Size quantization effect

In solid-state physics or quantum chemistry, size quantization refers to the increase in the size of the band gap with decreasing semiconductor particle size . The effect is only noticed with particles with a diameter in the nanometer range. The band gap distance is then no longer a material constant, but depends on the particle size. Generally speaking, nanoparticles are used when they have a diameter of less than 100 nm.

Nanoparticles in the single-digit nanometer range are so small that they limit the wave function of the electrons inside them. The particle in the box model can therefore be used as a model for a nanoparticle. The wave function of the electron can only propagate within the nanoparticle, as the potential at the edges of the particle is very high. The De Broglie wavelength of the electron is approx. 7.6 nm in a vacuum, i. This means that with nanoparticles smaller than 7.6 nm there is a limitation of the wave function. The wavelength of the wave function must now decrease with decreasing particle size in order to still fit into the nanoparticle. The equation for the De Broglie wavelength shows that the momentum of the electron must increase. Now we consider the energy value of the particle in the box:

${\displaystyle E={\frac {n^{2}\cdot h^{2}}{8mL^{2}}}}$

A small calculation example quickly shows that the LUMO (the energetically lowest unoccupied state, here with the main quantum number n  = 2) is more destabilized than the HOMO (the energetically highest occupied state, here with the main quantum number n  = 1) and the band gap energy increases with decreasing particle size.

Big Part:

${\displaystyle E\sim {\frac {n^{2}}{L^{2}}}={\frac {1^{2}}{1^{2}}}=1}$

${\displaystyle E\sim {\frac {n^{2}}{L^{2}}}={\frac {2^{2}}{1^{2}}}=4}$

The difference between 4 and 1 gives the band gap energy 3

Small particle:

${\displaystyle E\sim {\frac {n^{2}}{L^{2}}}={\frac {1^{2}}{0{,}1^{2}}}=100}$

${\displaystyle E\sim {\frac {n^{2}}{L^{2}}}={\frac {2^{2}}{0{,}1^{2}}}=400}$

The difference between 400 and 100 gives the band gap energy 300

Another small effect arises from the small number of atoms in a nanoparticle. The band is formed by the overlap of many atomic orbitals. If the number of atomic orbitals decreases, the orbitals only overlap at higher energies, since the atomic orbital distances become smaller at higher energies.

The following equation ( Brus formula ) describes the increase in the band gap energy for nanoparticles in comparison to that of the (infinitely extended) solid, whereby one imagines the excitation of an electron from the valence band into the conduction band , leaving a hole behind :

${\displaystyle \delta E={\frac {h^{2}}{8R^{2}}}\cdot \left({\frac {1}{m_{\mathrm {e} }}}+{\frac {1}{m_{\mathrm {h} }}}\right)-{\frac {1{,}8\,e^{2}}{4\pi \varepsilon \varepsilon _{0}R}}}$

It is R , the particle size, e is the electron charge, the dielectric constant in vacuum, the effective mass of the electron and the hole of the. The last term of the equation stands for the stabilization of the exciton (electron-hole pair) through the Coulomb interaction of the hole with the electron. Stabilizing terms have the sign “-”. The first term corresponds to the energy of the particle in the box. The limitation of the wave function leads to a destabilization. Therefore this term has a "+" as a sign. ${\displaystyle \varepsilon _{0}}$ ${\displaystyle m_{\mathrm {e} }}$ ${\displaystyle m_{\mathrm {h} }}$

Web links

• University of Osnabrück: Properties of crystalline substances ("solids")