 In this session, we are going to study the band theory of solids. Learning outcomes are by the end of this session, students will be able to discuss about band theory formation in solids and can classify solids depending on the band theory. The contents include formation of energy bands and classification of solids. X-ray diffraction studies show that a solid is an ordered structure. Therefore, atoms in a solid interact strongly and set up an internal electric field, which is periodic in nature. This periodic electric field affects the motion of free electrons. The application of quantum mechanics to the motion of electrons in solid shows that the allowed values of electron energy are distributed into bands. Each band consisting of a sequence of closely spaced energy levels which are arranged in a manner akin to the steps of a ladder. In 1928, Felix Blotch developed zone theory for the electrons moving in a periodic field that is provided by the crystal ladders. This theory is popularly known as band theory of solids. Knowledge of the formation of energy bands and the consequent restrictions imposed on electron motion in a solid are obtained from the band theory. To understand the mechanism of conduction in solids, let us discuss quantitatively the formation of energy bands in solids. Let us start with the atom. The most fundamental unit of matter which is capable of independent existence is called as atom. An atom consists of a central body called as nucleus about which a number of smaller particles called electrons move in orbits. The nucleus contains two types of particles called protons and neutrons. The number of electrons is the same as the number of protons. The electrons are arranged in different orbits or shells. A shell can contain a maximum of 2n square electrons where n is the number of shell. The electrons present in the outermost orbit are called valence electrons. Each atom has a certain number of orbits and each orbit represents a particular energy level. The electrons of an isolated atom can have certain definite energies. Pauli's explosion principle determines the maximum number of electrons which can be accommodated in each energy level. An energy level consists of several quantum states and n quantum state can contain more than one electron. When atoms group together to form a solid, there are millions of electrons belonging to each orbit of atoms. Each of them has different energy. As long as the atoms are widely separated, their interactions are negligible. Every atom has the same energy level diagram. As atoms come together to create a close packed periodic structure, they interact strongly due to their proximity to each other. By interaction, we mean that the positive nucleus of one atom attracts the electrons and repels the nucleus of the adjacent atom and vice versa. As a result, instead of energy level the same for all n isolated atoms, there arise n closely spaced separate levels which fall into groups. The transformation of a single energy level into two or more separate energy levels is known as energy level splitting. When two atoms come close, one energy level split into two energy levels. When three atoms approach each other closely, the original level splits into three levels. Four atoms produce four levels and slow on. In general, n interacting atoms cause a particular energy level to split into n levels. This group of energy levels resulting from splitting is so closely spaced that is called as an energy band. The individual valence electrons no longer belong to individual atoms but they now belong to all nuclei in the solid. Students, now please pause the video and try to answer these questions. Check for the correct answers. So, in an atom an orbit can contain a maximum of two n square electrons and the valence electrons are placed in the outermost orbit. So, we have seen that there are discrete energy levels for the isolated atoms. When a huge number of atoms are combined to form a solid, however these discrete energy levels are replaced by discrete ranges of energy or energy bands within which there are so many individual allowed energy values that within the bands the distribution can be considered to be continuous. This idea is seen in this figure. The electrons are distributed in the energy levels according to the Pauli's exclusion principle. The motion of an electron corresponds to its transition from a lower energy level to an upper vacant energy level. While occupying an energy band, electrons start from the lowest energy level in the band and feel the levels one after the other in the ascending order of energy. When two n electrons occupy the n levels available in the band, the band is said to be completely filled. In case of non-availability of two n electrons, the energy band gets partially filled and when there are no electrons to occupy the levels, the energy band remains vacant. So, this implies that two conditions are to be fulfilled for the electrical condition to take place in a solid. That is, there should be free electrons available in the solid and there should be vacant energy levels which are immediately above the levels occupied by the free electrons. Now consider that if a band has vacant energy levels but is lacking of electrons, there will be no carriers to move through the vacant levels when energy is supplied to the solid. Hence, the current conduction is not possible. On the other hand, if all the energy levels within the band are completely occupied by the electrons and there will be no energy level to which the electron can jump, then again the electron conduction is not possible. So, if an energy band is partially filled, then the electrons will have vacant upper energy levels into which they can jump. On acquiring energy from the electric field applied across that solid, the electrons move into the successive upper energy levels and cause the electrical conduction. Thus, partially filled energy band is required for electrical conduction in a solid. An energy band is a graphic representation of the energy levels associated with the top energy band and next lower energy band in solid. Out of all energy bands, the important energy bands in the solids are valence band. The range of energies possessed by valence electrons is known as valence band. In a normal atom, valence band has the electrons of highest energy. This band may be completely or partially filled. These valence electrons are responsible for forming atomic bonds. The number of valence electrons in an atom also determine the valence of the element whose atom it is. For example, boron is trivalent whereas silicon and germanium are tetravalent. That is, they are having four valence electrons each. Then another type of band is conduction band. The range of energies possessed by the conduction electrons is known as conduction band. The electrons which are responsible for the conduction of current in a conductor are called as conduction electrons. Next energy band is forbidden energy gap. The separation gap between the conduction band and valence band is known as forbidden energy gap. The width of the forbidden energy gap is the size of the bondage of valence electron to the atom. The larger the energy gap, more closely the valence electrons are bound to the nucleus. In direction to push an electron from valence band to the conduction band, outer energy equal to forbidden energy gap must be provided. The concept of energy bands helps us in understanding the division of solids into three groups. According to band theory, the electrical conductivity of a solid is characterized by energy gap. That is separating the outermost energy bands, namely valence band and conduction band. The electrical conductivity of a material depends on the magnitude of the energy gap. Some solids have energy gaps that are very wide. It is greater than 3 electron volt. It would require the attainment of very large amount of energy to cause an electron jump from the valence band to the conduction band. A small number of electrons can acquire such large amount of energy to jump from valence band to conduction band. Therefore, very few electrons would be present in the conduction band. On application of external voltage across the solid, negligible current flows through it. And these solids are called as insulators. In some solids, the band gap is narrow. That is of the order of 2 electron volt or less. Gaining of small amounts of energy from the vibrations of atom can raise electrons from the valence band to the conduction band. The conduction band is then partially filled. If a potential is applied across the material, it causes the electrons in the conduction band to move to upper levels. As a result, current flows in a modest major in the solid and such solids are called semi-conductors. Then, in some solids, an upper vacant band overlaps the valence band as shown in the figure. It means that, electrons in the valence band have easy access to the levels in the upper vacant band. For this reason, very large number of electrons are available for conduction even at extremely low temperatures. When electric field is impressed across the solid, electrons readily jump into the upper unoccupied energy levels of the vacant band and current flows in a large major in the solid. Therefore, these solids exhibit good electrical conductivity and they are called as conductors. Students, now let us summarize the properties of solids on the basis of band theory. Thank you.