Question

Derive an expression for Hall voltage and Hall coefficient with neat labelled diagram.

Answer 1

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_hAevd. ................................ (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_dB .................................. (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_d8 ..................................... (5)
Using equation (4) in the equation (5)
V_H=V_dBd ............................ (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_hA
=J_xBd/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)`