Tutorial 3B - Hall Effect

Learning Objectives

To study the Hall Effect.

To derive the Hall equation.

Key Questions

What is the Hall Effect?

How is it used?

The Hall Effect

It was first discovered in 1879 by an American Physicist, Edwin Herbert Hall (1855 - 1938).  He found that when a current-carrying conductor or a semiconductor is placed in a magnetic field, a voltage occurs that is perpendicular to the flow of the current.  This is called the Hall Voltage.

The Hall Effect can be observed in conductors, semi-conductors, ionised gases, and plasmas.  We will be looking at the effect of magnetic fields in electrons in semi-conductors, because the nature of semi-conductors allow for a relatively high voltage which is easy to measure.  It is also easily reproduced in a school or college Physics laboratory.  So let's have a look at what happens.

The separation of charges leads to a potential difference, or voltage that is called the Hall voltage (VH).  This voltage also causes an electric field, E which is uniform.  The idea is shown in the picture below: We know that the electric field, E is given by: The Hall voltage set up in a semi-conductor by a current in a magnetic field is 0.35 V.  The semi-conductor slice is 5.0 mm wide.   Calculate the electric field strength.

We also know from the definition of electric field that the force is:

F = Eq

So we can write: We also know that F = Bqv  from Magnetism Tutorial 1, so we can write: The q terms obligingly cancel out and we can rearrange to give: Measuring the speed of individual electrons is not at all easy, but if we review , we know that:

I = nAvq

Where:

• n = number of charge carriers per unit volume (m-3);

• A = area of the conductor (m2);

• v = speed of charge flow (m s-1);

• q = charge (C)

• I = current (A)

So we rearrange to get: and then substitute: Now area, A = dt, so we can write: and the d terms cancel out to give us our final relationship: where:

• VH = Hall voltage;

• B = magnetic flux density (T);

• I = current (A);

• n = number of charge carriers per unit volume (m-3);

• t = thickness of the slice (m);

• q = charge (C). Note that n is the number of charge carriers per unit volume, not the number of turns per unit length.   The term dt in the argument above was distance × thickness to give area.  It has nothing to do with a time interval!

The number of charge carriers per unit volume for typical semi-conductors is shown in the table:

 Semi conductor n / m-3 Gallium arsenide 1.10 × 1025 Germanium 2.02 × 1021 Pure Silicon 1.50 × 1016

These figures are for a temperature of 300 K.  The number of free charge carriers per unit volume rises as the temperature rises.  We will use these, as college and university physics labs tend to have a temperature of about 300 K (27 oC).  You would be well boiling if the temperature in the lab was 400 K.

For metals:

 Metal n / m-3 Aluminium 6.02 × 1028 Copper 8.46 × 1028 Tungsten 3.43 × 1028

 A slice of germanium is 5.5 mm long, 3.3 mm wide, and 0.35 mm thick.  It is carrying a current of 0.067 A and is placed in a magnetic field of flux density 0.14 T.    Calculate the Hall voltage if the charge on each charge carrier is 1.602 × 10-19 C.

The Hall effect in theory can be observed in metals, but the Hall voltage would be so tiny that it's negligible.  A calculation using the data above in a copper wire gives a Hall voltage of about 2 × 10-9 V.

Uses of the Hall Effect

In the school or college lab, the Hall probe is used to measure the value of a magnetic field.  The picture shows two Hall probes: The Hall probes are calibrated using a magnetic field of known flux density to reduce uncertainty.

Hall probe sensors are used in motor control to detect the speed of the motor.  Hall probe sensors are used to detect wheels locking in anti-lock braking systems.