Scaling properties of monolayer graphene away from the Dirac point

The statistical properties of the carrier density profile of graphene in the ground state in the presence of particle-particle interaction and random charged impurity in zero gate voltage has been recently obtained by Najafi et al. [Phys. Rev. E 95, 032112 (2017)]. The nonzero chemical potential ($\mu$) in gated graphene has nontrivial effects on electron-hole puddles, since it generates mass in the Dirac action and destroys the scaling behaviors of the effective Thomas-Fermi-Dirac theory. We provide detailed analysis on the resulting spatially inhomogeneous system in the framework of the Thomas-Fermi-Dirac theory for the Gaussian (white noise) disorder potential. We show that the chemical potential in this system as a random surface destroys the self-similarity, and also the charge field is non-Gaussian. We find that the two-body correlation functions are factorized to two terms: a pure function of the chemical potential and a pure function of the distance. The spatial dependence of these correlation functions is double logarithmic, e.g., the two-point density correlation behaves like $D_2(r,\mu )\propto {\mu }^{2}exp\left[-{\left(-a_{D}lnln{r}^{{\beta }_D}\right)}^{{\alpha }_D}\right]$ (${\alpha }_D=1.82$, ${\beta }_D=0.263$, and $a_D=0.955$). The Fourier power spectrum function also behaves like $ln\left[S\left(q\right)\right]=-{\beta }_S^{-a_S}{\left(lnq\right)}^{a_S}+2ln\mu$ ($a_S=3.0\pm 0.1$ and ${\beta }_S=2.08\pm 0.03$) in contrast to the ordinary Gaussian rough surfaces for which $a_S=1$ and ${\beta }_S=\frac{1}{2}{(1+\alpha )}^{-1}$ ($\alpha$ being the roughness exponent). The geometrical properties are, however, similar to the ungated ($\mu =0$) case, with the exponents that are reported in the text.

Figure 1

(a) Semilog plot of the distribution function of density $P\left(n\right)$. Left inset: ${\epsilon }_{Pn}\left(\mu \right)$, which is defined as the (absolute value of the) difference between the slopes of two sides. Right inset: the position of the peak of the distribution function $n_0\left(\mu \right)$. (b) The distribution function of the roughness function $P\left(W_2\right)$ for the boxes of size $l=100$. Inset: the position of the peak in terms of $\mu$.

Figure 2

$D_2(r,\mu )=\frac{f_D\left(\mu \right)}{\lambda \left(r\right)}$ in terms of (a) $r$ and (b) $\mu$. The fitting is done for $\lambda \left(r\right)=exp\left[{\left(-a_{D}lnln{r}^{{\beta }_D}\right)}^{{\alpha }_D}\right]$, with the parameters ${\alpha }_D=1.82$, ${\beta }_D=0.263$, $a_D=0.955$, and $f_D\left(\mu \right)\sim {\mu }^2$. $W_2(r,\mu )=\frac{f_W\left(\mu \right)}{\lambda \left(L\right)}$ in terms of (c) $L$ and (d) $\mu$. The fitting is done for $\lambda \left(L\right)=exp\left[{\left(-a_{W}lnln{L}^{{\beta }_W}\right)}^{{\alpha }_W}\right]$ with the parameters ${\alpha }_W=1.35\pm 0.01$, ${\beta }_W=0.26\pm 0.01$, $a_W=1.375\pm 0.005$, and $f_W\left(\mu \right)\sim {\mu }^2$. $S_q=\frac{f_q\left(\mu \right)}{\lambda \left(q\right)}$ in terms of (e) $q$ and (f) $\mu$ with $\lambda \left(q\right)exp\left[{\left(ln\left(q\right)/{\beta }_S\right)}^{a_S}\right]$, $a_S=3.0\pm 0.1$, ${\beta }_S=2.1\pm 0.03$, and $f_S\left(\mu \right)\sim {\mu }^2$.

Figure 3

$C_2(b,\mu )$ in terms of (a) $b$ and (b) $\mu$, for which the fitting is done for $D_2(b,\mu )=\frac{f_C\left(\mu \right)}{\lambda \left(b\right)}$. The fitting parameters are $\lambda \left(b\right)\equiv exp\left[{\left(-a_{C}lnln{r}^{{\beta }_C}\right)}^{{\alpha }_C}\right]$, ${\alpha }_C=1.35\pm 0.02$, ${\beta }_C=0.25\pm 0.01$, $a_C=1.36\pm 0.01$, and $f_C\left(\mu \right)\sim {\mu }^2$. The same analysis for $C_4(b,\mu )$ in terms of (c) $b$ and (d) $\mu$, with the parameters ${\alpha }_C=1.85\pm 0.02$, ${\beta }_C=0.26\pm 0.01$, $a_C=1.86\pm 0.01$, and $f_C\left(\mu \right)\sim {\mu }^3$.

Figure 4

Non-Gaussian parameters (a) $C_3(b,\mu )$ and (b) ${\left[C_2(b,\mu )\right]}^2/C_4(b,\mu )$. The latter changes with the second power of $\mu$ which confirms that the system is non-Gaussian.

Figure 5

Charge pattern of the systems with chemical potentials: (a) $\mu =0.001$, (b) $\mu =0.005$, (c) $\mu =0.01$, (d) $\mu =0.05$, (e) $\mu =0.1$, and (f) $\mu =0.2$. The different colors show the connected components.

Figure 6

Log-log plot of (a) $l\text{−}r$ and (b) $S\text{−}r$ with the exponents ${\gamma }_{lr}=1.42\pm 0.02$ and ${\gamma }_{Sr}=1.8\pm 0.1$. (c) The distribution function of the area inside the cluster $a$ (inset: the distribution function of the loop length $l$). (d) The distribution function of the cluster mass $S$ (upper inset: the distribution function of the mass gyration radius $R$; lower inset: the distribution function of the loop gyration radius $r$). The corresponding exponents are ${\tau }_a=1.82\pm 0.06$, ${\tau }_l=2.28\pm 0.07$, ${\tau }_S=1.55\pm 0.03$, ${\tau }_R=1.90\pm 0.06$, and ${\tau }_r=2.43\pm 0.05$.