Empirical relation between gate voltage and electrostatic potential in the one-dimensional electron gas of a split-gate device

We study the magnetic depopulation of one-dimensional subbands in a narrow channel in GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As heterojunction. The channel is sufficiently short that transport through the channel is ballistic and the quantized resistance due to the subband structure is readily observed. We have made a detailed comparison between the experimental data and a variational calculation with a simple model potential. This potential is flat in the middle and parabolic at the edges and it is found to be clearly better than the commonly used parabolic potential. Both the channel thickness and the one-dimensional carrier concentration in the channel have been extracted as a function of gate voltage. The good agreement found with only two adjustable parameters indicates that this model potential gives a good description of the actual potential in the one-dimensional electron gas in a split-gate device.