Now, we will study with a little more detail,

the bond structure of a crystalline solid.

In particular, we will investigate the block states in the case of periodical potential.

Without going into too much detail,

we can use the other usual chemical approach for the semiconductor bond structure.

Starting from two atoms, typically,

two hydrogens or hydrogen like,

the two atoms each have one electron.

If the distance between them is large,

the states are united without any mutual interactions.

If the distance, R, between these atoms become small,

electrons which are each located out of nuclei, A,

will begin to feel the potential created by the nuclei, B.

So, if R is not too small,

this extra potential can be considered as a petrubation, lifting the [inaudible].

In this case, for instance,

the lower energy curve will therefore be populated by two opposite spin electrons.

This behavior can be extended to all excited states.

The decrease of the energy of the system due to the interaction

when the atoms are brought together is the origin of the covalent bonding.

Now, we can consider a microscopic solid,

consisting of a very large number of atoms.

Typically, 10 to the 22,

or 10 to the 23.

If the distance between atoms are decreasing,

in order to create the crystalline lattice,

so the degeneracy is high as previously in the case of two atoms,

but instead of two energy curves,

a continuum of energy states,

green and red energies will be obtained because of the high value of N on the principle.

A forbidden region will be obtained if the maximum energy of valence bond,

green, is smaller than the minimum energy of the conduction bond, red.

Now, we have to solve the Schrodinger equation in order to

obtain the wave function on energy levels of the system.

So, I remind you that in the case of

a free electron in a box with periodic boundary conditions,

the solutions of the Schrodinger equation is the plane wave,

associated with a wave number,

K. The energy values are therefore,

edge bar Square, K square, over 2m0.

Now, in a perfect crystalline solid,

the atoms are located at the node of a periodic network.

Because they are fixed,

we can neglect their coulomb interaction since it is constant.

Therefore, the Hamiltonian of an electron includes three contributions.

The kinetic energy of electrons,

the coulomb interaction with the nuclei,

and the coulomb repulsion between the electrons.

This equation cannot be solved because of the large number of atoms in the solid.

Typically, 10 to the 22. Therefore,

we'll use Hartree-fock approximation instead of solving the equation for the full system,

the Schrodinger equation is solve for a single atom using an average potential.

So the Hamiltonian of the system is replaced by N Hamiltonian, small h,

for independent electrons with the periodic potential V of R. Therefore,

the solution of the Schrodinger equation is given by the Blochs's theorem,

which is related to periodicity of the potential V. More precisely,

the wave function solution of the Schrodinger equation is given by exponential.

i,k,r, times a function u,

which displays a potential periodicity called, Bloch function.

The Schrodinger equation can have value solution as

function of the wave vector k. K varies continuously.

The energy appears as a continuous function of k. In summary,

it can be pointed out that the periodicity of the network

appears as an essential feature for solving the Schrodinger's equation.

Thank you.