hal effect

Hall Effect in Metals and Semiconductors

Meenu KumariY0911030

1

1 Introduction

In 1879 E. H. Hall observed that when an electrical current passes througha sample placed in a magnetic field, a potential proportional to the currentand to the magnetic field is developed across the material in a directionperpendicular to both the current and to the magnetic field. This effect isknown as the Hall Effect, and is the basis of many practical applicationsand devices such as magnetic field measurements, and position and motiondetectors. With the measurements he made, Hall was able to determine forthe first time the sign of charge carriers in a conductor. Even today, HallEffect measurements continue to be a useful technique for characterizingthe electrical transport properties of metals and semiconductors. Indeed,the failure of the simple model of metallic conductivity to account for manyexperimental measurements of the Hall effect has been one of the princi-pal motivators leading to a better understanding of electronic properties ofmaterials.

2 Objective :

1. To measure the Hall voltage in thin Zinc and Copper foils.

2. To determine the hall coefficient from measurements of the currentand the magnetic indution.

3. To investigate the temperature dependence of the Hall voltage on thecopper sample.

4. To measure the Hall voltage in p-Germanium and n-Germanium semi-conductors.

3 Equipments

1. Hall effect, Cu and zinc, carrier board

2. Hall effect, p-Ge and n-Ge, carrier board

3. Power supply 0-30 V DC, 20 A, stabilized

2

4. Power supply, universal

5. Universal measuring amplifier

6. Tesla meter, digital

7. Hall probe, tangential, protective cap

8. Coil, 300 truns

9. Iron core, U-shaped, laminated, pole pieces

10. Digital multimeter and Connecting cords

4 Theory :

When current, I, flows through a conductor placed at right angles to mag-netic field B, there is a Lorenz force,

~F = Q(~v × ~B)

acting on the charge carriers in the conductor.Due to this Lorenz force, the charge carriers accumulate on the edges, whichcreates a potential in the transverse direction current I. This potentialdifference is called hall voltage UH .One can derive hall voltage to be

UH =RHBI

d

where RH = Hall coefficientd = width of conducting strip.If RH is negative, it is ”Normal Hall Effect” and the charge carriers areelectrons. If RH is positive, it is ”Anomalous Hall Effect”, and chargecarriers are positive(holes). In metals both electrons and holes are present,and their mobilities decide the sign of RH .

3

5 Observations, Calculations and graphs :

5.1 For Zinc:

Measurements with constant current applied :

Table 1: Measurement of Hall voltage as a function of magnetic field forI=8A, for zinc

Magnetic Hall voltage Hall voltage Hall voltagefield B(mT) UBH(10−5V UHO(10−5V ) UH(10−5V )

250 1.5 1.11 0.39

200 1.42 1.15 0.27

150 1.36 1.13 0.23

100 1.32 1.18 0.14

50 1.24 1.17 0.07

-50 1.17 1.23 -0.06

-100 1.09 1.22 -0.13

-150 0.96 1.17 -0.21

-200 0.87 1.13 -0.26

-250 0.79 1.12 -0.33

4

Plot of Hall voltage vs magnetic field for I=8A,for Zn

UH(i

n μV

)

-4

-3

-2

-1

0

1

2

3

4

-4

-3

-2

-1

0

1

2

3

4

B(in mT)-300 -200 -100 0 100 200 300

-300 -200 -100 0 100 200 300

Slope = (0.014±0.0003) μV/mTRH = (4.37±0.09)*10-11 m3/(A-sec)

Table 2: Measurement of Hall voltage as a function of magnetic field forI=12A, for zinc


-250 0.68 1.1 -0.42

-200 0.71 1.06 -0.35

-150 0.72 1.01 -0.29

-100 0.78 0.99 -0.21

-50 0.84 0.93 -0.09

50 1.18 1 0.18

100 1.22 0.97 0.25

150 1.31 0.91 0.4

200 1.46 0.97 0.49

250 1.52 0.96 0.56

5

Plot of Hall voltage vs magnetic field for I=12A,for Zn

UH(i

n μV

)

-6

-4

-2

0

2

4

6

-6

-4

-2

0

2

4

6

B(in mT)-300 -200 -100 0 100 200 300

-300 -200 -100 0 100 200 300

Slope = (0.0209±0.0007) μV/mTRH = (4.35±0.14)*10-11 m3/(A-sec)

Measurements with constant magnetic field applied :

Table 3: Measurement of Hall voltage as a function of current for differentmagnetic fields for zinc

B=-250mT B=-200mT B=200mT

Current UBH UHO UH UBH UHO UH UBH UHO UH

I(A) 10−5V 10−5V 10−5V 10−5V 10−5V 10−5V 10−5V 10−5V 10−5V

1 1.27 1.31 -0.04 1.18 1.26 -0.08 1.01 0.96 0.052 1.2 1.33 -0.13 1.2 1.27 -0.07 1.09 1.01 0.083 1.21 1.3 -0.09 1.14 1.27 -0.13 1.18 1.11 0.074 1.14 1.27 -0.13 1.17 1.33 -0.16 1.31 1.11 0.25 1.09 1.33 -0.24 1.15 1.34 -0.19 1.34 1.2 0.146 0.96 1.3 -0.34 1.12 1.33 -0.21 1.39 1.17 0.227 0.97 1.36 -0.39 1.1 1.36 -0.26 1.46 1.19 0.278 0.99 1.33 -0.34 1.05 1.35 -0.3 1.51 1.18 0.339 0.88 1.3 -0.42 1.01 1.33 -0.32 1.5 1.21 0.29

10 0.82 1.19 -0.37 0.93 1.27 -0.34 1.52 1.14 0.38

6

Plot of Hall voltage vs current for B=-250mT,for Zn

UH(i

n μV

)

-5

-4

-3

-2

-1

0

-5

-4

-3

-2

-1

0

I (in A)0 2 4 6 8 10 12

0 2 4 6 8 10 12

Slope = -(0.432±0.06) μV/ARH = (4.32±0.6)*10-11 m3/(A-sec)

Plot of Hall voltage vs current for B=-200mT,for Zn

UH(i

n μV

)

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

I (in A)0 2 4 6 8 10

0 2 4 6 8 10

Slope = -(0.318±0.014) μV/ARH = (3.98±0.18)*10-11 m3/(A-sec)

7

Plot of Hall voltage vs current for B=200mT,for Zn

UH(i

n μV

)

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

1.5

2

2.5

3

3.5

4

I (in A)0 2 4 6 8 10 12

0 2 4 6 8 10 12

Slope = (0.365±0.039) μV/ARH = (4.56±0.48)*10-11 m3/(A-sec)

d = 25 ∗ 10−6m

Slope(UHB )(in µV/mT )) I (in A) RH (in 10−11m3A−1sec−1)

= UHB ∗ d/I

0.014± 0.0003 8 4.37±0.090.0209± 0.0007 12 4.35±0.14

Slope(UHI )(in µV/A)) B (in mT) RH (in 10−11m3A−1sec−1)

= UHI ∗ d/B

-(0.432± 0.06) -250 4.32±0.6-(0.318± 0.014) -200 3.98±0.18

0.365± 0.039 200 4.56±0.48

Mean RH = 4.316 ∗ 10−11m3A−1sec−1

Standard deviation = 0.21 ∗ 10−11m3A−1sec−1

Literature value of RH of Zinc = 10 ∗ 10−11m3A−1sec−1

Absolute error = (RHlit

−RHexp

RHlit∗ 100)% = 56.8%

8

5.2 For copper :

Measurements with constant current applied:

Table 4: Measurement of Hall voltage as a function of magnetic field forcopperB (mT) I=8A I=12A

UBH UHO UH UBH UHO UH

10−5V 10−5V 10−5V 10−5V 10−5V 10−5V

-250 1.85 1.48 0.37 1.67 1.17 0.5-200 1.77 1.48 0.29 1.61 1.21 0.4-150 1.66 1.45 0.21 1.52 1.2 0.32-100 1.6 1.44 0.16 1.44 1.22 0.22-50 1.53 1.44 0.09 1.34 1.25 0.0950 1.37 1.47 -0.1 1.2 1.24 -0.04

100 1.29 1.43 -0.14 1.11 1.29 -0.18150 1.22 1.42 -0.2 1.03 1.28 -0.25200 1.16 1.38 -0.22 0.97 1.33 -0.36250 1.06 1.43 -0.37 0.92 1.39 -0.47

Plot of Hall voltage vs magnetic field for I=8A,for Cu

UH(i

n μV

)

-4

-3

-2

-1

0

1

2

3

4

-4

-3

-2

-1

0

1

2

3

4

B(in mT)-300 -200 -100 0 100 200 300

-300 -200 -100 0 100 200 300

Slope = -(0.014±0.0005) μV/mTRH = -(3.15±0.11)*10-11 m3/(A-sec)

9

Plot of Hall voltage vs magnetic field for I=12A,for Cu

UH(i

n μV

)

-6

-4

-2

0

2

4

6

-6

-4

-2

0

2

4

6

B(in mT)-300 -200 -100 0 100 200 300

-300 -200 -100 0 100 200 300

Slope = -(0.0191±0.0003) μV/mTRH = -(2.87±0.04)*10-11 m3/(A-sec)

Measurements with constant magnetic field applied

Table 5: Measurement of Hall voltage as a function of magnetic field forcopperI(A) B=-250mT B=-200mT


10−5V 10−5V 10−5V 10−5V 10−5V 10−5V

1 1.3 1.23 0.07 1.44 1.41 0.032 1.34 1.27 0.07 1.47 1.39 0.083 1.38 1.26 0.12 1.53 1.43 0.14 1.51 1.35 0.16 1.59 1.43 0.165 1.59 1.36 0.23 1.62 1.42 0.26 1.68 1.41 0.27 1.64 1.43 0.217 1.7 1.42 0.28 1.66 1.44 0.228 1.76 1.44 0.32 1.72 1.44 0.289 1.78 1.4 0.38 1.73 1.41 0.32

10 1.82 1.4 0.42 1.75 1.4 0.35

10

Plot of Hall voltage vs current for B=-250mT,for Cu

UH(i

n μV

)

0

1

2

3

4

0

1

2

3

4

I (in A)0 2 4 6 8 10 12

0 2 4 6 8 10 12

Slope = (0.407±0.018) μV/ARH = -(2.93±0.13)*10-11 m3/(A-sec)

Plot of Hall voltage vs current for B=-200mT,for Cu

UH(i

n μV

)

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

1.5

2

2.5

3

3.5

4

I (in A)0 2 4 6 8 10 12

0 2 4 6 8 10 12

Slope = (0.342±0.016) μV/ARH = -(3.07±0.14)*10-11 m3/(A-sec)

11

Table 6: Measurement of Hall voltage as a function of magnetic field forcopperI(A) B=200mT B=250mT


10−5V 10−5V 10−5V 10−5V 10−5V 10−5V

1 1.32 1.38 -0.06 1.37 1.4 -0.032 1.33 1.39 -0.06 1.3 1.4 -0.13 1.37 1.5 -0.13 1.28 1.38 -0.14 1.34 1.53 -0.19 1.24 1.45 -0.215 1.35 1.51 -0.16 1.18 1.4 -0.226 1.32 1.51 -0.19 1.21 1.46 -0.257 1.31 1.52 -0.21 1.15 1.45 -0.38 1.27 1.5 -0.23 1.09 1.43 -0.349 1.23 1.52 -0.29 1.1 1.47 -0.37

10 1.13 1.48 -0.35 1 1.4 -0.4

Plot of Hall voltage vs current for B=200mT,for Cu

UH(i

n μV

)

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

I (in A)0 2 4 6 8 10

0 2 4 6 8 10

Slope = -(0.291±0.029) μV/ARH = -(2.62±0.26)*10-11 m3/(A-sec)

12

Plot of Hall voltage vs current for B=250mT,for Cu

UH(i

n μV

)

-4

-3

-2

-1

0

-4

-3

-2

-1

0

I (in A)0 2 4 6 8 10 12

0 2 4 6 8 10 12


Measurements with constant magnetic field applied, but at different tem-perature

Table 7: Measurement of Hall voltage as a function of current for B=250mTfor copper at T=750CCurrent Hall voltage Hall voltage Hall voltage

A(A) UBH(10−5V ) UHO(10−5V ) UH(10−5V )

1 -11.02 -10.98 -0.042 -11.21 -11.13 -0.083 -11.33 -11.19 -0.144 -11.35 -11.17 -0.185 -11.34 -11.07 -0.276 -11.36 -11.12 -0.247 -11.19 -10.85 -0.348 -11.24 -10.91 -0.339 -11.4 -10.95 -0.45

10 -11.35 -10.94 -0.41

13

Plot of Hall voltage vs I for B=250mT,for Cu at T=750C

UH(i

n μV

)

-5

-4

-3

-2

-1

0

-5

-4

-3

-2

-1

0

I (in A)0 2 4 6 8 10 12

0 2 4 6 8 10 12


d = 18 ∗ 10−6m

Slope(UHB )(in µV/mT )) I (in A) RH (in 10−11m3A−1sec−1)

= UHB ∗ d/I

-(0.0014± 0.0005) 8 -(3.15±0.11)-(0.0191± 0.0003) 12 -(2.87±0.04)

Slope(UHI )(in µV/A)) B (in mT) RH (in 10−11m3A−1sec−1)

= UHI ∗ d/B

0.407± 0.018 -250 -(2.93±0.13)0.342± 0.016 -200 -(3.07±0.14)

-(0.291± 0.029) 200 -(2.62±0.26)-(0.407± 0.02) 250 -(2.93±0.14)

Mean RH = −2.928 ∗ 10−11m3A−1sec−1


Literature value of RH of copper = −5.3 ∗ 10−11m3A−1sec−1

14


−RHexp

RHlit∗ 100)% = 44.7%

From measurement taken at T=750CRH = −(3.168 ± 0.25) ∗ 10−11m3A−1sec−1

This value is quite close to the mean value obtained (within error bars).Thus, RH is not temperature dependent.

15

5.3 For p-Germanium :

Measurements with constant current applied

Table 8: Measurement of Hall voltage as a function of magnetic field forI=30mA, for p-Ge


300 48.1 3.4 44.7280 45.7 3.5 42.2260 43 3.5 39.5240 40.3 3.5 36.8220 37.5 3.5 34200 34.7 3.6 31.1180 31.8 3.6 28.2160 29 3.6 25.4140 25.8 3.6 22.2120 22.9 3.7 19.2100 19.9 3.7 16.280 16.6 3.7 12.960 13.4 3.7 9.740 10.2 3.7 6.520 7 3.8 3.20 3.9 3.8 0.1

-20 0.4 3.3 -2.9-40 -2.8 3 -5.8-60 -5.9 2.9 -8.8-80 -9.1 2.8 -11.9

-100 -12.3 2.7 -15-120 -15.4 2.7 -18.1-140 -18.4 2.6 -21-160 -21.5 2.6 -24.1-180 -24.5 2.6 -27.1-200 -27.4 2.5 -29.9-220 -30.2 2.5 -32.7-240 -33.1 2.5 -35.6-260 -35.9 2.5 -38.4-280 -38.5 2.4 -40.9-300 -41.1 2.4 -43.5

16

Plot of Hall voltage vs B for I=30mA,for p-Ge

UH(i

n m

V)

-60

-40

-20

0

20

40

60

-60

-40

-20

0

20

40

60

B(in mT)-300 -200 -100 0 100 200 300

-300 -200 -100 0 100 200 300

Slope = (0.151±0.0005) mV/mTRH = (5.03±0.017)*10-3 m3/(A-sec)


Table 9: Measurement of Hall voltage as a function of magnetic field forp-GeI(A) B=250mT B=195mT


(mV ) (mV ) (mV ) (mV ) (mV ) (mV )

-30 -43.8 -3.4 -40.4 -35.7 -3.4 -32.3-25 -36.3 -2.8 -33.5 -30.7 -2.9 -27.8-20 -30.4 -2.3 -28.1 -24.8 -2.3 -22.5-15 -22.9 -1.7 -21.2 -18.6 -1.7 -16.9-10 -14.8 -1.1 -13.7 -13 -1.1 -11.9-5 -8.8 -0.6 -8.2 -7.1 -0.6 -6.50 -1.1 0 -1.1 0.1 0 0.15 6.7 0.6 6.1 5.5 0.6 4.9

10 13.2 1.1 12.1 11.6 1.2 10.415 20.1 1.7 18.4 16.7 1.7 1520 26.6 2.2 24.4 22.2 2.2 2025 33.7 2.8 30.9 28.6 2.8 25.830 40.9 3.4 37.5 34.3 3.4 30.9

17

Plot of Hall voltage vs current for B=250mT, for p-Ge

UH(i

n m

V)

-60

-40

-20

0

20

40

-60

-40

-20

0

20

40

I(in mA)-40 -30 -20 -10 0 10 20 30 40

-40 -30 -20 -10 0 10 20 30 40

Slope = (1.301±0.007) mV/mARH = (5.204±0.028)*10-3 m3/(A-sec)

Plot of Hall voltage vs current for B=195mT, for p-Ge

UH(i

n m

V)

-40

-20

0

20

40

-40

-20

0

20

40

I(in mA)-40 -30 -20 -10 0 10 20 30 40

-40 -30 -20 -10 0 10 20 30 40

Slope = (1.065±0.007) mV/mARH = (5.46±0.035)*10-3 m3/(A-sec)

18

Tabulation of calculated RH values:

d = 10−3m

Slope(UHB )(in mV/mT )) I (in mA) RH (in 10−3m3A−1sec−1)

= UHB ∗ d/I

(0.151±0.0005) 30 (5.03±0.017)

Slope(UHI )(in mV/mA)) B (in mT) RH (in 10−3m3A−1sec−1)

= UHI ∗ d/B

1.301± 0.007 250 5.204±0.0281.065± 0.007 195 5.46±0.065

Mean RH = 5.23 ∗ 10−3m3A−1sec−1


Literature value of RH of p-Ge = 4.17 ∗ 10−3m3A−1sec−1


−RHexp

RHlit∗ 100)% = 25.4%

19

5.4 For n-Germanium :

Measurements with constant current applied

Table 10: Measurement of Hall voltage as a function of magnetic field forI=30mA, for n-Ge


300 -40.6 6 -46.6280 -37.6 5.9 -43.5260 -34.5 5.9 -40.4240 -31.3 5.8 -37.1220 -28.2 5.8 -34200 -25.1 5.8 -30.9180 -22 5.8 -27.8160 -19 5.8 -24.8140 -15.8 5.7 -21.5120 -12.7 5.7 -18.4100 -9.7 5.7 -15.480 -6.6 5.7 -12.360 -3.6 5.7 -9.340 -0.5 5.6 -6.120 2.5 5.6 -3.10 5.5 5.6 -0.1

-20 8.8 6.3 2.5-40 11.8 6.3 5.5-60 14.9 6.5 8.4-80 18.1 6.6 11.5

-100 20.9 6.6 14.3-120 24 6.7 17.3-140 27.1 6.7 20.4-160 30.1 6.8 23.3-180 33 6.8 26.2-200 36 6.8 29.2-220 39.1 6.8 32.3-240 42 6.9 35.1-260 44.9 6.9 38-280 47.9 6.9 41-300 50.9 6.9 44

20

Plot of Hall voltage vs B for I=30mA,for n-Ge

UH(i

n m

V)

-60

-40

-20

0

20

40

60

-60

-40

-20

0

20

40

60

B(in mT)-300 -200 -100 0 100 200 300

-300 -200 -100 0 100 200 300

Slope = -(0.1504±0.0004) mV/mTRH = -(5.01±0.013)*10-3 m3/(A-sec)


Table 11: Measurement of Hall voltage as a function of magnetic field forn-GeI(A) B=250mT B=195mT


(mV ) (mV ) (mV ) (mV ) (mV ) (mV )

30 -31.9 5.8 -37.7 -24.4 5.9 -30.325 -27.3 4.9 -32.2 -20.1 4.7 -24.820 -21.3 3.7 -25 -16.6 3.8 -20.415 -16.1 2.6 -18.7 -12.1 2.6 -14.710 -11.1 1.6 -12.7 -8.2 1.5 -9.75 -5.2 0.4 -5.6 -4.1 0.4 -4.50 -0.3 -0.5 0.2 0.2 -0.7 0.9

-5 6 -1.9 7.9 4.3 -1.8 6.1-10 11.6 -3 14.6 8.2 -3 11.2-15 16.9 -4.2 21.1 12.4 -4.1 16.5-20 22 -5.3 27.3 15.9 -5.2 21.1-25 28.2 -6.6 34.8 20.3 -6.4 26.7-30 33.3 -7.7 41 25.3 -7.7 33

21

Plot of Hall voltage vs current for B=250mT, for n-Ge

UH(i

n m

V)

-60

-40

-20

0

20

40

60

-60

-40

-20

0

20

40

60

I(in mA)-30 -20 -10 0 10 20 30

-30 -20 -10 0 10 20 30

Slope = -(1.323±0.007) mV/mARH = -(5.29±0.028)*10-3 m3/(A-sec)

Plot of Hall voltage vs current for B=195mT, for n-Ge

UH(i

n m

V)

-40

-30

-20

-10

0

10

20

30

40

-40

-30

-20

-10

0

10

20

30

40

I(in mA)-30 -20 -10 0 10 20 30

-30 -20 -10 0 10 20 30

Slope = -(1.043±0.006) mV/mARH = -(5.34±0.031)*10-3 m3/(A-sec)

22

Tabulation of calculated RH values:

d = 10−3m

Slope(UHB )(in mV/mT )) I (in mA) RH (in 10−3m3A−1sec−1)

= UHB ∗ d/I

(-0.1504±0.0004) 30 -(5.01±0.013)

Slope(UHI )(in mV/mA)) B (in mT) RH (in 10−3m3A−1sec−1)

= UHI ∗ d/B

-1.323± 0.007 250 -5.29±0.028-1.043± 0.006 195 -5.34±0.031

Mean RH = −5.11 ∗ 10−3m3A−1sec−1


Literature value of RH of n-Ge = −4.8 ∗ 10−3m3A−1sec−1


−RHexp

RHlit∗ 100)% = 6.6%

6 Results and discussion :

Sample RH , experimental value RH , theoretical value Absolute error(in 10−11m3A−1sec−1) (in 10−11m3A−1sec−1)

Zinc (4.316± 0.21) 10.0 56.8%Copper (-2.928± 0.18) -5.3 44.7%

Copper (at T = 750C) (-3.168± 0.25)

Value of RH obtained at T = 750C is quite close to the value at room tem-perature (lies within the error bars). So, RH is independent of temperature.

23

Sample RH , experimental value RH , theoretical value Absolute error(in 10−3m3A−1sec−1) (in 10−3m3A−1sec−1)

p-germaium (5.23± 0.22) 4.17 25.4%n-germaium (-5.11± 0.15) -4.8 6.6 %

7 Precautions :

• All the connections should be made properly.

• Probe and pins should be connected with correct polarities.

• The temperature of the Cu sample shouldn’t be more than 1400C, elsethe circuit board might get damaged.

• Compensation of the initial hall voltage should be done properly.

24

hal effect

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