Biexciton in Strongly Oblate Ellipsoidal Quantum Dot with Relativistic Corrections
الموضوعات : فصلنامه نانوساختارهای اپتوالکترونیکیArezu Jahanshir 1 , Ekwevugbe Omugbe 2
1 - 1Department of Physics and Engineering Sciences, Buein Zahra Technical University, Buein Zahra, Iran
2 - Department of Physics, University of Agriculture and Environmental Sciences Umuagwo, Imo State, Nigeria
الکلمات المفتاحية: Exciton, Ellipsoidal QD, Ground State Energy, Oblate Ellipsoid, Relativistic Correction,
ملخص المقالة :
Recent progress in high-technology equipment enables the fabrication of quantum dots such as GaAs, and GaAlAs confining a finite number of excitons and allowing for control of the properties of quantum dots. Biexciton quantum dots are the simplest example that can be used to upgrade optoelectronics technologies. This theoretical research investigates a model of the biexciton state in the strongly oblate ellipsoidal quantum dot with the relativistic corrections of mass and Hamiltonian in the framework of the quantum field theory due to the importance of the relativistic effect for this type of quantum dot shapes. The Sturmian function transformation and Wick ordering method to calculate the vacuum state energy eigenvalue of the biexciton system are utilized. Based on the relativistic behavior of interactions, the mass corrections to the Hamiltonian are defined. Dependence of the relativistic mass on the distances between electrons and the constituent mass to the coupling constant is obtained. The results show that as increasing quantum dot size, the relativistic mass and Hamiltonian corrections terms decrease.
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Islamic Azad University
| Journal of
Spring 2024 / Vol. 9, No. 2 |
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Research Paper
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Biexciton in Strongly Oblate Ellipsoidal Quantum Dot with Relativistic Corrections
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Arezu Jahanshir*,1, Ekwevugbe Omugbe2 1Department of Physics and Engineering Sciences, Buein Zahra Technical University, Buein Zahra, Iran 2Department of Physics, University of Agriculture and Environmental Sciences Umuagwo, Imo State, Nigeria
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Received: 18 Feb. 2024 Revised: 21 Mar. 2024 Accepted: 5 Mar. 2024 Published: 15 Jun. 2024
| Abstract Recent progress in high-technology equipment enables the fabrication of quantum dots such as GaAs, and GaAlAs confining a finite number of excitons and allowing for control of the properties of quantum dots. Biexciton quantum dots are the simplest example that can be used to upgrade optoelectronics technologies. This theoretical research investigates a model of the biexciton state in the strongly oblate ellipsoidal quantum dot with the relativistic corrections of mass and Hamiltonian in the framework of the quantum field theory due to the importance of the relativistic effect for this type of quantum dot shapes. The Sturmian function transformation and Wick ordering method to calculate the vacuum state energy eigenvalue of the biexciton system are utilized. Based on the relativistic behavior of interactions, the mass corrections to the Hamiltonian are defined. Dependence of the relativistic mass on the distances between electrons and the constituent mass to the coupling constant is obtained. The results show that as increasing quantum dot size, the relativistic mass and Hamiltonian corrections terms decrease. |
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Keywords: Exciton, Ellipsoidal QD, Ground State Energy, Oblate Ellipsoid, Relativistic Correction
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1. INTRODUCTION
The analytical study on the multi particles in the strongly oblate ellipsoidal semiconductor quantum dots (SOEQD) with relativistic mass correction is an active area of advanced high-technology research [1-3]. The significance of SOEQD in optoelectrical equipment, communication technology, and quantum information has driven the interest in studying the multi-particle quantum dot based on the relativistic behavior of a system. We choose the multi-particle quantum dots as multi-exciton systems, which play a crucial role in many tasks related to advanced nano-technology also the control of multi-exciton bound states within quantum dots holds great importance for this type of technology. Hence, the theoretical investigation has contributed to understanding the behavior of bound states in quantum optics and nanoelectronics. In this theoretical research, we use biexciton systems in quantum dots and define the energy eigenvalue and relativistic mass corrections based on SOEQD shape [3-5]. The characteristics and properties of biexciton as the electrons and holes bound states within the confinement and relativistic effects are investigated. The potential confinement leads to a change in the biexciton constituent particles overlap, and the relativistic effect results in a difference in the constituent mass of particles, which is different from the rest mass and effective mass. Due to the relativistic mass corrections on the energy spectrum, the biexciton system in SOEQD demonstrates a larger emission band, luminescence spectrum, and luminescence decay times compared to non-nano dimensional semiconductors. Multi-excitons in SOEQD have an important property based on face-to-face interactions and lead to multi-particle correlation such as exciton-exciton, biexciton-exciton, and biexciton-biexciton interactions [4-6]. These interactions show the new quantum-optical nonlinearities behavior and entangled states [6-8]. The measure and features of relativistic mass correction can depend on Multiple elements and factors, including the size and shape of the strongly ellipsoidal semiconductor quantum dots, the composition of the quantum dots material, and the external environment. This theoretical study's results are predicted to offer guidelines for tuning properties of strongly ellipsoidal quantum dots and thus facilitating the advancement of practical nano-dimensional electrical appliances and devices. Consequently, in this research for simplicity of calculations, we choose just biexciton as a four-particle state (electron-hole) of SOEQD and show the relativistic correction effects in the field of nano-quantum physics and optoelectronics [6-8]. The remainder of this theoretical investigation is organized in the following format Section 2, determination of relativistic mass correction in quantum field theory. In Section 3, a transformation of intertwined spaces is introduced and describes the behavior of SOEQD. In Section 4, the biexciton mass spectrum and energy eigenvalue of the vacuum state are explained and calculated. Finally, in Section 5, concluding comments are given.
A. Research Aim
Biexcitons can arise in certain semiconductor quantum dots and nanostructures under specific conditions. We investigate a biexciton Coulomb system in SOEQD, consisting of two excitons bound states (two electron and two holes bound state). Our analytical approach is based on the Sturmian function transformation and Wick ordering method [9] applied to the quantum oscillator system [7-14]. We determine and calculate the relativistic mass correction to the energy eigenvalue and other properties of a nano-scales 
semiconductor that is confined in SOEQD.
B. Sturmian Function and Transformation
One of the mathematical techniques and methods in quantum mechanics and quantum field theory is the Sturmian function transformation [14, 16]. This technique allows us to describe a quantum system in a Sturmian form simplifying the theoretical analysis and calculation of specific properties. It is widely used in semiconductor quantum dots, wells, and nanostructures. The Sturmian function transformation is beneficial to determine bound state behavior in atomic, molecular, quantum chemistry, solid-state physics, and particle physics. By applying the Sturmian function transformation, the bound state problem can be represented in the second-order linear differential equation that provides a powerful mathematical tool for studying various aspects of nano-scale systems such as wavefunctions, mass spectrum, binding energy and energy eigenvalues, and quantum-optical properties. Hence, we present the Sturmian function transformation to study the biexciton in SOEQD within the relativistic mass correction. The radial Schrödinger equation (RSE) can describe the biexciton properties based on the Sturmian representation and the set of eigenfunctions. The Sturmian function 
is the solution of RSE
(1)
This function has a notable benefit over RSE when used as a basis for expansion when choosing a different basis of intertwined spaces for the biexciton states. For instance, quantum oscillator wave functions satisfy the needed conditions. The mechanism of transforming the independent parameter has long been a valuable strategy for solving RSE with various types of potentials. In this manuscript, the study of the biexcitons system involves using the Coulomb potential in the strongly ellipsoidal semiconductor quantum dots. A more general equivalence emerges as a change of variable in the wave function of intertwined spaces
,
, (2)
and this transformation can map the nonrelativistic RSE and its solutions for given potentials [17, 18]. We know that for 
, 
can be obtained for certain classes of potentials such as Coulomb or Yukawa-type potentials 
. We study the biexciton states within Coulomb potential with a modifiable number of power factors 
, and have used a variable change to interpret the Sturmian equation and two intertwined spaces in the context of conventional physical RSE. We also mention that the biexciton states in SOEQD can be addressed using algebraic methods, such as the Wick ordering technique, and solve RSE due to define and determine mass spectrum based on the relativistic mass correction.
2. Theoretical model of the ellipsoidal quantum dots
We investigate the four-particle bound state (biexciton) in the ellipsoidal quantum dot with the relativistic mass correction. As we know, ellipsoidal quantum dots have two different types of geometry (prolate and oblate). The strongly prolate ellipsoidal quantum dots explained in the 
-dimensional space and SOEQD described in the 
-dimensional space [3-6]. We will present our theoretical findings on strongly oblate ellipsoidal quantum dots in semiconductors. It is schematically plotted in Fig. 1. The confinement potential and interaction Coulomb potential of a strongly oblate ellipsoidal quantum dots with condition 
in 
The dimensional Cartesian coordinate system is considered as follows
, 
(3)
where 
are semiaxes of the ellipsoid, respectively. the Hamiltonian of biexciton bound state (two electrons and two holes) based on the Coulomb potential of interactions 
(as it is shown in Fig.1), in natural unite 
, reads
(4)
where
is the charge of particles, 
is the relative permittivity of semiconductor, 
is the vacuum permittivity, corresponds to the electron-hole, hole-hole, and electron-electron interaction. The Biexciton system is a four-body problem that can be constructed in H-type and K-type. H-type coordinates begin by defining electron-electron and hole-hole the relative vector between the centers-of-mass of the biexcitons Fig. 2.
Fig. 1. Schematic plot of SOEQD and Biexciton bound state inside the red cycle.
Fig. 2. Jacobi coordinates 
of biexciton in the four-body problem.
-channel represents the correlation matrix within two interacting electron-holes, and then the exciton-exciton correlation. 
-type coordinates are formed by sequentially establishing the relative vector between the center-of-mass of a subgroup of particles and one extra particle [19]. We choose the -type coordinates to describe biexciton properties in SOEQD. Therefore, the three sets of linearly independent Jacobi coordinates 
for particles in the four-body problem as it is shown in Fig. 2, reads
Jacobi Coordinate for particle 
Jacobi Coordinate for Particle 
Jacobi Coordinate for Particle 
Jacobi Coordinate for Particle 
(5)
and
the position vector of Particle
the position vector of Particle
the position vector of Particle
the position vector of Particle
(6)
where 
represents the position vector of the 
-particle with the rest mass 
, and 
is the coordinate of the center of mass. Hence Eq. (4) reads
Now for simplicity, we consider 
and 
in the electrostatic field, and
Then Eq. (7) is realized by the following parameters
We assume that the motion of the electrons and holes clusters are limited by themselves and do not move freely in quantum dots. These clusters move only in the biexcitonic structure, hence
therefore, the radial part of RSE with Hamiltonian Eq. (8) in the cylindrical coordinate system 
presents as follows
(9)
Where 
is the magnetic quantum number and the wavefunction has the form 
. Now, based on SOEQD
. And finally, Eq. (21) will have the following form
(10)
As we have two clusters, we choose 
, and after some mathematical calculation, the Hamiltonian of relative motion of exciton-exciton interaction as a biexciton bound state with the radius vectors 
in SOEQD reads
(11)
3. Auxiliary symplectic space in quantum theory
In the context of quantum mechanics, an auxiliary symplectic space refers to an additional space that is introduced to simplify the calculation and analysis of a system. This subject deals with symplectic manifolds, which are mathematical objects that capture the geometric properties of dynamical quantum systems, and often are described using position and momentum variables [20, 21]. This method is essential and beneficial in solving the RSE of the bound states. It allows different aspects of the bound state's properties and characteristics to be presented and clarified in the position and momentum spaces. By introducing an auxiliary symplectic space as a powerful mathematical tool and technique, one can provide a clearer understanding of the dynamics of bound states. Dimension of the auxiliary symplectic space depends on the particular bound states and the goals of the investigations. Now, we introduce the auxiliary symplectic space in quantum theory for solving the mass spectra and energy eigenvalues of the biexciton bound state using Eq. (15) in SOEQD. For this reason, we identify canonical operators in Eq. (
(11), by changing 
, and always map 
into 
and map 
into 
. Two intertwined spaces can be modified by connections
and 
. (12)
Hence, the radial Laplacian in a 
dimensional Riemannian space to the intertwined space
, (13)
is transformed, and 
. Therefore, under the relative motion of particles in the biexciton bound state, to determine the energy eigenvalue and mass spectra of biexciton in SOEQD, we change the variables 
and define the wavefunction
, (14)
and then Eq. (
(11) reads
(15)
where 
is the dimension of the auxiliary intertwined space. The biexciton bound state as a quantum oscillating system can be presented by the Wick ordering technique in the auxiliary symplectic space. It formulates the momentum (
) and position (
) operators (canonical variables [21]) in terms of
and
(16)
operators, where 
, 
, and 
is the vacuum state frequency of biexciton. Substituting operators 
into the Eq. (15) and ordering by operators 
. The Wick ordering product [8]over the canonical operators is 
, and 
, then the interaction Hamiltonian of four particles bound state is realized by the following equation
(17)
Where 4. Relativistic corrections Increasing the Coulomb interaction coupling constant in semiconductor quantum dots can lead to various properties, such as relativistic behavior of interaction behavior. By decreasing the size and shape of the semiconductor quantum dots the electrons and holes become more closely, leading to a stronger Coulombic coupling constant. By increasing the number of electrons, holes or both the carrier density within the semiconductor quantum dot Coulombic coupling constant can be enhanced. By choosing materials or modifying the dielectric medium based on connecting two or more semiconductor layers the strength of the Coulomb interaction can be increased. Applying a strong electromagnetic field on a system the coupling constant strength can be increased. Hence, we choose theoretical modification to increase the Coulomb interaction in the strong oblate ellipsoidal The Green function in the form of path integral can be written as where (20) where the functional where Using Eq. (7) and relation where with The term where 5. Results and discussion One of the main results and interesting effects of SOEQD is the potential confinement of particles, which is achieved through photon exchange. This potential confinement leads to distinct electrical and optical properties that are important and valuable in quantum-optical devices, nano-electronic components, and optoelectronic equipment. As it is widely recognized, the optical properties of SOEQD are an essential aspect of nanophysics. Therefore, exciton and biexciton states in SOEQD are an interesting subject. As we show in previous sections the relativistic corrections on the total Hamiltonian can increase the accuracy of results in calculating energy, mass, and spin interactions. In this section, we will calculate the relativistic corrections to the mass and Hamiltonian of biexciton in SOEQD, which are performed theoretically. Two-center approximation method is a universal method to describe the biexciton bound states of the electron-hole (exciton) interacting clusters. This tool simplifies system interactions by treating the clusters independently. This approximation assumes that the biexciton bound state’s wavefunction can be explained as separate wavefunctions for each excitonic cluster. The biexciton properties can be solved using RSE or the relativistic Bethe-Salpeter equation. This approach allows us to describe the biexciton analytically and numerically, compared to considering the interactions between and within the excitonic bound states. Now, to determine the biexciton mass spectrum and properties, we use the theoretical effective mass of an exciton in a where utilizing and applying correlations and then, considering that the distance between two clusters is 
is the potential, 
is he Hamiltonian of the free oscillator, 
is the energy of the vacuum or the ground state energy, and 
is the interaction Hamiltonian. 
is the normal ordering (Wick ordering[8]) symbol and we consider that in the symplectic intertwined spaces the interaction Hamiltonian doesn’t contain 
and 
. These terms are included in the vacuum state energy. Hence, we define a specific condition that determines the frequency of the biexciton bound state 
=0. Now, using conditions 
and 
for creating the bound state (i.e., the confined state exists at the lowest point of oscillator frequency and energy eigenvalue) [17], one can define the ground state energy eigenvalue and mass spectrum of biexciton system in SOEQD.
quantum dot and describe the relativistic conditions on the biexciton bound state. The mass spectrum and energy eigenvalue of the bound states in SOEQD are present based on the quantum mechanics approaches and Feynman path integral method in the quantum field theory [22-26]. As we know, the long-range and small-range behaviors of the propagator function of the related system’s current with the different quantum numbers can determine the main properties of the bound states in SOEQD. The Feynman functional path integral method allows us to average over the field and describe the Green function 
of bound state and interactions of particles in quantum field theory, similar to the Schrödinger picture in quantum physics. We explain the related current of charged particles in the system and obtained current through averaging over the external field for bounded particles [23, 24, 26]. This method defines the kernel function of two charged particles in SOEQD with the specific rest masses. Hence, one can determine the 
-point function by averaging over the field 
. The Green function 
for the scalar particle with the rest mass 
and coupling constant 
in the external field 
is defined from
. (18)
(19)
, 
, and 
is the normalization condition with
with boundary conditions 
. Then using the variational method and Eq. (19) and , the polarization function of -point function
reads
(21)
resembling the behavior of 
particles with masses 
in the non-relativistic quantum mechanics and with the potential and nonpotential interaction that includes in 
in the form of the Feynman path integral and at 
, it is analogous to the relation
(22)
is an eigenvalue of 
for the motion of 
bounded particles with masses 
, it depends on the parameters 
, the rest mass, the constituent mass, and the coupling constant, respectively. The constituent masses of interacting particles mean the mass of particles inside the bounded system. Then, the Green’s function at 
can determine the bound state total mass 
of bounded particles as follows [25, 27]
(23)
one can demonstrate the polarization function 
. Hence, the bound state mass spectrum is defined as 
. We can determine the bound state mass 
based on the method of steepest descent in the following manner for
(24)
(25)
, 
is the reduced mass of the system, 
is the constituent mass of particles, which determines the relativistic mass correction of each particle [17, 18, 27]. Now, we define the relativistic correction on the Hamiltonian of biexciton bound state. The total Hamiltonian of the biexciton can be further developed with the guidance of relativistic effects such as the relativistic correction to the mass, kinetic energy, spin interaction, and other perturbative terms. The relativistic effects result from a change in the biexciton bound states’ energy brought on by relativistic mass. The other one is spin interaction, which is contained in spin-spin, spin-orbit, and spin tensor interaction. Coupling [27, 28]. These relativistic corrections affect the ground and excited energy levels of the biexciton in SOEQD, which have a considerable effect and meaningful contribution to the electrical and optical characteristics of quantum dots. The total Hamiltonian 
describes all interaction potentials [29] that are included in the term
of Eq. (21), and all energy spectrum reads 
. The potential of interaction has the form
(26)
corresponds to the contribution of one photon exchange, and the term 
gives only cross term. 
is the speed of light, and 
is the coupling constant of one photon exchange. We suppose that in the initial moment, particles are in the rest and interact among themselves only by electrostatic field. In this case, we have 
. From 
, 
, it follows that the asymptotic behavior of must depend on 
in a linear way [30]. According to this, the dependence of 
is Euclidean time on t is the proper time of the composite particle is chosen as 
, 
and 
. After performing a series of mathematical calculations and substitutions, we get the nonperturbative relativistic correction on the total Hamiltonian as a relativistic correction term which is defined by the equation
(27)
is the reduced mass of biexciton, 
is the distance between the clusters, 
is the potential interaction, 
is the scalar term and 
is the vector term of potential interaction of the biexciton SOEQD [29]. The non-relativistic results of confined states are no longer sufficient for the description of the Large Hadron Collider, experimental results in particle physics, and nano-scale interactions. Therefore, it is necessary to consider the relativistic corrections on mass, energy, spins, and nonperturbative and perturbative interactions. As we are aware, the real relativistic corrections are very small, therefore within the framework of quantum field theory, the theoretical approach outlined in Eq. (27) simplifies deriving the relativistic corrections for a nonrelativistic interaction potential. Hence, the relativistic Hamiltonian of biexciton with the mass 
, in the strongly oblate ellipsoidal 
quantum dot within the Coulomb interaction in the natural unit has the form 
, 
is the Bohr radius.
quantum dot. It typically considers the reduced mass of the exciton (electron-hole) system 
. Once 
theoretically obtained, it can be used to describe the exciton dynamics within the quantum dot system using the two-center approximation method. Therefore, due to the present relativistic behavior of the system, we present the method based on quantum mechanics and quantum field theory approaches. Hence, using the Bethe-Salpeter equation 
and Eq. (15) we will describe relativistic processes and the behavior of the system within the strong Coulomb interaction with the coupling constant 
. Relativistic effects are accounted for by including relativistic energy and expanding the term 
, hence 
, 
in the axillary symplectic space is obtained in the following way 
, and then Eq. (15) reads
(28)
is considered as the rest mass of (exciton) electron-hole confined state in the two-point approximation method, 
is the reduced mass of biexciton and 
is the constituent mass of exciton (electron-hole), and 
in the natural unit. These masses describe the mass of the electron and hole inside the biexciton bound state and differ from the rest mass and effective mass of particles in the strongly oblate ellipsoidal quantum dot. We consider using conditions 
and 
, and 
, 
are oscillator frequencies of clusters in a biexciton-bound state (for more detail see [31] ). We determine oscillator frequencies for conditions 
and 
and define 
, and Eq. (28) reads
(29)
(30)
(31)
, where 
, after performing specific mathematical substitutions, the energy eigenvalue can be defined from Eq. (31) in the form
(32
Then, taking advantage of the condition 
, by minimizing the Eq. (32), we define the equation for describing oscillator frequency 
of biexciton in strongly oblate ellipsoidal quantum dot
(33)
Using 
, Eq. (33) reads
(34)
and, then, with the condition , oscillator frequency 
of biexciton is defined and then this formula will help us to calculate the reduced mass 
of the biexciton system based on Eq. (25). We calculate the result for the ground states of energy levels and oscillator frequency, respectively. As a result, Eqs. (32) and (33)) are described in the form
(35)
and
(36)
Also, we determine the energy spectrum of the relativistic correction based on Eq. (27) is defined by the following expression (
)
(37)
Since we know that the relativistic corrections in the absence of a strong external electromagnetic field can be neglected, therefore, by modifying Eq.(22) based on the theoretical calculation from Eq. (18), we can determine the relativistic mass correction of an electron-hole cluster inside the biexciton system, and the resulting equation is solved as an equation 
, the reduced mass of the biexciton system is defined using the relation 
. The results are calculated within the semiconductor quantum dot parameters for the , and the effective mass of electron and hole are taken as 
with the relative permittivity 
13.1. We consider specific parameters for quantum dots due to tuning the strong electrostatic coupling constant. The coupling constant depends on the size, shape, charge distribution, and dielectric environment of quantum dots and potential confinement. In this research, we consider the distance between electrons
: (0.1-5) 
, where is the Bohr radius. It is a characteristic length scale between electrons and in a biexciton system. The biexciton properties with the relativistic corrections on the total Hamiltonian, mass, and energy eigenvalue are calculated and presented in Table 1, Fig. 3, and Fig. 4.
TABLE I
Biexciton to electron mass ratio in the strongly oblate ellipsoidal quantum dots with coupling constants (50, 80, 100) times the Coulomb interaction (
).
|
|
|
|
|
|
|
| 1.7287 | 1.7317 | 1.7350 | 1.7391 | 1.7480 | 1.7640 |
| 1.7547 | 1.7559 | 1.7573 | 1.7592 | 1.7635 | 1.7724 |
| 1.7824 | 1.7826 | 1.7828 | 1.7831 | 1.7840 | 1.7860 |
Corresponding author: Arezu Jahanshir Address: Department of Physics and Engineering Sciences, Buein Zahra Technical University, Buein Zahra, Iran. Tell: 00982833894492 Email: jahanshir@bzte.ac.ir |
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