HALL EFFECT

Hall effect

Consider a rectangular slab that carries a current I in the X-direction. A uniform magnetic field of flux density B is applied along the Z-direction. The current carriers experience a force (Lorents force) in the downward direction. This leads to an accumulation of electrons in the lower face of the slab. This makes the lower face negative. Similarly the deficiency of electrons make the upper face positive. As a result, an electric field is developed along Y-axis. This effect is called Hall effect and the emf thus developed is called Hall voltage VH. The electric field developed is called Hall field EH.
If ‘e’ is the charge of carrier and ‘v’ the velocity, then Lorentz force acting on it due to magnetic field is given by,
Fm = -e(vxB)
Since v and B are perpendicular to each other,
Fm = -evB ---------- (1)
The electric field exerts a force on charge carriers. It is given by,
Fe = EHe --------- (2)
At equilibrium,
Fm = Fe
Ie, -evB = EHe
Or EH = -Bv -------- (3)
But current density J=nev where n is the electron density.
Or v = J/ne
Substituting this in equation (3),
EH = -BJ/ne
Or EH/BJ = -1/ne = RH --------- (4)
RH is called Hall coefficient.
If the carriers are holes,
RH = 1/pe --------- (5)
Where p is the hole density.
If ‘d’, ‘w’ and ‘A’ represent the thickness, width and area of cross section of the slab respectively, then,
EH = VH/d --------- (6)
J = I/A = I/wd --------- (7)
Substituting (6) and (7) in equation (4),
RH = EH/BJ = VHw/BI
Or VH = BIRH/w = BI/wne ------------ (8)

Measurement of Hall voltage and Hall coefficient
A rectangular slab of the given material having a thickness ‘d’ and width ‘w’ is placed between the pole pieces of an electromagnet with magnetic flux density B coinciding z-axis. A current I that coincides with X-axis is allowed to pass through the sample by connecting it to a battery.
The Hall voltage is measured by placing two probes at the centres of the bottom and top surfaces of the sample.
Hall coefficient RH = VHw/BI

Importance of Hall effect
Hall effect proved that band theory of solids is more accurate than free electron theory. Hall effect proved that electrons are the majority carriers in all the metals and n-type semiconductors. In p-type semiconductors, holes are the majority carriers.
Applications of Hall effect
To determine the type ( n-type or p-type) of semiconductors.
To determine the concentration of the carriers.
In nondestructive testing.
In Hall generators.