PARTICLE-HOLE PAIR AND BEELECTRON STATES IN ZnO/(Zn,Mg)O QUANTUM WELLS AND DIRAC MATERIALS

In this paper a theoretical studies of the space separation of electron and hole wave functions in the quantum well O Zn Mg ZnO 0.73 0.27 / are presented. For this aim the self-consistent solution of the Schrödinger equations for electrons and holes and the Poisson equations at the presence of spatially varying quantum well potential due to the piezoelectric effect and local exchange-correlation potential is found. The one-dimensional Poisson equation contains the Hartree potential which includes the one-dimensional charge density for electrons and holes along the polarization field distribution. The three-dimensional Poisson equation contains besides the one-dimensional charge density for electrons and holes the exchange-correlation potential which is built on convolutions of a plane-wave part of wave functions in addition. The shifts of the Hartree valence band spectrums and the conduction band spectrum with respect to the flat band spectrums as well as the Hartree-Fock band spectrums with respect to the Hartree ones are found. An overlap integrals of the wave functions of holes and electron with taking into account besides the piezoelectric effects the exchange-correlation effects in addition is greater than an overlap integral of Hartree ones. The Hartree particles distribute greater on edges of quantum well than Hartree-Fock particles. It is found that an effective mass of heavy hole of O Zn Mg 0.73 0.27 under biaxial strain is greater than an effective-mass of heavy hole of ZnO. It is calculated that an electron mass is less than a hole mass. It is found that the Bohr radius is grater than the localization range particle-hole pair, and the excitons may be spontaneously created. Schrödinger equation for pair of two massless Dirac particles when magnetic field is applied in Landau gauge is solved exactly. In this case the separation of center of mass and relative motion is obtained. Landau quantization l B    for pair of two Majorana fermions coupled via a Coulomb potential from massless chiral Dirac equation in cylindric coordinate is found. The root ambiguity in energy spectrum leads into Landau quantization for beelectron, when the states in which the one simultaneously exists are allowed. The tachyon solution with imaginary energy in Cooper problem ( 0 2   ) is found. Indexing terms/

/ are presented. For this aim the self-consistent solution of the Schrödinger equations for electrons and holes and the Poisson equations at the presence of spatially varying quantum well potential due to the piezoelectric effect and local exchange-correlation potential is found. The one-dimensional Poisson equation contains the Hartree potential which includes the one-dimensional charge density for electrons and holes along the polarization field distribution. The three-dimensional Poisson equation contains besides the one-dimensional charge density for electrons and holes the exchange-correlation potential which is built on convolutions of a plane-wave part of wave functions in addition. The shifts of the Hartree valence band spectrums and the conduction band spectrum with respect to the flat band spectrums as well as the Hartree-Fock band spectrums with respect to the Hartree ones are found. An overlap integrals of the wave functions of holes and electron with taking into account besides the piezoelectric effects the exchange-correlation effects in addition is greater than an overlap integral of Hartree ones. The Hartree particles distribute greater on edges of quantum well than Hartree-Fock particles. It is found that an effective mass of heavy hole of strain is greater than an effective-mass of heavy hole of ZnO. It is calculated that an electron mass is less than a hole mass. It is found that the Bohr radius is grater than the localization range particle-hole pair, and the excitons may be spontaneously created.
Schrödinger equation for pair of two massless Dirac particles when magnetic field is applied in Landau gauge is solved exactly. In this case the separation of center of mass and relative motion is obtained. Landau quantization

Introduction
There has been widely studied in the ultraviolet spectral range lasers based on direct wide-bandgap hexagonal würtzite crystal material systems such as ZnO [1][2][3][4][5][6]. Significant success has been obtained in growth ZnO quantum wells with (ZnMg)O barriers by scrutinized methods of growth [7,8]. The carrier relaxation from (ZnMg)O barrier layers into a ZnO quantum well through time-resolved photoluminescense spectroscopy is studied in the paper [9]. The time of filling of particles for the single ZnO quantum well is found to be 3 ps [9].
In the paper we present a theoretical investigation of the intricate interaction of the electron-hole plasma with a polarization-induced electric fields. The confinement of wave functions has a strong influence on the optical properties which is observed with a dependence from the intrinsic electric field which is calculated to be 0.37 MV/cm [10], causing to the quantum-confined Stark effect (QCSE). In this paper we present the results of theoretical studies of the space separation of electron and hole wave functions by self-consistent solution of the Schrödinger equations for electrons and holes and the Poisson equations at the presence of spatially varying quantum well potential due to the piezoelectric effect and the local exchange-correlation potential.
In addition large electron and hole effective masses, large carrier densities in quantum well ZnO are of cause for population inversions. These features are comparable to GaN based systems [11,12].
A variational simulation in effective-mass approximation is used for the conduction band dispersion and for quantization of holes a Schrödinger equation is solved with würtzite hexagonal effective Hamiltonian [13] including deformation potentials [14]. Keeping in mind the above mentioned equations and the potential energies which have been included in this problem from Poisson's equations we have obtained completely self-consistent band structures and wave functions.
So in this paper we present a self-consistent calculation an above mentioned equations in würtzite ZnO quantum well taking into account the piezoelectric effect and the exchange-correlation potential for bandgap renormalization and engineering of localized Hartree-Fock wave functions. The energy shifts as well as the localization range of exchangecorrelational wave functions with respect to Hartree energy shifts and Hartree localization range of wave functions require a scrutiny study.
We consider the pairing between oppositely charged particles with complex dispersion. The Coulomb interaction leads to the electron-hole bound states scrutiny study of which acquire significant attention in the explanations of superconductivity. If the exciton binding energy is grater than the localization range particle-hole pair, the excitons may be spontaneously created.

A. Effective Hamiltonian
It is known [13,15] that the valence-band spectrum of hexagonal würtzite crystal at the  point originates from the sixfold degenerate 15  state. Under the action of the hexagonal crystal field in würtzite crystals, 15  splits and leads to the formation of two levels: for the square of wave vector In the low-energy limit the Hamiltonian of würtzite ), ). , the Hamiltonian may be transformed to the diagonal form indicating two spin degeneracy [18]: From Kane model one can define the band-edge parameters such as the crystal-field splitting energy cr  , the spinorbit splitting energy so  and the momentum-matrix elements for the longitudinal ( z e ) z-polarization and the transverse ( z e  ) polarization : Here we use the effective-mass parameters, energy splitting parameters, deformation potential parameters as in papers [14,16,17].
We consider a quantum well of width w in ZnO under biaxial strain, which is oriented perpendicularly to the growth direction (0001) and localized in the spatial region In the ZnO/MgZnO quantum well structure, there is a strain-induced electric field. This piezoelectric field, which is perpendicular to the quantum well plane (i.e., in z direction) may be appreciable because of the large piezoelectric constants in würtzite structures.
The transverse components of the biaxial strain are proportional to the difference between the lattice constants of materials of the well and the barrier and depend on the Mg content x:

ZnO/(Zn,Mg)O quantum well
We take the following wave functions written as vectors in the three-dimensional Bloch space: The Bloch vector of  -type hole with spin m is a natural number. Thus the hole wave function can be written as The valence subband structure can be determined by solving equations system: The wave function of electron of first energy level with accounts QCSE [19]:

ISSN 2347-3487
2588 | P a g e J u l y 30, 2 0 1 5 One can find the functional, which is built in the form: where c H is a conduction band kinetic energy including deformation potential: ).
The potential energies ) (z V can look for as follows: are the conduction and valence bandedge discontinuities which can be represented in the form [21]: is exchange-correlation potential energy which is found from the solution of three-dimensional Poisson's equation, using both an expression by Gunnarsson and Lundquist [22], and following criterions. At carrier densities      (26) The solution of equations system (13), (17), (22) as well as (13), (17), (23) does not depend from a temperature.
Solving one-dimensional Poisson's equation (23) one can find screening polarization field and Hartree potential energy by substituting her in the Schrödinger equations. From Schrödinger equations wave functions and bandstructure are found. The conclusive determination of screening polarization field is determined by iterating Eqs. (13), (17), (22) until the solutions of conduction and valence band energies and wave functions are converged: The complete potential which describes piezoelectric effects and local exchange-correlation potential in quantum well one can find as follows 38) J u l y 30, 2 0 1 5 is the Coulomb potential of the quantum well,  is the dielectric permittivity of a host material of the quantum well, and A is the area of the quantum well in the xy plane.
The Hamiltonian of the interaction of a dipole with an electromagnetic field is described as follows:       The light absorption spectrum presented in the paper in Fig. 7, reflects only the strict TE (x or y) light polarization. J u l y 30, 2 0 1 5   Hence the Bohr radius is grater than the localization range particle-hole pair, and the excitons may be spontaneously created. We have calculated carriers population of the lowest conduction band and the both heavy hole and light hole valence band. Solving (13) for holes in the infinitely deep quantum well and finding the minimum of functional (17) for electrons in a quantum well with barriers of finite height, we can find the energy and wave functions of electrons and holes with respect to Hartree potential and exchange-correlational potential in a piezoelectric field at a carriers concentration 12 10 * 4 = = p n cm 2  . The screening field is determined by iterating Eqs. (13), (17), (22) until the solution of energy spectrum is converged. The squares of Hartree and Hartree-Fock wave functions for electrons, heavy holes and light holes are presented in Fig. 5. From Fig. 5 one can conclude that an overlap integrals of the wave functions of holes and electron taking into the account besides the piezoelectric effects the exchange-correlation effects in addition are greater than an overlap integrals of Hartree ones. Hartree charge density distribution and Hartree-Fock charge density distribution are presented in Fig. 6. Comparing charge density distributions presented in Fig. 6 one can conclude that Hartree particles distribute greater on edges of quantum well than Hartree-Fock particles. J u l y 30, 2 0 1 5 It is found that the localization range particle-hole pair 11  If the Bohr radius is grater than the localization range particle-hole pair, the excitons may be spontaneously created.

Results and discussions
6 Creation of beelectron of Dirac cone: the tachyon solution in magnetic field.
The graphene [24][25][26] presents a new state of matter of layered materials. The energy bands for graphite was found using "tight-binding" approximation by P.R. Wallace [27]. In the low-energy limit the single-particle spectrum is Dirac cone similarly to the light cone in relativistic physics, where the light velocity is substituted by the Fermi velocity The graphene is the single graphite layer, i. e. two-dimensional graphite plane of thickness of single atom. The graphene lattice resembles a honeycomb lattice. The graphene lattice one can consider like into the composite of two triangular sublattices. In 1947 Wallace in "tight-binding" approximation consider a graphite which consist off the graphene blocks with taken into account the overlap only the nearest  -electrons.
The two-dimensional nature of graphene and the space and point symmetries of graphene acquire of the reason for the massless electron motion since lead into massless Dirac equation (Majorana fermions) [27,28]. At low-energy limit the single particle spectrum forms with  -electron carbon orbital and consist off completely occupation valence cone and completely empty conduction cone, which have cone like shape with single Dirac point. In Dirac point the existing an electron as well as a hole is proved. The state in Dirac cone is double degenerate with taken into account a spin.
The existing of the massless Dirac fermions in graphene was proved based on the unconventional quantum Hall effect. The reason of creation the integer Hall conductivity [29][30][31][32] is derived from Berry phase [33,34].
When the magnetic field is applied perpendicularly into graphene plane the lowest (n=0) Landau level has the energy   in two nonequivalent cones  K , correspondingly [35]. In the paper [35] the Dirac mass via a splitting value is found when Zeeman coupling is absence. These properties of the lowest Landau level which distribute between particles and antiparticles in equal parts are base of the integer quantum Hall effect in graphene [35]. For Hence the Coulomb potential may be found in the form [36]:  When magnetic field is applied perpendicularly into graphene plane in z axis along field distribution. The vector potential in the gauge [20] The solution Eq. (50) can look for in the form: Substituting the solution in Eq. (50), one can find for the radial function the following equation l is a number of natural numbers set [37]. The root ambiguity in energy spectrum at the solution of the problem about quantization with relativistic invariance lead in quantum field theory into the creation of a pairs of particles (particles+antiparticles) [38]. When l is a number of complex numbers set the tachyon solutions are provided by arising the complex energy in spectrum of quantization of Landau for pair of two Majorana fermions coupled via a Coulomb potential.
For graphene with strong Coulomb interaction the Bethe-Salpeter equation for the electron-hole bound state was solved and a tachyonic solution was found [39].

Conclusions
In this paper a theoretical studies of the space separation of electron and hole wave functions in the quantum well O Zn Mg ZnO 0.73 0.27 / by the self-consistent solution of the Schrödinger equations for electrons and holes and the Poisson equations at the presence of spatially varying quantum well potential due to the piezoelectric effect and local exchange-correlation potential are presented. The exchange-correlation potential energy is found from the solution of three-dimensional Poisson's equation, using both an expression by Gunnarsson and Lundquist [22], and following criterions. The criterion /4 > n k F at carrier densities 12 10 * 4 cm 2  , at a temperature T=0 K is carried as The criterion does not depend from a width of well. The solution of equations system (13), (17), (23) as well as (13), (17), (22) does not depend from a temperature. The ratio of Coulomb potential energy to the Fermi energy is under biaxial strain is greater than an effective-mass of heavy hole of ZnO. It is calculated that an electron mass is less than a hole mass. It is found that the Bohr radius is grater than the localization range particle-hole pair, and the excitons may be spontaneously created.
Schrödinger equation for pair of two massless Dirac particles when magnetic field is applied in Landau gauge is solved exactly. Landau quantization l B  =  for pair of two Majorana fermions coupled via a Coulomb potential from massless chiral Dirac equation in cylindric coordinate is found. In this case the separation of center of mass and relative motion is derived. The root ambiguity in energy spectrum leads into Landau quantization for beelectron, when the states in which the one simultaneously exists are allowed. The tachyon solution with imaginary energy in Cooper problem ( 0 < 2  ) is found. The wave function are shown to be expressed via the associated Laguerre polynomial. In the paper the Cooper problem in superconductor theory is solved as quantum-mechanical problem for two electrons unlike from the paper [39] where the Bethe-Salpeter equation was solved for electron-hole pair.