Derive an Expression for Hall Voltage and Hall Coefficient with Neat Labelled Diagram. - Applied Physics 1

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=nhAevd. ................................ (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
Fm=evdB .................................. (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
EH=VH/d ................................ (4)
Where VH 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)
EH=Vd8 ..................................... (5)
Using equation (4) in the equation (5)
VH=VdBd ............................ (6)
From equation (7) and (2),the drift velocity of holes
Vd=l/enhA = Jx/enh .......................... (7)
Hence,hall voltage can be written as

VH=lBd/enhA
=JxBd/enh

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)

Concept: Hall Effect
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