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Chapter 1_Part10_Conclusion describes the theoretical basis of Energy Band Theory of Solids by analyzing Kronig-Panney Model which idealizes 1-D crystalline array and determines the Schrodinger Equation Solution in 1-D array of square wells.

SSPD_Chapter 1_Part10_concluded_ ENERGY BAND THEORY IN A SOLID BASED ON KRONIG-PENNEY MODEL.

Kronig-Penney model is 1-D array of square wells as shown in Figure 1.44.. This is 1-D idealization of a linear array of atoms in a single crystal lattice structure. The solution of Schrodinger Equation using this array of Square wells becomes more tractable and it still brings out the important features of the quantum behavior of electrons in real life crystalline periodic lattice.

There are four assumptions in Kronig-Penney Model analysis namely:

  1. Electron interaction with the core is purely coulombic ;
  2. Electron to electron interaction is precluded;
  3. Non-ideal effects, such as collisions with the lattice and the presence of impurities, are neglected;
  4. Atoms are fixed in position whereas they are having thermal vibrations.

Figure 1.44. Kronig-Penny Model of a linear array of atoms in a single crystal solid.

The solution of Schrodinger Equation can be arrived at mathematically but for simplicity of presentation we will discuss the problem in qualitative terms only.

Study of electron in a crystalline structure is really the study of an electron in a periodically varying potential field. For simplicity of analysis we assume a linear array of

atoms . The crystal length is L cm. Let Z-axis be the longitudinal axis and let the crystal be repeated along the Z-axis with a period of L cm from - ∞ to + ∞ . Along X-axis and Y-axis it is of infinite length. So we have a semi-infinite crystal of finite length L cm in Z-axis. For the ease of calculation we assume that crystal is repeated along z-axis at L cm.

Since we have assumed a periodic crystalline structure along Z axis therefore the solution of the Schrodinger Equation is applicable only in the bulk and not at the boundaries of the crystal.

We will assume that L cm = 1cm =1×10 -2 m. The crystal structure is referred to as the lattice. The atoms of the crystal are referred to as the lattice centers. The distance between two consecutive lattice centers is referred to as the lattice constant ‘a’ Å. A typical lattice constant is 2 Å. Therefore the linear array contains L/a = 1×10 -2 m/2×10 -10 m = 5×10 7 atoms in one period. Let this number be N i.e. N= 5×10 7 .

We have a periodically varying potential field along the linear array with a periodicity of ‘a’ Å hence the Fourier Series Expansion of the potential is:

1.83

The potential field has a period of ‘a’ Å hence 2π/a is the fundamental periodicity and the harmonics are 2(2π/a) , 3(2π/a) , 4(2π/a) …………………..m(2π/a)

In the periodic potential field following is the Schrodinger Equation for time independent part:

2 ψ/∂z 2 + [{2m(E-V(z))}/ћ 2 ]ψ = 0............ 1.84

If we had assumed that our very wide potential well was flat bottomed with V(z) = 0 everywhere along the potential box then the solution of the Schrodinger Equation would be a progressive wave as would be obtained for free space:

Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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Source:  OpenStax, Solid state physics and devices-the harbinger of third wave of civilization. OpenStax CNX. Sep 15, 2014 Download for free at http://legacy.cnx.org/content/col11170/1.89
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