Theory of the valley-valve effect in graphene nanoribbons

A potential step in a graphene nanoribbon with zigzag edges is shown to be an intrinsic source of intervalley scattering—no matter how smooth the step is on the scale of the lattice constant $a$. The valleys are coupled by a pair of localized states at the opposite edges, which act as an attractor and/or repellor for the edge states propagating in valley $K/K^{\prime }$. The relative displacement $\Delta$ along the ribbon of the localized states determines the conductance $G$. Our result $G=\left(e^2/h\right)\left[1-\text{cos}\left(N\pi +2\pi \Delta /3a\right)\right]$ explains why the “valley-valve” effect (the blocking of the current by a $p\text{−}n$ junction) depends on the parity of the number $N$ of carbon atoms across the ribbon.

Figure 1

(Color online) Conductance $G$ of a zigzag nanoribbon containing a potential step $U=\frac{1}{2}{U}_0\left[1+\text{tanh}\left(2x/d\right)\right]$. The red and blue curves are obtained by a computer simulation of the tight-binding model of graphene, with parameters $d=10a$ and $E_F=0.056t$, where $a$ is the lattice constant and $t$ is the nearest-neighbor hopping energy. Upon varying $U_0$, the conductance abruptly switches to zero when the Fermi level $E_F$ is crossed and a $p\text{−}n$ junction is formed (red solid curve; the deviation from an ideally quantized step function is $\lesssim {10}^{-7}$). This valley-valve effect occurs only for an even number $N$ of carbon atom rows (zigzag configuration). When $N$ is odd (antizigzag configuration), the conductance remains fixed at $2e^2/h$ (blue dotted curve, again quantized within ${10}^{-7}$).

Figure 2

Nanoribbons in the zigzag configuration (left panel, $N$ even) and in the antizigzag configuration (right panel, $N$ odd). In both cases, the atoms at opposite edges belong to different sublattices (indicated by the black and white dots).

Figure 3

(Color online) Top panel: Zigzag nanoribbon containing a $p\text{−}n$ interface from $x_{-}$ to $x_{+}$ (dotted line). The spatial extension of the edge states in the lowest mode is indicated by the gray areas. Incoming edge states are in the valley $K$, while outgoing edge states are in the valley $K^{\prime }$. The arrows indicate the direction of propagation: in the conduction band (solid) and valence band (dashed). The corresponding dispersion relations are plotted in the lower panel. The localized (dispersionless) edge state, which is responsible for the intervalley scattering, is indicated in red.

Figure 4

(Color online) Conductance for a tilted $p\text{−}n$ interface, with potential $U=\frac{1}{2}{U}_0\left\{1+\text{tanh}\left[2\left(x-y\text{tan}\theta \right)/d\right]\right\}$, at fixed $U_0=2E_F$ as a function of the relative displacement $\Delta \equiv x_{+}-x_{-}=W\text{tan}\theta$ of the intersection of the interface with the edges of the nanoribbon. The parameters are the same as those in Fig. 1, which corresponds to $\Delta =0$. The data from this computer simulation is described by the analytical result [Eq. (2)].

Figure 5

(Color online) Comparison between results of computer simulations (dots) and numerical evaluation of Eq. (12). The parameters are $N=20$ (solid line) and $N=21$ (dashed line), $U_0=2E_F=0.0058t$, and $d\equiv E_F/U_x=100a$.

Figure 6

(Color online) The conductance of a tilted $p\text{−}n$ interface with next-nearest neighbor hopping included $\left(t^{\prime }/t=0.1\right)$. The parameters of the ribbon are $E_F=0.19t$, $U_0=0.16t$, and $d=100a$. The number of atoms across the ribbon is $N=40$ (solid line) and $N=41$ (dashed line).