# Fractional incommensurately modulated superstructure at phase transition and quantum Hall effects in Weil 2D half layered materials: intrinsic graphene and Polar Oxide Heterostructures

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In the article the temperature dependencies of incommensurately modulated superstructures as well as domain walls are found by interplay between the stimulated emission processes and the quasielastic energy relaxation and are specified by nonequilibrium energy distribution. Due to an high recombination effects at low-energy carriers their energy distribution exhibits an incommensurate morphology shape. Hence we have found neutrino-like shape with respect to pumping intensity response. The photoconductivity appears to be proportional to the concentration of the photogenerated carriers for the case of short-range momentum relaxation. This concentration as well as the photoconductivity decrease with the temperature and the energy of the photoexcitation. In the article \cite{LoLiubov3:2020qr} we have found that the Weyl Hall semiconductor (WHS) states allow the topological phase transition happened between two quantum anomalous Hall (QAH) insulator phase with opposite Chern numbers and we have based on phonon dispersion of Weil 2D half semiconductor monolayers group like PtCl$_{3}$. Landau-Ginzburg-Devonshire theory of thin ferroelectric polar-active nanofilms in incommensurate phases and semiconductor heterostructures is presented. The self-consistent solutions of the Euler-Lagrange equation for the polarization vector and the Maxwell equations for light which propagates along Oz axis in thin ferroelectric polar-active nanofilms have been found. Quantized solutions of one-dimensional Maxwell equations for thin ferroelectric films in Incommensurate phase with space dispersion have been specified. The analytical solutions of the Maxwell wave equations as well as natural optical gyrotropy effects are found in Rb$_{2}$ZnBr$_{4}$ as well as K$_{2}$SeO$_{4}$ Incommensurate phases crystals connected with giant light velocity as well as via interaction with coherent phonon oscillations. In the framework of the superspace symmetry group theories the Maxwell wave equations are solved which are shown to be connected with the symmetry group of $D_{2h}^{16}$ or isomorphic groups. In the paper \cite{LoLiubov3:2020qr} the non-zero gyration $g_{33}$ and gyrotropic birefringence $\epsilon_{12}$ tensors of K$_{2}$SeO$_{4}$ and Rb$_{2}$ZnBr$_{4}$ materials based on $D_{2h}^{16}$ space symmetry group were found. The values of natural optical gyrotropy as well as Rashba spin splitting are shown to be specified like $(k^{(0)}\pm\tilde{k}^{(2)})^{2}$ as displacements of two symmetrically allocated parabolas from Brillouin zone center. Hence if sine-type edges of Group-VI Dichalcogenides and Rb$_{2}$ZnBr$_{4}$, Ca$_{2}$RuO$_{4}$, Cr$_{2}$O$_{3}$, MnTiO$_{3}$, La-doped BiFeO$_{3}$, PtCl$_{3}$, PdBr$_{3}$, RuCl$_{3}$, PtI$_{3}$, and carbon nanoribbons may be approximated armchair or zigzag edges of incommensurately modulated superstructures then in the sections 5 are found that structure phase transitions to be related with giant spin-orbit interaction (SOI). In the article a creation of giant spin-orbit splitting ($\sim\,2.5$ eV) coupled with the carbon Dirac cones (K and K') are shown to be related with structure phase transition in carbon nanoribbon with armchair or zigzag edges of of incommensurately modulated superstructure of Group-VI Dichalcogenides and Rb$_{2}$ZnBr$_{4}$, Ca$_{2}$RuO$_{4}$, Cr$_{2}$O$_{3}$, MnTiO$_{3}$, La-doped BiFeO$_{3}$, PtCl$_{3}$, PdBr$_{3}$, RuCl$_{3}$, PtI$_{3}$, and carbon nanoribbons. The giant SOI effects coupled with armchair or zigzag edges of incommensurately modulated superstructure were not found in Ref. \cite{{Vasko:2008jd},{Linnik:2014jd}} in which the currents of the two graphene valleys are mutually compensated providing zero net electric current as well as strain-induced valley current coupled with warping of the Dirac cones (K and K') $j_{x}^{(G)}=2\times\,10^{-3}$ pA/$\mu$m has been estimated. In the article \cite{LoLiubov3:2020qr} the natural optical gyrotropy effects are shown to be found with light velocity like $\epsilon=\hbar\,ck/eV=14.0798$ eV, $\epsilon=\hbar\,ck/eV=27.6009$ eV, $\epsilon=\hbar\,ck/eV=7.5726$ eV, with the corresponding considered in Tables 4,5,6 in Ref. \cite{LoLiubov3:2020qr} of gyrotropic birefringences $\Delta\,n_{11}$ and gyrotropies $g$. In the article the structure phase transition in the effective masses framework are shown to be coupled with temperature dependencies of the edges shape in the incommensurately modulated superstructure.