Coherent Jetting from a Gate-Defined Channel in Bilayer Graphene

Graphene has evolved as a platform for quantum transport that can compete with the best and cleanest semiconductor systems. Here, we report on the observation of distinct electronic jets emanating from a narrow split-gate-defined channel in bilayer graphene. We find that these jets, which are visible via their interference patterns, occur predominantly with an angle of $60°$ between each other. This observation is related to the trigonal warping in the band structure of bilayer graphene, which, in conjunction with electron injection through a constriction, leads to a valley-dependent selection of momenta. This experimental observation of electron jetting has consequences for carrier transport in two-dimensional materials with a trigonally warped band structure in general, as well as for devices relying on ballistic and valley-selective transport.

Figure 1

(a) Measurement setup schematic. The metallic tip is scanned above the sample in the areas denoted by the red rectangles. (b) Conductance $G(V_{\mathrm{SG}},V_{\mathrm{BG}})$. Charge neutrality (CNP) underneath the split gates and hence the formation of a channel is achieved along the diagonal conductance minimum. These data have also been published in a previous work [27]. (c) Conductance as a function of the displacement field $D$ along the CNP (cf. red dotted line). (d) Filtered conductance $G_f(x,y)$ [28] on both sides of the channel for $V_{\mathrm{tip}}=-10\mathrm{V}$. Clear jets with interference patterns emanate from the channel predominantly with a 60° angle between them. The black arrow marks an example of a secondary jet. Because of rapid oscillations in the data, the color scale appears purple. (e),(f) Basic scattering processes possible in the sample. (e) Valley-conserving scattering off a smooth scattering potential. (f) Valley-conserving (black arrows) and valley-altering (orange arrows) atomic-scale scattering.

Figure 2

(a) Numerical derivative of the conductance $G(x,V_{\mathrm{tip}})$ along the black dotted line in Fig. 1. Interferences are observed below the black-dashed guide to the eye. (b) Normalized conductance $\mathrm{\Delta }{G}_n(x,y)$ [28] measured within the black rectangle in Fig. 1. Each line is vertically offset with respect to the previous line. A periodicity of ${\lambda }_F/2\approx 29\mathrm{nm}$ is observed (cf. black arrow).

Figure 3

Top: Fermi line for the BLG dispersion in the $K^{\pm }$ valleys in the single-gated region outside the channel for parameters $v=1.02\times {10}^6\mathrm{m}/\mathrm{s}$, $v_3\approx 0.12v$, and ${\gamma }_1=0.38\mathrm{eV}$. $k_{\parallel }$ ($k_{\perp }$) is the traveling momentum parallel (perpendicular) to the channel [cf. Fig. 1]. The gray bar is a density plot of the momentum distribution $|\mathrm{\Psi }(k_{\perp }){|}^2$ for electrons in the channel, ranging from white (no occupation) to gray (substantial occupation). We compute this distribution from the lowest channel subband state $\mathrm{\Psi }(k_{\perp })$ in momentum space. The charge carrier’s real space trajectory points along the normal vector, prescribing the propagation direction at an angle $\alpha$. Bottom: effective angular propagation probability $\mathcal{P}$ of electrons injected from the channel, obtained by combining the angular velocity distribution originating from the bulk dispersion with the weight factors $|\mathrm{\Psi }(k_{\perp }){|}^2$. Two valley-polarized jets are predicted for generic orientations of the channel (here, the armchair edge of BLG is misaligned by 5° with respect to the channel axis). Longer, thinner jets imply a higher probability for electrons to propagate in a well-defined direction.