Coulomb Interaction in quantum structures - Effects of Georgian - German School and Workshop in Basic Science Coulomb Interaction in quantum structures - Effects of space and dielectric confinement Tamar Tchelidze Ivane Javakhishvili Tbilisi State University, Faculty of exact and Natural Sciences

Georgian - German School and Workshop in Basic Science 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

Georgian - German School and Workshop in Basic Science 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.

Georgian - German School and Workshop in Basic Science There are two types of excitons Wannier-Mott excitons Frenkel exciton

Georgian - German School and Workshop in Basic Science 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 A striking resemblance with the hydrogen atom is already evident at first sight; the role of the proton is played here by the hole. If we regard the electron and hole as point charges characterized by their charge and effective masses (the so-called effective mass approximation), we can apply a modified Bohr model of the hydrogen atom. We shall see that this illustrative approximation can explain the majority of the principal features observed in the optical spectra of Wannier excitons in semiconductors

Georgian - German School and Workshop in Basic Science Binding Energy Binding Energy o electron in Hydrogen atom Ry(H) = 13.6 eV 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 ∼ kBT

Georgian - German School and Workshop in Basic Science kBT = 10 meV , T ≈ 110 K

Georgian - German School and Workshop in Basic Science 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

Georgian - German School and Workshop in Basic Science “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

Georgian - German School and Workshop in Basic Science 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.

Georgian - German School and Workshop in Basic Science 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.

Georgian - German School and Workshop in Basic Science Quantum wells – 1D confinment, 2D excitons

Georgian - German School and Workshop in Basic Science 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

Georgian - German School and Workshop in Basic Science 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.

Georgian - German School and Workshop in Basic Science Discontinuity of dielectrical constant induces polarization charges at the interface If the semiconductor layer is very thin exciton problem becomes two dimensional Coulomb problem with dielectric constant of surrounding barrier

Georgian - German School and Workshop 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.

Georgian - German School and Workshop in Basic Science 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

Georgian - German School and Workshop in Basic Science 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

Georgian - German School and Workshop in Basic Science Thank you for attention