Hall Effect

Physics 322L: Report 1

Tory Johansen

2/25/2013

AbstractThis lab was an investigation into the Hall effect for semiconductors. The requirements for satisfactory

completion of this lab include the determination of Hall coefficient for a particular semiconductor sample, determination of the sign and density of charge carriers within the sample, and the determination of the mobility of the charge carriers.

IntroductionIn the presence of a magnetic field applied perpendicularly to current within a conductor, the charge

carriers of the current experience a Lorentz force causing accumulation of charge on conductor surface. This accumulation leaves equal and opposite charges on opposite surfaces of the conductor, thereby creating a transverse electric potential, the Hall voltage VH. The Hall voltage can be given as

V H=RHIBt

Where t is the length of the semiconducting sample parallel to the applied magnetic field B⃗.RH is the Hall coefficient, a property of the material describing its conductivity. RH can be given as

RH=∓ 1ne

Where its sign is determined by the sign of its charge carriers, e is the magnitude of its charge carriers, and n(carriers/unit volume) is the magnitude of the charge carrier density. Within a semiconductor, the charge carriers can be either electrons or positive holes.The mobility μ can be given as

μ=vd

→

E⃗or

μ= 1ρne

Where μ is the mobility of the charge carriers withinE⃗, v⃗d is their drift velocity, and ρ is the material resistivity. Finally, a known current I and sample with known RH can be used to measure unknown magnetic fields by measuring VH. A probe of this type has sensitivity defined as

Sensitivity=V H

IB

Procedure1. Setup Hall probe, power supply, electromagnet, and circuits as in diagram 1. Zero the gaussmeter,

voltmeter, and ammeter.2. Measure dimensions of sample with traveling microscope.

Experiment 1: Relationship between VH and I⃗

1. Establish a magnetic field of 4 kilogauss in the electromagnet by increasing the current.2. Establish a current in circuit 1 (given by diagram 1) of around 20 mA. 3. Record the resulting VH from the voltmeter.4. Repeat step 2 and 3 by incrementing the current by 20 mA until reaching around 380 mA.

Experiment 2: Relationship between VH and B⃗

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1. Establish a current of around 200 mA in circuit 1.2. Establish a magnetic field of around 1 kilogauss.3. Record resulting VH from voltmeter.4. Repeat 2 and 3 by incrementing the magnetic field by approximately .2 kilogauss, until reaching around 3.6

kilogauss.

Experiment 3: Determination of Resistance

1. Remove electromagnet component of setup and rewire circuit to measure the voltage from selected currents.

2. Establish a current of around 50 mA.3. Record corresponding V.4. Repeat 2 and 3, iterating by approximately 20 mA, until reaching 230 mA.

Experiment 4: Temperature Dependence

1. Reset the circuit to its experiment 1 state.2. Setup thermocouple, attach to sample.3. Establish a magnetic field of 4 kilogauss and a current of 120 mA.4. Wrap sample in thin layer of tissue to promote slower heating5. Reduce temperature by application of liquid nitrogen to sample.6. Once the temperature stabilizes, record the induced VH at each corresponding temperature.

DataThe following figures appear to be missing error bars. This is due to their very small size.

19.95 4039.97

60.4680.15

100.92

120.44

139.72

159.82

179.77198.9

219.8240.1

260.5280.1

301.7309.6

321.5341.4

361.4380.4

0

50

100

150

200

250

300

f(x) = 11.8994805194805 x − 1.3966666666667R² = 0.99588681819698

Experiment 1

Current (mA)

Volta

ge (m

V)

Figure 1: B = 4 KGauss

3

99.8 120140.1

159.7 180202

220240

260280

300320

340360

0

20

40

60

80

100

120

140

f(x) = 6.63259340659341 x + 26.5791208791209R² = 0.999924761872445

Experiment 2

B (mT)

V (m

V)

Figure 2: I = 200 mA

49.9368.9

90.3

110.42

130.143

149.86

170.14

190.72210.5

230.3249.9

269.70

100

200

300

400

500

600

f(x) = 39.5899300699301 x + 55.0221212121212R² = 0.999941460620094

Experiment 3

I (mA)

V (m

V)

Figure 3

4

26 26 27 27 27 28 28 28 28 29 29 29 30 3077

78

79

80

81

82

83

84

85

f(x) = − 0.154909688013136 x + 83.8233333333333R² = 0.97556703236765

Experiment 4: Trial 1

T (K)

V (m

V)

Figure 4: B = 4 KGauss, I = 120 mA

Results and AnalysisThe purpose of experiment 1 was to establish a relationship between the current and VH. A plot of the

recorded data (Figure 1) yields a direct linear relationship given by

V H=11.899∗I−1.3967

Where VH is in mV and I is in mA. RH can then be determined from the data and sample dimensions as

RH=V H t

IB

And

σ R H=√1−Rs2

Where Rs is the R2 value from Figure 1. Upon calculation of RH, it was found that the particular sample has an average RH = 0.04491∓0.0043548 (Table 1). The actual RH varied between 0.4357 for I = 0.3804 and 0.045907 for I=0.03997. These values fall outside of the calculated uncertainty in RH. This error could have arisen from numerous sources. One source was the instability of the circuit. Often it was noted that a large change in potential appeared for no apparent reason. Another error is that the magnetic field, while set at a constant value, actually steadily decayed as the experiment progressed. This would lead to direct error in the calculation of RH.

For the predominantly negative charge carriers, the charge density can be calculated as

n= −1RH e

And

σ n=n√(σ RH

RH

)2

=nσ R H

RH

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Where RH is the previously calculated value and e = 1.602*10-19 is the magnitude of the electron’s charge. The average value of the charge density was found to be n = -1.39034*1020 ∓1.34896 E19carriers/m3 (Table 1). This charge density is considerably lower than that for metals such as Na, where n=-2.49688E+25.

The average sensitivity was found to be 1.6633, determined as

Sensitivity=V H

IB

The purpose of the second experiment was to establish a relationship between VH and the magnetic field. A plot of this data (figure 2) shows another direct and linear relationship given by

V H=6.6326∗B+26.579

Where VH is in mV and B is in T. The average sensitivity was found to be 1.6629 (Table 2), which agrees well with experiment 1. Similar to experiment 1, a noticeable decay in the current was found by the time the experiment completed, again giving rise to considerable error.

The purpose of the third experiment was to determine the resistance of the sample. The plotted data in Figure 3 gives

V=39.59∗I +55.022

Where VH is in mV and I is in mA. This experiment eliminates the Hall effect and so the resistance can be given as

R=VI

And

σ R=R √(σV

V)

2

+(σ I

I)2

Where σ V∧σR correspond to apparatus uncertainties (∓ .005mA or mV). The average resistance was found to be

R=1.925∓0.01(Table 3). The determination of the resistance allows the calculation of the sample’s resistivity. Returning to the data from experiment 1, the average resistivity ρ = 19.1678∓0.5678 ohm-m was determined from

ρ=RAL

And

σ ρ= ρ√(σ R

R)

2

+(σ A

A)

2

+(σ L

L)

2

Where A is the cross-sectional area of the sample, L is the length of the sample, and σ’s correspond to uncertainties (σ L=∓ .05 mm¿. The resistivity of copper is approximately 1.68*10−8 ohm-m which is much less than the calculated resistivity.

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The average conductivity of the material was determined to be σ = 0.05217∓.0015 S/m (Table 3) from

σ=1ρ

And

σ cond=σ √(σρ

ρ)

2

An approximant value for the conductivity of copper would be σ = 5.96*107 S/m, which is far greater than the calculated conductivity.

The mobile carrier mobility can then be calculated as

μ= 1ρne

And

σ μ=μ√(σ ρ

ρ)

2

+(σ n

n)

2

The average mobility was found to be μ = -0.002349∓0.0002276 (Table 1).

The purpose of the final experiment was to determine if RH is dependent upon temperature. Figure 4 gives the relationship between VH and T as

V H=−.01549∗T+83.823

Using the equations as in experiment 1, RH values for each temperature data point were calculated. These values were then compared to the standard room temperature average RH value from experiment 1. As Table 4 shows, the reduction in temperature increases the expected RH value.

ConclusionIn conclusion the average value of RH was found to be RH= 0.04491∓0.0043548. The charge density of

the sample was n = -1.39034*1020 ∓1.34896 E19carriers/m3 . The average resistance was R=1.925∓0.01 . The

average resistivity was ρ = 19.1678∓0.5678 ohm-m and average conductivity was therefore σ = 0.05217∓.0015 S/m. The average mobility was found to be μ = -0.002349∓0.0002276.

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Appendix

Diagram 1

Table 1

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