All-strain based valley filter in graphene nanoribbons using snake states

A pseudomagnetic field kink can be realized along a graphene nanoribbon using strain engineering. Electron transport along this kink is governed by snake states that are characterized by a single propagation direction. Those pseudomagnetic fields point towards opposite directions in the $K$ and $K^{\prime }$ valleys, leading to valley polarized snake states. In a graphene nanoribbon with armchair edges this effect results in a valley filter that is based only on strain engineering. We discuss how to maximize this valley filtering by adjusting the parameters that define the stress distribution along the graphene ribbon.

Figure 1

(a) Sketch of the strained graphene ribbon. Strain is defined by two circles of radius $R$ (red dashed), which smoothly decay to zero towards the input and output leads, and the width of the strained region is defined by $\beta$. The color map indicates smaller (brighter regions) and larger (darker regions) local displacements. Central atoms (green dashed line along the $x$ axis) are always unstrained in this configuration. (b) Top: Contour plot of the induced pseudomagnetic field profile for a representative strain configuration characterized by the parameters ($\beta , R_0$). Bottom: A cross view of the pseudomagnetic field along the lines placed in $y>0$ (purple short-dashed) and $y<0$ (blue long-dashed) regions of the system in the top panel, for two different maximum radii $R_0^2>R_0^1$.

Figure 2

Energy states for a Dirac particle in the presence of a (pseudo)magnetic kink. Results for $K$ and $K^{\prime }$ valleys are the same for an external magnetic field, whereas for a pseudomagnetic field, the $K^{\prime }$ spectrum (red dashed curves) differs from that from the $K$ valley (black solid curves).

Figure 3

Examples of calculated trajectories of electron wave packets propagating with momenta around $K$ (black solid curves) and $K^{\prime }$ (red dashed curves) valleys, starting at ($x,y$) points (indicated by blue solid dots) given by (1250 and 300 Å) and ($-1250$ and $-300\text{Å}$ ), respectively. Arrows indicate the direction of propagation along the trajectories.

Figure 4

Probability densities, as a function of time, of finding the electron before ($P_1$), within ($P_2$), and after ($P_3$) the $\beta =900$ and $R_0=10000\text{Å}$ strained region, for a wave packet with $k=0.06\text{Å}{}^{-1}$ around the $K$ point of the Brillouin zone. Results for $P_3$ considering a wave packet around $K^{\prime }$ are shown for comparison.

Figure 5

Valley polarization of the outgoing wave packet, with $k=0.06\text{Å}{}^{-1}$, as a function (a) of the width of the strained region $\beta$, for different radii $R_0$, and (b) as function of the strain radius $R_0$, for different $\beta$ values.

Figure 6

Valley polarization of the outgoing wave packet as a function of its energy, considering (a) $\beta =900$ and different radii $R_0$, and (b) $R_0=6000\text{Å}$, for different values of $\beta$.