Energy - band theory of solids

Energy bands:

In case of single isolated atom,there will be single energy level.For a gaseous element,the electronic energy levels are same for a single free atom because the individual atom in a gas are well apart and has negligible influence on each other.When coming to solids,most of the conductors and semi conductors are crystalline in nature.So let us study about the energy band theory of solids in crystalline nature.

Energy band theory of solids:

In general.,the atoms in as solid form crystal structure.Any crystal is made up of a space array of atoms of molecules in regular repetition of some fundamental building blocks.In a crystal the individual atoms are so closely packed that the resulting energy levels are modified due to interaction between the atoms.When an atom forms crystal,the energy levels of inner-shell electrons are not appreciably affected,however the levels of outer electrons are considerably altered as these electrons are shared by the adjacent atoms in crystal.As a result of this electrons,the energy levels spread up to form a band as shown in fig1.,

In order to understand the formation of energy bands in crystals,let us consider a silicon crystal made up of N atoms.We shall also assume that the inter atomic spacing can be varied without affecting the fundamental crystal structure.So for large inter atomic spacing say 'd', the interaction between adjacent atoms is negligible,and the energy levels are same as that of an isolated atoms.In silicon,the outermost sub shells,namely 3s and 3p contain 2 electrons each.Hence,in a silicon crystal which consists of N atoms,the outermost sub shells 3s and 3p consist of 2N electrons each.Thus the 3s sub shells has 2N electrons occupying the available 2N states,and the 3p sub shells has only 2N electrons partially occupying the available 6N states,all at the same energy level.This has been illustrated in below fig2.,

fig.2., Energy band in crystal as afunction of interatomic spacing

If the inter atomic spacing is gradually  decreased i.e., moving from right to left,there will be gradual increase in the interaction between the neighboring atoms.Due to this interaction,the atomic wave functions overlap,and the crystal becomes electronic system which should obey Pauli's exclusion principle.Hence,the 2N states spread out to form a band of energy. Actually,the separation between the levels is small but since N is very large.This spread will have several electron volts of energy and is referred to as energy band and is indicated by lower shaded region as in fig2.,.The 2N states in this band are completely filled with 2N electrons.Similarly,at the same value of inter atomic spacing at p-level,6N p states spread up to form a band.This band is shown as the upper shaded region as in fig.2.,

Tough a total of  6N states are available in this band,only 2N states are occupied and 4N states remains unoccupied.An energy gap exists between the two bands.This energy gap is called forbidden energy gap,as no electron can occupy states in this gap.

Valence Band:
This forbidden gap decreases as the atomic spacing is decreased and becomes zero with further reduction inter atomic spacing say at 'c' as in fig.1., That is,the two energy bands will overlap when the inter atomic spacing is small enough.Under such circumstances,the 6N p states in the upper band merge with 2N s states in the lower band,giving total of 8N states.Half of these 8N states are occupied by 4N available electrons.These 4N electrons now no longer belong to either p sub shell but belong to the crystal as a whole.Thus at this inter
atomic spacing, each atom in the crystal can contribute 4 electrons to the crystal.the band occupied by these contributed electrons is called valence band.

Conduction Band:
If the inter atomic spacing is reduced further,say 'b',the interaction between the atoms becomes extremely large,and the band structure assume the shape as shown in fig.,The exact band structure depends upon
 (i)the orientation of the atoms relative to one another in space
(ii)the atomic number of the atom
and may also be obtained from Schrodinger's wave equation.For spacing 'c',4N states in the valence band are completely filled by 4N electrons and this valence band is separated from the upper unfilled or empty band by a forbidden energy gap contains no allowed states and is referred to as conduction band.



Energy band Structure:
Energy band of a solid contains three bands i.e., the valence,the conduction band and the forbidden energy gap.So depending upon the forbidden energy gap between the two bands,there are three possible energy band structures.They are insulators,semiconductors  and conductors.

 

Insulators:
In case of insulators forbidden energy gap is very wide.Due to this fact,the electrons are very tightly bounded to the parent atoms.For example., in case of materials like glass,valence band is completely filled at zero degree kelvin and the energy gap between the valence band and conduction band is of the order of 10eV.Even in the presence of high electric field,the electrons donot move from valence band to conduction band.
When a high electric field is applied,some of the electrons may acquire energy which is required to move from valence band to conduction band due to increase in temperature.This explains the reason for why certain materials which are insulators at room temperatures becomes conductors at high temperatures.

Semi Conductors:
In semi conductors,the forbidden gap is very small.Germanium and Silicon are the examples of semiconductors.Forbidden energy gap for Germanium is of the order of 0.7eV and for Silicon is of the order of 1.1eV.The electrical properties of semi conductor lie in between insulators and good conductors.At zero degree kelvin there are no electrons in conduction band and the valence band is completely filled. when a small amount of energy is supplied,the electrons can easily jump from valence band to conduction band.

Conductors:
In case of conductors,there is no forbidden band.Here the valence band and conduction band overlap with each other and plenty of free electrons are available for electric conduction.The electrons from valence band freely enter into conduction band.Due to overlapping of valence band and conduction band ,a slight potential difference across the conductor causes the free electrons to constitute electric current.The most important point in conductors is that due to the absence of forbidden band,there is no structure to establish holes.The total current in conductor is simply a flow of electrons.

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I have created this blog for educational purpose,so for that i have written the content by referring many books,web pages.I have also uploaded google images and you tube videos for the better understanding of concept and I would also like to inform you that I am not responsible for the ads which are being posted in my blog.