Quantum Hall effect in narrow graphene ribbons

The edge states in the integer quantum Hall effect are known to be significantly affected by electrostatic interactions leading to the formation of compressible and incompressible strips at the boundaries of Hall bars. We show here, in a combined experimental and theoretical analysis, that this does not hold for the quantum Hall effect in narrow graphene ribbons. In our graphene Hall bar, which is only 60 nm wide, we observe the quantum Hall effect up to Landau level index $k=2$ and show within a zero-free-parameter model that the spatial extent of the compressible and incompressible strips is of a similar magnitude as the magnetic length. We conclude that in narrow graphene ribbons the single-particle picture is a more appropriate description of the quantum Hall effect and that electrostatic effects are of minor importance.

Figure 1

SEM image of a plasma patterned six-terminal graphene Hall bar with $w=60$ nm and $l=250$ nm.

Figure 2

Hall conductivity ${\sigma }_{xy}$ as a function of gate voltage $V_g$ and Landau level filling factor $\nu =h/\left(e^{2}B\right)C_g(V_g-V_g^D)$ at $B=11$ T and 30 mK. (Figure adapted from Ref. 9.)

Figure 3

Schematic of the compressible and incompressible strips in graphene, showing the electron density $\propto$$n\left(x\right)$ and the Landau levels with their local fillings. In (a) the ribbon is in a $C$ state with a compressible strip at its center, whereas in (b) it is in an $I$ state with an incompressible strip in the middle. The regions near the edges of the ribbon where the Landau levels increase very sharply due to hard-wall confinement are not shown. In these regions of size $\sim$$l_B$ away from the edges, the electrostatic analysis is no longer applicable.[14]

Figure 4

Top: Two-terminal conductance plateau corresponding to Landau level index $k=0$, plotted over the back-gate voltage $V_g$. The charge neutrality point is at $V_g^D$, and the dashed line indicates $G_0$. Bottom: The position and width of the compressible and incompressible strips are plotted, which, in Fig. 3, correspond to the occupation level $4(k+1/2)n_L$ with $k=0$. The dashed-dotted line shows $a$, the outer edge position; the solid line shows $a^{\prime }$, the inner edge position of the incompressible strips. In the upper graph the suppressed contribution of distant incompressible strips to the conductance at the $I$-$C$ transition is incorporated, whereas in the lower graph it is neglected, resulting in a slight mismatch in width of the shaded areas.