# BAND THEORY : Overlapping energy bands in metals

I have been studying Energy bands and suddenly I encountered a diagram in which the two bands(valence and conduction) overlaps in case of metals. What is the meaning of this overlapping? Suppose they overlap from $\mathrm{-2\ eV \ to \ 4 \ eV}$, it means that electrons belonging to these energy levels belongs to both valence as well as conduction band? Isn't it senseless? How an electron in the valence shell can be a valence shell electron too and a conducting/free electron too?

Any help would be higly appreciated from any brother/sister.

• There is more to the electronic structure than just energy. The electron wave functions have different wave numbers (reciprocal of wavelength) on which the energy depends. These dispersion relations are different in different directions in the crystal. But the decisive thing for conductivity is whether there is an energy gap or whether there is overlap of partially filled bands.
– user137289
Commented Jan 11, 2018 at 20:54
• I'm in Grade $12$. I don't know higher quantum mechanical physics! Please explain it as per the level of Class $12$ Students. Commented Jan 11, 2018 at 20:56
• It is very difficult to explain. The concept of reciprocal space ($k$-space, k is the wave vector) is very abstract. It is a Fourier transform of the crystal, so $k$-space is also a thing in three dimensions. And it is not necessary to understand diodes etc.
– user137289
Commented Jan 11, 2018 at 21:05
• But what about my this question Sir? This is from my class 12 Book, how can i explain it? Commented Jan 11, 2018 at 21:08
• Rather than thinking of them as overlapping, just think of it as one band which is not full. Therefore electrons can "move around". Commented Jan 11, 2018 at 21:27

## 1 Answer

The energy bands are allowed values of energy for the electrons. In the picture the energy levels of carbon atoms are shown. If you have a large interatomic distance between atoms (the right of the picture) the atoms won't interact much. The energies are spaced far apart and many atoms can share the same energy level because they are not in danger of violating the Pauli exclusion principle.

If the atoms get closer together the exclusion principle comes into play and the energy levels split into many lines. So many that it looks like a band. In this band the electrons can hop to unoccupied levels because they so are close together. Thermal energy is sufficient to change between levels.

If the atoms get even closer together the band splits again into a conductance band and valence band. This time the electrons can still move inside the bands but the gap is so large that the chance of changing between valence and conduction is very slim.

Metals fall into this middle regime where the 'conduction band and valence band overlap'. You can't actually tell these apart in this regime so when you say the valence band and conduction band overlap you just mean that they are joined.

Note: to predict the actual conductivity of a material you also need to consider how the energy levels are filled.

• $\mathrm{4th}$ para was actually helpful! Thanku! Commented Jan 12, 2018 at 10:16