Dear Samares, and colleagues:

welcome,

I would like to add my comment on the answer of the question:

It is so as the lattice constant decreases the interatomic distance will be reduced. As a consequence binding forces between the valence electrons and  and the parent atoms will increase. These valence electrons will occupy the valence band. Since the valence electrons are bound, they have to be supplied with energy to make them moving free inside the material and become conduction electrons. The minimum energy that must be given to valence electrons to become a conduction electron is the energy gap.

So, as the valence electrons get more bound by decreasing the interatomic distance, the more energy required to make them free in the conduction band. 

So, A direct consequence of decreasing the lattice constant is the increase in the energy gap. As a rule of thumb the energy gap is inversely proportional to the interatomic distance. 

An other factor affecting the energy gap is the dielectric constant, which depends on the density of atoms and their polarizability. The dielectric constant is proportional to N the density of atoms per cm^3 and the alpha the polarizability. The polarizability depends on the electronic structure of the atom.  It is so that the energy gap is inversely proportional to  the dielectric constant which in  turn is inversely proportional to the inter-atomic distance. 

Therefore, there are to competing effects to the lattice constant on the energy gap.

the binding energy and the dielectric screening.

Best wishes

Phenomenologically, the bands are represented as a function of the crystal momentum, so k=2pi/a, small k, means larger a, and vice versa. The lowest conduction band energy and the largest valence band energy, is where the band gap is measured theoretically. Thus, the difference in these energies for k being large, means a is smaller.

Dear Prof  Kar,

Does this type of correlation also exist in case of elemental crystalline semiconductors of IV group which have indirect band gap?

Dear Samares,

There are some parameters that have effect on band gap of materials. Such as bond nature of materials (difference in electronegativity of atoms ), lattice constant (value of wave function overlap), ionic radii of atoms.

In addition to the suggestions of the colleagues, please visit for example http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_3.htm to visually figure out the deoendence of bandgap on lattice constant.

If you see Fig. 2.3.2 (depicted in many solid-state physics books), performed by applying Block function on a periodic structure (i.e. the Crystal lattice) for a pure covalent bonded crystal (i.e. diamond), a reduction of the lattice constant a induces a larger bandgap. Why? The reason is in the reply of Dr. de Araujo.

I think this plot is useful for visualizing and remembering this dependence.

Maybe a broad and phenomenological answer is that the smaller the atomic spacing, the stronger the interatomic interaction and hence the larger the energy states splitting.

But of course it has to do with the nature of the interacting atoms as well.

I deeply appreciate the response from colleagues. Some direct replies first.

Suresh Sharma: The "a" - "Eg" correlation is observed in the case of group IV elemental semiconductors; also in the case of II-VI compound and ternary I-III-VI2 and II-IV-V2 semiconductors. For example, C (Diamond) and Si, have band-gaps of 5.46, 1.11 eV, and lattice constants of 0.356, 0.543 nm respectively. 

Jaafar Jalilian: I agree that many other factors need to be considered; one of these is certainly electronegativity difference between the cation and the anion in the case of compound semiconductors. All compound semiconductors will necessarily be partly ionic and not completely covalent solids; the amount of ionicity will depend upon the cation/anion electronegativity difference. Other parameters being the same, a larger ionicity may reflect in a larger band-gap. 

Ahmed Morshed: I agree with you, as elaborated in the following:

All chemical bonds are basically electrostatic; the difference between covalent and ionic bond is that in the case of the latter, cations and anions are fixed, whereas, in the case of the former these interchange. For semiconductors with the same crystal structure, e.g. tetrahedral, small lattice constant means small interatomic distance, and hence, a strong electrostatic attraction. The band-gap represents the energy needed for bond-breaking, hence reflects the strength of the attractive force. 

We need to be careful, when we compare the "a" and "Eg" of semiconductors. We need to compare semiconductors with the same crystal structure. Next important parameter is the percentage of ionic bonding. Hence, it will not be proper to compare III-V with II-VI semiconductors, because II-VI compounds will in general be more ionic than III-V compounds, or, group IV elemental semiconductors with any compound semiconductors. 

Different Ionic properties may differ the electrostatic behavior whereas lattice constant or  spacing also influences the interactive energy function or electrostatic behavior thus two factors mostly influence for the variation of energy gap of semiconductor. Another factor partially acts to influence both the ionic and electrostatic behaviors is the dielectric constant of the materials. Its also vary with frequency thus we found bit variation of band gap at high frequency.

Dear Samares, and colleagues:

welcome,

I would like to add my comment on the answer of the question:

It is so as the lattice constant decreases the interatomic distance will be reduced. As a consequence binding forces between the valence electrons and  and the parent atoms will increase. These valence electrons will occupy the valence band. Since the valence electrons are bound, they have to be supplied with energy to make them moving free inside the material and become conduction electrons. The minimum energy that must be given to valence electrons to become a conduction electron is the energy gap.

So, as the valence electrons get more bound by decreasing the interatomic distance, the more energy required to make them free in the conduction band. 

So, A direct consequence of decreasing the lattice constant is the increase in the energy gap. As a rule of thumb the energy gap is inversely proportional to the interatomic distance. 

An other factor affecting the energy gap is the dielectric constant, which depends on the density of atoms and their polarizability. The dielectric constant is proportional to N the density of atoms per cm^3 and the alpha the polarizability. The polarizability depends on the electronic structure of the atom.  It is so that the energy gap is inversely proportional to  the dielectric constant which in  turn is inversely proportional to the inter-atomic distance. 

Therefore, there are to competing effects to the lattice constant on the energy gap.

the binding energy and the dielectric screening.

Best wishes