The Hall Effect

The Hall effect is due to the nature of the current in a conductor. Current consists of the movement of many small charge carriers, typically electrons, holes, ions or all three. When a magnetic field is present that is not parallel to the direction of motion of moving charges, these charges experience a force, called the Lorentz force. When such a magnetic field is absent, the charges follow approximately straight, 'line of sight' paths between collisions with impurities, phonons, etc. However, when a magnetic field with a perpendicular component is applied, their paths between collisions are curved so moving charges accumulate on one face of the material. This leaves equal and opposite charges exposed on the other face, where there is a scarcity of mobile charges. The result is an asymmetric distribution of charge density across the Hall element that is perpendicular to both the 'line of sight' path and the applied magnetic field. The separation of charge establishes an electric field that opposes the migration of further charge, so a steady electrical potential is established for as long as the charge is flowing.

When a specimen (metal or semiconductor) carrying a current I is placed in a transverse magnetic field B,then an electric field is induced in the direction perpendicular to both I and B. This phenomena is called Hall Effect.

fig.shows a semiconductor bar carrying a current I in the positive X direction. Let a magnetic field B is applied in the positive Z direction. Now a force is exerted on the charge carries (whether electrons or holes) in the negative Y direction.

Due to this force,moving charges accumulate on one face of the material. This leaves equal and opposite charges exposed on the other face, where there is a scarcity of mobile charges. The result is an asymmetric distribution of charge density across the Hall element that is perpendicular to both the 'line of sight' path and the applied magnetic field. The separation of charge establishes an electric field that opposes the migration of further charge, so a steady electrical potential is established for as long as the charge is flowing.

In equilibrium state,the electric field intensity E due to Hall effect must exerts a force on the carrier which just balances the magnetic force, i.e.,

          eE=Bev,          (1)

where e is the magnitude of charge on electron or hole and v is the drift velocity.

Now,

       E=VH/w

or      vH=E*w

since     E=B*v

hence      VH=Bvw     (2)

where w is the distance between two surfaces

We know the current density J is given by

    J=I/A=I/tw=nev

      v=I/netw             (3)

where t is the thickness of the plate and w is the distance between two surfaces

from (2) and (3),we have.

Hall Effect measurement setup for electrons. Initially, the electrons follow the curved arrow, due to the magnetic force. At some distance from the current-introducing contacts, electrons pile up on the left side and deplete from the right side, which creates an electric field ?_y. In steady-state, ?_y will be strong enough to exactly cancel out the magnetic force, so that the electrons follow the straight arrow (dashed).

where I is the current across the plate length, B is the magnetic field, t is the thickness of the plate, e is the elementary charge, and n is the charge carrier density of the carrier electrons.

The Hall coefficient is defined as

where j is the current density of the carrier electrons, and  is the induced electric field. In SI units, this becomes

(The units of R_H are usually expressed as m³/C, or O�cm/G, or other variants.) As a result, the Hall effect is very useful as a means to measure either the carrier density or the magnetic field.

One very important feature of the Hall effect is that it differentiates between positive charges moving in one direction and negative charges moving in the opposite. The Hall effect offered the first real proof that electric currents in metals are carried by moving electrons, not by protons. The Hall effect also showed that in some substances (especially p-type semiconductors), it is more appropriate to think of the current as positive "holes" moving rather than negative electrons. A common source of confusion with the Hall Effect is that holes moving to the left are really electrons moving to the right, so one expects the same sign of the Hall coefficient for both electrons and holes. This confusion, however, can only be resolved by modern quantum mechanical theory of transport in solids.