Page 22 - Electrical Properties of Materials

The Hall effect 5

It may be clearly seen that by taking ζ =1/τ eqn (1.17) agrees with
eqn (1.6); hence we may regard the two models as equivalent and, in any given
case, use whichever is more convenient.

1.4 The Hall effect

Let us now investigate the current flow in a rectangular piece of material, as
shown in Fig. 1.3. We apply a voltage so that the right-hand side is positive.
Current, by convention, flows from the positive side to the negative side, that
is in the direction of the negative z-axis. But electrons, remember, flow in a
direction opposite to conventional current, that is from left to right. Having
sorted this out let us now apply a magnetic field in the positive y-direction. The
force on an electron due to this magnetic field is

e(v × B). (1.18)

To get the resultant vector, we rotate vector v into vector B. This is a clock-
wise rotation, giving a vector in the negative x-direction. But the charge of the
electron, e, is negative; so the force will point in the positive x-direction; the
electrons are deflected upwards. They cannot move farther than the top end of
the slab, and they will accumulate there. But if the material was electrically Equilibrium is established when
neutral before, and some electrons have moved upwards, then some positive the force due to the transverse
ions at the bottom will be deprived of their compensating negative charge. electric field just cancels the force
Hence an electric field will develop between the positive bottom layer and the due to the magnetic field.
negative top layer. Thus, after a while, the upward motion of the electrons will
be prevented by this internal electric field. This happens when

E H = vB. (1.19)

Expressed in terms of current density, R H is called the Hall coefficient.

1
E H = R H JB, R H = . (1.20)
N e e

In this experiment E H , J, and B are measurable; thus R H , and with it the density
of electrons, may be determined.

Voltmeter Electrons • B I x

+ + + + + + + + + + + z
J
y B
+
Fig. 1.3
Schematic representation of the measurement of the Hall effect.