The Hall Effect
This section derives the equation of the hall effect & also explains it.
The Hall Effect:
If a magnetic field is applied perpendicular to the direction in which holes drift in a p-type bar, the path of the holes tends to be deflected (Fig. given below).
Using vector notation, the total force on a single hole due to the electric and magnetic fields is
In the y-direction the force is equ (2)
The important result of Eq. (2) is that unless an electric field ξy is established along the width of the bar, each hole will experience a net force (and therefore an acceleration) in the -y-direction due to the qvxBz product. Therefore, to maintain a steady state flow of holes down the length of the bar, the electric field ξy must just balance the product vxBz: equ (3)
so that the net force Fy is zero. Physically, this electric field is set up when the magnetic field shifts the hole distribution slightly in the -_y-direction. Once the electric field ξy becomes as large as vxBz, no net lateral force is experienced by the holes as they drift along the bar. The establishment of the electric field ξy is known as the Hall effect, and the resulting voltage VAB = ξyw is called the Hall voltage.
The drift velocity (using q and p0for holes), the field ξybecomes
Thus the Hall field is proportional to the product of the current density and the magnetic flux density. The proportionality constant RH = (qpo)-1 is called the Hall coefficient. A measurement of the Hall voltage for a known current and magnetic field yields a value for the hole concentration p0
Since all of the quantities in the right-hand side of Eq. (5) can be measured, the Hall effect can be used to give quite accurate values for carrier concentration.
If a measurement of resistance R is made, the sample resistivity p can be calculated:
Since the conductivity σ = 1/p is given by qμpPo, the mobility is simply the ratio of the Hall coefficient and the resistivity:
Measurements of the Hall coefficient and the resistivity over a range of temperatures yield plots of majority carrier concentration and mobility vs.temperature. Such measurements are extremely useful in the analysis of semiconductor materials.