Semiconductor Physics Hall Effect

1. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Introduction:
Materials are classified as conductors, semiconductors and insulators on the
basis of conductivity and resistivity.
Semiconductor acts as an insulator at absolute zero and as a conductor at high
temperatures and in the presence of impurities.
Properties of a semiconductor:
 Resistivity of a semiconductor is in the order of 10-4 to 0.5 ohm-metre.
 AT 0 K, it behaves like an insulator.
 When the temperature is raised or when impurities are added, the conductivity
increases. (This is converse to conductors where conductivity decreases with
raise in temperature)
 Semiconductors have negative temperature co-efficient of resistance.
 In conductors, the electrons are the charge carriers and will take part in
conduction. But, in semiconductors both electrons and holes are charge carriers
and will take part in conduction. (Hole is produced by the vacancy of electrons)

2. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Energy bands in solids:
Millions of atoms constitute a solid. In every atom electrons are distributed in
different orbits called shells. The electrons first occupy the lower energy shells
and then distributed to higher energy shells. The distribution is governed by
Pauli’s Exclusion principle.
Energy levels of an atom is changed into energy band in a solid. The group of
electrons in a particular shell forms an energy band.
The electrons in the outer most orbit, which may not be fully filled are called
valence electrons. These valence electrons constitute the valence band.
The next energy band called conduction band is formed at the next permissible
higher energy level, with a difference in energy level equal to ΔE above the
valence band i.e., there exists an energy gap ΔE between the valence band and
the conduction band. This is called forbidden energy gap or band gap.

3. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Conductors, Insulators and Semiconductors:
 In conductors (Ex. Metals), the conduction and valence band overlaps each
other and has no forbidden energy gap. Hence, ΔE = 0.
 In insulators, the forbidden energy gap is very large. ΔE = 5 eV or more. (For
Diamond ΔE = 7 eV). Hence it is not possible for electron to jump from valence
band to the conduction band. Since the conduction band cannot have free
electrons, insulators behave as bad conductor of electricity.
 In semiconductors, the forbidden energy is less than 2 eV. The energy
provided by heat at room temperature is enough to lift the electrons from the
valence band to the conduction band. Therefore, at room temperature,
semiconductors are capable of conducting some electric current. (For
silicon, forbidden energy gap is 1.1 eV and for germanium 0.72 eV).

4. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Elemental and Compound Semiconductors:
Based on the composition, semiconductors are classified as Elemental and
Compound Semiconductors.
Elemental Semiconductors:
 They are made of single element. Ex. Si, Ge
 They are called Indirect band gap semiconductors. Electron-hole
recombination takes place through traps, which are present in the band gap
 Here heat is produced through recombination. Used for the making diodes
and transistors
Current amplification is more
Life time of charge carrier is more
Compound Semiconductors:
 They are made compounds. Ex. GaAs, GaP, InP, CdS
 They are called Direct band gap semiconductors. Electron-hole recombination
takes place directly with each other
 Here heat is produced through recombination. Used for the making LEDs,
LASER diodes and ICs
Current amplification is less
Life time of charge carrier is less

5. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Types of Semiconductors:
Based on the purity, semiconductors are classified as Intrinsic and Extrinsic type
Semiconductors
Intrinsic Semiconductors:
 A semiconductor in a extremely pure form is called intrinsic semiconductor.
 As there are no impurities, the number of free electrons must be equal to the
number of holes.
 They have low electrical conductivity.
 At 0 K, the Fermi level exactly lies between the conduction band and valence
band.
Extrinsic Semiconductors:
 Semiconductor which is doped with impurity is called extrinsic semiconductor.
The process of adding pentavalent or trivalent impurities to a pure or intrinsic
semiconductor is doping (respectively called N-type and P-type semiconductor)
 Due to the presence of impurities, the number of free electrons will not be
equal to the number of holes.
 Doping reduces the band gap. So, they have high electrical conductivity.
 At 0 K, the Fermi level lies closer to the conduction band in N-type
semiconductor and lies near valence band in P-type semiconductor.

6. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
N-type and P-type Semiconductors:
Based on the type of impurity added, extrinsic N-type and P-type semiconductors
N-type Semiconductor:
 It is obtained by doping an intrinsic semiconductor with pentavalent (5 valence
electrons) impurity atoms like phosphorous, arsenic, antimony etc.,
 So, the number of free electrons is more than the number of holes
 Majority carriers are electrons and minority carrier are holes
 Since electrons are donated in this type of semiconductor, the energy level of
these donated electrons is called donor energy level (Ed). Ed is very close to the
conduction band. Also, Fermi level lies exactly between donor energy level and
conduction band.
P-type Semiconductor:
 It is obtained by doping an intrinsic semiconductor with trivalent (3 valence
electrons) impurity atoms like boron, gallium, indium, etc.,
 So, the number of hole is more than the number of free electrons
 Majority carriers are holes and minority carrier are electrons
 Since electrons are accepted in this type of semiconductor, the energy level of
these accepted electrons is called acceptor energy level (Ea). Ea is very close to
the valence band. Also, Fermi level lies exactly between acceptor energy level
and valence band.

7. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Fermi Energy level:
Fermi energy level is the energy level which distinguishes the filled and empty
states (or) it is the maximum energy level up to which electrons are filled.
Fermi energy at 0 K:
 Fermi energy level of an intrinsic semiconductor lies exactly between the
lowest energy level of conduction band and highest energy level of valence band
 Fermi energy level of a N-type semiconductor lies exactly between the
minimum energy level of conduction band and donor energy level
 Fermi energy level of P-type semiconductor lies exactly between the
acceptor energy level and the maximum energy level of valence band
Fermi energy when the temperature is increased:
 When the temperature is increased for an N-type semiconductor, some
electrons are shifted from the donor energy level to the conduction band and
hence the Fermi level is shifted down
 When the temperature is increased for an P-type semiconductor, some
electrons in the valence band goes to the acceptor energy levels and hence the
Fermi level is shifted in the upward direction

8. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Fermi Energy level:
N-type semiconductor P-type semiconductor

9. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
INTRODUCTION TO HALL EFFECT:
We know, in semiconductor, both electrons and holes are charge carriers and will
take part in conduction. Measurement of conductivity will not determine whether
the conduction is due to electron or holes and therefore will not distinguish
between N-type and P-type semiconductor.
N-type and P-type semiconductors can be distinguished by determining the Hall
co-efficient using Hall effect.
Hall effect:
When a current carrying conductor is placed in a transverse magnetic field
(perpendicular), an electric field is produced inside the conductor in a direction
normal to both the current and the magnetic field. This effect is called Hall effect.
Hall voltage:
The generated voltage called Hall voltage.
Hall co-efficient:
Hall field per unit current density per unit magnetic induction is called Hall co-
efficient.
If the Hall co-efficient is negative, then the material is an N-type semiconductor.
If the Hall co-efficient is positive, then the material is a P-type semiconductor.

10. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Hall effect in N-type semiconductor:
Theory:
Consider an N-type semiconductor in which the current is allowed to pass along
x-direction from left to right and the magnetic field is applied in z-direction, as a
result Hall voltage is produced in y-direction.
Since the direction of current is from left to right, the electrons moves from right
to left in x-direction.
Due to the magnetic field applied, the electrons move towards downward
direction with the velocity ‘v’ and cause the negative charge to accumulate at the
bottom face of the material. Therefore, a potential difference is established
between the top face and the bottom face of the semiconductor which gives rise
to field EH in the negative y-direction.

11. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Derivation of Hall co-efficient in N-type semiconductor:
Here, the force due to potential difference = - e EH --------------- (1)
Force due to magnetic field = - Bev --------------- (2)
At equilibrium, equation (1) = equation (2)
- e EH = - Bev
(or) EH = Bv --------------- (3)
We know that the current density Jx in the x direction is
Jx = -neev
(or) v = - Jx / nee --------------- (4)
Substituting equation (4) in equation (3),
EH = - B Jx / nee
(or) EH = RH Jx B --------------- (5)
Where, RH is known as the Hall co-efficient and is given by
RH = - (1 / nee) --------------- (6)
The negative sign indicates that the field is developed in the negative y-direction

12. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Hall effect in P-type semiconductor:
Theory:
Consider a P-type semiconductor in which the current is allowed to pass along x-
direction from left to right and the magnetic field is applied in z-direction, as a
result Hall voltage is produced in y-direction.
Since the direction of current is from left to right, the holes will also move in the
same direction.
Due to the magnetic field applied, the holes move towards downward direction
with the velocity ‘v’ and cause the positive charge to accumulate at the bottom
face of the material. Therefore, a potential difference is established between the
top face and the bottom face of the semiconductor which gives rise to field EH in
the positive y-direction.

13. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Derivation of Hall co-efficient in P-type semiconductor:
Here, the force due to potential difference = e EH --------------- (7)
Force due to magnetic field = Bev --------------- (8)
At equilibrium, equation (1) = equation (2)
e EH = Bev
(or) EH = Bv --------------- (9)
We know that the current density Jx in the x direction is
Jx = nhev
(or) v = Jx / nhe --------------- (10)
Substituting equation (4) in equation (3),
EH = B Jx / nhe
(or) EH = RH Jx B --------------- (11)
Where, RH is known as the Hall co-efficient and is given by
RH = (1 / nhe) --------------- (12)
The positive sign indicates that the Hall field is developed in the positive y-
direction

14. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Hall co-efficient in terms of Hall voltage:
If the thickness of the sample is ‘t’ and the voltage developed is VH, then
Hall voltage VH = EH t --------------- (13)
Substituting equation (5) in equation (13),
VH = RH Jx B t --------------- (14)
If ‘b’ is the width of the sample, then, Area of the sample = b t
Therefore, Current density Jx = Ix / (b t) --------------- (15)
Substituting equation (15) in equation (14),
VH = (RH Ix B t) / (b t)
(or) VH = (RH Ix B) / (b) --------------- (16)
Therefore, Hall co-efficient RH = (VH b) / (Ix B) --------------- (17)
Note: The sign of VH will be opposite for N-type and P-type semiconductors

15. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
Experimental determination of Hall co-efficient:
A semiconductor slab of thickness ‘t’ and breadth
‘b’ is taken and current is passed using the battery.
This slab is placed between the pole pieces of an
electromagnet, so that, the current direction
coincides with x-axis and magnetic field coincides
with z-axis. The Hall voltage (VH) is measured by
placing two probes at the centre of the top and
bottom faces of the slab.
If ‘B’ is the magnetic field applied and VH is the Hall
voltage produced, then the Hall co-efficient can be
calculated from the formula
RH = (VH b) / (Ix B)

16. UNIT – IV: SEMICONDUCTOR PHYSICS
APPLIED PHYSICS FOR ENGINEERS by Umayal
References:
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4. Palanisamy P. K., "Engineering Physics", Scitech Publications (India) Pvt. Ltd, Chennai.
5. Arumugam M., "Engineering Physics", Anuradha Publishers, 2010.
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