Helical scattering and valleytronics in bilayer graphene

We describe an angularly asymmetric interface-scattering mechanism which allows to spatially separate the electrons in the two low-energy valleys of bilayer graphene. The effect occurs at electrostatically defined interfaces separating regions of different pseudospin polarization, and is associated with the helical winding of the pseudospin vector across the interface, which breaks the reflection symmetry in each valley. Electrons are transmitted with a preferred direction of up to $60°$ over a large energetic range in one of the valleys, and down to $-60°$ in the other. In a Y-junction geometry, this can be used to create and detect valley polarization.

Figure 1

(Color online) Helical scattering of electrons in bilayer graphene at an interface between regions of different pseudospin polarization. Panel (a) sketches in a side view how such an interface is obtained by sandwiching a bilayer graphene flake between positively and negatively charged gates (light orange and dark blue, respectively; not drawn to scale). Panel (b) shows the rotation of the pseudospin vector in the valleys around the K and ${\text{K}}^{\prime }$ points. In the polar plots of panels (c)–(h), solid (blue) lines show the angle-resolved transmission probability $T\left(\phi \right)$ in the K valley while dashed (red) lines show this probability in the ${\text{K}}^{\prime }$ valley. Each panel pertains to a different Fermi energy $E_F$, ranging from values just above the gap where the transmission in each valley is angularly asymmetric but small [panels (c) and (d)], over a regime where angular asymmetric scattering occurs around a transparent direction [panels (e)–(g)] to the regime far above the gap, where one approaches valley-independent, angularly symmetric scattering [panel (h)]. Panels (i) and (j) contrast these finding to the case of Klein tunneling at an np junction (Refs. 6, 7), which does not distinguish the valleys.

Figure 2

(Color online) Absence of intervalley scattering. Panel (a): top-view sketch of a (zigzag) graphene nanoribbon geometry with a pseudospin interface, and definitions of reflection and transmission coefficients (summed over all channels) when a K-polarized current is injected into lead 1. Panels (b) and (c): numerical results for the valley-resolved reflection and transmission coefficients (nanoribbon width $W\simeq 150\hslash /\sqrt{2m\left|U_0\right|}$).

Figure 3

(Color online) Valleytronics applications. Panel (a): top-view sketch of a valley splitter, consisting of a Y junction where electrons are injected through lead 1, and collected in leads 2 and 3. In this setup, the contact region has the same gate polarity as lead 1, and the incoming current in this lead is valley unpolarized. Panels (b) and (c): valley-resolved conductance of electrons in leads 2 and 3. Panel (d): ratio of the larger to the smaller conductance in these two leads. Analogously, panels (e)–(h) demonstrate that valley polarization can be created in lead 1 by injecting unpolarized currents into lead 2 or 3. Panels (i)–(l) show how the total current in leads 2 and 3 can be used to the detect the valley polarization of an incoming current in lead 1.

Figure 4

(Color online) Alternative design of the valley splitters, polarizers, and detectors of Fig. 3, where the contact region has the same gate polarity as leads 2 and 3. Qualitatively, the results for valleys K and ${\text{K}}^{\prime }$ are now interchanged; however, the influence of edge states between leads 2 and 3 result in a notable lead asymmetry of the conductance ratios.

Figure 5

(Color online) (a) Tight-binding model of a flake of bilayer graphene, consisting of a top and a bottom layer where each layer accommodates a honeycomb lattice of carbon atoms. Black dots denote dimer sites, where carbon atoms of both layers lie on top of each other and are coupled with interlayer coupling constant ${\gamma }_1$. Light (red and blue) dots denote sites of isolated carbon atoms (either on the top or the bottom layer). Lines denote intralayer nearest-neighbor hopping with coupling constant ${\gamma }_0$. (b) Effective model for the two low-energy bands, obtained after decimation (algebraic elimination) of the dimer sites. Thick lines denote coupling with strength $2{\gamma }^{\prime }$ while thin lines denote coupling with strength ${\gamma }^{\prime }$, where ${\gamma }^{\prime }={\gamma }_0^2/{\gamma }_1$.