Quantum Hall effect in a high-mobility two-dimensional electron gas on the surface of a cylinder

The quantum Hall effect is investigated in a high-mobility two-dimensional electron gas on the surface of a cylinder. This special topology leads to a spatially varying filling factor along the current path. The resulting inhomogeneous current-density distribution gives rise to additional features in the magnetotransport, such as resistance asymmetry and modified longitudinal resistances. We experimentally demonstrate that the asymmetry relations satisfied in the integer filling factor regime are valid also in the transition regime to noninteger filling factors, thereby suggesting a more general form of these asymmetry relations. A model is developed based on the screening theory of the integer quantum Hall effect that allows the self-consistent calculation of the local electron density and thereby the local current density including the current along incompressible stripes. The model, which also includes the so-called “static skin effect” to account for the current-density distribution in the compressible regions can explain the main experimental observations. Due to the existence of an incompressible-compressible transition in the bulk, the system behaves always metal-like in contrast to the conventional Landauer-Büttiker description, in which the bulk remains completely insulating throughout the quantized Hall plateau regime.

Figure 1

(a) Sketch of a Hall bar on the periphery of a cylinder. (b) Schematic of such a Hall bar indicating the gradient of the magnetic field $k$, the imposed current $I_{EA}$ and imposed chemical potentials ${\mu }_i$ at leads $i$. The magnetic field is perpendicular at the position $y_{\perp }$ ($y=0$ is defined to be at the center Hall leads $C-G$). The 1DLS are shown schematically. The arrows indicate their chirality.

Figure 2

Longitudinal resistances with a magnetic field $B_0$ normal to the surface at the center Hall leads $y_{\perp }=0$. (a) $R_{DC}^L$, the inset shows the orientation of the Hall bar, (b) $R_{CB}^L$, the inset shows the second derivative of $R_{CB}^L$, as a function of the reciprocal of the magnetic field, (c) $R_{DB}^L$.

Figure 3

Fundamental frequencies of the Shubnikov–de Haas oscillations $B_{\text{SdH}}^{-1}$ in units of $2e{\left(hn\right)}^{-1}$ calculated from $R_{DC}^L$ (dots) and $R_{FG}^L$ (squares). Open and filled symbols indicate the different branches. The low-field Hall resistances $R_{DF}^H$ in units of ${\left(en\right)}^{-1}$ are shown by small triangles, which are connected by a line.

Figure 4

(Color online) (a) Hall resistances $R_{CG}^H$ and $R_{BH}^H$, (b) longitudinal resistance $R_{GH}^L$ for $y_{\perp }=-9.4\mu \text{m}$ vs $B_0$. Quantization in $R_{GH}^L$ at ${\nu }_2{\nu }_1/\left({\nu }_2-{\nu }_1\right)$ is indicated by the shaded rectangles for those regions where $R_{CG}^H$ and $R_{BH}^H$ are at integer filling factors. We plot the calculated resistance $R_{CG}^H-R_{BH}^H$ by a thin line, which coincidences mostly with $R_{GH}^L$.

Figure 5

(Color online) (a) Hall resistances $R_{CG}^H$, $R_{DF}^H$, and $R_{BH}^H$, (b) longitudinal resistance $R_{CB}^L$ for $y_{\perp }=-2.1\mu \text{m}$ vs $B_0$. Peaks in $R_{CB}^L$ appear at the nearest Hall lead pair at the high magnetic field end of the quantized Hall plateau.

Figure 6

(Color online) The calculated spatial distribution of local filling factors with integer values, $\nu \left(x,y\right)=2$ incompressible (yellow/gray) and compressible (white) for three selected values of the magnetic field corresponding to a filling factor at the $y_{\perp }=0$ position $\nu =\left(\text{a}\right)$ 2.5, (b) 2.1, and (c) 1.9 at 1.6 K. The unit cell is chosen to be $1\times 2\mu {\text{m}}^2$, spanning $48\times 96$ mesh points in our numerical simulation.

Figure 7

(Color online) The measured longitudinal resistance $R_{GH}^L$ (black line) vs $B_0$ and the calculated theory curve $R^{\text{SSE}}$ (red/gray line).