Hall Effect in Semiconductor

In this lecture, we are going to learn about the Hall effect, the principle of the Hall effect, the theory & derivation of the Hall effect, the derivation of the Hall coefficient, and the application of the Hall effect in a very detailed manner. So let’s start with the knowledge of the principle of the Hall effect.

Hall Effect Definition

• If a specimen ( metal or semiconductor) carrying a current I is placed in a transverse magnetic field B, an electric field E is induced in the direction of perpendicular to both I and B. This phenomenon is known as Hall Effect and the generated voltage is called Hall voltage.

Hall Effect Derivation

• Consider a rectangular slab of an n-type semiconductor material that carries a current I along the positive X-direction, as shown in the figure below.

• In an n-type semiconductor, electrons are the majority carriers.

• Let a 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,

F_L=-Bev_d ……(1)

Where e is the magnitude of the charge of the electrons and V_d is the drift velocity.

• 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.

• 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.

• This electric field exerts an upward force on the electron and is given by,

F_E=-eE_H ……(2)

• At equilibrium, the Lorentz force and electric force get balanced. Hence,

F_E=F_L

• Therefore, from equations (1) and (2)

-eE_H=-Bev_d

or E_H=Bv_d ……(3)

• If b is the width (i.e., the distance between the top and bottom surface) of the specimen, then

E_H=\frac{V_H}{b} ……(4)

or V_H=E_Hb …….(5)

and V_H=Bv_db ……(6)

• Let t be the thickness of the specimen along the Z-direction. Therefore, its area of cross-section normal to the direction of current is bt.

• If J is the current density then,

J=\frac{I}{bt} ……(7)

• But J can also be expressed as

J=-n_eev_d ……(8)

where n_e is the density of electrons.

\therefore \; v_d=\frac{J}{n_ee} ……(9)

• Hence, substituting equation (9) in (6),

V_H= -Bb\; \frac{J}{n_ee} ……(10)

• But V_H is also equal to E_Hb

\therefore \; E_Hb= -Bb\;\frac{J}{n_ee}

or E_H = \frac{BJ}{n_ee} ……(11)

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Also Read

• Zener Diode – Explanation, Working, Applications, Circuit Symbol
• Josephson Effect – Definition, Application, AC and DC Effect
• SOLAR CELL – Photovoltaic Cell

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Hall Coefficient formula

• The Hall Effect is described by means of the Hall Coefficient R_H. It is given by,

R_H=\frac{1}{ne}

where n is, in general, the carrier concentration.

R_H for n-type and p-type material

• A negative sign is used while denoting the Hall coefficient for an n-type material, i.e., it is given by

R_H=-\frac{1}{ne} ……(12)

where n_e is the density of electrons.

• But for p-type material, a positive sign is used to denote the Hall coefficient, i.e., it is given by

R_H=\frac{1}{ne} ……(13)

where n_h is the density of holes.

• Therefore, equation (11) can be written as

E_H=BJR_H

\therefore \; R_H=\frac{E_H}{JB} ……(14)

• But we Know E_H=\frac{V_H}{b} \;and \; J = \frac{I}{bt}. Hence equation (14) becomes,

R_H=\frac{V_Hbt}{IBb}

\therefore \; \boxed{\mathbf{R_H=\frac{V_Ht}{IB}}} ……(15)

• Since the quantities V_H, t, I, and B are measurable, the Hall coefficient R_H can be determined.

Mobility Determination from Hall Effect

• For n-type material the conductivity is given by,

\sigma_e=n_ee\mu_e

where \mu_e is the mobility of electrons.

\therefore \; \mu_e=\frac{\sigma_e}{n_ee} ……(16)

or \therefore \; \mu_e=-\sigma_e R_H ……(17)

• Similarly, for p-type material, the conductivity is given by,

\sigma_h=n_he\mu_h

where \mu_h is the mobility of holes.

\therefore \; \mu_h=\frac{\sigma_h}{n_he} ……(18)

or \therefore \; \mu_h=-\sigma_h R_H ……(19)

• In the above discussion, it is assumed that all the charge carriers travel with average velocity. But actually, the charge carriers have a random thermal distribution in velocity.

• With this distribution taken into consideration, R_H is defined in general as,

R_H = \frac{3\pi}{8ne}\;=\frac{1.18}{ne} ……(20)

Therefore, equations (16) and (18) can be written as,

\boxed{\mathbf{\mu_e \;= \frac{-\sigma_eR_H}{1.18}}} ……(21)

\boxed{\mathbf{\mu_h \;= \frac{\sigma_hR_H}{1.18}}} ……(22)

Application of Hall Effect

• The Hall effect can be used for:

1. Determining whether a semiconductor is n-type or p-type.
2. Determining the carrier concentration and mobility.
3. Determining the magnetic field B in terms of Hall voltage V_H.
4. Designing the Gauss meter and electronics meters based on Hall voltage.

Practice problems

Q. Calculate the Hall voltage when a conductor carrying a current of 100 A, is placed in a magnetic field of 1.5 T. The conductor has a thickness of 1 cm, and the number density of charges inside the conductor is 5.9 ×10²⁸ /m3.

A.

I=100 A,\; B=1.5 T,\; n= 5.9 \times10^{28} /m^3,\;t=1 cm

V_H=\frac{IB}{n e t} = \frac{100 \times 1.5}{ (5.9 \times10^{28})(1.6 \times 10^{-19}) (10^{-2}) }

V_H = 15.89 \times 10^{−7} V

Frequently Asked Questions on Hall Effect

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