Creation of bielectron of Dirac cone: the tachyon solution in magnetic field

Schr\"odinger 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 $\epsilon=\pm\,B\sqrt{l}$ 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 bielectron, when the states in which the one simultaneously exists are allowed. The tachyon solution with imaginary energy in Cooper problem ($\epsilon^{2}<0$) is found. The continuum symmetry of Dirac equation allows perfect pairing between electron Fermi spheres when magnetic field is applied in Landau gauge creating a Cooper pair.


I. INTRODUCTION
There has been widely studied in the ultraviolet spectral range lasers based on direct widebandgap hexagonal würtzite crystal material systems such as ZnO 1-6 . 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 an dependence from the intrinsic electric field which is calculated to be 0.37 MV/cm 10  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 exchangecorrelation potential for bandgap renormalization and engineering of localized Hartree-Fock wave functions. The energy shifts as well as the localization range of exchange-correlational wave functions with respect Hartree energy shifts and Hartree localization range of wave functions require a scrutiny study.

II. THEORY
We take the following wave functions written as vectors in the three-dimensional Bloch space: (1) The Bloch vector of ν-type hole with spin ς v = ± and momentum k t is specified by its three 15 , known as spherical harmonics with the orbital angular momentum l = 1 and the eigenvalue m l its z component. The envelope Z-dependent part of the quantum well eigenfunctions can be specified from the boundary conditions ψ m (Z = 0) = ψ m (Z = 1) = 0 of the infinite quantum well as where Z = ( z w + 1 2 ), m is a natural number. Thus the hole wave function can be written as The valence subband structure E ςv ν (k t ) can be determined by solving equations system: where i = 1, 2, 3.
The wave function of electron of first energy level with accounts QCSE 16 : where From bond conditions 16,17 where A is the area of the quantum well in the xy plane, ρ is the two-dimensional vector in the xy plane, k t = (k x , k y ) is in-plane wave vector. The constant multiplier C is found from normalization condition: One can find the functional, which is built in the form: where where H c is a conduction band kinetic energy including deformation potential.
From Kane model one can define the band-edge parameters such as the crystal-field splitting energy ∆ cr , the spin-orbit splitting energy ∆ so and the momentum-matrix elements for the longitudinal (e z) z-polarization and the transverse (e ⊥ z) polarization : Here we use the effective-mass parameters, energy splitting parameters, deformation potential parameters as in papers 14,20,21 .
The potential energies V (z) can looked for as follows: where Φ H (z) is the solution of one-dimensional Poisson's equation with the strain-induced electric field in the quantum well, δ U c,v (z) are the conduction and valence bandedge discontinuities which can be represented in the form 18 : Φ xc (z) is exchange-correlation potential energy which is found from the solution threedimensional Poisson's equation, using both an expression by Gunnarsson and Lundquist 19 , and following criterions. At carrier densities 4 * 10 12 cm −2 , k F > √ n/4 at a temperature T=0 K as 1 > 0.1 has been carried. k F is Fermi wave vector. The criterion is independent from a width of well. The solution of equations system (4), (8), (11), (12) as well as (4), (8), where Solving one-dimensional Poisson's equation (12) where Z = z w + 1 2 , g ν and g 1 correspond to the degeneration of the ν hole band and the first quantized conduction band, respectively, e is the value of electron charge, κ is the permittivity of a host material, and f ν,p (k t ), f 1n (k t ) are the Fermi-Dirac distributions for holes and electrons.
Exchange-correlation charge density may be determined as: using the expansion of plane wave At the condition [Ψ α,ν,n (k F , z) sin k F ρ] << 1, the solution eq. (13) may be found as The solution the three-dimensional Poisson's equation may be presented in the form: The complete potential which describes piezoelectric effects and local exchange-correlation potential in quantum well one can find as follows