You can find details for such dependence in the following power point given by

M. S. Omar

"Solid Surface and Nanoscale materials Structure"

July 2016

DOI10.13140/RG.2.2.31947.59684

Conference: CharmoAt: University of CharmoAffiliation: University of Salahaddin

Thank you dear Prof. Mustafa, the PPT is very useful, thank you.

I think they are inversely related.

Generally, band gap energy increases with decrease of particle sizes, which is called quantum confinement effect. You can observe this effect through UV-vis spectroscopy, which shows a blue shift in the spectrum by decreasing the particle size. Please search about this title by using “quantum confinement effect”.

Band gap increases with decrease in size due to electron confinement at nano-scale so called "quantum size effect".

In a simple words electrons are confined i.e occupied less space than bulk, hence VBM and CBM potentials are shifted more +ve and -Ve respectively, resulting high band gap. Brus equation explains the mathematical relation or quantum mechanics of this phenomenon. For Brus equation please find the attachment.

Dear Dr Aziz, Thank you for your valuable explanation and suggestion.

Dear Dr. Kumar, thank you very much for your kind cooperation, your explanation is very easy to understand, thank you.

The energy confinement is inversely proportional to the size of nano particles. (radius). As the size of nano particles increase, the band gap of nano particle decrease, but never approaches to zero.

means. Lower the nanoparticles size, higher is the band gap due to quantization effect. In general, the size of nano particles and band gap are inversely proportional to each other.

Thank you very much Jagdeep.

The band gap energy increases with decrease of particle sizes

Thank you dear Dr Waleed

One treats the electronic properties of a single semiconducting  nanoparticle (quantum dot, for example) as that of a particle in a box. This yields the discretisation of energy levels, and the presence of degeneracy depending upon the number of dimensions that are subject to quantum confinement. For instance, a quantum dot is considered to be quantum-confined in all the three dimensions, whereas a quantum wire would have confinement in two of the three dimensions.

The relationship between energy band gap and the particle size is given by the Effective mass approximation:

E=Eg+h2π2/2R2(1/me+1/mh)-1.8e2/4π(ϵ0)R+smallerterms.

Here, E is the energy band gap of semiconducting nanoparticle, Eg is the energy band gap of the bulk semiconducting phase, me and mh are the effective masses for electrons and holes in the semiconducting phase, and R is the radius of the nanoparticle (assuming spherical symmetry as an approximation). h is the Planck's constant and ϵ0 is the permitivitty of free space.

Reference: Nanotechnology: Principles and Practices | Sulabha K. Kulkarni | Springer , Ch. 14, pp. 362).

In a single atom of materials (i.e. semiconductor) the energy band gap is equal to the distance between ground state and first excited state, while in bulk material, both levels are broadened. This broadening leads to the narrowing of energy band gap. In nanoparticles, broadening should be less than that in the bulk, and narrower energy band gap is expected. As we shrink the semiconducting nanoparticles (quantum dots), the size of particle approaches the size of electron-hole distance known as the Bohr radius. For a 3D spherical particle, we consider the energy of a particle in a "infinite potential well" to describe the energy band gap.

E_n=(h^2 n^2)/(8 m_c R^2)

This is the energy band gap for a spherical box (same lengths in all the three dimensions), n is the energy level, h is the planck's constant, m_c is the effective mass of a point charge, and R is the radius of box (or the size of particle). As the particle increases in size, the energy band gap decreases. Therefore, as size varies in quantum dots, the energy changes because the exciton in quantum dots behaves like a "particle in a box."

Thank you dear Muhammad for you valuable explanation.

Simply told both are inversely proportional to each other.

Band gap energy is inversely proportional to the size of the nanoparticles due to the 'quantum confinement effect'.

Thank you Rohini

The band bap decrease or increase with respect to the particles size is actually depends on the principal host inwhich the metal nanoparticles embedded.

But, in most of the cases, the energy band gap decreases with rise in size of the metal nanoparticles due the red-shift of absorption peak of nanoparticles .

However, for silver nanoparticles the converse results (in below said paper) are available in literature. For indepth understanding refer the fallowing article..

"Plasmonic resonance of Ag nanoclusters diffused

in soda-lime glasses" Phys.Chem.Chem.Phys.,

2015, 17, 8596.

and also

Mostafa A. El-Sayed articles

You will get mathematical relations and also you can understand as well

I agree with Dr. Jannath G Reddy.

Hello,

In general, the band gap decreases with decreased the crystal size (10-100 nm)..... sometime the verse is occurred if the crystal size (1-10 nm) that beyond to quantum size effect .

Hello,

• In general, the band gap decreases with decreased the crystal size (10-100 nm).....
• sometime the verse is occurred if the crystal size (1-10 nm) that beyond to quantum size effect .

Hello,

Band gap increases with decrease in size due to electron confinement at nano-scale so called "quantum size effect".

Please refer the following link:

https://www.researchgate.net/post/Why_does_the_band_gap_increase_when_decreasing_the_size_of_nanostructures

Due to the Moss-Bustein effect, as the particle size decreases the optical band gap can be widened by shifting towards higher frequency side gradually.

Ref: L.S. Rao et al. / Materials Chemistry and Physics 203 (2018) 133-140

I am quite agree with L. M. Ahmed, In general, the band gap decreases with decreased the crystal size ( higher than 10nm). The verse is occurred if the crystal size (1-10 nm) that beyond to quantum size effect .