconnect it to measure the voltage drop across the length L of the wafer (∆ V L ). 10.) Replace the wafer in the magnetic field. Vary the current from 0mA to 50mA and record the values of ∆ V L at each step with associated uncertainty. Be sure that you take data at the same current values as you did in step 8 so that at each value of the current you have both a measurement of ∆ V H and ∆ V L . 11.) Remove the wafer from the magnetic field, disconnect all cables, and turn off all devices. Analysis 1.) Using the sign of the hall voltage which you observed, and comparing your experimental setup with your work on the pre-lab questions, determine whether this germanium wafer is n -type or p -type. Page 5 of 7

Toronto Metropolitan University - PCS224 Distribution, sale or profit without the authorization of the owner of this material is expressly prohibited Report Labs will be completed in groups, you will enroll in a group with your lab partner at the beginning of each lab session. Each group will submit a single report through the assignment section on D2L. • Introduction – What is the experiment’s objective? • Theory – You may be able to show a derivation of the physics you’re investigating, or you may want to reference a source that provides a description/equation representing the physics you’re investigating. – You may want to provide graphs that illustrate or predict how you expect the system under study to behave. • Procedure – Explain the systematic steps required to take any measurements. • Results and Calculations – Tabulate your measurements in an organized manner. – Based on your procedure, you should know what your tables – Provide examples of any calculations. • Discussion and Conclusions – Discuss the main observations and outcomes of your experiment. – Summarize any significant conclusions. • References • (Appendices) Page 7 of 7