3.23 Sspd_chapter 1_part10_ continued_the theoretical basis of band  (Page 2/3)

In a solid if electrons are excited to an energy level greater than 0eV then the electron escapes out of the solid and is now present in free space where it can occupy a continuum of energy states. Hence we show a continuum of energy state above 0 eV.

SSPD_Chapter 1_Part10_continued_EXPLANATION OF ENERGY BAND THEORY ON THE BASIS OF PAULI’S EXCLUSION PRINCIPLE.

A single Atom and a 3-D array of atoms arranged in a crystalline solid have a fundamental difference.

A single atom has a single potential well which acts as the bounded space surrounding the nucleus. Within this bounded space only those electrons are permitted which fulfill the boundary conditions for standing waves. Since these electrons are fermions they obey Pauli’s Exclusion Principle. In an elemental phase space only two electrons of opposite spin can be accommodated. What this means is that no two electrons can be at the same energy level at the same spatial coordinate. If two electrons are at the same energy level, spatially they must spread out. If spatially they are squeezed to the same coordinates then they must spread out in energy sense.

When there are two atoms well separated, electrons in their respective host atoms can continue to be in the ground state and the two ground states will be equal. Here the two systems are non-interacting and their phase spaces are well separated.

In a crystalline solid, the 3-D array of potential wells constitute a common phase space bounded by the surface potential barriers of the solid and having a periodic potential variation throughout the three dimensional phase space. Now the electrons of the respective atoms belong to the common phase space and these electrons need to fulfill the boundary condition of standing waves as set by the common phase space and not by the potential well of individual atoms. The individual potential wells are drastically modified when the atoms are three dimensionally arranged into crystalline solid as shown in Fig(1.41. b).

The electrons do not belong to the individual host atoms but belong to the whole lattice space. Electrons instead of being localized are non-localized but their localization is destroyed to definite degree depending upon the orbital position of the electrons.

In a crystalline solid with a large atomic number, the inner orbital electrons are tightly bound to their respective nuclei and hence their localized nature is retained. The energy bands corresponding these electrons are narrow.

As we move outward, the localized nature is progressively destroyed and progressively they are mutually interacting. These leads to wider and wider broadening of the energy bands.

In Figure(1.41), we show the nature of an individual potential well and also the nature of the much wider potential well formed by the combination of the linear array of potential wells. We have also shown the periodic potential variation in the wide potential box caused by the total solid.

Within the single crystal solid, the creation of the much wider potential box and the periodic potential variation within the box are the underlying cause of the energy bands.