Hall Effect in N-Type Silicon Steve Kim and Lawrence Sulak Boston University (Dated: November 12, 2018) The Hall Effect refers to the process by which a potential difference is caused across an electrical conductor along the direction of current when subject to a magnetic field perpendendicular to the current. The effect can be used to study the properties of metals and semiconductors. Our experiment studied phosphorus doped silicon using the Hall Effect. We found that phosphorus doped silicon has a mobile charge carrier density of (N = 3:1 ± 2:5) × 1022m−3 I. INTRODUCTION applied magnetic field. Solving equation 3 for VH results in equation 4. The Hall Effect was discovered by and is named after ~ Edwin Hall. An overall visual description of the Hall VH = wj ~vdjjBj (4) Effect is shown in figure 1. The ribbon sample, like the one in figure 1, will have thickness d and length l as well. It will have a charge car- rier density, N, which tells the amount of charge carriers per volume. This leads us to equation 5. I = qNwdvd (5) Combining equations 4 and 5, gives one a model for the Hall Effect, which is expressed in equation 6. BI BIR V = = H (6) FIG. 1. A slab of metal undergoes the Hall Effect. Electrons H qNd d are deflected towards the negative x axis and accumulate. The linear relationship between I and VH suggests that As shown in figure 1, the Hall Voltage is apparent the Hall Coefficient, RH , can be found through the slope. across W , which is perpendicular to the current flow- Doped semiconductors also exhibit the Hall Effect. ing along L. The presence of a magnetic field causes the Doping refers to the process by which impurities are Lorentz force, which is shown in equation 1. added to a semiconductor in order to manipulate its elec- trical properties. Doping silicon with phosphorus causes the presence of excess electrons, which are responsible F~ = q ~v × B~ (1) for being charge carriers. Phosphorus doped silicon can be referred to as n-type silicon. Using the Hall Effect, one can characterize n-type silicon by finding the Hall This causes the flowing charge to move towards the Coefficient and N. sides of the slab along L. This accumulation of charge is what causes the Hall Voltage. Note that voltage can be expressed through equation 2. II. EXPERIMENTAL SETUP I V = − E~ · d~l (2) To develop a model of the Hall Effect, we consider equation 1 and the equation for the force on a charge due to an electric field, F~ . qV F~ = qE~ = H = q ~v × B~ (3) w d FIG. 2. Our experimental setup is shown above. q is the charge of the carrier, E~ is the electric field, ~vd is the carrier drift velocity, VH is the Hall Voltage, Figure 2 shows a detailed set up of our experiment. w is the width of the sample, and B~ is the externally The n-type silicon was mounted on a printed circuit 2 N. These parameters are shown in equations 7 and 8. −4 3 RH = (3:4 ± 2:9) × 10 m =C (7) N = (3:1 ± 2:5) × 1022m−3 (8) The errors in these parameters were found by taking the standard deviation of the calculated Hall Coefficients and charge densities. To better understand the error in our linear regression line, we found the χ2 values for them. The expected χ2 values are listed in equations 9-11. 2 χ500G = 9:0 ± 4:2 (9) 2 FIG. 3. A linear plot depicting the three trials of VH vs. I χ1000G = 10:0 ± 4:5 (10) for our experiment. 2 χ1500G = 9:0 ± 4:2 (11) The χ2 for the three plots are shown in equations 12-14. board (PCB). Through the magnet power supply, we 2 were able to alter the power of the externally applied χ500G = 52:0 ± 6:0 (12) 2 3 magnetic field. The magnetic field present at the sample χ1000G = (1:9 ± :2) × 10 (13) was quantified by the gaussmeter probe. The Hall Volt- χ2 = (5:8 ± :6) × 102 (14) age was measured by a microvoltmeter, and the applied 1500G Hall Current was measured by a microammeter. V. CONCLUSION III. PROCEDURE As shown in figure 3, our trials produced a linear re- lationship between VH and I. This matches our expec- Upon varying the Hall power supply to measure Hall tations due to the model for the Hall Effect presented current, We did not decrease the applied Hall current at in equation 6. However, it is apparent that our re- any point in order to prevent hysteresis. In addition, the sults are flawed in other ways. According to equation voltmeter was zeroed due to uncontrollable noise. 6, one would expect a higher magnetic field to result in Once the ammeter stabilized, we recorded the Hall a steeper slope. Figure 3 shows the opposite of what voltage. Since the microvoltmeter used to measure the we expected, as the slope of the best fit line gets less Hall voltage had fluctuations, we took the average of ten steep as the magnetic field increases. Furthermore, we readings and its standard deviation as the error bars. see an extremely large discrepancy between our expected All the aforementioned steps were repeated at different χ2 values, in equations 9-11, and actual, in equations 12- magnetic fields in order test the dependence of the Hall 14. This can be attributed to our unreasonably small voltage on the applied magnetic field. error bars. To account for this, future experimental- ists should account for other sources of error and for a wider range of fluctuations in the voltmeter. Other IV. RESULTS AND ERROR ANALYSIS sources of error could have come from fluctuations in line voltage, temperature dependence, and a large RC Three trials were conduted at 500G, 1000G, and time constant of our set up. The first can be fixed with 1500G. A MATLAB plot of our results is shown in figure a Variac Transformer, the next by measuring tempera- 3. The thickness of the n-type silicon ribbon was mea- ture, and the last by waiting an appropriate amount of sured in Advanced Lab and listed by the manufacturer time before a measurement is taken. Despite our large −4 3 to be d = 0:38mm. Knowing these paramters, we took errors, we found that RH = (3:4 ± 2:9) × 10 m =C and our data and used the MATLAB R2018b linear regres- N = (3:1 ± 2:5) × 1022m−3. We have reason for these sion function and obtained a best fit line. This returned to believe these to be somewhat sensible values based on the slope of the line, which we used to determine RH and past Advanced Lab efforts to conduct this experiment. 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