Georgian - German School and Workshop in Basic Science Tamar Tchelidze Ivane Javakhishvili Tbilisi State University, Faculty of exact and Natural Sciences

Georgian - German School andWorkshop in Basic Science


Tamar TchelidzeIvane Javakhishvili Tbilisi State University,

Faculty of exact and Natural Sciences

Coulomb Interaction in quantum structures - Effects of

space and dielectric confinement



•Concept of Exciton •Exciton in Low Dimensional Structures, effect of Space confinement•Coulomb Interaction in Ideal 2D structure•Effect of Dielectric Confinement •Effect of mass confinement



The Concept of Exciton

The absorption of photon by an interband transition in a semiconductor or insulator create an electron in the conduction band and hole in the valence band. This oppositely charged particles attract each other though Coulomb interaction, and there may be the probability of the formation of neutral electron-hole pair called an Exciton.



There are two types of excitons

Wannier-Mott excitons

Frenkel exciton



We’ll consider Wannier-Mott excitons are mainly observed in semiconductors

To study the exciton we can apply a moddified Bohr model of the hydrogen atom

this illustrative approximation can explain the majority of the principal features observed in the optical spectra of excitons



Binding Energy

Ry(H) = 13.6 eVBinding Energy o electron in Hydrogen atom

• different ratio of the effective masses. Unlike the pair of a light electron and a very heavy proton, the exciton is composed of two light quasi-particles with comparable masses me mh which entails a lower stability of the exciton in comparison with the hydrogen atom• electron and hole are in medium with dielectric constant ranging between 10-30, which also reduces exciton binding energy

For stability of Excitons binding energy must be higher than k∼ BT



kBT = 10 meV , T ≈ 110 K



on the one hand Excitons in most of semiconductors are not observable at room temperature, because of the low binding energy

on the other hand excitonic emission is very important for opto-electronic application, as it is •Narrow •High energetic



“Coulomb Interaction Engineering” using low dimensional structures.

Confinement of exciton in nanostructures with linear size small compared to exciton Bohr radius – ax seems to be an evident solution of these problem



Restriction of motion of electrons and holes in some directions by the interval of length d results in quantization of

the energy with difference between the energy levels.

the Coulomb interaction can result only in a small correction to this energy and to this restricted motion.



It can affect essentially only the motion in the remaining unrestricted degrees of freedom. Thus, the problem of an exciton as a bound state of electron and hole due to Coulomb interaction becomes low D=3-I dimensional. Here I is the number of freedom degrees in which the free motion of electrons and holes is confined to intervals small compared to the exciton radius and D the effective dimensionality of the exciton.



Quantum wells – 1D confinment, 2D excitons



In (x,y) plane the motion of electron and hole is governed by Coulomb interaction, while in z direction it is governed by space confinement

Energy levels for 2D Coulomb problem



Dielectric confinement

Dielectric confinement is effective reduction of the dielectric constant due to the penetration of electric field into barrier medium with a small dielectric constant.

Dielectric confinement reduces the effective dielectric constant of the system, screening of the e-h Coulomb interaction, and consequently, increases the binding energy.



If the semiconductor layer is very thin exciton problem becomes two dimensional Coulomb problem with dielectric constant of surrounding barrier

Discontinuity of dielectrical constant induces polarization charges at the interface

Georgian - German School andWorkshop in Basic SciencewitGeorgian - German School andWorkshop in Basic Sciencewit

Mass confinement

Mass confinement is increase of effective mass of carriers through penetration of wave function into the barrier region with higher effective mass, which increases binding energy.



Question: how many times is increased binding energy of exciton in 1 nm layer which is embedded in material with dielectric constant 5 times smaller than dielectric constant of layer material



Method of image charges

Discontinuity of dialectical constant induces polarization charges at the interface, which can be incorporated using the method of image charges

Potential in well region is given by placing image charge e1 at some position in the barrier and regarding that the whole structure has dielectric constant ε1

Potential inbarrier region is given by placing image charge e2 at the same position in the barrier and regarding that the whole structure has dielectric constant ε2



Thank you for attention

Documents

Georgian - German School and Workshop in Basic Science Tamar Tchelidze Ivane Javakhishvili Tbilisi State University, Faculty of exact and Natural Sciences