Switchable valley filter based on a graphene $p-n$ junction in a magnetic field

Low-energy excitations in graphene exhibit relativistic properties due to the linear dispersion relation close to the Dirac points in the first Brillouin zone. Two of the Dirac points located at opposite corners of the first Brillouin zone can be chosen as inequivalent, representing a new valley degree of freedom, in addition to the charge and spin of an electron. Using the valley degree of freedom to encode information has attracted significant interest, both theoretically and experimentally, and gave rise to the field of valleytronics. We study a graphene $p-n$ junction in a uniform out-of-plane magnetic field as a platform to generate and controllably manipulate the valley polarization of electrons. We show that by tuning the external potential giving rise to the $p-n$ junction we can switch the current from one valley polarization to the other. We also consider the effect of different types of edge terminations and present a setup where we can partition an incoming valley-unpolarized current into two branches of valley-polarized currents. The branching ratio can be chosen by changing the location of the $p-n$ junction using a gate.

Figure 1

Three-terminal device used as a switchable valley filter. The three leads of horizontal size $d$, with zigzag edge termination, upper lead $L_0$, lower-left lead $L_1$, and lower-right lead $L_2$, are attached to the rectangular scattering region of length $L$ and width $W$. Top and bottom edges of the scattering region are of armchair type, while the left and right edges are of zigzag type. The $p-n$ interface of thickness $2\ell$ (white color gradient) is modeled by an $x$-dependent on-site potential $V\left(x\right)$. In an out-of-plane magnetic field with $n$ Landau levels occupied ($n=0,1,2\cdots$), there are $2n+1$ edge states with opposite chirality in the $n$- and $p$-doped region. Along the $p-n$ interface there are $2(2n+1)$ copropagating snake states.

Figure 2

Case when $V_0=0$ and $E=\delta /2$. (a) State in the scattering region due to incoming mode from lead $L_0$ with $k_{y}a=2.08$. The inset table lists the transmissions $T_{10}$ and $T_{20}$ and polarizations $P_1$ and $P_2$. (b) Band structure of leads $L_0$ and $L_2$. The Fermi energy is indicated by the horizontal dashed line. The red arrow indicates the incoming mode on lead $L_0$ which has velocity $v<0$ and is chosen to be plotted in (a).

Figure 3

Case when $V_0=\delta /2$ and $E=0.001t$. (a) State in the scattering region due to incoming mode from lead $L_0$ with $k_{y}a=2.08$. The inset table lists the transmissions $T_{10}$ and $T_{20}$ and polarizations $P_1$ and $P_2$. (b) Top: Band structure of lead $L_0$. The Fermi energy is indicated by the horizontal dashed line. The red arrow indicates the incoming mode chosen to plot (a). Bottom: Band structure of lead $L_2$.

Figure 4

Polarization $P_1$ in lead $L_1$ (blue, left axis) and $P_2$ in lead $L_2$ (green, left axis) as a function of $V_0$ for the device shown in Fig. 1. For $0<V_0<\delta$ only the $n=0$ Landau level (LL) is occupied and $P_2\approx -1$. As soon as higher LLs get involved ($V_0>\delta$), where the edge states are not valley polarized, the valley polarization in $L_1$ and $L_2$ decreases towards 0 with increasing $V_0$. The sum of the transmissions $T_{10}+T_{20}$ (red steplike curve, right axis) exhibit quantization due to LLs in the scattering region. The device is a good valley filter for $V_0<\delta$, i.e., when only the $n=0$ LL is occupied. Vertical (gray dashed) lines mark the LL energies $E_n=\sqrt{2n}\hslash v_F/{\ell }_B$ in the $n$-doped region calculated for a linear Dirac dispersion. The parameters chosen for this figure are $L=2080a, W=984a, \ell =40a$, and $d=320a$.

Figure 5

Transmissions $T_{10}$ (solid lines), $T_{20}$ (dotted lines), and $T_{10}+T_{20}$ (dashed line) as a function of $V_0$ for different system sizes. The inset shows that the amplitude of the oscillations tend to vanish as we increase the system size, which is accomplished by multiplying the parameters $L,W,d,$ and $\ell$ by a factor of $\alpha =1$ (blue), 1.5 (green), and 4 (red) while $\varphi /{\varphi }_0=0.003$ is kept constant.

Figure 6

Geometry of a device with a tilted staircase edge. Upper panel: Zoom-in onto a part of the tilted upper edge displaying the staircase. The step length (length of the region of constant width $W$) is denoted by ${\ell }_{\text{step}}$. Lower panel: A schematic of the device. The slope of the tilted edge $k$ is related to ${\ell }_{\text{step}}$, e.g., $k=0.003$ corresponds to ${\ell }_{\text{step}}\approx 166a$, while $k=0.03$ corresponds to ${\ell }_{\text{step}}\approx 16a$.

Figure 7

Transmission $T_{20}$ in a device with a tilted staircase edge as a function of the position of the $p-n$ interface $x_0$ for different values of the slope $k$ [see Fig. 6]. (a) Plateaulike behavior of $T_{20}$, as expected for ${\ell }_{\text{step}}\gg 2{\ell }_B$. (b) Sinelike behavior of $T_{20}$ for ${\ell }_{\text{step}}\lesssim 2{\ell }_B$. In this figure $L=780a$.