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# Energy bands in solids

## Energy bands in solids

The energy levels of an atom take discrete values. When atoms are brought together to form molecules, additional energy levels are created due to the interactions between atoms.

The number of energy levels of a molecule is proportional to the number of atoms it contains.

A solid body can be thought of as a giant molecule with a massive number of atoms. Consequently, the number of energy levels in a solid becomes extremely large.

The difference in energy between the levels becomes so small that the adjacent levels are almost indistinguishable.

As a result, they form an approximately continuous distribution of energy. However, there are some intervals of energy where no energy levels are present.

This means that the energy levels of a solid are arranged into a number of continua called energy bands, with regions between them called band gaps.

These bands are not gaps in terms of distance between electrons but a range of energies that electrons cannot take (recall that atomic orbitals are regions around an atom and higher energy orbitals have larger volumes).

## Electrons in energy bands

The electrons of a solid are arranged in energy bands. Electrons fill the lower level energy bands before the higher level ones.

The highest fully occupied band in a solid is called the valence band. Electrons in this band are primarily responsible for bonding together adjacent atoms. These are called bonding or valence electrons.

In chemistry, the outermost shell of an atom is known as the valence shell.

The electrons in the valence band are comprised of those in the valence shells of all of the solid atoms. These shells are fully filled by the electrons shared by adjacent atoms in the solid structure.

In a solid, the band with energy directly above that of the valence band is the conduction band. In general, electrons move to the conduction band only if they are excited and the conduction band only contains excited electrons.

## Holes in the valence band

At any temperature above absolute zero, bonding electrons have a chance to become excited and cross over to the conduction band. This leaves a "vacancy" in the valence band, known as a hole.

This hole is considered to be a positive charge.

Electrons and holes are normally considered to be paired up in electron-hole pairs. Electrons in the conduction band and holes in the valence band are able to move freely within their respective bands.

Electrons and holes flow in opposite directions when a current is passed through a material.

## Electrical properties of solids

The resistivity and conductivity of a solid depends on the band gap between the valence band and the conduction band.

The gap is large in insulators (several $\text{eV}$) and moderate in semiconductors ($\approx 1\text{ eV}$). The two bands actually overlap in conductors such as metals.

In other words, the energy needed for an electron to jump from the valence band to the conduction band is very high in an insulator, lower in a semiconductor and no energy is needed for a jump in a conductor.

Given that electrons need to be in the conduction band to travel (for an electric current to pass), the differences in band gaps imply that resistivity is lowest in a conductor and highest in an insulator.

Similarly the conductivity is highest in a conductor and lowest in an insulator.

Band structure of solids.