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Explain Hall effect & its significance. With a neat diagram derive the expression for the hall voltage & Hall co-efficient.
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Hall effect & its significance:

• A metal or semiconductor carrying a current I placed in a transverse magnetic field B, an electric field E is induced in the direction perpendicular to both current and magnetic field. This phenomenon is called as Hall effect and the electric field or voltage induced is called Hall voltage (VH).
• 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 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.
• In equilibrium condition, electric field intensity EH due to Hall effect exerts a force on charge carriers which balances the magnetic force.
q EH= B qv
where q is the magnitude of charge carrier and v is the drift speed.
• EH = VH /d where d is the distance between upper and lower surfaces. If J is the current density, ne is the charge density and w is the width of the specimen in the direction of the magnetic field then, J = nev = I / wd, Thus VH = EH d = Bvd = BJd / ne = BI / new ... (1)
As EH = VH / d = Bv = BJ / ne
now J = nev = I / A
$\sigma \ E$Also, J =
$\sigma = J / E = I / A. 1 / E$
$E_H= \dfrac {BJ}{ne}=\dfrac {B}{ne}\dfrac {I}{A}=\dfrac {B}{ne}.\dfrac {I}{A}.\dfrac {E}{E}$

Hence,$\dfrac {E_H}{E}=\dfrac {B}{ne}.(\dfrac {I}{A}.\dfrac {1}{E})$

$\dfrac {E_H}{E}=\dfrac {B}{ne}.\sigma=\dfrac {B}{ne\rho}$
where$\rho = \dfrac1 \sigma, \ hence \dfrac {E_H}{E} = \dfrac {B} {ne\rho} \cdots (2)$ * By measuring$V_H, I, B \ and \ w, the \ charge$ density ne can be calculated from equation (1). The Hall coefficient is defined as$R_H= \dfrac 1 {ne}=\dfrac {V_H. w}{BI} \cdots (3)$

• If conduction through the specimen is due to single type of charge carriers, the conductivity$\sigma$ and mobility$\mu$ are related as$\sigma = \mu \ ne$
• If the conductivity and Hall coefficient are measured, then mobility can be determined as$\mu = \sigma \ R_H$
• Since$V_H= \dfrac {BI}{new}$
$= \dfrac {BI.d}{new d}=\dfrac {BI.d}{ne \ A} =\dfrac {BJd}{ne}$

• Hall field/unit current density/unit magnetic induction is called Hall coefficient$R_H$.
Hence,$R_H=\dfrac {E_H}{JB}=\dfrac {V_H/d}{JB}$
Substitute,$V_H = \dfrac {BJd}{ne}$

Hence,$R_H=\dfrac {BJd}{ne.JB.d} = \dfrac 1 {ne}$

• For given directions of B and current, the sign of the Hall voltage is positive. For an n-type semiconductor, the Hall voltage will be negative for the same setup, hence the sign of the majority carrier can be found out.

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