Probing the features of electron dispersion by tunneling between slightly twisted bilayer graphene sheets

Abstract

Tunneling conductance between two bilayer graphene (BLG) sheets separated by 2 nm-thick insulating barrier was measured in two devices with the twist angles between BLGs less than 1°. At small bias voltages, the tunneling occurs with conservation of energy and momentum at the points of intersection between two relatively shifted Fermi circles. Here, we experimentally found and theoretically described signatures of electron-hole asymmetric band structure of BLG: since holes are heavier, the tunneling conductance is enhanced at the hole doping due to the higher density of states. Another key feature of BLG that we explore is gap opening in a vertical electric field with a strong polarization of electron wave function at van Hove singularities near the gap edges. This polarization, by shifting electron wave function in one BLG closer to or father from the other BLG, gives rise to asymmetric tunneling resonances in the conductance around charge neutrality points, which result in strong sensitivity of the tunneling current to minor changes of the gate voltages. The observed phenomena are reproduced by our theoretical model taking into account electrostatics of the dual-gated structure, quantum capacitance effects, and self-consistent gap openings in both BLGs.

Figures (5)

• Figure 1: (a) Schematic of a dual-gated structure consisting of top and bottom BLGs separated by the hexagonal boron nitride (hBN) barrier. (b) Electron dispersions in two BLGs twisted on the angle $\theta$ (greatly exaggerated), whose Fermi lines (red and green circles) intersect in the black points where the energy and momentum conserving tunneling occurs.
• Figure 2: Maps of tunneling conductance (in arbitrary units) as functions of top and bottom gate voltages for (a,c) Device 1 with the combination of twist angles $\theta=0.05^\circ$ and $0.13^\circ$, and (b,d) Device 2 with the twist angle $\theta=0.73^\circ$. The upper panels (a,b) show experimental results and the lower panels (c,d) show theoretical calculations.
• Figure 3: Tunneling conductance $G$ along the diagonal of equal carrier densities $n_1=n_2$ in (a) Device 1 and (b) Device 2. Theoretical calculations (orange curves) demonstrate the same hierarchy as the experiment (blue curves): $G$ is higher on the hole side ($n_{1,2}<0$) than on the electron side ($n_{1,2}>0$). In contrast, the calculations with neglecting electron-hole asymmetry of BLG spectra (green curves) provide the symmetric picture. The dips in experimental curve in (b) at $n_{1,2}\sim\pm3\times10^{12}\,\hbox{cm}^{-2}$ are caused by secondary Dirac points due to an adjacent hBN layer.
• Figure 4: Signatures of gap opening and sublayer polarization of van Hove singularities in BLG, theoretically calculated for Device 2. (a,b) Field-induced interlayer potential differences $U_{1,2}$ in top and bottom BLGs as functions of the gate voltages (gaps are equal to their absolute values). Solid lines are charge neutrality lines $n_1=0$ or $n_2=0$, and dotted lines bound the tunneling regions. (c) Matrix element of the tunneling (\ref{['matr_el']}) at Fermi level.
• Figure 5: (a) Experimental map of tunneling conductance $G$ for Device 2 and its cross-sections 1-4 shown on the inset plots by blue curves. Theoretically calculated $G$ along the same cross-sections is shown by orange curves. (b) Schematic of electron localization in the upper sublayer of top BLG at the point where $G$ is suppressed by the matrix element, because the Fermi level is positioned at the ceiling of the top BLG valence band (small red circle). (c) The same, but for the point of maximum $G$ due to electron localization in the lower sublayer of top BLG, when its Fermi level is at the floor of conduction band. In both cases (b,c), the Fermi level of bottom BLG (green circle) is deep in the valence band.