Currents in the compressible and incompressible regions of the two-dimensional electron gas

We derive a general expression for the low-temperature equilibrium orbital current distribution in a two-dimensional electron gas, subjected to a perpendicular magnetic field and in a confining potential that varies slowly on the scale of the magnetic length l. The analysis is valid within a self-consistent one-electron description, such as the Hartree or standard Kohn-Sham equations. Our expression, which correctly describes the current distribution on scales larger than l, has two components: One is an ‘‘edge current,’’ which is proportional to the local density gradient, and the other is a ‘‘bulk current,’’ which is proportional to the gradient of the confining potential. The direction of these currents generally displays a striking alternating pattern. In a compressible region at the edge of the nth Landau level, the edge current is simply j=-e${\mathrm{\omega }}_{\mathit{c}}$ ${\mathit{l}}^2$(n+1/2)∇ρ×${\mathbf{e}}_{\mathit{z}}$, where ${\mathrm{\omega }}_{\mathit{c}}$ is the cyclotron frequency and ρ is the electron sheet density. The bulk component, a Hall current, dominates in the incompressible regions. In the ideal case of perfect compressibility and incompressibility, only one type of current contributes to a given region, and the integrated orbital currents in these regions are universal, independent of the widths, positions, and geometry of the regions. The integrated orbital current in the nth edge channel is (n+1/2)e${\mathrm{\omega }}_{\mathit{c}}$/2π, whereas in an incompressible strip with integral filling factor ν it is νe${\mathrm{\omega }}_{\mathit{c}}$/2π with the opposite sign.