# Behavior of the Energy Spectrum and Electric Conduction of Doped Graphene

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## Abstract

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## 1. Introduction

## 2. Theoretical Model

_{xx}·d enters the formula for the electrical resistance of a graphene layer

_{xx}·d are given in units of ${e}^{2}\cdot {\hslash}^{-1}$.

## 3. Results

_{xx}(μ) as functions of the energy ε and the Fermi level μ, respectively; d means the thickness of a graphene layer. The calculations of g(ε) and σ

_{xx}(μ) are carried out by Equations (9) and (10). The values of the energy ε and the Fermi level μ are given in units of the energy zone half-width $w$. The substitutional impurity concentration $y=0.2$, the order parameter $\eta =0.3$, the parameter of binary interatomic correlations ${\mathit{\epsilon}}_{}^{\mathit{B}\mathit{B}}=\mathbf{0}$, the scattering potential $\delta /w=-0.2$ (Figure 2), and $\delta /w=-0.6$ (Figure 3).

_{xx}(μ) for the Fermi level located in the gap is equal to zero. For the Fermi level outside the gap, the electrical conductance of graphene is nonzero and increases with the density of states on the Fermi level.

_{xx}on the scattering potential $\delta $ and the order parameter η, we present the dependence of the electrical conductance of graphene σ

_{xx}on the order parameter of impurity atoms η for different values of the scattering potential $\delta $ in Figure 4 and Figure 5. The number of electrons per atom, whose energies are in the energy zone, is equal to $\langle Z\rangle =1.01$. For such value of $\langle Z\rangle $, the Fermi level μ(η) calculated by Equation (14) lies to the right of the energy gap. In Figure 4b and Figure 5b, we show the Fermi level μ(η) as a function of the order parameter of an impurity η. In Figure 4c and Figure 5c, we give the dependence of the partial density of states g

_{i}(μ) at the Fermi level on the order parameter of an impurity η; i = 1, 2 is the number of a sublattice. Figure 4d and Figure 5d present the dependence of the imaginary part of the coherent potential ${\sigma}_{i}^{\u2033}\left(\mu \right)$ at the Fermi level on the order parameter of impurity atoms η.

_{xx}agree qualitatively with the formula obtained in the limiting case of weak scattering $\left|\delta /w\right|\ll 1$ [14] and with Equation (25). In this case, we took into account that the dependence of the electrical conductance of graphene, which was obtained in the limiting case of weak scattering [14], on the ordering of an admixture is applicable only in the case of such concentration at which the Fermi level lies in the vicinity of the Dirac point. In the present work, we give the results of numerical calculations of the dependence of the electrical conductance of graphene σ

_{xx}on the ordering parameter η. We have considered such values of concentrations and scattering potentials, at which the Fermi level lies in the vicinity of the Dirac point, as well as outside it.

## 4. Conclusions

_{xx}increases with the order parameter $\eta $ and tends to infinity, as the order parameter $\eta \to {\eta}_{max}$.

_{xx}of graphene with the order parameter η of impurity atoms is caused by an increase in the density of states at the Fermi level, $g\left(\mu \right)$, and by an increase in the relaxation time of electron states, $\tau \left(\mu \right)$, tending to infinity, as $\eta \to {\eta}_{max}$.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Sun, J.; Marsman, M.; Csonka, G.I.; Ruzsinszky, A.; Hao, P.; Kim, Y.-S.; Kresse, G.; Perdew, J.P. Self-consistent meta-generalized gradient approximation within the projector-augmented-wave method. Phys. Rev. B
**2011**, 84, 035117. [Google Scholar] [CrossRef] [Green Version] - Yelgel, C.; Srivastava, G.P. Ab initio studies of electronic and optical properties of graphene and graphene–BN interface. Appl. Surf. Sci.
**2012**, 258, 8338–8342. [Google Scholar] [CrossRef] - Denis, P.A. Band gap opening of monolayer and bilayer graphene doped with aluminium, silicon, phosphorus, and sulphur. Chem. Phys. Lett.
**2010**, 492, 251. [Google Scholar] [CrossRef] - Deng, X.; Wu, Y.; Dai, J.; Kang, D.; Zhang, D. Electronic structure tuning and band gap opening of graphene by hole/electron codoping. Phys. Lett. A
**2011**, 365, 3890–3894. [Google Scholar] [CrossRef] - Skrypnyk, Y.V.; Loktev, V.M. Impurity effects in a two-dimensional system with the Dirac spectrum. Phys. Rev. B
**2006**, 73, 241402. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.-Y.; Tsai, W.-F.; Chang, K.; An, X.-T.; Zhang, G.-P.; Xie, X.-C.; Li, S.-S. Electron delocalization in gate-tunable gapless silicene. Phys. Rev. B
**2013**, 88, 125431. [Google Scholar] [CrossRef] [Green Version] - Pershoguba, S.S.; Skrypnyk, Y.V.; Loktev, V.M. Numerical simulation evidence of spectrum rearrangement in impure graphene. Phys. Rev. B
**2009**, 80, 214201. [Google Scholar] [CrossRef] [Green Version] - Radchenko, T.M.; Shylau, A.A.; Zozoulenko, I.V. Influence of correlated impurities on conductivity of graphene sheets: Time-dependent real-space Kubo approach. Phys. Rev. B
**2012**, 86, 035418. [Google Scholar] [CrossRef] [Green Version] - Radchenko, T.M.; Tatarenko, V.A.; Sagalianov, I.Y.; Prylutskyy, Y.I.; Szroeder, P.; Biniak, S. On adatomic-configuration-mediated correlation between electrotransport and electrochemical properties of graphene. Carbon
**2016**, 101, 37–48. [Google Scholar] [CrossRef] [Green Version] - Radchenko, T.M.; Tatarenko, V.A.; Sagalianov, I.Y.; Prylutskyy, Y.I. Effects of nitrogen-doping configurations with vacancies on conductivity in grapheme. Phys. Lett. A
**2014**, 378, 2270–2274. [Google Scholar] [CrossRef] [Green Version] - Radchenko, T.M.; Shylau, A.A.; Zozoulenko, I.V.; Ferreira, A. Effect of charged line defects on conductivity in graphene: Numerical Kubo and analytical Boltzmann approaches. Phys. Rev. B
**2013**, 87, 195448. [Google Scholar] [CrossRef] [Green Version] - Radchenko, T.M.; Shylau, A.A.; Zozoulenko, I.V. Conductivity of epitaxial and CVD graphene with correlated line defects. Solid State Commun.
**2014**, 195, 88–94. [Google Scholar] [CrossRef] [Green Version] - Radchenko, T.M.; Tatarenko, V.A.; Sagalianov, I.Y.; Prylutskyy, Y.I. Configurations of structural defects in graphene and their effects on its transport properties. In Graphene: Mechanical Properties, Potential Applications and Electrochemical Performance; Edwards, B.T., Ed.; Nova Science Publ.: Hauppauge, NY, USA, 2014; Chapter 7; pp. 219–259. [Google Scholar]
- Los’, V.F.; Repetsky, S.P. A theory for the electrical conductivity of an ordered alloy. J. Phys. Condens. Matter
**1994**, 6, 1707–1730. [Google Scholar] [CrossRef] - Repetsky, S.P.; Vyshyvana, I.G.; Kruchinin, S.P.; Bellucci, S. Influence of the ordering of impurities on the appearance of an energy gap and on the electrical conductance of grapheme. Sci. Rep.
**2018**, 8, 9123. [Google Scholar] [CrossRef] [Green Version] - Merino-Díez, N.; Garcia-Lekue, A.; Carbonell-Sanroma, E.; Li, J.; Corso, M.; Colazzo, L.; Sedona, F.; Sanchez-Portal, D.; Pascual, J.I.; de Oteyza, D.G. Width-dependent band gap in armchair graphene nanoribbons reveals Fermi level pinning on Au(111). ACS Nano
**2017**, 11, 11661–11668. [Google Scholar] [CrossRef] - Saroka, V.A.; Shuba, M.V.; Portnoi, M.E. Optical selection rules of zigzag graphene nanoribbons. Phys. Rev. B
**2017**, 95, 155438. [Google Scholar] [CrossRef] [Green Version] - Chung, H.C.; Lee, M.H.; Chang, C.P.; Lin, M.F. Exploration of edge-dependent optical selection rules for graphene nanoribbons. Opt. Express
**2011**, 19, 23350–23363. [Google Scholar] [CrossRef] [Green Version] - Chung, H.-C.; Lin, Y.-T.; Lina, S.-Y.; Ho, C.-H.; Chang, C.-P.; Lin, M.-F. Magnetoelectronic and optical properties of nonuniform graphene nanoribbons. Carbon
**2016**, 109, 883–895. [Google Scholar] [CrossRef] [Green Version] - Chung, H.-C.; Chang, C.-P.; Lin, C.-Y.; Lin, M.-F. Electronic and optical properties of graphene nanoribbons in external fields. Phys. Chem. Chem. Phys.
**2016**, 18, 7573. [Google Scholar] [CrossRef] [Green Version] - Si, C.; Sun, Z.; Liu, F. Strain engineering of graphene: A review. Nanoscale
**2016**, 8, 3207–3217. [Google Scholar] [CrossRef] - Gui, G.; Li, J.; Zhong, J. Band structure engineering of graphene by strain: First-principles calculations. Phys. Rev. B
**2008**, 78, 075435. [Google Scholar] [CrossRef] [Green Version] - Pereira, V.M.; Castro Neto, A.H.; Peres, N.M.R. Tight-binding approach to uniaxial strain in grapheme. Phys. Rev. B
**2009**, 80, 045401. [Google Scholar] [CrossRef] [Green Version] - Ducastelle, F. Analytic properties of the coherent potential approximation and its molecular generalizations. J. Phys. C Solid State Phys.
**1974**, 7, 1795–1816. [Google Scholar] [CrossRef] - Velicky, B. Theory of electronic transport in disordered binary alloys: Coherent potential approximation. Phys. Rev.
**1969**, 184, 614–627. [Google Scholar] [CrossRef] - Slater, J.C.; Koster, G.F. Simplified LCAO method for the periodic potential problem. Phys. Rev.
**1954**, 9, 1498–1524. [Google Scholar] [CrossRef] - Repetsky, S.; Vyshyvana, I.; Nakazawa, Y.; Kruchinin, S.; Bellucci, S. Electron transport in carbon nanotubes with adsorbed chromium impurities. Materials
**2019**, 12, 524. [Google Scholar] [CrossRef] [Green Version] - Repetsky, S.P.; Shatnii, T.D. Thermodynamic potential of a system of electrons and phonons in a disordered alloy. Theor. Math. Phys.
**2002**, 131, 832–851. [Google Scholar] [CrossRef]

**Figure 1.**Dependence of the density of electron states $g\left(\epsilon \right)$ on the energy $\epsilon $ at the concentration of a substitutional impurity $y=0.2$, for different values of the parameter of binary interatomic correlations on the first coordinate sphere ${\mathit{\epsilon}}_{\mathit{l}\mathit{j}\mathbf{0}\mathit{i}}^{\mathit{B}\mathit{B}}={\mathit{\epsilon}}_{}^{\mathit{B}\mathit{B}}$, $\eta =0$, and values of the scattering potential: (

**a**) $\delta /w=-0.2$, (

**b**) $\delta /w=-0.6$. The density of electron states calculated in the approximation of coherent potential is shown by a continuous curve with regard to the processes of scattering on the pairs of atoms in the limits of the first coordinate sphere; the dotted curve corresponds to ${\mathit{\epsilon}}_{}^{\mathit{B}\mathit{B}}=\mathbf{0}$, the dashed line to ${\mathit{\epsilon}}_{}^{\mathit{B}\mathit{B}}=-\mathbf{0.05}$, and the dash-dotted curve to ${\mathit{\epsilon}}_{}^{\mathit{B}\mathit{B}}=-\mathbf{0.1}$.

**Figure 2.**Dependence of (

**a**) the density of electron states g(ε) on the energy ε; (

**b**) the electrical conductance σ

_{xx}(μ) on the Fermi level μ, where d is the thickness of a graphene layer; and (

**c**) the electrical conductance σ

_{xx}(μ) on the Fermi level μ (shown on a larger scale). The substitutional impurity concentration $y=0.2$, the scattering potential $\delta /w=-0.2$, the order parameter $\eta =0.3$, and the parameter of binary interatomic correlations ${\mathit{\epsilon}}_{}^{\mathit{B}\mathit{B}}=\mathbf{0}$. The blue curve shows the results of calculations in the approximation of coherent potential, while the red curve shows those with regard to the processes of the scattering of electrons on the pairs of atoms of the first coordinate sphere.

**Figure 3.**Dependence of (

**a**) the density of electron states g(ε) on the energy ε; (

**b**) the electrical conductance σ

_{xx}(μ) on the Fermi level μ, where d is the thickness of a graphene layer; and (

**c**) the electrical conductance σ

_{xx}(μ) on the Fermi level μ (shown on a larger scale). The substitutional impurity concentration $y=0.2$, the scattering potential $\delta /w=-0.6$, the order parameter $\eta =0.3$, and the parameter of binary interatomic correlations ${\mathit{\epsilon}}_{}^{\mathit{B}\mathit{B}}=\mathbf{0}$. The blue curve shows the results of calculations in the approximation of the coherent potential, while the red curve shows those with regard to the processes of the scattering of electrons on the pairs of atoms of the first coordinate sphere.

**Figure 4.**Dependence of (

**a**) the electrical conductance of graphene σ

_{xx}; (

**b**) the Fermi level μ; (

**с**) the partial density of states ${g}_{i}\left(\mu \right)$ at the Fermi level; and (

**d**) the imaginary part of the coherent potential ${\sigma}_{i}^{\u2033}\left(\mu \right)$ at the Fermi level on the order parameter of impurity atoms η. The substitutional impurity concentration $y=0.2$ and the scattering potential $\delta /w=-0.2$. Circles correspond to ${g}_{1}\left(\mu \right)$ and ${\sigma}_{1}^{\u2033}\left(\mu \right)$ of the first sublattice, in which the impurity atoms are located in the case of full order. Filled circles show ${g}_{2}\left(\mu \right)$ and ${\sigma}_{2}^{\u2033}\left(\mu \right)$ for the second sublattice.

**Figure 5.**Dependence of (

**a**) the electrical conductance of graphene σ

_{xx}; (

**b**) the Fermi level μ; (

**с**) the partial density of states ${g}_{i}\left(\mu \right)$ at the Fermi level; and (

**d**) the imaginary part of the coherent potential ${\sigma}_{i}^{\u2033}\left(\mu \right)$ at the Fermi level on the order parameter of impurity atoms η. The substitutional impurity concentration $y=0.2$ and the scattering potential $\delta /w=-0.6$. Circles correspond to ${g}_{1}\left(\mu \right)$ and ${\sigma}_{1}^{\u2033}\left(\mu \right)$ of the first sublattice, in which the impurity atoms are located in the case of full order. Filled circles show ${g}_{2}\left(\mu \right)$ and ${\sigma}_{2}^{\u2033}\left(\mu \right)$ for the second sublattice.

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**MDPI and ACS Style**

Bellucci, S.; Kruchinin, S.; Repetsky, S.P.; Vyshyvana, I.G.; Melnyk, R.
Behavior of the Energy Spectrum and Electric Conduction of Doped Graphene. *Materials* **2020**, *13*, 1718.
https://doi.org/10.3390/ma13071718

**AMA Style**

Bellucci S, Kruchinin S, Repetsky SP, Vyshyvana IG, Melnyk R.
Behavior of the Energy Spectrum and Electric Conduction of Doped Graphene. *Materials*. 2020; 13(7):1718.
https://doi.org/10.3390/ma13071718

**Chicago/Turabian Style**

Bellucci, Stefano, Sergei Kruchinin, Stanislav P. Repetsky, Iryna G. Vyshyvana, and Ruslan Melnyk.
2020. "Behavior of the Energy Spectrum and Electric Conduction of Doped Graphene" *Materials* 13, no. 7: 1718.
https://doi.org/10.3390/ma13071718