Unconventional Correlation between Quantum Hall Transport Quantization and Bulk State Filling in Gated Graphene Devices

We report simultaneous transport and scanning microwave impedance microscopy to examine the correlation between transport quantization and filling of the bulk Landau levels in the quantum Hall regime in gated graphene devices. Surprisingly, a comparison of these measurements reveals that quantized transport typically occurs below the complete filling of bulk Landau levels, when the bulk is still conductive. This result points to a revised understanding of transport quantization when carriers are accumulated by gating. We discuss the implications on transport study of the quantum Hall effect in graphene and related topological states in other two-dimensional electron systems.

Figure 1

(a) Device structure and circuit schematic of MIM. (b) Landau fan diagram of device MLG No. 1 at $T=4.7\mathrm{K}$. (c) Typical response curves of MIM-Im and MIM-Re as a function of 2D resistivity in a 2DES structure. The MIM-Im signal increases monotonically as the 2DES becomes more conductive and thus provides better screening, and the MIM-Re signal peaks at an intermediate resistivity value which maximizes power dissipation. (d) MIM-Im signals of repeated scans along the same line across MLG No. 1 as the carrier density is tuned between $p$ and $n$ types at $B=9\mathrm{T}$ and $T=4.7\mathrm{K}$. The suppression of the MIM-Im signal in the bulk corresponds to transitions through various Landau levels, matching the established Landau level structure in MLG. The scale bar is $2\mu \mathrm{m}$.

Figure 2

(a) Comparison of the longitudinal resistance (red curve in the upper panel), normalized transverse Hall conductance (green curve in the upper panel), and MIM-Im signal averaged over a small region inside the graphene bulk (blue curve in the lower panel), measured at $T=4.7\mathrm{K}$ and $B=9\mathrm{T}$ for device MLG No. 1. The shaded regions mark the transport plateau (green) and the incompressible bulk transition (blue), respectively. The difference in gate voltage between the center of the $R_{xx}$ minimum and the center of the MIM-Im minimum is 1.6 V. (b), (c) Real space images of MIM-Im signal at gate values corresponding to the center of the transport plateau and the minimum of the bulk MIM-Im signal, as indicated by the arrows. The scale bar is $2\mu \mathrm{m}$.

Figure 3

(a) Simultaneous measurement of $R_{xx}$ (upper panel) and $d(\mathrm{MIM}\text{−}\mathrm{Im})/dz$ (lower panel) for MLG No. 1 at $B=7\mathrm{T}$ and $T=4.7\mathrm{K}$. (b) $(V_g-V_{g0})/\nu B$ plotted as a function of Landau level filling factor $\nu$. (c) $\mathrm{\Delta }{V}_g$ plotted as a function of $V_{\mathrm{LL},\mathrm{MIM}}$, which is proportional to the bulk density.

Figure 4

Simulated carrier density profiles corresponding to the transport plateau (red) of $\nu =2$, which is at 90% filling of the bulk LLs, and the completely filled bulk (blue). The density is normalized by the density value corresponding to the complete filling of the bulk LLs, in this case, $2{\rho }_0$. The upper inset shows the electric field distribution. An incompressible strip forms near the position where the density crosses the complete filling value. The distance of the incompressible strip from the graphene edge is $\sim 250\mathrm{nm}$, independent of the particular LL. The lower inset shows the real-space conductivity configuration corresponding to the transport plateau: The conductive edge region is separated from the conductive bulk by the incompressible strip.