Magnetoresistance in Bismuth – Experimental Investigations (Part 1)

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By Timothy Raney…Bald Engineer Guy with Glasses
Illustrations by the author.

Introduction
When an experimenter places a current carrying conductor in a magnetic field, its resistance usually increases. This effect is called magnetoresistance. It is due to the Lorentz force acting on the electrons combined with a distribution in the electrons' velocities[1]. The purpose of these experiments was to investigate an increase in the bulk electrical resistivity (r) of a bismuth (Bi) specimen by measuring a change in its resistance when placed in a magnetic field. Elemental bismuth exhibits a considerable magnetoresistance[2]. This is a specific material property of bismuth and its magnitude is the highest among the elements[3]. These experiments focused on longitudinal magnetoresistance exhibited by a metallic bismuth specimen as a current carrier in a transverse magnetic field[4]. I selected this particular galvanomagnetic effect since I could potentially make a magnetic field sensor from a bismuth sample. I also wanted to demonstrate magnetoresistance effect to my satisfaction. All resistance measurements are shown in ohms (Ω). Bismuth specimen geometry did not strictly conform to expressing resistivity (r) as a product of r = l/a; where “l” is the length and “a” is the cross-sectional area, with r expressed in ohm-meters.

Theory
Magnetoresistance is a subordinate effect under galvanomagnetic effects. In theory, longitudinal magnetoresistance is due to the magnetic field deflecting conduction (free) electrons within a metallic conductor. In this specific case, the magnetic flux density (B) is perpendicular to the current density (J) and the magnetoresistive effect is measured along the direction of J as in the experimental circuit in figure 1. Under the influence of the combined electric and magnetic fields, conduction electrons drift through and experience random collisions with the ions in the bismuth’s crystal lattice. The consequential Lorentz force deflects each moving electron from its path. These electrons accumulate against an inside face of the sample since they are constrained to move within its boundaries. This condition then creates an electric field that opposes the Lorentz force. Consequently, the sample’s resistivity increases as the electrons are deflected from their forward motion under an applied potential[5],[6],[7].

Goals.

1. Prepare bismuth specimens suitable for conducting these investigations.

2. Measure bismuth’s change in resistance (DR) as a function of its resistivity (r) when placed in a known static magnetic flux (B) provided by a permanent magnet.

3. Determine the relationship between an increase in magnetic flux and resistance.

4. Determine if a simple bismuth strip and a digital ohmmeter can measure an unknown magnetic flux accurately.

5. Demonstrate magnetoresistance to show the effect does exist.

Assumptions.

1. Bismuth’s bulk resistivity increases when subjected to a sufficiently strong magnetic flux (arbitrarily above 1 kilogauss (kG).

2. A change in resistivity (Dr) is proportional to a change in resistance (DR) and is measurable when using a suitable specimen, measurement technique and appropriate apparatus.

3. The DR is proportional to magnetic flux above a certain value.

4. One can observe DR with a conventional digital multimeter (DMM) set to measure resistance (in ohms – Ω).

 

Magnetic Polarity Convention
These experiments will use the polarity convention attributed to the National Institute of Standards and Technology (NIST). This convention states the magnet’s “north-seeking pole” is the pole attracted to the geographic North Pole. This is a relative definition of magnetic polarity[8]. The magnets used in these experiments were marked accordingly.

 

Bismuth Specimen Orientation and Preparation
Bismuth specimens were oriented with their plane perpendicular to the magnetic flux. The specimen’s where placed within the geometric center of radar type gap magnets (figures 2 and 4). Specimens were formed in a flat rectangle with a minimal cross section to increase its resistance (figure 3).

Apparatus
This work used apparatus suitable (for the most part) for the experiment. Each summary lists the salient apparatus and materials used for that experiment. The following items are some examples.

  • Bench Multimeter, Model 5491A, BK Precision®.
  • Electrometer, Model 610A, Keithley Instruments, Inc.
  • Wheatstone Bridge, Test Set# 5300, Leeds & Northrup Company.
  • DC Magnetometer Gaussmeter, Model DCM, AlphaLab, Inc.
  • Gaussmeter, locally fabricated by T.E. Raney.
  • Standard Cell, Model 100 (#740102), The Eppley Laboratory, Inc.
  • Resistance Box, Model 829, Shallcross Manufacturing Company.
  • Bismuth metal samples, locally prepared by T.E. Raney.
  • Magnet#1 and Magnet#3 - radar magnetron type gap magnets.
  • Laboratory jacks, clamps, stands and connecting wire.

Procedures
The procedures remained fairly constant throughout the investigation. Where applicable, each summary lists procedures pertaining to that experiment. In general, the procedures included:

  • Selecting the appropriate apparatus for the experiment.
  • Using the apparatus and/or equipment according to accepted practice.
  • Following equipment instructions according to its technical manual.
  • Recording the ambient temperature or any other pertinent conditions.
  • Making careful observations.
  • Recording the data for later analysis.

Bismuth Specimen Preparation
The samples were prepared by pouring a few grams of molten bismuth on a glass plate. Another glass plate was immediately pressed onto the bismuth before it could solidify. The resulting bismuth sheet was allowed to cool, removed from the glass carefully and then cut to the desired shape with a sharp utility knife. It is important to recognize that elemental bismuth is very brittle, especially when prepared as a thin sheet or wire. It can crumble in your fingers. Thus, the specimens needed protection from breaking into small fragments. Bismuth specimen #1 was essentially a section of specimen #2 as shown on the right (fig. 5). Its resistance was not measurable with any accuracy given the instrumentation available at the time. Therefore, it was not used further. Specimen #2 consisted of five pieces of bismuth laminated between protective plastic adhesive tape. This configuration maximized its length (8.89cm total) for the space between the gap magnet poles.

The specimen was then mounted on an index card to protect it further. The specimen’s resistance was ~8.0Ω at 23OC (earth ambient flux). A measurement was made with Magnet #3 providing a 5.1kilogauss (kG) flux. However, specimen #2’s wire leads were pressed against the bismuth legs and taped in place. This attachment method resulted in poor electrical contact. Thus, the poor electrical continuity prevented making accurate resistance measurements with a Wheatstone bridge or a new BK Precision® multimeter. Therefore, the DR for specimen #2 was inconclusive, nor was it used again since disassembly would have broken the brittle bismuth. Soldering each joint would have been preferable as done with later specimens.

Bismuth Specimen #3
The purpose of this experiment was to determine the change in resistance (DR) with the plane of specimen #3 placed perpendicular to the flux within the magnet gap. Specimen #3 was 0.48mm x 17.46mm x 7.94mm with two buss wire connections soldered to each end as shown (fig. 6). The brittle bismuth specimen was also laminated with protective clear plastic tape. The specimen is oriented with its plane perpendicular to the magnetic flux (B) as shown. The resistance was measured with the Wheatstone bridge. The bridge’s rheostats were adjusted until the galvanometer did not deflect (null point)[9],[10]. The initial resistance (Ri) was 0.119Ω (earth ambient field) and final resistance (Rf) was 0.121Ω in the ~8.6kG B field (Magnet #1). DR was then 2mΩ. Ambient temperature was 24OC. Magnet #1 had adjustable pole pieces for producing varying flux values dependent on the gap width.

These results indicate the specimen’s resistance increased slightly (2mΩ) when placed in the B field. If DR is divided by B, the resistance increase equals 2.33 X 10-7 W/gauss. The result might not compare well with published values. A comparison is valid only if both values were derived using similar apparatus, conditions and sample geometry. Supporting data to verify any agreement with known values was not readily available. The useful range of this Wheatstone bridge under 1Ω is uncertain[11],[12]. However, the results show a DR given a sufficiently dense flux. A subsequent experiment with a modified Magnet #3 (details below) yielded a similar result. In this case, specimen#3’s DR was ~2mΩ. Ri was 0.119Ω and changed to 0.122Ω in a ~9.6kG B field. Ambient temperature was 28OC. The next experiment was essentially identical, but included a specimen with a smaller cross sectional area.

Bismuth Specimen #4
I prepared bismuth specimen #4 as before, with two soldered connections and laminated with tape as shown. The specimen was oriented with its plane perpendicular to the flux. Nominal dimensions were 0.483mm x 2.38mm x 19mm. Ri was 0.048Ω and Rf was 0.059Ω in the ~8.6kG flux (Magnet #1). The DR was then 0.011Ω. Ambient temperature was 26OC.

Magnet #3 Modification and Calibration. See Part 2.


Footnotes:

[1] R.G. Lerner and G.L. Trigg (Eds.), Encyclopedia of Physics (2nd Ed.), VCH Publishers, Inc., New York, 1991, pp. 686-687. See W.A. Reed - Magnetoresistance.

[2] L.B. Loeb, Fundamentals of Electricity and Magnetism (2nd Ed.), John Wiley & Sons, Inc., New York, 1938, pg. 309.

[3] W.M. Haynes and D.R. Lide (Eds.), CRC Handbook of Chemistry and Physics (92nd Ed.), CRC Press - Taylor & Francis Group, Boca Raton, FL, 2011, pg. 4-6.

[4] R.G. Lerner and G.L. Trigg, pg. 421. See D.J. Sellmyer and C.M. Hurd - Galvanomagnetic and Related Effects.

[5] S. Chikazumi (S.H. Charap, trans.), Physics of Magnetism, John Wiley & Sons, Inc., New York, 1964, pp. 420-421.

[6] L.S. Lerner, Physics for Scientists & Engineers, Jones & Bartlett Publishers, Inc., Sudbury, MA, 1996, pp. 737-739.

[7] R.G. Lerner and G.L. Trigg, pg. 422. See Sellmyer & Hurd.

[8] L.R. Moskowitz, Permanent Magnet Design and Application Handbook, Cahners Books International, Inc., Boston, MA, 1976, pg. 179.

[9] M.B. Stout, Basic Electrical Measurements, Prentice-Hall, Inc., New York, 1950, pg. 75 and pp. 385-386.

[10] V. Karapetoff and B.C. Dennison, Experimental Electrical Engineering and Manual for Electrical Testing (vol. I - 4th Ed.), John Wiley & Sons, Inc., New York, 1933, pp. 34-37.

[11] Ibid, Stout.

[12] Ibid, Karapetoff and Dennison.

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