<< 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.

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 does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor 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
Good
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?

Ask