BAND GAP &THE KRONIG-PENNEY MODEL PART 1 BLOCH THEORM

Electronic is most important branch of engineering . Electronics is used by companies for automation we know that semiconductor devices are heart of today computer because we use computer we have to know what going on inside its our duty so lets go to topic

The free electron model or drude model failed not failed completely but can't tell us about the nature or the properties of semiconductor The failure of the free electron model is due to the oversimplified
assumption that a conduction electron in a metal experiences a constant or
zero potential due to the ion cores and hence i$free to move about in a crystal which is over simplified by kronig-penny model the basic assumption for this model as following

1. .the ion cores may be considered at rest and the potential experienced by an electron in a crystal is assumed to be periodic with period equal to the lattice constant

2). the quantum mechanical wave function also must be periodic

Mathematical model

ion, the total wave function for the system is obtained from a combination of wave functions each one of which involves the coordinates of one electron. In other words, the field experienced by a particular electron is assumed to be the resultant of the field due to fixed nuclei and the average field due to the charge distribution of all nuclei affect the electron inside crystal as we going to see but the motion of electron in periodic potential yield the following result

a). There exist allowed energy bands separated by forbidden regions or band gaps.

b). The electron energy E(k) is periodic in k i.e wave vector related to brillouin zone .

The Bloch Theorm

The one-dimensional schrodinger equation is given by

the solution to these wave equation are plane wave of the type .

where

for an electron moving in one dimensional periodic potential , v(x) the schrodinger equation is written as

since the potential is periodic with period equal to lattice constant a we have

now the bloch theorm state that .the solution to the schrodinger equation is plane wave solution of the type which are modulated by a function Uk(x) having the same periodicity as that of the lattice thus the solution are of form

the wave function is called bloch function

with