<< Chapter < Page Chapter >> Page >
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.


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:


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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Solid state physics and devices-the harbinger of third wave of civilization' conversation and receive update notifications?