# 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 Vd 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 VH 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 ne 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 VH is also equal to EHb

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

or E_H = \frac{BJ}{n_ee} ……(11)

## Hall Coefficient formula

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

R_H=\frac{1}{ne}

where n is, in general, the carrier concentration.

### RH 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 ne 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 nh 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 VH, t, I, and B are measurable, the Hall coefficient RH 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, RH 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 VH.
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

1. ### Name one practical use of the Hall effect.

Hall effect is used to determine if a substance is a semiconductor or an insulator. The nature of the charge carriers can be measured.

2. ### How is Hall’s potential developed?

When a current-carrying conductor is in the presence of a transverse magnetic field, the magnetic field exerts a deflecting force in the direction perpendicular to both magnetic field and drift velocity. This causes charges to shift from one surface to another thus creating a potential difference.

3. ### What is a Hall effect sensor?

A Hall effect sensor is a device that is used to measure the magnitude of a magnetic field.

4. ### In the Hall effect, the direction of the magnetic field and electric field are parallel. True or False?

False. The magnetic field and electric field are perpendicular to each other.

Hello friends, my name is Trupal Bhavsar, I am the Writer and Founder of this blog. I am Electronics Engineer(2014 pass out), Currently working as Junior Telecom Officer(B.S.N.L.) also I do Project Development, PCB designing and Teaching of Electronics Subjects.

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