Hall Effect | Hall Effect Derivation

What is Hall Effect?

When a sample of conductor carrying, current is placed in a uniform magnetic field perpendicular to the direction of the current, a transverse field will be set up across the conductor. This was first experimentally observed by Edwik H. Hall in 1879. The field developed across the conductor is called Hall field and corresponding potential difference is called Hall voltage and its value is found to depend on the magnetic field strength, nature of the materials and applied current. This type of effect is called Hall effect. The principle of Hall effect is based on the simple dynamics of charges moving in electromagnetic fields.

Explanation of Hall Effect

To explain Hall effect, consider a sample of a block of conductor of length l, width d and thickness t, through which electric current I is supplied along x-axis as shown in figure 1. The flow of electron is in the opposite direction to the conventional current. If the conductor is placed in a magnetic field B along z-axis perpendicular to the direction of current, a force Bev then acts on each electrons in the direction from top surface to the bottom of the sample. Thus electrons accumulate on the bottom surface of the conductor which will make the surface negatively charged and top surface will be charged positively. Hence a potential difference opposes the flow of electrons.

hall-effect

The flow ceases when the potential difference across the conductor along y-axis reaches a particular value i.e. Hall voltage (VH), which may be measured by using a high impedance voltmeter. If d be the width of the slab of the sample, then the electric field or the Hall Field (EH) will be setup across the sample. Hence at equilibrium condition, the force downwards due to magnetic field will be equal to the upward electric force, i.e.

eE_H = Bev</p> <p>\dfrac{eV_H}{d} = Bev</p> <p>V_H = Bvd            (i)

Here v is drift velocity, which can be expressed by the relation

I = -nevA                         (ii)

Where n is number of electrons per unit volume and A is the area of cross-section of the conductor. The area of the cross-section in the sample is A = td. So from equation (i) and (ii) we get

V_H = -\dfrac{Bi}{net}          (iii)

We can take some typical values for copper and silicone to see the order of magnitude of VH. For copper n=1029m-3 and for Si, n = 1=25 m-3. Hence the Hall voltage at B = 1T and i=10A and t = 1 mm for copper and Silicone are, 0.6µV and 6 mV respectively. The Hall voltage is much more measurable in semiconductor than in metal i.e. Hall effect is more effective in semiconductor.

Recalling equation (iii) and expressing in terms of current density and Hall field we get,

\dfrac{E_H}{JB} = -\dfrac{1}{ne}

 

Where \dfrac{E_H}{JB} is called Hall Coefficient (RH). It is negative for free electron and positive for holes in semiconductors. In some cases, it has been found that RH is positive for metal. It also implies that the charge carriers are positive rather than negative. Hence we have

 R_H = -\dfrac{1}{ne}       (iv)

The Table below gives the Hall coefficients of a number of metals and semiconductors at room temperature with number of electrons per unit volume. The unit of RH is m3/Coulomb.

Table 1

Hall Coefficients and Number of electrons per unit volume of Materials

Materialsn(1028 m3)RH X 1011
Na2.50-25.0
K1.50-42.0
Cu11.00-5.5
Ag7.40-8.4
Al21.00-3.0
Sb0.31+2300.0
Be2.60+244.0
Zn19.00+3.3
Si1.50 X 10-7

If the steady electric field E is maintained in a conductor by applying a external voltage across it, the carriers of current attains a drift velocity v. The drift velocity acquired in unit applied electric field is known as the mobility of the carrier and is denoted by µH and is also called Hall mobility. So we have

\mu_H = \dfrac{v}{E} = \dfrac{J}{neE} = \sigma R_H = \dfrac{R_H}{\rho} (v)

Thus by measuring the resistivity of the materials and knowing the Hall coefficient, density along y-axis and current density along x-axis. This ratio is called Hall angle. Hence we have,

\dfrac{J_y}{J_x}= \sigma \dfrac{E_y}{J_x} = \mu_H B_Z = \sigma R_H B_z         (vi)

The Hall angle measures the average number of radians traversed by a particle between collisions.

Again, from equation (iii), we get

 R = \dfrac{V_H}{i} = \dfrac{B}{net}     (vii)

The quantity R has dimension of resistance, through it is not resistance in conventional sense. It is commonly called Hall resistance. From this relation it is expected to increase Hall resistance linearly with the increase of magnetic field, however, German Physicist Klaus Von Klitizing in 1980 in his experiment showed that the Hall resistance did not increase linearly with the field, instead, the plot showed a series of stair steps as shown in figure 2. Such effect has become known as the quantized Hall effect and Klaus was awarded the 1985 Nobel Prize in Physics for his discovery.

quantized-hall-effect

Application of Hall Effect

The Hall effect has many applications. It is used to accurate measurement of magnetic field, Hall mobility etc. It is also used to determine the nature of materials. It is also used to determine whether the specimen is metal, semiconductor or insulator.