Scaling properties of induced density of chiral and nonchiral Dirac fermions in magnetic fields

We find that a repulsive potential of graphene in the presence of a magnetic field has bound states that are peaked inside the barrier with tails extending over $\ell (N+1)$, where $\ell$ and $N$ are the magnetic length and Landau level index. We have investigated how these bound states affect scaling properties of the induced density of filled Landau levels of massless Dirac fermions. For chiral fermions, we find, in strong coupling regime, that the density inside the repulsive potential can be greater than the value in the absence of the potential, while in the weak coupling regime, we find negative induced density. Similar results also hold for nonchiral fermions. As one moves from weak to strong coupling regimes, the effective coupling constant between the potential and electrons becomes more repulsive, and then it changes sign and becomes attractive. Different power laws of induced density are found for chiral and nonchiral fermions.

Figure 1

We have schematically plotted electron density for a filled Landau level. The electron density in the absence of the potential is $\frac{1}{2\pi {\ell }^2}$, represented by the dashed horizontal line. In graphene LLs, under certain conditions, induced density can be positive (a), in contrast to the usual case, where it is negative (b).

Figure 2

$Y$ axis represents values of $g_0\left(\frac{R}{\ell }\right)$ obtained by data collapse for $N=0$ and $\frac{V}{E_c}<0.7$. Note that this quantity is proportional to $-{\alpha }_N(\frac{V}{E_c},\frac{R}{\ell })$. Values $(\frac{R}{\ell },\frac{V}{E_c})$ used in the data collapse are shown in the inset. Note that the induced density changes sign near $R/\ell =1$. Crosses represent the results obtained by treating $V\left(r\right)$ in the second-order perturbation theory. These results demonstrate that perturbative methods cannot be applied in the intermediate and strong coupling regimes.

Figure 3

Single-particle energy spectrum of massless Dirac fermions in the presence of repulsive potential and magnetic field. Energies of five Landau levels are shown for $N=-2,-1,0,1,2$. Dashed lines represent unperturbed energies, and those that deviate from these are energies of bound states (at $B=20$ T, $R=10$ nm, $R/\ell =1.74$ and $V=0.1$ eV).

Figure 4

Bound states of strong coupling limit: $|{\Psi }_1^{1/2}{\left(r\right)|}^2$ for $V/E_c=1.82$, $R/\ell =0.01$, $R=1$ nm, $V=1.2$ eV, and $E=0.009$ eV. Inset: $A$ ($B$) component is dashed (dotted) line.

Figure 5

Strong coupling regime: the parameters are $E_c=0.1316$ eV, $E_m=0.0395$ eV, $V=0.26$ eV, $V/E_c=1.98$, and $R/\ell =0.3$. (a) Solid line is for $N=0$ and $E=0.017$ eV. Dashed line is for $N=1$ and $E=0.061$ eV. Inset: $B$ component of $|{\Psi }_0^{-1/2}{\left(r\right)|}^2$ and $A$ (dashed) and $B$ (solid) components of $|{\Psi }_1^{-1/2}{\left(r\right)|}^2$ in the absence of potential. (b) Solid line is for $N$=2 and $E=0.083$ eV. Dashed line is for $N=3$ and $E=0.101$ eV. Inset: $A$ and $B$ components of $|{\Psi }_2^{-1/2}{\left(r\right)|}^2$ and $|{\Psi }_3^{-1/2}{\left(r\right)|}^2$ in the absence of potential.

Figure 6

(a) Weak coupling regime: $|{\Psi }_0^{-1/2}{\left(r\right)|}^2$ for $R/\ell =2.0$, $V/E_c=2.172$, $R=11$ nm, $V=0.13$ eV, and $E=0.110$ eV. $A$ ($B$) component is solid (dashed) line. Inset: dashed line is without potential and solid is with potential. (b) Intermediate coupling regime: $|{\Psi }_0^{-1/2}{\left(r\right)|}^2$ for $R/\ell =0.6$, $V/E_c=0.607$, $R=5$ nm, $V=0.08$ eV, and $E=0.013$ eV.

Figure 7

(a) ${\ell }^{2}n(r)$ for $N=0$ and $R/\ell ⩽1$: $(\frac{R}{\ell },\frac{V}{E_c})=(1,1.215)$ (solid), $(\frac{R}{\ell },\frac{V}{E_c})=(0.60,0.607)$ (dashed), and $(\frac{R}{\ell },\frac{V}{E_c})=(0.30,0.334)$ (dotted). Dashed-dotted line is ${\ell }^{2}n(r)$ in the absence of the potential. (b) For $N=0$ and $R/\ell >1$: $(\frac{R}{\ell },\frac{V}{E_c})=(1.74,2.582)$ (solid), $(\frac{R}{\ell },\frac{V}{E_c})=(1.50,1.823)$ (dashed), $(\frac{R}{\ell },\frac{V}{E_c})=(1.22,1.063)$ (dotted).

Figure 8

Approximate data collapse of $g_0\left(\frac{R}{\ell }\right)$ is obtained for a larger range of $\frac{V}{E_c}$ than the one used in Fig. 2. Values of $(\frac{V}{E_c},\frac{R}{\ell })$ used are shown in inset.

Figure 9

$N=0$ LL states must not overlap with $N=\pm$ 1 LLs. Induced density at $r=0$ is positive for $R/\ell <1$.

Figure 10

${\ell }^{2}n\left(r\right)$ for $N=1$. $(\frac{R}{\ell },\frac{V}{E_c})=(1.5,3.038)$ (dashed), $(\frac{R}{\ell },\frac{V}{E_c})=(0.6,0.607)$ (dotted), and $(\frac{R}{\ell },\frac{V}{E_c})=(0.01,0.607)$ (dashed dotted). Solid line is ${\ell }^{2}n\left(r\right)$ in the absence of the potential.

Figure 11

$g_1(R/\ell )$ is obtained by data collapse for $N=1$ and $\frac{V}{E_c}\ll 1$. Values of $(\frac{V}{E_c},\frac{R}{\ell })$ used are shown in inset.

Figure 12

$N=1$ LL states must not overlap with $N=0,2$ LLs. The range of $R/\ell$, where the induced density at $r=0$ is positive, increases with increasing $V/E_c$.