Hall effect BY :HEBA BAKRY

IntroductionThe Hall effect describes the behavior of the free carriers in a semiconductor when applying an electric as well as a magnetic field.

One important characterization tool in the measurement of the Halleffect to measure mobilities and carrier concentrations in a givensemiconductor material. In this experiment, we will make suchmeasurements, observing the Hall effect for one semiconductor sample.

The temperature dependence of the Hall coefficient and resistivity willallow us to distinguish between p-type and n-type samples, and tocalculate the electron and hole mobilities and concentrations.

Hall effect Is a phenomenon that occurs when a conductor or semiconductor isplaced in the magnetic field and a voltage is applied through the materialperpendicular to the magnetic field. While this voltage induces currentflow along the electric field direction, the charge carriers also experiencea magnetic deflection from their path. This results a separation of positiveand negative carriers, and thus the generation of an electric fieldperpendicular to the direction of current flow. Note that, at sufficienttemperature, the net current in a semiconductor is made up ofcounteracting currents of p-type and n-type carriers

The Hall effect describes the behaviorof the free carriers in a semiconductorwhen applying an electric as well as amagnetic field. The experimental setupshown in Figure, depicts asemiconductor bar with a rectangularcross section and length L. A voltageVx is applied between the twocontacts, resulting in a field along thex-direction. The magnetic field isapplied in the z-direction.

the holes move in the positive x-direction. Themagnetic field causes a force to act on themobile particles in a direction dictated by theright hand rule. As a result there is a force, 𝐹𝑌along the positive y-direction, which moves theholes to the right. In steady state this force isbalanced by an electric field, 𝐸𝑌, so that there isno net force on the holes. As a result there is avoltage across the sample, which can bemeasured with a high impedance voltmeter.This voltage, 𝑉𝐻 , is called the Hall voltage.Given the sign convention as shown in Figure ,the hall voltage is positive for holes.

The electrons travel in the negative X-direction. Therefore the force, 𝐹𝑌, is in the positive Y-direction due to the negative charge and the electrons move to the right, just like holes. The balancing electric field, 𝐸𝑌, now has the opposite sign, which results in a negative Hall voltage.

Hall effect experimentConsider a current-carrying conductor in the form of a rectangular strip is placed in a magnetic field with the lines of force at right angles to the current, a transverse e.m.f. – the so called Hall voltage – is set up across the strip. This phenomenon is due to the Lorentz force: the charge carriers which give rise to the current flow through the specimen are deflected in the magnetic field B as a function of their sign and of their velocity v:

𝐹𝐵 = 𝑞𝐵𝑉𝑑 …(1)

Where 𝑉𝑑 is Drift velocity which is the average velocity in the direction of an applied electric field of the all conducting charge carriers in the sample

As more and more carriers are deflected, the accumulation of charge produces a “Hall field” EH that exerts a force opposite to the Lorentz force.

So the electric force:

𝐹𝐸 = 𝑞𝐸 …(2)

Equilibrium is reached when these two opposing forces are equal in magnitude, which allows us to determine the drift speed (v)

𝑞𝐵𝑉𝑑 = 𝑞𝐸

∴ 𝑉𝑑 =𝐸

𝐵=

𝑉𝐻

𝐵𝑑

but the current in the conductor is given by…

I=n e Vd A

so Vd=𝑰

𝑨 𝒏𝒆…(2)

From equation (1) & (2) we get that…

𝑩𝑰

𝑨𝒏𝒆=

𝑽𝑯

𝒅…(3)

And it is customary to define the “Hall coefficient” (RH) in terms of the

measured quantities: 𝑹𝑯 =𝟏

𝒏𝒆

From equation (3) we get that…

𝑽𝑯

𝑰=

𝑩𝒅

𝑨𝒏𝒆= 𝑹

As R is the resistance of the rectangular strip

But 𝑹 = 𝝆𝒅

𝑨∴ 𝝆 =

𝑩

𝒏𝒆Where ρ is the resistivity

And 𝝈 =𝒏𝒆

𝑩Where σ is the conductivity

And from this equation we can Determine the mobility of the charge carriers which given by

𝝁 =𝝈

𝒏𝒆

Consider a semiconductor sample placedin the magnetic field B that is in the z-direction (refer to figure). Suppose wepass a current I through that sampleperpendicular to the magnetic field, say inthe x-direction. Then the "free" electronsand "holes" in the sample experience aforce given by the Lorentz equation,

(1)Hall Effect Geometry

Applications of Hall effect(1) Determination of N-type of semiconductor

For a N-type semiconductor, the Hall coefficient is negative whereas for a P-type semiconductor, it is positive. Thus from the direction of the Hall voltage developed, one can find out the type of semiconductor.

(2) Calculation of carrier concentration

Once Hall coefficient RH is measured, the carrier concentration can be obtained,

HH eRpor

eRn

1)(

1

(3) Determination of mobility

We know that, conductivity, = e q (or)

Thus by measuring `’ and RH, ’ can be calculated.

(4) Measurement of magnetic flux density.

Using a semiconductor sample of known `RH’, the magnetic flux density can be deduced from,

Hp

p

n Rpe

.

IR

tVB

H

H