Coherent electron focusing with quantum point contacts in a two-dimensional electron gas

Transverse electron focusing in a two-dimensional electron gas is investigated experimentally and theoretically for the first time. A split Schottky gate on top of a GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$As heterostructure defines two point contacts of variable width, which are used as injector and collector of ballistic electrons. As evidenced by their quantized conductance, these are quantum point contacts with a width comparable to the Fermi wavelength. At low magnetic fields, skipping orbits at the electron-gas boundary are directly observed, thereby establishing that boundary scattering is highly specular. Large additional oscillatory structure in the focusing spectra is observed at low temperatures and for small point-contact size. This new phenomenon is interpreted in terms of interference of coherently excited magnetic edge states in a two-dimensional electron gas. A theory for this effect is given, and the relation with nonlocal resistance measurements in quantum ballistic transport is discussed. It is pointed out, and experimentally demonstrated, that four-terminal transport measurements in the electron-focusing geometry constitute a determination of either a generalized longitudinal resistance or a Hall resistance. At high magnetic fields the electron-focusing peaks are suppressed, and a transition is observed to the quantum Hall regime. The anomalous quantum Hall effect in this geometry is discussed in light of a four-terminal resistance formula.