ISSN 2321-807X Exciton formed from spatially separated electron and hole in dielectric quantum dots Sergey I. Pokutnyi Chuіko Institute of Surface Chemistry of National Academy of Sciences of Ukraine, 17 General Naumov Str., UA - 03164 Kyiv, Ukraine; E-mail:
[email protected]
Abstract
A quasi-zero-dimensional nanosystem involving a hole moving in the volume of a dielectric quantum dot
(QD) oxide aluminum and an electron localized on the outer spherical interface between the QD and a dielectric matrix has been studied. It is shown that by changing the parameters of nanoheterostructures containing dielectric nanoparticles of aluminum oxide (QD radius, QD and the dielectric constant of the matrix, as well as the ratio of the effective masses of the electron and hole can be directed to change the position of the energy levels of an exciton binding energy of excitons, the width of the exciton bands, as well as the energy of exciton transitions in exciton bands. The latter allows to use such nanostructures as active regions nanolasers working on exciton transitions.
Key words: Exciton, Spatially separated electron and hole, Binding energy, Coulomb interaction energy, Quantum dots
Council for Innovative Research Peer Review Research Publishing System
Journal: Journal of Advances in Chemistry Vol. 11, No. 10 www.cirjac.com
[email protected],
[email protected] 3848 | P a g e
September 07, 2015
ISSN 2321-807X 1. Introduction The use of semiconductor nanosystems based on spherical nanocrystals called quantum dots (QDs) as the active regions of nanolasers is hindered by a small binding energy of exciton (where a is the QD radius) in these nanosystems [1- 4]. In this context, investigations aimed at finding nanostructures with significantly increased
values are of
considerable importance. In this paper, on a significant increase in the ground - state binding energy of an exciton formed by a spatially separated hole (moving in a QD volume oxide aluminum) and electron (localized at the QD–dielectric matrix interface) as compared to the exciton binding energy in single crystal of oxide aluminum (Al2O3). It is shown that by changing the parameters of nano heterostructures containing dielectric nanoparticles of aluminum oxide (QD radius, QD and the dielectric constant of the matrix, as well as the ratio of the effective masses of the electron and hole can be directed to change the position of the energy levels of an exciton binding energy of excitons, the width of the exciton bands, as well as the energy of exciton transitions in exciton bands.The latter allows to use such nanostructures as active regions nanolasers working on exciton transitions.
2. Binding energy of the exciton of a spatially separated electron and hole in dielectric quantum dots Let us consider a model quasi-zero-dimensional nanosystem comprising a spherical dielectric QD of radius a (containing a dielectric with permittivity ) surrounded by a dielectric matrix with permittivity . Let a hole (h) with effective mass move in the QD volume, an electron (e) with effective mass
occur in the matrix, and an infinitely high potential
barrier exist at the QD–matrix interface. Thus, in the model under consideration, hole h cannot leave the QD volume, while electron e cannot enter this volume [5]. For the sake of simplicity, while not losing generality, we can assume that the hole occurs at the QD center. As the QD radius a increases so that
, (1) where аех = ε2 ћ2/ µ е2 (is the Bohr exciton radius in the dielectric with permittivity
(2)
, e is the elementary charge,
µ= mе (2) mh /(mе(2)+ mh )
(2)
is the reduced effective mass of the exciton, mе is the effective mass of an electron in the dielectric with the permittivity ε2), the spherical QD – matrix interface transforms into a flat interface between the dielectric with permittivity and the dielectric matrix with permittivity
. The fact that all characteristic dimensions of the problem are significantly larger than
the interatomic distances a0,
a , aex a0
(3)
allows us to consider the electron and hole motion in the quasi - zero - dimensional nanosystem in the effective mass approximation [5]. Also the conditions (1) and (3) allows us to describe the studied nanosystem using macroscopic parameters (dielectric with permittivity matrix and QD, effective masses of an electron and a hole in the matrix and QD) [5]. The exciton formed by the spatially separated electron and hole becomes two-dimensional (2D). The contribution of the energy of polarization interaction of the electron and hole with the QD surface to the exciton Hamiltonian in the first approximation can be ignored. Then, the potential energy term of the exciton Hamiltonian contains only the energy of the Coulomb interaction between the hole and electron [5]:
(4) where r is the electron distance from the QD center. The energy spectrum of 2D exciton formed by the spatially separated electron and hole is described as an expression of [6]:
,
3849 | P a g e
(5)
September 07, 2015
ISSN 2321-807X where n = 0, 1, 2, … is the main quantum number of the exciton,
is the reduced mass of
the exciton, and Ry0 = 13.606 eV is the Rydberg constant. According to Eq. (5), the Bohr radius and ground-state binding energy of this 2D exciton can be expressed as follows:
,
(6) (7)
Let us find total energy
, and the exciton ground-state binding energy
consideration using the variation method. Exciton ground_state binding energy
in the nanosystem under in this nanosystem can be
expressed as follows [5]:
, where the term hole (
(8)
accounts for the average energies of interaction of the
) and electron (
) with their images on variation functions
(9) where A is the normalization coefficient and μ(a) is the exciton reduced mass that serves the variation parameter. Since the average energies of interaction of the hole and electron with their images contribute to in Eq. (8) with opposite signs, these contributions compensate each other to a significant extend. For this reason, the exciton groundstate binding energy differs but slightly from the corresponding total energy of the exciton ground state in these nanosystems [5].
3. Comparison of theory and experiment Experimental investigations [3] of nanosystems with oxide aluminum quantum dots showed that the electron may be localized in a polarization well near the QD surface, while the hole moves in the QD volume. Variational calculations of total energy
and binding energy
(8) for the ground state of exciton
formed by the spatially separated electron and hole have been performed for nanosystems with oxide aluminum (Fig. 1) QDs synthesized in a vacuum oil, analogous to those experimentally studied in [3]. It was reported that oxide aluminum quantum dots (
= 10) synthesized in the vacuum oil ( 1
1.96 ), exhibited low-temperature absorption
at T = 4.5 K.
These QDs had radii within
3.1 ≤ а ≤ 19.1 nm and effective hole masses about (1) e
(m
m0 0,537 )
(mh /m0) = 6.2);
(10)
the effective electron mass calculated for this vacuum oil
is
[7]. The samples studied in [3] were characterized by low QD concentrations (~0.003–0.06%) in
the vacuum oil. The optical properties of these nanosystems are mostly determined by the energy spectra of electrons and holes localized at the spherical surface of separate QDs synthesized in the vacuum oil. Fig.1, which shows the dependence of the total energy Е0(а) (8), and the binding energy Еех(a) (8) of the ground state of an exciton (from space - separated electrons and holes) in nanosystems containing aluminum oxide nanoparticles the radius a (10), it follows that the bound states of the electron-hole pairs occur near the surface of the spherical nanoparticles, starting from the critical radius of nanoparticles of since the radius of the nanoparticles
aa
(1) c
3.1 nm
a ac(1) 3.1 nm.
State electron-hole pair,
are in the negative energy (measured from the ceiling of the
band gap Еg = 3.7 eV nanoparticles of aluminum oxide), which corresponds to a bound state of an electron and a hole. In this case, the Coulomb interaction energy Veh(r) (4) between the electron and the hole, and the energy of the polarization interaction of an electron and a hole with a spherical interface (nanoparticle - matrix) prevail over the size quantization energy of an electron and a hole in nanosystems An increase in QD radius a is accompanied by growth in total energy of the exciton ground state. In the interval of radii a (10) QDs, binding energy
3850 | P a g e
and binding energy
(8)
(8) of the exciton ground state is
September 07, 2015
ISSN 2321-807X significantly (by a factor of 3—49) greater than the exciton binding energy crystal. Beginning with QD radii
aa
asymptotically approach the values of
( 2) c
Eex2D
19.1nm
Eex3D
total energy
≈ 51.16 meV [3] in aluminum oxide single and binding energy
(8)
= - 2.504 eV (7), which determine the ground_state binding energies of 2D
excitons formed by the spatially separated electron and hole (see Fig. 1). In the context of the implementation of experiments [3] Bohr exciton radius аех (2) in QD is аех = 1.41 nm (2) (because the values of the effective masses of the electron (mе /m0) = 0.4 and hole (mh /m0) = 6.2 ). Therefore, the conditions (1) and (3) allows us to describe quasi-zero nanosystem in which the radius a of the QD varies in a range of (10), using the macroscopic parameters (permittivity matrix and QD, the effective masses of the electron and the hole in the matrix and QD) and to consider the motion of an electron and a hole in nanosystems in the effective mass approximation. In experimental studies [3] studied the optical properties of heterogeneous liquid nanophase composites based on high dielectric nanoparticles of alumina (placed in the vacuum oil), with an average radius does not exceed a = 25 nm. Such nanostructures at low concentrations (x = 0.03%) nanoparticle interaction between the nanoparticles can be neglected. Optical properties of nanosystems mainly determine the energy spectrum of electrons and holes localized near the surface of single nanoparticles are placed in a dielectric matrix [3-5]. In [3] it was shown that the aluminum oxide nanoparticles in a of vacuum oil have broad absorption band (1.4 eV to 3.7 eV). In [3], the bandgap nanoparticles also detected level Ed = -2,3 eV donor type (with a width (0.3 - 0.4 eV), the counted from the bottom of the conduction band of the nanoparticles. Since the experimental conditions [3] and the average of the radii a of nanoparticles less than 25 nm, then it follows from the results of the variational calculation of the ground state energy Е0(а) (8) exciton in nanosystems containing nanoparticles of aluminum oxide, changing radii a QD in the interval (10) (see. Fig.1) in the band gap of oxide aluminum nanoparticles occurs zone exciton states (of spatially separated electrons and holes), with a width not exceeding
Eex2D
= 2.504 eV, located at the bottom of the conduction band (see . Fig. 1). Thus, the level of Ed = -2.3 eV
(with a width (0.3 - 0.4 eV), the experimentally observed in [1] falls within this zone exciton nanoparticles of aluminum oxide (see Fig. 1). Significant interval (1.4 eV to
Eex2D
= 2.504 eV) absorption band was observed in [1], also falls into
the area of the exciton states of aluminum oxcide nanoparticles (see. Fig.1). Electron transitions in an area of the exciton states cause significant absorption in the visible and near-infrared wavelengths and cause significant blurring of the experimentally observed absorption edge [3].
4. Conclusion By varying the parameters nanoheterostructures containing dielectric nanoparticles of aluminum oxcide (radii a QD, QD and dielectric constant of the matrix, as well as the ratio of the effective mass of the quasiparticles (mh / ), can be directed to vary the position of the energy levels of an exciton, the exciton binding energy, the width exciton bands, as well as the energies of exciton transitions in exciton bands. The latter circumstance, apparently opens up new possibilities to use such as nanoheterostructures nanolasers active region. I hope that this work will stimulate experimental investigations into nanostructures, which can be used as the active regions of nanolasers operating on exciton transitions.
References [1] L.Gao, F.Gao. Applied Physics Letters, 103 (5), 0531011 (2013). [2] S.A.Khairallah, A.Anderson, A.M.Rubenchik, et al. AIP Advances, 5, 047120 (2015). [3] Yu.N.Kulchin, V.P.Dzyuba, V.A.Milichko. Advanced Materials Research A, 677, 36 (2013). [4] Letters A, 342 (5), 347 (2005).
S.I.Pokutnyi. Physics
[5] S.I.Pokutnyi, Semiconductors, 47 (6), 791 (2013). [6] Yu.E.Lozovik, V.N.Nishanov, Sov. Phys. Solid State, 18 (11), 1905 (1976). [7] S.I.Pokutnyi. Physics Letters A, 203 (5), 388 (1995).
3851 | P a g e
September 07, 2015
ISSN 2321-807X
Fig.1. Dependence of the ( solid curves) exciton ground-state energy (Е0(а) - Eg ) (5), and (dashed curves) exciton ground-state binding energy (Еех(a) - Eg ) (5) on the radius a QD aluminum oxcide . Here Eg = 3.7 eV 2D
2D
- bandgap in QD aluminum oxcide, values Eex = 2.504 eV, and aex = 0.352 nm - the binding energy of the ground state and two-dimensional exciton Bohr radius (from spatially separated electrons and holes).
3852 | P a g e
September 07, 2015
