Band Theory of Solid Materials

In solid-state physics, band theory is a theoretical model that describes the states of electrons in solid materials that can only have energy values within specified particular ranges. The behaviour of an electron (and thus its energy) in a solid is dependent on the behaviour of all other particles in the vicinity. This is in sharp contrast to how an electron behaves in free space, where it can have any energy. Allowable bands are the ranges of electron energy that can be used in a solid. Permitted bands are energy ranges between two allowed bands in which electrons within the solid may not have these energies. The band theory is the foundation of solid-state electronics technology, accounting for many of the electrical and thermal properties of solids. The discrete permissible energies—the energy levels—of single, isolated atoms correspond to the band of energies permitted in a solid. These discrete energy levels are disrupted by quantum mechanical phenomena when the atoms are gathered together to form a solid, and the many electrons in the collection of individual atoms occupy a band of levels in the solid termed the valence band. The conduction band is formed when empty states in a single atom broaden into a band of levels that are ordinarily empty. 

Walter Heitler and Fritz London discovered the energy bands

Pauling was one of just three young Americans in the late 1920s — the others were MIT’s John Slater and University of Chicago’s Robert Mulliken — who could combine a thorough understanding of modern physics with a great interest in addressing chemical problems. Pauling was the one with the most rudimentary chemistry training of the three.

Two young German physicists, Walter Heitler and Fritz London, presented a more direct challenge to Pauling’s goal of describing the chemical bond in quantum mechanical terms. Heitler and London were the first to use Erwin Schrödinger’s wave equations to solve the problem of the simplest chemical bond, that of two hydrogen atoms, in collaboration with Schrödinger.

They used Heisenberg’s new concept of “exchange energy,” which he coined. According to the idea, when two atoms get closer, the chances of a negatively charged electron from one attracting the positively charged nucleus of the other increases, and vice versa. The two electrons would eventually start hopping back and forth between the two nuclei at a pace of billions of times per second, resulting in an electron exchange. The two electrons would be unable to determine which nucleus they belonged to in some ways.

Heitler and London estimated that the attraction created by the electron exchange would be balanced at some point by the repulsion of the two positively charged nuclei, resulting in a chemical bond with a certain length and strength, by combining that concept with Schrödinger’s wave equation. They showed the concept by using mathematics to show how two hydrogen atoms can link together.

It had been a huge victory. When Pauling visited Europe, he became convinced that their strategy was sound. However, this was simply the first step in what might be a massive area, and it was only done for the simplest molecule conceivable. There were a slew of other issues that needed to be addressed. And it would be Pauling’s job to figure out how to fix them.

The electrons in an atom are present in different energy levels

Energy levels (also known as electron shells) are defined by predetermined distances from an atom’s nucleus where electrons can be located. Higher-energy electrons have greater energy as you get further away from the nucleus.

Electrons are always added to the lowest energy level first, until it contains the maximum number of electrons conceivable, and then to the next higher energy level, and so on. The number of orbitals determines the maximum number of electrons at a particular energy level. Per orbital, there can only be two electrons.

Valence electrons are electrons in the atom’s outermost energy level. Many of an atom’s properties, including how reactive it is, are determined by them.

This splitting up of sharp and tightly packed energy levels for Energy Bands

Consider a single silicon atom, which has quantized energy levels (see the Bohr model for Hydrogen). Because of their reciprocal interaction, the quantized energy levels of two identical atoms hybridize and split into two distinct levels when they are brought closer together. In general, the energy levels split into N levels when N atoms are brought closer together until they reach the equilibrium interatomic distance d. If N is enormous (as it is in a crystal), these N levels are relatively close to each other, forming a continuous energy band.

Consider silicon atoms organized in a periodic lattice with a large lattice parameter (or inter-atomic distance) so that each atom can be treated as separate. E1 and E2 refer to the two levels having the most energy. Let’s decrease the atom lattice hypothetically now: energy levels split into two continuous bands, the conduction band CB and the valence band VB.

Two continuous energy bands (CB and VB) exist in a silicon crystal, separated by an electron-inaccessible forbidden band. The restricted area is referred to as the “gap,” and its length is equal to the breadth of the gap. The substance has a property called Eg. The lowest energy level of the conduction band is designated EC, whereas the highest energy level of the valence band is designated EV, forming a connection Eg=EC-EV. The energies available to electrons, or the energies of the states potentially occupied with electrons, are represented by the conduction and valence bands CB and VB, respectively. They provide no information regarding the effective occupation of energy states by electrons.

Conclusion

The band structure of a solid in solid-state physics explains the energy bands that an electron within the solid can have (“allowed bands”) and the energy gaps that it cannot have (“forbidden bands”). The existence of energy bands is hypothesised in band theory to characterise electron activity in materials. Many physical features of solids may be explained using the band structure of the material. The large-scale limit of molecular orbital theory can also be visualised as bands.

Atomic orbitals are a discrete collection of energy levels occupied by the electrons of a single solitary atom. When two or more atoms are combined to form a molecule, their atomic orbitals split into separate molecular orbitals, each with a different energy. As a result, the number of molecular orbitals is proportional to the number of valence electrons in the molecule. The number of orbitals grows extremely big when a huge number of atoms (1020 or more) are brought together to form a solid. As a result, the energy gap between them narrows significantly. As a result, instead of the discrete energy levels of individual atoms, the levels in solids create continuous bands of energy. Some energy intervals, on the other hand, are devoid of orbitals, resulting in band gaps. In the case of semiconductors and insulators, this idea becomes even more crucial.