Derive an expression for Hall voltage and Hall coefficient with neat labelled 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'l'

starts flowing through it in x direction.

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

l=n_{h}Aevd. ................................ (J)

where nFdensity of holes

A = wxd = cross sectional area of the specimen

Vd = drift velocity of the holes.

The current density is

J=I/A =nheVd .......................... (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_{H}=V_{H}/d ................................ (4)

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

In equilibrium condition,the force due to the magnetic field 8 and the force due to the electric

field EHacting on the charges are balanced. So from the equation (3)

E_{H}=V_{d}8 ..................................... (5)

Using equation (4) in the equation (5)

V_{H}=V_{d}Bd ............................ (6)

From equation (7) and (2),the drift velocity of holes

V_{d}=l/enhA = Jx/en_{h} .......................... (7)

Hence,hall voltage can be written as

VH=lB_{d}/en_{h}A

=J_{x}Bd/en_{h}

An imporram paramerer IS rr1e hall coefficient defined as the hall field per unit current density per unit magnetic induction and is written as

`R_h = (V_hA)/(lB)`