Modeling electronic structure and transport properties of graphene with resonant scattering centers

We present a detailed numerical study of the electronic properties of single-layer graphene with resonant (hydrogen) impurities and vacancies within a framework of noninteracting tight-binding model on a honeycomb lattice. The algorithms are based on the numerical solution of the time-dependent Schrödinger equation and applied to calculate the density of states, quasieigenstates, ac and dc conductivities of large samples containing millions of atoms. Our results give a consistent picture of evolution of electronic structure and transport properties of functionalized graphene in a broad range of concentration of impurities (from graphene to graphane), and show that the formation of impurity band is the main factor determining electrical and optical properties at intermediate impurity concentrations, together with a gap opening when approaching the graphane limit.

Figure 1

(Color online) The lattice structure of a graphene sheet. Each carbon is labeled by an coordinate $\left(m,n\right)$, where $m$ is along the zigzag edge and $n$ is along the armchair edge. Each carbon (red) has three nearest neighbors (yellow) and six next-nearest neighbors (blue).

Figure 2

(Color online) Comparison of the analytical DOS (in units of $1/t$, black solid) with the numerical results of a sample contains $512\times 512$ (red dashed) or $4096\times 4096$ (green dotted) carbon atoms.

Figure 3

(Color online) Density of states (in units of $1/t$) as a function of energy $E$ (in units of $t$) for different resonant impurity $\left({\epsilon }_d=-t/16,V=2t\right)$ or vacancy concentrations: $n_i\left(n_x\right)=0.1\mathrm{\%}$, 0.2%, 0.3%, 0.4%, 0.5%, 1%, and 5%. Sample size is $4096\times 4096.$

Figure 4

(Color online) Density of states (in units of $1/t$) as a function of energy $E$ (in units of $t$) for the vacancies with large concentrations: $n_x=5\mathrm{\%}$, 10%, 20%, 30%, 50%, and 90%. Sample size is $4096\times 4096$ for $n_x\le 50\mathrm{\%}$ and $8192\times 8192$ for $n_x=90\mathrm{\%}$.

Figure 5

Typical atomic structures of most favorable isolated carbon groups in graphene with large concentration of vacancies. The energy eigenvalues of each group are listed in the central column (in units of $t$), and $P$ is the probability of a particular group to be found in a graphene sample.

Figure 6

Density of states (in units of $1/t$) as a function of energy $E$ (in units of $t$) for the resonant impurities $\left({\epsilon }_d=-t/16,V=2t\right)$ with large concentrations: $n_i=50\mathrm{\%}$, 90%, 99%, 99.5%, and 100%. Sample size is $2048\times 2048$.

Figure 7

(Color online) Density of states (in units of $1/t$) as a function of energy $E$ (in units of $t$) in the presence of a uniform perpendicular magnetic field $\left(B=60\text{T}\right)$ with vacancy concentration $n_x=0.01\mathrm{\%}$. The red curves are Gaussian fits of Eq. (19) centered about each Landau levels, with $w=7.09\times {10}^{-4}$ for $E_N=0$ $\left(N=0\right)$, $w=7.03\times {10}^{-4}$ for $E_N=0.0909t\left(N=1\right)$, and $w=7.87\times {10}^{-4}$ for $E=0.0232t$ (between zero and first Landau levels). Sample size is $8192\times 8192$.

Figure 8

(Color online) Density of states (in units of $1/t$) as a function of energy $E$ (in units of $t$) in the presence of a uniform perpendicular magnetic field $\left(B=20\text{T}\right)$ with different vacancy concentrations: $n_x=0\mathrm{\%}$, 0.01%, 0.05%, 0.1%, and 0.5%. Sample size is $4096\times 4096$.

Figure 9

(Color online) Density of states (in units of $1/t$) as a function of energy $E$ (in units of $t$) in the presence of a uniform perpendicular magnetic field $\left(B=50\text{T}\right)$ with different hydrogen concentrations $n_i=0\mathrm{\%}$,0.01%,0.05%,0.1%,0.5%, and 1%. Sample size is $4096\times 4096$.

Figure 10

(Color online) The error ($\delta$ and $\sigma$) of the approximation of $|\Psi \left(\epsilon \right)⟩$ of a quasieigenstate in a graphene sample $\left(4096\times 4096\right)$ with vacancies or hydrogen $(\left({\epsilon }_d=-t/16,V=2t\right)\phantom{)}$ impurities. The concentration of the defeats is 0.1%.

Figure 11

(Color online) Position of hydrogen impurities (black dots in the top left panel) and contour plot of the amplitudes of the quasieigenstates in the central part of a graphene sample $\left(4096\times 4096\right)$ with different energies. The concentration of the hydrogen impurities $\left({\epsilon }_d=-t/16,V=2t\right)$ is 0.1%.

Figure 12

(Color online) Contour plot of the amplitudes of the quasieigenstates in the central part of a graphene sample $\left(4096\times 4096\right)$ with different energies. The concentration of the vacancy impurities (indicated by black dots) is 0.1%.

Figure 13

(Color online) Comparison of the numerically calculated optical conductivity ($\mu =0$ or 0.2 eV, $T=300\text{K}$) with Eq. (36) (analytical I) and Eq. (37) (analytical II). The size of the system is $M=N=8192$.

Figure 14

(Color online) Comparison of DOS (in units of $1/t$) and optical conductivity $\left(\mu =0,\text{T}=300\text{K}\right)$ with symmetrical random $\left(v_r\right)$ or antisymmetrical fixed $\left(\pm v_d\right)$ potential on sublattices A and B. The size of the system is $M=N=4096$.

Figure 15

(Color online) Comparison of optical conductivity $\left(\mu =0,T=300\text{K}\right)$ with different concentration of hydrogen impurities. The size of the system is $M=N=4096$, except for the clean graphene $\left(M=N=8192\right)$.

Figure 16

(Color online) Conductivity $\sigma$ (in units of $e^2/h$) as a function of charge carrier concentration $n_e$ (in units of electrons per atom) for different resonant impurity $\left({\epsilon }_d=-t/16,V=2t\right)$ or vacancy concentrations $\left(n_x\right)$: (a) $n_i=n_x=0.1\mathrm{\%}$, (b) 0.2%, (c) 0.3%, (d) 0.5%, (e) 1%, and (f) 5%. Numerical calculations are performed on samples containing (a) $8192\times 8192$ and (b)–(f) $4096\times 4096$ carbon atoms. The charge carrier concentrations $n_e$ are obtained by the integral of the corresponding density of states represented in Fig. 3.

Figure 17

(Color online) Red dots: conductivity $\sigma$ (in units of $e^2/h$) as a function of $K_F$ (in units of ${\text{Å}}^{-1}$) for resonant impurity (top panel, ${\epsilon }_d=-t/16$, $V=2t$) or vacancy (bottom panel). The concentration of the impurities is $n_i=n_x=0.01\mathrm{\%}$. Numerical calculations are performed on samples containing $4096\times 4096$ carbon atoms. Black lines: fit of Eq. (44) with $q_0=0.01{\text{Å}}^{-1}$, $R=1.25\text{Å}$ for $n_i=0.01\mathrm{\%}$, and $q_0=0$, $R=1.28\text{Å}$ for $n_x=0.01\mathrm{\%}$.