Band Theory of Metals

Band theory is a theoretical model in solid-state physics that defines the states of electrons in solid materials that can only have energy values within defined ranges. In a solid, the behaviour of an electron (and consequently its energy) is influenced by the behaviour of all other particles. In contrast, an electron’s behaviour in free space, where it can have any energy, is quite different. The ranges of electron energies that can exist in a solid are known as allowable bands. Permitted bands are energy ranges between two allowed bands; electrons within the solid are not allowed to have these energies. The band theory is the foundation of solid-state electronics technology and accounts for many of the electrical and thermal properties of solids. The range of energies permissible in a solid is connected to the discrete authorised energies—the energy levels—of single, isolated atoms.

The discrete energy levels of individual atoms are disrupted by quantum mechanical phenomena when they are joined to create a solid, and the many electrons in the collection of individual atoms occupy a band of levels in the solid known as the valence band. The conduction band is a band of typically empty levels formed by empty states in each individual atom. Electrons in a solid can transfer from one energy level in a given band to another in the same band or another band, usually crossing a prohibited energy gap in the process, just as electrons at one energy level in an individual atom might shift to another vacant energy level.

Examples : 

Sodium :

An isolated atom has electrons with distinct energies that differ from one another. If two isolated atoms are brought into very close contact, the electrons in the orbits of the two atoms will interact with one another. Consequently, in the combined system, the energy levels of electrons will not remain constant, but will fluctuate, with the energies being somewhat lower and higher than the original values. As a result, at the location of each energy level, there are two energy levels that are closely spaced. In the case of a solid consisting of a ‘N’ number of atoms, and the electrons of these atoms interact and produce a ‘N’ number of closely spaced energy levels in place of discrete energy levels, this is referred to as a band of permissible energies. There are empty energy zones between the bands of permissible energies, which are referred to as the banned band of energies. It is supported by the Kronig-Penney model that there are these bands of energies present in the universe (allowed bands and forbidden bands). Even while finding the mathematical solution to Schrödinger’s wave equation is time-consuming, it does provide a hint as to the origin of energy bands in quantum mechanics.

The development of energy bands has been discussed using the example of sodium (Na) metal as a case study. It is discovered that the energy levels of the valence electrons in solitary sodium atoms expand into bands when the atoms are brought together to create a solid. The electron energies of the 3S and 3P orbitals are depicted in the figure. At sodium’s interatomic gap, it is seen that these bands strongly overlap each other.

Glass : 

Glass is an insulating substance that can be transparent to visible light for reasons linked to its electrical insulating properties. The quantum energy of visible light photons is insufficient to cross the band gap and bring electrons up to an available energy level in the conduction band. Glass’ visual properties can also provide information into the effects of “doping” on solid qualities. Colour is imparted to glass by a small fraction of impurity atoms in the form of specific accessible energy levels that absorb specific colours of visible light. The ruby mineral (corundum) contains aluminium oxide with a trace amount of chromium (approximately 0.05 percent) that absorbs green and blue light to give it its characteristic pink or red colour.

Silicon : 

Silicon is a semiconductor material with a lower number of free electrons than a conductor but a higher number than an insulator. Silicon has a wide range of applications in the realm of electronics because of this unique property. Conduction and valence bands are the two types of energy bands found in silicon. The valence band in a solid is made up of a sequence of energy levels with valence electrons. The energy levels of the valence band are filled with electrons at absolute 0K temperature. When electrons are in the valence band, this band holds the most energy, and no current flows because of them.

The conduction band has a greater energy level and contains the least amount of energy. This band is partially occupied by free electrons, which can flow wherever in the solid and are known as free electrons. The flow of current is controlled by these electrons. Between the conduction band and the valence band, there is an energy gap. The forbidden energy gap is the name given to this energy disparity. The character of a solid is determined by this gap.

The amount of forbidden energy gap determines whether a solid is metal, insulator, or semiconductor in nature. Metals have a small gap, while insulators have a huge gap. The gap in semiconductors is neither big nor do the bands overlap. At 300 degrees Celsius, silicon has a forbidden gap of 1.2 eV.

Covalent bonding is known to exist in silicon crystals. Silicon is a non-conducting material. A hole is generated behind an electron when it breaks free from its covalent bond. As the temperature rises, more electrons enter the conduction band, while more holes form in the valence band.

Germanium : 

A computer software called Quantum Espresso was used to calculate the electron band structure of germanium (also known as the diamond structure). Ge is a semiconductor having a direct bandgap of 0.67 eV and a bandgap of 0.67 eV. The calculated bandgap was tweaked in order to achieve the desired gap. When compared to Ge, the computed bands follow a path that corresponds to comparable band structures.

Conclusion : 

Band theory is a theoretical model in solid-state physics that defines the states of electrons in solid materials that can only have energy values within defined ranges. In a solid, the behaviour of an electron (and consequently its energy) is influenced by the behaviour of all other particles.Glass is an insulating substance that can be transparent to visible light for reasons linked to its electrical insulating properties. The quantum energy of visible light photons is insufficient to cross the band gap and bring electrons up to an available energy level in the conduction band.