Spin-Orbital Effect of Diexciton in Quantum Dots

Great efforts have been made to understand how bound state arise in the formalis m of quantum field theory and to work out effective methods to calculate all characteristics of these bound states, especially their masses and binding energy. The analysis of a bound state is simplest when constituent particles can be considered to be nonrelativistic, i.e. when they travel at speeds considerably less than "c". We studied an exotic system (state) on the basis of investigation of asymptotic behavior of the loop function for the scalar particles in the external gauge field and determined it for the scalar particles in the with relativistic feature of interaction and have been able to obtain the mass spectrum of bound state, the constituent mass of diexciton and spin interactions and find out that determining the mass of the bound state systems requires; first of all, determin ing the Eigenvalue of the Hamiltonian with Coulomb potential and then calculating the mass and binding energy or spin interactions of diexciton system which we could achieve. The current paper has calculated energy spectrum and constituents mass of particles in limited core mass conditions and we could conclude that constituent mass of particle in the system is different from mass in free states. So having below conditions form inhomogeneous multi-layer environment in nanostructure ) ( 5 1 5 3 B C A B A x x − in mind, and using oscillator representation method, spin Hamiltonian coefficient of diexciton system with effective confinement potential, Coulomb effect between particles, and spin effect, will be d iscussed. Relation between distances properties of diexciton system and distance between two electrons in quantum dots and dependency relation will be determined in conclusion.


Introduction
Electrons and holes in microscopic semiconductor structures can display astounding quantum behavior. These structures, namely exciton are expected to be the basis of new generation of electronic and optoelectronic and nano-structures [1,2]. Therefore description of the bound states of electron and hole is one of the general problems in quantum mechanics and nano quantum dots. This problem is studied by many authors and it is well known but in this article we try to present new approaches which help us to describe and determine better and more useful. One of the corrections in particle physics can be classified as relativ istic.
At present the technical achievements make it possible to create the exotic state with electron and exciton so in this quantum-relativ istic state, the calculat ion of relativ istic corrections becomes necessary. However, the conventional prescription of the calculation of the relativistic nature interaction within the framework of the phenomenological potential model of is absent at the moment and the theoretical models intended for describing the relativistic corrections to the spectrum are limited to the lowest order on the coupling constant. The other powerful stimulus for further development of the bound state theory is provided by the spectacular experimental progress in precise measurements of atomic energy levels. Calculat ion and analysis of energy spectrum in Coulo mb potential of electron and exciton in relativistic conditions due to requirement of using higher grades of relativistic corrections have attracted physics theoreticians.
We will consider this problem, according to the asymptotic behavior of the loop function in the scalar electrodynamics field and method oscillator representation in quantum physics [3]. The method presented in this paper considers relativistic effect that it concepts energy spectrum, mass and constituent mass in the exotic system and analytical is calculated with Coulo mb interaction. Th is work is devoted to study the bound states problems, wh ich will be carried out on the research basis asymptotical behavior of two scalar part icles in external gauge field. In this case, loop function of two scalar part icles with different masses 1 m and 2 m , with average on external statistical field is considered and the Green function is used. The polarizat ion operator in an external electro magnetic field looks like [4]: The Green function ) | , ( A y x G for scalar part icles in an external field is determined fro m the equation: Where m is the mass of a scalar particle, and α is the coupling constant of interaction. The gauge field averaging defines as follows: Here ) (x J α is a real current, and (4) By using the form of the functional integral we can represent green function as follow [5]: where used the notations with the conditions: The mass of the bound state in the extreme limit is determined by the equation [4,5]: After averaging over the field we have (for details of an evaluation see [5]): here: Functional integral in (8) is similar to the Feynman Path integral trajectories [5] in nonrelat ivistic quantum mechanics [9] for the motion of t wo part icles with masses 1 µ and 2 µ . The interaction of these particles is described by the nonlocal functional ij W , in wh ich are contained both potential and non-potential interaction (see appendix 1). Taking into account (8) and (9) in the extreme limit fro m (7) for the mass of the bound state we get (for detail see [5][6][7]): where the parameter µ is determined fro m the equation: and E(µ) -is the eigenvalue of the nonrelativistic Hamiltonian. Parameters 1 µ and 2 µ are mass components of the bound state (constituent mass of particles), wh ich are d ifferent fro m the masses 2 1 , m m of free condition and also can be found with the help of conditions:

Exciton's Hamiltonian in Relativistic Corrections
As we know one of the basic problems of nonrelativ istic quantum mechanics is to determine the energy spectrum and eigenvalue of the nonrelat ivistic Hamiltonian of system described by the Schrödinger equation with a special potential. Co mp lete solution of this equation has been found in 1965 (for a detailed historical literature see [8] and Landau 1977[9], for the Coulo mb potential. The oscillator representation method imp lies that a wave function, being a bound ground state of quantum system with a good looking potential, is expanded over the oscillator basis in which coordinate and mo mentu m are exp ressed through the creation and annihilat ion operators. The oscillator representation method in the nonrelativistic Schrödinger equation is proposed to calculate the energy spectrum for central symmetric potential allo wing the existence of bound state. (for details see [10]). It is well known that for determin ing the mass of the bound state systems first of all one should determine the eigenvalue of the Hamiltonian with Coulo mb potential. For this aim we consider the following Schrödinger equation [11,12]:  (14) and at large distance, for the Coulomb potential the asymptotic behavior of wave function is well known [12,13].
We considered this equation as the condition for the determination of the energy spectrum of the in itial system. By using oscillator representation method and techniques of quantum filed theories represent the canonical variables in terms of the creation ( + a ) and annihilation ( − a ) operators in the R d space: . ,...  (15) here ω is the oscillator frequency which has been known as Hamiltonian of free oscillator and can write as follow (for more detail see [11]): We have to mod ify the variables in the starting Schrödinger equation. In this case, we used the substitution After simple calcu lations we obtain Schrödinger equation in oscillator representation method (for the Coulo mb poten- Now we try to describe interactions in nano-quantum dots. Energy is absorbed by electrons presented in semiconductor, which means as activating electron fro m capacity band to conductive ban. Electron moves to conductive band and establish a connection with its position in capacity band (hole: hole with positive charge) under Coulo mb potential gravity.
Electron and the hole start to spin around themselves and create an exciton system. Exciton system is relatively a stable structure, which in this field is considered as a long time life. Interaction and mutual effect of each layer with other layer in inho mogeneous environment is explained through electrostatic forces, electron relat ive potential (resulted fro m dissemination of charge and electron flow, which result to formation of related electro magnetic field between electrons), spin effects and particles' complementary effects [16].
In an effective mass approximation the Hamiltonian of exciton is ( Electrostatic force on assumed positive charges (visual charges) in distance of h from the border of two environments with different dielectric coefficient will be as follows [19]: e z e z (23)

Four-particle Spin Interaction in Quantum Dots
As we know all Hamilton of part icles should be determined through surveying spin-spin and spin-orbital effect between particles in quantum dots. So final potential in quantum dots will be equal to vector potential resulted fro m uni-photon transaction and the potential resulted from electrons' confinement and spin effect: (co mplementary Hamilton's effect ( Ĥ ∆ ) and relative potential ( ) (x U s ) have been omitted due to simplicity): Hamiltonian of spin-orb ital interaction is: Hamiltonian of spin-spin interaction is: S i -spin of particles and we can determine by table 1. Table 1. Spin-orbital and spin-spin ,matrix elements.

S=2
S=0 Election of distance between to Fermions' particles (electron) ( ) 0 x in d iexciton system for calculating spin-spin and spin-orbital effects will be so that total vector potential effect resulted fro m uni-photon exchange ) (x Uv s have overlap with the potential resulted from electron converge ) (x U s . Using oscillator representation method and moving to Hamilton sphere coordination, system would be calculated (oscillat ion representation method and calculations resulted fro m it are discussed in this article [20]). Doing required replacements, for two potential we would have (we d id this calculation for exotic atom with Coulo mb potential [21]): r -is effective radius of system. In 1991, Schoberl-Louch-Gro mes [22], found spin-orbital equations, while these equations are in force for quantum points and we will use them here. Suppose 2 1Ŝ S S + = + for sum of t wo electrons' spin in diexciton, and for spin-orbital and spin-spin equations we will have: Having little reform and replacing spin-orbit and spin-spin Hamilton constant coefficient, the system would be: Using oscillating method in spherical coordination and changing parameters in equation (5), parameters related to particles' d istances in d iexciton system for electrostatic coulomb potential ru ling in quantum dots and overlap of spin potential would be calculated [23], and we have: Where, 3 x -exciton radius (distance between mass center of two holes and two electrons). 2 x -distance between two holes. 0 x -distance between two electrons. As we know, exciton radius will be calcu lable through separating sections related to mass center moves and relative movement through Wannier Equation [24] that is: Now using equation (7,8) and (12), we will determine constant coefficient of spin Hamiltonian and exciton radius: Above equations clearly determine that there is a direct relation between spin Hamiltonian constant coefficient with visual load su m near to layers' border and an indirect relation between system reduced mass and exciton radius.

Conclusions
In this study we worked on spin effects in Nano-crystals constructed by diexciton system in space between ) ( nanostructures. Oscillating calcu lation method has been used to calculate this effect and results of moving form general coordination to Jacobean coordination have been fully discussed. The method based on the investigation of the asymptotic behavior of the polarization loop function for diexciton in an external electro-magnetic field and oscillator representation method. We determined the spin-orbital interaction Hamiltonian with the relat ivistic corrections and defined spin-spin and spin-orbital Hamilton constant coefficient. In addition we have shown that how spin Hamiltonian constant coefficients SS LS σ σ , of exciton system between inhomogeneous environments are related to exciton rad ius of homogeneous environment. Diagrams for some of nanostructures show that spin effect increase with increase of visual charges (z), and get to steep slop along decrease in dielectric relat ive coefficient in potential well ε and electron distance decreases due to their confinement in quantum dot along with increase of visual charges (z). r r r r r