EXCITON INSULATOR STATES IN MATERIALS WITH ‘MEXICAN HAT’ BAND STRUCTURE DISPERSION AND IN GRAPHENE

In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in graphene and in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral twodimensional Schrödinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection is exactly solved. The solutions of Schrödinger equation in momentum space in studied materials by projection the twodimensional space of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral twodimensional Schrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. In studied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and light absorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole are separated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materials allows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone and hence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a width of band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of the equation of motion for a microscopic dipole of graphene where 1 > s r . The absorption spectrums for the zinc-blende GaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with available experimental data shows good agreement. Indexing terms/


Introduction
The graphene [1][2][3][4][5] presents a new state of matter of layered materials. The energy bands for graphite was found using "tight-binding" approximation by P.R. Wallace [6]. 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 6

=
F v m/s and is described by the massless Dirac equation.
In the paper we present a theoretical investigation of excitonic states as well as their wave functions in graphene and in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN. An integral form of the twodimensional Schrödinger equation of Kepler problem in momentum space is solved exactly by projection the twodimensional space of momentum on the three-dimensional sphere in the paper [7].
The integral Schrödinger equation was analytically solved by the projection the three-dimensional momentum space onto the surface of a four-dimensional unit sphere by Fock in 1935 [8].
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 greater than the flat band gap in narrow-gap semiconductor or semimetal then at sufficiently low temperature the insulator ground state is instable with respect to the exciton formation [9][10][11][12]. And excitons may be spontaneously created. In a system undergo a phase transition into a exciton insulator phase similarly to Bardeen-Cooper-Schrieffer (BCS) superconductor. In a single-layer graphene (SLG) the electron-hole pairing leads to the exciton insulator states [13].
In the paper an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with complex dispersion is analytically solved. A complex dispersions lead to fundamental difference in exciton insulator states and their wave functions.
We analytically solve an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection.  [14]. It means the prominent Coulomb effects [15].
It is known that the Coulomb interaction leads to the semimetal-exciton insulator transition, where gap is opened by electron-electron exchange interaction [9,[16][17][18]. The perfect host combines a small gap and a large exciton binding energy [9,12].
It is known [24] in the weak-coupling limit [25], exciton condensation is a consequence of the Cooper instability of materials with electron-hole symmetry of reflection inside identical Fermi surface. The identical Fermi surfaces is a consequence of the particle-hole symmetry of massless Dirac equation for Majorana fermions.
A deeper understanding of the influence of band structures on optical properties should help one to answer many questions. In addition, the interesting effects of strong electron-hole Coulomb interaction are presented in these materials. Many-body interactions lead to effects, which consist the screening, dephasing, bandgap renormalization, and phasespace filling [26][27][28][29].
A general phenomenon of Coulomb enhancement may be explained as follows. Due to the Coulomb attraction, an electron and a hole have a larger tendency to be located near each other, than that in the case of noninteracting particles. This increase of the interaction duration leads to an increase of the optical transition probability.  (2) where p is the transversal quasimomentum of carriers in the plane of the quantum well,

Equation of motion for a microscopic dipole.
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: is a microscopic dipole due to an electron-hole pair with the electron (hole) momentum p (-p) and the subband number , is the matrix element of the electric dipole moment, which depends on the wave vector k and the numbers of subbands, between which the direct interband transitions occur, e is a unit vector of the vector potential of an electromagnetic wave, p is the momentum operator. Subbands are described by the wave functions j is the number of a subband from the conduction band,   is the electron spin, j is the number of a subband from the valence band, and  is the hole spin. We consider one lowest  [18]).
In this section, we derive the equation of motion for the mean value of the product p p a b for a microscopic dipole, which specifies of a medium polarization, which becomes macroscopic due to the applied external field.
The average value of a certain physical magnitude F , which corresponds to the operator F can be expressed through the spur of a matrix, which is a certain statistic operator obeying the Heisenberg equation: One can find the expectation value from the convolution of two operators:  Using the operator algebra and the density matrix formalism as well as diagram method based on cyclic geometry of diagrams, we have n n n n n n (12) The functions We have replaced the bare Coulomb potential energy with the screened one:  .
The coefficient of the sum in the second term of series (15) is In the third term of the series, the coefficient is Then the series can be rewritten as a infinitely decreasing geometric progression By summing all terms of the series, we obtain For the dielectric function, we use the static Lindhard formula: Since the cyclic diagrams are the basic type of diagrams in the scattering processes at a high density of the electron-hole gas, the diagram method is equivalent of the self-consistency method, as well as the random phase approximation.
The answer how to derive the integro-differential equation (11) for a microscopic dipole is given by the scheme , , : plus the expression, whose graphic representation reminds a binary blister, , , : plus the expression corresponding to the plot in the form of an oyster.
The sum over momenta in the polarization equation, which includes the carrier-carrier correlations of higher orders than Hartree-Fock ones, can be found if the self-energy in the equation is added by the term, which is presented in the equation in the Hartree-Fock approximation, by replacing the Coulomb potential energy with the screened one and the ,..., , 2 1 vertices, correspondingly, is the sum of all closed diagrams of the m -th order [30,31].
The absorption spectrums for the zinc-blende GaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are shown in Figures 1-4 at a temperatures of 300K, 10K, 4.4K if a concentrations or particles in a quantum well are as follows 11 10 cm 2  , 11 10 5 cm 2  , 11 10 9 cm 2  , as well as if a concentrations or particles in a bulk 16 10 cm 3  , 17 10 cm 3  .
The interesting effects of strong electron-hole Coulomb interaction are presented in these materials. Many-body interactions lead to effects, which consist of the screening, dephasing, bandgap renormalization, and phase-space filling [26][27][28][29]32]. It is known [32] the continuum spectra agrees well with Elliot's formula. With increasing free particles concentrations continuum spectra shifts rapidly to lower energies, whereas the energy of lowest exciton remains a constant value. The reason of constant energy of the lowest exciton line consists in compensation of the weakening of the e-h pair binding energy due to screening its of free particles by reduction of edge of absorption (see in Figures 1-4). The cause of this remarkable sight consists in the charge neutrality of an exciton [32].
From Figures 3,4 one can see an arising of negative range at lower wavelength from the lowest exciton line at increasing of particles densities and at constant temperature or at decreasing of temperature at constant of particles densities. Its may be explained by the exchange renormalization of band gap spectra and the Coulomb renormalization of electric dipole momenta. It should be noted that many-particles interactions are of cause the both an arising of negative range at lower wavelength from the lowest exciton line and an disappearance of 2s exciton line. As well as an insignificant enhancement of the lowest exciton is of an arising effect. Our theoretical studies have well agreement with the experimental results [33]. O c t o b e r 2 2 , 2 0 1 5

Exciton insulator states with "Mexican hat" band structure dispersion
The "Mexican hat" band structure dispersion is found in the form The Schrödinger equation for the calculating of exciton states can be written in the general form q ,  is a quantized energy. We look for the bound states and hence the energy will be negative.
An integral form of the two-dimensional Schrödinger equation in momentum space for the gapped graphene is solved exactly by projection the two-dimensional space of momentum on the three-dimensional sphere.
For the "Mexican hat" shape band structure dispersion 1, where an each point on sphere is defined of two spherical angles  ,  , which are knitted with a momentum q [7,8]. A space angle  may be found as surface element on sphere dq q q [7,8]. A spherical angle  and a momentum q are shown [7,8] to be knitted as   Since [7] ), , ( (30) then substituting (28), (30) in (25) The integral equations for "Mexican hat" shape band structure based on Eq. (26) may be found in the form Since [35,36] ), cos (

Quantized spectral series of the excitonic states of valence Dirac cone.
In the honeycomb lattice of graphene with two carbon atoms per unit cell the space group is 1 3h The direct production of two irreducible presentations of wave function and wave vector of difference In the low-energy limit the single-particle spectrum is Dirac cone described by the massless Dirac equation for a massless Dirac fermions (Majorana fermions). The Hamiltonian of graphene for a massless Dirac fermions [6] ), The dispersion of energy bands may be found in the form [6] , The Schrödinger equation for the calculating of exciton states can be written in the general form q ,  is a quantized energy. We look for the bound states and hence the energy will be negative. When an each point on sphere is defined of two spherical angles  ,  , which are knitted with a momentum q [7,8]. A space angle  may be found as surface element on sphere q d q q q d d d Since [7] ), , ( The integral equations for SLG based on Eq. (41) may be found in the form

Figure 6: (Color online) Single-particle spectrum of graphene for massless Dirac fermions (Majorana fermions).
Since [35,36] ), cos (  Quantized spectral series of the excitonic states distribute in valence Dirac cone. The energies of bound states are shown to be found as negative, i. e. below of Fermi level. Thus if the electron and hole are separated, their energy is higher than if they are paired.

Elliott formula and light absorption rates of the excitonic states of valence Dirac cone.
The intervalley transitions probability caused intervalley photoexcitations taken into account Coulomb interaction of electron-hole pair one can obtain from Fermi golden rule in the form Considering the case of relatively weak excitation the total rate of increase of the number of photons in the fixed mode one can obtain in the form

Results and discussions
The integral Schrödinger equation for a parabolic bands was analytically solved by the projection the three-dimensional momentum space onto the surface of a four-dimensional unit sphere by Fock in 1935 [8]. In integral form of the twodimensional Schrödinger equation of Kepler problem in momentum space is solved exactly by the projection of the twodimensional space of momentum on the three-dimensional sphere in the paper [7].
In graphene the existing of bound pair states are still subject matter of researches [13].
In the paper [37,38] the possibility of extremum loop or toroidal energy surfaces in würtzite semiconductor is discussed. The Bychkov-Rashba model [39] describes the electron motion in two-dimensional system with applied electric potential gradient field perpendicularly to the plane of system. The Rashba and Dresselhaus spin-orbit couplings in GaAs quantum wells were studies in the paper [40].
The bismuth tellurohalides BiTeCl, BiTeBr and BiTeI have hexagonal crystal structures and giant Rashba spin splitting in two-dimensional systems [41,42]. These materials in the low-energy limit are described "Mexican hat" shape dispersion.
In the paper an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with complex dispersion is analytically solved. A complex dispersions lead to fundamental difference in exciton insulator states and their wave functions.
We analytically solve an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection.
It is known that the Coulomb interaction leads to the semimetal-exciton insulator transition, where gap is opened by particle-particle exchange interaction [12]. The perfect host combines a small gap and a large exciton binding energy [12].
We consider the pairing between oppositely charged particles in a materials with "Mexican hat" shape band structure dispersion or in materials with giant Rashba spin splitting and in graphene. The Coulomb interaction leads to the electronhole bound states scrutiny study of which acquire significant attention in explanations of hight-temperature superconductivity.
It is known [9,12] if the exciton binding energy is greater than the flat band gap in narrow-gap semiconductor or semimetal then at sufficiently low temperature the insulator ground state is instable concerning to the exciton formation with follow up spontaneous production of excitons. In a system undergo a phase transition into a exciton insulator phase similarly to BCS superconductor. In a semiconductors with "Mexican hat" shape band structure dispersion as well as in graphene the electron-hole pairing leads to exciton insulator states.
The integral Schrödinger equation for a parabolic bands was analytically solved by the projection the three-dimensional momentum space onto the surface of a four-dimensional unit sphere by Fock in 1935 [8].
In the paper an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with complex dispersion is analytically solved. A complex dispersion leads to fundamental difference in the energy of exciton insulator states and their wave functions.
We analytically solve an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection.
It is known that the Coulomb interaction leads to the semimetal-exciton insulator transition, where gap is opened by electron-electron exchange interaction [9,[16][17][18]. The perfect host combines a small gap and a large exciton binding energy [9,12].
The particle-hole symmetry of Dirac equation of layered materials allows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence band and conduction band and hence driving the Cooper instability. In the weakcoupling limit in graphene with the occupied conduction-band states and empty valence-band states inside identical Fermi surfaces in band structure, the exciton condensation is a consequence of the Cooper instability.
It is shown that the solutions of Coulomb problem of electron-hole pair does not depend from a width of band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of the equation of motion for a microscopic dipole of graphene where 1 > s r .

Conclusions
In this paper we found the solution the integral Schrödinger equation in a momentum space of two interacting via a Coulomb potential Dirac particles that form the exciton in graphene and in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN.
In low-energy limit this problem is solved analytically. We obtained the energy dispersion and wave function of the exciton in graphene and in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN. The excitons were considered as a system of two oppositely charge Dirac particles interacting via a Coulomb potential.
We solve this problem in a momentum space because on the whole the center-of-mass and the relative motion of the two Dirac particles can not be separated.
We analytically solve an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. An integral form of the two-dimensional Schrödinger equation in momentum space for graphene and in materials with "Mexican hat" band structure dispersion is solved exactly by projection the twodimensional space of momentum on the three-dimensional sphere.
Quantized spectral series of the excitonic states distribute in valence Dirac cone. The energies of bound states are shown to be found as negative, i. e. below of Fermi level. Thus if the electron and hole are separated, their energy is higher than if they are paired. In the SLG as well as in materials with "Mexican hat" band structure dispersion the electron-hole pairing leads to the exciton insulator states. O c t o b e r 2 2 , 2 0 1 5 It is shown that the solutions of Coulomb problem of electron-hole pair does not depend from a width of band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of the equation of motion for a microscopic dipole of graphene where  ), 2 cos ( cot = ) 2 cos ( ) sin In order to find a light absorption rates necessarily to solve the integral ).
which can be rewritten as follows The solution the following integral We find the solution of the integral Then substituting (73) into (70) we obtain the integral in the form which can be expressed via a hypergeometric functions as follows which can be rewritten in the form In order to find the solution of the integral (79) it is necessarily to consider the integral of form: O c t o b e r 2 2 , 2 0 1 5 which can be transformed into the integral of form: . ) The solution the integral (108) one can find using the binomial theorem and following replacements , ) So integral (82) may be rewritten as follows The solution of the integral (111) one can find by replacement We obtain the following expression for the sought for integral: The solution the integral (86) one can find using the binomial theorem We find the solution of the sought for integral as follows: