Contiguous two-dimensional regions in the quantized Hall regime

The integer quantum Hall effect is characterized by the observance of plateaus or steps in the Hall resistance for certain regimes of density and magnetic field. These steps are centered about the midpoints between adjacent Landau levels. The value of the Hall resistance at these steps is found to be equal to h/${\mathrm{ie}}^2$. The two-dimensional (2D) systems that have been used previously to investigate this effect have been relatively homogeneous within their boundaries. In the quantized Hall regime the boundaries of these systems are equipotentials, with potential discontinuities occurring at the boundary with the resistive contacts where current enters or leaves the sample. The investigations discussed here explore the interaction between differently characterized macroscopic 2D regions in a given sample. These different regions are either quantized Hall regions of integer Landau-level indices or nonquantized regions. The results show that the boundary of a 2D quantized region is not necessarily an equipotential and that the interaction between different regions at the boundary between two 2D regions is determined predominantly by the values of the Hall resistance that characterize each region. As a result, the equipotential distributions for an inhomogeneous sample can be predicted. It is also shown that the current may cross the boundary between regions anywhere along its length. This leads to the solution of the ‘‘two-terminal’’ quantized resistance problem as potential lines are allowed to cross the boundary between a 2D quantized region and the end contacts along the entire length of the boundary.