Snake states in graphene quantum dots in the presence of a $p$-$n$ junction

We investigate the magnetic interface states of graphene quantum dots that contain $p$-$n$ junctions. Within a tight-binding approach, we consider rectangular quantum dots in the presence of a perpendicular magnetic field containing $p$-$n$ as well as $p$-$n$-$p$ and $n$-$p$-$n$ junctions. The results show the interplay between the edge states associated with the zigzag terminations of the sample and the snake states that arise at the $p$-$n$ junction due to the overlap between electron and hole states at the potential interface. Remarkable localized states are found at the crossing of the $p$-$n$ junction with the zigzag edge having a dumb-bell-shaped electron distribution. The results are presented as a function of the junction parameters and the applied magnetic flux.

Figure 1

(a) Number of nearly-zero-energy states within the interval $\left|E\right|<0.1$ meV as a function of armchair ($N_a$) and zigzag ($N_z$) edge atoms. (b) Number of zero energies as a function of $N_a$ for $N_z=5,10,20$.

Figure 2

Energy levels of a system with a $p$-$n$ junction parallel to the zigzag edges in a rectangular GQD with $L_x=7.52$ nm, $L_y=8.6$ nm, and ${\Phi }_c/{\Phi }_0=0$ as a function of applied voltage $U_b$. The system is illustrated in the inset. The different colors represent the different gate voltages ($+U_b$ for $p$ type and $-U_b$ for $n$ type). The red dashed lines correspond to the energy states given by Eq. (3) and the green dashed lines represent the energy levels obtained from Eq. (4).

Figure 3

Probability densities corresponding to the points indicated in Figs. 2a, 2b, 2c, 2d, 2e, 2f, 2g, 2h. The black vertical line indicates the position of the $p$-$n$ junction.

Figure 4

Energy levels of a rectangular GQD with a $p$-$n$ junction parallel to the armchair edges (see the inset of the figure) as a function of applied electrostatic potential $U_b$ for the same parameters as Fig. 2.

Figure 5

Probability densities corresponding to the points indicated in Figs. 4a, 4b, 4c, 4d, 4e, 4f. In (a), $E_{\left(8\right)}=0$ indicates the eighth degenerate zero energy. The black horizontal line indicates the position of the $p$-$n$ junction.

Figure 6

Upper panel: The same as Fig. 2, but with nonzero magnetic flux ${\Phi }_c/{\Phi }_0=0.1$. The red dashed lines are the LLs of an infinite graphene sheet which are shifted up (down) by $U_b$ ($-U_b$). Lower panels: Probability densities corresponding to the points indicated by (a)–(d) in the upper panel. The black vertical line indicates the position of the $p$-$n$ junction.

Figure 7

Current density profile corresponding to the states indicated in Figs. 6a, 6b, 6c, 6d. The black line indicates the position of the $p$-$n$ junction. The inset in (b) shows a streamline plot of the enlargement region.

Figure 8

The same as Fig. 4, but in the presence of magnetic flux ${\Phi }_c/{\Phi }_0=0.1$. The red dashed lines are the LLs of an infinite graphene sheet which are shifted up (down) by $U_b=0.25$ eV ($-U_b$).

Figure 9

(a) Energy levels of a rectangular GQD subjected to a $p$-$n$ junction parallel to the zigzag edges (see the inset of Fig. 2) as a function of magnetic flux threading one carbon hexagon ${\Phi }_c$ for $U_b=0.25$ eV. The green dashed curves are the Landau levels of an infinite graphene sheet which are shifted up (down) by $U_b$ ($-U_b$). (b) Persistent current corresponding to the first electron state [gray solid curve, labeled as (1)] and the first hole state [red dashed curve, labeled as (2)] as a function of external magnetic flux ${\Phi }_c$.

Figure 10

Probability densities corresponding to the points indicated by (a)–(d) in the enlarged circle in Fig. 9a. The black line indicates the position of the $p$-$n$ junction.

Figure 11

The same as Fig. 9, but for a rectangular GQD subjected to a $p$-$n$ junction parallel to the armchair edges (see the inset of Fig. 4).

Figure 12

The position of the oscillations in Figs. 9b and 11b and the number of flux quanta through the surface area $S=L_x{L}_y,L_x{L}_y/2,S=L_x{L}_y/1.37$ as a function of magnetic flux.

Figure 13

Probability densities corresponding to the points indicated by (a)–(d) in the enlarged circle of Fig. 11a. The black line indicates the position of the $p$-$n$ junction.

Figure 14

Current density profile corresponding to the points indicated by (a)–(c) in the enlarged circle of Fig. 11a. The black horizontal line indicates the position of the $p$-$n$ junction.

Figure 15

Energy levels of a rectangular GQD with a $p$-$n$-$p$ junction parallel to the zigzag edges as a function of external magnetic flux for the same parameters as Fig. 9. The lower inset illustrates the system. The number of $p$-type atoms is twice as large as the number of $n$-type atoms. The green dashed curves are the LLs of an infinite graphene sheet which are shifted up (down) by $U_b$ ($-U_b$).

Figure 16

(a) Electronic probability density and (b) the corresponding current profile for the points indicated in Figs. 15a and 15b. The black vertical lines indicate the position of the $p$-$n$ junctions.

Figure 17

The same as Fig. 15, but for a rectangular GQD with a $p$-$n$-$p$ junction along the armchair edges. The system is illustrated in the lower inset.

Figure 18

Probability densities corresponding to the points indicated by (a)–(d) in the enlarged region of Fig. 17. The black horizontal line indicates the position of the $p$-$n$ junctions.

Figure 19

Current density profile corresponding to the states shown in Figs. 18a and 18b. The black horizontal lines indicate the position of the $p$-$n$ junctions.

Figure 20

Energy levels of a rectangular GQD with a triangular-shaped $p$-$n$ junction as a function of external magnetic flux for the same parameters as Fig. 9. The lower inset shows schematically the system. The green dashed curves are the LLs of an infinite graphene sheet which are shifted up (down) by $U_b$ ($-U_b$).

Figure 21

Probability densities corresponding to the points indicated by (a)–(d) in the enlarged region of Fig. 20. The black line indicates the position of the $p$-$n$ junction.

Figure 22

Energy levels of a rectangular GQD with a triangular-shaped $p$-$n$-$p$ junction as a function of external magnetic flux for the same parameters as Fig. 9. The lower inset shows the system schematically, where the yellow regions with zigzag edges indicate the $n$-type atoms. The green dashed curves are the LLs, which are shifted up (down) by $U_b$ ($-U_b$).

Figure 23

Probability densities corresponding to the points indicated by (a)–(h) in the enlarged region of Fig. 22. The black lines indicate the position of $p$-$n$ junctions.