# What is a Hall Effect in Metals and Semiconductor

Hall Effect was introduced by an American Physicist Edwin H.Hall in the year 1879. It is based on the measurement of the electromagnetic field. It is also named as ordinary Hall Effect. When a current-carrying conductor is perpendicular to a magnetic field, a voltage generated is measured at right angles to the current path. Where current flow is similar to that of liquid flowing in a pipe. Firstly it was applied in the classification of chemical samples. Secondly, it was applicable in Hall Effect Sensor where it was used to measure DC fields of the magnet, where the sensor is kept stationary.

## Principle of Hall Effect

Hall Effect is defined as the difference in voltage generated across a current-carrying conductor, is transverse to an electrical current in the conductor and an applied magnetic field perpendicular to the current.

**Hall Effect = induced electric field / current density * the applied magnetic field** –(1)

### Theory of Hall Effect

Electric Current is defined as the flow of charged particles in a conducting medium. The charges that are flowing can either be Negative charged – Electrons ‘e- ‘/ Positive charged – Holes ‘+’.

**Example**

Consider a thin conducting plate of length L and connect both ends of a plate with a battery. Where one end is connected from the positive end of a battery to one end of the plate and another end is connected from the negative end of a battery to another end of the plate. Now we observe that currently starts flowing from negative charge to the positive end of the plate. Due to this movement, a magnetic field is generated.

### Lorentz Force

For instance, if we place a magnetic bare nearby the conductor the magnetic field will disturb the magnetic field of charge carriers. This force which distorts the direction of charge carriers is known as Lorentz force.

Due to this, the electrons will move to one end of the plate and holes will move to another end of the plate. Here Hall voltage is measured between two sides of plates with a multimeter. This effect is also known as the Hall Effect. Where the current is directly proportional to deflected electrons in turn proportional to the potential difference between both plates.

Larger the current larger is the deflected electrons and hence we can observe the high potential difference between the plates.

Hall Voltage is directly proportional to the electric current and applied magnetic field.

**VH = I B / q n d** —– ( 2 )

I – Current flowing in Sensor

B – Magnetic Field Strength

q – Charge

n – charge carriers per unit volume

d – Thickness of the sensor

### Derivation of Hall Coefficient

Let current IX is current density, JX times the correctional area of the conductor wt.

**IX = JX wt = n q vx w t** ———-( 3 )

According to Ohms law, if current increases the field also increases. Which is given as

**JX = σ EX** , ————( 4 )

Where σ = conductivity of the material in the conductor.

On considering the above example of placing a magnetic bar right angle to the conductor we know that it experience Lorentz force. When a steady state is reached there will be no flow of charge in any direction which can be represented as,

**EY = Vx Bz**, ————–( 5 )

EY – electric field / Hall field in the y-direction

Bz – magnetic field in the z-direction

VH = – ∫0w EY day = – Ey w ———-( 6 )

VH = – ( (1/n q ) IX Bz ) / t , ———– ( 7 )

Where RH = 1/nq ———— ( 8 )

Units of Hall Effect: m3 /C

**Hall Mobility**

**µ p or µ n = σ n R H** ———— ( 9 )

Hall mobility is defined as µ p or µ n is conductivity due to electrons and holes.

**Magnetic Flux Density**

It is defined as the amount of magnetic flux in an area taken right angles to the magnetic flux’s direction.

**B = VH d / RH I** ——– ( 1 0 )

### Hall Effect in Metals and Semiconductor

According to the electric field and magnetic field the charge carriers which are moving in the medium experience some resistance because of scattering between carriers and impurities, along with carriers and atoms of material which are undergoing vibration. Hence each carrier scatters and loses its energy. Which can be represented by the following equation

**F retarded**** = – mv/t** , ————– ( 1 1 )

t = average time between scattering events

According to Newtons seconds law ,

**M (dv/dt )= ( q ( E + v * B ) – m v) / t** ——( 1 2 )

m= mass of the carrier

When a steady state occurs the parameter’ v ‘ will be neglected

If ’B’ is along z-coordinate we can obtain a set of ’ v ‘ equations

**vx = ( qT Ex) / m + (qt BZ vy ) / m** ———– ( 1 3 )

**vy = (qT Ey ) / m – (qt BZ vx) / m** ———— ( 1 4 )

**vz = qT Ez / m** ———- ( 1 5 )

We know that **Jx = n q vx** ————— ( 1 6 )

Substituting in the above equations we can modify it as

**Jx = ( σ/ ( 1 + (wc t)2)) ( Ex + wc t Ey )** ———– ( 1 7 )

**J y = ( σ * ( Ey – wc t Ex ) / ( 1 + (wc t)2**) ———- ( 1 8 )

**Jz = σ Ez** ———— ( 1 9 )

We know that

**σ n q2 t / m** ———– ( 2 0 )

σ = conductivity

t = relaxation time

and

**wc q Bz / m** ————– ( 2 1 )

**wc = cyclotron frequency**

Cyclotron Frequency is defined as in a magnetic field frequency of rotation of a charge. Which is the strength of the field.

Which can be explained in the following cases to know if it is not strong and/or “t” is short

Case (i) : If wc t << 1

It indicates a weak field limit

Case (ii) : If wc t >> 1

It indicates a strong field limit.

### Advantages

The advantages of the hall-effect include the following.

- Speed of operation is high i.e, 100 kHz
- Loop of operations
- Capacity to measure large current
- It can measure Zero speed.

### Disadvantages

The disadvantages of the hall-effect include the following.

- It cannot measure the flow of current greater than 10cm
- There is a large effect of temperature on carriers, which is directly proportional
- Even in the absence of a magnetic field small voltage is observed when electrodes are at centered.

### Applications of Hall Effect

The applications of the hall-effect include the following.

- Magnetic Field senor
- Used for multiplication
- For direct current measuring, it uses Hall Effect Tong Tester
- We can measure phase angles
- We can also measure Linear displacements transducer
- Spacecraft propulsion
- Power supply sensing

Thus, the Hall Effect is based on the Electro-magnetic principle. Here we have seen the derivation of Hall Coefficient, also Hall Effect in Metals and Semiconductors. Here is a question, How is Hall Effect applicable in Zero speed operation?