Quantum confinement effects take over when we shrink an object to the size-scale of an emergent phenomenon. As we shrink a semiconducting particle (such as quantum dots), the size of the particle approaches the size of the electron-hole distance known as the Bohr radius. CdSe has a Bohr radius of 56 Angstroms, for example. For a 3D spherical particle, we consider the energy of a "particle in a box" (as chemists see it), or the energy of a particle in a "infinite potential well" (as physicists like to think of it), to describe the band gap energy.

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

This is the band gap energy for a spherical box (same lengths in all 3 dimensions), n is energy level, h is planck's constant, m_c is the effective mass of a point charge, and R is the radius of our box (or the size of the particle). We see from this that as our particle increases in size, the band gap energy decreases. Therefore, as size varies in QDs, the energy changes because the exciton in the QDs behaves like a "particle in a box."

Quantum confinement effects take over when we shrink an object to the size-scale of an emergent phenomenon. As we shrink a semiconducting particle (such as quantum dots), the size of the particle approaches the size of the electron-hole distance known as the Bohr radius. CdSe has a Bohr radius of 56 Angstroms, for example. For a 3D spherical particle, we consider the energy of a "particle in a box" (as chemists see it), or the energy of a particle in a "infinite potential well" (as physicists like to think of it), to describe the band gap energy.

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

This is the band gap energy for a spherical box (same lengths in all 3 dimensions), n is energy level, h is planck's constant, m_c is the effective mass of a point charge, and R is the radius of our box (or the size of the particle). We see from this that as our particle increases in size, the band gap energy decreases. Therefore, as size varies in QDs, the energy changes because the exciton in the QDs behaves like a "particle in a box."

For bulk material, we do not have discrete energy states. But, as the particle becomes smaller, the energy levels become discrete.

Now if you imagine a quantum dot according to "a particle in a box" concept, then, you know that the energy gap between different energy states is inversely proportional to the square of the length of the box. For a quantum dot, the length of the box is actually its size. So, with decrease in size (i.e., with decrease in the length of the box), the energy gap will increase and with increase in size the reverse will happen.

This is primarily the reason behind the dependence of the band gap energy of quantum dots on its size.

All these definitions from books don't approach to real understanding. Has anyone ever seen linear spectra ("atom-like" spectra) for QD? I haven't

Hello Dmytro,

The image below shows the change in band gap with change in size of a QD. All of them are the same material (CdSe in this case), with varying size. The emission (as well as absorption) wavelength changes with the size, and that is why they have different colours.

http://www.umt.edu/ethics/Debating%20Science%20Program/ODC/imx/Fluorescence_of_CdSe-CdS.jpg

An atom-like spectra is rarely seen in the case of QDs, as the typical QDs reported have 2- 50 nm size, which contains 10000 + atoms per QD and the spectra is no longer atom-like. Also these QDs need to be highly monodisperse to get sharp spectral features.

You can probably find atomic like spectra reported for very small clusters (mostly in gas phase) in the literature.

In a nutshell, you need vary small and highly monodisperse particles to observe atom-like spectra.

As the particles get smaller, you also need to take more care of surface effects. The spectra of small quantum dots is quite well known from theory (albeit inexactly). Look up papers on "jellium sphere"s by John Perdew and others.

Is it possible to prepare quantum dots with different color (different sizes) for TiO2?

Quantum size effect in TiO2 nanoparticles. Does it exist?

To Susana López Ayala

Band gap energy of TiO2 is too wide to absorb visible light.so I think Quantum size effect in TiO2 nanoparticles doesnt exist .

it seems that if there is, at least this article suggests

Self-ordered TiO2 quantum dot array prepared via

anodic oxidation.

Drbohlavova et al. Nanoscale Research Letters 2012, 7:123

Since TiO2 band gap does not lie in visible region you will not observe size dependent tunable visible emissions but there is certainly quantum confinement effect once the size approaches Bohr exciton radius limit. This will be reflected in their optical absorption spectra.

The band gap energy of any semiconductor is given by

E = hc/λ

As for nanoparticles λ depends on size, the band gap energy increases as λ decreases.

I agree with all the comments given above, however, the exact reason for increase of the band gap for nanoscale materials is related to the overlap of the atomic orbitals involved in a specific material. In the bulk state, the number of atoms involved in bonding are high and each atom shares some atomic orbitals in bonding to overlap. Therefore, we will have many molecular orbitals( ignoring the orbitals with low energy) half of them are bonding molecular orbitals (valence band) and the other half are antibonding orbitals (conduction band) and the gap between these two will be small. In the nanoscale materials, however, the number of atoms and also atomic orbitals involved in the overlap are much less than the bulk counterpart and the band gap will increase. In any molecule, the more atomic orbitals take part in bonding, the less energy gap will be between the molecular orbitals.

Why does band gap energy of quantum wire vary with its diameter? Also what is the mathematical relationship between diameter and energy gap.   

Dear Gone Rajender, could you please provide the source of this equation?

As it was mentioned above, by analogy to the Hydrogen atom the most probable distance between the electron and hole in an exciton is given by the so-called exciton Bohr radius (a0). The exciton Bohr radius provides a very useful length scale to describe the spatial extension of excitons in semiconductors, and ranges from *2 to *50 nm depending on the semiconductor. The exciton Bohr radius a0 and the band gap of the semiconductor are correlated, so that materials with wider band gaps possess smaller a0 (e.g., Eg and a0 are, respectively, 0.26 eV and 46 nm for PbSe, 1.75 eV and 4.9 nm for CdSe, and 3.7 eV and 1.5 nm for ZnS).

But because of the effect, known as quantum confinement, the electronic structure of semiconductor NPs is strongly size dependent. As discussed above, confinement begins to affect the exciton wave function as the size of the NP approaches a0

There are two different approaches to understand it:

In the first approach (“top-down”), the NP is treated as a small piece of semiconductor material in which the exciton is spatially confined. In this approach,

the NP can better be represented as a spherical potential box with the quantized energy levels :

Econf= (h^2 n^2)/(8 mD^2)

It is obvious that decreasing the dimension of the quantum well causes an increase in the amount of energy of each level. Resulting in a larger band gap between the energy levels.

The second method (“bottom-up”) involves a quantum chemical (molecular) approach, in which the NP is build up atom by atom and is treated as an increasingly larger molecular cluster that eventually evolves into a bulk semiconductor crystal.

As the molecule (e.g, a small CdSe cluster) becomes larger, the number of (Atomic orbitals) AOs that are combined to form (Molecular orbitals) MOs (bonding and anti-bonding) increases, leading to an increasingly larger number of energy levels and decreasing the HOMO-LUMO energy gap. For a sufficiently large number of combining atoms (i.e., when the bulk limit is reached) the energy levels become so numerous and so closely spaced that a quasi-continuum (i.e., an energy band) is formed, analogous to the conduction and valence bands described above. The HOMO level is the top of the VB, whereas the LUMO is the bottom of the CB.

The electronic structure of a semiconductor NP consisting of a few tens to a few thousand atomic valence orbitals, forming as many MOs will be characterized by energy bands with a large density of levels at intermediate energy values and discrete energy levels near the band edges, where the density of MO states is small. Moreover, the HOMO-LUMO energy gap will be larger than for bulk and size-dependent, increasing with decreasing size of the NC.

With reference to TiO2 Quantum Dots I recommend the recent article by Stengl et al: Fast and Straightforward Synthesis of Luminescent Titanium(IV) Dioxide Quantum Dots; Journal of Nanomaterials: Volume 2017, Article 3089091 (https://doi.org/10.1155/2017/3089091)

(Excitation 350-400nm, fluorescence maxima at 450-500nm)

Article Structural and Photoluminescent Properties of a Composite Ta...

i have doubt from Ruslan's answer. Yes it is true that the electron localization (position) is inversely size dependent but how the kinetic energy increase when the potential energy increase as QD is decreases. To my opinion, this is not consistent with Heisenberg uncertainty principle.

The bandgap energy of the quantum dot increases with the reduction of its size because of the quantum confinement. In the coming comment i will explain how this occurs:

The confinement of the electrons means limiting their movement  in a space with dimensions of their de Broglie   wavelength . The consequence of this confinement in space is the quantization of their energy and momentum. In this case they are subjected to principles of the quantim mechanical motion rather than classical mechanics.

The quantum dot is a an assembly of atoms of specific material that has few nanometer dimensions. It is termed as a virtual atom.

Because of this microscopic size the electrons are confined inside the dot. This is the same for the electrons in an atom they are confined and localized in the atomic space.

In order to obtain the possible energy levels in the atom or in the dot because of the space confinement one has to use quantum mechanical laws. That is one has to solve the Schrodinger equation  with relevant boundary condition. 

The motion of electrons in the confined space can be modeled by the motion of a particle in a potential well with infinite walls. 

While the atom is modeled by one well with infinite wall with size of the atom, the electrons in the quantum dot can be modeled by a potential well with infinite walls with the minimum energy level is that of the conduction band. Since we are interested in the confinement in the conduction band. Like wise we are interested in the holes in the valence band therefore, there will be an inverted well for the holes in the valence band. 

So, the picture is now is that we have electrons confined in the conduction band and holes in the valence band.

If there is no confinement as in big crystal, there will be no confinement and the electrons in the conduction band will occupy the bottom of the conduction band and the holes will reside at the top of the valence band.

When we solve Schrodinger equation in potential well with infinite wall we find that the electrons will have only discrete energy levels in the valence band and so doe the holes in the valence band.

The energy level diagram in the conduction band will be similar to that of an atom and so the energy levels of the holes in the valence band will have also discrete energy levels as the shown in the energy level diagram  given in the attached link: archive.cnx.org/.../optical-properties-of-group-12-16-ii-vi-semiconductor-nanoparticl

The energy levels above the conduction band has an energy measured from the conduction band edge Ene = h^2 n^2/ 8 pi^2 me d^2,

where h is the Planck constant, n the order of the level, me is the effective mass of electrons in the quantum dot and d is the length of the dot.

The same equation holds for the holes energy Enh in the valence band with mh the hole effective mass instead of  of me.

So, the energy gap for the quantum dot becomes Egqd= Eg + E1e + E1h,

with E1e is the excess energy of the first level in the quantum dot E1e and E1h is that corresponding level for the holes.

It is clear that one can tune the effective energy gap in the quantum dot by changing its size d.

Best wishes 

To get more insight in the subject please follow the book:https://www.researchgate.net/publication/236003006_Electronic_Devices

Best wishes

Book Electronic Devices