Quasi-zero-dimensional semiconductor nanosystems consisting of spherical semiconductor nanocrystals, i.e., quantum dots with radii of a =1-10 nm and containing cadmium sulphide and selenide, gallium arsenide, germanium, silicon, and zinc selenide in their volume, and synthesized in a borosilicate glass matrix currently attract particular research attention due to their unique photoluminescent properties, i.e., the ability to efficiently emit light in the visible or near infrared ranges at room temperature [1-10]. The optical and electro-optical properties of such quasi-zero dimensional nanosystems are to a large extent controlled by the energy spectrum of the spatially confined electron-hole pair (exciton) [4-16]. In most theoretical models for calculating the energy spectra of quasiparticles in quantum dots (QDs), the effective mass approximation is used, which was considered applicable for QDs by analogy with bulk single crystals [11-13]. However, the problem concerning the applicability of the effective mass approximation to the description of semiconductor QDs remains unsolved [4-18]. In [14], a new modified effective mass method is proposed to describe the exciton energy spectrum in semiconductor QDs with radii of a ≈ aex (aex is the exciton Bohr radius in the semiconductor material contained in the QD volume). It was shown that, within a model in which the QD is represented as an infinitely deep potential well, the effective mass approximation can be applied to the description of an exciton in QDs with radii a comparable to the exciton Bohr radius a_ex, assuming that the reduced effective exciton mass is a function of the radius a, μ = μ(a). In the adiabatic approximation and within the modified effective mass method [14], an expression for the binding energy of an exciton, whose electron and hole move within the semiconductor QD volume, was derived in [15]. In [15], the effect of significantly increasing the exciton binding energy in cadmium selenide and sulphide QDs with radii a, comparable to the exciton Bohr radii a_ex and relative to the exciton- binding energy in cadmium selenide and sulphide single crystals (by factors of 7.4 and 4.5, respectively) was also detected.

In the experimental study [7], it was found that excess electrons produced during interband excitation of the cadmium sulphide QD had a finite probability of overcoming the potential barrier and penetrating into the borosilicate glass matrix, into which the QD is immersed. In experimental studies [10, 19] (as well as in [7]) using glass samples with cadmium-sulphide and zinc selenide QDs, it was found that the electron can be localized in the polarization well near the outer QD surface, while the hole moves within the QD volume. In [10, 19], the optical properties of borosilicate glass samples containing QD zinc selenide were experimentally studied. The average radii of such QDs were in the range a ≈ 2.0-4.8 nm. In this case, the values of a are comparable to the exciton Bohr radius aex ≈ 3.7 nm in a ZnSe single crystal. At low QD concentrations, when the optical properties of the samples are mainly controlled by those of individual QDs in the borosilicate glass matrix, a shift of the peak of the low temperature luminescence spectrum to the short wavelength region (with respect to the band gap Eg of the zinc selenide-single crystal) was observed. The authors of [10] assumed that this shift was caused by quantum confinement of the energy spectra of the electron and the hole localized near the spherical surface of the QD. In this case, the following problem remained open: the quantum confinement of the state of which electron and hole (the hole moving in the QD volume and the electron localized at the outer spherical QD-dielectric matrix interface or the electron and hole localized in the QD volume) caused such a shift in the luminescence spectrum peak? The use of semiconductor nanosystems as the active region of nanolasers is prevented by the low binding energy of the QD exciton [8, 9, 13]. Therefore, studies directed at the search for nanostructures in which a significant increase in the binding energy of QD excitons can be observed are of importance.

Currently, the theory of exciton states in quasi- zero- dimensional semiconductor nanosystems has not been adequately studied. In particular, no theory exists for an exciton with a spatially separated electron and hole in quasi- zero- dimensional nanosystems. Therefore, in this study, we developed the theory of an exciton formed from a spatially separated electron and hole (the hole is in the semiconductor QD volume and the electron is localized at the outer spherical surface of the QD-dielectric matrix interface) [20-22]. It was shown that the short wavelength shift of the peak of the low temperature luminescence spectrum of samples containing zinc selenide QDs, observed under the experimental conditions of [10], was caused by quantum confinement of the ground state energy of the exciton with a spatially separated electron and hole. The effect of significantly increasing the binding energy of an exciton (with a spatially separated electron and hole) in a nanosystem containing zinc selenide QDs, compared with the binding energy of an exciton in a zinc selenide single crystal (by a factor of 4.1-72.6), was detected [20-22]. In [10, 19], a shift of the spectral peak of the low temperature luminescence was also observed for samples with a QD concentrations from x = 0.003-1%. It was noted [10, 19] that at such a QD content in the samples, the interaction between charge carriers localized above the QD surfaces must be taken into account. Therefore, in [23, 24], we develop the theory of excitonic quasimolecules (biexcitons) (formed from spatially separated electrons and holes) in a nanosystem, which consists of ZnSe QDs synthesized in a borosilicate glass matrix.

Spectroscopy of excitons in Quasi – Zero – Dimensional nanosystems

Let us consider the simple model of a quasi-zero-dimensional system, i.e., a neutral spherical semiconductor QD of the radius a, which contains semiconductor material with the permittivity ε2 in its volume, surrounded by a dielectric matrix with permittivity ε₁. A hole h with the effective mass mhmoves in the QD volume, while an electron e with the effective mass mе⁽¹⁾ lies in the matrix (re and rhare the distances from the QD centre to the electron and hole). Let us assume that the QD valence band is parabolic. Let us also assume that there is an infinitely high potential barrier at the spherical QD-dielectric matrix interface; therefore, the hole h cannot leave the QD volume and the electron e cannot penetrate into the QD volume in the model under study [20-22].

The results of the variational calculation of the energy of the ground state of an exciton E₀(a, ε) (17) in the nanosystem under study containing zinc selenide QDs of the radius a (20) are shown in the Figure 1[20-22]. Here, the values of function μ(a) (19) and the results of the variational calculation of the exciton ground-state energy E₀(a, ε) (17) are obtained for a nanosystem containing zinc selenide QDs, synthesized in a borosilicate glass matrix, as studied in the experimental works [10, 19]. In the experimental work [10], borosilicate glass samples doped with zinc selenide with concentrations ranging from x = 0.003-1%, obtained by the sol-gel method, were studied. According to X-ray diffraction measurements, the average radii a of ZnSe QDs formed in the samples were within a ≈ 2.0-4.8 nm. In this case, the values of a˜a˜ were comparable to the exciton Bohr radius aex ≈ 3.7 nm in a zinc selenide single crystal. At low QD concentrations (x = 0.003 and 0.06%), their interaction can be disregarded. The optical properties of such nanosystems are primarily controlled by the energy spectra of electrons and holes localized near the spherical surface of individual QDs synthesized in the borosilicate glass matrix.

In [10, 19], a peak in the low-temperature luminescence spectrum at an energy of E1 ≈ 2.66 eV was observed at the temperature T = 4.5 K in samples with x = 0.06%; this energy is lower than the band gap of a zinc selenide single crystal (Eg = 2.823 eV). The shift of the peak of the low-temperature luminescence spectrum with respect to the band gap of the ZnSe single crystal to the short-wavelength region is ΔE1 = (E1–Eg) ≈ –165 meV. The authors of [10] assumed that the shift ΔE1 was caused by quantum confinement of the energy spectra of electrons and holes localized near the spherical surface of individual QDs, and that it was associated with a decrease in the average radii a of zinc-selenide QDs at low concentrations (x = 0.06%). In this case, the problem of the quantum confinement of which electron and hole states (the hole moving within the QD volume and the electron localized at the outer spherical QD-dielectric matrix interface or the electron and hole localized in the QD volume) caused such a shift of the luminescence-spectrum peak remained open. Comparing the exciton ground-state energy (E₀(a,ε) – Eg) (17) with the energy of the shift in the luminescence-spectrum peak ΔE1 ≈ –165 meV, we obtained the average zinc selenide QD radius a1 ≈ 4.22 nm (see Figure 1) [20-22]. The QD radius a1 may be slightly overestimated, since the variational calculation of the exciton ground-state energy can yield slightly overestimated energies [33, 34]. The determined average QD radius a1 was found to be within the range of the average radii of zinc selenide QDs (a a  ≈ 2.0-4.8 nm) studied under the experimental conditions of [10, 19]. It should be noted that the average Coulomb interaction energy V¯¯¯ehV¯eh (a, ε) = ψ0(re,rh,r,a)|Veh(r)|ψ0(re,rh,r,a)ψ0(re,rh,r,a)|Veh(r)|ψ0(re,rh,r,a) between the electron and hole primarily contributed to the ground-state energy (17) of the exciton in the nanosystem containing zinc selenide QDs with radii a1 comparable to the exciton Bohr radius in a zinc-selenide single crystal (aex ≈ 3.7 nm). In this case, the average energy of the interaction of the electron and hole with self- and “foreign” images,

In [20-22], within the framework of the modified effective mass method [14], the theory of artificial atoms formed from spatially separated electrons and holes (holes moving in the volume of a semiconductor (dielectric) QD and an electron localized on the outer spherical interface between the QD and a dielectric matrix) is developed. The energy spectrum of superatoms (excitons of spatially separated electrons and holes) from QD radius a ≥ ac (about 4 nm) is fully discrete [20-22, 29, 30]. This is referred to as a hydrogen-superatom and is localized on the surface of a valence electron QD. The energy spectrum of the superatom consists of a quantum-dimension of discrete energy levels in the band gap of the dielectric matrix. Electrons in superatoms are localized in the vicinity of the nucleus (QD). The electrons move in well-defined atomic orbitals and serve as the nucleus of QD, containing in its volume semiconductors and insulators. Ionization energy superatoms take on large values (of the order of 2.5 eV), which is almost three orders of magnitude higher than the binding energy of the excitons in semiconductors [20-22, 29, 30].

We will briefly discuss the possible physical and chemical effects that are relevant for the results of this paper. In our proposed [20-22, 29, 30] model of a hydrogen superatom localized on the surface of the QD is a valence electron. In quasi-atomic structures of the outer valence, electrons can participate in a variety of physical and chemical processes, similar to the atomic valence electrons in atomic structures. Artificial atoms have the ability to connect to the electron orbitals of electrons N (where N can vary from one to several tens). At the same time, the number of electrons N can take values of the order of a few tens or even surpass the serial numbers of all the known elements found in Mendeleev’s table [20-22, 29, 30]. This new effect allows for attaching to the electronic orbitals of artificial atoms N electrons, causing a high reactivity and opening up new possibilities for superatoms related to their strong oxidizing properties, increasing the possibility of substantial intensity in photochemical reactions during catalysis and adsorption, as well as their ability to form many new compounds with unique properties (in particular, the quasi-molecule and quasicrystals) [24, 29, 30]. Therefore, studies aimed at the theoretical prediction of the possible existence of artificial new atoms (not listed in the Mendeleev table) and to their study in terms of experimental conditions are very relevant.

Quantum discrete states of the individual atoms of alkali metals are determined by the movement of only one, i.e., the outermost valence electron, around a symmetric atomic core (containing the nucleus and the remaining electrons) [35]. In the hydrogen superatom formed quantum-energy spectra of discrete energy levels of the valence electron [20-22, 29, 30]. Thus, the observed similarity of the spectra of discrete electronic states and individual superatoms alkali metal atoms, as well as the similarity of their chemical activity [20-22, 29, 30, 35].

In Section 4, on the basis of a spectroscopic analogy of electronic states of artificial atoms and individual alkali metal atoms, a new artificial atom is theoretically predicted, which is similar to the new alkali metal atom.

In [20-22, 29, 30], a new model of an superatom is proposed, which is a quasi-zero–dimensional nanosystem consisting of a spherical QD (nucleus superatom) with radius a and which is included within its scope as a semiconductor (dielectric) with a dielectric constant ε₂, surrounded by a dielectric matrix with a dielectric constant ε₁. A hole h with the effective mass mh moves in the QD volume, while an electron e with the effective mass mе⁽¹⁾ lies in the dielectric matrix. In such a nanostructure, the lowest electronic level is situated in the matrix and the humble hole level is the volume QD. Large shift of the valence band (about 700 meV) is the localization of holes in the volume QD. A large shift of the conduction band (about 400 meV) is a potential barrier for electrons (electrons move in the matrix and do not penetrate into the volume QD). The Coulomb interaction energy of an electron and a hole, and the energy of the electron polarization interaction with the surface section (QD-matrix) (since the permittivity ε₂ is far superior to QD permittivity ε₁ matrix) cause localization of the electron in the potential well above the surface of QD [20-22, 29, 30].

With increasing radius a QD, so that а >> аех⁰ (where аех⁰ (14) two-dimensional Bohr radius of the electron) spherical surface section (QD- matrix) transforms into a flat surface section. In this artificial atom, electrons localized on the surface (QD-matrix) become two-dimensional. In this case, the potential energy in the Hamiltonian describing the motion of an electron in a superatom, the main contribution to the energy of the Coulomb interaction Veh(r) (11) between an electron and a hole [20-22]. Polarization interaction energy of the electron and the hole with a spherical surface section (QD-matrix) delivers a much smaller contribution to the potential energy of the Hamiltonian and thus, contributions to a first approximation can be neglected [20-22]. In this regard, the two-dimensional electron energy spectrum En in the artificial atom takes the form (13).

Depending on the binding energy Еех(a,ε) of an electron in the ground state superatom (QD containing zinc-selenide radius a and surrounded by a matrix of borosilicate glass [10]) as obtained in [20-22] by the variational method, it follows that the bound state of an electron occurs near the spherical interface (QD-matrix), starting with the value of the critical radius QD a ≥ ac⁽¹⁾ = 3.84 nm, when this hole moves in the volume QD and the electron is localized on the surface of the spherical section (QD-matrix). In this case, the Coulomb interaction energy Veh(r) (11) between the electron and the hole, and the energy of the polarization interaction of electrons and holes with a spherical surface section (QD-matrix) prevail over the size quantization of the energy of electrons and holes in the artificial atom. Thus, [20-22] found that the occurrence of superatoms had a threshold and was only possible if the radius of QD КТ а ≥ ас⁽¹⁾ = 3.84 nm.

With the increasing radius of a QD scan, an increase in the binding energy of the electron in the ground state superatom was observed. In the range of radii 4.0 ≤ а ≤ 29.8 nm, the binding energy of the electron in the ground state superatom significantly exceeded (in (4,1-76,2) times) the value of the exciton binding energy Ẽ⁰ex ≈ 21.07 meV in a single crystal of zinc-selenide [20-22]. Beginning with a radius QD а ≥ ас⁽²⁾ = 29.8 nm, the energy of the ground state of an electron in a superatom asymptotically follow the value E⁰ex = -1.5296 eV, which characterized the energy of the ground state of two-dimensional electrons in an artificial atom (15) [20-22]. The effect of significantly increasing the energy of the ground state of an electron in a superatom was primarily determined by two factors [20-22]: 1) a significant increase in the Coulomb interaction energy |Veh(r)| (2) electron-hole (the “dielectric enhancement” [34]); 2) the spatial limitations on the quantization volume QD, while with an increasing radius of a QD, since the radius of QD a ≥ ac⁽²⁾ = 52a⁰ex = 29.8 nm, superatoms became two-dimensional with a binding energy of the ground state E⁰ex(15), the value of which exceeded the exciton binding energy in a single crystal of zinc-selenide by two orders. The effect of “dielectric enhancement” as a result of the dielectric constant ε₁ of the matrix was much lower than the dielectric constant of QD ε₂, which played an essential role in the interaction between the electron and the hole in the superatom playing field produced by these quasi-particles in a matrix. Thus, the interaction between the electron and the hole in the superatom was significantly larger than in a semiconductor permittivity ε₂ [34]. Quantum discrete states of the individual atoms of alkali metals were determined by the movement of only one, the outermost valence electron, around a symmetric atomic core (containing the nucleus and the remaining electrons) [35]. Where large distances were the case between r electron and the nucleus (so that r >> a0, where a0 = 0.053 nm – the Bohr radius of the electron in a hydrogen atom), the field of the atomic core was described by the Coulomb field [35]:

           The similarity of the individual series of neutral alkali metal atoms with the hydrogen Balmer series suggests that the energy spectra of neutral alkali metal atoms can be labelled valence electron radiation in transition from higher levels to the level of principal quantum number n = 2 [35].

In a single atom of an alkali metal valence electron moving in the Coulomb field of the atomic core (26) having the same functional dependence on r as the Coulomb field (11), in which the valence electron in hydrogen-like model of artificial atom. This leads to the fact that the energy spectra of the valence electron in a single atom of an alkali metal (27) and in the artificial atom (13) describe the spectrum of hydrogen-type. At the same time, the number of possible quantum states of valence electrons in a hydrogen-like artificial atom model is the same as the number of quantum states of discrete valence electrons in a single atom of an alkali metal [20-22, 29, 30].

Table 2 shows the position of the valence electron energy levels in the atoms of alkali metals (K, Rb, Sc) [35] and the new artificial atom X, as well as the level shifts of the valence electron (ΔEкRbΔERbк, ΔERbScΔEScRb, ΔEScxΔExSc) relative to the adjacent level. Assume that the shift of the energy level Ex artificial atom X (relative to the energy level Esc of the atom Sc) will be the same as the shift of the energy level ERb of the atom Rb (relative energy level Esc of the atom Sc), (i.e., ΔEScxΔExSc = ΔERbScΔEScRb). In this case, the level of the valence electron artificial atom will be Ex = –593 meV. Using the dependence of the binding energy Еех(a,ε) of the ground state of an electron in an artificial atom [20-22] (QD containing zinc-selenide radius a and surrounded by a matrix of borosilicate glass [10]), we found the radius QD zinc-selenide a1 = 5.4 nm, which corresponded to the Ex = – 593 meV. It should be noted that the energy levels of a valence electron in the individual atoms of alkali metals (K, Rb, Sc) [35] and the new artificial atom X are located in the infrared spectrum.

Thus, we propose a new model of an artificial atom that is a quasi-atomic heterostructure consisting of a spherical QD (nucleus superatom) radius a and which contains in its scope, zinc- selenide, surrounded by a matrix of borosilicate glass (in volume QD, h hole effectively moves mass mh, e and the electron effective mass mв⁽¹⁾ is located in the matrix), thus allowing for finding a new artificial atom X (absent in the Mendeleev periodic system), which is similar to a new single alkali metal atom. This new artificial atom of a valence electron can participate in various physical [20-22, 29, 30] and chemical [30, 35] processes that are analogous to atomic valence electrons in atomic systems (in particular, the selected alkali metal atoms [35]). Such processes are unique as a result of the new properties of artificial atoms: strong oxidizing properties that increases the possibility of substantial intensity in photochemical reactions during catalysis and adsorption, as well as their ability to form a plurality of the novel compounds with unique properties (in particular, the quasi-molecule and the quasicrystals [23, 24]). The application of semiconductor nanoheterostructures as the active region nanolasers prevents small exciton binding energy in QD. Therefore, studies aimed at finding nanoheterostructures, which will yield a significant increase in the binding energy of the local electronic states in QDs, are relevant [20-22]. The effect of significantly increasing the energy of the electron in a hydrogen superatom [20-22, 29, 30] allows for better experimental detection of the existence of such superatoms at room temperatures and will stimulate experimental studies of nanoheterostructures containing superatoms, which can be used as active region nanolasers when working with optical transitions.

Conclusions

The theory of an exciton with a spatially separated electron and hole was developed within the framework of the modified effective mass method [14], in which the reduced effective exciton mass is a function of the semiconductor QD radius a. The average zinc-selenide QD radius was determined by comparing the dependence of the exciton ground-state energy (17) on the QD radius, obtained by the variational method within the modified effective mass method [14] and using the experimental peak of the low-temperature luminescence spectrum [10, 19]. It was shown that the short-wavelength shift of the peak of the low-temperature luminescence spectrum of the samples containing zinc selenide QDs, which was observed under the experimental conditions noted in [10, 19], was caused by renormalization of the electron-hole Coulomb interaction energy (11), as well as the energy created by the polarization interaction (5) of the electron and hole with the spherical QD-dielectric matrix interface, related to spatial confinement of the quantization region by the QD volume. In this case, the hole moves in the QD volume and the electron is localized at the outer spherical QD-dielectric matrix interface [20-22, 29-32].

To apply semiconductor nanosystems containing zinc-selenide QDs as the active region of lasers, it is required that the exciton binding energy |Eex(a, ε)| (21) in the nanosystem be at the order of several kT₀and at room temperature T₀ (k is the Boltzmann constant) [13]. Nanosystems consisting of zinc-selenide QDs grown in a borosilicate glass matrix can be used as the active region of semiconductor QD lasers. In the range of zinc selenide QD radii a (22), the parameter |Eex(a, ε)/kT₀| take on significant values ranging from 3.1 to 56 [20-22, 29-32].

The effect of significantly increasing the binding energy (21) of the exciton ground state in a nanosystem containing zinc selenide QDs with radii a (22) was detected; compared to the exciton binding energy in a zinc selenide single crystal, the increase factor was 4.1-72.6. [20-22, 29-32]. It was shown that the effect of significantly increasing the binding energy (21) of the exciton ground state in the nanosystem under study was controlled by two factors [20-22, 29-32]: (i) a substantial increase in the electron-hole Coulomb interaction energy (11) and an increase in the energy of the interaction of the electron and hole with “foreign” images (8), (9) (the “dielectric enhancement” effect [34]); (ii) spatial confinement of the quantization region by the QD volume; in this case, as the QD radius a increased, starting from a ≥ a(2)cac(2) ≈ 29.8 nm, the exciton became two-dimensional with a ground-state energy (15), which exceeded the exciton binding energy in a zinc selenide single crystal by almost two orders of magnitude. A review devoted to the theory of excitonic quasimolecules (biexciton) (made up of spatially separated electrons and holes) in a nanosystem that consists of ZnSe QDs synthesized in a borosilicate glass matrix was developed within the context of the modified effective mass approximation. Using the variational method, we obtained the total energy and the binding energy of the biexciton singlet ground state in such a system as functions of the spacing between the QD surfaces and the QD radius. It was established that, in a nanosystem composed of ZnSe QDs with the average radii a¯1a¯1, the formation of a biexciton (exciton quasimolecule) was of the threshold character and possible in a nanosystem where the spacing D between the QD surfaces is defined by the condition D(1)c≤D≤D(2)cDc(1)≤D≤Dc(2) [23, 24]. Moreover, the exciton quasimolecule (biexciton) can exist only at temperatures below a certain critical temperature, i.e., Tс≈49 KTс≈49 K [23, 24]. It was established that the spectral shift of the low temperature luminescence peak [10, 19] in such a nanosystem resulted due to quantum confinement of the energy of the biexciton singlet ground state.

Thus, we propose a new model of an artificial atom, which is a quasi-atomic heterostructure consisting of a spherical QD (nucleus superatom) with radius a and which contains in its scope zinc selenide, surrounded by a matrix of borosilicate glass (in volume QD moves h hole effective mass m_h, e and the electron effective mass m_e⁽¹⁾ is located in the matrix), and which is allowed to find a new artificial atom X (absent in the Mendeleev periodic system), which is similar to a new single alkali metal atom. This new artificial atom of valence electron can participate in various physical [20-22, 29, 30] and chemical [30, 35] processes that are analogous to atomic valence electrons in atomic systems (in particular, the selected alkali metal atoms [35]). Such processes are unique due to the new properties of artificial atoms: strong oxidizing properties that increase the possibility of substantial intensity in photochemical reactions during catalysis and adsorption, as well as their ability to form plurality among novel compounds with unique properties (in particular, the quasi-molecule and the quasicrystals [23, 24]).