Length estimation of a finite length quantum wire from finite size effects

When tunneling between two parallel quantum wires the current is proportional to |M(k)|², where M(k) is the Fourier-transform of the product of the participating wave functions in the two wires. Since the wave-vectors in both wires are generally different, M(k) will be strongly peaked at k^±=k₁±k₂. Here k_1,2 are the Fermi wave-vector in wires 1 or 2.

To measure |M(k)|² a central gate-voltage and a magnetic field are varied. This momentum-resolved-tunneling maps the dispersion of the cleaved-edge-overgrowth wires used in this experiment. Since these measurements are carried out with a central gate that does not span the length of the whole wire, we create regions of varying density along the wires. These density inhomogeneities contribute in various ways to |M(k)|², which allows for the extraction of the length of wire sections with different densities.

Here we show that in certain wave-vector ranges , k>|k₁-k₂| or k2 , |M(k)|² contains no contributions from inhomogeneities in the wire due to the applied gate voltage, which are quite pronounced outside of this range. Yet there are visible oscillations of conductance as a function of gate-voltage and magnetic field. Their period is governed by the full length of the quantum wire and provides an independent measure of the length. Together with experimental data we present numerical simulations that support our findings.

Supported by Swiss NSF, Swiss Nano Institute, and European Microkelvin Platform