Hall effect | Derivation, Importance of the Hall Effect

The conductivity measurements are not sufficient for the determination of the number of conducting charge and their mobility. Moreover, these measurements do not give any information about the sign of the prominent charge carrier.  The Hall Effect supplies the information on the sign of the charge carrier.
    When a magnetic field is applied perpendicular to a conductor carrying current, a voltage is developed across the specimen in the direction perpendicular to both the current and magnetic field.  This phenomenon is known as Hall Effect.

    Consider that an external electric field is applied along the axis of the specimen, and then the electrons will drift in opposite direction. When a magnetic field is applied perpendicular to the axis of the specimen, the electrons will tend to be deflected to one side. Of course, the electrons will not drift into space but a surface charge is developed.  The surface charge then gives rise to a transverse electric field which causes a compensating drift such that the carriers remain in the specimen.  This effect is known as Hall Effect. 
The Hall Effect is thus observed when a magnetic field is applied at a right angle to a conductor carrying a current.

    Consider a slab of material subjected to an external electric field Ex along the x-direction and a magnetic field Hz along the z-direction as shown in Fig. 13.  Due to the electric field, a current density Ix will flow in the direction of Ex.  Let us consider the case in which the current is carried by electrons of charge -e.  Under the influence of the magnetic field, the electron will be subjected to a Lorentz force such that the upper surface collects a positive charge while the lower surface has a negative charge.

The accumulation of charge on the surface of the specimen continues until the force on moving charges due to the electric field associated with it is large enough to cancel the force exerted by the magnetic field.

    Ultimately a stationary state is reached when the current along y-axis vanishes, and a field Ey is set up.  If the charge carriers are holes then the case will be reversed, i.e., the upper surface would become negative while the lower surface is positive.  Thus by measuring the Hall voltage in the y – direction, information about the sign of charges may be obtained.  In this way, the measurement of Hall voltage gives information about the charge carriers.

Hall voltage and Hall Co-efficient

Importance of the Hall Effect

1.   The sign of the current carrying charges is determined.

2.  From the magnitude of the Hall coefficient the number of charge carriers per unit volume can be calculated.

3.  The mobility is measured.

4. It can be used to decide whether a material is a metal, semiconductor, or insulator.

 Here we should remember that not all the metals have a negative Hall constant but some metals have a positive hall constant. (i.e., charge carriers are holes) and if both holes and electrons contribute to conductivity then RHall can be positive or negative depending upon the relative densities and mobilities of the carriers.

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