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
Georgian - German School and
Workshop in Basic Science
Binding Energy
Ry(H) = 13.6 eV
Binding 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 1030, 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
structures.
Interaction
Engineering”
using
low
dimensional
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
levels.
between the energy
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