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Transcript
The Hall Effect Declan Cockburn: 02038633. Lab partner: William Walshe. November 2004 Abstract: This experiment involves the investigation into the Hall effect, properties of a magnetic coil, examination of the properties of germanium in a magnetic field and the learning of and how to use the lock in amplifier. We found that the charge carriers in germanium was (unlike most other semi-conductors) actually electrons. Thus the sign of R was negative. For the first sample: 3 3 RH The hall coefficient was found to be: 12.2 10 0.2 10 Tm VA 20 20 3 The carrier density was found to be: 5.12 10 .08 10 m The conductance was found to be 2.14 103 0.06 103 1.m 1 For the second sample 3 3 1 1 RH The hall coefficient was found to be: 11.4 10 .2 10 TmV A The carrier density was found to be: 5.48 1020 .01 1020 m 3 The conductance was found to be: 7.6 103 .8 103 1.m 1 Introduction: In 1879 E. H. Hall discovered what is today called the Hall effect. It is a rather simple phenomenon whereby a conducting plate has a DC current and a transverse magnetic field going through it as in Fig. 1. What happens is: there is a Lorentz force ev B caused by the magnetic field, which tends to deflect the moving charge carriers to the left hand side in the case of Fig.1. Due to this a build up of charge occurs the sides of the plate, which creates EH a transverse electric field that quickly balances out the Lorentz force. This means that the carriers can now return to traveling straight. This is all pretty fundamental, the interesting thing about the Hall effect however, is the complete lack of Fig.1 the free electron theory’s ability to predict the polarity of the electric field in some conductors. Why is this? Well in the majority of cases it is the electrons which are the charge carriers. In some cases though it is positive holes that carry charge. A stationary magnetic field affects all moving charges, regardless of the sign of the charge. Then the (+) holes are pushed to the same side as the (-) electrons would be if they were the carriers, even though the conventional current remains the same! The magnitude of the electric field is given by: E H RH B j Where: j =I and is the current RH is called the Hall coefficient. It is defined as being 1 Nq Where N is the number of charge carriers per unit density and q is the charge on said carrier. But the Hall voltage is given by: VH E H w RH BI H BI t Another term of interest is the carrier mobility given by: Nq As can be seen it is proportional to the conductivity which is merely the reciprocal of the resistivity. Experimental Details: (i) The aim of this experiment was to measure the magnetic field between magnetic coils as a function of the current applied to the coils. This was very straightforward and required us to slowly and consistently increment a measurable a current -2.5A +2.5A and back again while tabulating the readings with the associated field between the coils (ii) The aim of this experiment was to show that the Hall voltage was proportional to the product of the magnetic field and the current between contact points 1&2. Secondly to determine the sign of RH (and hence know what are the charge carriers) and lastly to measure the conductivity of the sample. The circuit was setup as in Fig. 2. With the magnetic field B perpendicular to the page. There also existed a 10 kΩ potentiometer not shown that was used to remove the reading of V0 (extra independent voltage caused by the misalignment of contacts) as much as possible. V0 is caused by a slight misalignment between the contacts of 3&4. Readings were taken of V34 against B for three different values of I (10, 20, 30 mA) and plotted. Also I was measured against the voltage drop between contacts 1&2 Fig.2 (iii) The aims of this experiment were to determine the carrier type/sign of RH . To measure conductivity and obtain carrier mobility. To estimate the misalignment of two transverse contacts. To learn about the characteristics and operation of a Lock In Amplifier (LIA) and finally to determine RH using a rotating field. This experiment is quite a long one and involves the use of a magic cylinder, which far from being magic, uses 8 permanently magnetic segments. Their poles are pointed in such directions as to have a transverse magnetic field through it’s core. See Figure 3. Also a very “interesting” and sensitive instrument was used called an LIA. For the first part of this experiment, using a DMM: V34 was measured against I = 5, 10, 15, 20, 25 mA with B = 170mT and repeated for B = -170mT. Next the above experiment was repeated except with the cylinder rotating, and the measurements being taken by an oscilloscope. For the next part of the experiment the LIA was required. A lot of reading, trial and error was required to figure out how to operate this unfamiliar tool. Once our understanding of it was brought to fruition we began by hooking it up to detect V34. Set the cylinder rotating, set the gain to achieve a suitable deflection and finally chose a suitable time constant. The phase setting was then investigated where the phase was measured against the voltage given on the LIA (which is the rms value of V34). A maximum was found. For I = 5, 10, 15, 20 mA, V34 was measured at the max value of the phase setting which was found each time. As a result of the LIA’s noise reduction capability, it can give a much better readings at low voltages than the oscilloscope. To examine this V34 was measured for I= 1, 0.5 and 0.1 mA. Again the phase setting was corrected to be at its maximum each time. The results were then tabulated. Results and Analysis: (i) To investigate the properties of an electromagnet: Investigating the properties of an electromagnet: Magnetic Field vs. Current Through Coils 200 150 100 B (mT) 50 0 -3 -2 -1 0 1 2 3 -50 -100 -150 -200 Ic (Amps) Fig. 4 As can be seen in Fig. 4, the magnetic field, though proportional to the current, has another factor affecting it: what state it was in just previously. Even though the magnet we were using has a soft iron core, this shows that nevertheless it is affected by a residual magnetic field. This is significant as it shows that there is no unique value of B verses Ic. This means that B must be measured directly for each value of V34. (ii) Measurements of VH using the electromagnet: V43 vs B at 10mA y = 0.1211x + 5.6185 30 25 20 V34 (mV) 15 10 5 0 -200 -150 -100 -50 -5 0 50 -10 -15 -20 B (mT) C m Value 5.62 0.1211 Fig. 5.1 Error 0.07 0.0006 V43 H BI I m 0.1211 0.0006 T V H I I 10 .1mA t 1 .02mm RH 1 H t mt 12.1 10 3 .4 10 3 Tm VA I 100 150 200 250 V43 vs B at 20mA y = 0.2454x + 10.476 60 50 40 V34 (mV) 30 20 10 0 -200 -150 -100 -50 -10 0 50 -20 -30 -40 B (mT) C M Value 10.5 0.245 Error Fig. 5.2 0.1 0.001 V43 H BI I m 0.245 0.001 T V H I I 20 .1mA t 1 .02mm RH 2 H t mt 12.3 10 3 .4 10 3 Tm VA I 100 150 200 V43 vs B at 30mA y = 0.3682x + 14.123 100 80 60 V34 (mV) 40 20 0 -200 -150 -100 -50 -20 0 50 100 150 200 -40 -60 B (mT) Fig. 5.3 C M Value 14.1 0.368 Error 0.2 0.002 V43 H BI I m 0.368 0.002 T V H I I 30 .1mA t 1 .02mm RH 3 H t mt 12.3 10 3 .3 10 3 Tm VA I As can be seen from the three graphs above (Figs. 5.1, 5.2 and 5.3) it is quite plain to see that V43 (and therefore VH) is directly proportional to B. Also it can be seen from the associated calculations that RH is the same for each (and thus independent of) I. RH Since 1 Nq then: N 1 RH q where RH RH 1 RH 2 RH 3 3 Thus: 12.1 12.3 12.3 .4 2 .4 2 .32 RH 3 3 3 3 3 10 12.2 10 0.2 10 TmVA And: N 1 5.12 1020 .08 1020 m 3 3 19 (12.2 10 0.2 10 )(1.6 10 ) 3 Where |q| was taken to be |e| = 1.6x10-19C Determining the sign of RH requires more information than above. It requires knowledge of the direction of the magnetic field using a compass. So for a value of V34 say +40mV at I = 30mA. The direction of B was as shown in Fig.6 Fig. 6 Thus it is shown in Fig. 7 according to the Lorentz force that the charge carriers are obviously electrons, which means that the sign of RH in this case is negative. Fig. 7 The conductivity of the sample is given in this case by: Where resistance R Vdrop I l Rwt Voltage drop vs. I Voltage drop (mV)) 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 I (mA) Fig. 8 Value -0.18 0.933 C m Error 0.06 0.003 Fig 8. allows us to get an accurate measurement of R the resistance across the sample. Where: R m 0.933 0.003 (10 10 3 .02 10 3 ) 2.14 103 0.06 103 1 .m 1 3 3 3 3 (.933 .003)(1 10 .02 10 )( 5 10 .02 10 ) The carrier mobility can now be determined: Since (assuming): Then: Nq 2.14 103 0.06 103 26 1m 2 1C 1 20 20 19 (5.12 10 .08 10 )(1.6 10 ) (iii) Measurements of VH using a “magic cylinder” permanent magnet: This part of the experiment uses the magic cylinder (see Fig.3 for apparatus) which here is set to it’s arbitrary +/- 170mT. The results are tabulated below in Fig. 9 with “a” denoting +B and “b” denoting a -B. VH and V0 were found by: 1 VH {V34 ( B, I ) V34 ( B, I )} 2 1 V0 {V34 ( B, I ) V34 ( B, I )} 2 I (mA) 5 10 15 20 25 error=.1 V34a (mV) 22.4 44 64.4 83.8 102.6 error=.1 V12a (mV) 1.09 2.53 4.04 5.47 6.86 error=.01 V34b (mV) 3 4.8 5.9 6.5 7.4 error=.1 V12 (mV) 1.12 2.55 4.08 5.45 6.86 error=.01 VH (mV) 9.7 19.6 29.25 38.65 47.6 error=.1 V0 (mV) 12.7 24.4 35.15 45.15 55 error=.1 Fig. 9 RH and its associated error was found from the following equation for each of the above values. RH VH t , but because of the direction of the magnetic field for VH, RH will be negative BI and thus charge carriers electrons. RH 0.0114 0.0115 0.0115 0.0114 0.0112 RH 0.0006 0.0004 0.0003 0.0003 0.0003 The mean turning out to be: N V0/V12 11.6 9.64 8.7 8.25 8.02 (V0/V12) 0.2 0.09 0.05 0.04 0.03 RH 11.4 103 .2 103 TmV 1 A1 1 5.48 1020 .01 10 20 m 3 RH q R turns out to be .26 +/- .02Ω l 7.6 10 3 .8 10 3 1.m 1 Rwt And the carrier mobility: Nq 7.6 103 .8 103 87 9m 2 1C 1 (5.48 1020 .01 1020 )(1.6 1019 ) Next for the same values of I, V34 was measured again except this time with the digital oscilloscope and with the cylinder rotating. The oscillator has to be DC coupled because of the two components in V34: the Hall voltage which varies with time and the constant V0. If it wasn’t DC coupled then we would be unable to measure V0. RH cannot be determined. The following values were obtained I (mA) 5 10 15 20 25 error=.1 V34 (mV) 20 40 60 85 98 error=1 V12 (mA) 1.09 2.42 3.92 5.33 6.6 error=.01 These correlate well with the above values of V34. Now the LIA is used to measure a value of V vs. the phase setting at I=10mA to investigate the maximum. Output voltage vs Phase setting (g) Vout (centivolts) 1 0.5 0 0 50 100 150 200 -0.5 -1 -1.5 Phase (degrees) Fig. 10 250 300 350 From Fig 9. it is obvious that if we were to have continued taking readings they would have formed a sine-like function. Fig.10 also shows the max value of V to be 12 +/- .5 mV which approx corresponds to the rms value of the input signal which is VH. The values obtained were: Vrms (mV) 6 12 17.5 23 I (mA) 5 10 15 20 Vrms 0.2 0.5 0.5 1 B (mT) 170 170 170 170 error=.1 Thus: VH (mV) 8.5 17 24.7 32 And since: RH 0.01 0.01 0.0096 0.0094 VH 0.3 0.7 0.7 1 RH VH t BI RH 0.0008 0.0007 0.0005 0.0005 Thus the mean is: 10 10 9.6 9.4 .82 .7 2 .52 .52 3 3 3 RH 10 9.8 10 0.3 10 TmVA 4 4 Thus: N 1 6.4 1020 .2 1020 m 3 3 19 (9.8 10 0.3 10 )(1.6 10 ) 3 The sign of RH cannot be determined. Finally, we investigate the accuracy of the LIA. The rms (as in ) value of VH is measured for small values of current to investigate the relationship. The values are as such: Vrms (mA) I (mA) VH (mA) VH/I 1 1.25 1.77 1.77 0.5 0.68 0.96 1.92 0.1 0.36 0.51 5.1 The most startling thing is that VH/I is not a constant. This is because it is very difficult to get precise measurements of VH at such low currents in-spite of the LIA’s ability to practically detect the brainwave of an amoeba. Conclusions: Properties of an electromagnet were investigated. VH was shown to be proportional to For the first sample: RH The hall coefficient was found to be: 12.2 103 0.2 103 Tm VA 3 The carrier density was found to be: 5.12 10 .08 10 m The conductance was found to be 2.14 103 0.06 103 1.m 1 And the mobility was found to be 26 1m 2 1C 1 20 20 The second sample RH The hall coefficient was found to be: 11.4 10 3 .2 103 TmV 1 A1 Using the LIA the hall coefficient was found to be: 9.8 10 3 0.3 103 Tm VA 3 The carrier density was found to be: 5.48 10 .01 10 m Using the LIA the carrier density was found to be 6.4 1020 .2 1020 m 3 The conductance was found to be: 7.6 103 .8 103 1.m 1 And the mobility was found to be: 87 9m 2 1C 1 20 20 For the second sample the LIA’s N and RH do not agree with the conventional method. I would trust the conventional method more as I am not entirely confident with my evaluation of VH using LIA.