CURRENTS IN THE COMPRESSIBLE AND INCOMPRESSIBLE REGIONS OF THE 2-DIMENSIONAL ELECTRON-GAS

We derive a general expression for the low-temperature equilibrium orb ital current distribution in a two-dimensional electron gas, subjected to a perpendicular magnetic field and in a confining potential that v aries slowly on the scale of the magnetic length l. The analysis is va lid within a self-consistent one-electron description, such as the Har tree or standard Kohn-Sham equations. Our expression, which correctly describes the current distribution on scales larger than l, has two co mponents: One is an ''edge current,'' which is proportional to the loc al density gradient, and the other is a ''bulk current,'' which is pro portional 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 cu rrent is simply j = -e omega(c)l(2)(n+1/2)del pXe(z), where omega(c) i s the cyclotron frequency and p is the electron sheet density. The bul k component, a Hall current, dominates in the incompressible regions. In the ideal case of perfect compressibility and incompressibility, on ly one type of current contributes to a given region, and the integrat ed orbital currents in these regions are universal, independent of the widths, positions, and geometry of the regions. The integrated orbita l current in the nth edge channel is (n+1/2)e omega(c)/2 pi, whereas i n an incompressible strip with integral filling factor nu it is nu e o mega(c)/2 pi with the opposite sign.