Transmission phase of an electron in a quantum point contact

Here we calculate the electron transmission phase through a quantum point contact (QPC). The QPC is considered in the saddle-point approximation in the single-electron picture. We show that when the electron energy is close to the height of the potential barrier, the transmission phase depends linearly on the energy. The coefficient in the linear dependence is logarithmically enhanced by the ratio of the height over the curvature of the barrier potential. We compare the calculated transmission phase with the first experimental measurements of the phase.

Figure 1

The effective potential is sketched by the solid line. The horizontal dashed-dotted line indicates the energy level. The red dashed line shows the parabolic approximation to the potential.

Figure 2

Left panel: Transmission probability versus the gate voltage. Right panel: Transmission phase versus the gate voltage. Solid black lines show the data from Ref. [5]. Red dashed lines show our theoretical fit at $\hslash {\omega }_x=0.2$ meV. Blue dashed-dotted lines show our theoretical fit at $\hslash {\omega }_x=0.5$ meV. Vertical lines in both panels indicate the region where the transmission probability varies from 5% to 95%.

Figure 3

The Gaussian scattering potential.

Figure 4

Transmission probability versus $\epsilon$.

Figure 5

The energy-dependent part of the scattering phase versus $\epsilon$. The solid black line shows results of the numerical calculation. The red dashed line shows the phase given by the analytical formula (18). Blue dots shows the phase given by the adjusted analytical formula.

Figure 6

The phase deviation from the linear dependence.

Figure 7

The plot of $\arg \left[\Gamma (\frac{1}{2}+i\epsilon )\right]$ versus $\epsilon$.