The Hall Effect

The magnetic force on a moving charge is given by the vector multiple of charge with the vector product of velocity and magnetic field: F = q.v x B Sinɵ. Such magnetic force is perpendicular to both the velocity of the charge and the magnetic field and its direction is given by the right hand rule.


If a metal slab of rectangular cross-section carrying a current Jx in +X direction is placed in uniform magnetic field Bz, in z-direction, an emf develops along the y-axis (perpendicular to the current as well as the magnetic field). This voItage is called Hall Voltage and this phenomenon is called Hall Effect.


In the absence of magnetic field the current flowing in +X direction means electrons drifting with velocity vx in negative X direction. The magnetic field creates Lorentz force FL (= e.vx.Bz), which causes the electrons to bend downwards and accumulate on lower surface of the conductor, producing a net negative charge there. Simultaneously a net positive charge appears on the upper surface. This combination of +ve and –ve charges creates a downward field called Hall Field FH.


The field created by the surface charges produces a force which opposes this Lorentz force. The accumulation process continues till the Hall force completely cancels the Lorentz force. Thus in steady state FH = FL.

So e.EH = e.vx.Bz or EH = vx.Bz 
The current density    Jx = – n.e.vx
 Divide the two equations to get     (EH / Jx) = – (Bz / n.e)

 Thus the electrical field produced by Hall Effect is: EH = – (1 / n.e) Bz.Jx

 

The polarity of Hall voltage depends on whether the conduction is by the electrons or by the holes. Thus the measurement of Hall voltage helps us to find the sign of predominant charge carrier, the charge density and the mobility of charge carriers.

The same treatment is valid for semiconductors where the majority of charge carriers is either the electrons (e) or the holes (h). If the semiconductor has both the electrons and holes in it, and they move in opposite direction in an external electric field, the Lorentz magnetic force (F = q.v.B) will deflect them in the same direction, and the current jointly produced by them will be: J = e.Ex ( nμe + p μh)

The value of Hall constant for n-type and for p-type semiconductors is given by
      RH = – (1 / n.e) for n-type and RH = – (1 / p.e) for p-type.
      And the two-carrier Hall mobility is            μH= (p.μh2 – n.μe2) / (n.μe+ p.μh)

For an intrinsic semiconductor, where n = p = ni, the above eqn. becomes: μH = μh – μe Usually μe > μh Hence μH is negative for an intrinsic (undoped) semiconductor. The sign of Hall coefficient is determined by the carriers with larger mobility. And if an intrinsic p- type goes over to extrinsic type, the sign of Hall coefficient changes.