Hall Effect and Measurement of Hall Coefficient

13–20th January 2010

1 | H a l l E f f e c t

Hall Effect & Measurement of Hall Coefficient

Harsh Purwar (07MS-76) Piyush Pushkar (07MS-33)

Amit Nag (07MS-19) Sibhasish Banerjee (07MS-55) VIth Semester, Integrated M.S.

Indian Institute of Science Education and Research, Kolkata Experiment No. 1 Condensed Matter Physics (PH – 314)

Abstract: In this experiment Hall’s Effect was studied/observed and various parameters like Hall’s coefficient, carrier density, mobility etc were measured/calculated. The experiment was done for two types of semi-conductor crystals of Germanium (Ge) {3833 & 3911}, one having electrons as the majority charge carrier and other holes. The dependence of Hall voltage on the magnetic field and the current passing through the probe is also studied.

Introduction Hall Effect is a phenomenon that occurs in a conductor carrying a current when it is placed in a magnetic field perpendicular to the current. The charge carriers in the conductor become deflected by the magnetic field and give rise to an electric field (Hall Field) that is perpendicular to both the current and magnetic field. If the current density, 𝐽𝑥 , is along 𝑥 and the magnetic field, 𝐵, is along 𝑧, then Hall field, 𝐸𝑦 , is either

along +𝑦 or – 𝑦 depending on the polarity of the charge carriers in the material (conductor). It was E. H. Hall who first observed the above mentioned event in 1879 (1). Hall Effect is the basis of many practical applications and devices such as magnetic field measurements, and position and motion detectors. Also, Hall Effect measurement is a useful technique for characterizing the electrical transport properties of metals and semiconductors. Hall Effect sensors are readily used in various sensors such as rotating speed sensors, fluid flow sensors, current sensors, and pressure sensors. Other applications may be found in some electric airsoft guns and on the triggers of electropneumatic paintball guns, as well as current smart phones, and some global positioning systems.

Theory As mentioned earlier, the reason for existence of Hall Field in 𝑦 direction is because of charge accumulation caused by Lorentz forces on movement of charge carriers. In equilibrium this transverse Hall Field, 𝐸𝑦 , will balance the Lorentz force and current will flow only in the 𝑥 – direction. From the Drude

theory of conduction it is obvious that applied electric field, 𝐸𝑥 , and the current density, 𝐽𝑥 , should be related as,

𝐸𝑥 = 𝜌(𝐻𝑧). 𝐽𝑥 where 𝜌 𝐻𝑧 is the magneto-resistance which is field independent. And also, for the transverse

field, 𝐸𝑦 , which balances the Lorentz force, one might expect it to be proportional to both the applied

magnetic field, 𝐻𝑧 , and current density, 𝐽𝑥 , as,



𝐸𝑦 = 𝑅𝐻 .𝐻𝑧 . 𝐽𝑥

here, 𝑅𝐻 is called as the Hall coefficient which is negative for negative charge carriers and vice versa. In the presence of electric field, 𝐸𝑥 and 𝐸𝑦 and magnetic field, 𝐻𝑧 , the equation of motion of a

negative charge carrier can be written as, 𝑑𝑝

𝑑𝑡= −𝑒 𝐸 +

𝑝

𝑚𝑐× 𝐻 −

𝑝

𝜏

In steady state the current is independent of time, and therefore 𝑝𝑥 and 𝑝𝑦 will satisfy,

−𝑒𝐸𝑥 − 𝜔𝑐 . 𝑝𝑦 −𝑝𝑥𝜏

= 0

−𝑒𝐸𝑦 − 𝜔𝑐 . 𝑝𝑥 −𝑝𝑦

𝜏= 0

where

𝜔𝑐 =𝑒𝐻

𝑚𝑐

Now applying 𝑝𝑦 = 0 and 𝐽 = −(𝑛𝑒/𝑚).𝑝 we get,

𝑅𝐻 =𝐸𝑦

𝐽𝑥 .𝐵𝑧= −

1

𝑛𝑒𝑐

It asserts that the Hall coefficient depends on no parameters of the metal except the density of charge carriers (2).

Sample Details For n-type Germanium (Ge) crystal

o Thickness 𝑧 : 5 × 10−2 𝑐𝑚 o Resistivity 𝜌 : 10 Ω𝑐𝑚 o Conductivity 𝜎 : 10 𝐶𝑉−1𝑠−1𝑚−1

For p-type Germanium (Ge) crystal o Thickness 𝑧 : 5 × 10−2 𝑐𝑚 o Resistivity 𝜌 : 10 Ω𝑐𝑚 o Conductivity 𝜎 : 10 𝐶𝑉−1𝑠−1𝑚−1

Figure 1: Schematic diagram showing various fields acting on a p-type

semiconductor crystal attached to the probe.



Experimental Procedure Calibration of the Magnetic field with current

The magnetic field produced by the electromagnets was calibrated with the current flowing through it using an Indium Arsenide Hall probe for measuring magnetic field and an ammeter for measuring current. The following protocol was implemented in order.

1. The constant current power supply (DPS - ***) connected to the electromagnet (EMU - ***) and digital gauss-meter (DGM – 102) connected to the indium arsenide Hall probe were switched on after making appropriate connections.

2. The indium arsenide hall probe was covered with the metallic sheath and was placed away from the electromagnet and other apparatuses.

3. The digital gauss-meter was set at 1x and the reading was adjusted to zero using the zero adjustment knob of the gauss-meter.

4. The probe was then uncovered and placed at the center of the two electromagnets

with the help of a wooden stand. 5. The current through the electromagnet was

slowly increased via constant current power supply and corresponding magnetic field readings displayed by the digital gauss-meter were noted and are listed in Table 1.

NOTE: The current supplied by the power supplies should never be increased or decreased rapidly. It may lead to electric shocks and burn the apparatuses.

Dependence of Hall Voltage on Magnetic Field The Hall voltage across the semiconductor (probe) was measured varying the magnetic field around it and keeping the current through the probe constant. The following protocol was implemented.

Appropriate connections in Hall Effect set-up (DHE – 21) consisting of a constant current power supply and a digital milli-voltmeter were made and the apparatus was switched on. The widthwise contacts of the Hall probe were connected to the terminal marked ‘voltage’ and lengthwise contacts to the terminal marked ‘current’ as shown in the figure.

The current flowing through the probe was set to a fixed value say 3 mA.

The Hall probe was then placed away from the magnetic field and the Hall voltage was set close to zero by aligning the contact pins properly.

The magnetic field was then switched on and the Hall voltage was noted varying the current flowing through the electromagnet slowly, say in steps of 0.2 amperes.

Above was repeated for 2 different probes and for 3 values of probe current 𝐼 for each of the two probes as listed below in Table 2-7.

Dependence of Hall Voltage on Current through the Hall Probe In this part of the experiment we vary the current through the Hall probe or 𝐽𝑥 and see its impact on the Hall voltage keeping the probe in a constant magnetic field. The following protocol was implemented.

The current through the electromagnet was fixed to some value say 1.0 ampere. This fixes the magnetic field around the coil.

Placing the Hall probe in between the two electromagnets as mentioned earlier, the current through it was varied and the corresponding Hall voltage was noted.

Above was repeated for 2 different probes and for 3 values of magnetic field 𝐻 for each of the two probes as listed below in Table 8-13.

Figure 2: The two electromagnets; Hall probe is placed in between two of them.



Observation / Graphs Table 1: For calibrating Magnetic field (H) through the coil.

Obs. No. Current

(I) {Ampere} Magnetic Field

(H) {Gauss}

1 0.00 73

2 0.11 292

3 0.23 506

4 0.34 712

5 0.46 933

6 0.53 1077

7 0.61 1245

8 0.70 1425

9 0.84 1688

10 0.97 1966

11 1.04 2100

12 1.11 2240

13 1.21 2460

14 1.30 2630

15 1.42 2890

16 1.52 3090

17 1.62 3300

18 1.70 3470

19 1.82 3710

20 2.00 4070

21 2.22 4470

22 2.35 4770

23 2.42 4900

24 2.52 5080

25 2.62 5280

26 2.73 5470

27 2.83 5640

28 2.91 5770

29 3.02 5950

30 3.16 6150

31 3.23 6240

32 3.33 6360

33 3.46 6510

34 3.59 6650

35 3.66 6720

Above data was plotted and fitted linearly. Last four data points corresponding to the high currents were excluded during fitting.



Plot 1: Calibration of magnetic field (H) with current (I).

Table 2: Measurement of Hall voltage developed across the probe – 1 (3833) by varying magnetic field, for constant current of 2.99 mA passing through it.

Obs. No.

Current through the Electromagnet (A)

Magnetic Field {H} (Gauss)

Hall Voltage {VH} (mV)

Hall Coefficient {R} (m3/C)

1 0.00 74 -1.7 -0.0383

2 0.20 467 -5.6 -0.0200

3 0.40 860 -9.2 -0.0179

4 0.62 1292 -13.7 -0.0177

5 0.83 1704 -17.4 -0.0171

6 1.03 2097 -21.2 -0.0169

7 1.21 2451 -24.4 -0.0166

8 1.40 2824 -27.5 -0.0163

9 1.62 3256 -30.7 -0.0158

10 1.83 3668 -33.7 -0.0154

11 2.02 4042 -36.3 -0.0150

12 2.18 4356 -38.4 -0.0147

13 2.41 4808 -40.9 -0.0142

14 2.60 5181 -43.3 -0.0140

15 2.80 5574 -45.5 -0.0137

16 3.00 5966 -47.0 -0.0132

17 3.21 6379 -48.8 -0.0128

18 3.42 6791 -50.5 -0.0124

19 3.59 7125 -51.4 -0.0121

20 3.80 7538 -52.7 -0.0117




Obs. No.





1 0.00 74 -2.5 -0.0336

2 0.19 447 -8.6 -0.0192

3 0.39 840 -14.7 -0.0175

4 0.59 1233 -21.1 -0.0171

5 0.81 1665 -28.3 -0.0170

6 0.99 2019 -33.7 -0.0167

7 1.20 2431 -40.2 -0.0165

8 1.39 2804 -45.1 -0.0161

9 1.60 3217 -50.7 -0.0158

10 1.80 3610 -55.4 -0.0153

11 2.00 4002 -60.3 -0.0151

12 2.20 4395 -64.2 -0.0146

13 2.40 4788 -68.1 -0.0142

14 2.61 5200 -72.2 -0.0139

15 2.78 5534 -75.0 -0.0136

16 3.02 6006 -78.8 -0.0131

17 3.21 6379 -81.5 -0.0128

18 3.41 6772 -83.9 -0.0124

19 3.61 7164 -86.3 -0.0120

20 3.82 7577 -88.1 -0.0116


Obs. No.





1 0.00 74 -0.5 -0.0042

2 0.20 467 -10.0 -0.0134

3 0.39 840 -20.7 -0.0154

4 0.59 1233 -31.1 -0.0158

5 0.81 1665 -42.0 -0.0158

6 0.99 2019 -52.0 -0.0161

7 1.20 2431 -62.5 -0.0161

8 1.39 2804 -70.5 -0.0157

9 1.60 3217 -78.9 -0.0153

10 1.80 3610 -87.6 -0.0152

11 2.00 4002 -94.4 -0.0148

12 2.20 4395 -101.1 -0.0144

13 2.40 4788 -107.8 -0.0141

14 2.61 5200 -113.7 -0.0137

15 2.78 5534 -120.1 -0.0136

16 3.02 6006 -125.6 -0.0131



17 3.21 6379 -129.3 -0.0127

18 3.41 6772 -133.4 -0.0123

19 3.61 7164 -136.9 -0.0120

20 3.82 7577 -140.0 -0.0116

Plot 2: Variation of Hall Voltage with Magnetic field for different values of probe current for probe- 1.


Obs. No.





1 0.00 74 2.6 0.0583

2 0.25 565 7.6 0.0224

3 0.46 978 12 0.0205

4 0.65 1351 16.2 0.0200

5 0.90 1842 21.7 0.0196

6 1.10 2235 25.8 0.0192

7 1.25 2529 29.0 0.0191

8 1.43 2883 32.5 0.0188

9 1.66 3335 37.1 0.0185

10 1.82 3649 40.1 0.0183

11 2.01 4022 43.5 0.0180

12 2.28 4552 48.2 0.0176

13 2.48 4945 51.3 0.0173

14 2.64 5259 53.5 0.0170

15 2.81 5593 56.4 0.0168

16 3.07 6104 58.2 0.0159




Obs. No.





1 0.00 74 4.4 0.0592

2 0.23 526 12.1 0.0230

3 0.41 880 17.9 0.0204

4 0.62 1292 25.4 0.0197

5 0.81 1665 32.3 0.0194

6 0.95 1940 37.3 0.0192

7 1.15 2333 44.1 0.0189

8 1.37 2765 51.7 0.0187

9 1.61 3236 59.8 0.0185

10 1.83 3668 66.7 0.0182

11 2.11 4218 74.9 0.0178

12 2.32 4631 80.7 0.0174

13 2.51 5004 85.6 0.0171

14 2.71 5397 90.0 0.0167

15 2.91 5790 93.6 0.0162

16 3.08 6123 96.3 0.0157


Obs. No.





1 0.00 74 7.6 0.0641

2 0.21 487 17.8 0.0229

3 0.37 801 26.7 0.0209

4 0.61 1272 39.9 0.0196

5 0.81 1665 52.5 0.0198

6 1.04 2117 64.0 0.0189

7 1.20 2431 73.4 0.0189

8 1.40 2824 83.2 0.0185

9 1.62 3256 94.8 0.0163

10 1.82 3649 104.9 0.0180

11 2.02 4042 114.6 0.0178

12 2.21 4415 123.0 0.0175

13 2.43 4847 131.9 0.0171

14 2.59 5161 137.8 0.0167

15 2.85 5672 145.6 0.0161

16 3.05 6065 150.7 0.0156



Plot 3: Variation of Hall Voltage with Magnetic field for different values of probe current for probe - 2.

Table 8: Measurement of Hall voltage developed across the probe – 1 (3833) by varying current passing through it, for constant magnetic field of 2038 Gauss corresponding to 1.00 ampere of current through the electromagnet.

Obs. No. Current through the Hall Probe {I}

(mA) Hall Voltage {VH}

(mV) Hall Coefficient {R}

(m3/C)

1 0.12 -1.1 -0.0225

2 0.6 -5.3 -0.0217

3 0.9 -8.0 -0.0218

4 1.2 -11.0 -0.0225

5 1.64 -14.6 -0.0218

6 2.1 -18.7 -0.0218

7 2.62 -23.3 -0.0218

8 3.1 -27.6 -0.0218

9 3.62 -32.2 -0.0218

10 4.14 -36.7 -0.0217

11 4.74 -41.9 -0.0217

12 5.55 -48.8 -0.0216

13 6.22 -54.5 -0.0215

14 7.04 -61.4 -0.0214

15 7.56 -65.7 -0.0213

16 8.23 -71.2 -0.0212

17 8.65 -74.6 -0.0212

18 9.1 -78.2 -0.0211



19 9.6 -82.3 -0.0210

20 10.22 -87.3 -0.0210

21 10.7 -91.1 -0.0209

22 11.15 -94.6 -0.0208

23 11.85 -99.8 -0.0207

24 12.24 -102.8 -0.0206

25 12.74 -106.4 -0.0205

26 13.19 -109.7 -0.0204

27 13.81 -114.2 -0.0203

28 14.37 -118.0 -0.0201

29 15.01 -121.7 -0.0199

Table 9: Measurement of Hall voltage developed across the probe – 1 (3833) by varying current passing through it, for constant magnetic field of 4002 Gauss corresponding to 2.00 amperes of current through the electromagnet.

Obs. No. Current through the Hall probe {I}



(m3/C)

1 0.12 -1.8 -0.0187

2 1.08 -15.7 -0.0182

3 2.02 -29.4 -0.0182

4 3.04 -44.2 -0.0182

5 4.04 -58.9 -0.0182

6 5.08 -73.7 -0.0058

7 6.01 -87.0 -0.0181

8 7.08 -102.0 -0.0180

9 8.08 -116.0 -0.0179

10 9.14 -130.4 -0.0178

11 10.08 -143.3 -0.0178

12 11.06 -156.0 -0.0176

13 12.14 -170.0 -0.0175

14 13.1 -182.0 -0.0174

15 14.1 -194.4 -0.0172





(m3/C)

1 0.12 -2.2 -0.0154

2 1.04 -18.8 -0.0151

3 2.04 -36.9 -0.0152

4 3.14 -56.6 -0.0151

5 3.94 -70.9 -0.0151

6 5.08 -91.2 -0.0150



7 6.07 -108.7 -0.0150

8 7.11 -126.8 -0.0149

9 8.18 -143.6 -0.0147

10 9.15 -160.0 -0.0147

11 10.12 -176.0 -0.0146

12 10.94 -189.7 -0.0145

Plot 4: Variation of Hall voltage with current passing through it, for different values of magnetic field for probe - 1.

Table 11: Measurement of Hall voltage developed across the probe – 2 (3911) by varying current passing through it, for constant magnetic field of 2038 Gauss corresponding to 1.00 ampere of current through the electromagnet.




(m3/C)

1 0.12 1.1 0.0225

2 1.56 13.3 0.0209

3 2.17 18.5 0.0209

4 3.15 26.7 0.0208

5 3.89 32.8 0.0207

6 4.74 39.8 0.0206

7 5.13 43 0.0206

8 5.92 49.3 0.0204

9 6.59 54.6 0.0203

10 7.34 60.5 0.0202

11 8.2 67.1 0.0201



12 9.15 74.3 0.0199

13 10.15 81.8 0.0198

14 11.26 90 0.0196

15 12.33 97.4 0.0194

16 13.12 103 0.0193

17 14.28 110.9 0.0191

18 15.33 118.1 0.0189

19 16.65 126.8 0.0187

20 17.57 132.3 0.0185

21 18.71 138.6 0.0182

22 19.78 144.6 0.0179





(m3/C)

1 0.12 2 0.0208

2 0.82 12.8 0.0195

3 1.46 22.6 0.0193

4 2.11 32.7 0.0194

5 2.6 40.1 0.0193

6 3.11 47.8 0.0074

7 3.94 60.2 0.0191

8 4.52 69 0.0191

9 5.38 81.7 0.0190

10 5.86 88.6 0.0189

11 6.15 92.7 0.0188

12 6.74 101.2 0.0188

13 7.21 107.8 0.0187

14 7.8 116.3 0.0186

15 8.5 126 0.0185

16 9.02 133 0.0184

17 9.49 139.5 0.0184

18 10.03 146.9 0.0183





(m3/C)

1 0.12 2.5 0.0176

2 1.04 20.8 0.0169

3 1.35 26.9 0.0169



4 2.2 43.7 0.0168

5 2.77 54.8 0.0167

6 3.15 62.2 0.0167

7 3.94 77.4 0.0166

8 4.28 83.9 0.0166

9 4.75 93 0.0166

10 5.31 103.5 0.0165

11 5.66 110.1 0.0165

12 6.43 124.2 0.0163

13 7 134.8 0.0163

14 7.78 148.9 0.0162

15 8.48 161.4 0.0161

16 9.07 171.9 0.0160

17 9.28 175.5 0.0160

18 9.88 186.1 0.0159

Plot 5: Variation of Hall voltage with current passing through it, for different values of magnetic field for probe - 2.



Calculations As mentioned above in the theory section, that,

𝑅 =𝐸𝑦

𝐽𝑥𝐻=𝜇𝐸𝑥𝐽𝑥

=𝜇

𝜎=

1

𝑛𝑒

where 𝜇 is mobility of the charge carriers and 𝜎 is the conductivity. 𝑛 is the negative carrier density. Hence for fixed magnetic field and fixed input current, the Hall voltage is proportional to 1 𝑛 . It follows that,

𝜇 =𝑅

𝜎

Calculation of Hall Coefficient 𝑹 : The Hall coefficient 𝑅 or 𝑅𝐻 has already been calculated for the two types of Germanium (Ge) semi-conductor crystals in the tables above. As mentioned Hall’s coefficient depends only on the number density or carrier density therefore we can collect all the data and find out a collective mean and standard deviation. For Ge semi-conductor probe – 1 (3833):

Mean Hall coefficient: −0.0171 𝑚3 𝐶 Standard deviation: 0.0042

For Ge semi-conductor probe – 2 (3911): Mean Hall coefficient: 0.0197 𝑚3/𝐶

Standard deviation: 0.0072 Calculation of carrier density 𝒏 : For Ge semi-conductor probe – 1 (3833):

𝑛 =1

𝑅𝑒

⟹ 𝑛1 =1

−0.0171 × −1.6 × 10−19

𝒏𝟏 = 𝟑.𝟔𝟓 × 𝟏𝟎𝟐𝟎 𝒎−𝟑 Similarly, for Ge semi-conductor probe – 2 (3911):

𝑛2 =1

0.0197 × 1.6 × 10−19

𝒏𝟐 = 𝟑.𝟏𝟕 × 𝟏𝟎𝟐𝟎 𝒎−𝟑 Calculation of carrier mobility 𝝁 : For Ge semi-conductor probe – 1 (3833):

𝜇 = 𝑅𝜎 ⟹ 𝜇1 = 0.0171 × 10 𝝁𝟏 = 𝟎.𝟏𝟕𝟏 𝒎𝟐𝑽−𝟏𝒔−𝟏

For Ge semi-conductor probe – 2 (3911):

𝜇2 = 0.0197 × 10 𝝁𝟐 = 𝟎.𝟏𝟗𝟕 𝒎𝟐𝑽−𝟏𝒔−𝟏



Inference The Hall voltage depends on the magnetic field and the current flowing through the probe.

The Hall coefficient whereas, is independent of these two factors and depends only upon the density of the charge carriers.

The two probes, Probe – 1 (3833): n-type semiconductor Probe – 2 (3911): p-type semiconductor

The measured/calculated Hall coefficient for the two given probes, Probe – 1 (3833) = −0.0171 ± 0.0042 𝑚3/𝐶 Probe – 2 (3911) = 0.0197 ± 0.0072 𝑚3/𝐶

Using the above result we also calculated the charge carrier density and carrier mobility for the two probes which are found to be,

Carrier density for Probe – 1 (3833) = 3.65 × 1020 𝑚−3 Carrier density for Probe – 2 (3911) = 3.17 × 1020 𝑚−3

Carrier mobility for Probe – 1 (3833) = 0.171 𝑚2𝑉−1𝑠−1 Carrier mobility for Probe – 2 (3911) = 0.197 𝑚2𝑉−1𝑠−1

Sources of Error The following may account for the errors associated with this experiment.

Due to temperature fluctuation thermo – EMF and corresponding heating – current are generated. Hence these affect the reading of Hall voltage.

The calibration of the magnetic field with current is done at 10x scale for larger values of current. Thus errors creep in due to measurement in this scale.

When the hall probe is inserted manually the probe may be relatively tilted with the axis of magnetic field coils.

The contact pins on the semiconductor surface should be adjusted properly to completely remove the zero error in Hall voltage or should be noted and taken care of.

References 1. On a new action of the magnet on electric currents. Hall, E.H. 3, 1879, American Journal of Mathematics, Vol. 2, pp. 287-292. 2. Ashcroft, Neil W. and Mermin, N. David. [ed.] Dorothy Garbose Crane. Solid State Physics. s.l. : Harcourt College Publishers, 1976. 3. Department of Physics, Indian Institute of Science Education & Research, Kolkata. Roorkee : Scientific Equipment & Services. User's Manual.

Documents

Hall Effect and Measurement of Hall Coefficient