Focusing of electron flow in a bipolar graphene ribbon with different chiralities

The focusing of electron flow in a symmetric $p\text{−}n$ junction (PNJ) of graphene ribbon with different chiralities is studied. Considering the PNJ with the sharp interface, in a armchair ribbon, the electron flow emitting from $\left(-L,0\right)$ in $n$ region can always be focused perfectly at $\left(L,0\right)$ in $p$ region in the whole Dirac fermion regime, i.e., in whole regime $E_0<t$, where $E_0$ is the distance between Dirac-point energy and Fermi energy and $t$ is the nearest hopping energy. For the bipolar ribbon with zigzag edge, however, the incoming electron flow in $n$ region is perfectly converged in $p$ region only in a very low energy regime with $E_0<0.05t$. Moreover, for a smooth PNJ, electrons are backscattered near PNJ, which weakens the focusing effect. But the focusing pattern still remains the same as that of the sharp PNJ. In addition, quantum oscillation in charge density occurs due to the interference between forward and backward scatterings. Finally, in the presence of weak perpendicular magnetic field, charge carriers are deflected in opposite directions in the $p$ region and $n$ region. As a result, the focusing effect is smeared. The lower the energy $E_0$, the easier the focusing effect is destroyed. For the high-energy $E_0$ (e.g., $E_0=0.9t$), however, the focusing effect can still survive in a moderate magnetic field on order of 1 T.

Figure 1

(Color online) Schematic diagram of the graphene PNJ (a) in an armchair ribbon and (b) in a zigzag ribbon. The graphene ribbon is along $x$ direction and sharp $p\text{−}n$ interface is located at $x=0$. The electron flow injected at $\left(-L,0\right)$ (blue area) in the $n$ region (orange lattice region) is focused around the symmetric site $\left(L,0\right)$ (red area) in the $p$ region (green lattice region).

Figure 2

(Color online) After injecting from the source lead in the $n$ region, the right-going electrons are scattered by the PNJ and boundary of the bipolar graphene ribbon.

Figure 3

(Color online) Distribution of local particle density $\delta \rho \left(x,y\right)/\delta V$ in a graphene ribbon with a single sharp PNJ at $x=0$. (a) Zigzag ribbon with ribbon width $W=500\times 3a$. The source flow is injected from the honeycomb unit cell at $\left(-800b,0\right)$ and focused around the spot located at $\left(800b,0\right)$. (b) Armchair ribbon with ribbon width $W=501b$. The source flow is injected from the honeycomb unit cell at $\left(-320\times 3a,0\right)$. The other parameter used: $E_0=0.05t$.

Figure 4

(Color online) Contour of dispersion relation $E\left(k_x,k_y\right)$ of graphene sheet. The energy interval between nearest contour lines is $\delta E=0.1t$. (a) $E\left(k_x,k_y\right)$ of the graphene sheet with the carbon-carbon bond is along the $x$ direction corresponding to armchair ribbon or Fig. 1a. (b) $E\left(k_x,k_y\right)$ of the graphene sheet with the carbon-carbon bond is along the $y$ direction corresponding to zigzag ribbon or Fig. 1b.

Figure 5

(Color online) Scattering momenta $k_{x,r/t}^{intra}$ and $k_{x,r/t}^{inter}$ vs injecting momentum $k_{x,in}$ for the zigzag ribbon with sharp PNJ. In the interband case, $k_{x,r/t}^{inter}=-k_{x,in}$ for all conserved $k_y$ (red line). While in the intraband case (black lines), $k_{x,r/t}^{inter}$ is not equal to $-k_{x,in}$. Different black lines along the black arrow correspond to $k_y=0$, $0.1/b$, $0.15/b$, and $0.2/b$, respectively.

Figure 6

(Color online) Contour of local particle density in zigzag ribbon with a sharp PNJ for (a) $E_0=0.1t$ and (b) $E_0=0.2t$.

Figure 7

(Color online) Contour of local particle density in armchair ribbon with a sharp PNJ for (a) $E_0=0.1t$ and (b) $E_0=0.2t$.

Figure 8

(Color online) Contour of local particle density in armchair ribbon with a sharp PNJ for (a) $E_0=0.5t$ and (b) $E_0=0.9t$.

Figure 9

(Color online) Instead of contour of local particle density in Fig. 7b, the quiver of local current-density vector around convergence spot is plotted.

Figure 10

(Color online) Instead of contour of local particle density in Fig. 8b, the quiver of local current-density vector around convergence spot is plotted.

Figure 11

(Color online) (a) Sketch of edge in the scattering region with zigzag edge. (b) Sketch of edge in the scattering region with armchair edge.

Figure 12

(Color online) Contour of local particle density in the armchair ribbon with a sharp PNJ for $E_0=0.5t$. (a), (b), (c), and (d) correspond to random on-site potential strengths $w=0$, 0.2, 0.5, and 1.0, respectively.

Figure 13

(Color online) Contour of local particle density in armchair ribbon with a sharp PNJ for $E_0=0.5t$. (a), (b), (c), and (d) correspond to $p=0$, 0.1, 0.2, and 0.5, respectively.

Figure 14

According to Figs. 12, 13, maximum values (the value at the focusing spot central $\left[L,0\right]$) of the focusing spot ${\text{LDOS}}_{max}$ vs the strength of random potential $w$ [in panel (a)] and probability of one missing atom $p$ [in panel (b)] are plotted, respectively.

Figure 15

(Color online) (a) Smoothly changed $U\left(x\right)$ forming smooth PNJ used in panels (b)–(d), in which $x_0=0.25\ast 3a$, $0.75\ast 3a$ and $3\ast 3a$, respectively. [(b)–(d)] Contour of local particle density in armchair ribbon for $E_0=0.1$ [same to Fig. 7a where sharp PNJ is used]. For different panels, smooth PNJs shown in panel (a) are used, respectively.

Figure 16

(Color online) Contour of local particle density in the $p$ region for an armchair ribbon in the perpendicular weak magnetic filed. The sharp PNJ is located at $x=0$. The magnetic field $B{S}_0=0.0001{\varphi }_0/\pi$, $E_0=0.2t$ in panel (a), and $E_0=0.9t$ in panel (b).