Hall Effect: Theory, Definition, Formula, Derivation & Applications

---

Principle

When a current carrying semiconductor is placed in a transverse magnetic field, a voltage is induced in the semiconductor direction perpendicular to both the current and magnetic field. This phenomenon is called Hall effect. The induced voltage is called Hall voltage (V_H). It was discovered by Hall in 1879.

Figure 1. Schematic diagram of the Hall effect experiment.

Theory

Consider a rectangular slab of an n-type semiconductor material of width (w), thickness (t) and that carries a current I along the positive X-direction and magnetic field B be applied along the positive Z-direction. Under the influence of this magnetic field, the electron experience a force called Lorentz force given by

\[ \vec{F}_M = q (\vec{v}_d \times \vec{B}_z) \tag{1} \]

This Lorentz force is exerted on the electrons in the negative Y-direction. The direction of this force is given by Fleming’s left-hand rule. Thus, the electrons are, therefore, deflected downwards and collect at the bottom surface of the specimen.

\[ \vec{F}_M = -e (\vec{v}_d \times \vec{B}_z) \]

\[ \vec{F}_M = -e (\vec{v}_d \vec{B}_z) \sin 90^\circ \]

\[ \vec{F}_M = -e \vec{v}_d \vec{B}_z \tag{2} \]

On the other hand, the top edge of the specimen becomes positively charged due to the loss of electrons. Hence, a potential called the Hall voltage V_H is developed between the upper and lower surfaces of the specimen, which establishes an electric field E called the Hallfield across the specimen in the negative Y-direction

The force acts on the charges in the presence of generated electric field,

\[ \vec{F}_E = -e \vec{E}_H \tag{3} \]

Under the equilibrium condition, the force due to electric component will counter balance that of magnetic force.

\[ F_E = F_M \] \[ v_d = \frac{E_H}{B_z} \tag{4} \]

The current density the semiconductor will be,

\[ J_x = -ne v_d \tag{5} \]

Substitute v_d value in Eqn 5,

\[ J_x = \frac{-ne E_H}{B_z} \tag{9} \]

Where \( R_H = - 1/ne \) is the Hall co-efficient.

We can write \( R_H \) as

\[ R_H = \frac{E_H}{J_x B_z} = \frac{V_H}{t J_x B_z} \tag{11} \]

where \( E_H = \frac{V_H}{t} \)

We know current density \( J_x = \frac{I}{A} = \frac{I}{(w \times t)} \). Hence \( R_H \) will be

\[ R_H = \frac{V_H \times w \times t}{t \times B_z \times I} \tag{12} \] \[ R_H = \frac{V_H \times w}{I \times B_z} \tag{13} \]

If \( R_H \) is positive, it is a p-type semiconductor. If \( R_H \) is negative, it is an n-type semiconductor.

Applications of Hall effect

1. To asses the type of semiconductor, whether it is a p-type or n-type.
2. The evaluation of concentration of charge carriers.
3. The calculation of mobility of charge carriers. \[ \sigma = n e \mu; \mu = \sigma R_H \]
4. To measure the strength and direction of magnetic field.