Why is there a forbidden energy gap in solid state electronics?

[sachin123]

I am reading about solid state electronics(semiconductors).

I've read that when atoms of many many atoms are brought close each other,as in a solid,
the energy levels of energy states of different atoms get split up into different closely lying energy states collectively forming a band.

For example when silicon atoms are brought close together,the 3s and 3p orbitals which were earlier separated join to form a continuous band containing all the electrons in the band.

As the distance between atoms further decreases,the band is "split up" into 2 parts(one the conduction,other the valence) each containing half the total number of electrons.
The energy gap between the 2 bands now formed is called Band gap or Forbidden energy gap?


Now why does that split happen?
Why is the energy gap forbidden?Why can't an electron have an energy in the gap?

Can you please explain?
Thank You

 

[grzz]

Of course Quantum Mechanics explains all that.

But let me give an analogy.

The behaviour of electrons is similar to that of waves. Consider a standing wave in a string. The string only resonates to a particular set of frequencies.

So also an electron in an atom can only take a particular set of values of energy. Other values of energy are forbidden to the electron just like other frequencies are forbidden to the string.

 

[sachin123]

but those frequencies are integral multiples of some fundamental frequency.
There are no bands.
Here every possible energy in a band is possible,but energies between 2 bands isn't possible.
This,I don't understand.

 

[grzz]

... energies between 2 bands isn't possible ...

The physical properties of the string, for example length, mass, tension and its boundary conditions will dictate which frequencies are possible and which are not possible. Similarly the physical conditions of the electron will allow/prohibit some energies.

When an electron finds itself in the close neighbourhood of a set of atoms, the isolated energy values allowed to it when the electron is bound to only one atom will broaden into a band reflecting the physical conditions of the state in which the electron finds itself.

Of course, as I said before, QM explains all this.

 

[tom.stoer]

Think about an atom with one energy level which can be occupied with zero, one or two electrons (in the case of two electrons they must have anti-parallel spin). Now think about two such atoms and bring them together. Instead of having two energy levels at exactly the same energy as before the orbitals will be deformed and the energy levels will change slightly. So instead of the two unperturbed energy levels you will have one with a slightly lower energy and a second one with a sightly higher energy. If you now have two electrons occupying these two levels (in total you now could have four electrons b/c you have two such levels) they will usually occupy the lower level which results in a bound state. Of course the perturbed energy depends on the distance between the two atoms.

What I have explained so far is just forming a H² molecule (bound state) starting with two isolated H atoms. The distance between the two protons is adjusted such that the energy of the two electrons in the lower level plus the Coulomb repulsion minimize the total energy.

If you now consider some specific orbitals e.g. of iron and bring together 10^N (N large!) instead of two atoms, for each new atom the levels will split again. Adding more atoms the split will be smaller, so new atoms will result in smaller perturbations of the energy levels. In the limit of infinitly many atoms the original unperturbed levels will split into a continuous energy band.

Up to now we discussed one energy level. If you start with several different levels (orbitals) you get several bands; typically in condensed matter physics one discusses the valence band and the conduction band. Contributations of lower levels (or bands) to the electronic properties of the solid are usually neglected.

http://en.wikipedia.org/wiki/Electronic_band_structure

(to all condensed matter physicists: don't crucify me for this explanation; I know that it's much more complicated but that it can be formulated rigorously ...)

 

[cbetanco]

The Band gap is just a result of solving shrodingers equation. When you have a periodic potential, Blochs theroem says that the solution will be plane waves. The gaps occur where no such solutions exist, because of restrictions on the wave number k, and hence the energy.

 

[sachin123]

Okay,thank you all.