Hall Effect and Its Applications | Semiconductors | Electrical Engineering

If a piece of conductor (metal or semiconductor) carrying a current is placed in a transverse magnetic field, an electric field is produced inside the conductor in a direction normal to both the current and the magnetic field. This phenomenon is known as the Hall Effect and the generated electric field as the Hall field.

Consider a rectangular piece of conductor carrying a current I in the positive x-direction and subjected to a magnetic flux density b in the positive z-direction (Fig. 7.22). The current carriers will experience a Lorentz force in the negative y-direction. As a result, the carriers are deflected towards the bottom surface of the sample and are accumulated.

If the current carriers are electrons, as in the case of an n-type semiconductor, this accumulation will make the bottom surface negatively charged with respect to the top surface. Therefore, an electric field, called the hall field, will be developed along the -ve y-direction.

The force on the current carriers due to this Hall field will oppose the Lorentz force. An equilibrium is established when these two forces balance each other. At this stage, no further accumulation of electrons takes place on the bottom surface and the Hall field reaches a steady value.

If the current carriers are holes, i.e., when the conductor is a p-type semiconductor the accumulation of carriers on the bottom surface will make this surface positively charged relative to the top surface. In this case, the Hall field is produced along the positive y-direction. The force on the holes due to the Hall field opposes the Lorentz force and balances it under equilibrium conditions preventing further accumulation of holes. The Hall field then attains its steady value.

If E_H is the Hall field in the y-direction, the force due to this field on a carrier of charge e is eE_H. The average Lorentz force on a carrier is evB, where v is the drift velocity in the x-direction. In equilibrium these two forces balance, i.e.,

eE_H = evB

If J is the current density in the x-direction, we have;

J = n_cev

where n_c is the concentration of current carriers. From equation (i) and (ii) we get;

E_H = BJ/n_c e

The Hall Effect is described by means of the Hall coefficient R_H defined in terms of the current density J by the relation;

E_H = R_H J B … (xxv)

∴ We obtain R_H = 1/n_c e … (xxvi)

A vigorous treatment shows that R_H is actually given by;

R_H = r/n_c e

where r is a numerical constant. In most of the cases the value of r does not differ much from unity. The error occurring from setting r = 1 is, therefore, not large.

If the current carriers are electrons, the charge on the carrier is negative and hence;

R_H = 1/ne … (xxvii)

where n represents the concentration of electrons.

If the current carrier are holes, the carrier charge is positive and therefore,

R_H = 1/pe … (xxviii)

where p represents the hole concentration.

Thus, the sign of the Hall coefficient tells us whether the sample is an H-type or a p-type semiconductor.

The Hall coefficient is determined by measuring the Hall voltage that generates the Hall field. If V_H is the Hall voltage across a sample of thickness d, then

V_H = E_H d

Using equation (xxv), we can write;

V_H = R_H J B D … (xxix)

If w is the width of the sample then its cross-sectional area is dw and the current density J is given b;

J = 1/dw

where I is the current flowing through the sample.

From equations (xxix) and (xxx), we get;

V_H = R_H IB/w

Hence R_H = V_H w/IB … (xxxi)

The polarity of V_H will be opposite for n-type and p-type semiconductors. Therefore, the sign of R_H will be different for the two types of semiconductors.

It will be noticed that the net electric field in the semiconductor, which is the vector sum of applied field (let it be E_x) and Hall field E_H, is not directed along the axis but it is some angle to it, known as Hall angle and is given by-

Thus, a simultaneous measurement of R₁₁ and σ can lead to an experimental value for the carrier drift mobility. We have assumed that the conduction process is by means of one type of carrier only. In some semiconductors, e.g., extrinsic material, the assumption is not valid and the Hall coefficient must be modified to account for the presence of two types of charge carriers.

Both types of carriers will drift under the influence of the applied field E_x with drift velocities v_h = _h E_x in the +ve x-direction and v_e = _eE_x in the -ve x-direction using the same co-ordinate system as in Fig. 7.22. As a consequence, the holes and electrons each experience a Lorentz-force given by-

F_h = e(v_h × B) = -e v_h B_z

and F_e = e(v_e × B) = -e v_e B_z

Both deflecting forces are in the same direction, deflecting electrons and holes to the front face, as before. The carriers recombine at the surface and the net charge there produces an electric field in the y-direction, E_y, i.e., E_H. In equilibrium, the current due to the deflected electrons and holes must be exactly cancelled out by the current flowing in the opposite direction due to the Hall field. If the transverse velocities of the deflected carriers are v_hy and v_ey, the current in the transverse field is (epv_hy – env_ey) and, therefore, in equilibrium;

σE_H = e(pv_hy – nv_ey)

Expression for the transverse velocities can be obtained from the Hall angle equations, which gives;

The generalized version of Ohm’s law is used to eliminate E_x from this equation, which can then be rearranged to give the Hall coefficient;

Finally, we include the general expression for the conductivity σ = e(p _h + n _e) to give;

Comparison of this expression with equations (xxvii) and (xxviii) shows that the Hall coefficient and hence the Hall voltage are generally smaller for near intrinsic materials than for the more highly doped extrinsic materials.

Applications of Hall Effect:

1. Determination of Semiconductor Type:

For an n-type semiconductor the Hall coefficient is -ve whereas for a p-type semiconductor it is +ve. Thus, the sign of the Hall coefficient can be used to determine whether a given semiconductor is n or p-type.

2. Determination of Carrier Concentration:

By measuring the Hall coefficient the carrier concentration of a semiconductor can be determined from the relations;

3. Determination of Mobility:

If the conduction is due to one type of carrier, e.g., electrons we have;

σ = n e μ

where is the mobility of the electrons.

Using equation (xxvii) we get;

μ = σ │R_H│ … (xxxii)

i.e., knowing σ and determining R_H, the mobility can be determined. Because of the presence of the term r in R_H = r/n_ce the mobility determined from the Hall coefficient will be different from the mobility defined by = v/E. The mobility as determined from equation (xxxii) is, therefore, called the Hall mobility to distinguish it from the actual drift mobility. However, since the factor r is not much different from unity, the two mobilities are nearly the same.

4. Measurement of Magnetic Flux Density:

Since the Hall voltage V_H is proportional to the magnetic flux B for a given current I through a sample, the Hall Effect can be used as the basis for the design of a magnetic flux density meter.

5. Measurement of Power in an em Wave:

In an electromagnetic wave in free space, the magnetic field H and the electric field E are the at right angles. Thus, if a semiconductor is placed parallel to E it will drive a current I in the semiconductor. The semiconductor is subjected simultaneously to a transverse magnetic field H producing a Hall voltage across the sample. The Hall voltage will be proportional to the product E_H, i.e., to the magnitude of the pointing vector of the em wave. Thus, Hall Effect can be used to determine the power flow of an electromagnetic wave.

6. Hall Effect Multiplier:

If the magnetic flux density B is produced by passing a current I₁ through an air core coil, B will be proportional to I₁. The Hall voltage is thus proportional to the product of I and I₁. This forms the basis for a multiplier.