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In the one-band model of strong coupling, the influence of substitutional impurity atoms on the energy spectrum and electrical conductance of graphene is studied. It is established that the ordering of substitutional impurity atoms on nodes of the crystal lattice causes the appearance of a gap in the energy spectrum of graphene with width  η | δ | centered at the point  yδ , where  η  is the parameter of ordering,  δ  is the difference of the scattering potentials of impurity atoms and carbon atoms, and  y  is the impurity concentration. The maximum value of the parameter of ordering is    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\eta }}}_{{\boldsymbol{\max }}}{\boldsymbol{=}}{\bf{2}}{\boldsymbol{y}}{\boldsymbol{,}}\,{\boldsymbol{y}}{\boldsymbol{\le }}{\bf{1}}/{\bf{2}}$$\end{document}η    max    =  2  y  ,    y  ≤  1  /  2. For the complete ordering of impurity atoms, the energy gap width equals    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${\bf{2}}{\boldsymbol{y}}{\boldsymbol{|}}{\boldsymbol{\delta }}{\boldsymbol{|}}$$\end{document}   2  y   |  δ  |. If the Fermi level falls in the region of the mentioned gap, then the electrical conductance    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\sigma }}}_{{\boldsymbol{\alpha }}{\boldsymbol{\alpha }}}{\boldsymbol{\to }}{\bf{0}}$$\end{document}σ    α  α    →  0at the ordering of graphene, i.e., the metal–dielectric transition arises. If the Fermi level is located outside the gap, then the electrical conductance increases with the parameter of order  η  by the relation    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\sigma }}}_{{\boldsymbol{\alpha }}{\boldsymbol{\alpha }}}{\boldsymbol{ \sim }}{{\boldsymbol{(}}{{\boldsymbol{y}}}^{{\bf{2}}}{\boldsymbol{-}}\frac{{\bf{1}}}{{\bf{4}}}{{\boldsymbol{\eta }}}^{{\bf{2}}}{\boldsymbol{)}}}^{{\boldsymbol{-}}{\bf{1}}}$$\end{document}σ    α  α    ~    (y   2   −   1  4η   2    )    −  1. At the concentration    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${\boldsymbol{y}}{\boldsymbol{=}}{\bf{1}}{\boldsymbol{/}}{\bf{2}}$$\end{document}   y  =  1  /  2, as the ordering of impurity atoms  η  →1, the electrical conductance of graphene    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$${{\boldsymbol{\sigma }}}_{{\boldsymbol{\alpha }}{\boldsymbol{\alpha }}}{\boldsymbol{\to }}{\boldsymbol{\infty }}$$\end{document}σ    α  α    →  ∞, i.e., the transition of graphene in the state of ideal electrical conductance arises.

Influence of the ordering of impurities on the appearance of an energy gap and on the electrical conductance of graphene