Energy Gap Induced by Friedel Oscillations Manifested as Transport Asymmetry at Monolayer-Bilayer Graphene Boundaries

We show that Friedel charge oscillation near an interface opens a gap at the Fermi energy for electrons with wave vectors perpendicular to the interface. If the Friedel gaps on two sides of the interface are different, a nonequilibrium effect—shifting of these gaps under bias—leads to asymmetric transport upon reversing the bias polarity. The predicted transport asymmetry is revealed by scanning tunneling potentiometry at monolayer-bilayer interfaces in epitaxial graphene on SiC(0001). This intriguing interfacial transport behavior opens a new avenue toward novel quantum functions such as quantum switching.

Popular Summary

Electrons, by their quantum nature, are also waves. When scattered by static defects in solids such as metals and semiconductors, they form standing waves that can be seen on the solid surfaces. Such an interference pattern, often called Friedel oscillation, is not expected to impact how electrons conduct in the material because the large electron density can easily dwarf the Friedel oscillation. The situation can be different in materials such as recently discovered graphene and topological insulators, where the electron density is often low and the electronic interaction can become important. In this paper, we demonstrate that the Friedel oscillation can indeed open an energy gap for electron transport in graphene, which in turn can lead to asymmetric transport behavior across the interface in a composite monolayer-bilayer graphene system.

Our new finding is that the Friedel gap is opened because the charge oscillation, occurring at the interface, couples the right- and left-going electron waves near the Fermi energy. This gap opens both in the monolayer and in the bilayer of the composite graphene system formed epitaxially on SiC (0001), and it represents an extra energy cost for electron transmission across the monolayer-bilayer interface. The different strengths of the Coulomb interaction in the monolayer and bilayer make their gap sizes different, and that difference is accentuated by the bias voltage and manifested as asymmetric electrical transport across the interface. With our multiprobe scanning-tunneling-potentiometry measurements, we have demonstrated experimentally such a transport asymmetry.

In addition to the fundamental interest of our demonstration of the Friedel-gap-opening phenomenon, the sensitivity of the transport asymmetry to the scattering boundary conditions that we have shown makes scanning-tunneling potentiometry a suitable tool to probe the electron wave functions with respect to their chirality, Berry’s phase, and pseudospin polarization.

Figure 1

(a) Schematic illustration of the Friedel gap at $E_F$ for wave vectors perpendicular to an interface, assuming a linear unperturbed dispersion (e.g., that of graphene). (b) Schematics showing the Friedel gap shifting on two sides of an interface. Upper panel: Friedel gaps open at the respective chemical potentials, without considering the nonequilibrium effect of Friedel gap shifting. Lower panel: Both gaps completely exist below their respective chemical potentials.

Figure 2

(a) Schematic of the STP measurement setup. STM images of (b) ML and (c) BL graphene. Scale bar: 1 nm.

Figure 3

(a) STM image of a step-up boundary. Scale bar: 6 nm. (b) Schematic of the step-up boundary. (c) Potential profiles measured across the boundary with forward-and reversed-bias conditions (222-$\mu A$ current). (d) STM image of a step-down boundary. Scale bar: 10 nm. (e) Schematic of the step-down boundary. (f) Potential profiles measured across the boundary with forward- and reversed-bias conditions (313-$\mu A$ current). (g) STM image of a ML-ML graphene boundary. Scale bar: 15 nm. (h) Schematic of a ML-ML boundary. (i) Potential profiles measured across the boundary with different current values and directions.

Figure 4

(a) The asymmetry of potential drops ($\mathrm{\Delta }V$) as a function of the averaged potential drop ($\overline{V}$) at the junction between forward and reverse biases. Inset: Measured and calculated upper bound of $\mathrm{\Delta }V$ at low $\overline{V}$. (b) The potential drop at reverse bias ($V_{-}$) as a function of potential drop at forward bias ($V_{+}$) at the junction. Data points shown in blue color correspond to those measured with a bias voltage of less than 0.5 mV.