Current density distribution for the quantum Hall effect

TL;DR

The paper investigates the microscopic current distribution in Hall bars under the integer quantum Hall regime, where the Hall resistance is quantized as $R_H=h/(e^2 \\nu)$. Building on the screening theory, it introduces a complete current density $j = j_\\text{impo} + j_\\text{pers}$ that includes a macroscopic persistent current $I_\\text{pers}$. The analysis shows that on the plateaus the current is chiral: $j$ flows with opposite sign on opposite edges due to the local gradient of the Landau level energy $\\varepsilon(y)$, while $I_\\text{impo}$ concentrates on the edge with higher potential and vanishes on the opposite edge. Moreover, $I_\\text{pers}$ decreases as $|I_\\text{impo}|$ grows, yielding a consistent picture of edge-to-bulk evolution and reconciling experimental results that observe bulk-like current portions. The framework clarifies how nonlocal screening shapes current flow in the QHE and provides a practical basis for interpreting Hall-bar transport on plateaus.

Abstract

Our microscopic understanding of the integer quantum Hall effect is still incomplete. For decades, there has been a controversial discussion about "where the current flows" if the Hall resistance is quantized. Here, we qualitatively analyze the current density distribution in a Hall bar based on the screening properties of a two-dimensional electron system in the quantum Hall regime. Beyond previous publications, we include a closed loop persistent current that exists inside a Hall bar if the Hall resistance is quantized. We find, that the persistent current density decreases with increasing Hall voltage. Accounting for this dependence, we find, that the current flows in the opposite directions along opposite edges of the Hall bar, while the imposed current flows unidirectionally and only on the side of the Hall bar connected with its higher electrical potential edge.

Figures (3)

• Figure 1: Qualitative sketches of the predictions of the screening theory for the first plateau ($\nu=1$) of the QHE featuring the case of two ICSs near the edges of the Hall bar at $y=\mp b$. In (a), we assume $I_\text{impo}=0$, such that $\mu=\mu_0$ is constant. In (b), we illustrate the effect of $I_\text{impo}\ne0$ in the linear approximation, cf. main text. Energy $\varepsilon(y)$ of the first LL (solid line), the second LL (dashed line), and the chemical potential $\mu(y)$ (dotted red line). Only inside the ICSs (dark gray background) $d\varepsilon/dy\ne0$ and $d\mu/dy\ne0$.
• Figure 2: Extension of Fig. \ref{['fig:ICSs']} featuring the current density for two different magnetic fields along the first plateau, as depicted in the insets at the upper end of the figure. Panels (a) and (c) on the left hand side illustrate the case of two separate ICSs, which occurs for $B$ near the low magnetic field end of the first plateau. Panels (b) and (d) on the right show one single ICS extending through most of the bulk of the Hall bar, which occurs for $B$ near the high magnetic field end of the first plateau. Shown are the energy $\varepsilon(y)$ of the first LL (solid line), the chemical potential $\mu(y)$ (red dotted line), the current density directions along $\pm x$ inside the ICSs (dark gray background) and, in the line plots the actual current densities $j(y)$ in arbitrary units. Negative currents (flowing in the $-x$-direction) correspond to the integral $\int_{j<0}dy j$, positive currents to $\int_{j>0}dy j$. Regions with $\varepsilon<\mu_0$ contribute to $I_\text{pers}$, regions with $\varepsilon\ge\mu_0$ to $I_\text{impo}$. Finite $I_\text{impo}$ causes $I_\text{pers}$ to decrease.
• Figure 3: Illustration of geometry in (b) of the ICS corresponding to the non-equilibrium situation with $I_\text{impo}\ne0$ of Fig. \ref{['fig:electrostatics']}(c); (a) and (c) are energy diagrams of the ICS along the upper and lower edges, repsecively. States contributing to $j_\text{pers}$ are indicated in blue and those of $j_\text{impo}$ in red. The persistent current describes a closed path inside the Hall bar as all contributing states have energies below the chemical potentials of both leads, $E<\mu_\text{S},\mu_\text{D}$. The imposed current is carried by states with energies $\mu_\text{S}\ge E>\mu_\text{D}$. Its electrons enter the Hall bar from the source contact, move along its high potential (lower) edge and leave it into empty states of the drain contact.