Band theory is a method that calculates the energy gap in electrons of a specific solid to separate two kinds of bands, namely Valence bands and Conduction bands. Any solid can be classified among Conductors, Semiconductors, and Insulators based on its properties. Contrary to the discrete energy of free atoms, the quasi-visible bands are the energy source in a solid molecule. In conductors, like metals, the valence bands overlap the conduction bands while there remains a small gap between those two kinds of bands in the semiconductors like silicon. On the other extreme conductors are insulators that do not transfer heat or electricity and maintain a large gap between electrons of conduction bands and valence bands. This peculiar positioning of the conduction and valence bands determines whether some objects can transmit heat and carry electricity.

Features of Band Theory

• Solids are gigantic molecules with many atoms in a close-knit structure.
• In solids, when valence shells of each atom make contact, atomic orbitals together form molecular orbitals.
• These outer shells work as the only system of electrons for both atoms. Every other atom in the crystal follows the exact pattern. 

Formation of bands

Valence shells are the outermost shells of an atom that are a set of orbitals that bond with another atom. The electrons in these shells are called valence electrons. According to the Aufbau principle, lower energy orbitals fill up before higher energy orbitals. Corollary to this, empty orbitals go higher up energy levels.

• Overlapping bands: Overlapping bands are two different energy states where electrons can move from one another, provided no external energy is supplied. Higher energy bands might overlie lower energy bands. 

Eg.: Valence 3S band in silicon coincides with an empty 3P band. 

• Non-overlapping bands: Non-overlapping bands are energy states ranked in a hierarchy where the other band is either above or below one particular energy band. The number of valence electrons does not make a difference.

Eg.: Valence 3S band does not overlap with 3P band with no electrons.

• Valence band: Bands with valence electrons bear lower energy and are called valence bands. It can be partially or entirely filled with electrons.

Eg.: 

1. Partially filled with electrons: Fe, Cu
2. Entirely filled with electrons: Mg

• Conduction band: The band just higher the valence bond is termed as the conduction band. It can be partially filled or empty of electrons. If it has electrons, they are called free electrons.
• Forbidden gap: The gap between the conduction band and valence band is called the forbidden gap as no electrons can reside between these two bands. In semiconductors, the Fermi level is situated in the forbidden gap, at an equal distance from the two kinds of bands.
• Fermi level: The probability of finding electrons at a given energy level and temperature, introduced by the Fermi-Dirac distribution function is termed f(E). At absolute zero, no electron can pass the valence bond to the conduction bond as f(E) = 1 meaning the orbits are filled. In quantum mechanics, the Pauli extension principle proves that every allowed energy status can only house two electrons of opposite spin. This particular energy level at low temperatures is known as the Fermi level. With added temperature, some electrons can move beyond this level, and high temperature induces more electrons to shift, thus creating an electrical flow. 

Types of solids

• Insulator: The solids in which the electrons cannot move from valence band to conduction band because of the massive forbidden gap between them are called insulators. These items are used as heat-resistant accessories like kettle-handle or shockproof articles to protect from electricity. Wood, glass, plastic are some examples. When the forbidden gap is ≥5 eV, the said item can be used for the aforementioned tasks.
• Conductor: For the conductor, the electrons can move easily from the valence band to the conduction band as the forbidden gap verges upon zero. Metals are common examples of such elements.
• Semiconductor: Semiconductors have a forbidden gap of ≤3 that makes some electrons leave the valence band for the conduction band.

1. Intrinsic semiconductors: Silicon, Geranium, etc. are intrinsic semiconductors that are pure and consist of four valence electrons. The current is produced by the opposite movement of electrons from the conduction band to the valence band and vice versa simultaneously.
2. Extrinsic semiconductor: Induced with little impurities known as doping, intrinsic semiconductors turn into extrinsic one. The conductivity can increase as high as 10000 times. Depending on the minuscule addition of the substitute element, this can further be divided into two kinds.
3. N-type conductor: N-type semiconductors have a pentavalent impure element like phosphorus that produces an extra electron that initiates the energy flow. The phosphorus atom creates four covalent bonds.
4. P-type conductor: When a trivalent impurity like Boron is introduced to an intrinsic semiconductor, it creates three covalent bonds and one positive hole that circulates the energy.
5. Organic semiconductors: Some conjugated organic polymers have a high level of conductivity. Their intrinsic semiconductivity maintains a gap of 1-2eV. Some examples are polyphenyl and polyacetylene. While doped with oxidising agents, these also form n-type and p-type semiconductors.

Conclusion

The Band theory is an extension of Molecular orbital theory concerning covalent bonds in solids. It is comparable yet distinct from the electron exchange for protons that creates electrical bonds. Band theory itself attempts to elucidate the physical workings of the transference of heat and energy. Even though there is no material proof of this theory, it supports building up further experiments and explanations more minutely. The categorisation of solids into conductors, semiconductors, and insulators has resulted in effective technological progress in variegated industries and benefitted theoretical science.