Theory of nonlinear ballistic transport in quasi-one-dimensional constrictions

We report on quantum-mechanical calculations of the differential conductance G of a ballistic and uniform constriction in the nonlinear-response regime of transport. The constriction has been connected to two semi-infinite two-dimensional electron gases, which serve as emitter and collector when a source-drain voltage ${\mathit{V}}_{\mathrm{sd}}$ is applied. In accordance with the fact that there should not be any electron backscatterings in the idealized constriction, it is assumed, in the calculations, that the electrostatic potential is flat in the constriction along the transport direction and the voltage drops of the applied source-drain voltage occur only at the ends of the constriction. Our calculations show that in addition to the conductance plateaus, which are quantized at multiples of 2${\mathit{e}}^2$/h in the linear-response regime of transport, new plateau structure develops as the source-drain voltage is increased. We predict that the edges of the conductance plateaus are shifted linearly with the applied source-drain voltage. Based on this prediction we discuss a method proposed for measuring the energy spacings of the sublevels of the constriction. Under the assumption that the transmission of an electron depends only on the difference between the energy of the electron incident on the constriction and the bottom of the electrostatic confining potential, a simple analytical form is derived for the nonlinear differential conductance of the constriction. This form complements our exact calculational results and shows that at zero temperature the differential conductance at a given finite ${\mathit{V}}_{\mathrm{sd}}$ and a given Fermi energy ${\mathit{E}}_{\mathit{F}}$ is a weighted average of two zero-${\mathit{V}}_{\mathrm{sd}}$ conductances at Fermi energies of ${\mathit{E}}_{\mathit{F}}$+(1-α)${\mathit{eV}}_{\mathrm{sd}}$ and ${\mathit{E}}_{\mathit{F}}$-α${\mathit{eV}}_{\mathrm{sd}}$, where α describes the fraction of the source-drain voltage that drops on the connection between the constriction and the drain reservoir.