State the Hall effect. Derive the expression for Hall voltage and Hall coefficient with neat diagram.

State the Hall effect. Derive the expression for Hall coefficient with neat diagram.

#### Solution

if a current carrying conductor or semiconductor is placed in a transverse magnetic field, a potential difference is developed across the specimen in a direction perpendicular to both the current and magnetic field. The phenomenon is called HALL EFFECT.

As shown consider a rectangular plate of a p-type semiconductor of width ‘w’ and thickness ‘d’ placed along x-axis. When a potential difference is applied along its length ‘a’ current ‘I’ starts flowing through it in x direction.

As the holes are the majority carriers in this case the current is given by

I = n_{h} Aev_{d} ……………………………(1)

where n_{h }= density of holes

A = w×d = crosss ectional area of the specimen

V_{d }= drift velocity of the holes.

The current density is

`J = I/ A = n_h ev_d `…………………..(2)

The magnetic field is applied transversely to the crystal surface in z direction. Hence the holes experience a magnetic force

F_{m} = ev_{d}B …………………………….(3)

In a downward direction. As a result of this the holes are accumulated on the bottom surface of the specimen.

Due to this a corresponding equivalent negative charge is left on the top surface.

The separation of charge set up a transverse electric field across the specimen given by,

`E_ = (V_H)/ d` …………………………..(4)

Where V_{H }is called the HALL VOLTAGE and E_{H} the HALL FIELD.

In equilibrium condition the force due to the magnetic field B and the force due to the electric field

E_{H }acting on the charges are balanced. So the equation (3)

`eE_H = ev_d B`

`E_H = v_d B` ……………………………….(5)

Using equation (4) in the equation (5)

`V_H = v_d B d` ………………………….(6)

From equation (1) and (2), the drift velocity of holes is found as

`v_d =I/ (en_h A) = J/(en_h)` ……………………..(7)

Hence hall voltage can be written as `V_H =(IBd)/( en. A) = ( J_xBd)/( en.)` as

An important parameter is the hall coefficient defined as the hall field per unit current density per unit magnetic induction.

`R_H = (E_H)/(J_xB)`