Magneto Resistance in Semi-conductors

Indian Institute of Science Education and Research, Kolkata

17th – 24th March 2010

Magneto-resistance in Semi-Conductors Harsh Purwar (07MS-76) Amit Nag (07MS-19) th VI Semester, Integrated M.S. Indian Institute of Science Education and Research, Kolkata Experiment No.: 5 Condensed Matter Physics Lab (PH – 324)

Objectives of the Experiment:  

To study the variation of the magneto-resistance of a sample with the applied magnetic field. To measure the magneto-resistance of the given semiconductor sample.

Introduction/Theory: Magneto-resistance is the property of a material to change the value of its electrical resistance when an external magnetic field is applied to it. The effect was first discovered by William Thomson (more commonly known as Lord Kelvin) in 1856, but he was unable to lower the electrical resistance of anything by more than 5%. This effect was later called ordinary magneto-resistance (OMR). The phenomenon is due to the fact that the drift velocity of all the charge carriers is not same. When the magnetic field is turned on the Hall voltage, 𝑉𝐻 = 𝐸𝑦 𝑡 = 𝑣 × 𝐻 compensates or cancels the Lorentz force for the carriers with the average velocity, 𝑣. But as said all the charge carriers do not move with the same velocities and so the carriers moving with velocities less than 𝑣 are over compensated and those moving with velocities greater than 𝑣 are Figure 1: Schematic diagram showing various fields acting on a p-type undercompensated, resulting in trajectories that are not along the semiconductor crystal attached to the applied electric field 𝐸𝑥 . This results in an effective decrease of four probe. the mean free path and hence an increase in resistivity or resistance of the sample. Now as the applied magnetic field is increased the trajectories of the charge carriers are more deviated which increases the magneto-resistance further. To understand this lets consider a conducting annulus with perfectly conducting rims as shown in the figure above. Without a magnetic field, the battery drives a radial current between the rims. When a magnetic field parallel to the axis of the annulus is applied, a circular component of current flows as well, due to the Lorentz force. In a simple model, supposing the response to the Lorentz force is the same as for an electric field, the carrier velocity 𝑣 is given by, 𝑣 = 𝜇 𝐸 + 𝑣 × 𝐵 where 𝜇 is the carrier mobility. Solving for the velocity, we find, 𝜇 𝑣= 𝐸 + 𝜇𝐸 × 𝐵 1 + 𝜇2 𝐵2 where the reduction in mobility due to the magnetic field 𝐵 is apparent. Figure 2: Corbino Disc.

1|Magneto-resistance, Expt. - 5

Indian Institute of Science Education and Research, Kolkata

17th – 24th March 2010

Experimental Setup: The setup consists of the following:  Four probe arrangement  Sample (Ge, n-type)  Magneto-resistance set-up  Electromagnet  Constant current power supply  Digital Gauss-meter Four Probe Arrangement: It consists of four collinear, equally spaced (2 mm) and individually spring loaded probes mounted on a PCB strip. The two outer probes supply constant current to the sample and the two inner probes measure the voltage developed across them. This eliminates the error due to the contact resistance which is particularly serious in semiconductors. The sample is placed on a platform provided for the purpose and the four probe is mounted on it. Current Probes

Voltage Probes Figure 3: Schematic diagram of a four probe.

Sample: Germanium (Ge) crystal, n-type dimensions: 10 × 10 × 0.5 mm.

Procedure: The following protocol was implemented in order for measuring the magneto-resistance of the given sample.  All necessary connections were made and various components were switched on.  The Four probe with sample and the Hall probe were both placed side by side in the space between the two iron cores of the electromagnets.  For a fixed value of probe current (lets’ say 4.04 mA), the voltage developed across the two inner probes of the four probe was measured and noted along with the magnetic field which was increased by increasing the current though it.  Above was done for three different values of the probe current (4.04 mA, 3.12 mA and 2.02 mA). The recorded data is presented in a tabular fashion in the following section.

2|Magneto-resistance, Expt. - 5

17th – 24th March 2010

Indian Institute of Science Education and Research, Kolkata

Results/Observations: The electrical resistance of the sample was found to be 𝑅 = 44.21 Ω. This was calculated using Ohm’s Law for 𝑉 = 149.0 𝑚𝑉, and 𝐼 = 3.37 𝑚𝐴. Table 1: For measurement of magneto-resistance of a Germanium n-type crystal for probe current 𝐼 = 4.04 mA.

Obs. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Current through the electromagnet (Ampere) 0.00 0.20 0.40 0.60 0.80 1.01 1.20 1.41 1.62 1.80 2.01 2.22 2.40 2.60 2.80 3.00 3.22 3.40 3.60 3.80 4.00 4.13

Magnetic Field (Gauss)

Voltage 𝑉𝑚 (mV)

80 260 440 630 820 1040 1220 1430 1650 1830 2040 2250 2430 2620 2810 3000 3230 3380 3560 3740 3900 4010

177.6 177.6 177.6 177.7 177.7 177.9 178.0 178.2 178.4 178.5 178.8 179.1 179.4 179.7 180.0 180.4 180.7 181.0 181.4 181.9 182.3 182.5

𝑅𝑚 =

𝑉𝑚 𝐼

Ω 43.96 43.96 43.96 43.99 43.99 44.03 44.06 44.11 44.16 44.18 44.26 44.33 44.41 44.48 44.55 44.65 44.73 44.80 44.90 45.02 45.12 45.17

Rm − R 𝑅 -0.01 -0.01 -0.01 -0.01 -0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02

Table 2: For measurement of magneto-resistance of a Germanium n-type crystal for probe current 𝐼 = 3.12 mA.

Obs. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Current through the electromagnet (Ampere) 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.41 1.60 1.80 2.01 2.20 2.41 2.60

Magnetic Field (Gauss)

Voltage 𝑉𝑚 (mV)

70 260 440 630 820 1020 1220 1430 1630 1830 2040 2230 2430 2620

137.6 137.6 137.6 137.6 137.6 137.7 137.8 137.9 138.0 138.2 138.4 138.6 138.9 139.1

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𝑅𝑚 =

𝑉𝑚 𝐼

Ω 44.10 44.10 44.10 44.10 44.10 44.13 44.17 44.20 44.23 44.29 44.36 44.42 44.52 44.58

Rm − R 𝑅 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01

Indian Institute of Science Education and Research, Kolkata 15 2.81 2820 16 3.00 3010 17 3.20 3190 18 3.40 3390 19 3.60 3560 20 3.81 3740 21 4.01 3910 22 4.12 4000

139.4 139.6 139.9 140.2 140.5 140.8 141.1 141.3

17th – 24th March 2010 44.68 0.01 44.74 0.01 44.84 0.01 44.94 0.02 45.03 0.02 45.13 0.02 45.22 0.02 45.29 0.02

Table 3: For measurement of magneto-resistance of a Germanium n-type crystal for probe current 𝐼 = 2.02 mA.

Obs. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Current through the electromagnet (Ampere) 0.00 0.20 0.40 0.60 0.80 1.00 1.21 1.40 1.60 1.80 2.00 2.21 2.40 2.60 2.81 3.01 3.20 3.40 3.61 3.80 4.00 4.11

Magnetic Field (Gauss)

Voltage 𝑉𝑚 (mV)

70 260 430 630 830 1020 1240 1430 1620 1820 2030 2240 2420 2620 2820 3010 3190 3380 3570 3730 3910 4000

88.7 88.7 88.7 88.7 88.7 88.8 88.8 88.9 89.0 89.1 89.2 89.4 89.5 89.7 89.8 90.0 90.2 90.4 90.6 90.9 91.1 91.1

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𝑅𝑚 =

𝑉𝑚 𝐼

Ω 43.91 43.91 43.91 43.91 43.91 43.96 43.96 44.01 44.06 44.11 44.16 44.26 44.31 44.41 44.46 44.55 44.65 44.75 44.85 45.00 45.10 45.10

Rm − R 𝑅 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 -0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02

Indian Institute of Science Education and Research, Kolkata

17th – 24th March 2010

Graphs/Plots: The following plots were drawn from the above data.

Plot 1: Magneto-resistance of the sample versus applied magnetic field.

Plot 2: Log-log plot of relative increase in the resistance of the sample (relative to its resistance at zero magnetic field) due to the application of the magnetic field.

5|Magneto-resistance, Expt. - 5

Indian Institute of Science Education and Research, Kolkata

17th – 24th March 2010

Conclusions: 

The resistance of the sample first decreases for low magnetic fields as observed in the following graph and then increases with the increase in the magnetic field.

Plot 3: Relative increase in the resistance of the sample (relative to its resistance at zero magnetic field) due to the application of the magnetic field.



Within the experimental limits the decrease in the resistance of the sample is not significant and so we conclude that the magneto-resistance does exist and increases with the magnetic field for a constant probe current.

Sources of Errors:  

The most common error in these experiments arises due to magnetic hysteresis and therefore it is preferred not to calibrate the electromagnets for magnetic field but to measure it simultaneously using a Hall probe. Loose connections and contact resistances should be taken care off using proper soldering etc.

6|Magneto-resistance, Expt. - 5