Electrical Engineering Material Third semester

Infinite Potential Well

Engineering stimulates the mind.

Q.1: Starting from the suitable equation, prove that the energy of an electron that is confined in an infinite potential well of width 'L' is quantize (2075 Chaitra BEl 2 year-First Pert(II/I). 

Q.1.b) Prove that the energy of a particle confined in an infinite potential well is quantized. Also, find the expression for the normalized wave function(Regular II/I BEL.BEX 2074 Chaitra)[8 marks]

Let us consider an electron is confined in a region defined by 0<x<L; where ‘L’ denotes the width of infinite potential well in which an electron is confined.

The potential can be defined as :

V_(x) = 0;  at 0 <x< L

V_(x) = ∞;  at L <x< 0

It is seen that the potential energy of the particles inside the well is zero and infinite outside the well. Thus the electron inside the well needs infinite potential energy to leave the well hence particles cannot leave the well.

Schrodinger wave equation is

But inside the well  V = 0

Where 

The solution of eq(1) can be written as

Where A and B are constants whose values are to be determined by using boundary conditions.

Here A is constants so A can’t be zero so, sine must be Zero

From eq(2) and eq(4), we can write

Here E is the total energy of the electron when potential energy is equal to Zero.

The energy of the electron that is confined in the region of an infinite potential well cannot be arbitrary nut must be defined by the parameter ’n’. Alternatively, the energy of an electron confined in an infinite potential well is quantize.   

Extrinsic Semiconductor

The conductivity of semiconductor risen when some element belonging to either group iii or group v in Mendeleev periodic table are added on it. These elements are called impurities. The extrinsic semiconductors doped with these impurities is called extrinsic semiconductor. These conductor are of two type :

1. P- Type semiconductor
2. N- Type semiconductor

N –Type semiconductor:

When a pentavalent impurity like Arsenic, Potassium, and Antimony is added to a pure semiconductor a n-type semiconductor is formed. The four electrons out of five valance electron of these impurities make covalent bond with four valance electron of si so one electron remains free on every add of impurities. This extra electron continuously revolves around the impurities in core.

                                                                                In this type of semiconductor the concentration of electron increases with every add of impurities so it is called negative type(n-type) semiconductor since the pentavalent atom donate electrons for conduction so it is called donor. In n-type semiconductor e- are majority charge carrier and holes are minty charge carrier.  If n and p represents extrinsic electron and hole concentration of semiconductor. The donor concentration then supposing all donor side are ionized. 

 P-Type semiconductor

When a trivalent impurity likes Boron (B), Al, Ge, In is added to pure Si, a p-type semiconductor is formed. The valence electron of these impurities can form only three complete covalent bonds with silicon which has four valance electrons there is deficiency of one electron to four bonds. This give rise to holes.  In this type of semiconductor the concentration of hole increases with every add of impurities so is called positive type of semiconductor (P-type). Since trivalent atom creates a hole and this hole has tendency to accept an electrons so trivalent impurities is called as accepter.

Let n & p are electrons and hole in semiconductor Na is accepter concentration. The doping of trivalent impurities will increases the hole concentration in valance band but does not increases the electron concentration in conduction band so.

Importance of Fermi –energy level:

1. For intrinsic semi conductor Fermi energy level.
2. It is important for4 understanding and determining electric and thermal properties of certain material.
3. They allows us to make calculation as to the density of electron and holes in material or relative amount of each depending on temperature.
4. This is crucial to our understanding of current flow throw semiconductor.
5. In metals the Fermi energy gives us information about velocity of electrons which participates in ordinary electron conduction.
6. For metals the density of conduction electrons can be implied from the Fermi energy
7. It is used to describe insulators metals and semiconductors
8. The Fermi level play an important role in bond theory of solids
9. In doped semiconductor p-type and n-type, the Fermi level is shifted by the impurities illustrated by their band gaps.

 

PN- Junction:

In n-type semiconductor electron are majority charge carrier and holes are minority charge carriers and in p-type semiconductor holes are majority charge carrier and electrons are minority charge carriers.

                                when we bring p-type semiconductor in contact with n-type semiconductor then there form an abrupt discontinuity at the junction. This abrupt junction is known as metallurgical junction.

If there two n-type and p-type semiconductor are brought together to form a junction  

Type of polarization

1.       Electronic polarization

2.       Ionic polarization

3.       Dipole orientation polarization

4.       Interfacial polarization

Electronic Polarization  

In an atom in an absence of electric field both charge center lies at same point so there is no polarization but if the external field is applied electron cloud move toward the opposite of applied field and charge separates so there is polarization and induced dipole moment . This type of polarization is known as electronic polarization.   

Induced dipole moment is given by :-

P_ind =

the actual field experienced by dielectric material in between capacitor plate is called local electric field . it depends not only on the free charge of the capacitor plate but also on the polarization. In a material of a cubic structure local electric field is given by

 

For more, please visit by most popular searches

• engineering notes nepal : Find here most important and useful for short time
• ioe civil engineering notes
• pulchowk campus notes
• ioe solutions
• engineering mathematics 2 pdf ioe
• solution of engineering mathematics volume 2 ioe
• civil engineering notes nepal