Hall Effect in Thin Films

HALL EFFECT IN THIN FILMS

When a current-carrying conductor is placed in a transverse magnetic field, the Lorentz force on the moving charges produces a voltage perpendicular to both the field and the current. This is known as the Hall Effect. Measurements of the Hall voltage are used to determine the density and sign of charge carriers in a conductor. When this is known, the effect can used as a probe for magnetic field measurement.

In this experiment, samples of chromium and silver in the form of thin films of various thicknesses are available.

Consider the slab of conductor shown in Fig. 1.

Fig. 1

Geometry of Hall Effect

If the slab passes a current I at right angles to an applied magnetic field B, an electric field arises in a direction perpendicular to both I and B. The Hall coefficient is

where t is the thickness of the sample, I is the current, J is the current density and VH is the Hall voltage. Satisfy yourself that VH, and hence RH, are positive for positive charge carriers, and negative for negative charge carriers.

The density of charge carriers, n, the drift velocity, vd, and the conductive mobility, μ, follow from the measured values of RH, and ρ, the resistivity of the sampleassuming the free-electron model of metals. (For details consult Ref. 1, p. 217; Ref. 2, p. 827). According to this model,

The drift velocity is given by

where A is the cross-sectional area of the conductor, and e the electronic charge. (Work out how to extend this equation to account for positive charge carriers). The conductive mobility defined by vd/E, is given by

.

EH/E|| is the tangent of the Hall Angle. Recently, the Quantum Hall effect, in the limit of very high magnetic fields, has been used for a new determination e2/h (See Physics Today June 1981 p. 17) and as the new standard for resistance.

Experiment

The set up for measuring RH is sketched in Fig. 2.

Fig. 2

Hall Chip and Circuit

The magnetic field is measured Hall Probe Gaussmeter. It is recommended that the field is measured at every magnet current at which Hall measurements are made. Make sure you understand where the best place to position the probe is.

It is suggested you begin measurements on one of the Ag samples. Use the potentiometer at A1A2 (see Fig. 2) to zero out the voltage across AB at zero field. Why does this voltage arise at all? Make sure the potentiometer gives a zero even when high currents (30 mA) are applied at CD. Always use the limiting resistor on the power supply and never exceed 40 mA current through the sample. Otherwise the sample will be destroyed. Check the potentiometer setting throughout the experiment.

Place the sample in the centre of the magnet gap with its broad side perpendicular to the field. Determine RH and its sign by using Eq. (1) and making the appropriate plots.

The thickness of the Hall samples can be determined from the interferometric photographs provided by the manufacturers. In our case,

See Note Below.

For general details on interference see Ref. 3.

In order to obtain the resistivity ρ, first measure the resistance between V1 and V2 with the HP DVM using the 4-wire method. Why is it crucial to use the 4-wire method as opposed to the 2-wire method? The width of the sample w, and the distance between V1 and V2 can be measured using the travelling microscope.

Having obtained RH and ρ, the quantities n, vd, and μ can be calculated. Comment on the magnitudes of n and vd.

Now take measurements on a Cr sample. How does the conduction Cr differ from that in Ag?

One of your samples can now be used to investigate the homogeneity of the magnet. Run a profile across the central diameter. How does the homogeneity change with the magnet current?

Brief Note on the Hall Sample Thickness Measurements

The sample thickness was obtained by interference measurements before the electrodes were attached. A thin semi-reflective layer was deposited on them and another semi-reflective flat was placed over, and parallel to, the substrate and sample to form a thin parallel air gap. Interference fringes were observed in light of wavelength 5890 Å, as in Fig. 3. The fringes are displaced at the edge of the sample due to the change in the spacing in the air gap. It can be shown (try it yourself) that this fringe displacement or "fringe step" is given by the relation:

provided that the fringes are observed nearly normal to the plane of the sample and where λ = 5890 Å is the wavelength of the light used. Thus the interferograms may be used to find the sample thickness.

Fig. 3

Arrangement of sandwiched films on a microscope stage to

determine film thickness by an optical interference method.

References

1. Purcell, Electricity and Magnetism, p. 217. See also Chapter IV.

2. D. Halliday and R. Resnick, Physics (Wiley and Sons, N.Y., 1962).

3. F.A. Jenkins and H.E. White, Fundamentals of Optics (McGraw-Hall, N.Y., 1957).

4. Whittle and Yarwood, Experimental Physics for Students (Chapman & Hall, 1973).

5. S. Tolansky, Multiple Beam Interferometry of Surfaces and Films.

Rev. Nov. '84 GMG, Feb/02 BWS

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