Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд
In review, deals with the theory of exciton quasimolecules (formed of spatially separated   electrons and holes) in a nanosystems that consists of semiconductor and dielectric colloidal quantum dots (QDs) synthesized in a dielectric and semiconductor matrixs. It has been sh...
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Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine
2022
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Surface| _version_ | 1869291910998261760 |
|---|---|
| author | Покутній, С. І. Терець, А. Д. |
| author_facet | Покутній, С. І. Терець, А. Д. |
| author_institution_txt_mv | [
{
"author": "С. І. Покутній",
"institution": "Інститут хімії поверхні ім. О.О.Чуйка Національної академії наук України \/ Вроцлавський науково-технічний університет"
},
{
"author": "А. Д. Терець",
"institution": "Інститут хімії поверхні ім. О.О.Чуйка Національної академії наук України"
}
] |
| author_sort | Покутній, С. І. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2023-04-20T10:24:32Z |
| description | In review, deals with the theory of exciton quasimolecules (formed of spatially separated   electrons and holes) in a nanosystems that consists of semiconductor and dielectric colloidal quantum dots (QDs) synthesized in a dielectric and semiconductor matrixs. It has been shown that the exciton  quasimolecule formation is of the threshold character and possible in a nanosystem, where the distance D between the  surfaces of QD is given by the condition    (where and are some critical distance).
We have shown that in such a nanoheterostructures acting as “exciton molecules” are the QDs with excitons localizing over their surfaces. The position of the quasimolecule state energy band depends both on the mean radius of the QDs, and the distance between their surfaces, which enables one to purposefully control it by varying these parameters of the nanostructure.
It was found that the binding energy of singlet ground state of exciton quasimolecules, consisting of two semiconductor and dielectric QDs is a significant large values, larger than the binding energy of the biexciton in a semiconductor and dielectric single crystals almost two orders of magnitude. It is shown that the major contribution to tue binding energy of singlet ground state of exciton quasimolecule is made by the energy of the exchange interaction of electrons with holes and this contribution is much more substantial than the contribution of the energy of the Coulomb interaction between the electrons and holes.  It is established that the position of the exciton quasimolecule energy band depends both on the mean radius of the QDs and the distance between their surfaces.
It is shown that with increase in temperature above the threshold (), a transition can occur from the exciton  quasimolecule to exciton state. It has been found that at a constant concentration of excitons (i.e. constant concentration of QD) and temperatures Т below  , one can expect a new luminescence band shifted from the exciton band by the value of the exciton quasimolecule binding energy. This new band disappears at higher temperatures (). At a constant temperature below , an increase in exciton concentration (i.e. in QD concentration) brings about weakening of the exciton luminescence band and strengthening of the exciton quasimolecule.
These exciton quasimolecules are of fundamental interest as new quasi-atomic colloidal nanostructures; they may also have practical value as new nanomaterials for nanooptoelectronics. The fact that the energy of the ground state singlet exciton  quasimolecule is in the infrared range of the spectrum, presumably, allow the use of a quasimolecule  to create new infrared sensors in biomedical research. |
| doi_str_mv | 10.15407/Surface.2022.14.049 |
| first_indexed | 2025-09-24T17:25:28Z |
| format | Article |
| fulltext |
Поверхня. 2022. Вип. 14(29). С. 49–62 49
PASC: 73.21 – b doi: 10.15407/Surface.2022.14.049
EXCITON QUASIMOLECULES IN NANOSYSTEMS WITH
SEMICONDUCTOR AND DIELECTRIC COLLOIDAL
QUANTUM DOTS: A REVIEW
S. I.1,2 Pokytnyi, A.D.1 Terets
1 Chuiko Institute of Surface Chemistry of National Academy of Sciences of Ukraine, 17 General
Naumov Str., Kyiv 03164, Ukraine, E-mail: pokutnyi.serg@gmail.com;
2 Wrocław University of Science and Technology, 27 Wyb. Wyspiańskiego,
Wrocław 50-370, Poland, E-mail: terets.andrey@gmail.com
In review, deals with the theory of exciton quasimolecules (formed of spatially separated
electrons and holes) in a nanosystems that consists of semiconductor and dielectric colloidal
quantum dots (QDs) synthesized in a dielectric and semiconductor matrixs. It has been shown
that the exciton quasimolecule formation is of the threshold character and possible in a
nanosystem, where the distance D between the surfaces of QD is given by the condition
(1) (2)
c cD D D (where (1)
cD and (2)
cD are some critical distance).
We have shown that in such a nanoheterostructures acting as “exciton molecules” are
the QDs with excitons localizing over their surfaces. The position of the quasimolecule state
energy band depends both on the mean radius of the QDs, and the distance between their
surfaces, which enables one to purposefully control it by varying these parameters of the
nanostructure.
It was found that the binding energy of singlet ground state of exciton quasimolecules,
consisting of two semiconductor and dielectric QDs is a significant large values, larger than the
binding energy of the biexciton in a semiconductor and dielectric single crystals almost two
orders of magnitude. It is shown that the major contribution to tue binding energy of singlet
ground state of exciton quasimolecule is made by the energy of the exchange interaction of
electrons with holes and this contribution is much more substantial than the contribution of the
energy of the Coulomb interaction between the electrons and holes. It is established that the
position of the exciton quasimolecule energy band depends both on the mean radius of the QDs
and the distance between their surfaces.
It is shown that with increase in temperature above the threshold ( cT T ), a transition
can occur from the exciton quasimolecule to exciton state. It has been found that at a constant
concentration of excitons (i.e. constant concentration of QD) and temperatures Т below cT , one
can expect a new luminescence band shifted from the exciton band by the value of the exciton
quasimolecule binding energy. This new band disappears at higher temperatures ( cT T ). At a
constant temperature below cT , an increase in exciton concentration (i.e. in QD concentration)
brings about weakening of the exciton luminescence band and strengthening of the exciton
quasimolecule.
These exciton quasimolecules are of fundamental interest as new quasi-atomic colloidal
nanostructures; they may also have practical value as new nanomaterials for
nanooptoelectronics. The fact that the energy of the ground state singlet exciton quasimolecule
is in the infrared range of the spectrum, presumably, allow the use of a quasimolecule to create
new infrared sensors in biomedical research.
50
Keywords: exciton quasimolequle, electrons, holes, quantum dots, binding energy, exchange
interaction, Coulomb interaction
Introduction
The studies of quasi – zero dimensional (0-D) nanosystems made up of spherical
colloidal quantum dots (QDs) with the mean radius a in the range of 1 to 10 nm consisting of
semiconductor (germanium, cadmium sulfide and selenide, gallium arsenide, zinc selenide) and
dielectric (aluminum oxide) materials synthesized in dielectric (semiconductor) matrices attract
considerable interest due to their unique photoluminescence properties, ability to effectively emit
light in the visible and near infrared spectral range at room temperatures [1–11]. Optical and
electro-optical properties of such quasi 0-D nanosystems are to a large extent governed by the
energy spectrum of the spatially confined electron-hole pair (exciton) [1, 2, 5– 21].
In Ge/Si heterostructures, structures that have self-assembled Ge/Si nanoislands are
promising in the implementation of effective sources of infrared radiation, since the
photoluminescence signal of such nanostructures, in the spectral range (0.25 to 1.14 eV) is
observable all the way to room temperature [1,2,6]. To create new Ge/Si - based heterostructures
with new effective opto-electronic devices, the mechanism of light absorption in such
nanoheterostructures must be studied [1–6]. Ge/Si hetero structures with germanium QDs are
classified as heterostructures of the second type. Such nanoheterostructures are characterized by
the presence of significant gaps in the valence and conduction bands. The ground electron state
therein is located in the silicon matrix, and the ground hole level is in the volume of the
germanium QD. The significant gap in the valence band (in the order of 610 meV) causes hole
localization in the QD volume. A significant gap in the conduction band (in the order of
340 meV) is the potential barrier for electrons (electrons move throughout the matrix and do not
penetrate into the QD volume) (see Fig. 1) [1–6].
Fig. 1. Band diagram of the QD–matrix nanoheterostructure. In the nanoheterostructure, the QD
is a potential well for a hole and a barrier for an electron. The energies (1) (1), ,c vE E and
(2) (2),c vE E correspond to the positions of the bottom of the conduction band and the top
of the valence band of the matrix and QD, respectively
The idea of a superatom (or artificial atom) was fruitful for the development of
nanophysics [22, 23]. A superatom (quasiatomic nanoheterostructures) consists of a spherical
QD with radius a, the volume of which contains a semiconductor (or dielectric) material. The
QD is surrounded by a dielectric (semiconductor) matrix. In this nanosystem, the lowest
51
electronic levelis in the matrix, and the lowest hole level is within the volume of the QD. A
electron can be localized in a potential well above the QD surface and, at the same time, the hole
moves in the volume of the QD. The electrons move in the matrix and do not penetrate the
volume of the QD. The energy of the Coulomb interaction of electrons with holes and the energy
of the polarization interaction of electrons with the interface (QD-matrix) form a potential well
in which the electron is localized over the surface of the QD. Certain orbitals, localized to the
surrounding QD correspond to the electrons in a superatom [15–21]. A substantial increase in the
binding energy of the ground state of an electron in a superatom in comparison with the binding
energy of an exciton in zinc selenide, cadmium sulfide and selenide, germanium, aluminum
oxide and single crystals was detected in Refs. [15–21].
The convergence of two (or more) QDs up to a certain critical value between surfaces
of the QD leads to the overlapping of electron orbitals of superatoms and the emergence of
exchange interactions. In this case, the overlap integral of the electron wave functions takes a
significant value. As a result, the conditions for the formation of quasimolecules from QDs can
be created [24–28]. One can also assume that the above conditions for the formation of
quasimolecules can be provided by external physical fields [29,30]. This assumption is
evidenced by the results of Refs. 29 and 30, in in which the occurrence of the effective
interaction between QDs at considerable distances under conditions of an electro magnetic field
was experimentally observed. In Refs. 31, energies of the ground state of “vertical” and
“horizontal” located pairs of interacting QDs (“molecules”from two QDs) were determined as a
function of the steepness of the confining potential and the magnetic field strength. The quantum
part of a nanocomputer which was implemented on a pair of QDs (“molecules”from two QDs)
with charge states is n qubits [32].
Ref. 33, theoretically analyzed the exciton transitions in the double vertically conjugated
QD of germanium, separated by a silicon layer of d thickness. The QD of germanium was in the
form of pyramids, in which the ratio of the height h to the lateral dimension l = (10, 15, and 20)
nm was (≈ 0.1). The spatial structure of excitons and the oscillator was theoretically studied [33].
The ground state of the exciton in a single germanium QD corresponded to the configuration in
which the hole was in the ground state in germanium QD, and an electron moving in the silicon
matrix was localized near the apex of the pyramidal QD. It is shown that for small distances
(d < 3 nm) among QDs, the electron configuration is analogous to the case of single QDs. It is
found that with the increase of the distance d to the values (d = (3,0 to 3.5) nm) for l = (10, 15)
nm, the oscillator strength for the interband transition with the formation of the ground state of
the exciton can be much larger (up to a factor of 5) than the analogous value in a single QD.
In [4] using the method of electron-beam lithography obtained heterostructures which are
linear chains of QDs germanium on Si substrates. The average sizes of the QD Ge is less than 60
nm. It was noted [27] that, at such a QD content in the samples, one must take into account
the interaction between charge carriers localized above the QD surfaces.
In Ref. 33, in contrast to work [27], when studying the occurrence of exciton states
between the QD surfaces, the exchange interaction between electrons was not taken into account.
Such exchange interactions, as was shown in [27], made the main contribution to the binding
energy of the biexciton.
In Refs. 19 and 20, the theory of the exciton formed by spatially separated electron and
hole is developed (the hole moves in the bulk of a germanium QD and the electron is localized
above the spherical interface between the QD and the silicon matrix). It was found that the
binding energy of an exciton in such a nanosystem is much higher (almost an order of
magnitude) than the binding energy of an exciton in the silicon single crystal. In Refs. 19 and 20,
in the framework of classical electrodynamics, an expression was obtained that describes the
Coulomb interaction between an electron and hole. In Refs. 19 and 20, the exciton appeared as a
result of the Coulomb interaction between the electron and hole, which, in contrast to Refs. 1, 2
52
and 33, was dependent on the permittivities of the QD and the matrix. The energy spectrum of
the exciton, as a function of the radius of the germanium QD, was obtained in Ref. 19 and 20, for
QDs with radii exceeding 3 nm. This is due to the fact that the study of exciton states in the
nanosystem containing germanium QDs with radii a less than 3 nm in the framework of the
approach in which the expression describing the Coulomb interaction between the electron and
hole was obtained by the methods of classical electrodynamics is incorrect.
When there were large concentrations of cadmium sulfide, zinc selenide, germanium QDs
in the samples (from x ≈ 0.6% to x ≈ 1% ) a maximum, interpreted on the basis of the appearance
of bonded QD states, was detected in the low-temperature absorption spectra. In order to explain
the optical characteristics of such nanosystems, we proposed a model of a quasimolecule
representing two cadmium sulfide, zinc selenide and germanium QDs that form an exciton
quasimolecule as a result of the interaction of electrons and holes [24–28].
It was noted [24–28] that at such levels of QD content in the samples (from 6%0.x to
1%x ), one must take into account the interaction between charge carriers localized above the
QD surfaces. Therefore, in Refs. [24–28], we developed the theory of an exciton quasimolecule
(or biexciton) (formed from spatially separated electrons and holes) in a nanosystem that consists
of zinc selenide, cadmium sulfide and selenide, germanium, aluminum oxide QDs synthesized in
a dielectric and semiconductor matrixs. Using the variational method, we obtained the total
energy and the binding energy of the exciton quasimolecule (or biexciton) singlet ground state in
such a systems as functions of the spacing D between the QD surfaces and of the QD radius a.
We showed that the exciton quasimolecule formation is of a threshold character and is possible
in a nanosystem, in which the spacing between the QD surfaces exceeds a certain critical
spacing.
It was found that the binding energies of the exciton quasimolecule consisting of two
zinc selenide, cadmium sulfide and selenide, germanium, aluminum oxide colloidal QDs
acquired an anomalously high value that exceeds the binding energy of the biexciton in zinc
selenide, cadmium sulfide and selenide, germanium, aluminum oxide by almost two orders of
magnitude. This effect consisted in a significant increase in the binding energy of the ground
singlet state exciton quasimolecule due to the fact that because of their presence in nanoscale
interfaces (QD-matrix), the energy of the exchange interaction of the electrons with the holes
(renormalized Coulomb interaction between electrons and holes) in the exciton quasimolecule
was much greater than the energy exchange interaction between electrons and holes in a single
crystal [24–36]. It is established that the spectral shift of the low-temperature luminescence peak
[8] in such a nanosystems is due to quantum confinement of the energy of the exciton
quasimolecule ground state.
In review, deals with the theory of exciton quasimolecules (formed from spatially
separated electrons and holes) in nanosystems consisting of semiconductor and dielectric
colloidal QDs is insufficiently studied. Therefore, in this review, deals with the theory of exciton
quasimolecules (formed of spatially separated electrons and holes) in a nanosystems that consists
of semiconductor and dielectric QDs synthesized in a dielectric and semiconductor matrixs. It is
shown that exciton quasimolecules formation is of the threshold character and possible in
nanosystems, in with the spacing between the QDs surfaces is larger than a certain critical
spacing. It was found that the binding energy of singlet ground state of exciton quasimolecules,
consisting of two semiconductor and dielectric QDs is a significant large values, larger than the
binding energy of the biexciton in a semiconductor and dielectric single crystals almost two
orders of magnitude. It is shown that the major contribution to tue binding energy of singlet
ground state of exciton quasimolecule is made by the energy of the exchange interaction of
electrons with holes and this contribution is much more substantial than the contribution of the
energy of the Coulomb interaction between the electrons and holes. It is established that the
position of the biexciton state energy band depends both on the mean radius of the QDs and the
53
distance between their surfaces. This circumstance allows one to purposefully control the
position of the biexciton state energy band by varying these parameters of the nanostructure.
Energy of the exciton quasimolecule ground singlet state
Consider a model of a nanosystem that consists of two spherical semiconductor (or
dielectric) QDs [24–28]: QD(A) and QD(B) with radius a, grown in a matrix of semiconductor
(or dielectric) with a dielectric constant . Let the spacing between the spherical QD surfaces
be D, and the spacing between the spherical QD centers be L. Each QD is formed from a
semiconductor (or dielectric) material with dielectric constant For simplicity and without loss
of generality, we assume that holes ( )h A and ( )h B with effective masses hm are localized in
centers of QD(A) and QD(B) and electrons (1)e nd (2)e with effective masses (1)
em are localized
near the spherical surfaces of QD(A) and QD(B), respectively ( (1)Ar is the distance of the electron
е(1) from the QD(A) center; (2)Br is the distance of the electron е(2) from the QD(B) center; (2)Ar
is the distance of the electron е(2) from the QD(A) center; (1)Br is the distance of the electron
е(1) from the QD(B) center; 12r is the distance between the electron е(1) and е(2) (see Fig. 2).
The above assumption is reasonable, since the ratio between the effective masses of electron and
hole in the nanosystem is much smaller than unity (1) / 1 .e hm m Let us assume that there is
an infinitely high potential barrier on a spherical interface (QD – matrix). In the nanosystem the
holes do not therefore escape from the volume of the QD while the electrons do not enter the
QD.
Fig. 2. Schematic representation of a nanosystem consisting of two spherical semiconductor (or
dielectric) QDs: QD(A) and QD(B) of radii a. The holes h(A) and h(B) are located in the
QD(A) and QD(B) centers, and the electrons e(1) and e(2) are localized near the QD(A)
and QD(B) surfaces. (1)Ar is the distance of the electron е(1) from the QD(A) center;
(2)Br is the distance of the electron е(2) from the QD(B) center; (2)Ar is the distance of
the electron е(2) from the QD(A) center; (1)Br is the distance of the electron е(1) from the
QD(B) center, and is the distance between the electrons L is the
spacing between the QD centers, and D is the spacing between the QD surfaces.
Let us now use this model to consider the possibility of the formation of an exciton
quasimolecule from spatially separated electrons and holes (the holes are located at the centers of
QD(A) and QD(B) and electrons are localized near their spherical surfaces). Using adiabatic
approximation and the effective mass approximation, the Hamiltonian of the exciton
quasimolecule (of spatially separated electrons and holes) can be written in the form [24–28]:
54
(1) (2) intA BH H H H (1)
where (1)AH and (2)BH are the Hamiltonians of the excitons of spatially separated hole h(A) and
electron е(1) and hole h(B) and electron е(2), respectively. The contribution of the energy of
polarization interaction with the surface of QD to the Hamiltonians of the excitons (1)AH and
(2)BH can be, as a first approximation neglected [24–28]. Thus the exciton Hamiltonian (1)AH
takes the form The contribution of the energy of polarization interaction with the surface of QD
to the Hamiltonians of the excitons (1)AH and (2)BH can be, as a first approximation neglected
[24–28]:
2
(1) (1) (1) ( ) (1) ( ) (1) ( ) (1) (1) ( ) ( )( , ) ( , , ) ( ) ( ) ,
2
A e h A A h A A h A e A h A h A g
h
H V r r U r r a V r V r E
(2)
In (2), the first term is the kinetic energy operator of the exciton; the energy of Coulomb
interaction (1) ( )e h AV between the electron e(1) and the hole h(A) is described by the formula
[15]:
2
(1) ( )
(1) ( )
e h a
e h A
e
V
r r
(3)
where (1) (1)/e h e hm m m m is the reduced mass of the 2D exciton (of spatially separated
electron and hole) [6,7], 1 2 1 22 / ( ) is the dielectric constant of nanosystem.The
potentials
( )
( ) ( )
( )
0,
( )
,
h A
h A h A
h A
r a
V r
r a
(4)
(1) (1) (1)( ) ,e A eV r r a (5)
describe the motion of quasiparticles in the nanosystem in the model of an infinitely deep
potential well; and is the band gap in the semiconductor (or dielectric) material with
dielectric constant
The Hamiltonian is of the same form as (2). In the first approximation we can
neglect the contributions to the Hamiltonian of the interaction energies of the electrons e(1)
and e(2) and the holes and with polarization fields induced by these charge carriers on
the surfaces of QD (A) and QD (B) [5]. Thus the Hamiltonian incorporates only the
energies of Coulomb interaction of electron e(1) with hole h(B), and electron е(2) with hole h(A),
as well as that between electrons е(1) and е(2), and holes h(A) and h(B).
Under the assumption that the spins of the electrons е(1) and е(2) are antiparallel let us
write down the normalized wave function of the ground singlet state of the exciton
quasimolecule as a symmetric linear combination of wave functions 1 (1) (2),A Br r and
2 (2) (1),A Br r [24–28],
1
2 2
(1) (2) (1) (2) 1 (1) (2) 2 (2) (1)( , , , ) 2 (1 ( , ) , , ,S A A B B A B A Br r r r S D a r r r r
(6)
55
where S(D, a) is the overlap integral of single-electron wave functions. Assuming that the
electrons e(1) and e(2) move independently from each other, let us represent the wave functions
1 (1) (2),A Br r and 2 (2) (1),A Br r (6) as a product of single-electron wave functions (2) (1)( )A Ar
and (2) (1)( )B Br , as well as (2) (2)( )A Ar and (2) (1)( )B Br , respectively. Let us also represent the
single-electron wave functions as variational functions of Coulomb type [15–21]:
2
(1) (1) (1)( ) exp ( )( / ) ,D
A A A exr A a r a
2
(2) (1) (2)( ) exp ( )( / ) ,D
B B B exr A a r a , (7)
2
(2) (2) (2)( ) exp ( )( / ) ,D
A A A exr A a r a
2
(1) (1) ( 1)( ) exp ( )( / ) ,D
B B B exr A a r a
where is a variational parameter, 2 2 2( / )D
exa e is the Bohr radius of two –
dimensional (2D) exciton localized over the flat interface between the zinc selenide (or
aluminum oxide) and the matrix of dielectric.
In the framework of the variational method, the energy of the exciton quasimolecule
ground singlet state, as a first approximation, is given by the mean value of the Hamiltonian
(1) over the states described by the wave functions of the zeroth approximation
(1) (2) (1) (2) (1) (2) (1) (2), , , , , , ( , , ,S A A B B S A A B BE D a D a r r r r H r r r r
, (8)
where is a variational parameter.
With the explicit form of the wave functions (5), (6), the energy functional of the exciton
quasimolecule singlet ground state takes the form [24–28]
0 2
( , ( , ( , ( , ), )
( , ( , ), 2 ( , ( )) ,
1 ( , ( , ), )ex
j D a D K D a D a
E D a D a E a a
S D a D a
(9)
Here, ( , ( ))exE a a is the energy functional of the exciton ground state (for the exciton formed
from an electron and a hole spatially separated from the electron):
(1)(1) (1) (1) (1)( , ( ) ( ) ( )Aex A A A AE a a r H r (10)
The second term in (9)
2
( , ( , ( , ( , ), )
( , ( , ), )
1 ( , ( , ), )b
j D a D K D a D a
E a a D a
S D a D a
(11)
is a functional of the binding energy of singlet ground state of excitonic quasimolecule. In the
functional determined by formula (9), ( , ( , ), )j D a D a is determined by the expression.
int(1) (1) (2) (2) (1) (1) (2) (2)( , ( , ), ) ( ) ( ) ( ) ( )A A B B A A B Bj D a D a r r H r r (12)
56
The functional ( , ( , ), )j D a D a (12) can be represented as the algebraic sum of the
functionals of the average energies of Coulomb interaction. In the functional described by (9),
( , ( , ), )K D a D a is determined by the formula
int(1) (1) (2) (2) (1) (1) (2) (2)( , ( , ), ) ( ) ( ) ( ) ( )B B A A A A B BK D a D a r r H r r (13)
The functional ( , ( , ), )K D a D a (13) can be represented as the algebraic sum of the
functionals of the average energies of the exchange interaction.
Within the framework of the variational method at the first approximation the total
energy of ground singlet state of excitonic quasimolecule is determined by average value of the
Hamiltonian (1) for states, which are described by wave functions of the zero approximation
(1) (2) (1) (2)( , , , )S A A B Br r r r [24–28]
0 ( , ) 2 ( ) ( , )ex bE D a E a E D a (14)
where 0 ( , )E D a is the binding energy of the ground singlet state of the excitonic quasimolecule
and ( )exE a is the binding energy of the ground state of the exciton (consisting of spatially
separated electron and hole) localized over the surface of QD, which was worked out in [15
]. In (14) energy is measured in units
0 2
0 04( / ) / ( )exE m Ry (15)
where 0 13.606 eVRy is the Rydberg constant.
In [24, 25] presents the results of the variational calculations of the binding energy
( , )bE D a of the exciton quasimolecule ground state in a nanosystem with a QD of zinc selenide
with the mean radius 3.9nm,a (permittivity 2 9.25, effective mass of the hole in QD
0( / ) 0.7,hm m the reduced exciton mass (with a spatially separated electron and hole)
0( / ) 0.30,m the dielectric constant of nanosystem 3.29, synthesized in a dielectric
matrix of the borosilicate glass (permittivity 1 2 , the effective mass of the electron in the
matrix (1)
0( / )bm m was deduced in Ref. 34 and amounts to 0.537), as well as in the nanosystem
with a QD of aluminum oxide with the mean radius 1 3.2 nm,a (permittivity 2 10 , effective
mass of the hole in QD ( ) = 6.2, the reduced exciton mass (with a spatially separated
electron and hole) 0( / ) 0.49,m the dielectric constant of nanosystem 3.28, synthesized
in a dielectric matrix of the vacuum oli VM - 4 (permittivity 1 1.96 , the effective mass of the
electron in the matrix (1)
0( / ) 0,537.bm m The variational method that we used for the calculation
of the exciton quasimolecule ground state binding energy ( , )b D a is applicable provided that
it is much smaller than the binding energy of the exciton ground state ( )ex a , i.e. the
following condition must be fulfilled [24–28]
( ( , ) / ( )) 1b exE D a E a . (16)
The binding energy ( , )bE D a of the exciton quasimolecule ground state in the
nanosystem containing QDs of zinc selenide with average radii 1 3.9 nm,a has a minimum of
57
(1)
1 1( , 4.2 meVbE D a (at the distance 1 3.2 nmD ) ( corresponds to the
temperature ) [24]. As it follows form [24], the exciton quasimolecule appears in the
nanosystem at distances (1)
1 2.4 nmcD D between the surfaces of QD. The formation of such
a exciton quasimolecule is of threshold character and can occur in a nanosystem with QDs of the
mean radius , where the distance D between the surfaces of QD exceeds a certain critical
value (1)
cD . The existence of such distance (1)
cD arises from quantum size effects in which the
decrease in the energies of interaction of the electrons and holes entering into the Hamiltonian
(Eq. (1)) of the exciton quasimolecule with decrease of the distance D between the surfaces of
the QD cannot compensate for the increase in the kinetic energy of the electrons and holes [24].
The binding energy of the exciton ( )ex a amounts to 1( ) 54 meVex a [15], with the
energy of the exciton quasimolequle ground state (8) taking the value
0 1 1( , ) 112.2 meVD a . It should be emphasized that the criterion (16) of the applicability of
the variational method for the calculation of the exciton quasimolecule binding energy
( , )b D a is fulfilled (1)
1 1 1( , / ( ) 0.08.b exE D a E a At larger distances D between the surfaces
of QD: (2) 16.4 nm,cD D the exciton quasimolecule breaks down into two excitons
(consisting of spatially separated electrons and holes), localized over QD surfaces. Thus a
exciton quasimolecule can be formed in a nanosystem where (1) (2)
c cD D D [24]. Furthermore,
a exciton quasimolecule can exist only at temperatures lower than the critical temperature
= 49 K. In the zinc selenide single crystal, the biexciton is formed with the binding energy
0,45 meVbE corresponding to the temperature 5.2 K. At the same time, the exciton
quasimolecule binding energy (1)
bE in the nanosystem is about an order of magnitude higner than
bE
The binding energy ( , )bE D a of the exciton quasimolecule ground state in the
nanosystem containing QDs of aluminum oxide with average radii 1 3.2 nma has a minimum
of (1)
1 1( , ) 7.03 meVbE D a (at the distance 1 2.9 nmD ) (Fig. 3) (
corresponds to the temperature 82cT K ). In such a nanosystem, an exciton quasimolecule can
be formed where (1) (2)
c cD D D (where (1) 2.1 nmcD and (2) 4.4 nmcD ) [25] (see Fig. 3).
The binding energy of the exciton ( )ex a amounts to 1( ) 100.8 meVex a [18], with the
energy of the exciton quasimolequle ground state (8) taking the value
0 1 1( ) 208.63 meVD a . It should be emphasized that the criterion (16) of the applicability
of the variational method for the calculation of the exciton quasimolecule binding energy
( , )bE D a is fulfilled (1)
1 1 1( , ) / ( ) 0.07b exE D a E a [25]. Furthermore, a exciton quasimolecule
can exist only at temperatures lower than the critical temperature = 82 K. In the aluminum
oxide monocrystal, the biexciton is formed with the binding energy bE = 0.61 meV
corresponding to the temperature 7.04 K. The exciton quasimolecule binding energy (1)
bE in the
nanosystem is more than an order of magnitude higner than
58
Fig. 3. The dependence of the binding energy of the ground singlet state 1( , )bE D a of the
excitonic quasimolecule (continuous line) in a nanosystem made up of two spherical
aluminum oxide quantum dots QD(А) and QD(В) with the mean radius 1 3.2 nm,a on
the distance D between the surfaces of QD(А) and QD(В). Dependence of the exchange
interaction energy of the electrons and holes (dashed line) and the energy of the Coulomb
interaction between electrons and holes (dot dashed line) on the distance D between the
surfaces of the quantum dots
The values of the binding energy ( , )bE D a of the exciton quasimolecule, as well as the
total energy 0 1( , )E D a (14) of the excitonic quasimolecule in the nanosystem consisting of QD of
of aluminum oxide significantly exceed the corresponding values of ( , )bE D a and
0 1( , )E D a (14) in the nanosystem containing QD of zinc selenide. This is due to the fact that,
according to formulas (14) and (15), the values of the binding energy ( , )bE D a of an excitonic
quasimolecule and the total energy 0 1( , )E D a (14) of an excitonic quasimolecule with increasing
value ( ) the reduced exciton mass (with a spatially separated electron and hole) increase (~
0( / )m . The values of ( , )bE D a and 0 1( , )E D a (14) also increase 2( ( ) with a decrease in
the dielectric constant of the nanosystem.
As follows from the results of the variational calculations, the major contribution to the
exciton quasimolecule binding energy is from the energy of exchange interaction of electrons
and holes, which by far surpasses that from their Coulomb interaction (i.e., the ratio ≤ 0.08)
Upon an increase in the distance D between the surfaces of the QDs, the overlap
integral S (D,a) of single-electron wave functions (7) decreases. This leads to the fact that
starting from the values of , the exciton quasimolecule breaks down into two excitons
(consisting of spatially separated electrons and holes), localized over QD surfaces (see Fig. 2 and
Fig. 3). Since the calculations of the biexciton ground state binding energy ( , )bE D a in the
nanosystem are variational, the values of ( , )bE D a and 1
bE can be somewhat underestimated
[24–28].
Correlation of the theory with experiments
In [8] observed a low-tempeature luminescence peak at ≈ 2.716 eV (at a temperature
of T = 4.5 K) in samples with a zinc selenide QD content of x ≈ 0.6%; this peak was below the
band gap ( 2,823 eVgE ) of the zinc selenide single crystal. The shift of the 1E low-
temperature luminescence peak with respect to the band gap of the zinc selenide single crystal
was 1 1 103 eVgE E E .
59
Comparing the energy (14) of the biexciton ground state with the shift 1 103 eVE of
the luminescence spectral peak, we obtain the average spacing 1D ≈ 4.5 nm between the QD(A)
and QD(B) surfaces (at an average QD radius of 1 3.9 [24].a nm In this case, the biexciton binding
energy is 2 3bE meV (corresponding to a temperature of 2 35T K ), and the value of 1a = 3.9 nm is
in the range of the average zinc selenide QD radii 0( 2.0 4.8a nm) studied in the experimental
conditions of [8].
Thus, the experimentally observed shift 1E of the low-temperature peak of the
luminescence spectrum in the samples containing zinc selenide QDs with a content of x ≈ 0.6%
[8] is due to the dependence of the energy (14) of the singlet ground state of a biexciton
(formed from spatially separated electrons and holes) on the spacing D between the QD surfaces
and on the QD radius a [24].
Thus, the proposed model of the biexciton gives the explanation of the optical properties
of the nanosystem consisting of zinc selenide QDs grown in a dielectric matrix [24], in particular
the occurrence of a maximum of 1E in the low-temperature luminescence spectra of the
samples [8]. Since the values of the binding energies 1E of the biexciton singlet
ground state are negative, this means that additional energy is required to excite the second
electron–hole pair in the nanosystem. The latter circumstance leads to a shift of the absorption
edge of the nanosystem at high excitation intensities, as observed in experiments [8]
Conclusions
Thus it has been shown that the exciton quasimolecule formation is of the threshold
character and possible in a nanosystem, where the distance D between the surfaces of
colloidal QD is given by the condition (1) (2)
c cD D D (where and (2)
cD are some critical
distance). A significant increase in the binding energy of the singlet ground state of exciton
quasi-molequle (of spatially separated electrons and holes) in a nanosystem that consist of zinc
selenide QDs and aluminum oxide QDs and grown in a dielectric matrixs has been predicted; the
effect is almost two orders of magnitude larger than the binding energy of biexciton in a zinc
selenide and aluminum oxide monocrystals. It is shown that the major contribution to the
exciton quasi-molequle binding energy is made by the energy of the exchange interaction of
electrons with holes and this contribution is much more substantial than the contribution of the
energy of Coulomb interaction between the electrons and holes [24–28].
It is shown that the major contribution to tue binding energy of singlet ground state of
exciton quasimolecule is made by the energy of the exchange interaction of electrons with holes
and this contribution is much more substantial than the contribution of the energy of the
Coulomb interaction between the electrons and holes. It is established that the position of the
exciton quasimolecule energy band depends both on the mean radius of the QDs and the distance
between their surfaces [24–28].
It is shown that with increase in temperature above the threshold (Т ), a transition can
occur from the exciton quasimolecule to exciton state. It has been found that at a constant
concentration of excitons (i.e. constant concentration of QD) and temperatures Т below 9 cT ,
one can expect a new luminescence band shifted from the exciton band by the value of the
exciton quasimolecule binding energy bE . This new band disappears at higher temperatures
( ( )cT T . At a constant temperature below cT , an increase in exciton concentration (i.e. in QD
concentration) brings about weakening of the exciton luminescence band and strengthening of
the exciton quasimolecule one [25–28].
60
These exciton quasimolecules are of fundamental interest as new quasi-atomic
nanostructures; they may also have practical value as new nanomaterials for
nanooptoelectronics. The fact that the energy of the ground state singlet exciton quasimolecule is
in the infrared range of the spectrum, presumably, allow the use of a quasimolecule to create new
infrared sensors in biomedical research.
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62
ЕКСИТОННІ КВАЗІМОЛЕКУЛИ В НАНОСИСТЕМАХ З
НАПІВПРОВІДНИКОВИМИ ТА ДІЕЛЕКТРИЧНИМИ
КОЛОЇДНИМИ КВАНТОВИМИ ТОЧКАМИ: ОГЛЯД
С. І. Покутній1,2 , А. Д. Терець1
1 Інститут хімії поверхні ім. О.О. Чуйка Національної академії наук України, вул.
Генерала Наумова, 17, Київ, 03164, Україна, pokutnyi.serg@gmail.com
2 Вроцлавський науково-технічний університет, наб. Виспінскійого 27,
Вроцлав 50-370, Польща
У огляді розглядається теорія екситонних квазимолекул (утворених із просторово
розділених електронів і дірок) у наносистемі, що складається з напівпровідникових та
діелектричних колоїдних квантових точок (КТ), синтезованих у діелектричній та
напівпровідниковій матрицях. Показано, що утворення екситонної квазимолекули має
пороговий характер і можливе в наносистемі, в якої відстані D між поверхнями КТ
визначаються умовою (де та деякі критичні відстані).
Показано, що в такій наногетероструктурі «екситонні квазімолекули» являють
собою дві КТ, на поверхні яких локалізуються екситони. Положення смуги енергії такого
стану екситонної квазімолекули залежить як від середнього радіусу КТ, так і від
відстані між їх поверхнями. Така залежність дозволяє цілеспрямовано керувати
величиною енергії екситонної квазімолекули, змінюючи ці параметри наноструктури.
Встановлено, що енергії зв’язку основного синглетного стану екситонних
квазімолекул приймають суттєві значення, значно більші (майже на два порядки) за
енергію зв’язку біекситону в напівпровідникових і діелектричних монокристалах.
Показано, що основний внесок в енергію зв'язку основного синглетного стану екситонної
квазімолекули вносить енергія обмінної взаємодії електронів з дірками, і цей внесок є
значно більшим, ніж внесок енергії кулоновської взаємодії між електронами і дірками.
Встановлено, що з підвищенням температури вище порога (Т ), може
відбуватися перехід зі стану екситонної квазімолекули в екситонний стан. Показано, що
при постійній концентрації екситонів (тобто постійній концентрації КТ) і
температурах Т нижче можна очікувати нову смугу люмінесценції, зміщену від
екситонної смуги на величину енергії зв’язку екситонної квазимолекули. Ця нова смуга
зникає при більш високих температурах (Т ≥ ). При постійній температурі нижче
збільшення концентрації екситонів призводить до ослаблення смуги екситонної
люмінесценції та посилення смуги екситонної квазімолекули.
Такі екситонні квазімолекули становлять фундаментальний інтерес як нові
квазіатомні наноструктури; вони також можуть мати практичну цінність як нові
наноматеріали для нанооптоелектроніки. Оскільки енергія основного синглетного стану
квазімолекули знаходиться в інфрачервоному діапазоні спектра, то це дозволяє
використовувати такі квазімолекули для створення нових інфрачервоних датчиків у
біомедичних дослідженнях.
Ключові слова: екситонна квазімолекула, електрони, дірки, квантові точки, енергія
зв'язку, обмінна взаємодія, Кулонівська взаємодія
|
| id | oai:ojs.pkp.sfu.ca:article-745 |
| institution | Surface |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-24T17:45:57Z |
| publishDate | 2022 |
| publisher | Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | surfacezbircomua/8d/61d096960a8951fafa6fcf5584d76d8d.pdf |
| spelling | oai:ojs.pkp.sfu.ca:article-7452023-04-20T10:24:32Z Exciton quasimolecules in nanosystems with semiconductor and dielectric colloidal quantum dots: a review Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд Покутній, С. І. Терець, А. Д. exciton quasimolequle electrons holes quantum dots binding energy exchange interaction Coulomb interaction екситонна квазімолекула електрони дірки квантові точки енергія зв'язку обмінна взаємодія Кулонівська взаємодія In review, deals with the theory of exciton quasimolecules (formed of spatially separated&nbsp;&nbsp; electrons and holes) in a nanosystems that consists of semiconductor and dielectric colloidal quantum dots (QDs) synthesized in a dielectric and semiconductor matrixs. It has been shown that the exciton&nbsp; quasimolecule formation is of the threshold character and possible in a nanosystem, where the distance D between the&nbsp; surfaces of QD is given by the condition&nbsp; &nbsp;&nbsp;(where and are some critical distance). We have shown that in such a nanoheterostructures acting as “exciton molecules” are the QDs with excitons localizing over their surfaces. The position of the quasimolecule state energy band depends both on the mean radius of the QDs, and the distance between their surfaces, which enables one to purposefully control it by varying these parameters of the nanostructure. It was found that the binding energy of singlet ground state of exciton quasimolecules, consisting of two semiconductor and dielectric QDs is a significant large values, larger than the binding energy of the biexciton in a semiconductor and dielectric single crystals almost two orders of magnitude. It is shown that the major contribution to tue binding energy of singlet ground state of exciton quasimolecule is made by the energy of the exchange interaction of electrons with holes and this contribution is much more substantial than the contribution of the energy of the Coulomb interaction between the electrons and holes. &nbsp;It is established that the position of the exciton quasimolecule energy band depends both on the mean radius of the QDs and the distance between their surfaces. It is shown that with increase in temperature above the threshold (), a transition can occur from the exciton&nbsp; quasimolecule to exciton state. It has been found that at a constant concentration of excitons (i.e. constant concentration of QD) and temperatures Т below&nbsp; , one can expect a new luminescence band shifted from the exciton band by the value of the exciton quasimolecule binding energy. This new band disappears at higher temperatures (). At a constant temperature below , an increase in exciton concentration (i.e. in QD concentration) brings about weakening of the exciton luminescence band and strengthening of the exciton quasimolecule. These exciton quasimolecules are of fundamental interest as new quasi-atomic colloidal nanostructures; they may also have practical value as new nanomaterials for nanooptoelectronics. The fact that the energy of the ground state singlet exciton&nbsp; quasimolecule is in the infrared range of the spectrum, presumably, allow the use of a quasimolecule&nbsp; to create new infrared sensors in biomedical research. У огляді розглядається теорія екситонних квазимолекул (утворених із просторово розділених електронів і дірок) у наносистемі, що складається з напівпровідникових та діелектричних колоїдних квантових точок (КТ), синтезованих у діелектричній та напівпровідниковій матрицях. Показано, що утворення екситонної квазимолекули має пороговий характер і можливе в наносистемі, в якої відстані D між поверхнями КТ визначаються умовою&nbsp; &nbsp;(де &nbsp;та деякі критичні відстані). Показано, що в такій наногетероструктурі «екситонні квазімолекули» являють собою дві КТ, на поверхні яких локалізуються екситони. Положення смуги енергії такого стану екситонної квазімолекули залежить як від середнього радіусу КТ, так і від відстані між їх поверхнями. Така залежність дозволяє цілеспрямовано керувати величиною енергії екситонної квазімолекули, змінюючи ці параметри наноструктури. Встановлено, що енергії зв’язку основного синглетного стану екситонних квазімолекул приймають суттєві значення, значно більші (майже на два порядки) за енергію зв’язку біекситону в напівпровідникових і діелектричних монокристалах. Показано, що основний внесок в енергію зв'язку основного синглетного стану екситонної квазімолекули вносить енергія обмінної взаємодії електронів з дірками, і цей внесок є значно більшим, ніж внесок енергії кулоновської взаємодії між електронами і дірками. Встановлено, що з підвищенням температури вище порога (Т ), може відбуватися перехід зі стану екситонної квазімолекули в екситонний стан. Показано, що при постійній концентрації екситонів (тобто постійній концентрації КТ) і температурах Т нижче &nbsp;можна очікувати нову смугу люмінесценції, зміщену від екситонної смуги на величину енергії зв’язку екситонної квазимолекули. Ця нова смуга зникає при більш високих температурах (Т ≥ ). При постійній температурі нижче&nbsp; &nbsp;&nbsp;збільшення концентрації екситонів призводить до ослаблення смуги екситонної люмінесценції та посилення смуги екситонної квазімолекули. Такі екситонні квазімолекули становлять фундаментальний інтерес як нові квазіатомні наноструктури; вони також можуть мати практичну цінність як нові наноматеріали для нанооптоелектроніки. Оскільки енергія основного синглетного стану квазімолекули знаходиться в інфрачервоному діапазоні спектра, то це дозволяє використовувати такі квазімолекули для створення нових інфрачервоних датчиків у біомедичних дослідженнях. Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine 2022-11-30 Article Article application/pdf https://surfacezbir.com.ua/index.php/surface/article/view/745 10.15407/Surface.2022.14.049 Surface; No. 14(29) (2022): Surface; 49-62 Поверхность; № 14(29) (2022): Поверхня; 49-62 Поверхня; № 14(29) (2022): Поверхня; 49-62 3154-8091 3154-8083 10.15407/Surface.2022.14 en https://surfacezbir.com.ua/index.php/surface/article/view/745/739 Авторське право (c) 2022 С. І. Покутній, А. Д. Терець |
| spellingShingle | екситонна квазімолекула електрони дірки квантові точки енергія зв'язку обмінна взаємодія Кулонівська взаємодія Покутній, С. І. Терець, А. Д. Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд |
| title | Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд |
| title_alt | Exciton quasimolecules in nanosystems with semiconductor and dielectric colloidal quantum dots: a review |
| title_full | Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд |
| title_fullStr | Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд |
| title_full_unstemmed | Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд |
| title_short | Екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд |
| title_sort | екситонні квазімолекули в наносистемах з напівпровідниковими та діелектричними колоїдними квантовими точками: огляд |
| topic | екситонна квазімолекула електрони дірки квантові точки енергія зв'язку обмінна взаємодія Кулонівська взаємодія |
| topic_facet | exciton quasimolequle electrons holes quantum dots binding energy exchange interaction Coulomb interaction екситонна квазімолекула електрони дірки квантові точки енергія зв'язку обмінна взаємодія Кулонівська взаємодія |
| url | https://surfacezbir.com.ua/index.php/surface/article/view/745 |
| work_keys_str_mv | AT pokutníjsí excitonquasimoleculesinnanosystemswithsemiconductoranddielectriccolloidalquantumdotsareview AT terecʹad excitonquasimoleculesinnanosystemswithsemiconductoranddielectriccolloidalquantumdotsareview AT pokutníjsí eksitonníkvazímolekulivnanosistemahznapívprovídnikovimitadíelektričnimikoloídnimikvantovimitočkamioglâd AT terecʹad eksitonníkvazímolekulivnanosistemahznapívprovídnikovimitadíelektričnimikoloídnimikvantovimitočkamioglâd |