Single-particle probing of edge-state formation in a graphene nanoribbon

We investigate the effect of a perpendicular magnetic field on the single-particle charging spectrum of a graphene quantum dot embedded inline with a nanoribbon. We observe uniform shifts in the single-particle spectrum which coincide with peaks in the magnetoconductance, implicating Landau level condensation and edge state formation as the mechanism underlying magnetic field-enhanced transmission through graphene nanostructures. The experimentally determined ratio of bulk to edge states is supported by single-particle band-structure simulations, while a fourfold beating of the Coulomb blockade transmission amplitude points to many-body interaction effects during Landau level condensation of the $\nu =0$ state.

Figure 1

(a) Atomic force micrograph of the GNR ($W\approx 90$, $L\approx 800$ nm) with an inline QD of diameter $D\approx 200$ nm. A functioning GNR intervenes between the QD and the plunger gate, but does not play any role in the results of the present work. (b) Conductance through the GNR as a function of back-gate voltage at $T=100$ mK. The insets show the position of the Fermi level relative to the Dirac cone over each range of back-gate voltage. (c) Logarithmic conductance as a function of source-drain and back-gate voltages. (d) Conductance as a function of plunger- and back-gate voltages obtained during a different cool down of the device. The white arrows indicate broad resonances stemming from localized states within the constriction.

Figure 2

(a) Conductance as a function of back gate and magnetic field. The white dashed line indicates the back-gate voltage used when performing the measurements shown in Fig. 3a. (b) Line profiles of (a) at 1 (blue, dotted) and 9 T (green, solid). The inset shows a close-up of the highlighted (pink) region I. Highlighted regions show ranges of back-gate voltage where the conductance either oscillates (I) or is uniformly enhanced (II).

Figure 3

(a) Conductance as a function of plunger-gate voltage and $B$ at $V_{BG}=-0.38$ V [compare with Fig. 2a]. Superimposed trace (white, solid) shows the conductance as a function of the magnetic field at $V_{PG}=-5$ V. Green (solid) line indicates fields where line profiles shown in Fig. 4a are extracted. (b) Coulomb blockade resonances (black) extracted from the raw $G\left(V_{PG}\right)$ data in (a) by removing the modulations in the background conductance. Two resonances have been traced over in order to emphasize the shift to a lower energy at a critical field $B_c$, whose dependence on occupancy follows the red (dashed) line. The relation between $B_c$ and the maximum occupancy $N_{\max }$ of LL${}_0$ has also been indicated. (c) Schematic band structure of a graphene bilayer with zig-zag edges at different magnetic fields. Numbers next to diagram correspond to the regions indicated in (a). The position of the Fermi level in each case is indicated by the black line. (d) Close-up of the region outlined by the purple box in (b) showing three illustrative pairs of resonances sharing the same kink structure, with the implied spin filling sequence shown below.

Figure 4

(a) Line sections showing the conductance as a function of estimated occupancy at $B$ between 8.5 and 9 T, in the vicinity of the green (solid) line in Fig. 3a. (b) High resolution plot of the conductance trace shown in (a) at 8.9 T with a background removed by subtracting a smoothed background. (c) Fourier spectrum of data shown in (b). (d) Tight binding band structure of LL${}_0$ (blue) and LL${}_{-1}$ (red/green) of a 1221-atom-wide (150-nm) armchair GNR in a field of 10 T.