1. 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 relativistic.

At present the technical achievements make it possible to create the exotic state with electron and exciton so in this quantum- relativistic state, the calculation of relativistic 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. Calculation and analysis of energy spectrum in Coulomb 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 Coulomb interaction. This work is devoted to study the bound states problems, which will be carried out on the research basis asymptotical behavior of two scalar particles in external gauge field. In this case, loop function of two scalar particles with different masses and , with average on external statistical field is considered and the Green function is used. The polarization operator in an external electromagnetic field looks like[4]:

(1)

The Green function for scalar particles in an external field is determined from the equation:

(2)

Where m is the mass of a scalar particle, and is the coupling constant of interaction. The gauge field averaging defines as follows:

(3)

Here is a real current, and

(4)

By using the form of the functional integral we can represent green function as follow[5]:

(5)

where used the notations

(6)

with the conditions:

The mass of the bound state in the extreme limit is determined by the equation[4,5]:

(7)

After averaging over the field we have (for details of an evaluation see[5]):

here:

(8)

(9)

Functional integral in (8) is similar to the Feynman Path integral trajectories[5] in nonrelativistic quantummechanics[9] for the motion of two particles with masses and. The interaction of these particles is described by the nonlocal functional, in which are contained both potential and non-potential interaction (see appendix 1). Taking into account (8) and (9) in the extreme limit from (7) for the mass of the bound state we get (for detail see[5-7]):

(10)

where the parameter μ is determined from the equation:

(11)

and E(μ) - is the eigenvalue of the nonrelativistic Hamiltonian. Parameters and are mass components of the bound state (constituent mass of particles), which are different from the masses of free condition and also can be found with the help of conditions:

(12)

2. Exciton's Hamiltonian in Relativistic Corrections

As we know one of the basic problems of nonrelativistic quantum mechanics is to determine the energy spectrum and eigenvalue of the nonrelativistic Hamiltonian of system described by the Schrödinger equation with a special potential. Complete solution of this equation has been found in 1965 (for a detailed historical literature see[8] and Landau 1977[9], for the Coulomb potential. The oscillator representation method implies 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 momentum are expressed through the creation and annihilation operators.

The oscillator representation method in the nonrelativistic Schrödinger equation is proposed to calculate the energy spectrum for central symmetric potential allowing the existence of bound state. (for details see[10]). It is well known that for determining the mass of the bound state systems first of all one should determine the eigenvalue of the Hamiltonian with Coulomb potential. For this aim we consider the following Schrödinger equation[11,12]:

(13)

- is the coupling constant of electromagnetic interaction. z – electron and hole charge number.

This equation for the wave function of orbital excitation is:

(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 initial system. By using oscillator representation method and techniques of quantum filed theories represent the canonical variables in terms of the creation () and annihilation () operators in the R^d space:

(15)

here is the oscillator frequency which has been known as Hamiltonian of free oscillator and can write as follow (for more detail see[11]):

(16)

We have to modify the variables in the starting Schrödinger equation. In this case, we used the substitution, and found the following conditions[14,15]:

(17)

After simple calculations we obtain Schrödinger equation in oscillator representation method (for the Coulomb potential ):

(18)

Now we try to describe interactions in nano- quantum dots. Energy is absorbed by electrons presented in semiconductor, which means as activating electron from 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 Coulomb 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 inhomogeneous environment is explained through electrostatic forces, electron relative potential (resulted from dissemination of charge and electron flow, which result to formation of related electromagnetic field between electrons), spin effects and particles’ complementary effects[16].

In an effective mass approximation the Hamiltonian of exciton is ():

(19)

- effective mass of electron and hole. Making the co-ordinate transformation¹ the Hamiltonian may be written[17]:

(20)

μ- is the reduce mass.

Hamiltonian for diexciton as we now the complete Schrödinger equation for diexciton so rewrite[18]:

(21)

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]:

(22)

Therefore, this assumed positive charge (visual charge) on the border of assumed inhomogeneous system will be:

(23)

3. Four-particle Spin Interaction in Quantum Dots

As we know all Hamilton of particles 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 from uni-photon transaction and the potential resulted from electrons’ confinement and spin effect: (complementary Hamilton’s effect () and relative potential ()have been omitted due to simplicity):

(24)

Hamiltonian of spin-orbital interaction is:

(25)

Hamiltonian of spin-spin interaction is:

(26)

where

(27)

S_i- spin of particles and we can determine by table 1.

Table 1. Spin-orbital and spin-spin ,matrix elements

Election of distance between to Fermions’ particles (electron) (x₀) in diexciton system for calculating spin-spin and spin-orbital effects will be so that total vector potential effect resulted from uni-photon exchange Uv_s(x) have overlap with the potential resulted from electron converge. U_s(x) Using oscillator representation method and moving to Hamilton sphere coordination, system would be calculated (oscillation representation method and calculations resulted from it are discussed in this article[20]). Doing required replacements, for two potential we would have (we did this calculation for exotic atom with Coulomb potential[21]):

(28)

Where -oscillator frequency of electrons and- is effective radius of system. In 1991, Schoberl-Louch-Gromes[22], found spin-orbital equations, while these equations are in force for quantum points and we will use them here. Suppose for sum of two electrons’ spin in diexciton, and for spin-orbital and spin-spin equations we will have:

(29)

And

(30)

Having little reform and replacing spin-orbit and spin-spin Hamilton constant coefficient, the system would be:

(31)

Using oscillating method in spherical coordination and changing parameters in equation (5), parameters related to particles’ distances in diexciton system for electrostatic coulomb potential ruling in quantum dots and overlap of spin potential would be calculated[23], and we have:

(32)

Where, x₃ - exciton radius (distance between mass center of two holes and two electrons). x₂ - distance between two holes. x₀ - distance between two electrons. As we know, exciton radius will be calculable through separating sections related to mass center moves and relative movement through Wannier Equation[24] that is:

(33)

Where, - Bohr radius of exciton or relative distance between electron and hole, - constituent mass of exciton system, - effective mass of electron and hold. Relation of exciton radius resulted from Wannier equation in homogeneous environment with dielectric constant coefficient and exciton radius founded for quantum dots between inhomogeneous environments with the same constant dielectric coefficient in oscillator method is:

(34)

Now using equation (7, 8) and (12), we will determine constant coefficient of spin Hamiltonian and exciton radius:

(35)

and

(36)

Above equations clearly determine that there is a direct relation between spin Hamiltonian constant coefficient with visual load sum near to layers’ border and an indirect relation between system reduced mass and exciton radius.

4. Conclusions

In this study we worked on spin effects in Nano-crystals constructed by diexciton system in space between nanostructures. Oscillating calculation 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 relativistic corrections and defined spin-spin and spin-orbital Hamilton constant coefficient.

In addition we have shown that how spin Hamiltonian constant coefficients of exciton system between inhomogeneous environments are related to exciton radius 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 relative coefficient in potential well and electron distance decreases due to their confinement in quantum dot along with increase of visual charges (z).

Appendix

The important point of calculations is the definition of Hamiltonian's structure of interaction. Interactions between composite particles are carried out through interchanging of gauge fields, therefore the propagator we shall note in a standard way:

(A.1)

for a potential of interaction after integration on we have[25]:

(A.2)

Where

(A.3)

Here and - is considered as proper time particles 1 and 2. We suppose that in initial moment particle rest and interact among themselves only by electric field. In this case we have . On the other hand, we consider that - Euclid time is connected with - proper time quarks as follows:

(A4)

Taking into account (3.3) and (3.4), and making out integration on the variables ds, after some simplifications from (3.2) is obtained:

(A.5)

Where

(A.6)

Notes

¹Jacobian transformation.